이 기술 요약은 Rupayan Saha가 2017년 West Virginia University에 제출한 논문 “Prediction of Maximum Scour Depth Using Scaled Down Bridge Model in a Laboratory”를 기반으로 합니다. 이 자료는 STI C&D에 의해 기술 전문가들을 위해 분석 및 요약되었습니다.
키워드
Primary Keyword: 교량 세굴 예측
Secondary Keywords: 최대 세굴 깊이, CFD, 수리 실험, 압력 흐름, 축소 모형, 교각 세굴
Executive Summary
도전 과제: 기존의 교량 세굴 예측 방법은 상호작용하는 세굴 과정을 분리하여 계산하고 극한의 흐름 조건을 고려하지 않아, 비안전적이거나 과도한 설계로 이어지는 부정확성을 가집니다.
연구 방법: 실제 하천 교량의 1:60 축소 물리 모델을 제작하여, 압력 흐름 및 월류(overtopping)를 포함한 다양한 유동 조건에서 발생하는 세굴을 측정했습니다.
핵심 돌파구: 이론적인 교각 세굴에 유량 수축과 직접적으로 연관된 ‘추가 세굴’ 항을 결합하는 새로운 통합 방정식을 개발하여, 자유 수면 흐름과 압력 흐름 조건을 구분하여 최대 세굴 깊이를 예측합니다.
핵심 결론: 이 연구는 최대 교량 세굴을 더 정확하고 신뢰성 있게 예측하는 방법을 제공하여, 더 안전하고 경제적인 교량 설계를 가능하게 합니다.
도전 과제: 이 연구가 CFD 전문가에게 중요한 이유
교량 세굴(Bridge Scour)은 교량 기초 주변의 하상 재료가 물의 흐름에 의해 침식되는 현상으로, 교량 붕괴의 가장 흔한 원인 중 하나입니다. 세굴 깊이를 정확하게 예측하는 것은 교량의 안전과 직결되지만, 기존의 예측 모델들은 종종 현장과 큰 오차를 보입니다.
현재 널리 사용되는 FHWA(미국 연방 고속도로국)의 가이드라인은 하천 폭이 좁아지며 발생하는 ‘수축 세굴(Contraction Scour)’과 교각 주변에서 국부적으로 발생하는 ‘국부 세굴(Local Scour)’을 독립적인 현상으로 간주하고 각각 계산한 뒤 합산합니다. 하지만 실제로는 이 두 과정이 동시에 발생하며 서로에게 영향을 미칩니다. 또한, 대부분의 예측 공식은 이상적인 직사각형 수로에서의 실험을 기반으로 하여, 실제 하천의 복잡한 지형이나 홍수 시 발생하는 교량 상판 잠김(압력 흐름) 또는 월류(overtopping)와 같은 극한 상황을 제대로 반영하지 못합니다. 이러한 한계는 결국 과도한 안전율 적용으로 인한 비경제적인 설계 또는 예측 실패로 인한 구조적 위험을 초래합니다.
접근 방식: 연구 방법론 분석
본 연구는 이러한 한계를 극복하기 위해 미국 조지아주 메이컨에 위치한 Towaliga 강 교량의 1:60 축소 수리 모형을 실험실 수조 내에 제작했습니다. 이 모델은 실제 하천의 복잡한 단면 형상(복단면)과 유역 지형을 정밀하게 재현했습니다.
연구팀은 다양한 유량 조건에서 실험을 수행했으며, 특히 극한 홍수 상황을 모사하기 위해 세 가지 주요 흐름 유형을 분석했습니다. 1. 자유 흐름 (Free Flow): 교량 하부 구조물이 물에 잠기지 않는 일반적인 흐름 상태 2. 잠긴 오리피스 흐름 (Submerged Orifice Flow): 교량 상판이 물에 잠겨 압력 흐름이 발생하는 상태 3. 월류 흐름 (Overtopping Flow): 유량이 더 증가하여 물이 교량 상판 위로 넘어가는 상태
실험 중 유속과 세굴 후 하상 변화는 음향 도플러 유속계(Acoustic Doppler Velocimeter, ADV)를 사용하여 3차원으로 정밀하게 측정되었습니다. 이 접근법을 통해 이상적인 실험실 환경이 아닌, 실제와 유사한 복합적인 조건에서 세굴이 어떻게 발생하는지에 대한 신뢰도 높은 데이터를 확보할 수 있었습니다.
돌파구: 주요 발견 및 데이터
발견 1: 통합된 최대 세굴 예측 모델의 개발
연구팀은 수축 세굴과 국부 세굴을 분리하지 않고, 최대 세굴 깊이를 하나의 통합된 방식으로 예측하는 새로운 개념을 제안했습니다.
최대 세굴 깊이 = 이론적 교각 세굴 깊이 + 흐름 수축으로 인한 추가 세굴 깊이
여기서 ‘흐름 수축으로 인한 추가 세굴 깊이’는 실험적으로 측정한 흐름 수축비(교량 구간과 상류 접근부의 단위 폭당 유량비, q₂/q₁)와 직접적인 상관관계를 가집니다. Figure 4.9에서 볼 수 있듯이, 흐름 수축비(q₂/q₁)가 증가할수록 정규화된 추가 세굴 깊이(Ym-csu/Y₁)가 체계적으로 증가하는 것을 확인했습니다. 이는 흐름이 교량에서 가속될수록 국부적인 최대 세굴이 더 깊어진다는 것을 정량적으로 보여줍니다. 이 모델은 두 세굴 메커니즘의 상호작용을 효과적으로 반영합니다.
발견 2: 압력 흐름 조건에서 세굴 심화 현상 규명
본 연구의 가장 중요한 발견 중 하나는 압력 흐름(잠긴 오리피스 및 월류 흐름)이 자유 흐름에 비해 세굴을 현저히 심화시킨다는 것입니다. Figure 4.9의 회귀 분석 결과, 압력 흐름 조건의 데이터(SO, OT)는 자유 흐름(F)보다 더 가파른 기울기를 보였습니다.
이는 교량 상판이 물에 잠기면서 측면 수축(Lateral Contraction)뿐만 아니라 수직 수축(Vertical Contraction) 효과가 더해지기 때문입니다. 교량 상판이 흐름의 ‘뚜껑’ 역할을 하여 물을 하상 쪽으로 강하게 밀어내고 가속시켜, 하상을 침식시키는 힘(전단 응력)을 극대화합니다. 동일한 측면 수축 조건이라도 수직 수축이 더해지면 최대 세굴 깊이가 훨씬 더 깊어지는 것을 실험적으로 증명했으며, 이는 기존 모델들이 놓치고 있던 중요한 물리 현상입니다.
R&D 및 운영을 위한 실질적 시사점
토목/수리 엔지니어: 본 연구는 교량 기초 설계를 위한 최대 세굴 깊이를 더 정확하게 예측할 수 있는 실용적인 절차를 제공합니다. 제시된 방정식을 통해 자유 흐름과 압력 흐름을 구분하여 설계에 반영함으로써, 과설계를 줄이고 안전성을 높일 수 있습니다.
교량 안전 점검팀: 압력 흐름이나 월류가 발생하는 극한 홍수 시 세굴 위험이 급격히 증가한다는 본 연구 결과는, 이러한 조건에서 교량 주변의 모니터링을 강화해야 할 필요성을 시사합니다.
CFD 모델러: 실제 하천 지형을 정밀하게 모사한 이 실험 데이터(Figure 4.4는 실험실과 현장 데이터 간의 높은 일치도를 보여줌)는 복잡한 교량 세굴 시나리오에 대한 CFD 시뮬레이션의 검증(Validation)을 위한 귀중한 자료로 활용될 수 있습니다.
논문 상세 정보
Prediction of Maximum Scour Depth Using Scaled Down Bridge Model in a Laboratory
1. 개요:
제목: Prediction of Maximum Scour Depth Using Scaled Down Bridge Model in a Laboratory
저자: Rupayan Saha
발행 연도: 2017
학술지/학회: West Virginia University, Graduate Theses, Dissertations, and Problem Reports
키워드: Maximum scour depth prediction, Physical model, Bridge scour
2. 초록:
최근 미국 웨스트버지니아, 텍사스, 루이지애나 등지에서 발생한 치명적인 홍수로 인해 여러 교량이 붕괴되었습니다. 그중 교량 세굴은 교량 붕괴의 주요 원인 중 하나로, 인명 및 재산 피해를 유발합니다. 1960년대부터 많은 세굴 연구가 수행되었음에도 불구하고, 세굴 예측은 여전히 어려운 과제입니다. 현재의 세굴 예측 방식은 과대 또는 과소 예측되는 경향이 있는데, 이는 수축 세굴과 국부 세굴이 실제로는 동시에 발생함에도 불구하고 개별적으로 추정하여 합산하기 때문일 수 있습니다. 또한, 기존 세굴 공식들은 이상적인 직사각형 수로에서의 자유 수면 흐름 실험을 기반으로 하지만, 실제 극한 홍수 시에는 교량 월류와 잠긴 오리피스 흐름이 복합적으로 발생하며 세굴 깊이는 현장 특이적입니다. 본 연구에서는 조지아 공과대학 수리학 연구실에서 실제 하천 지형을 포함한 1:60 축소 교량 모델을 사용하여 다양한 흐름 조건(자유, 잠긴 오리피스, 월류)에서 실험을 수행했습니다. 실험 결과 분석을 통해, 널리 사용되는 경험적 세굴 추정 방법(CSU 교각 세굴 공식, Melville-Sheppard 공식 등)과 실험 결과를 결합하여, 맑은 물 세굴 조건에서의 최대 세굴 깊이를 예측하는 포괄적인 방법을 제안했습니다. 분석 과정에서 국부 세굴에 대한 흐름 수축의 영향을 평가했으며, 이는 개별적인 세굴 깊이 추정 대신 단일 예측 방법의 필요성을 확인시켜 주었습니다. 또한, 국부 세굴 주변의 주변 하상고를 이용한 면적 평균 수축 세굴 깊이 예측 방법을 제안하고, 측정된 흐름 수축비로 분석했습니다. 수직적 흐름 수축과 교대 근처 교각의 존재가 최대 세굴 깊이에 미치는 영향도 조사했습니다. 결과적으로 압력 흐름에서는 측면 및 수직 수축의 조합이 최대 세굴 깊이를 증폭시켰으며, 교각의 존재 유무는 최대 세굴 깊이의 위치에는 영향을 미치지 않지만, 유량 재분배로 인해 세굴량에는 차이를 보였습니다.
Figure 1.1. Ellsworth Barranca Bridge experiencing problem due to scour in Ventura County, California (California Department of Transportation)
3. 서론:
세굴은 하천의 흐름과 지형학적 과정의 복합적인 작용으로 하상 재료가 제거되는 현상입니다. 특히 교량 세굴은 교량 개구부를 통과하는 물에 의해 모래나 암석 같은 하상 재료가 제거되는 것을 의미합니다. 교량이 건설되면 교각 및 교대 주변에 독특한 흐름장이 형성되고, 제방이나 교대로 인한 단면적 수축은 유속을 증가시킵니다. 이러한 흐름은 교량 기초의 매립 깊이를 감소시켜 심각한 손상을 유발할 수 있습니다. 교량 붕괴는 지진, 홍수 등 여러 원인으로 발생하지만, 교량 세굴은 미국에서 가장 흔한 교량 붕괴 원인으로 지목되어 왔습니다. 예를 들어, 1950년 이후 미국 전체 교량 붕괴의 약 60%가 교량 기초 세굴과 관련이 있었습니다. 경제적 관점에서도 1993년 한 해에만 2,500개 이상의 교량이 세굴로 파괴되거나 심각한 손상을 입어 약 1억 7,800만 달러의 복구 비용이 발생했습니다. 이러한 직접적인 비용 외에도 상업 활동 중단으로 인한 간접 비용은 5배 이상으로 추정됩니다. 이처럼 교량 세굴은 전 세계적으로 중요한 교량 안전 문제 중 하나입니다.
4. 연구 요약:
연구 주제의 배경:
교량 세굴은 교량 붕괴의 주요 원인이지만, 세굴 깊이를 정확하게 예측할 수 있는 방정식은 아직 부족합니다. 이로 인해 엔지니어들은 과도한 안전율을 적용하여 기초를 깊게 설계하게 되고, 이는 비경제적인 교량 건설로 이어집니다. 반면, 안전율이 부족하면 홍수 시 기초가 노출될 위험이 커져 안전에 치명적입니다.
기존 연구 현황:
기존의 세굴 예측 공식들은 대부분 이상적인 직사각형 수로와 같은 단순화된 실험실 환경에서 개발되었습니다. 이는 실제 하천의 복잡한 지형과 교량 구조물 주변의 흐름을 제대로 재현하지 못하는 한계가 있습니다. 또한, 현재 FHWA 가이드라인은 수축 세굴과 국부 세굴을 독립적인 과정으로 보고 각각 계산 후 합산하도록 권장하지만, 여러 연구에서 두 과정이 상호작용하며 단순 합산 시 과대 예측될 수 있음을 보여주었습니다.
연구의 목적:
본 연구의 주된 목적은 수축 세굴과 국부 세굴을 별도로 계산하지 않고, 단일 방정식을 사용하여 최대 세굴 깊이를 예측하는 방법을 개발하는 것입니다. 이를 위해 다음과 같은 세부 목표를 설정했습니다. – 다양한 흐름 유형(자유 흐름, 잠긴 오리피스 흐름, 월류 흐름)이 최대 세굴 깊이에 미치는 영향을 평가합니다. – 흐름 수축이 국부 세굴에 미치는 영향을 정량화하는 방법을 개발합니다. – 측면 수축과 수직 수축의 차이를 규명합니다. – 기존에 확립된 세굴 공식을 활용하여 최대 세굴 깊이를 예측하는 개선된 방법론을 개발합니다.
핵심 연구:
연구의 핵심은 실제 하천 지형을 재현한 1:60 축소 교량 모델을 이용한 수리 실험입니다. 실험은 맑은 물 세굴(Clear-water scour) 조건에서 수행되었으며, 다양한 유량과 수심 조건에서 세 가지 흐름 유형(자유, 잠긴 오리피스, 월류)을 모사했습니다. 실험을 통해 얻은 유속 및 하상고 데이터를 분석하여, 흐름 수축비(q₂/q₁)를 핵심 변수로 사용하여 최대 세굴 깊이를 예측하는 경험적 관계식을 도출했습니다. 이 과정에서 널리 사용되는 CSU 공식과 Melville-Sheppard(M/S) 공식을 기준 세굴 깊이로 활용하여, 흐름 수축에 의한 ‘추가 세굴’ 효과를 정량화했습니다.
5. 연구 방법론
연구 설계:
본 연구는 실제 교량(Towaliga 강 교량)의 축소 모형을 이용한 실험적 접근법을 채택했습니다. 실험실 수조 내에 1:60 비율로 축소된 교량 및 하천 지형 모델을 제작하고, 제어된 조건 하에서 다양한 수리 시나리오를 재현했습니다. Froude 수 상사법칙을 사용하여 실험실 모델과 실제 현상 간의 동적 유사성을 확보했습니다.
데이터 수집 및 분석 방법:
데이터 수집: 음향 도플러 유속계(ADV)를 사용하여 흐름 단면의 3차원 유속 분포를 측정했습니다. 세굴 전후의 하상고는 ADV와 포인트 게이지를 이용하여 정밀하게 측정되었습니다.
데이터 분석: 측정된 유속과 수심 데이터를 이용하여 상류 접근부와 교량부의 단위 폭당 유량(q₁ 및 q₂)을 계산하고, 이를 통해 흐름 수축비(q₂/q₁)를 도출했습니다. 최대 세굴 깊이와 흐름 수축비 간의 관계를 규명하기 위해 최소자승법을 이용한 회귀 분석을 수행했습니다.
Figure 3.4. Tailgate of the flume to adjust the flow depth downstream
연구 주제 및 범위:
연구는 맑은 물 세굴 조건에 국한되었습니다. 실험은 총 8개의 주요 시나리오(Run 1~8)로 구성되었으며, 유량, 수심, 흐름 유형(자유, 잠긴 오리피스, 월류)을 변화시켰습니다. 또한, 수직 수축의 효과를 명확히 보기 위해 교량 상판이 없는 조건(Run 5, 6)과 교대 근처 교각의 영향을 보기 위해 해당 교각을 제거한 조건(Run 7, 8)도 실험에 포함되었습니다.
6. 주요 결과:
주요 결과:
최대 세굴 깊이는 교량의 상류나 하류 단면이 아닌, 교량 구간 내부 중간 지점에서 발생하는 것으로 관찰되었습니다.
최대 세굴 깊이는 ‘이론적 교각 세굴’과 ‘흐름 수축에 의한 추가 세굴’의 합으로 표현될 수 있으며, ‘추가 세굴’은 흐름 수축비(q₂/q₁)와 강한 양의 상관관계를 가집니다.
압력 흐름(잠긴 오리피스, 월류)은 동일한 측면 수축 조건의 자유 흐름보다 훨씬 더 깊은 세굴을 유발합니다. 이는 교량 상판에 의한 수직 수축 효과가 더해지기 때문입니다.
CSU 공식과 M/S 공식을 기준 세굴 깊이로 사용했을 때, ‘추가 세굴’ 항의 크기가 다르게 나타났으며, 이는 각 공식이 고려하는 변수(예: 유속 강도, 입자 크기)의 차이에서 기인합니다.
교대 근처 교각의 부재는 최대 세굴 깊이의 위치에는 영향을 주지 않았으나, 유량 재분배로 인해 세굴의 총량은 소폭 증가시켰습니다.
Figure 목록:
Figure 1.1. Ellsworth Barranca Bridge experiencing problem due to scour in Ventura County, California (California Department of Transportation).
Figure 1.2. Scour around bridge piers on the Tinau River, Nepal (Shrestha, 2015).
Figure 3.1. Laboratory model of Towaliga River bridge.
Figure 3.2. Location and view of Towaliga River Bridge
Figure 3.3. Entrance section of the flume
Figure 3.4. Tailgate of the flume to adjust the flow depth downstream
Figure 3.5. Plan view of flume for model construction
Figure 3.6. Geometry of compound channel for (a) plan view; (b) cross section view at bridge when looking downstream
Figure 3.7. Sediment size distribution of the bed material for this study
Figure 4.1. Plan view of velocity measurement locations
Figure 4.2. Approach flow velocity distributions for run 1 when looking downstream.
Figure 4.3. Velocity distributions at upstream face of bridge section for run 1 when looking downstream
Figure 4.4. Comparison between measured laboratory data and observed field data.
Figure 4.5. Bridge cross-section comparison after scour for run 2
Figure 4.6. Photographs of bed after scour for run 2
Figure 4.7. Schematic diagram of notations to calculate maximum scour depth
Figure 4.8. Schematic diagram for calculation of contraction scour using flow depth
Figure 4.9. Normalized additional scour depth using CSU equation as a function of q2/q1.
Figure 4.10. Evaluation of vertical contraction effect using normalized additional scour depth as a function of q2/q1
Figure 4.11. Normalized additional scour depth using M/S equation as a function of q2/q1.
Figure 4.12. Theoretical pier scour depth ratio using CSU and M/S equation in terms of flow intensity.
Figure 4.13. Adjusted ambient bed level after scouring of run 3
Figure 4.14. Normalized additional scour depth using ambient method as a function of q2/q1.
Figure 4.15. Normalized local scour depth due to contraction as a function of q2/q1
Figure 4.16. Normalized area average contraction scour depth as a function of q2/q1.
7. 결론:
본 연구는 기존 교량 세굴 예측 방법의 한계를 극복하기 위해 실제 하천 지형을 반영한 축소 모형 실험을 수행했습니다. 실험 결과, 최대 세굴 깊이는 이론적 교각 세굴과 흐름 수축에 의한 추가 세굴의 합으로 설명될 수 있음을 밝혔습니다. 특히, 교량 상판이 물에 잠기는 압력 흐름 조건에서는 수직 수축 효과로 인해 자유 흐름보다 세굴이 현저히 심화되는 것을 규명했습니다. 이를 바탕으로, 흐름 유형(자유/압력)과 흐름 수축비를 고려하여 최대 세굴 깊이를 단일 방정식으로 예측할 수 있는 새로운 방법론을 개발하고 제안했습니다. 이 방법은 기존의 분리된 계산 방식보다 더 정확하고 신뢰성 있는 예측을 가능하게 하여, 교량 설계의 안전성과 경제성을 향상시키는 데 기여할 수 있습니다.
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전문가 Q&A: 주요 질문에 대한 답변
Q1: 왜 일반적인 직사각형 수조 대신 특정 교량의 1:60 축소 모델을 사용했나요?
A1: 일반적인 직사각형 수조는 실제 하천의 복잡한 흐름 패턴을 재현할 수 없습니다. 본 연구에서는 실제 하천의 복단면 형상과 지형을 정밀하게 모사하여, 현장에서 발생하는 3차원적이고 복합적인 흐름 특성을 실험실에서 구현하고자 했습니다. 이를 통해 실험 결과의 신뢰도를 높이고, Figure 4.4에서 보듯이 실제 현장 관측 데이터와 비교 검증하여 모델의 타당성을 확보할 수 있었습니다.
Q2: 연구에서 제안한 ‘추가 세굴’ 항을 더하는 방식은 기존에 수축 세굴과 국부 세굴을 더하던 방식과 근본적으로 어떻게 다른가요?
A2: 기존 방식은 두 세굴을 완전히 독립적인 현상으로 보고 각각의 공식으로 계산한 뒤 산술적으로 합산했습니다. 반면, 본 연구의 ‘추가 세굴’ 항은 흐름 수축비(q₂/q₁)라는 단일 물리 변수에 대한 경험적 함수로 도출되었습니다. 이는 수축이 국부 세굴에 미치는 ‘상호작용’과 ‘증폭 효과’를 직접적으로 반영하는 항으로, 두 현상을 분리하지 않고 통합된 결과로서 최대 세굴 깊이를 예측한다는 점에서 근본적인 차이가 있습니다.
Q3: Figure 4.10은 압력 흐름이 세굴 깊이 대 수축비 그래프에서 더 가파른 기울기를 만드는 것을 보여줍니다. 물리적인 이유는 무엇인가요?
A3: 이는 측면 수축과 수직 수축의 복합 효과 때문입니다. 자유 흐름에서는 물이 측면으로만 수축되지만, 압력 흐름에서는 교량 상판이 ‘뚜껑’처럼 작용하여 흐름 단면을 수직으로도 제한합니다. 이로 인해 흐름은 측면뿐만 아니라 수직 방향(하상 방향)으로도 강하게 가속되어, 자유 흐름 조건일 때보다 훨씬 더 큰 침식력을 발생시킵니다. 이 추가적인 수직 가속이 그래프에서 더 가파른 기울기로 나타나는 물리적 원인입니다.
Q4: CSU 공식과 M/S 공식을 기준으로 사용했을 때 ‘추가 세굴’ 요소가 다르게 계산되는 이유는 무엇입니까?
A4: 두 공식이 이론적 교각 세굴을 계산할 때 고려하는 변수가 다르기 때문입니다. M/S 공식은 유속 강도(V₂/Vc)나 퇴적물 입자 크기 같은 변수를 이미 포함하고 있어, 본 연구와 같은 맑은 물 세굴 조건에서는 더 보수적인 CSU 공식보다 초기 교각 세굴 깊이를 작게 예측하는 경향이 있습니다. 따라서 동일한 총 측정 세굴 깊이에 도달하기 위해서는, 더 작은 기준값(M/S 공식)을 사용할 때 나머지 ‘추가 세굴’ 항이 상대적으로 더 커져야 합니다.
Q5: 논문에서 언급된 ‘주변 하상고(ambient bed level)’ 방법의 의의는 무엇인가요?
A5: 이 방법은 교각 주변의 국부적인 세굴 구멍과 하상 전체가 낮아지는 수축 세굴을 물리적으로 분리하는 직접 측정 기법입니다. 세굴 발생 후 교각 양쪽의 하상고를 선형으로 연결(보간)하여 국부 세굴이 없었을 때의 가상 하상면을 설정합니다. 이를 통해 두 세굴 요소를 더 물리적으로 타당하게 정량화할 수 있었고, 압력 흐름 조건에서 전체적인 수축 세굴이 실제로 더 크다는 것을 확인하는 데 사용되었습니다.
결론: 더 높은 품질과 생산성을 위한 길
부정확한 교량 세굴 예측은 교량의 안전을 위협하고 불필요한 건설 비용을 초래하는 오랜 난제였습니다. 본 연구는 실제 하천의 복잡한 조건과 극한 홍수 상황을 모사한 정밀한 실험을 통해, 흐름 수축과 국부 세굴의 상호작용을 통합적으로 고려하는 새로운 예측 모델을 제시했습니다. 특히 압력 흐름 시 수직 수축 효과가 세굴을 크게 증폭시킨다는 사실을 규명함으로써, 더 안전하고 경제적인 교량 설계의 길을 열었습니다.
(주)에스티아이씨앤디에서는 고객이 수치해석을 직접 수행하고 싶지만 경험이 없거나, 시간이 없어서 용역을 통해 수치해석 결과를 얻고자 하는 경우 전문 엔지니어를 통해 CFD consulting services를 제공합니다. 귀하께서 당면하고 있는 연구프로젝트를 최소의 비용으로, 최적의 해결방안을 찾을 수 있도록 지원합니다.
연락처 : 02-2026-0442
이메일 : flow3d@stikorea.co.kr
저작권 정보
이 콘텐츠는 “Prediction of Maximum Scour Depth Using Scaled Down Bridge Model in a Laboratory” (저자: Rupayan Saha) 논문을 기반으로 한 요약 및 분석 자료입니다.
이 기술 요약은 精密工学会誌/Journal of the Japan Society for Precision Engineering에 발표된 徳永 仁史, 岡根 利光, 岡野 豊明의 논문 “高速な流れ解析手法を統合した流路設計のための設計インタフェース -湯流れ解析下におけるダイカスト湯道設計への適用一” (2016)을 기반으로, STI C&D의 기술 전문가에 의해 분석 및 요약되었습니다.
키워드
Primary Keyword: 다이캐스팅
Secondary Keywords: SPH (Smoothed Particle Hydrodynamics), GPU, CFD, 유동 해석, 런너 설계, 설계 인터페이스, 실시간 시뮬레이션
Executive Summary
The Challenge: 기존의 CFD 시뮬레이션은 계산 시간이 길어 반복적인 다이캐스팅 런너 설계에 비효율적이며, 이는 최적화되지 않은 설계와 품질 문제로 이어집니다.
The Method: 본 연구는 고속 GPU 가속 SPH(Smoothed Particle Hydrodynamics) 유동 해석 기법과 FFD(Free-Form Deformation) 형상 변형 기술을 통합했습니다.
The Key Breakthrough: 이 통합 기술을 통해 엔지니어는 시뮬레이션이 실행되는 동안 런너 형상을 실시간으로 수정하고 용탕 유동 거동에 미치는 영향을 즉시 확인할 수 있습니다.
The Bottom Line: 이 양방향 설계 접근법은 설계-평가 주기를 획기적으로 단축하여, 향상된 주조 품질을 위한 유로의 신속한 최적화를 가능하게 합니다.
The Challenge: Why This Research Matters for CFD Professionals
제품의 성능과 품질은 설계 단계의 결정에 크게 좌우됩니다. 이는 다이캐스팅과 같은 제조 공정의 설계(방안 검토)에서도 마찬가지입니다. 효율적인 설계 및 제조 사이클을 위해서는 프로토타입 제작과 같은 물리적 검증 이전에, 설계 단계에서 반복적으로 설계안을 검증하고 개선하는 과정이 필수적입니다.
이를 위해 다양한 공학 해석(CAE) 도구가 사용되어 왔지만, 기존의 유동 해석 기법 대부분은 계산 시간이 방대하여 설계안을 반복적으로 검증하고 개선하기에는 한계가 있었습니다. 특히 다이캐스팅 공정에서 런너와 게이트의 미세한 형상 변화가 제품 품질에 결정적인 영향을 미치지만, 긴 해석 시간 때문에 데이터 기반의 신속한 최적화보다는 설계자의 경험과 직관에 의존하는 경우가 많았습니다. 이러한 비효율성은 개발 기간을 지연시키고 잠재적인 품질 문제를 야기하는 핵심 원인이었습니다.
The Approach: Unpacking the Methodology
본 연구는 설계와 해석 프로세스를 통합하여 설계 효율을 극대화하는 새로운 인터페이스를 제안합니다. 이 방법론의 핵심은 고속 유동 해석 기술과 실시간 형상 변형 기술의 결합입니다.
고속 유동 해석 (GPU-Accelerated SPH): 해석 기법으로는 입자 기반의 SPH(Smoothed Particle Hydrodynamics)를 채택했습니다. SPH는 자유 표면 유동 해석에 강점을 가지며, 형상이 변형될 때마다 복잡한 격자를 재생성할 필요가 없어 본 연구에 이상적입니다. 특히, 계산 과정을 GPU(Graphics Processing Unit)에서 병렬 처리하여 기존의 CPU(1코어) 계산 대비 120배 이상의 압도적인 속도 향상을 달성했습니다 (Table 1).
실시간 형상 변형 (Free-Form Deformation, FFD): 설계자가 유로 형상을 직관적으로 수정할 수 있도록 FFD 기법을 도입했습니다. FFD는 수정하려는 형상 주위에 제어 격자점을 설정하고, 이 제어점들을 이동시켜 내부 형상을 부드럽게 변형시키는 방식입니다 (Fig. 3).
통합 설계-해석 워크플로우: 본 연구의 핵심은 이 두 기술을 통합한 양방향 워크플로우입니다 (Fig. 2).
초기 런너 형상(STL 데이터)을 SPH 경계 입자로 변환합니다.
설계자는 FFD 제어점을 조작하여 런너 형상을 실시간으로 수정합니다.
시스템은 변경된 형상에 대해 즉시 고속 SPH 유동 해석을 수행합니다.
설계자는 용탕의 유동 변화를 시각적으로 확인하고, 만족스러운 결과를 얻을 때까지 형상 수정과 해석을 반복합니다.
이러한 접근법은 설계자의 아이디어가 즉각적으로 시뮬레이션에 반영되는 실시간 피드백 루프를 구축하여 설계 최적화 과정을 혁신적으로 단축시킵니다.
The Breakthrough: Key Findings & Data
본 연구는 제안된 기법을 다이캐스팅 런너 설계에 적용하여 그 유효성을 입증했습니다.
Finding 1: 복잡한 런너 유동 현상의 정확한 재현
새로운 해석 기법의 신뢰성을 검증하기 위해, 유동 특성이 잘 알려진 T형 런너와 V형 런너의 용탕 충전 거동을 시뮬레이션했습니다. 그 결과, 기존의 실험 및 상용 소프트웨어 해석 결과와 정성적으로 일치하는 결과를 얻었습니다. – T형 런너: 용탕이 런너를 완전히 채우기 전에 게이트에서 먼저 사출되고, 게이트를 통과한 용탕이 넓게 퍼지는 현상(사출각 β가 90°에 미치지 못함)이 정확하게 재현되었습니다 (Fig. 7). 이는 공기 혼입의 원인이 될 수 있습니다. – V형 런너: 용탕이 런너 형상을 따라 부드럽게 유동하며, 게이트에서 거의 90°에 가까운 안정적인 사출각(β)을 유지하는 모습이 확인되었습니다 (Fig. 8).
Finding 2: 실시간 설계 최적화 및 즉각적인 피드백
본 연구의 가장 핵심적인 성과는 실시간 형상 변경을 통한 설계 개선 가능성을 입증한 것입니다. 문제가 있는 T형 런너를 기반으로 시뮬레이션을 실행하는 도중에 FFD 제어점을 이용해 게이트의 위치를 상하로 이동시켰습니다.
게이트 상향 이동: 게이트 위치를 6.36mm 위로 이동시키자, 사출각(β)이 기존 86.71°에서 90.00°로 개선되었습니다 (Table 2). 이는 용탕의 흐름을 안정시켜 T형 런너의 설계 결함을 실시간으로 해결했음을 의미합니다 (Fig. 10a).
게이트 하향 이동: 반대로 게이트 위치를 5.45mm 아래로 이동시키자, 사출각(β)은 79.01°로 악화되어 유동이 더욱 불안정해지는 것을 즉각적으로 확인할 수 있었습니다 (Fig. 10b).
약 84만 개의 입자를 사용한 이 시뮬레이션은 NVIDIA GeForce GTX 980 GPU 환경에서 초당 약 85 프레임의 속도로 실행되어, 설계자가 지연 없이 상호작용하며 설계안을 탐색할 수 있음을 보여주었습니다.
Practical Implications for R&D and Operations
공정 엔지니어: 이 연구는 런너 및 게이트 형상을 실시간으로 수정하며 공기 혼입을 최소화하고 금형 충전 패턴을 개선하는 등 공정 최적화를 신속하게 수행할 수 있는 가능성을 제시합니다.
품질 관리팀: 논문의 [Table 2]와 [Figure 10] 데이터는 게이트 위치라는 특정 형상 변화가 사출각(β)이라는 핵심 품질 지표에 미치는 영향을 명확하게 보여주므로, 불량의 근본 원인을 파악하고 새로운 검사 기준을 수립하는 데 정보를 제공할 수 있습니다.
설계 엔지니어: 이 결과는 설계자가 고가의 금형을 제작하기 전에 훨씬 더 넓은 설계 공간을 단시간에 탐색할 수 있음을 의미합니다. 단순한 T형이나 V형을 넘어, 유동에 최적화된 새로운 형태의 런너를 발견할 수 있는 강력한 도구가 될 수 있습니다.
Paper Details
高速な流れ解析手法を統合した流路設計のための設計インタフェース -湯流れ解析下におけるダイカスト湯道設計への適用一 (Design Interface for Flow Channel Design Integrated with Highly Efficient Fluid Flow Analysis Method – Application to Runner Design of Die-Casting during Casting Flow Simulation -)
There are a number of useful fluid flow analysis methods that support designers to design flow channels of engineering products or to design flow channels used in manufacturing processes. It is important to derive better design by the iteration of evaluation and refinement of the design proposal so that the resulting product could achieve the required performance. However, most of the conventional methods are not so efficient that the evaluation and refinement cannot be executed enough. In order to make the evaluation part of the iteration process efficient, our previous paper presented a highly efficient fluid flow analysis method that adopted smoothed particle hydrodynamics (SPH) method, and that accelerated its calculation using graphics processing unit (GPU). Furthermore, in order to support designers more efficiently, this paper presents a new method for flow channel design based on form deformation techniques integrated with the analysis method, which enables the modeling of flow channel shape during simulating the flow behavior in it. In order to confirm the usefulness of the method, it is applied to an example of runner design of die-casting during casting flow simulation.
3. Introduction:
제품의 설계 단계에서의 결정은 최종 제품의 성능이나 품질에 큰 영향을 미친다. 제품의 제조 공정에서도 방안 검토라는 프로세스 설계 단계가 존재하며, 이는 제조의 성패와 제품의 품질을 결정한다. 효율적인 설계 및 제조 사이클을 실현하기 위해서는, 물리적 제조 이전에 설계 단계에서 반복적으로 설계안을 검증하고 개선하는 것이 중요하다. 이를 지원하기 위해 다양한 공학 해석 기법이 제안되었으나, 기존 기법 대부분은 계산 시간이 방대하여 반복적인 검증 및 개선을 지원하는 도구로는 부적합했다. 이러한 문제에 대해 저자들은 이전 연구에서 유로를 가진 제품 설계 및 주조/다이캐스팅 공정 방안 검토를 대상으로 고속의 간편한 유동 해석 기법을 제안했다. 이는 해석 프로세스를 고속화하여 효율화를 꾀하는 것이었다(Fig. 1b). 본 연구에서는 한 걸음 더 나아가 설계와 해석 프로세스의 통합을 통해(Fig. 1c) 추가적인 효율화를 실현하고자 한다. 구체적으로는, 저자들이 제안한 해석 기법에 해석 중 실행 가능한 형상 변형 기법을 도입하여 양방향 유로 설계 기법을 제안한다.
4. Summary of the study:
Background of the research topic:
제품 및 제조 공정의 설계 단계에서 반복적인 검증과 개선은 최종 품질을 위해 매우 중요하지만, 기존 공학 해석(CAE) 도구의 긴 계산 시간으로 인해 비효율적이다.
Status of previous research:
저자들은 이전 연구에서 SPH(Smoothed Particle Hydrodynamics) 입자법을 GPU를 이용해 고속화하는 유동 해석 기법을 제안하여, 설계-해석 반복 과정 중 해석 부분의 시간을 단축시키는 연구를 수행했다.
Purpose of the study:
본 연구의 목적은 이전 연구를 발전시켜, 고속 유동 해석 기법에 실시간 형상 변형 기법을 통합함으로써 설계와 해석 프로세스 자체를 통합하는 것이다. 이를 통해 설계자가 시뮬레이션 중에 직접 형상을 수정하며 유동 변화를 즉각적으로 확인할 수 있는 양방향(interactive) 설계 인터페이스를 제안하고, 그 유효성을 검증하고자 한다.
Core study:
제안된 양방향 설계 기법을 다이캐스팅 공정의 런너(탕도) 형상 설계 문제에 적용한다. T형 런너를 기반으로 시뮬레이션 중에 FFD(Free-Form Deformation)를 이용해 게이트 형상을 실시간으로 변형시키고, 이에 따른 용탕의 사출 거동(사출각 등) 변화를 분석하여 설계 개선 가능성을 평가한다.
5. Research Methodology
Research Design:
본 연구는 고속 유동 해석 기법과 형상 변형 기법을 통합한 새로운 설계 인터페이스를 개발하고, 이를 다이캐스팅 런너 설계라는 구체적인 사례에 적용하여 유효성을 검증하는 방식으로 설계되었다. 초기 형상(T형 런너)을 기준으로 실시간 변형을 가했을 때의 유동 거동 변화를 상용 해석 소프트웨어 결과와 비교하여 정성적 일치성을 확인한다.
Data Collection and Analysis Methods:
유동 해석: 입자법의 일종인 SPH(Smoothed Particle Hydrodynamics)를 사용. 지배 방정식으로는 질량 보존, 운동량 보존, 열전도/열전달 방정식을 사용하며, 다이캐스팅 공정의 고압 환경을 고려한 상태 방정식과 반발력 모델을 적용.
고속화: 모든 SPH 계산을 GPU(NVIDIA GeForce GTX 980) 상에서 CUDA 7.0을 이용해 병렬 처리.
형상 변형: FFD(Free-Form Deformation) 기법을 사용하여 제어점 이동을 통해 경계 입자들의 위치와 법선 벡터를 실시간으로 재계산.
사례 연구: 알루미늄 합금 ADC12를 용탕으로 사용하고, 직경 70mm의 슬리브 내에서 플런저를 1m/s 속도로 이동시켜 폭 20mm, 두께 2mm의 게이트를 통해 용탕을 사출하는 조건을 설정.
Research Topics and Scope:
연구의 범위는 GPU 가속 SPH 유동 시뮬레이션 환경 하에서 FFD를 이용한 실시간 형상 변형을 구현하고, 이를 다이캐스팅 런너 형상 설계에 적용하여 그 가능성을 탐색하는 데에 중점을 둔다. 공기 혼입이나 응고와 같은 복잡한 물리 현상은 고려하지 않으며, 정성적인 유동 경향을 신속하게 파악하는 것을 목표로 한다.
6. Key Results:
Key Results:
GPU를 이용한 SPH 계산은 CPU(1코어) 대비 120배 이상의 속도 향상을 보였다 (Table 1).
제안된 기법은 T형 런너와 V형 런너의 특징적인 유동 거동(런너 내 충전 양상, 게이트 사출각 등)을 상용 소프트웨어 결과와 유사하게 재현했다 (Fig. 7, 8).
시뮬레이션 중 T형 런너의 게이트 위치를 실시간으로 상향 이동(6.36mm)시키자, 사출각(β)이 86.71°에서 90.00°로 개선되는 것을 확인했다 (Table 2, Fig. 10).
반대로 게이트 위치를 하향 이동(-5.45mm)시키자, 사출각(β)이 79.01°로 악화되는 것을 즉각적으로 확인했다 (Table 2, Fig. 10).
약 84만 개 입자 모델에 대해 초당 약 85 프레임의 계산 및 렌더링 속도를 달성하여 원활한 양방향 조작이 가능함을 입증했다.
Fig.9 Control points of FFD set to T shape runner
Figure List:
Fig.1 Basic idea of increase in efficiency of design/analysis iterative process
Fig.2 Outline of interactive design/analysis process proposed in this paper
Fig.3 Free-form deformation (FFD) applied to shape defined with particles
Fig.4 Die-casting process using die-casting machine
Fig.5 Evaluation of runner shape by injected molten metal behavior
Fig.6 Example design of runner and gates in a die-casting machine
Fig.7 Simulated behavior of molten metal injected through T shape runner
Fig.8 Simulated behavior of molten metal injected through V shape runner
Fig.9 Control points of FFD set to T shape runner
Fig.10 Simulated behavior in the process of form deformation by user
7. Conclusion:
본 연구에서는 그래픽스 디바이스(GPU)를 통해 고속화된 SPH법 유체 시뮬레이션 기법에 형상 변형 기법을 도입하고, 그 연계 기법을 제안함으로써 유동 해석 하에서의 양방향 유로 형상 변경에 기반한 유로 설계 기법을 제안했다. 이 기법을 다이캐스팅의 런너 형상 검토에 적용 가능함을 보임으로써, 본 기법의 유효성을 나타냈다. 향후, 해석 결과의 정량적 평가 기법, 더 큰 변형 조작에의 대응, 더 복잡한 문제에의 적용, 다른 제조 공정 설계 및 제품 설계에의 적용 등을 검토하고자 한다. 또한, 응고 등을 포함한 더 상세한 해석 기법과의 연계도 검토할 계획이다.
8. References:
H. Tokunaga, T. Okane, and T. Okano: Application of GPU-Accelerated SPH Fluid Simulation to Casting Design, Proceedings of the 2012 Asian Conference on Design and Digital Engineering (ACDDE2012), 100042, (2012).
例えば、J. J. Monaghan: Simulating Free Surface Flows with SPH, Journal of Computational Physics, 110, (1994) 399.
T. W. Sederberg and S. R. Parry: Free-Form Deformation of Solid Geometric Models, Proceedings of SIGGRAPH’86, 20, 4, (1986) 151.
例えば、M. Müller, D. Charypar and M. Gross: Particle-Based Fluid Simulation for Interactive Applications, Proceedings of Eurographics/SIGGRAPH Symposium on Computer Animation, (2003).
N. Umetani, K. Takayama, J. Mitani, T. Igarashi: A Responsive Finite Element Method to Aid Interactive Geometric Modeling, Computer Graphics and Applications, IEEE, 31, 5, (2010) 43.
A. Ferrari, M. Dumbser, E. F. Toro, and A. Armanini: A New 3D Parallel SPH Scheme for Free Surface Flows, Computers & Fluids, 38, (2009) 1203.
P. W. Cleary, J. Ha, M. Prakash, T. Nguyen: 3D SPH Flow Predictions and Validation for High Pressure Die Casting of Automotive Components, Applied Mathematical Modelling, 30, (2006) 1406.
Q1: 이 양방향 접근법을 위해 유한요소법(FEM)과 같은 전통적인 격자 기반 방식 대신 SPH를 선택한 이유는 무엇입니까?
A1: SPH는 격자(mesh)가 없는 입자 기반 방법론이기 때문입니다. FEM과 같은 격자 기반 방법은 형상이 변형될 때마다 복잡하고 시간이 많이 소요되는 격자 재생성(remeshing) 과정이 필요합니다. 이는 실시간 상호작용에 큰 걸림돌이 됩니다. SPH는 이러한 과정이 필요 없어 형상이 동적으로 변하는 환경에 매우 적합하며, 본 연구가 목표하는 양방향 설계 인터페이스 구현에 이상적인 선택이었습니다.
Q2: 논문에서 GPU를 사용하여 120배 이상의 속도 향상을 언급했는데, 이 성능은 입자 수에 따라 어떻게 변합니까?
A2: 논문의 [Table 1]에 따르면, 입자 수가 증가함에 따라 프레임당 계산 시간은 늘어나지만, CPU 대비 GPU의 속도 향상 비율(CPU/GPU)은 약 15만 개에서 79만 개의 입자 수 범위에서 120~130배 수준으로 일관되게 높게 유지됩니다. 이는 제안된 GPU 병렬화 기법이 다양한 문제 크기에 걸쳐 효과적으로 작동함을 시사합니다.
Q3: 다양한 형상 변형 기법 중 FFD(Free-Form Deformation)를 채택한 특별한 이유가 있습니까?
A3: 논문에 따르면, FFD는 형상 표면뿐만 아니라 그 주변 공간 전체의 변형을 다룰 수 있기 때문에 채택되었습니다. SPH에서는 경계면을 표현하는 입자들이 단순히 표면 위에만 있는 것이 아니라, 그 주변에 여러 층으로 배치될 수 있습니다. FFD는 이러한 공간적 변형을 자연스럽게 처리할 수 있는 가장 기본적인 기법 중 하나로, 입자 기반 모델링에 적합하다고 판단되었습니다.
Q4: 다이캐스팅의 고압 환경에서 입자들이 경계를 뚫고 나가는 문제없이 어떻게 안정적인 해석을 수행했습니까?
A4: 본 연구에서는 다이캐스팅의 고압 환경을 고려하여 수정된 상태 방정식(Eq. 7)과 반발력 모델(Eq. 8)을 사용했습니다. 이 식들은 최대 유속(Vmax)을 명시적으로 고려하여 압력과 반발력을 계산합니다. Vmax 값을 적절히 설정함으로써, 고압으로 인해 발생할 수 있는 계산 불안정성이나 입자의 경계 투과 현상을 효과적으로 방지할 수 있었습니다.
Q5: [Table 2]의 해석 결과를 보면, 제안된 기법과 기존 상용 소프트웨어의 사출각(β) 값에 차이가 있습니다. 사용자는 이 차이를 어떻게 해석해야 합니까?
A5: 논문의 고찰(Discussion) 부분에서 언급하듯이, 서로 다른 해석 기법들은 정량적인 결과에서 차이를 보이는 것이 일반적입니다. 본 연구 기법의 주된 목표는 유동 거동의 정성적 경향을 빠르고 정확하게 파악하는 것입니다. 결과적으로 게이트를 올리면 사출각이 개선되고 내리면 악화된다는 경향성은 두 방법에서 동일하게 나타났습니다. 따라서 이 도구는 설계 초기 단계에서 다양한 아이디어를 신속하게 탐색하고 경향을 파악하는 데 매우 유용하며, 최종적인 정량 검증은 실험이나 고정밀 시뮬레이션을 통해 보완할 수 있습니다.
Conclusion: Paving the Way for Higher Quality and Productivity
기존의 긴 해석 시간은 다이캐스팅 공정에서 최적의 런너 설계를 찾는 데 큰 장벽이었습니다. 본 연구는 GPU 가속 SPH 해석과 실시간 형상 변형 기술을 통합하여 이 문제를 해결하는 혁신적인 돌파구를 제시했습니다. 설계자가 시뮬레이션 중에 직접 형상을 수정하고 그 결과를 즉시 확인함으로써, 설계-평가 주기를 획기적으로 단축하고 데이터에 기반한 신속한 의사결정을 내릴 수 있게 되었습니다.
이러한 양방향 설계 환경은 다이캐스팅 부품의 품질을 향상시키고 개발 기간을 단축하는 데 기여할 강력한 잠재력을 가지고 있습니다.
“STI C&D에서는 고객이 더 높은 생산성과 품질을 달성할 수 있도록 최신 산업 연구를 적용하는 데 전념하고 있습니다. 이 논문에서 논의된 과제가 귀사의 운영 목표와 일치한다면, 저희 엔지니어링 팀에 연락하여 이러한 원칙을 귀사의 부품에 어떻게 구현할 수 있는지 논의해 보십시오.”
(주)에스티아이씨앤디에서는 고객이 수치해석을 직접 수행하고 싶지만 경험이 없거나, 시간이 없어서 용역을 통해 수치해석 결과를 얻고자 하는 경우 전문 엔지니어를 통해 CFD consulting services를 제공합니다. 귀하께서 당면하고 있는 연구프로젝트를 최소의 비용으로, 최적의 해결방안을 찾을 수 있도록 지원합니다.
연락처 : 02-2026-0450
이메일 : flow3d@stikorea.co.kr
Copyright Information
This content is a summary and analysis based on the paper “高速な流れ解析手法を統合した流路設計のための設計インタフェース -湯流れ解析下におけるダイカスト湯道設計への適用一” by “徳永 仁史, 岡根 利光, 岡野 豊明”.
이 기술 요약은 Rupayan Saha, Seung Oh Lee, Seung Ho Hong이 2018년 ‘water’ 저널에 발표한 논문 “A Comprehensive Method of Calculating Maximum Bridge Scour Depth”를 기반으로 하며, STI C&D가 기술 전문가를 위해 분석하고 요약했습니다.
문제점: 기존의 교량 세굴 깊이 산정 공식은 극한 기상 현상으로 인한 교각 월류(overtopping)나 잠김 흐름(submerged flow)과 같은 복잡한 유동 조건을 정확히 반영하지 못하며, 국부 세굴과 수축 세굴을 독립적인 현상으로 간주하여 예측 정확도가 떨어집니다.
연구 방법: 실제 하천(Towaliga River)의 지형을 1:60으로 축소한 복단면 수로 수리 모형을 제작하고, 자유 수면 흐름, 잠김 오리피스 흐름, 월류 흐름 등 다양한 조건에서 세굴 실험을 수행했습니다.
핵심 발견: 최대 세굴 깊이는 ‘이론적 교각 세굴’과 ‘흐름 수축에 의한 추가 세굴’의 합으로 구성된다는 가설을 세우고, 추가 세굴량이 유량 수축비와 명확한 상관관계가 있음을 실험적으로 증명했습니다.
핵심 결론: 본 연구는 복잡한 흐름 조건에서도 최대 교량 세굴 깊이를 더 정확하게 예측할 수 있는 통합적이고 포괄적인 방법을 제시하여, 교량의 구조적 안전성 설계를 크게 향상시킬 수 있습니다.
문제점: CFD 전문가에게 이 연구가 중요한 이유
교량 붕괴의 가장 큰 원인은 교량 기초 주변의 하상 재료가 유실되는 ‘세굴’ 현상입니다. 미국에서는 1950년 이후 발생한 교량 붕괴의 약 60%가 세굴과 관련 있을 정도로 교량 안전에 치명적입니다. 특히 최근 빈번해지는 극한 기상 현상은 설계 기준을 초과하는 홍수를 유발하며 교량의 안전을 심각하게 위협합니다.
기존의 세굴 깊이 예측 공식들은 대부분 단순화된 사각 수로에서의 자유 수면 흐름 실험을 기반으로 개발되었습니다. 이로 인해 실제 하천의 불규칙한 지형이나, 극한 홍수 시 발생하는 교량 상판 월류 및 잠김 흐름과 같은 복잡한 수리 현상을 제대로 모사하지 못하는 한계가 있습니다. 또한, 현재 설계 실무에서는 흐름 단면 축소로 인한 ‘수축 세굴’과 교각 주변의 와류로 인한 ‘국부 세굴’을 별개의 현상으로 보고 각각 계산한 뒤 합산하지만, 실제로는 두 현상이 동시에 상호작용하며 발생하기 때문에 예측에 오차가 발생합니다. 이러한 부정확성은 교량 설계의 과잉 또는 과소 평가로 이어져 비경제적이거나 위험한 결과를 초래할 수 있습니다.
Figure 1. Towaliga River bridge in the field and model in the laboratory.
연구 접근법: 방법론 분석
본 연구는 이러한 한계를 극복하기 위해 미국 조지아주 메이컨에 위치한 Towaliga 강 교량을 대상으로 1:60 축척의 정밀한 물리적 수리 모형을 제작했습니다. 이 모형은 실제 하천의 복잡한 지형(복단면 형상)을 그대로 재현했으며, 세굴 실험을 위해 중앙부에 이동상(mobile bed) 구간을 설치하고 0.53mm의 중간 입경(d50)을 가진 모래를 사용했습니다.
연구팀은 다양한 홍수 시나리오를 모사하기 위해 세 가지 주요 흐름 조건에서 실험을 수행했습니다. 1. 자유 수면 흐름 (Free Flow): 일반적인 홍수 조건 2. 잠김 오리피스 흐름 (Submerged Orifice Flow): 수위가 교량 상판 하단까지 상승한 조건 3. 월류 흐름 (Overtopping Flow): 수위가 교량 상판을 넘어 흐르는 극한 홍수 조건
각 실험에서 유량과 수위를 정밀하게 제어했으며, 세굴이 평형 상태에 도달할 때까지 5~6일간 실험을 지속했습니다. 세굴 전후의 하상 고도는 음향 도플러 유속계(Acoustic Doppler Velocimeter, ADV)와 포인트 게이지를 사용하여 상세하게 측정되었고, 이를 통해 최대 세굴 깊이와 위치를 정확하게 파악했습니다.
핵심 발견: 주요 결과 및 데이터
연구팀은 실험 결과를 분석하여 기존 세굴 예측 방식의 한계를 극복할 새로운 종합적 방법을 제안했습니다.
결과 1: 최대 세굴 깊이의 새로운 구성 = 이론적 세굴 + 흐름 수축에 의한 추가 세굴
본 연구는 최대 세굴 깊이가 기존의 이론적 교각 세굴 깊이(CSU 또는 M/S 공식으로 계산)에 ‘흐름 수축으로 인한 추가적인 세굴 깊이’가 더해진 결과라는 핵심적인 가설을 제시했습니다. 실험 데이터를 분석한 결과, 이 ‘추가 세굴’ 성분은 교량을 통과하는 흐름의 수축 정도를 나타내는 ‘유량 수축비(q2/q1)’와 매우 강한 양의 상관관계를 보였습니다. 그림 4에서 볼 수 있듯이, 유량 수축비가 증가함에 따라 추가 세굴 깊이(Ym-csu/Y1)가 선형적으로 증가하는 경향이 뚜렷하게 나타났습니다. 이는 흐름이 교량 구간에서 더 많이 압축될수록 세굴이 더 심각해진다는 것을 정량적으로 입증한 것입니다.
결과 2: 압력 흐름 조건에서 세굴 효과 증폭
그림 4의 데이터는 또 다른 중요한 사실을 보여줍니다. 자유 수면 흐름(F)에 비해 잠김 오리피스 흐름(SO)이나 월류 흐름(OT)과 같은 압력 흐름(Pressure Flow) 조건에서 유량 수축비 증가에 따른 추가 세굴 깊이의 증가율(그래프의 기울기)이 훨씬 더 가파릅니다. 이는 압력 흐름 조건에서는 기존의 수평적 흐름 수축뿐만 아니라 교량 상판에 의한 수직적 흐름 수축이 추가로 발생하여 유속이 더 크게 증가하고, 결과적으로 세굴 현상이 증폭되기 때문입니다. 이는 극한 홍수 시 교량 안전성 평가에 반드시 고려해야 할 핵심 요소입니다.
R&D 및 운영을 위한 실질적 시사점
이 연구 결과는 교량 설계 및 안전 관리 분야의 전문가들에게 다음과 같은 실질적인 시사점을 제공합니다.
수리 및 교량 설계 엔지니어: 본 연구에서 제안된 5단계 최대 세굴 깊이 예측 절차(현장 데이터 수집 → 흐름 변수 계산 → 이론적 교각 세굴 계산 → 흐름 수축에 따른 추가 세굴 추정 → 합산)는 기존 방식보다 훨씬 더 정확하고 신뢰성 있는 교량 기초 설계를 가능하게 합니다. 특히 극한 홍수 조건에 대한 안전성을 크게 향상시킬 수 있습니다.
위험 평가팀: 압력 흐름(월류 등)이 세굴을 크게 증폭시킨다는 그림 4의 결과는, 100년 또는 500년 빈도의 극한 홍수 시 월류 가능성이 있는 기존 교량들의 안전성을 재평가해야 할 필요성을 강력하게 시사합니다.
CFD 모델링 전문가: 이 연구에서 측정된 복잡한 흐름 조건(잠김, 월류) 하의 상세한 실험 데이터는 교량 세굴에 대한 수치 모델링(CFD)의 정확도를 검증하고 개선하는 데 매우 귀중한 자료로 활용될 수 있습니다.
논문 상세 정보
A Comprehensive Method of Calculating Maximum Bridge Scour Depth
1. 개요:
제목: A Comprehensive Method of Calculating Maximum Bridge Scour Depth (최대 교량 세굴 깊이 계산을 위한 종합적 방법)
최근 극한 기상 현상의 반복적인 발생으로 교량 주변의 세굴 문제가 두드러지고 있습니다. 따라서 교량은 이러한 극한 기상 현상 동안 겪을 수 있는 높은 유량에 대한 세굴로 인한 붕괴를 방지하기 위해 적절한 보호 조치를 갖추어 설계되어야 합니다. 그러나 여러 권장 공식에 의한 현재의 세굴 깊이 추정은 높은 유량에서 부정확한 결과를 보여줍니다. 한 가지 가능한 이유는 현재의 세굴 공식이 자유 표면 흐름을 이용한 실험에 기반하고 있지만, 극한 홍수 사건은 잠김 오리피스 흐름과 결합된 교량 월류 흐름을 유발할 수 있다는 점입니다. 또 다른 가능한 이유는 최대 세굴 깊이에 대한 현재의 관행이 국부 세굴과 수축 세굴과 같은 다른 유형의 세굴 간의 상호작용을 무시한다는 점인데, 실제로는 이러한 과정들이 동시에 발생합니다. 본 논문에서는 축소된 교량 모델을 사용하여 복합 단면 수로에서 다양한 흐름 조건(자유, 잠김 오리피스, 월류 흐름) 하에 실험실 실험을 수행했습니다. 실험실 실험 결과와 널리 사용되는 경험적 세굴 추정 방법을 결합하여, 다른 세굴 깊이의 개별적 추정과 다른 세굴 구성 요소의 상호작용에 관한 문제를 극복하는 최대 세굴 깊이를 예측하는 포괄적인 방법을 제안합니다. 또한, 최대 세굴 깊이에 대한 교각 벤트(교대에 가깝게 위치)의 존재 효과도 분석 중에 조사되었습니다. 결과는 최대 세굴 깊이의 위치는 교각 벤트의 존재와 무관하지만, 최대 세굴 깊이의 양은 교각 벤트가 있을 때보다 없을 때 유량 재분배로 인해 상대적으로 더 높다는 것을 보여줍니다.
3. 서론:
교량이 강에 건설되면, 교각과 교대 주변에 국부적으로 독특한 유동장이 발달하기 때문에 교량 주변의 흐름 패턴이 바뀝니다. 또한, 강 양쪽 또는 한쪽에 있는 제방/교대로 인해 흐름 면적이 줄어들어 가속으로 인한 유속이 빨라집니다. 더 높은 속도를 가진 이 독특한 유동장은 교량 기초에 심각한 손상을 줄 수 있습니다. 따라서 기초의 깊이가 충분히 깊지 않으면 교량 붕괴의 가능성이 높아집니다. 교량은 지진, 바람, 홍수 등 여러 원인으로 붕괴될 수 있습니다. 그중에서도 교량 세굴은 교량 붕괴의 가장 큰 원인입니다. 예를 들어, 1950년 이후 미국에서 발생한 전체 교량 붕괴 중 약 60%가 교량 기초의 세굴과 관련이 있습니다. 콜로라도 교통부(CDOT)는 2013년 홍수로 최소 30개의 주 고속도로 교량이 파괴되고 20개가 심각하게 손상되었다고 추정했습니다. 네팔에서는 2014년 홍수 동안 하상 재료의 퇴화로 인해 티나우 강 위의 고속도로 교량 기초가 심각하게 노출되었습니다. 위 예에서 설명한 바와 같이, 교량 세굴은 전 세계적으로 주요 교량 안전 문제 중 하나라고 말하는 것이 정당합니다. 따라서 교량 기초에서의 정확한 세굴 예측은 교량 안전을 위한 엔지니어의 주요 목표가 됩니다.
4. 연구 요약:
연구 주제의 배경:
교량 세굴은 교량 붕괴의 주된 원인으로, 특히 극한 홍수 시 그 위험성이 커집니다. 기존의 세굴 예측 공식은 실제 하천의 복잡한 흐름 조건과 세굴 메커니즘의 상호작용을 제대로 반영하지 못해 정확도에 한계가 있었습니다.
이전 연구 현황:
1950년대 후반부터 수많은 연구가 진행되어 평형 세굴 깊이 추정 공식이 개발되었습니다. 그러나 대부분의 연구는 단순화된 직사각형 수로와 자유 수면 흐름 조건에서 수행되었습니다. 또한, 국부 세굴과 수축 세굴을 독립적인 과정으로 가정하여 각각을 계산 후 합산하는 방식을 사용해왔습니다.
연구 목적:
본 연구의 주된 목적은 다양한 유형의 세굴이 동시에 발생하는 상황에서 최대 세굴 깊이를 예측하는 데 사용할 수 있는 단일 방정식을 개발하는 것입니다. 이를 위해 서로 다른 세굴 구성 요소 간의 상호 작용을 규명하고, 널리 사용되는 세굴 공식(CSU, M/S)과 비교하여 최대 세굴 깊이를 계산하는 개선된 방법을 제안하고자 합니다.
핵심 연구:
실제 하천 지형을 모사한 1:60 축소 수리 모형을 이용하여 자유 흐름, 잠김 오리피스 흐름, 월류 흐름 조건에서 실험을 수행했습니다. 실험을 통해 측정한 최대 세굴 깊이와 기존 이론 공식을 비교 분석하여, ‘이론적 교각 세굴’과 ‘흐름 수축에 의한 추가 세굴’의 합으로 최대 세굴 깊이를 표현하는 새로운 접근법을 제시하고, 그 유효성을 검증했습니다.
5. 연구 방법론
연구 설계:
본 연구는 실제 교량(Towaliga River bridge)의 1:60 축소 물리 모형을 이용한 실험적 접근법을 채택했습니다. 복단면 형상의 수로에 이동상 구간을 설치하고, 다양한 수리 조건(자유 흐름, 잠김 오리피스 흐름, 월류 흐름)을 재현하여 세굴 현상을 관찰하고 측정했습니다.
데이터 수집 및 분석 방법:
하상 변동 측정: 음향 도플러 유속계(ADV)와 포인트 게이지를 사용하여 실험 전후의 하상 고도를 정밀하게 측정하고, 이를 통해 세굴 깊이와 범위를 분석했습니다.
유속 측정: ADV를 사용하여 접근부 및 교량 단면에서 3차원 유속 분포를 측정했습니다.
데이터 분석: 측정된 유량, 수위, 유속, 세굴 깊이 등의 변수를 사용하여 기존 세굴 공식(CSU, M/S)과 본 연구에서 제안한 새로운 모델을 비교 분석했습니다. 특히, ‘추가 세굴 깊이’와 ‘유량 수축비’ 간의 상관관계를 회귀 분석을 통해 도출했습니다.
Figure 3. Schematic diagram for calculation of maximum scour depth.
연구 주제 및 범위:
주요 연구 주제: 복잡한 흐름 조건(특히 압력 흐름)에서 발생하는 최대 교량 세굴 깊이의 종합적인 예측 방법 개발.
연구 범위: 단일 교량을 대상으로 한 축소 모형 실험에 국한됩니다. 실험은 청수 세굴(clear-water scour) 조건에서 수행되었으며, 퇴적물 입경은 0.53mm로 고정되었습니다. 교각 벤트의 유무에 따른 영향을 질적으로 분석했습니다.
6. 주요 결과:
주요 결과:
최대 세굴 깊이는 이론적 교각 세굴 깊이와 흐름 수축에 의한 추가 세굴 깊이의 합으로 표현될 수 있습니다.
흐름 수축에 의한 추가 세굴 깊이는 유량 수축비(q2/q1)와 강한 양의 상관관계를 가집니다. 즉, 유량 수축비가 클수록 추가 세굴이 더 깊어집니다.
압력 흐름(잠김 및 월류) 조건에서는 자유 수면 흐름 조건에 비해 추가 세굴 효과가 더 크게 나타납니다. 이는 수직 흐름 수축이 추가되기 때문입니다.
최대 세굴 깊이의 발생 위치는 인접한 교각의 유무와 무관하지만, 인접 교각이 없을 경우 유량 재분배로 인해 최대 세굴 깊이가 약간 더 깊어지는 경향을 보입니다.
기존 공식 중 CSU 공식이 M/S 공식보다 동일 조건에서 더 큰 세굴 깊이를 예측하며, 이는 M/S 공식이 청수 세굴 조건을 고려하는 유속 강도 인자(V2/Vc)를 포함하기 때문입니다.
Table 2. Summary of experimental results to calculate maximum scour depth.
그림 목록:
Figure 1. Towaliga River bridge in the field and model in the laboratory.
Figure 2. Geometry of compound channel for (a) plan view with velocity measurement locations; (b) cross section view at bridge.
Figure 3. Schematic diagram for calculation of maximum scour depth.
Figure 4. Effect of flow contraction on additional scour components using (a) Colorado State University (CSU) and (b) Melville-Sheppard (M/S) equations.
Figure 5. Comparison of CSU and M/S pier scour depth in terms of flow intensity.
Figure 6. Comparison of cross-sections for runs 3 and 8.
7. 결론:
많은 연구가 교량 교각 주변의 최대 세굴 깊이를 추정하고 세굴 메커니즘을 이해하기 위해 이루어졌습니다. 대부분의 이전 연구는 자유 흐름 하의 직사각형 수로를 사용한 실험실 실험에 기반했습니다. 그러나 최근의 극한 강우 사건으로 인해 교량에서는 잠김 오리피스 흐름과 월류 흐름이 빈번하게 발생하며, 이때 교량 하부 구조 주변의 유동장은 기존의 측면 흐름 수축에 더해 수직 흐름 수축 때문에 자유 흐름보다 더 복잡합니다. 또한, 대부분의 자연 하천 형태는 직사각형이 아닙니다. 현재 HEC-18에서 권장하는 지침은 수축 세굴과 국부 세굴 과정이 독립적이어서 별도로 결정하고 합산하여 총 세굴 깊이를 추정할 수 있다고 가정했습니다. 그러나 대규모 홍수 사건 동안 국부 세굴과 수축 세굴은 동시에 발생하며, 국부 세굴과 수축 세굴을 별도로 계산하면 부정확한 세굴 깊이를 초래합니다. 현재 방법론이 가진 약점을 극복하기 위해, 축소된 물리적 모델에서 실험실 실험을 수행하고 압력 흐름뿐만 아니라 자유 흐름 사례에서도 다른 유형의 세굴 구성 요소를 별도로 계산하지 않고 사용할 수 있는 최대 세굴 깊이를 예측하기 위한 단일 방정식이 개발되었습니다.
8. 참고 문헌:
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전문가 Q&A: 자주 묻는 질문
Q1: 일반적인 사각 수로가 아닌 특정 강(Towaliga River)의 1:60 축소 모형을 사용한 이유는 무엇인가요?
A1: 실제 하천은 본류와 홍수터로 구성된 복단면 형상과 불규칙한 지형을 가지고 있습니다. 단순화된 사각 수로는 이러한 복잡성을 재현할 수 없습니다. 실제 하천 지형을 그대로 모사함으로써, 본 연구의 결과가 이상적인 실험실 조건을 넘어 실제 현장에 더 가깝게 적용될 수 있도록 신뢰도를 높이기 위함입니다.
Q2: 그림 4에서 자유 흐름과 압력 흐름의 추세선 기울기가 다르게 나타나는 물리적 이유는 무엇인가요?
A2: 압력 흐름(잠김 및 월류) 조건에서는 교량 상판으로 인해 흐름이 수직 방향으로도 압축됩니다. 이는 기존의 수평적 흐름 수축에 더해 추가적인 유속 증가를 유발합니다. 따라서 동일한 유량 수축비(q2/q1)에서도 압력 흐름 조건일 때 ‘추가 세굴’ 효과가 더 크게 나타나 그래프의 기울기가 더 가파르게 되는 것입니다.
Q3: 논문에서 비교한 CSU 공식과 M/S 공식 중, M/S 공식이 지속적으로 더 낮은 세굴 깊이를 예측하는 이유는 무엇인가요? (그림 5 참조)
A3: M/S 공식은 유속과 한계유속의 비(V2/Vc)인 ‘유속 강도 인자’를 포함하여 청수 세굴(clear-water scour) 조건을 고려합니다. 반면, CSU 공식은 주로 이동상 세굴(live-bed scour)을 기반으로 개발되어 이 인자를 1로 가정합니다. 본 연구는 청수 세굴 조건에서 수행되었으므로, M/S 공식이 유속 강도 인자를 반영하여 CSU 공식보다 더 낮은 세굴 깊이를 예측하게 됩니다.
Q4: 실험 7과 8에서 교각 #7을 제거한 것의 의미는 무엇인가요?
A4: 이는 교각 간의 상호작용과 인접한 교각의 존재가 최대 세굴 깊이에 미치는 영향을 질적으로 분석하기 위함이었습니다. 실험 결과, 최대 세굴이 발생하는 ‘위치’는 교각 #7의 유무와 상관없이 교각 #6에서 동일했습니다. 하지만 최대 세굴의 ‘깊이’는 교각 #7이 없을 때 유량 재분배 현상으로 인해 약간 더 깊게 나타났습니다.
Q5: 이 연구에서 ‘흐름 수축에 의한 추가 세굴’은 어떻게 정의되고 계산되었나요?
A5: 이는 측정된 총 최대 세굴 깊이에서 표준 이론적 교각 세굴 공식으로 설명되지 않는 부분을 의미합니다. 구체적으로, 세굴이 가장 깊은 지점의 총 수심(Ym)에서 CSU 공식(dcsu) 또는 M/S 공식(dms)으로 계산된 이론적 교각 세굴 깊이를 빼서 계산했습니다. 이는 논문의 식 (7)과 (8)에 명시되어 있습니다.
결론: 더 높은 품질과 생산성을 위한 길
기존의 교량 세굴 깊이 예측 방법은 극한 홍수와 같은 복잡한 실제 상황을 제대로 반영하지 못하는 명백한 한계를 가지고 있었습니다. 본 연구는 ‘이론적 세굴’과 ‘흐름 수축에 의한 추가 세굴’을 결합하는 포괄적인 접근법을 제시함으로써 이 문제를 해결하는 중요한 돌파구를 마련했습니다. 특히 압력 흐름 조건에서 세굴이 증폭된다는 사실을 정량적으로 밝혀내어, 교량 설계 및 안전 진단의 정확성을 한 차원 높일 수 있는 실질적인 통찰력을 제공합니다.
(주)에스티아이씨앤디에서는 고객이 수치해석을 직접 수행하고 싶지만 경험이 없거나, 시간이 없어서 용역을 통해 수치해석 결과를 얻고자 하는 경우 전문 엔지니어를 통해 CFD consulting services를 제공합니다. 귀하께서 당면하고 있는 연구프로젝트를 최소의 비용으로, 최적의 해결방안을 찾을 수 있도록 지원합니다.
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저작권 정보
이 콘텐츠는 “Rupayan Saha” 외 저자의 논문 “A Comprehensive Method of Calculating Maximum Bridge Scour Depth”를 기반으로 한 요약 및 분석 자료입니다.
FLOW-3D HYDRO는 3D 전산 유체 역학(CFD) 소프트웨어로, 특히 자유 표면(free surface) 애플리케이션에 특화되어 있습니다. 이 문서는FLOW-3D HYDRO의 기본 모델 설정 과정을 검토하고, 주요 기능과 활용 사례를 제시합니다. 수치 모델은 설계 최적화, 미래 조건에서의 성능 예측, 기존 문제의 원인 조사, 위험 완화 및 의사 결정에 대한 신뢰도 향상에 기여합니다. CFD는 문제의 3D 유동 특성, 높은 수준의 정확도 요구, 높은 위험도 프로젝트, 상호 작용하는 복잡한 물리 현상 포함 등의 경우에 유용합니다. 이 소프트웨어는 유체 흐름을 시뮬레이션하여 엔지니어링 판단을 보완하고 설계 최적화에 기여하는 강력한 도구입니다.
1. 📝 FLOW-3D HYDRO의 기본 모델 설정 및 활용 개요
이 문서는 FLOW-3D HYDRO 소프트웨어의 기본 모델 설정 과정을 다룬다.
소프트웨어의 주요 기능, 활용 사례, 그리고 중요 개념을 요약하여 제시한다.
독자는 이 문서를 통해 FLOW-3D HYDRO의 활용 목적과 범위에 대한 전체적인 이해를 얻을 수 있다.
2. 🚀 FLOW-3D HYDRO의 3D 유동 해석과 자유 표면 특화 기능
FLOW-3D HYDRO는 Navier-Stokes 방정식을 3D로 풀어 유체 흐름을 시뮬레이션하는 소프트웨어이다.
이 소프트웨어는 특히 공기-물 인터페이스, 예를 들어 하천, 개수로, 댐, 수문, 교량 등에서의 자유 표면 문제 해결에 특화되어 있다.
횡단면 심도 평균 가정 없이 정확한 3D 유동 문제를 다룰 수 있다.
움직이는 객체나 퇴적물 수송 등 다양한 물리 현상과 연동하여 복합 분석이 가능하다.
메시 생성 및 형상 처리가 간단하며, 단일 유체 체적(Volume of Fluid) 접근 방식을 활용하여 계산 효율성이 높다.
자유 표면은 대체로 공기와 물 사이의 경계를 의미한다.
3. 🚀 FLOW-3D HYDRO를 활용한 수치 모델의 목적과 CFD 필요성
수치 모델은 통찰력을 얻고, 설계 및 운영 계획을 최적화하며, 미래 조건에서의 성능 예측, 기존 문제의 원인 조사, 위험 완화 및 의사 결정 신뢰도 향상을 위해 활용된다.
FLOW-3D HYDRO는 단순히 도구일 뿐이므로, 최종 판단은 엔지니어가 직접 적용해야 한다.
하지만 이 모델은 위험을 줄이고, 올바른 결정을 내리고 있다는 확신을 부여하는 중요한 수단이다.
CFD(전산유체역학)는 문제의 3D 유동 특성, 높은 정확도 요구, 위험도가 높은 프로젝트, 상호 작용하는 복잡한 물리 현상 등이 포함될 때 최적의 선택이다.
3D CFD 모델은 클라이언트 및 이해관계자와의 효과적인 의사소통 도구이고, 1D·2D·물리모델의 보완 도구로 활용될 수 있다.
4. 🏗️ 위어 유동 모델 설정 및 시뮬레이션 목표
FLOW-3D HYDRO를 사용하여 단순 위어 유동 모델을 설정하는 과정을 예시로 설명한다.
이 모델 설정 과정은 스필웨이 모델 등 다양한 복잡한 모델에도 똑같이 적용될 수 있다.
시뮬레이션의 목표는 스필웨이의 용량 평가, 유량 변화에 따른 통과 능력 확인, 에너지 소산 시설(dissipator)의 설계 최적화이다.
잠재적인 설계 문제(예: 벽이 너무 짧아 물이 옆으로 넘칠 수 있음)를 식별할 수 있다.
또한 수문(gates), 퇴적물 수송, 공기 혼입, 캐비테이션 등 추가적인 물리 현상도 연구 가능하다.
5. 🖥️ FLOW-3D HYDRO 모델 설정 및 시뮬레이션 준비 과정
Simulation Manager 창은 프로그램을 시작할 때 가장 먼저 보이며, 워크스페이스 생성으로 프로젝트를 정리할 수 있다.
모델 설정(Model Setup) 탭은 좌측에 위젯들이 배열되어 있고, 일반적으로 위에서 아래로 순서대로 작업을 진행한다.
전역 설정에서 시뮬레이션 종료 시간을 정의하고, 물리(Physics) 설정에서는 중력과 RNG 난류 모델이 기본 적용되며, 필요시 침식-퇴적, 열전달 등 추가 현상을 활성화할 수 있다.
유체 설정에서는 20°C 물이 자동 로드되며, 밀도 등 속성 변경이 가능하다.
형상(Geometry) 설정에서는 외부 CAD 소프트웨어의 3D STL 파일이나 FLOW-3D 내장 도형을 가져오고, 컴포넌트별 명칭/조직화, 표면 거칠기 적용이 가능하다; 메시는 형상에 맞게 추가 및 크기, 해상도 정의가 가능하고, FAVOR 기법으로 3D 솔리드 표현을 확인할 수 있다.
거친 메시는 빠른 실행이 가능하지만 표면 표현이 단순하며, 모델 검증 후 점차 미세한 해상도로 조정하는 것이 권장된다.
경계 조건에서는 각 메시 면에 조건을 지정하고, 상류·하류 유체의 고도 및 시간 시리즈 입력이 가능하며, 초기 조건으로 빠른 정상 상태 진입을 도울 수 있다.
출력 설정을 통해 저장 변수와 간격을 제어하여, 시뮬레이션 동안 결과 모니터링이 가능하다.
5.1. ️ FLOW-3D HYDRO 사용자 인터페이스와 기본 워크플로우
시뮬레이션 관리자(Simulation Manager) 창은 FLOW-3D HYDRO 실행 시 가장 먼저 보이는 첫 화면이다.
모델 설정(Model Setup) 탭은 주로 사용하는 작업 공간이며, 왼쪽에 위젯들이 배치되어 있고 위에서 아래로 순서대로 진행된다.
워크스페이스(Workspace)는 프로젝트 폴더와 유사한 개념으로, 모델 정리에 사용된다.
새 시뮬레이션을 추가할 때 이름 지정, 단위 선택, 그리고 사전 로드된 템플릿의 활용이 가능하다.
5.2. 기본 모델 설정 및 물리 환경 정의
전역(Global) 설정에서는 시뮬레이션 종료 시간을 사용자가 직접 정의하며, 예를 들어 30초로 설정할 수 있다.
물리(Physics) 설정에서는 템플릿을 통해 중력과 RNG 난류 모델이 기본적으로 활성화되어 있다.
필요에 따라 침식-퇴적, 열전달 등 추가적인 물리 현상을 옵션으로 활성화할 수 있다.
유체(Fluids) 설정에서 20°C의 물이 템플릿을 통해 기본적으로 로드된다.
5.3. ️ 외부 CAD 모델의 시뮬레이션 환경으로의 통합 및 속성 조정
밀도와 같은 속성을 변경할 수 있다.
형상(Geometry) 설정 시 외부 CAD 소프트웨어에서 생성된 3D STL 파일을 가져오는 것이 일반적이다.
외부에서 생성된 3D 모델을 시뮬레이션에 맞게 변환, 조정, 속성 설정 등을 수행해야 한다고 추정된다.
시뮬레이션의 정확도를 높이기 위하여 객체별로 속성 및 세부 설정이 필요하다.
5.4. FLOW-3D HYDRO 메시 생성 및 설정의 핵심
FLOW-3D HYDRO에서는 내장 프리미티브 도형, ASC 지형 파일, 여러 STL 파일 등 다양한 형상 파일의 불러오기 및 조작이 가능하다.
객체의 크기 조절이나 이동, 컴포넌트 분리와 이름 변경을 통해 형상 데이터를 효과적으로 조직화하고 속성을 정의할 수 있다.
각 컴포넌트별로 표면 거칠기 값을 다르게 지정함으로써, 예를 들어 지형은 0.01, 콘크리트는 더 부드럽게 설정 가능하다.
메시(Mesh)는 단일 균일 메시로 생성하며, 크기(예: 0.1m)와 해상도를 직접 정의하고, 메시의 확장 범위(extents) 또한 수정할 수 있다.
FAVOR(Fractional Area Volume Obstacle Representation) 기법을 통해 3D 솔리드가 메시 내에 어떻게 임베드되는지 확인할 수 있으며, 메시 해상도가 모델의 형상 표현에 중요한 영향을 미친다.
5.5. 시뮬레이션을 위한 경계 조건, 초기 조건, 출력 설정 요약
시뮬레이션 모델은 처음에 거친 메시로 테스트한 후, 기능이 확인되면 점진적으로 미세한 메시로 변경하여 사용한다.
경계 조건은 각 메시 면에 적용하며, 압력 경계에서는 유체 고도(예: 업스트림 4.5m, 다운스트림 2m)를 사용하고, 필요 시 시간에 따른 데이터(시간 시리즈) 입력도 가능하다.
초기 조건으로 모델 내에 초기 물을 설정하면 정상 상태에 더 빠르게 도달할 수 있으며, 전역 수위를 지정하거나 업스트림 유체 영역을 경계 조건에 맞게 추가로 지정할 수 있다.
출력 설정에서는 저장 변수와 저장 간격(예: 0.1초 간격) 등을 지정하여, 시뮬레이션 도중 결과를 모니터링한다.
경계 조건, 초기 조건, 출력 설정을 통해 시뮬레이션의 입력 및 출력 상태를 구체적이고 유연하게 제어할 수 있다.
6. 🖥️ 모델 실행과 실시간 결과 분석 방법
시뮬레이션 관리자의 “시뮬레이션(Simulate)” 버튼을 클릭해 모델 실행이 가능하다.
시뮬레이션은 로컬 컴퓨터에서 실행할 수 있다.
실행 중에는 시간 시리즈 플롯을 추가해 진행 상황(예: 상류 유량)을 모니터링할 수 있다.
“분석(Analyze)” 탭에서 3D 플롯으로 실행 중 실시간 결과를 확인할 수 있고, 투명도 조절 및 시간 단계별 확인이 가능하다.
새로운 시간 단계가 저장되면, 이를 다시 로드하여 최신 결과를 확인할 수 있다.
7. 🦾 FLOW-3D HYDRO의 사후 처리와 모델 활용 및 결론
모델 빌드 후 메시 크기, 경계 조건, 수위, 물리 현상, 수치 옵션, 형상 등 다양한 매개변수를 손쉽게 수정하여 여러 시나리오를 테스트할 수 있다.
이러한 매개변수 변경은 대부분 몇 번의 버튼 클릭만으로 간단히 이루어진다.
사후 처리에는 결과 시각화 및 비디오 생성이 포함되며, 이와 관련된 자세한 과정은 별도의 웨비나에서 다뤄질 예정이다.
FLOW-3D HYDRO는 3D 유동 문제와 특히 자유 표면 유동 모델링에 매우 강력한 도구이다.
체계적인 기본 모델 설정 과정을 통해, 한 번 모델 구축 후 다양한 설계 변경 및 시나리오 테스트를 효율적으로 수행할 수 있다.
이 소개자료는 “Numerical Model for a Nineteenth-Century Hydrometric Module”논문의 소개자료임.
연구 목적
본 연구는 19세기에 건설된 수량 측정 모듈의 작동을 연구하고, 수치 모델을 통해 원래의 유량 조절 목표를 충족했는지 여부를 확인하는 것을 목적으로 함.
연구 방법:
모델링 설정
FLOW-3D 전산 유체 역학(CFD) 소프트웨어를 사용하여 수량 측정 모듈의 수치 모델을 생성하였음.
19세기 수량 측정 모듈의 기하학적 형상 및 관련 유압 시스템을 모델에 반영하였음.
모듈의 동적 거동(과도 상태)을 재현하기 위한 시뮬레이션을 수행하였음.
모델 검증
실제 실험 측정값이 부족하기 때문에 문헌의 해석적 모델과 비교하여 수치 모델을 검증하였음.
모델이 수량 측정 모듈의 유량 조절 기능을 정확하게 예측하는지 평가하였음.
모델의 정확성을 확인하고 신뢰성을 확보하였음.
주요 결과:
흐름 특성 분석
수량 측정 모듈 내부의 흐름 속도, 수위 변화 등 흐름 특성을 FLOW-3D 모델을 통해 분석하였음.
모듈의 자동화 시스템 작동 시 유량 조절 과정을 시각적으로 제시하였을 것으로 예상됨.
설계 유량 조건에서 모듈의 유압적 성능을 평가하였을 것으로 예상됨.
구조물 영향 평가
수량 측정 모듈의 구조가 흐름 특성 및 유량 조절에 미치는 영향을 평가하였음.
19세기 자동화 시스템이 설정된 유량을 유지하는 능력을 분석하였음.
수치 모의실험 결과를 통해 역사적인 수량 측정 구조물의 작동 원리를 규명하였음.
결론 및 시사점:
FLOW-3D를 이용한 수치 모델은 역사적인 수량 측정 모듈의 동적 거동을 성공적으로 재현하였음.
19세기 자동화 시스템이 요구되는 유량 제한 값을 정확하게 유지하며 작동했음을 확인하였음.
수치 모델은 수리 공학 분야의 역사적 연구를 위한 유용한 도구로 활용될 수 있을 것으로 기대됨.
Fig. 2. Plan view of the project of the hydrometric module. (Adapted with permission from J. de Castro, unpublished data, 1863, CDAHCF Archives, Parc de la Sèquia, Manresa, Spain.)Fig. 7. Right-side view of model geometry.Fig. 9. Rear view at t ¼ 240 s and Q ¼ 1,300 L=s.
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이 소개자료는 “SIMULATION OF LOCAL SCOUR AROUND A GROUP OF BRIDGE PIER USING FLOW-3D SOFTWARE”논문에 대한 소개자료입니다.
Figure 4.18 scour development at time = 360 min and discharge 0.057 m3/sec
연구 목적
본 연구는 FLOW-3D 소프트웨어를 사용하여 교각 그룹 주변의 국부 세굴을 시뮬레이션하는 것을 목적으로 함.
연구 방법:
모델링 설정
FLOW-3D 소프트웨어를 사용하여 교각 그룹 주변의 국부 세굴 현상을 수치적으로 모의실험하였음.
교각의 기하학적 형상 및 하천 흐름 조건을 모델에 반영하였음.
다양한 교각 배열 및 흐름 조건에 대한 모델링을 수행하여 세굴 특성을 분석하였음.
모델 검증
수치 모델의 결과를 실험실 데이터 또는 현장 관측 자료와 비교하여 검증하였을 것으로 예상됨.
세굴 깊이, 세굴공의 형태 등 주요 세굴 변수에 대한 모델의 예측 성능을 평가하였을 것으로 예상됨.
모델의 신뢰성을 확보하기 위해 민감도 분석 및 불확실성 분석을 수행하였을 것으로 예상됨.
주요 결과:
흐름 특성 분석
교각 그룹 주변의 유속, 압력 분포 등 흐름 특성을 FLOW-3D 모델을 통해 분석하였을 것으로 예상됨.
교각 배열이 흐름 패턴 및 와류 형성에 미치는 영향을 시각적으로 제시하였을 것으로 예상됨.
세굴 발생 메커니즘과 관련된 흐름 특성을 파악하여 세굴 예측의 정확도를 높였을 것으로 예상됨.
구조물 영향 평가
교각 그룹의 배열 방식이 세굴 깊이 및 세굴공의 크기에 미치는 영향을 평가하였을 것으로 예상됨.
교각 주변의 세굴 특성을 분석하여 교각 기초 설계 시 고려해야 할 중요한 요소를 제시하였을 것으로 예상됨.
수치 모의실험 결과를 바탕으로 교량의 안정성을 평가하고 설계 개선 방안을 제시하였을 것으로 예상됨.
결론 및 시사점:
FLOW-3D 소프트웨어를 이용한 수치 모델링은 교각 그룹 주변의 세굴 현상을 분석하고 예측하는 데 효과적인 도구임이 확인되었을 것으로 예상됨.
본 연구 결과는 교각 기초의 안정성을 확보하고 교량 붕괴를 예방하는 데 기여할 수 있을 것으로 기대됨.
향후 다양한 교각 조건 및 하천 흐름 조건에 대한 추가적인 연구를 통해 모델의 적용성을 확대할 필요가 있음.
Figure 3.1 Laboratory layout
Figure 3.10 Computational domain and mesh setup around the bridge piers model
(4-10)
Figure 4.18 scour development at time = 360 min and discharge 0.057 m3/sec
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FLOW-3D를 이용한 다양한 정수지(Stilling Basin)에서의 수력 도약(Hydraulic Jump) 수치적 연구
Graphical Abstract
연구 배경 및 목적
문제 정의
Taunsa Barrage(파키스탄)의 정수지는 기존의 USBR Type-III Basin을 개량한 형태로, 충격 바플(Impact Baffle)과 마찰 블록(Friction Block) 포함.
하지만 운영 초기부터 바플 블록이 뽑히는 문제 발생 → 기존 사각형 바플 블록이 흐름 재부착(Flow Reattachment)과 낮은 항력(Drag) 문제를 가짐.
기존 연구에서는 쐐기형(Wedge-Shaped) 분리 블록(Splitter Blocks)의 사용이 제한적이었으며, 이들의 수력 도약(HJ) 및 에너지 소산 성능이 충분히 검토되지 않음.
연구 목적
FLOW-3D를 활용하여 USBR Type-III 및 쐐기형 바플 블록을 적용한 정수지에서의 수력 도약 및 유동 특성을 비교 분석.
자유 수면 프로파일, 롤러 길이(Roller Length), 상대 에너지 손실(Relative Energy Loss), 유속 분포 및 난류 운동 에너지(TKE) 분석.
새로운 정수지 설계가 HJ를 안정화하고 에너지 소산 성능을 향상시키는지 평가.
연구 방법
FLOW-3D 모델링 및 실험 검증
VOF(Volume of Fluid) 기법을 사용하여 자유 수면 추적.
RNG k-ε 난류 모델을 적용하여 유동장 해석 수행.
Taunsa Barrage의 USBR Type-III 및 개량된 쐐기형 바플 블록 정수지 모델을 구축하여 비교 실험.
수치 모델 설정
세 가지 정수지 유형 비교
Type-A: 기존 USBR Type-III 정수지
Type-B: 쐐기형 바플 블록 적용 정수지
Type-C: USBR 바플과 쐐기형 바플 블록을 혼합한 정수지
시험 조건
두 가지 유량 조건(44 m³/s, 88 m³/s)에서 실험 수행.
유입 Froude 수(Fr) 범위: 5.75까지 고려.
경계 조건: 유입부와 유출부는 압력(P), 벽면은 No-Slip 조건 적용.
주요 결과
자유 수면 프로파일 분석
Type-B 및 Type-C 정수지에서 수력 도약(HJ)이 더 짧고 안정적으로 형성됨.
유량 증가 시 HJ의 롤러 길이가 감소하는 경향을 보임.
Type-B 및 Type-C 정수지는 USBR Type-A보다 더 높은 상대 에너지 손실을 기록하여 효율적인 에너지 소산을 확인.
유속 및 난류 운동 에너지(TKE) 분석
Type-B 및 Type-C 정수지에서 난류 운동 에너지(TKE)가 빠르게 감소하여 난류 제어 효과가 우수함.
유속 분포 결과, Type-B 및 Type-C 정수지에서 바플 블록이 흐름을 효과적으로 분산시켜 유속 감소 효과를 제공.
전반적으로 Type-C(혼합형 정수지)가 가장 효과적인 유동 제어 및 에너지 소산을 제공함.
결론 및 향후 연구
결론
쐐기형 바플 블록을 포함한 Type-B 및 Type-C 정수지는 기존 USBR Type-III 모델보다 더 높은 에너지 소산 효과를 제공.
HJ 길이가 짧아지고, 전단 응력이 감소하여 침식 가능성이 줄어듦.
FLOW-3D를 이용한 시뮬레이션이 정수지 설계 최적화 및 유지보수 비용 절감에 기여할 수 있음.
향후 연구 방향
LES(Large Eddy Simulation) 및 더 정밀한 난류 모델을 적용하여 연구 정밀도를 향상.
보다 높은 유량(예: 100~500 m³/s)에서의 테스트 수행.
다양한 바플 블록 형상(예: 삼각형, 원형 등) 및 배열 최적화를 통한 추가 연구 진행.
연구의 의의
이 연구는 FLOW-3D를 활용하여 다양한 정수지 설계에서의 수력 도약(HJ) 및 에너지 소산 효과를 분석한 연구로, 기존 USBR Type-III 정수지의 문제점을 개선하고, 새로운 설계 방안을 제시함으로써 대형 수리 구조물의 안정성 향상 및 침식 저감에 기여할 수 있는 실질적인 데이터를 제공하였다.
Figure 12 At 44 m3 s, 2D illustration of the velocity contour after the HJ and at basin’s end in the Type-A stilling basin (a and b), Type-B stilling basin (c and d), and Type-C stilling basin (e and f)
Figure 14 At 88 m3 s, 2D illustration of the velocity contour after HJ and at basin’s end in the Type-A stilling basin (a and b), Type-B stilling basin (c and d), and Type-C stilling basin (e and f)
Figure 15 2D illustration of turbulent kinetic energy (TKE) and turbulent intensity (TI) at 44 m3 s discharge in (a and b) Type-A, (c and d) Type-B, and (e and f) Type-C stilling basins, respectively
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본 소개 논문은 Engineering Applications of Computational Fluid Mechanics에서 발행한 논문 “Numerical Modeling of Flow Over Trapezoidal Broad-Crested Weir”의 연구 내용입니다.
Fig. 9 Velocity vectors for Q = 0.0181 m3/s in the area of the broad-crested weir.
1. 서론
넓은 마루 위어(Broad-Crested Weir, BCW)는 수리학적 구조물로서 홍수 조절, 유량 측정 및 관개 시스템에서 활용됨.
BCW의 형상, 특히 사다리꼴 형태는 유량 및 에너지 손실에 영향을 미칠 수 있으며, 기존 실험적 연구와 함께 수치 모델링이 중요함.
본 연구에서는 FLOW-3D 및 SSIIM 2 소프트웨어를 사용하여 사다리꼴 BCW의 유동 특성을 분석하고, 수치 결과를 물리 실험 결과와 비교하여 모델링 정확도를 평가함.
2. 연구 방법
FLOW-3D 및 SSIIM 2 기반 CFD 모델링
VOF(Volume of Fluid) 기법을 사용하여 자유 수면을 추적.
Reynolds-Averaged Navier-Stokes (RANS) 방정식과 k-ε 난류 모델을 적용하여 난류 해석 수행.
FAVOR(Fractional Area/Volume Obstacle Representation) 기법을 활용하여 복잡한 구조물 형상을 반영.
SSIIM 2는 적응형(adaptive) 격자를 사용하며, Marker-and-Cell(MAC) 접근법을 적용하여 자유 수면을 계산.
경계 조건 설정:
유입부: 부피 유량(Volume flow rate) 조건 적용.
유출부: 자유 배출(Outflow) 조건 설정.
벽면: No-slip 조건 적용.
3. 연구 결과
FLOW-3D와 SSIIM 2 결과 비교
두 모델 모두 물리 실험 결과와 유사한 자유 수면 프로파일을 예측하였으며, 계산된 유량 계수(Discharge Coefficient, Cd)는 실험 값과 ±3% 이내의 차이를 보임.
FLOW-3D는 격자가 고정되어 있으며, 평균 435~550초의 계산 시간이 소요됨.
SSIIM 2는 적응형 격자를 사용하여 격자 수가 변하며, 계산 시간이 12,500~15,500초로 상대적으로 길었음.
유량 변화(Q = 0.0181 ~ 0.0055 m³/s)에 따른 자유 수면 프로파일 분석 결과, 두 모델 간 수위 차이는 1~1.5% 범위 내에 존재.
압력 및 유속 분포 분석
FLOW-3D의 결과에서는 위어 전면부에서 압력이 최대치를 기록하며, 후면부에서는 압력이 급격히 감소.
SSIIM 2에서도 유사한 압력 분포가 확인되었으나, 자유 수면 프로파일 계산에서 다소 차이가 발생.
속도 벡터 분석 결과, 위어 전면부에서 흐름이 가속되고 후면부에서 난류 강도가 증가하는 패턴이 관측됨.
4. 결론 및 제안
결론
FLOW-3D 및 SSIIM 2를 활용한 시뮬레이션은 사다리꼴 BCW 유동 해석에서 높은 신뢰도를 보였으며, 실험 결과와의 비교를 통해 모델의 타당성이 검증됨.
FLOW-3D는 고정 격자와 높은 계산 효율성을 제공하며, SSIIM 2는 적응형 격자를 활용하여 자유 수면의 변화를 보다 세밀하게 반영.
전체적인 Cd 값은 실험 데이터와 잘 일치하며, 실험과의 평균 오차율이 3% 이내임.
향후 연구 방향
3D 모델링을 활용하여 더욱 정밀한 유동 분석 수행.
LES(Large Eddy Simulation) 및 다른 난류 모델과의 비교 연구 필요.
자연 하천 환경에서의 적용 가능성을 평가하기 위한 추가 연구 필요.
5. 연구의 의의
본 연구는 FLOW-3D 및 SSIIM 2를 이용하여 사다리꼴 BCW에서의 유동 특성을 분석하고, 실험 결과와 비교하여 모델 신뢰성을 검증하였다. 이를 통해 수리 구조물 설계 및 유량 측정 기술 향상에 기여할 수 있는 실질적인 데이터 및 분석 방법을 제공한다.
Sketch of the orthogonal, structured and nonadaptive grid (hexahedral), used in Flow-3D. In the computations a finer grid is used.
Fig. 9 Velocity vectors for Q = 0.0181 m3/s in the area of the broad-crested weir.
6. 참고 문헌
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Figure 19. Streamlines from 3D model simulation for overall head works arrangement
1. 서론
네팔은 농업 현대화를 추진하고 있으며, 이에 따라 효율적인 관개 인프라 구축이 필요함.
Sunkoshi-Marin 유역 전환 프로젝트는 Bagmati 관개 계획을 위한 수자원을 공급하기 위해 설계됨.
헤드워크(headworks)는 하천에서 필요한 수량을 안정적으로 취수하고, 퇴적물 배출 및 홍수 방류를 위한 필수적인 수리 구조물임.
본 연구는 FLOW-3D를 활용하여 Sunkoshi-Marin 헤드워크의 수리학적 성능을 평가하고, 구조물의 효율성과 안정성을 분석하는 것을 목표로 함.
2. 연구 방법
FLOW-3D 기반 CFD 모델링
VOF(Volume of Fluid) 기법을 사용하여 자유 수면 추적.
RNG k-ε 난류 모델을 적용하여 난류 해석 수행.
FAVOR(Fractional Area/Volume Obstacle Representation) 기법을 사용하여 복잡한 지형을 정밀하게 반영.
경계 조건 설정:
유입부: 부피 유량(Volume flow rate) 조건 적용.
유출부: 자유 배출(Outflow) 조건 설정.
벽면: No-slip 조건 적용.
3. 연구 결과
유속 및 압력 분석
보(Barrage) 상부 평균 유속: 9 m/s 이상(완전 개방 시).
정지분지(Stilling Basin) 최대 유속: 10 m/s.
종방향 유속 프로파일에서의 최대 유속: 16.90 m/s.
음압(negative pressure) 발생 없음 → 공동(cavitation) 현상 없음.
최소 압력: 101.356 KPa(유입축 하류에서 관측됨).
방류 용량 분석
FSL(Full Supply Level)에서 보와 언더슬루이스(Under-sluice) 동시 운영 시 방류 용량: 10,086 m³/s.
100년 빈도 홍수량(9,241 m³/s) 안전하게 방류 가능.
479.5m 헤드워터 수위에서의 최대 방류 용량: 16,547 m³/s.
10,000년 빈도 홍수를 방류하기 위해 481.00m 데크(deck) 수준이 적절함.
4. 결론 및 제안
결론
FLOW-3D 기반 시뮬레이션을 통해 헤드워크의 수리학적 성능을 평가할 수 있음.
음압이 발생하지 않으며 공동현상이 우려되지 않음.
보와 언더슬루이스 구조가 퇴적물 배출 및 홍수 방류에 효과적으로 작용함.
수력 점프(hydraulic jump) 형성이 확인되며, 수위 변화에 따라 위치가 조정됨.
향후 연구 방향
다양한 수위 및 유량 조건에서 추가 시뮬레이션 수행.
다른 난류 모델(예: LES)과 비교 연구 필요.
현장 데이터와의 비교를 통해 모델 검증 수행.
5. 연구의 의의
본 연구는 FLOW-3D를 활용하여 헤드워크의 수리적 성능을 정량적으로 분석하고, 홍수 방류 및 취수 구조물 설계 최적화에 기여할 수 있는 정보를 제공함.
Figure 2. Weir axis at head works site
Figure 19. Streamlines from 3D model simulation for overall head works arrangement
6. 참고 문헌
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취수 구조물은 홍수 조절, 관개, 전력 생산, 상수 공급 등을 위해 사용되며, 이러한 구조물에서 발생하는 와류(Vortex)는 시스템 효율과 안정성을 저하시킬 수 있음.
와류는 공기 유입, 진동, 캐비테이션(cavitation), 방류량 감소 등 다양한 운영 문제를 유발할 수 있음.
본 연구는 FLOW-3D를 사용하여 3D 수치 모델을 구축하고, 취수 구조물에서 공기 유입형 와류(Air-entraining vortex)의 발생 조건을 분석하는 것을 목표로 함.
2. 연구 방법
FLOW-3D 기반 CFD 모델링
VOF(Volume of Fluid) 기법을 사용하여 자유 수면 추적.
RNG k-ε 난류 모델 및 LES(Large Eddy Simulation) 모델을 적용하여 난류 해석 수행.
FAVOR(Fractional Area/Volume Obstacle Representation) 기법을 사용하여 복잡한 구조물 형상을 반영.
경계 조건 설정:
유입부: 부피 유량(Volume flow rate) 조건 적용.
유출부: 자유 배출(Outflow) 조건 설정.
벽면: No-slip 조건 적용.
3. 연구 결과
와류 형성 조건 분석
취수구의 직경, 벽면 간격, 수위(Submergence depth), 유량 등의 변수에 따라 와류 형성 여부가 달라짐.
실험 결과와 비교 시, FLOW-3D는 와류를 효과적으로 예측하지만, 일부 임계 수위(Submergence depth)에서 실험 결과와 차이가 발생함.
벽면 간격(Sidewall clearance)이 작을수록 와류 형성이 더욱 뚜렷하게 나타나며, LES 모델이 laminar 모델보다 더 정확한 결과를 제공함.
4. 결론 및 제안
결론
FLOW-3D는 취수 구조물에서의 와류 형성을 정량적으로 예측하는데 효과적이며, 벽면 간격 및 유량 변화에 따른 흐름 특성을 분석할 수 있음.
LES 모델은 난류 효과를 보다 정밀하게 반영하여 보다 신뢰성 높은 결과를 제공.
반와류 장치(Anti-vortex plate) 사용 시, 와류 형성이 현저히 감소함을 확인.
향후 연구 방향
다양한 취수 구조물 형상 및 유속 조건에서의 추가 연구 필요.
LES 모델과 실험 데이터를 비교하여 모델의 정확도를 더욱 개선.
실제 현장 데이터를 기반으로 모델 검증 연구 수행.
5. 연구의 의의
본 연구는 FLOW-3D를 활용하여 취수 구조물에서의 와류 형성 조건을 분석하고, LES 모델을 이용하여 난류 효과를 보다 정밀하게 반영하는 방법을 제시하였다. 향후 취수 구조물 설계 및 운영 최적화에 기여할 수 있는 데이터 및 분석 방법을 제공한다.
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본 연구는 부유 플랩형 폭풍 해일 방어벽의 수리학적 특성을 수치적으로 분석하는 것을 목적으로 함.
폭풍 해일 제어 및 연안 홍수 완화에서 방어벽의 효과를 평가하기 위해 수행됨.
FLOW-3D 소프트웨어를 이용하여 방어벽의 유체역학적 거동을 모델링함.
2. 연구 방법
전산유체역학(CFD) 기법을 적용하여 부유 플랩형 방어벽을 모델링함.
수치 모델의 주요 구성 요소:
레이놀즈 평균 나비에-스토크스(RANS) 방정식을 이용한 난류 모델링.
VOF(Volume of Fluid) 기법을 사용하여 자유 수면 추적.
실제 조석(tidal) 및 폭풍 해일(storm surge) 조건을 반영한 경계 조건 적용.
기존 실험 데이터를 활용하여 모델 검증 수행.
3. 연구 결과
주요 연구 결과:
방어벽이 수위 감소 및 파랑 에너지 저감에 효과적임을 확인.
방어벽 각도에 따라 와류(vortex) 형성 및 난류 강도가 변화함.
파고, 방어벽 유연성, 유속에 따라 구조적 안정성이 영향을 받음.
실험 데이터와의 비교를 통해 모델의 예측 정확성이 높음을 확인함.
4. 결론
부유 플랩형 폭풍 해일 방어벽은 연안 홍수 완화에 효과적인 대안이 될 수 있음.
CFD 시뮬레이션을 통해 방어벽 설계 최적화에 유용한 정보를 제공할 수 있음.
향후 연구에서는 장기적인 구조적 내구성 및 실제 환경에서의 적용 가능성을 중점적으로 다뤄야 함.
Figure 1. Location of the study area
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유량 변화에 따른 Froude 수 범위: 1.1≤Fr≤201.1 \leq Fr \leq 201.1≤Fr≤20.
취수구 주변의 유동장 및 와류 발생을 고속 카메라와 유량 측정 센서를 이용하여 분석.
FLOW-3D 기반 수치 시뮬레이션 설정
VOF (Volume-of-Fluid) 기법을 활용하여 자유 수면 유동을 모델링.
RNG k−εk-\varepsilonk−ε 난류 모델을 적용하여 난류 특성을 해석.
FAVOR (Fractional Area/Volume Obstacle Representation) 기법을 적용하여 취수구 형상을 정밀 모델링.
격자(Grid) 설정 최적화:
외부 영역(Grid size = 0.1m).
내부 영역(Grid size = 0.07m, 취수구 주변 고해상도 적용).
LES(Large Eddy Simulation) 모델 적용 시 해석 정확도가 향상됨을 확인.
주요 결과
임계 침수 깊이(Critical Submergence Depth) 분석
Froude 수, Weber 수, Reynolds 수가 증가할수록 임계 침수 깊이(Sc/Di) 증가.
취수구 직경(Di)가 증가할수록 임계 침수 깊이가 감소하는 경향을 보임.
Sc/Di 값이 실험 결과와 비교했을 때 평균 오차 5~10% 이내로 높은 정확도 확인.
유동 패턴 및 난류 강도 분석
취수구 주변 와류 강도는 Froude 수 증가 시 급격히 증가.
LES 난류 모델 적용 시 난류 해석 정확도가 향상됨을 확인.
SSIIM 모델과 비교한 결과, FLOW-3D가 더 정밀한 유동 분석 결과를 제공.
최적 침수 깊이 산정 식 도출
실험 및 시뮬레이션 데이터를 기반으로 임계 침수 깊이(Sc)를 예측하는 새로운 경험식 도출:
Froude 수(Fr), Reynolds 수(Re), Weber 수(We) 등과 취수구 직경(Di) 간의 관계를 포함하는 포괄적인 모델을 제안.
결론 및 향후 연구
결론:
FLOW-3D 시뮬레이션이 수평 취수구에서 와류 형성을 방지하는 임계 침수 깊이 예측에 유용함을 입증.
Froude 수, Weber 수, Reynolds 수와 취수구 직경 간의 정량적 관계를 도출하여 설계 기준을 제공.
LES 난류 모델 적용 시 해석 정확도가 증가하므로, 향후 연구에서는 더욱 정밀한 난류 모델 활용 필요.
향후 연구 방향:
다양한 취수구 형상(수직, 사각형 등)에 따른 침수 깊이 변화 연구.
대형 취수구 및 댐 취수 시스템 적용을 위한 현장 검증 실험 수행.
AI 및 머신러닝 기법을 활용한 실시간 취수구 설계 최적화 연구.
연구의 의의
본 연구는 FLOW-3D를 활용하여 수평 취수구에서 와류 형성을 방지하기 위한 최적 침수 깊이를 도출하고, 기존 실험 데이터를 기반으로 정량적 관계를 검증하였다. 이를 통해 수력 발전소, 농업용 관개 및 산업용 취수 시스템 설계 시 실질적인 엔지니어링 데이터를 제공할 수 있음을 입증하였다.
Figure 2. Vortex strength scale used by Dargin and Anderson for classification of free surface vortices at intakes
Figure 4. Vortex formation and Critical submergence with coarse mesh
Figure 5b. Vorticity magnitude contours in Critical submergence depth
Reference
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Figure 3 The 3D computational domain model (50–18.6) slope change, and boundary condition for (50–30 slope change) model
연구 배경 및 목적
문제 정의
댐 구조물의 필수적인 요소 중 하나인 여수로(spillway)는 홍수 방류 시 댐을 보호하는 중요한 역할을 수행함.
기존의 오지(ogee)형 여수로와 달리, 계단형 여수로(stepped spillway)는 유체의 에너지 소산을 증가시켜 캐비테이션(cavitation) 위험을 감소시키는 장점이 있음.
계단형 여수로에서 유동 형태(nappe flow, transition flow, skimming flow)가 다르게 나타나며, 유속, 압력, 공기 유입 등의 변화가 발생함.
연구 목적
Flow-3D를 활용하여 계단형 여수로에서의 난류 유동을 수치적으로 해석하고 실험 결과와 비교.
여수로 경사의 변화가 공기 유입(air entrainment), 유속 분포(velocity distribution), 동압(dynamic pressure)에 미치는 영향을 분석.
다양한 유량 조건에서 수치 해석 결과와 기존 실험 결과를 비교하여 모델의 신뢰성을 검증.
연구 방법
수치 해석 설정
CFD(전산유체역학) 모델: Flow-3D 사용
난류 모델: RNG k-ε 모델 적용
자유 수면 추적: VOF(Volume of Fluid) 기법 활용
격자 설정: 직교 격자(orthogonal mesh) 사용, 셀 크기 0.015m
모델 실험 조건:
경사 변화: 50° → 30° 및 50° → 18.6°
단수(step height): 0.06m
유량 조건: 0.1 m³/s 및 0.235 m³/s
주요 결과
공기 유입(air entrainment) 및 유속 분포 분석
경사 변화 후 공기 유입 증가 → 동일 유량에서 계단 경사가 낮을수록 공기 함유량 상승.
실험 결과와 비교 시 모델의 공기 유입 예측이 높은 신뢰도를 가짐.
유속 분포 분석 결과, 계단 경사 감소 시 유속이 더 균일하게 분포하며 난류 발생이 감소.
동압(dynamic pressure) 분석
실험 데이터와 비교 시 경사 변화 전후 압력 분포가 유사한 경향을 보임.
경사 변화 후, 계단면의 압력 변동이 증가하나, 전체적인 패턴은 실험 결과와 잘 일치.
실험 데이터 대비 압력 차이는 10% 이내로 나타남.
결론 및 향후 연구
결론
Flow-3D를 활용한 계단형 여수로의 수치 해석이 실험 결과와 높은 일치도를 보이며, 신뢰성이 검증됨.
경사 변화가 공기 유입과 유속 분포에 큰 영향을 미치며, 유량 조건에 따라 최적 설계가 필요.
계단형 여수로 설계 시, 경사와 유량 조건을 고려하여 최적의 유동 상태를 확보하는 것이 중요함.
향후 연구 방향
LES(Large Eddy Simulation) 난류 모델 적용을 통한 세밀한 유동 해석.
다양한 계단 형상 및 유량 조건에서 추가적인 검증 수행.
실제 댐 및 홍수 방류 시스템 적용을 위한 현장 실험 데이터와 비교 연구.
연구의 의의
이 연구는 Flow-3D를 활용하여 계단형 여수로의 유동 특성을 정량적으로 분석하고, 수치 모델의 신뢰성을 검증하였다. 여수로 설계 최적화 및 댐 안전성 향상을 위한 기초 데이터를 제공하였다.
Figure 1 Sketch of the air concentration C and velocity V measured perpendicular to the pseudo-bottom used by Mirza (Ostad Mirza 2016)
Figure 3 The 3D computational domain model (50–18.6) slope change, and boundary condition for (50–30 slope change) model
Figure 5 Experimental and simulated air concentration distribution for steps number
References
Boes, R. M. & Hager, W. H. 2003a Hydraulic design of stepped spillways. Journal of Hydraulic Engineering 129 (9), 671–679.
Boes, R. M. & Hager, W. H. 2003b Two-Phase flow characteristics of stepped spillways. Journal of Hydraulic Engineering 129 (9), 661–670.
Chanson, H. 1994 Hydraulics of skimming flows over stepped channels and spillways. Journal of Hydraulic Research 32 (3), 445–460.
Chanson, H. 1997 Air Bubble Entrainment in Free Surface Turbulent Shear Flows. Academic Press, London.
Chanson, H. 2002 The Hydraulics of Stepped Chutes and Spillways. Balkema, Lisse, The Netherlands.
Felder, S. & Chanson, H. 2011 Energy dissipation down a stepped spillway with nonuniform step heights. Journal of Hydraulic Engineering 137 (11), 1543–1548.
Flow Science, Inc. 2012 FLOW-3D v10-1 User Manual. Flow Science, Inc., Santa Fe, CA.
Ghaderi, A., Daneshfaraz, R., Torabi, M., Abraham, J. & Azamathulla, H. M. 2020a Experimental investigation on effective scouring parameters downstream from stepped spillways. Water Supply 20 (5), 1988–1998.
Ghaderi, A., Abbasi, S., Abraham, J. & Azamathulla, H. M. 2020b Efficiency of trapezoidal labyrinth shaped stepped spillways. Flow Measurement and Instrumentation 72, 101711.
Gonzalez, C. A. & Chanson, H. 2008 Turbulence and cavity recirculation in air-water skimming flows on a stepped spillway. Journal of Hydraulic Research 46 (1), 65–72.
Gunal, M. 1996 Numerical and Experimental Investigation of Hydraulic Jumps. PhD Thesis, University of Manchester, Institute of Science and Technology, Manchester, UK.
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 39 (1), 201–225.
Matos, J. 2000 Hydraulic design of stepped spillways over RCC dams. In: Intl Workshop on Hydraulics of Stepped Spillways (H.-E. Minor & W. Hager, eds). Balkema Publ, Zurich, pp. 187–194.
Mohammad Rezapour Tabari, M. & Tavakoli, S. 2016 Effects of stepped spillway geometry on flow pattern and energy dissipation. Arabian Journal for Science & Engineering (Springer Science & Business Media BV) 41 (4), 1215–1224.
Ostad Mirza, M. J. 2016 Experimental Study on the Influence of Abrupt Slope Changes on Flow Characteristics Over Stepped Spillways. Communications du Laboratoire de Constructions Hydrauliques, No. 64 (A. J. Schleiss, ed.). Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland.
Roshan, R., Azamathulla, H. M., Marosi, M., Sarkardeh, H., Pahlavan, H. & Ab Ghani, A. 2010 Hydraulics of stepped spillways with different numbers of steps. Dams and Reservoirs 20 (3), 131–136.
Shahheydari, H., Nodoshan, E. J., Barati, R. & Moghadam, M. A. 2015 Discharge coefficient and energy dissipation over stepped spillway under skimming flow regime. KSCE Journal of Civil Engineering 19 (4), 1174–1182.
Takahashi, M. & Ohtsu, I. 2012 Aerated flow characteristics of skimming flow over stepped chutes. Journal of Hydraulic Research 50 (4), 427–434.
Versteeg, H. K. & Malalasekera, W. 2007 An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education, Harlow.
해일의 전파 속도는 초기 파고 및 하천 형상에 따라 달라지며, 좁은 수로에서 감쇠가 느려지는 경향을 보임.
측정 지점별 파고 감소율을 분석한 결과, 해일 감쇠율이 비선형적으로 나타남.
이는 수면 저항 및 흐름 분산에 따른 에너지 손실이 비균일하게 발생하기 때문으로 해석됨.
결론
산사태 유입각이 해일 발생의 주요 변수이며, 60°에서 최대 파고가 발생함.
수심이 깊을수록 해일 감쇠가 더 빠르게 진행되며, 초기 파고가 낮아짐.
FLOW-3D® 기반 시뮬레이션을 통해 선형 하천 저수지에서의 산사태 해일 전파 및 감쇠 메커니즘을 규명할 수 있음.
향후 연구에서는 다양한 하천 형상 및 실제 지형 조건을 반영한 추가 분석이 필요함.
Reference
Kiersch, G. A. 1964. “Vajont Reservoir Disaster.” Civil Engineering (ASCE) 34 (3): 32-39.
Hunan Hydro & Power Design Institute. 1983. Slope Engineering Geology. Beijing: Water Conservancy and Electric Power press.
Wiegel, R. L. 1995. “Laboratory Studies of Gravity Waves Generated by the Movement of A Submerged Body.” Transactions-American Geophysical Union 36 (5): 759-774.
Fritz, H. M., Moster, P. 2003. “Pneumatic Landslide Generator.” International Journal of Fluid Power 173 (2): 223-233.
Sander, J., Hutter, K. 1992. “Evolution of Weakly Non-linear Channelized Shallow Water Waves Generated by A Moving Boundary.”Acta Mechanic 91: 119-155.
Sander, J., Hutter, K. 1996. “Multiple Pulsed Debris Avalanche Emplacement at Mount St. Helens in 1980: Evidence form Numerical Continuum Flow Simulation.” Acta Mechanic 115:133-149.
Heinrich, Ph. 1992. “Nonlinear Water Waves Generated by Submarine and Aerial Landslides.” Journal of Waterway, Port, Coast, and Ocean Engineering, ASCE 118: 249-266.
Ataie-Ashtiani, B., Farhadi, I. A. 2006. “Stable Moving-particle Semi-implicit Method for Free Surface Flow.” Fluid Dynamic Research 38 (4): 241-256.
Monaghan, J. J. 1994. “Simulating Free Surface Flows with SPH.” Journal of Computational Physics 110: 399-406.
Ataie-Ashtiani, B., Shobeyri, G. 2001. “Numerical Simulation of Landslide Impulsive Waves by Incompressible Smoothed Particle Hydrodynamic.” International Journal for Numerical Method in Fluids 56: 209-232.
FLOW-3D®의 유한체적법(Finite Volume Method, FVM)을 적용하여 질량, 운동량, 에너지 보존 방정식을 해석함.
자유 표면 추적을 위해 VOF(Volume of Fluid) 기법을 활용함.
아크 상호작용(arc interaction), 전극 간 전압 차이, 용적력(arc pressure) 등을 반영하여 실험과 유사한 모델을 구축함.
FLOW-3D® 시뮬레이션 설정
전극 배치 및 전류 조건을 변경하며 용융지 거동을 분석함.
선도 전극(leading electrode)과 후속 전극(trailing electrode)의 전류 조합에 따라 용접 비드 형상이 어떻게 변하는지 평가함.
난류 모델 적용: k-ε 및 LES(Large Eddy Simulation) 모델을 비교 분석하여 용융지 내 난류 흐름을 해석함.
실험 데이터와 비교 검증
실험에서 얻은 온도 분포 및 용융지 형상 데이터를 시뮬레이션과 비교함.
고속 CCD 카메라 및 열화상 카메라를 이용하여 용융지 거동을 기록하고, 시뮬레이션과의 차이를 분석함.
CFD 결과와 실제 실험 결과 간의 오차율을 측정하여 모델의 정확성을 검토함.
주요 결과
전류 조건이 용접 비드 형상에 미치는 영향
선도 전극 전류가 높은 경우, 용접 비드의 침투 깊이(penetration depth)가 증가하였으며, 후속 전극 전류가 낮은 경우 용융지 크기가 상대적으로 작아짐.
동일한 총 열 입력(total heat input) 조건에서도 전류 조합에 따라 용접 비드 형상이 다르게 형성됨.
용융지가 형성되는 과정에서 선도 전극에 의해 형성된 용융지가 후속 전극에 의해 확장됨.
용융 흐름 및 아크 상호작용 분석
선도 전극의 높은 전류가 강한 아크 압력(arc pressure)을 유발하여 용융지를 깊게 형성함.
반면, 후속 전극이 용융지를 확장시키는 역할을 하며, 비드 폭(bead width)이 증가함.
후속 전극의 전압이 높을수록 용융지가 넓게 퍼지며, 전체 용접 품질이 향상됨.
시뮬레이션과 실험 비교 검증
FLOW-3D® 시뮬레이션 결과와 실험 데이터 간 유사도가 높음.
특히, 온도 분포 및 용융지 형상이 실험과 거의 일치하였으며, 용융 흐름의 주요 특성을 재현할 수 있었음.
그러나, 일부 실험에서는 예상보다 낮은 침투 깊이가 관찰됨 → 이는 모델에서 고려되지 않은 금속 증발 및 표면 장력 변화 때문으로 분석됨.
최적 용접 조건 도출
전극 간 최적 간격 및 전류 조합을 설정하면 용접 품질을 향상시킬 수 있음.
선도 전극의 전류가 후속 전극보다 높을 때, 깊은 침투와 균일한 용접 비드 형성을 유도할 수 있음.
후속 전극의 전압을 높여 용융지 확산을 최적화하면 용접 비드 균일성이 증가함.
결론
FLOW-3D® CFD 시뮬레이션을 활용하여 이중 SAW-T 공정에서의 용융지 거동을 성공적으로 해석함.
전극 간 전류 조합이 용접 비드 형상에 결정적인 영향을 미치며, 최적의 조합을 찾는 것이 중요함.
실험 데이터와 시뮬레이션 결과의 높은 일치도를 확인하였으며, 일부 미세한 차이는 추가적인 모델 보정이 필요함.
향후 연구에서는 금속 증발, 표면 장력 변화 등의 추가 물리 모델을 고려하여 더욱 정밀한 해석을 수행해야 함.
Reference
Cao, Z., Yang, Z., Chen, X.L., 2004. Three dimensional simulation of transient GMA weld pool with free surface. Welding Journal 6, 169s-176s. Cho, D.W., Na, S.J., Cho, M.H., Lee, J.S., 2013a. A study on V-groove GMAW for various welding positions. Journal of Materials Processing Technology 213, 1640-1652. Cho, D.W., Song, W.H., Cho, M.H., Na, S.J., 2013b. Analysis of submerged arc welding process by threedimensional computational fluid dynamics simulation. Journal of Materials Processing Technology 213, 2278-2291. Kiran, D.V., Cho, D.W., Song, W.H., Na, S.J., 2014. Arc behavior in two wire tandem submerged arc welding. Journal of Materials Processing Technology 214, 1546-1556. Cho, J.H., Na, S.J., 2009. Three dimensional analysis of molten pool in GMA-laser hybrid welding. Welding Journal 88, 35s-43s. Cho, W.I., Na, S.J., Cho, M.H., Lee, J.S., 2010. Numerical study of alloying element distribution in CO2 laser-GMA hybrid welding. Computational Materials Science 49, 792-800. Jaidi, J., Dutta, P., 2001. Modeling of transport phenomenon in a gas metal arc welding process. Numerical Heat Transfer Part A 40, 543-562. Kim, C.H., Zhang, W., DebRoy, T., 2003. Modeling of temperature field and solidification surface profile during gas metal arc fillet welding. Journal of Applied Physics 94, 2667-2679. Kim, J.W., Na, S.J., 1995. A study on the effect of contact tube to workpiece distance on weld pool shape in gas metal arc welding. Welding Journal 74, 141s-152s. Kiran, D.V., Basu, B., Shah, A.K., Mishra, S., De, A., 2010. Probing influence of welding current on weld quality in two wire tandem submerged arc welding of HSLA steel. Science and Technology of Welding and Joining 15, 111-116. Kiran, D.V., Basu, B., Shah, A.K., Mishra, S., De, A., 2011. Three dimensional heat transfer analysis of two wire tandem submerged arc welding. ISIJ International 51, 793-798. Kumar, S., Bhaduri, S.C., 1994. Three dimensional finite element modeling of gas metal arc welding. Metallurgical and Materials Transactions B 25B, 435-441. Mahapatra, M.M., Datta, G.L., Pradhan, B., Mandal, N.R., 2006. Three dimensional finite element analysis to predict the effects of SAW process parameters on temperature distribution and angular distortions in single pass butt joints with top and bottom reinforcements. International Journal of Pressure Vessels and Piping 83, 721-729. Pardo, E., Weckman, D.C., 1989. Prediction of weld pool and reinforcement dimensions of GMA welds using a finite element method. Metallurgical Transactions B 20B, 937-947. Shome, M., 2007. Effect of heat-input on austenite grain size in the heat-affected zone of HSLA-100 steel. Materials Science and Engineering A. 445, 454-460. Tsao, K.C., Wu, C.S., 1988. Fluid flow and heat transfer in GMA weld pools. Welding Journal 3, 70s-75s. Ushio, M., Wu, C.S., 1997. Mathematical modeling of three dimensional heat and fluid flow in a moving gas metal arc weld pool. Metallurgical and Materials Transactions B 28B, 509-+516. Wang, Y., Tsai, H.L., 2001. Impingement of filler droplets and weld pool dynamics during gas metal arc welding process. International Journal of Heat and Mass Transfer 44, 2067-2080.
본 연구는 FLOW-3D® CFD 시뮬레이션을 활용하여 Xiqiao Mountain(시차오산)의 버섯 돌(Mushroom Stone) 형성 과정을 규명함.
기존 연구에서는 유수 침식(stream water erosion)이 주요 형성 원인으로 제시되었으나, 본 연구에서는 CFD 분석을 통해 침식 외에도 화학적 및 물리적 풍화 작용이 결정적인 역할을 했음을 입증하고자 함.
광물 분석 및 현장 조사와 함께 컴퓨터 시뮬레이션을 수행하여 물리적, 화학적 풍화 작용과 유동 역학 간의 관계를 평가함.
연구 방법
현장 조사 및 샘플링
드론(DJI Phantom 4 RTK)을 활용하여 버섯 돌의 3D 지형 데이터를 정밀 측정.
암석 시료 7개를 서로 다른 위치에서 채취하여 **광물 분석(mineralogical analysis)**을 수행함.
지질 나침반을 사용하여 버섯 돌 곡면의 방향 및 침식 패턴을 기록함.
FLOW-3D® 기반 CFD 시뮬레이션
자유 표면 유동(Free Surface Flow)을 모델링하여 홍수 시 버섯 돌 주변의 유속 및 압력 분포를 분석.
난류 모델 적용: RANS(Reynolds-Averaged Navier-Stokes) 방정식을 사용하여 난류 효과를 고려함.
모의 홍수 실험을 진행하여 홍수 시기 물의 흐름이 버섯 돌에 미치는 영향을 평가함.
결과 비교 및 검증
광물 분석 데이터 및 현장 조사 결과를 CFD 시뮬레이션과 비교하여 풍화 및 침식 기작을 검증.
침식 패턴, 유속, 압력 분포 등을 종합 분석하여 버섯 돌 형성의 주요 기작을 도출함.
주요 결과
홍수 시 버섯 돌 주변 유동 특성
시뮬레이션 결과, 최고 유속은 버섯 돌의 측면에서 발생하며, 전·후면에서는 상대적으로 낮은 유속을 보임.
버섯 돌의 전면(상류 방향)에서는 고압력이 발생하여 아래쪽으로 흐름이 집중됨, 이는 하부 침식을 유도함.
그러나 시뮬레이션 결과, 버섯 돌의 좁은 하부 구조는 단순한 유수 침식만으로 형성될 수 없음을 보여줌.
버섯 돌 침식 패턴 및 풍화 작용
CFD 분석 결과, 버섯 돌 하부(풍하측)에 퇴적물이 집중적으로 형성되며, 이는 침식보다 퇴적 과정이 더 중요한 역할을 했음을 시사함.
실험 데이터와 비교 시, 유수 침식만으로는 현장에서 관찰된 곡면 구조를 재현할 수 없음.
대신, 장기간 퇴적물이 축적되면서 화학적 및 물리적 풍화 작용이 진행되었을 가능성이 높음.
광물 분석 결과 및 풍화 작용
XRD(X-ray diffraction) 분석 결과, 버섯 돌 하부의 암석은 석고(gypsum) 및 점토 광물 함량이 높으며, 이는 화학적 풍화가 활발하게 진행되었음을 의미함.
석고 크리스탈이 성장하면서 암석 내부 균열을 유발하는 할로클래스티(haloclasty) 현상이 관찰됨.
장기간 퇴적층 내에 존재했던 암석이 화학적 풍화 및 수분에 의한 연화 작용으로 약해진 후, 외부 퇴적물이 제거되면서 버섯 돌 하부의 곡면이 형성됨.
버섯 돌 형성 과정 및 주요 기작 정리
1단계: 버섯 돌이 퇴적물 속에 매립됨 → 장기간 퇴적물 내에서 화학적 풍화가 진행됨.
2단계: 퇴적물 제거 후, 풍화된 암석이 노출되면서 내부 곡면이 형성됨.
3단계: 추가적인 기계적 풍화 및 석고 결정 성장이 내부 균열을 유발하며 현재의 버섯 돌 형태가 완성됨.
결론
유수 침식만으로 버섯 돌이 형성되었다는 기존 가설은 CFD 시뮬레이션 결과와 일치하지 않음.
광물 분석 및 화학적 풍화 모델링 결과, 할로클래스티(haloclasty) 및 습윤 연화(softening due to moisture) 작용이 버섯 돌 형성의 주요 기작으로 확인됨.
CFD 시뮬레이션을 통한 수력학적 해석과 광물 분석을 결합하여 자연 암석 형성 기작을 정량적으로 분석하는 새로운 접근법을 제시함.
향후 연구에서는 장기적인 풍화 속도 및 추가적인 유체-암석 상호작용 모델링을 수행해야 함.
Reference
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FLOW-3D를 이용한 모닝 글로리(Morning Glory) 월류수문에서의 유동 수치 모델링
연구 배경 및 목적
문제 정의: 모닝 글로리(Morning Glory) Spillway는 댐의 수위 조절 및 홍수 방지를 위해 사용되는 원형 월류수문이다.
기존 설계에서는 부유물(Suspended Load)의 영향을 간과하는 경우가 많았으며, 이는 설계 가정에 큰 변화를 초래할 수 있다.
부유물 함유 흐름은 물의 밀도를 변화시켜 수문 성능에 영향을 미칠 수 있다.
연구 목적:
FLOW-3D 소프트웨어를 사용하여 모닝 글로리 수문에서의 부유물 농도 변화가 유량(Flow Discharge)에 미치는 영향을 평가.
3000, 6000, 9000, 12000 ppm의 부유물 농도를 적용하여 수문 상부에서 다양한 수위 조건에서의 유량 변화를 분석.
수치 모델 결과를 물리적 모델 실험 데이터와 비교하여 FLOW-3D의 예측 성능을 검증.
연구 방법
수치 모델링 및 시뮬레이션 설정
FLOW-3D 소프트웨어의 VOF(Volume of Fluid) 기법 및 FAVOR(Fractional Area-Volume Obstacle Representation) 기법을 사용하여 유동 및 고체 경계 시뮬레이션.
k-ε 및 RNG 난류 모델을 사용하여 난류 효과를 모델링.
모닝 글로리 수문 설계:
해라즈(Haraz) 댐의 모닝 글로리 Spillway를 모델링.
Solidworks 소프트웨어를 이용해 3D 모델링을 수행하고, FLOW-3D에 가져와 수치 시뮬레이션을 설정.
부유물 농도 설정:
3000, 6000, 9000, 12000 ppm의 부유물을 흐름에 추가하여 유량 변화 분석.
부유물 농도가 증가함에 따라 점도 및 유체의 물리적 특성이 변화함을 고려.
경계 조건 설정
입출구 및 벽면 경계 조건:
입구(Inlet): 유량 조건을 일정하게 유지.
출구(Outlet): 자유 유출 조건을 적용.
벽면(Wall): 비투과성(Impermeable) 경계 조건 설정.
공기-물 경계:
자유 수면(Free Surface) 조건을 적용하여 공기와의 접촉을 고려.
주요 결과
부유물 농도 증가에 따른 유량 변화
순수 물(부유물 없음) 상태에서의 평균 유량: 600 m³/s.
부유물 농도에 따른 유량 감소 효과:
3000 ppm: 평균 유량 605 m³/s, 유량 감소 3.8%.
6000 ppm: 평균 유량 575 m³/s, 유량 감소 87.12%.
9000 ppm: 평균 유량 575 m³/s, 유량 감소 7.18%.
12000 ppm: 평균 유량 483 m³/s, 유량 감소 26%.
부유물 농도가 증가할수록 수문을 통과하는 유량이 감소하며, 이는 부유물이 물의 점도 증가 및 밀도 변화에 따른 흐름 저항 증가에 기인.
유동 패턴 및 수문 성능 변화
FLOW-3D 시뮬레이션에서 부유물 농도가 증가할수록 유동의 안정성이 감소.
특히 터널 및 월류수문 목(Throat) 부분에서의 유량 변화가 뚜렷하게 나타남.
수문 상부에서의 월류 유속 감소와 혼합 층의 두께 증가가 관찰됨.
FLOW-3D 모델의 신뢰성 평가
FLOW-3D 시뮬레이션 결과와 실험 결과 간 높은 일치도 확인.
모델 검증 결과, 예측된 유량 변화가 물리적 실험과 평균 5% 이내의 오차율을 보임.
이는 FLOW-3D가 복잡한 부유물 흐름을 정확하게 모델링할 수 있음을 의미.
결론 및 향후 연구
결론:
FLOW-3D 소프트웨어는 모닝 글로리 월류수문의 부유물 농도 변화에 따른 유량 감소를 정확히 예측할 수 있음.
부유물 농도가 높을수록 유량 감소율이 증가하며, 특히 12000 ppm에서는 평균 26%의 유량 감소가 나타남.
이는 댐 설계 및 운영 시 부유물 농도를 고려해야 함을 시사하며, 월류수문의 성능을 보장하기 위한 설계 기준 마련 필요.
향후 연구 방향:
다양한 부유물 크기 및 형태에 따른 유량 변화 추가 연구 필요.
다양한 수문 형상 및 경사 조건에서 FLOW-3D 모델 검증.
AI 및 머신러닝을 활용한 부유물 농도 변화에 따른 유량 예측 모델 개발.
연구의 의의
이 연구는 FLOW-3D를 활용하여 모닝 글로리 월류수문의 부유물 농도 변화가 유량에 미치는 영향을 정량적으로 평가하고, 댐 안전성 및 수문 설계 최적화에 기여할 수 있는 실질적인 데이터를 제공한다.
Reference
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본 논문은 FLOW-3D®를 사용하여 선체 주변의 자유 표면 유동을 수치적으로 분석하고 실험 데이터를 기반으로 검증함.
DTNSRDC 5415 전투함 모델을 사용하여 난류 모델 및 수치 해석 기법을 검토함.
총 저항 예측, 파형 분석 및 난류 해석을 수행하여 모델의 신뢰성을 평가함.
CFD 시뮬레이션의 한계를 확인하고 메쉬 민감도 및 수치 기법 최적화 방향을 제시함.
연구 방법
실험 데이터 및 모델 선정
미국 해군이 개발한 DTNSRDC 5415 전투함 모델을 사용하여 수치 해석을 수행함.
프루드 수(Froude Number) 범위: 0.17 ~ 0.4로 설정하여 자유 표면 유동을 시뮬레이션함.
실험 데이터와 비교하여 시뮬레이션 결과의 정확성을 평가함.
FLOW-3D® 시뮬레이션 설정
VOF(Volume of Fluid) 방법을 사용하여 자유 표면 추적을 수행함.
난류 모델로 Reynolds-Averaged Navier-Stokes(RANS) 및 다양한 이류(advection) 기법을 비교 분석함.
메쉬 독립성 연구를 통해 최적의 격자 해상도를 결정함.
결과 비교 및 검증
실험 데이터와 비교하여 총 저항(Total Resistance) 예측 정확도를 평가함.
파형 분포 및 자유 표면 형상이 실험과 얼마나 일치하는지 분석함.
프루드 수(Froude Number)에 따른 저항 변화 및 난류 모델의 영향을 검토함.
주요 결과
총 저항 예측 및 비교 분석
1차 이류 기법(1st order upwind advection scheme) 사용 시, 실험 대비 총 저항이 약 3배 과대 예측됨.
ITTC-57 방법을 적용하여 마찰 저항을 보정하면, 실험과의 오차가 절반 수준으로 감소함.
2차 이류 기법(2nd order scheme)을 적용하면 총 저항 예측이 개선되었으나 여전히 약 2배 과대 평가됨.
파형 및 자유 표면 분석
시뮬레이션에서 자유 표면 형상 및 파형 패턴은 실험과 유사하게 나타남.
파랑 저항(Wave Resistance)은 메쉬 해상도가 높아질수록 실험값과 더 가까워짐.
그러나 경계층 해석이 부족하여 마찰 저항(Frictional Resistance) 예측이 부정확함.
메쉬 민감도 연구 결과
메쉬 독립성을 완전히 확보하지 못한 상태에서 총 저항이 65%까지 과대 평가됨.
메쉬 해상도를 증가시킬수록 저항값이 감소하지만, 연산 비용이 크게 증가함.
추가적인 연구를 통해 완전한 메쉬 독립성 확보 필요.
난류 모델 및 수치 기법 평가
2차 이류 기법 + 단조 유지(Monotonicity Preserving) 조합이 가장 적절한 결과를 제공함.
다중 블록 격자(Multi-Block Gridding)와 추가적인 난류 모델 적용이 필요함.
향후 연구에서는 경계층 및 마찰 저항 개선을 위한 고급 난류 모델 적용이 필수적임.
결론
FLOW-3D®는 선체 주변 자유 표면 유동의 질적(qualitative) 분석에 적합함.
총 저항 예측은 과대 평가되며, 마찰 저항 해석 능력이 제한적임.
2차 이류 기법 + 단조 유지 기법 적용 시, 실험과의 상관성이 가장 높음.
메쉬 독립성 확보 및 난류 모델 최적화가 추가 연구의 핵심 과제임.
Reference
Barkhudarov, M., “Multi-Block GriddingTechnique for FLOW-3D”, Technical Note #59-R2, FSI-00-TN59-R2, Flow Science Inc., 2004
Ferziger, H. J., and Peric, M., “ComputationalMethods for Fluid Dynamics”, 2001, SpringerVerlag
Lewis, E. V., “Principles of NavalArchitecture” SNAME, USA, 1989
Lin, A. C., “Bare Hull Effective PowerPredictions and Bilge Keel Orientation forDDG51 Hull Represented by Model 5415,”DTNSRDC/SPD-0200-03, 1982(http://conan.dt.navy.mil/5415/)
Ratcliffe, T. J., Muntick, I., Rice, J., “SternWave Topography and Longitudinal Wave Cutsobtained on Model 5415, With and WithoutPropulsion”, DTM, USA, 2001
Yao, G. F., “Development of New PressureVelocity Solvers in FLOW-3D”, Flow Science,Inc., USA, 2004
FLOW-3D를 이용한 저압 주조(LPC, Low Pressure Casting) 공정 시뮬레이션
연구 배경 및 목적
문제 정의: 저압 주조(LPC) 공정은 박벽(Thin-Walled) 및 중공(Hollow) 부품 제조에서 우수한 품질과 미적 요소를 충족할 수 있는 주조 기술로 인정받고 있다.
그러나 LPC 공정의 복잡한 처리 매개변수(예: 용탕 온도, 금형 온도, 압력 및 지속 시간)로 인해 최적화가 어렵다.
수치 시뮬레이션을 통해 이러한 매개변수를 효과적으로 제어할 수 있다.
연구 목적:
FLOW-3D 소프트웨어를 사용하여 LPC 공정 중 몰드 충진 및 응고 과정을 시뮬레이션.
금형 충진 시 발생 가능한 결함(예: 가스 포집, 수축 다공성)을 분석.
최적의 압력 시퀀스를 도출하고, 실험 결과와 비교하여 모델의 신뢰성을 검증.
연구 방법
수치 모델링 및 시뮬레이션 설정
FLOW-3D 소프트웨어의 유한 차분법(FDM, Finite Difference Method)을 사용.
FAVOR(Fractional Area/Volume Obstacle Representation) 기법을 통해 금형 형상 모델링.
VOF(Volume of Fluid) 기법을 활용하여 자유 표면 추적(Free Surface Tracking) 수행.
열전달(Heat Transfer) 및 상변화(Phase Change)를 포함한 에너지 방정식을 사용하여 응고 과정 시뮬레이션.
LPC 공정 설정 및 초기 조건
금형 설계 및 재료:
LM6 알루미늄 합금을 사용하여 H13 강철 금형에 주조.
금형 초기 온도: 350°C, 용탕 온도: 760°C 설정.
압력 시퀀스:
초기 대기압(Atmospheric Pressure)을 적용하고, 작은 벤트(Small Vent) 입구를 통해 점진적 압력 증가.
압력 램프(Pressure Ramp) 기법을 통해 몰드 충진 중 난류(Turbulence) 최소화.
격자 설정:
1.6백만 개의 보셀(Voxel)을 사용한 3.5 mm 메쉬 크기.
최적 메쉬 크기를 사용하여 모델 정확도 향상.
주요 결과
몰드 충진 시퀀스 분석
몰드 충진 시간: 약 6.5초 소요.
충진 시 층류 흐름(Laminar Flow)을 유지하여 가스 포집(Gas Entrapment)과 같은 결함 발생 억제.
금형 내 온도 분포:
용탕이 몰드에 진입 시 687°C의 온도를 유지.
냉각 패턴이 일관되게 나타나며, 응고 결함 최소화.
응고 과정 및 온도 분포
응고 완료 시간: 약 103.5초.
액상에서 고상으로의 변화가 선형적으로 진행.
고상 분율(Solid Fraction)이 0에서 1로 변환되는 동안 열전달이 균일하게 이루어짐.
실험적으로 얻은 주조물(Fig. 4)과 비교 시 완전 충진 및 결함 없는 주조물을 확인.
FLOW-3D 모델의 신뢰성 평가
FLOW-3D 시뮬레이션 결과와 실험 결과 간 높은 일치도 확인.
적용된 압력-시간 프로파일이 람inar 충진 시퀀스를 제공하여 충진 결함 최소화.
온도 분포가 합금의 고상온도(Solidus Temperature) 이상을 유지하여 주조물의 완벽한 충진 가능.
결론 및 향후 연구
결론:
FLOW-3D 소프트웨어가 LPC 공정 시 주조물의 몰드 충진 및 응고 과정을 정확하게 예측할 수 있음.
적절한 압력 시퀀스 설정을 통해 람inar 흐름을 유지하고 가스 포집 및 응고 결함을 억제할 수 있음.
온도 분포와 금형 온도 설정이 적절하여완벽히 충진된 주조물 제조 가능.
향후 연구 방향:
다양한 합금 재료 및 금형 재질을 대상으로 모델 검증.
AI 및 머신러닝을 활용한 실시간 LPC 공정 최적화 시스템 개발.
산업 응용을 위한 최적 LPC 공정 설계 및 실증 실험 수행.
연구의 의의
이 연구는 FLOW-3D를 활용하여 저압 주조 공정의 몰드 충진 및 응고 과정을 정량적으로 평가하고, 주조물의 품질을 향상시킬 수 있는 최적의 공정 매개변수를 제시하며, 자동차 및 항공우주 산업의 생산성 증대와 비용 절감에 기여할 수 있다.
Reference
Alan A Luo, Anil K Sachdev, Bob R Powell, Advanced casting technologies for lightweight automotive application, China Fou. 7 (2010) 463-469.
Alan A Luo, Magnesium casting technology for structural applications, J. Mag. Aly, 1 (2013) 2-22.
A. Srinivasan, U. T. S. Pillai, J. Swaminathan, B. C. Pai, Enhanced high temperature properties of low pressure cast AZ91 Mg alloy, Inter. J. Cast Met. Res. 19 (2006) 265-268.
Penghuai Fu, Alan A Luo, Haiyan Jiang, Liming Peng, Yandong Yu, Chunquan Zhai, Anil K Shachdev, Low pressure die casting of magnesium alloy AM50: Response to process parameters, J. Mater. Proc. Tech. 205 (2008) 224-234.
A. Kermanpur, Sh. Mahmoudi, A. Hajipour, Numerical simulation of metal flow and solidification in the multi-cavity casting moulds of automotive components, J. Mater. Proc. Tech. 206 (208) 62-68.
교각 주변 최대 세굴 깊이에 대한 부유물(Debris)의 영향 분석: FLOW-3D 시뮬레이션
연구 배경 및 목적
문제 정의: 교각(Bridge Pier) 주변의 국부 세굴(Local Scour)은 하천 바닥의 침식을 유발하여 교량의 구조적 안전성을 위협하는 주요 요인 중 하나이다.
홍수 시에는 유량 증가와 부유물 증가로 인해 세굴 현상이 더욱 심화된다.
부유물은 교각 주변에 쌓여 흐름을 방해하고, 난류(Turbulence) 증가 및 전단응력(Shear Stress) 증대로 이어져 세굴을 악화시킨다.
연구 목적:
FLOW-3D CFD 모델을 사용하여 부유물 형태 및 위치가 원형 교각 주변의 최대 세굴 깊이에 미치는 영향을 평가.
삼각형 및 직사각형 부유물을 수면에 떠있는 경우(floating)와 하상(sand bed)에 놓인 경우로 구분하여 비교.
실험 결과와의 비교 검증을 통해 모델의 신뢰성을 평가.
연구 방법
수치 모델링 및 시뮬레이션 설정
FLOW-3D 소프트웨어를 활용한 3차원 CFD 해석 수행.
RNG k-ε 난류 모델을 사용하여 난류 흐름을 모델링.
VOF(Volume of Fluid) 기법을 통해 자유 수면(free surface)을 추적.
모델 검증:
Robalo (2014)의 실험 데이터를 사용하여 평균 속도 및 세굴 깊이 비교.
실험 채널은 길이 12.0m, 폭 0.83m, 높이 1.0m의 직사각형 콘크리트 플룸(fluvial hydraulic channel) 사용.
부유물(Debris) 유형 및 조건
부유물 형태: 삼각형(Triangular) 및 직사각형(Rectangular).
부유물 위치:
수면(floating debris): 부유물이 흐름을 막아 세굴을 증대시킴.
하상(sand bed debris): 세굴 억제(countermeasure) 역할을 하여 세굴 깊이를 줄이는 효과.
하상 재료:
천연 석영 모래(Quartz Sand) 사용, D50 = 0.86 mm, 밀도 2666 kg/m³.
평균 접근 유속(U) = 0.317 m/s, 수심 0.15 m 설정.
모델 검증 및 비교 분석
FLOW-3D 결과와 실험 결과 비교:
평균 속도 프로파일 비교 시, 실험과 유사한 흐름 발달을 보여 모델의 신뢰성 확보.
그러나 이동 가능한 하상(movable bed)을 적용했을 때, 평균 30%의 편차가 발생.
Shields (1936) 기준과의 비교:
실험에서는 Neil (1967) 및 Garde (1970) 공식을 사용하여 한계 유속(Uc) 0.314 m/s를 평가.
FLOW-3D에서는 Shields 기준(Uc 0.403 m/s)을 사용하여 20% 높은 값을 적용.
이는 세굴 깊이 과소 예측의 주요 원인으로 분석.
주요 결과
세굴 깊이 및 부유물 영향 비교
부유물 없는 경우:
세굴 깊이 0.06 m.
부유물 유형에 따른 세굴 깊이:
직사각형 부유물(수면): 세굴 깊이 0.07 m (가장 큰 세굴).
삼각형 부유물(수면): 세굴 깊이 0.05 m.
삼각형 부유물(하상): 세굴 깊이 0.04 m, 세굴 감소 효과 ≈ 40%.
FLOW-3D 모델의 신뢰성 평가
고정된 하상(fixed bed) 시뮬레이션에서는 실험과 높은 일치도를 보임.
그러나 이동 가능한 하상(movable bed) 적용 시, 실험 결과와 평균 30% 편차 발생.
Shields 기준의 한계:
Uc 평가 방법의 차이가 주요 원인으로 분석.
Shields 기준은 한계 전단응력(Threshold Shear Stress)을 과대 평가하여 세굴 깊이를 과소 예측.
결론 및 향후 연구
결론:
FLOW-3D 모델은 부유물 유형 및 위치에 따른 세굴 깊이 변화를 예측할 수 있음.
수면에 떠 있는 직사각형 부유물이 세굴을 가장 크게 증가시키며, 삼각형 부유물(하상)은 세굴을 줄이는 효과가 있음.
Shields 기준의 적용이 세굴 깊이 과소 예측의 원인으로, 국내 하천 조건에 맞는 보정 필요.
향후 연구 방향:
다양한 난류 모델(예: LES, k-ω 모델) 적용 및 비교.
다양한 하상 조건 및 교각 형상에 대해 추가 검증.
AI 및 머신러닝을 활용한 세굴 예측 모델 개발.
부유물의 재질, 크기 및 배열에 따른 세굴 영향 연구.
연구의 의의
이 연구는 FLOW-3D를 활용한 교각 주변 부유물의 세굴 영향 분석을 통해 교량 설계 및 유지보수 전략 수립에 중요한 기초 데이터를 제공하며, 홍수 시 구조물 안전성을 높이는 데 기여할 수 있다.
Reference
Baykal, C., Sumer, B.M., Fuhrman, D.R., Jacobsen, N.G. e FredsØe, J. (2015). “Numerical investigation of flow and scour around a vertical circular cylinder”. Philos Trans A MAth Phys Eng Sci, 373(2033): 20140115.
Dias, A.J.P. (2018). Study the impact of debris on cavities erosion at bridge piers”. MSc Thesis, University of Beira Interior (in Portuguese).
Dias, A.J.P., Fael, C.S. and Núñez-González, F. (2019). Effect of debris on local scour at bridge piers. IOP Conference Series: Materials Science and Engineering, 471–022024.
Franzetti, S., Radice, A., Rabitti, M. and Rossi, G. (2011). Hydraulic design and preliminary performance evaluation of countermeasure against debris accumulation and resulting local pier scour on River Po in Italy. Journal of Hydraulic Engineering, 137(5), 615–620.
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Li, G., Lang, L. and Ning, J. (2013). 3D Numerical Simulation of Flow and Local Scour around a Spur Dike. Proceedings of the 35th IAHR World Congress, Chengdu, China.
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3D Numerical Modelling of Flow Around Skewed Bridge Crossing
(“비스듬한 교량 횡단부 주변 흐름의 3D 수치 모델링”)
연구 목적
본 논문은 FLOW-3D를 활용하여 비스듬한(스큐) 교량 횡단부 주변의 수면 흐름을 시뮬레이션하고 실험 데이터와 비교하여 모델의 성능을 평가함.
실험실 규모에서 다양한 스큐 각도(30°, 45°)를 적용하여 수면 프로파일 변화를 분석함.
Reynolds-Averaged Navier-Stokes (RANS) 방정식을 기반으로 한 FLOW-3D의 수치 해석 결과와 실험 데이터를 비교하여 정확도를 검증함.
교량 설계 및 홍수 관리에 있어 3D 수치 모델의 활용 가능성을 탐구함.
연구 방법
실험 모델 설정
영국 버밍엄 대학교 수리 실험실에서 다양한 교량 구조(아치형, 평면형 등)를 대상으로 실험 수행함.
22m 길이, 1.213m 너비, 0.4m 깊이의 복합 수로(compound channel)에서 교량 흐름 실험을 진행함.
실험 데이터는 기존 연구에서 활용된 1D 및 2D 모델과 비교 검증함.
FLOW-3D 시뮬레이션 설정
FAVOR(유체 부피 기법)를 적용하여 교량 구조를 모델링함.
난류 모델로 k−εk-\varepsilonk−ε 방정식을 사용하여 수치 해석 진행함.
메쉬 독립성 연구를 수행하여 최적의 격자 크기를 결정함.
결과 비교 및 검증
실험실에서 측정한 자유표면 프로파일과 FLOW-3D 결과를 비교하여 모델 신뢰도를 평가함.
다양한 흐름 조건(유량 변화, 교량 구조 차이 등)에 따른 모델 성능을 분석함.
실험값과 계산값 간 오차를 정량적으로 분석하고, 오차의 주요 원인을 규명함.
추가 분석
1D, 2D, 3D 모델 간 비교를 수행하여 모델별 장단점을 평가함.
실험 데이터와 수치 모델의 차이를 최소화하기 위한 보정 기법을 검토함.
주요 결과
수면 프로파일 비교
FLOW-3D 시뮬레이션은 실험 데이터와 유사한 수면 프로파일을 재현함.
30°와 45° 스큐 각도에서 측정된 최대 백워터(afflux) 값이 유사하게 나타남.
교량 형상 및 흐름 조건 변화에 따른 수면 변화 패턴을 정확히 예측함.
스큐 각도와 유량의 영향
스큐 각도가 증가할수록 백워터 높이가 증가함.
유량 증가 시 백워터 영향이 커지며, 45° 각도에서는 30°보다 평균 7~23% 높은 백워터 발생함.
난류 특성이 강한 구간에서는 수치 모델이 일부 오차를 보임.
FLOW-3D의 정확성 평가
실험값과 모델 예측값의 평균 오차율은 30°에서 3.5%, 45°에서 2.2%로 나타남.
실험실 조건에서는 모델이 비교적 정확한 결과를 제공하나, 자연 하천 환경에서 추가 검증이 필요함.
메쉬 해상도와 난류 모델의 선택이 결과에 중요한 영향을 미침.
설계 및 적용성 평가
FLOW-3D는 복잡한 교량 주변 흐름 해석에 유용한 도구임.
3D 모델을 활용하면 1D, 2D 모델보다 높은 정확도로 수면 변화를 예측할 수 있음.
향후 연구에서는 침식 및 고유 유량 변화를 포함한 실험 검증이 필요함.
결론
FLOW-3D는 비스듬한 교량 횡단부 주변 흐름을 효과적으로 시뮬레이션할 수 있음.
스큐 각도가 증가할수록 백워터가 증가하는 경향이 확인됨.
수치 모델과 실험 데이터 간 평균 오차는 3.5%~2.2% 범위로 나타남.
향후 연구에서는 더 높은 난류 해상도 및 자연 하천 환경에서 추가적인 검증이 필요함.
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유체 유동 해석: 용융 풀(Molten Pool) 거동을 해석하기 위해 VOF(Volume of Fluid) 기법 적용.
열전달 및 용접 프로세스 시뮬레이션:
용융 금속의 열전달 및 응고 해석.
용접 과정에서 발생하는 표면 장력(Marangoni Effect) 분석.
아크 기반 용접 비드 모델링:
용접 비드 형상을 원형(Arc) 모델로 근사하여 예측.
비드 프로파일의 중심 좌표 및 반지름을 주요 특징으로 설정.
심층 학습을 활용한 용접 비드 예측
신경망 모델: ResNet 기반 CNN 모델을 사용하여 이미지에서 용접 비드 프로파일을 추출하고, 위치 및 반지름 예측.
입출력 데이터:
입력: 용접 이음매의 단면 이미지 + 용접 위치 정보.
출력: 예측된 용접 비드 프로파일 (중심 좌표 및 반지름).
학습 데이터: Flow-3D 시뮬레이션 데이터를 활용하여 대량의 학습용 데이터를 생성.
주요 결과
ADAP 모델 성능 평가:
용접 비드 중심 좌표 예측 오차: 평균 0.73mm
반지름 예측 오차: 평균 0.66mm
실시간 예측 속도: 15ms (NVIDIA RTX 3060 GPU 기준)
기존 방법과 비교:
기존의 경험적 모델보다 더 높은 정확도로 용접 비드 형상을 예측.
CFD 기반 시뮬레이션보다 계산 속도가 훨씬 빠르며, 실시간 용접 경로 조정이 가능.
MLMP 용접의 실용성 증대:
자동화 용접 공정에서 실시간 예측 모델로 활용 가능.
용접 품질 향상을 위한 최적의 경로 및 공정 변수 제어 가능.
결론 및 향후 연구
결론:
ADAP 모델은 심층 학습을 활용하여 MLMP 용접에서 실시간 이음매 프로파일을 정확하게 예측할 수 있음을 입증함.
Flow-3D 기반 시뮬레이션 데이터를 이용한 학습을 통해 실험 데이터 없이도 높은 정확도로 용접 형상을 예측 가능함.
향후 연구 방향:
더 다양한 용접 공정 변수(토치 각도, 와이어 공급 속도 등)를 포함하여 모델 성능 개선.
실제 산업 환경에서 로봇 용접 시스템과 통합하여 실증 실험 진행.
3D 스캐너와 결합하여 실시간 품질 모니터링 및 피드백 시스템 구축.
연구의 의의
본 연구는 AI 기반 데이터 중심 용접 품질 예측 모델을 제안함으로써, 기존 경험적 방식에서 벗어나 정확하고 실시간 대응이 가능한 자동화 용접 시스템 개발에 기여할 수 있음을 시사한다.
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선박 설계 시 추진 시스템의 효율성을 결정하는 핵심 요소 중 하나는 선박 저항(항해 중 발생하는 해양 저항)이다.
선박 저항은 선박의 연료 소비와 환경 영향을 좌우하며, 초기 설계 단계에서는 Holtrop-Mennen (HM)과 같은 통계적 방법을 주로 사용한다.
완성된 3D 선체 디자인이 마련되면 CFD 시뮬레이션이나 축척 모델 실험을 통해 보다 정밀한 저항 값을 산출할 수 있다.
본 연구는 RoPax 여객선을 대상으로 Flow-3D 소프트웨어를 활용하여 다양한 선박 속도에서의 저항을 계산하고, 이를 HM 방법과 비교·분석하는 데 목적이 있다.
연구 방법
CFD 시뮬레이션 수행
소프트웨어: Flow-3D를 사용하여 3차원 Navier-Stokes 방정식을 풀어 선박 주변의 자유 표면 유동을 해석.
메쉬 기법: FAVOR (Fractional Area-Volume Obstacle Representation) 기법을 이용한 ‘Free Gridding’으로 복잡한 선체 형상을 간단하게 모델링.
경계조건: 입구에 유속 조건(선박 속도에 해당하는 값)과 해수의 깊이를 설정하여 실제 해양 조건을 반영.
난류 모형: RANS k-ε 모델을 사용하여 난류 효과를 고려.
메쉬 민감도 분석
다양한 격자 크기를 적용하여 결과의 민감도를 평가하고, 최적의 해상도와 계산 시간을 확보함.
비교 분석
CFD 시뮬레이션 결과로 도출된 선박 저항 값을 Holtrop-Mennen (HM) 방법의 예측값과 비교.
낮은 선박 속도(10 knots)에서는 CFD 결과와 HM 방법 간의 차이가 미미하나, 속도가 증가할수록 CFD 결과가 HM 예측보다 크게 증가하는 경향을 분석.
주요 결과
저항 값 비교:
10 knots에서 CFD 시뮬레이션 결과는 HM 방법과 유사하였으나, 15 knots 이상에서는 CFD 결과가 HM 방법보다 현저히 높은 저항 값을 나타냄.
예를 들어, 20 knots에서는 HM 방법 대비 약 35% 높은 저항 값이 나타났으며, 24 knots에서는 약 32% 차이가 발생함.
메쉬 민감도:
더 미세한 메쉬(최종적으로 Mesh 3 사용)에서 시뮬레이션된 저항 값은 거친 메쉬에 비해 낮은 값을 보여, 격자 크기가 결과에 미치는 영향을 확인함.
선박 속도에 따른 변화:
선박 속도가 증가할수록 파 생성 및 파 부서짐으로 인한 추가 저항이 크게 기여하며, 이는 선박 저항의 비선형적인 증가로 나타남.
결론 및 향후 연구
Flow-3D를 활용한 CFD 시뮬레이션은 선박 저항을 예측하는 데 효과적인 도구임을 확인하였다.
특히, 고속 조건에서 CFD 결과는 HM 방법보다 높은 저항 값을 산출하며, 이는 파 저항의 기여를 반영한 결과로 해석된다.
향후 연구에서는 다른 난류 모형(예: Wilcox k-ω, RNG k-ε)과의 비교, 실제 모델 테스트(예: 축척 모델 실험)와의 추가 검증을 통해 CFD 해석의 정확성을 더욱 향상시킬 필요가 있다.
본 연구 결과는 선박 설계 및 최적 운항 속도 결정 등 실무에 유용한 참고 자료로 활용될 수 있다.
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날카로운 정상부를 가진 삼각형 허들(Sharp-Crested Triangular Hump) 위의 유동 특성 수치 모델링
연구 배경
문제 정의: 수리 구조물의 성능 및 수면 프로파일을 정확히 예측하는 것은 실험적으로 어렵고 비용이 많이 듦.
목표: CFD(Computational Fluid Dynamics)를 활용하여 삼각형 허들 위의 유동 특성을 보다 효율적이고 정확하게 분석.
접근법: FLOW-3D 기반 시뮬레이션을 수행하여 실험 데이터와 비교 검증.
연구 방법
삼각형 허들(Weir) 개요
위어(Weir)는 개수로에서 유량 조절과 방류 역할을 수행하는 중요한 수리 구조물.
본 연구에서는 크기가 50 cm × 30 cm × 7 cm인 Sharp-Crested Triangular Hump 모델을 사용.
수치 모델링
FLOW-3D를 사용하여 RANS(Reynolds-Averaged Navier-Stokes) 방정식과 VOF(Volume of Fluid) 방법을 적용.
FAVOR(Fractional Area-Volume Obstacle Representation) 기법을 사용하여 메쉬 내 장애물 영향을 반영.
총 1,920,000개의 격자 셀을 사용하여 시뮬레이션 수행.
실험 설정
Universiti Teknologi PETRONAS(UTP)의 수리 실험실에서 실험 수행.
30cm 폭, 60cm 높이, 10m 길이의 플룸(flume)에서 실험 진행.
4가지 유량 조건(30, 51.3, 75.3, 31 m³/h) 및 경사 조건(0, 0.006, 0.01)으로 실험 설계.
주요 결과
수치 시뮬레이션 vs 실험 데이터 비교
수치 시뮬레이션과 실험 결과 간의 차이는 4~5% 이내로 매우 높은 정확도를 보임.
수면 프로파일, 평균 유속, 프로우드 수(Froude Number) 등이 실험과 잘 일치.
유동 특성 분석
프라우드 수(Froude Number) 변화:
상류(Upstream)에서는 Froude Number < 1.0 → 서브크리티컬(Subcritical) 흐름.
하류(Downstream)에서는 Froude Number > 1.0 → 슈퍼크리티컬(Supercritical) 흐름.
유속(Flow Velocity) 변화:
하류로 갈수록 유속 증가, 삼각형 허들이 흐름을 방해하면서 압력 변화를 유발.
수심(Flow Depth) 변화:
상류에서는 높은 수심 유지, 하류에서는 급격한 감소 확인.
수치 시뮬레이션의 유용성
FLOW-3D가 삼각형 허들 및 수리 구조물의 유동 해석에 효과적임을 확인.
기존의 실험적 접근보다 비용이 낮고 신속한 설계 검토 가능.
결론 및 향후 연구
FLOW-3D 기반 CFD 시뮬레이션이 삼각형 허들의 유동 해석 및 설계 최적화에 효과적임을 검증.
실험 데이터와 비교했을 때 높은 정확도(오차 4~5%)를 나타내며, 초기 설계 검토에 유용함.
향후 연구에서는 다양한 난류 모델(k-ε, RNG, LES) 적용 및 추가적인 수리 구조물 연구가 필요.
연구의 의의
이 연구는 수리 구조물의 유동 해석을 위해 CFD 시뮬레이션을 실험적으로 검증하여, 위어 및 삼각형 허들 설계의 최적화 및 성능 예측을 위한 신뢰성 높은 방법론을 제시했다는 점에서 큰 의미가 있다.
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중간 수심에서 고정된 박스형 자유 수면 방파제의 유체역학적 성능 공식을 결정하기 위한 연구
Guoxu Niu, Yaoyong Chen, Jiao Lu, Jing Zhang, Ning Fan
Abstract
two-dimensional viscous numerical wave tank coded mass source function in a computational fluid dynamics (CFD) software Flow-3D 11.2 is built and validated. The effect of the core influencing factors (draft, breakwater width, wave period, and wave height) on the hydrodynamic performance of a fixed box-type free surface breakwater (abbreviated to F-BW in the following texts) are highlighted in the intermediate waters. The results show that four influence factors, except wave period, impede wave transmission; the draft and breakwater width boost wave reflection, and the wave period and wave height are opposite; the draft impedes wave energy dissipation, and the wave height is opposite; the draft and wave height boost the horizontal extreme wave force; four influence factors, except the draft, boost the vertical extreme wave force. Finally, new formulas are provided to determine the transmission, reflection, and dissipation coefficients and extreme wave forces of the F-BW by applying multiple linear regression. The new formulas are verified by comparing with existing literature observation datasets. The results show that it is in good agreement with previous datasets.
1. Introduction
A breakwater dissipates wave energy and reflects waves from the open sea, representing a crucial protective structure for the exploitation and utilization of marine resources. It is also an essential auxiliary marine structure that improves offshore engineering construction conditions and shortens ship berthing times [1,2,3]. With the development and utilization of ocean space and resources, the demand for breakwaters has also varied. The construction of breakwaters has shifted from onshore to offshore. Because most wave energy is concentrated near the water surface, a fixed box-type free surface breakwater (F-BW, Figure 1) was created [4,5]. The F-BW is a type of reflective breakwater with simple structure and high efficiency, which reduces the transmitted wave height by reflecting the incident wave energy [6,7]. Compared with the traditional bottom-founded breakwater, F-BW does not influence water exchange inside and outside the breakwater while maintaining a high wave attenuation efficiency, and has a high application prospect.
Figure 1. Two-dimensional schematic sketch of the F-BW models.
The hydrodynamic performance of the breakwater is important for the research and development of the F-BW, which mainly comprises two aspects. One aspect is the wave attenuation performance, including wave transmission coefficient Ct, reflection coefficient Cr, and dissipation coefficient Cd (hereinafter referred to as RTD coefficients). The other is the wave force, which concerns the safety and stability of the breakwater, including the horizontal wave force and vertical wave force. In terms of research on the RTD coefficients of an F-BW, some scholars have studied the influence of the breakwater width and draft on the reflection and transmission coefficients when energy dissipation was ignored. [8,9,10,11]. In order to provide some judgement for the needy practitioners, a closed-form formula has been created to predict the transmission coefficient in deep water [8,9,10]. A study by Kolahdoozan et al. [12] showed the poor prediction performance of the formula proposed by Macagno [8] for intermediate water. Therefore, it is necessary to explore a proposed formula for the transmission coefficient under intermediate-water conditions. Different from the analytical solution of potential flow theory, other scholars studied the influence of the draft, breakwater width, and wave height on the performance of wave reflection, transmission, and dissipation of the F-BW via experimental tests conducted in intermediate waters [13,14,15,16]. The computational fluid dynamics (CFD) technique provides us an alternative way to interpret the interaction between wave and F-BW. Koftis and Prinos [16] applied the two-dimensional unsteady Reynolds Averaged Navier–Stokes model to study the influence of the dimensionless draft on the transmission and reflection coefficients of an F-BW. Elsharnouby et al. [17] studied the influence of the draft on the wave transmission of the F-BW by using Flow-3D 11.2 software. Their results showed that the increasing draft impedes wave transmission. Some researchers carried out earlier work on wave force of F-BW due to concerns regarding safety and stability of the F-BW. Guo et al. [11] confirmed that draft, breakwater width and wave period also influenced the horizontal and vertical wave forces by adopting mathematical analysis based on linear potential flow theory. Chen et al. [18] investigated the effects of wave height and wave period on the horizontal and vertical wave forces of F-BW through a series of experiments. The results showed that the horizontal and vertical wave forces increase with increasing wave height. Limited by the fact that the mathematical analysis tends to ignore flow viscosity [19,20,21] and the physical model test is complicated and costly, considerable effort has been devoted to studying the hydrodynamic performance of an F-BW through numerical simulation in recent years. Zheng et al. [22] and Ren et al. [23] used the smoothed particle hydrodynamics (SPH) method to numerically simulate the horizontal and vertical wave forces of F-BWs under regular waves. Unlike previous studies which overlooked the nonlinearity of wave forces, the positive and negative maximum wave force could be observed in the studies of Zheng et al. [22] and Ren et al. [23]. Human activities are less involved in deep water, and the cost-effectiveness of F-BW construction is poorer in deep water than intermediate water. Reflection coefficient Ct, and dissipation coefficient Cd are also an indispensable part of the wave attenuation performance of F-BW. The horizontal and vertical wave forces are related to the security of the F-BW. However, the prediction formulas based on tests or numerical simulations for horizontal and vertical wave forces of the F-BW in the above studies were rare. Therefore, an attempt is necessary to present a proposed formula for the prediction of RTD coefficients and wave forces, which will provide design judgments for the relevant practitioners in intermediate waters. The objective of this paper is to provide the prediction formulas for RTD coefficients and wave forces in the intermediate waters under the condition that waves do not overtop the breakwater. With the rapid development of the CFD technique, Kurdistani et al. [24] proposed a formula for submerged homogeneous rubble mound breakwaters based on a large dataset from the CFD model, and the proposed formula was verified by using the literature observation datasets. Inspired by their research method, a numerical wave flume is built through a grid convergence test and validated with the existing experimental results. The prediction equations of RTD coefficients and wave forces are provided by applying multiple linear regression and verified by comparing with existing literature observation datasets. The major conclusions are finally summarized, and some prospects are proposed.
2. Theoretical Introduction
2.1. Governing Equations
Flow-3D 11.2 is widely used in coastal engineering as a powerful CFD software program [25]. The interaction of waves and breakwaters is simulated in a numerical wave tank by using Flow-3D 11.2 software in this paper. The numerical wave tank adopts an incompressible viscous fluid in the wave and F-BW interaction. The Reynolds averaged Navier–Stokes (RANS) equation was applied as the governing equation for turbulent flow. Assuming that the Cartesian coordinate system o-xyz originates from the still water surface, the continuity equation is shown in Equation (1), and the momentum equation is expressed in Equation (2).
where i,j = 1,2 for two-dimensional flows, xi represents the Cartesian coordinate, and ui is the fluid velocity along the x- and z-axes. Ax and Az are the area fractions open to flow in the x and z directions, respectively, ρ is the fluid density, p is the pressure, v is the dynamic viscosity, and g is the gravity force. The Reynolds stresses term, 𝜌𝑢𝑖′𝑢𝑗, is modeled by the renormalized-group (RNG) turbulence model.
2.2. RNG Turbulence Model
The interaction of waves and an F-BW induces turbulence. The RNG turbulence model is adopted to close the governing equations [26], and the discrete governing equations are solved by the finite difference method. The transport equations of turbulent kinetic energy kT and dissipation rate εT in this model are as follows:
The volume of fluid (VOF) method was developed to track the evolution of the free surface [27]. The governing equation is shown as follows:
where F represents the fractional volume of water fluid, F = 1 indicates that the numerical cell is full of water, and F = 0 corresponds to the cell fully occupied by air. Numerical cells with a value of 0 < F < 1 represent a water surface.
Furthermore, the generalized minimal residual (GMRES) method was used to solve the velocity-pressure term [28], and the first-order upwind scheme and Split Lagrangian method were used to solve the volume of fluid advection. The structure of the F-BW is directly imported into Flow-3D 11.2 by the software built-in drawing function. The appearance of an F-BW depicted by the mesh could be viewed with the fractional area volume obstacle representation (FAVOR) method. All numerical simulations were run in parallel using an Intel Core (TM) i5-4460 processor (3.20 GHz). Furthermore, to ensure the accuracy of the numerical solution, the maximum iteration time step was set to 0.001 s, and the results were output at 0.01-s intervals.
2.3. Principle of Mass Source Wavemaker
The present study emerged from the interest shown in the use of F-BW in a specific zone at an actual project in East China Sea. The detailed structural design dimensions of F-BW and wave characteristics are shown in Table 1. All the incident waves are considered to be regular waves. The regular waves used in the study contain a large range of wave periods and wave heights, which represent the majority of wave parameters in real-world problems, making this study of great practical importance. The interaction between the second-order Stokes wave and the current is not considered in the twelve major wave parameters, due to the differing time and spatial scales between the wave and the current [29]. The twelve waves in this research are all in the range of either linear or nonlinear second-order Stokes waves. Figure 2 shows the suitability range of different wave theories. According to Figure 2, the F-BW at this project is located in intermediate waters. Equation (5) presents the wave elevation equation η of the second-order Stokes wave and the wave elevation equation of the linear wave is the first term on the right side of this equation.
where Hi is the incident wave height, k is the wavenumber, σ is the wave frequency, and h is the still water depth.
Figure 2. Wave parameter conditions analyzed in this study and their relations in the Le Méhauté diagram.
Table 1. Summary of the simulated scenarios.
The boundary wavemaker method produces re-reflection waves. Lin and Liu [30] proposed a popular mass source wave generation method [31,32,33,34,35,36]. In the present method, numerical wave generation is achieved by importing a given volume flow rate Vfr into the mass source model. The expression of the volume flow rate Vfr is as follows:
where C is the phase velocity, W is the tank width, η(t) is the wave surface elevation by solving Equation (5).
To effectively reduce the calculation divergence caused by excessive waves in the NWT at the initial stage, the volume flow rate Vfr is multiplied by an increasing envelope function to make the wave increase gradually in the first three wave periods. The equation of the increasing function is as follows:
where t is time and T is the wave period.
2.4. Principle of Numerical Solution
In this paper, the time series of wave elevations were recorded at five different locations (i.e., WG1–WG5) on the onshore and offshore sides of the F-BW (Figure 3a). Furthermore, the current WG spacings are selected according to the water depth and wave period. The distances between the wave source and WG1, WG1 and WG2, WG2 and F-BW, and F-BW and WG5 are set at 1.5 m, 0.2 m, 1.8 m, and 1.435 m, respectively. Note that the distance between wave gauges WG1 and WG2 is more than 0.05 L and less than 0.45 L, and the distances between wave gauges WG2 and F-BW and between WG5 and F-BW are less than 0.25 L and more than 0.2 L (wavelength), as recommended by the two-point method [37]. Two wave gauges (WG1 and WG2) are mounted in a line on the offshore side of the F-BW to separate the incident wave heights Hi and the reflected wave heights Hr by using this method. To prove that the horizontal wave force of the F-BW is related to the free surface onshore and offshore of the breakwater, probe WG3 is placed 0.02 m in front of the F-BW, while probe WG4 is placed 0.02 m behind the F-BW to measure the wave profile at the front (η3) and back (η4) of the F-BW. The wave gauge (WG5) is mounted on the onshore side of the F-BW to obtain the surface elevation of the transmitted wave heights Ht. The wave transmission, reflection, and wave energy dissipation coefficients are defined by solving Equation (8a)–(8c).
where Ct is the transmission coefficient; Cr is the reflection coefficient; and Cd is the wave energy dissipation coefficient.
Figure 3. Schematic layout and mesh sketch of the numerical wave tank for the F-BW.
Furthermore, the horizontal and vertical wave forces are simulated by the integration of the water pressure p at the wet surface of the F-BW. The two kinds of wave forces include the hydrostatic force and hydrodynamic force according to the FLOW-3D theory manual [25]. Because the F-BW is always fixed at the free surface, the vertical wave force needs to remove part of the hydrostatic force (the value up to ρVg, where ρ is the density of water and V is the volume of the F-BW). The shear stress is small enough to be ignored in this paper relative to the wave force. The horizontal wave force and the vertical wave force are denoted by Fx and Fz, respectively. The horizontal wave force is consistent with the direction of wave propagation, and the vertical wave force is vertically upward. To facilitate the research, obtaining the extreme value of the steady part of the wave force time series, we define the average value of the horizontal wave force positive and negative peak as the horizontal positive maximum wave force Fx+max and horizontal negative maximum wave force Fx−max, the vertical wave force positive and negative peak as the vertical positive maximum wave force Fz+max and vertical negative maximum wave force Fz−max. The representative time series of the dimensionless wave elevation, horizontal, and vertical wave forces are shown in Figure 4. The numerical results of Hi, Hr, Ht, Fx±max, Fz±max were acquired based on the stable elevations in this figure. To facilitate discussion, we define Fx±max/0.005 ρgh2 and Fz±max/0.005 ρgh2 as the dimensionless horizontal and vertical maximum wave forces on the F-BW, respectively. The crest and trough values of the time series of the wave forces are studied because the extreme values of the horizontal and vertical wave forces on the F-BW under the Stokes second-order wave have a slightly sharper crest and flatter trough.
Figure 4. Time histories of wave elevation η measured by WG1, WG2, and WG5 and horizontal and vertical wave forces of F-BW at Hi = 0.07 m, T = 1.4 s, B = 0.5 m, dr = 0.14 m, and h = 0.75 m.
The integral formula of the horizontal and vertical wave force is shown in Equation (9).
where 𝑛⃗ is the unit normal vector of the object surface s and the water pressure p is determined by the Bernoulli equation.
3. Model Setup and Validation
3.1. Numerical Wave Tank Setup
The detailed numerical wave tank (NWT) setup is shown in Figure 3b,c. The total length of the NWT was twenty-four wavelengths L long in the x-axis direction, 0.1 m wide in the y-axis direction, and 1 m deep in the z-axis direction. A scale ratio of 1:40 and a constant water depth h of 0.75 m are adopted based on the Froude similarity law. The mesh consisted of two distinct regions. The first region was the computation domain, four wavelengths length, with a width of 0.1 m and a depth of 1 m. The unit grid size of the total NWT was L/100~L/200 in the x and z directions, and ten grids were partitioned in the y directions in this domain. The second region was two identical damping domains with ten wavelength lengths. The Sommerfeld radiation method was employed to bate the secondary reflection of waves at both ends of the NWT. The grid size along the x-axis direction was gradually extended with an identical ratio of 1.01, and one grid was set for the lateral width of the NWT [38].
To describe the F-BW more accurately, nested grids were applied in the domain around the F-BW. The uniform nested grid was equal to half of the compute domain grid in the x, y and z-axis directions. Furthermore, the finer mesh resolution of 0.0035 cm in z direction was nested near the still water level (SWL), The region extends ±0.07 m from the SWL to ensure that the expected wave heights (0.03 m, 0.05 m, 0.07 m, 0.09 m) are encompassed within the region.
The boundary conditions of this NWT were set as follows: both ends of the NWT were defined as outflow boundaries, two sides of the domain were defined as symmetry boundaries, the atmospheric pressure was utilized at the upper boundary, and the lower surface of the computational domain was a no-slip wall boundary without surface roughness.
A mass source model with wide WS and high HS was added to the numerical flume. The symmetry boundaries were used overspreading with the mass source form, and the y-direction width of the mass source block was consistent with the width of the NWT. Pledging each edge of the mass block to coincide with the grid line of the NWT is shown in Figure 3b,c.
3.2. Numerical Model Validation
The present research is mainly implemented under the framework of CFD technology. To demonstrate the accuracy of the simulation results, it is essential to compare them with the extant results. The model is verified by the following three aspects in this section.
3.2.1. Grid Independent Verification
The mesh partition is a crucial procedure in CFD numerical simulation and needs much attention. The number and size of grids are essential criteria for evaluating the convergence of numerical results. Poor grid quality will directly affect the accuracy of numerical results and computation time. Consider that the proposed calculation cases Hi = 0.06 m, T = 1.2 s, and h = 1.2 m by Ren et al. are close to the target cases in this paper [23]. This wave condition is applied to complete the grid independence verification. Different grid arrangements can be seen in Table 2, and the time series of the wave profiles under the three grid sizes are compared with the theoretical results by solving Equation (5), as shown in Figure 5. The error of the numerical simulation results was calculated according to Equation (10). The wave profile deviations among the coarse mesh, medium mesh, and fine mesh are compared. The wave profiles under the medium mesh and the fine mesh are closer, and the deviation from the theoretical value is less than 5%, which meets the requirements of Det Norske Veritas (DNV) [39]. It can be judged that only medium meshes and refined meshes meet the requirements of numerical simulation. Considering the balance between calculation accuracy and calculation efficiency, the following numerical simulations always chose a medium mesh.
where Htheoretical is the wave height of the theoretical result and Hnumerical is the wave height of the numerical result.
Figure 5. Grid independent verification: influence of mesh size on wave profile.
Mesh Type
Computation Domain Grid Size (cm)
Nested Domain Grid Size (cm)
Cell Number
Elapsed Time (×104 s)
Wave Height (cm)
Error %
Coarse
2
1
701460
0.6496
5.642
5.96
Middle
1
0.5
3411180
7.6832
5.768
3.87
Fine
0.5
0.25
13350960
48.1437
5.769
3.85
Theoretical
–
–
–
–
6.000
–
Table 2. Mesh independence check results.
3.2.2. Validation of Wave Forces
In this section, to further inspect the accuracy of the numerical results of wave forces in this paper, according to the wave conditions of Hi = 0.06 m, T = 1.2 s, h = 1.2 m, and draft dr = 0.2 m, a rectangular box of width B = 0.8 m and wave height Hi = 0.4 m is fixed and semi-immersed, as proposed by Ren et al. [23]. The horizontal and vertical wave forces of the F-BW were verified by comparison with the theoretical results of Mei and Black [40] and the numerical simulation results of Ren et al. [23]. The time series of the wave forces are compared in Figure 6. The total simulation time of this case is 16 wave cycles. Since it takes some time for the progressive wave to arrive at the F-BW from the source, the horizontal and vertical wave forces begin to reach the stable state at t = 7 T seconds in Figure 6. By comparison, the simulated time series of horizontal and vertical wave forces are almost consistent with those presented by Ren et al. [23] and Mei and Black [40]. This result indicated that the present NWT could meet the calculation accuracy.
Figure 6. Comparison of the normalized wave force on an F-BW with previous studies (Mei and Black [40]; Ren et al. [23]). (a) Normalized horizontal wave force; (b) Normalized vertical wave force.
4. Results and Discussion
4.1. Influence Analysis of Four Factors on the Hydrodynamic Performance of F-BW
Among all the influencing factors (refer to Appendix A), the hydrodynamic performance of the F-BW is significantly affected by the following four factors: draft (dr/h), breakwater width (B/h), wave period (T*sqrt(g/h)), and wave height (Hi/h). For the mechanism analysis of the interaction between waves and breakwater, the mechanism study of the horizontal wave force is rather complicated. Since the breakwater is in a semisubmerged state, the Morison formula is no longer applicable to the guidance of the calculation of the horizontal wave force. The horizontal wave force is studied separately from the water particle velocity; see the free surface difference (η3–η4) in the front and back sides of the F-BW and the water particle streamline in Figure 7 and Figure 8 for details. Among them, five representative cases are selected from all cases in this article for comparative analysis corresponding to Figure 7a–e. Note that case (a) T1.2dr0.14B0.5Hi0.07 represents a wave period of 1.2 s, draft of 0.14 m, breakwater width of 0.5 m and incident wave height of 0.07 m. Due to the effect of water blockage, flow separation is generated at the bottom corner of the offshore side of the breakwater, and the generated clockwise vortex destroys the original motion path of the wave water particles without structure in Figure 8a and allows the free surface difference in the front and back of the F-BW to gradually reach a maximum. At time instant t0 in Figure 7, the horizontal wave force also reaches a maximum. It can be seen in Figure 8b that the vertical wave force is easier to analyze. When the vertical wave force is at its maximum, the streamline realizes complete diffraction, and no vortex is generated. Furthermore, to understand the mechanism and contribution of each influencing factor on the hydrodynamic performance of the F-BW in detail, the statistical results are shown in Figure 9, Figure 10, Figure 11 and Figure 12.
Figure 7. Comparative analysis of five different cases under the interaction between waves and breakwater: First column: numerically obtained snapshots of free surface profile and velocity field; Second column: time history of free surface and horizontal wave force.
Figure 8. Snapshots of the velocity streamline field: (a) Time instant of the horizontal positive maximum wave force; (b) Time instant of the vertical positive maximum wave force.
Figure 9. Effect of the draft dr on the hydrodynamic performance of the F-BW at wave heights Hi = 0.05 m and Hi = 0.07 m. (a) Horizontal positive and negative maximum wave forces Fx+max and Fx−max; (b) Vertical positive and negative maximum wave forces Fz+max and Fz−max; (c) Transmission coefficient Ct, reflection coefficient Cr, and dissipation coefficient Cd.
Figure 10. Influence of the breakwater width B on the hydrodynamic performance of the F-BW at wave heights Hi = 0.05 m and Hi = 0.07 m. (a) Horizontal positive and negative maximum wave forces Fx+max and Fx−max; (b) Vertical positive and negative maximum wave forces Fz+max and Fz−max; (c) Transmission coefficient Ct, reflection coefficient Cr, and dissipation coefficient Cd.
Figure 11. Influence of the wave period T on the hydrodynamic performance of the F-BW at wave heights Hi = 0.05 m and Hi = 0.07 m. (a) Horizontal positive and negative maximum wave forces Fx+max and Fx−max; (b) Vertical positive and negative maximum wave forces Fz+max and Fz−max; (c) Transmission coefficient Ct, reflection coefficient Cr, and dissipation coefficient Cd.
Figure 12. Influence of the wave height Hi on the hydrodynamic performance of the F-BW at draft dr = 0.14 m and dr = 0.28 m. (a) Horizontal positive and negative maximum wave forces Fx+max and Fx−max; (b) Vertical positive and negative maximum wave forces Fz+max and Fz−max; (c) Transmission coefficient Ct, reflection coefficient Cr, and dissipation coefficient Cd.
4.1.1. Effect of Draft
Figure 7 lists the distribution diagram of the free surface difference and water particle velocity under cases (a) and (b) at the time instant of the horizontal wave force maximum. Except for the draft being different, the two cases are consistent. Among them, case (a) has a wave period of 1.2 s, draft of 0.14 m, wave height of 0.07 m and breakwater width of 0.5 m. Case (b) has a period of 1.2 s, draft of 0.35 m, wave height of 0.07 m and breakwater width of 0.5 m.
In the second column of Figure 7a, when time t0 = 11.48 s, the maximum free surface difference is 0.068 m, and the maximum horizontal wave force is 7.98 N. In the second column of Figure 7b, when time t0 = 11.52 s, the maximum free surface difference is 0.083 m, and the maximum horizontal wave force is 15.91 N. Obviously, the increase in the draft enhances the water blockage action in front of the F-BW, weakens the diffraction effect of the wave, and delays the time for the horizontal wave force to reach its maximum. Figure 9a shows that Fx+max increases with increasing draft under wave heights of Hi = 0.05 m and Hi = 0.07 m. Similarly, the absolute values of Fx−max exhibit a similar law. The absolute values of Fz−max and Fz+max decrease with increasing draft under wave heights of Hi = 0.05 m and Hi = 0.07 m in Figure 9b, which is related to the exponential decay of the wave kinetic energy along the water depth. It is not difficult to see in Figure 7a,b that the wave hydrodynamic pressure on the lower surface of the F-BW decreases with decreasing wave kinetic energy as the water depth increases. The effective action area increases as the draft reduces the penetration of waves. Figure 9c shows that the transmission coefficient decreases with increasing draft under wave heights of Hi = 0.05 m and Hi = 0.07 m. Due to the increase in the interaction area between waves and F-BW, the reflected wave energy increases in Figure 7, and Figure 7b is more obvious than Figure 7a. The wave energy dissipation coefficient increases with decreasing draft in Figure 9c. Since the wave energy is mainly concentrated on the still water level, the fluid particle velocity maximum at the lower corner of F-BW is 0.30 m/s in Figure 7a is more than the 0.17 m/s in Figure 7b, more wave energy is dissipated when the fluid particle with higher velocity collides with F-BW due to decreasing draft.
Overall, the increasing draft impedes incident waves cross F-BW and promotes the increase in horizontal wave force and wave reflection, which threatens the stability of the structure.
4.1.2. Effect of Breakwater Width
To clarify the mechanism of the breakwater width effect on the hydrodynamic performance of the F-BW, except that the breakwater width is different, cases (a) and (c) in Figure 7 are consistent. In case (c), the period is 1.2 s, the draft is 014 m, the wave height is 0.07 m, and the breakwater width is 0.2 m.
The free surface difference and vortex in Figure 7a,c are similar. Figure 10a shows that the breakwater width effect on Fx−max and Fx+max is not obvious. When the vertical wave force is at its maximum, the streamline realizes complete diffraction, and no vortex is generated in Figure 8b. Therefore, the vertical wave force is related to the acting area of the F-BW lower surface. Figure 10b shows that the absolute values of Fz−max and Fz+max increase with increasing breakwater width. In the second column of Figure 7c, when time t0 = 11.42 s, the free surface difference and the horizontal wave force reach a maximum faster than in case (a). Obviously, the increase in the breakwater width increases the wave diffraction difficulty. Figure 10c shows that the transmission coefficient decreases with increasing breakwater width, and the reflection coefficient increases with increasing breakwater width. Due to fluid particle velocity maximum is similar between Figure 7a,c. The increase in breakwater width has little influence on wave energy dissipation.
In short, the increasing breakwater width is not conducive to incident wave cross F-BW, and promotes the increase of wave reflection and vertical wave force. Obviously, more weights need to be added to ensure the safety of the breakwater when breakwater width increases.
4.1.3. Effect of Wave Period
To clarify the mechanism of the wave period effects on the hydrodynamic performance of the breakwater, except that the wave period is different, cases (a) and (d) are consistent. Figure 7d shows that the wave period is 1.8 s, the draft is 0.14 m, the wave height is 0.07 m and the breakwater width is 0.5 m.
In the second column of Figure 7d, when time t0 = 11.13 s, the maximum free surface difference is 0.051 m, and the maximum horizontal wave force is 6.90 N. According to Equation (9), because the wave energy is more abundant on the two sides of the breakwater in case (4), the horizontal wave force is comparable even if the free surface difference is smaller than that in case (1). Figure 11a shows that Fx−max and Fx+max are weakly related to the wave period under wave heights of Hi = 0.05 m and Hi = 0.07 m. Because the long-period waves possess a large wave energy in Figure 7d, they increase the wave pressure on the lower surface of the F-BW. Therefore, the absolute values of Fz−max and Fz+max increase linearly with the wave period in Figure 11b. Figure 11c shows that the transmission coefficient increases with increasing wave period under wave heights of Hi = 0.05 m and Hi = 0.07 m. Long-period waves have a better diffraction ability at the same depth, and more wave energy passes through the F-BW. The decreasing ratio of the breakwater width to wavelength weakens the ability to block progressive waves, and the reflection coefficient decreases accordingly. The wave energy dissipation coefficient shows an alphabetic symbol “M” distribution with the wave period. This indicates that the wave energy dissipation is more complex and requires further study. When the dimensionless wave period is 5.06, both the transmission and reflection coefficients are close to 0.71, the dissipation coefficient is at the minimum by applying Equation (8c).
In brief, the increasing wave period plays a significant role in increasing the wave transmission and the reducing wave reflection. Although it has little effect on the horizontal wave force, it promotes an increase in the vertical wave force, which is unfavorable to the security of the breakwater.
4.1.4. Effect of Wave Height
To clarify the mechanism of the wave height effects on the hydrodynamic performance of the breakwater, except that the wave height is different, cases (a) and (e) are consistent. Figure 7e shows that the wave period is 1.2 s, the draft is 014 m, the wave height is 0.03 m and the breakwater width is 0.5 m.
In the second column of Figure 7e, when time t0 = 11.44 s, the maximum free surface difference is 0.031 m, and the maximum horizontal wave force is 3.43 N. Obviously, the increase in wave height increases the diffraction difficulty of the wave and delays the time when the horizontal wave force reaches its maximum. The higher the wave height, the more abundant the wave energy in Figure 7a,e. The water particle velocity maximum is 0.11 m/s in Figure 7e, which is much less than the water particle velocity maximum in Figure 7a. The larger wave height causes a larger wave elevation difference, and the larger horizontal wave force under other variable conditions is consistent by comparing Figure 7a,e. Therefore, Fx−max and Fx+max increase linearly with increasing wave height under drafts dr = 0.14 m and dr = 0.28 m in Figure 12a. The increase in wave height leads to increasing dynamic wave pressure, which in turn leads to increasing wave pressure on the F-BW lower surface and an increase in vertical wave force. Therefore, Fz−max and Fz+max increase linearly with increasing wave height under drafts dr = 0.14 m and dr = 0.28 m in Figure 12b. Figure 12c shows that the increasing wave height results in more wave reflection and less transmission due to the increasing blockage effect. The reflection ability weakens with decreasing interaction area (the ratio of the wetted surface height of the front wall of the F-BW to the wave height). The water particle velocity maximum of 0.11 m/s in Figure 7e is less than the water particle velocity maximum of 0.3 m/s in Figure 7a. The increasing water particle velocity with increasing wave height results in better vortex dissipation near the F-BW. Hence, the wave energy dissipation coefficient increases.
In conclusion, the increasing wave height reduces the wave reflection but increases horizontal and vertical wave forces, which is disadvantageous to the security of the breakwater.
4.2. Prediction Equations of F-BW Hydrodynamic Performance Parameters
To understand the contribution of each influencing factor to the hydrodynamic performance of the F-BW in detail, the factors affecting the RTD coefficients and wave force mainly include the wave period T, wave height Hi, draft dr, breakwater width B, and still water depth h. In Equation (11), the RTD coefficients Ct,r,d and wave force extremum Fx,z±max are expressed as follows:
Using the dimensionless analysis method and the numerical simulation results of 30 groups of simulated conditions in Table 1 based on the Origin 2019b software platform, multiple linear regression was performed by the least squares method, and the prediction equations of the RTD coefficients and wave force are given in Equation (12a–g). The detailed formulas are shown in Table 3.
Table 3. Statistics of prediction equation. Note that 0.0933 ≤ dr/h ≤ 0.4667, 0.26667 ≤ B/h ≤ 0.8, 3.6166 ≤ T*sqrt(g/h) ≤ 6.5099, and 0.04 ≤ Hi/h ≤ 0.12.
4.3. Deviation Analysis of the Prediction Equations
Inspired by Kurdistani et al.’s [24] research method, the current study uses their method to assess the reliability of each predictive formula. The literature observation datasets include the measured RTD coefficients from Koutandos [13] (three cases (R1, R2 and R3) in Figure 16 of his literature) and Liang et al. [14] (six cases in Figures 14a, 19a and 22a of their literature), the wave forces from Mei and Black [40] and Ren et al. [23] (a case in Figure 10 of their literature). The numerical results obtained by Flow-3D are plotted on the x-axis in Figure 13, and predicted values of the predictive equations are plotted on the y-axis in Figure 13. Figure 13a shows a 20% error for the application of Liang et al. [14] transmission coefficient datasets that are mostly lower-estimated values of transmission coefficient with respect to Equation (12a), an almost 10% error for the application of Liang et al.’s [14] reflection coefficient and dissipation coefficient datasets, and Koutandos’s [13] RTD coefficients datasets. Figure 13b shows an almost 10% error for the application of Mei and Black [40] and Ren et al. [23] maximum wave force, which indicates that the present prediction equations are valid.
Figure 13. Comparison of the results between previous studies and the numerical results of this study. (a) Transmission coefficient Ct, reflection coefficient Cr, and dissipation coefficient Cd (Koutandos [13], Liang et al. [14]) and (b) Maximum wave force (Mei and Black [40]; Ren et al. [23]).
It is clearly found that the distribution points of the reflection coefficient and wave energy dissipation coefficient of the F-BW are relatively concentrated in a particular region in Figure 13a, indicating that the F-BW is dominant in reflecting waves and has stable wave dissipation ability. In addition, the horizontal negative maximum wave force of the F-BW is similar to the vertical negative maximum wave force, and the horizontal positive maximum wave force of the F-BW is slightly larger than the vertical positive maximum wave force in Figure 13b.
5. Conclusions
The present study investigated a high-accuracy numerical wave tank (NWT) based on the Flow-3D platform. A series of numerical simulations in the intermediate waters were carried out at a constant water depth (h) of 0.75 m under regular wave conditions, with a wave height range between 0.03–0.09 m, a wave period range between 1–1.8 s, and a breakwater width range between 0.2–0.6 m. The effects of four influencing factors (dr, B, T, Hi) on the hydrodynamic performance (RTD coefficients and wave forces) are highlighted. The vital conclusions are as follows:
The performance of two-dimensional viscous numerical wave tanks (NWTs) with a mass source wave maker and small length scale (1:40) are analyzed. By comparison, the wave model employed in this paper is competent for the numerical simulation of the F-BW.
The results show that the increase in the four influence factors, except the wave period, benefits the decrease in the wave transmission. The increase in draft and breakwater width is beneficial to the increase in the wave reflection, and the wave period and wave height are opposite. The increase in draft benefits the decrease in wave energy dissipation, and the wave height is opposite.
The increase in the draft and wave height benefits the increase in the horizontal positive and negative maximum wave forces. In addition to the draft, the increase in the other three influence factors benefits the increase in the vertical positive and negative maximum wave forces.
Applying multiple linear regression presents the prediction equations of RTD coefficients and the extreme wave force. The prediction equations are verified by comparing them with literature observation datasets.
This study provides insight into the relation of RTD coefficients and wave forces with parameters such as draft, breakwater width, wave period and wave height. The simulated results of the given predicted equations can be generalized to the prototype scale by using Froude’s scaling law and can be used to guide the design of F-BWs in intermediate waters.
Appendix A
The wave period T, wave height Hi, draft dr, breakwater width B, and water depth h are the main factors that affect the wave dissipation performance and wave force of an F-BW in the intermediate waters. Therefore, the wave force of an F-BW can be expressed as a function of the above factors as follows:
Taking water depth h, gravity acceleration g, and water density ρ as the repetitive parameters, the three dimensionless parameters are expressed as follows:
[h] = [M0L1T0], [g] = [M0L1T−2], [ρ] = [M1L−3T0], where wave force per breakwater length in the vertical wave direction 𝐹F, expressed as [F = ρgh2], Equation (A1) can be written as follows:
According to wave theory, there is a nonlinear relationship between the wave force and the four dimensionless parameters in Equation (A2). The relationship between the dependent variable and independent variable in nature is exponential. It can be expressed as follows:
where α, x1, x2, x3, and x4 are the unknown coefficients.
Taking the natural logarithm of both sides of Equation (A3) to obtain the double logarithm function model, the equation can be written in linear form as follows:
Using multiple function linear regression analysis, each unknown coefficient in the equations can be obtained and then substituted into Equation (A3) to obtain the wave force equations. Similarly,
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지표 산사태로 발생한 파랑의 3차원 시뮬레이션: OpenFOAM과 FLOW-3D HYDRO 모델 비교
Ramtin Sabeti, Mohammad Heidarzadeh, Alessandro Romano, Gabriel Barajas Ojeda & Javier L. Lara
Abstract
The recent destructive landslide tsunamis, such as the 2018 Anak Krakatau event, were fresh reminders for developing validated three-dimensional numerical tools to accurately model landslide tsunamis and to predict their hazards. In this study, we perform Three-dimensional physical modelling of waves generated by subaerial solid-block landslides, and use the data to validate two numerical models: the commercial software FLOW-3D HYDRO and the open-source OpenFOAM package. These models are key representatives of the primary types of modelling tools—commercial and open-source—utilized by scientists and engineers in the field. This research is among a few studies on 3D physical and numerical models for landslide-generated waves, and it is the first time that the aforementioned two models are systematically compared. We show that the two models accurately reproduce the physical experiments and give similar performances in modelling landslide-generated waves. However, they apply different approaches, mechanisms and calibrations to deliver the tasks. It is found that the results of the two models are deviated by approximately 10% from one another. This guide helps engineers and scientists implement, calibrate, and validate these models for landslide-generated waves. The validity of this research is confined to solid-block subaerial landslides and their impact in the near-field zone.
1 Introduction and Literature Review
Subaerial landslide-generated waves represent major threats to coastal areas and have resulted in destruction and casualties in several locations worldwide (Heller et al., 2016; Paris et al., 2021). Interest in landslide-generated tsunamis has risen in the last decade due to a number of devastating events, especially after the December 2018 Anak Krakatau tsunami which left a death toll of more than 450 people (Grilli et al., 2021; Heidarzadeh et al., 2020a). Another significant subaerial landslide tsunami occurred on 16 October 1963 in Vajont dam reservoir (Northern Italy), when an impulsive landslide-generated wave overtopped the dam, killing more than 2000 people (Heller & Spinneken, 2013; Panizzo et al., 2005). The largest tsunami run-up (524 m) was recorded in Lituya Bay landslide tsunami event in 1958 where it killed five people (Fritz et al., 2009).
To achieve a better understanding of subaerial landslide tsunamis, laboratory experiments have been performed using two- and three-dimensional (2D, 3D) set-ups (Bellotti & Romano, 2017; Di Risio et al., 2009; Fritz et al., 2004; Romano et al., 2013; Sabeti & Heidarzadeh, 2022a). Results of physical models are essential to shed light on the nonlinear physical phenomena involved. Furthermore, they can be used to validate numerical models (Fritz et al., 2009; Grilli & Watts, 2005; Liu et al., 2005; Takabatake et al., 2022). However, the complementary development of numerical tools for modelling of landslide-generated waves is inevitable, as these models could be employed to accelerate understanding the nature of the processes involved and predict the detailed outcomes in specific areas (Cremonesi et al. 2011). Due to the high flexibility of numerical models and their low costs in comparison to physical models, validated numerical models can be used to replicate actual events at a fair cost and time (e.g., Cecioni et al., 2011; Grilli et al., 2017; Heidarzadeh et al., 2020b, 2022; Horrillo et al., 2013; Liu et al., 2005; Løvholt et al., 2005; Lynett & Liu, 2005).
Table 1 lists some of the existing numerical models for landslide tsunamis although the list is not exhaustive. Traditionally, Boussinesq-type models, and Shallow water equations have been used to simulate landslide tsunamis, among which are TWO-LAYER (Imamura and Imteaz,1995), LS3D (Ataie-Ashtiani & Najafi Jilani, 2007), GLOBOUSS (Løvholt et al., 2017), and BOUSSCLAW (Kim et al., 2017). Numerical models that solve Navier–Stokes equations showed good capability and reliability to simulate subaerial landslide-generated waves (Biscarini, 2010). Considering the high computational cost of solving the full version of Navier–Stokes equations, a set of methods such as RANS (Reynolds-averaged Navier–Stokes equations) are employed by some existing numerical models (Table 1), which provide an approximate averaged solution to the Navier–Stokes equations in combination with turbulent models (e.g., k–ε, k–ω). Multiphase flow models were used to simulate the complex dynamics of landslide-generated waves, including scenarios where the landslide mass is treated as granular material, as in the work by Lee and Huang (2021), or as a solid block (Abadie et al., 2010). Among the models listed in Table 1, FLOW-3D HYDRO and OpenFOAM solve Navier–Stokes equations with different approaches (e.g., solving the RANS by IHFOAM) (Paris et al., 2021; Rauter et al., 2022). They both offer a wide range of turbulent models (e.g., Large Eddy Simulation—LES, k–ε, k–ω model with Renormalization Group—RNG), and they both use the VOF (Volume of Fluid) method to track the water surface elevation. These similarities are one of the motivations of this study to compare the performance of these two models. Details of governing equations and numerical schemes are discussed in the following.
Numerical models
Approach
Developer
FLOW-3D HYDRO
This CFD package solves Navier–Stokes equations using finite-difference and finite volume approximations, along with Volume of Fluid (VOF) method for tracking the free surface
Flow Science, Inc. (https://www.flow3d.com/)
MIKE 21
This model is based on the numerical solution of 2D and 3D incompressible RANS equations subject to the assumptions of Boussinesq and hydrostatic pressure
Danish Hydraulic Institute (DHI) (https://www.mikepoweredbydhi.com/products/mike-21-3)
OpenFOAM (IHFOAM solver)
IHFOAM is a newly developed 3D numerical two-phase flow solver. Its core is based on OpenFOAM®. IHFOAM can also solve two-phase flow within porous media using RANS/VARANS equations
IHCantabria research institute (https://ihfoam.ihcantabria.com/)
NHWAVE
NHWAVE is a 3D shock-capturing non-Hydrostatic model which solves the incompressible Navier–Stokes equations in terrain and surface-following sigma coordinates
Kirby et al. (2022) (https://sites.google.com/site/gangfma/nhwave, https://github.com/JimKirby/NHWAVE)
GLOBOUSS
GloBouss is a depth-averaged model based on the standard Boussinesq equations including higher order dispersion terms, Coriolis terms, and numerical hydrostatic correction terms
Løvholt et al. (2022) (https://www.duo.uio.no/handle/10852/10184)
BOUSSCLAW
BoussClaw is a new hybrid Boussinesq type model which is an extension of the GeoClaw model. It employs a hybrid of finite volume and finite difference methods to solve Boussinesq equations
Clawpack Development Team (http://www.clawpack.org/)Kim et al. (2017)
THETIS-MUI
THETIS is a multi-fluid Navier–Stokes solver which can be considered a one-fluid model as only one velocity is defined at each point of the mesh and there is no mixing between the three considered fluids (water, air, and slide). It applies VOF method
TREFLE department of the I2M Laboratory at Bordeaux, France (https://www.i2m.u-bordeaux.fr/en)
LS3D
A 2D depth-integrated numerical model which applies a fourth-order Boussinesq approximation for an arbitrary time-variable bottom boundary
Ataie-Ashtiani and Najafi Jilani (2007)
LYNETT- Mild-Slope Equation (MSE)
MSE is a depth-integrated version of the Laplace equation operating under the assumption of inviscid flow and mildly varying bottom slopes
A simplified 3D Navier–Stokes model for two fluids (water and landslide material) using VOF for tracking of water surface
Horrillo et al. (2013)Kim et al. (2020)
(Cornell Multi-grid Coupled Tsunami Mode (COMCOT)
COMCOT adopts explicit staggered leap-frog finite difference schemes to solve Shallow Water Equations in both Spherical and Cartesian Coordinates
Liu et al. (1998); Wang and Liu (2006)
TWO-LAYER
A mathematical model for a two-layer flow along a non-horizontal bottom. Conservation of mass and momentum equations are depth integrated in each layer, and nonlinear kinematic and dynamic conditions are specified at the free surface and at the interface between fluids
Imamura and Imteaz (1995)
Table 1 Some of the existing numerical models for simulating landslide-generated waves
In this work, we apply two Computational Fluid Dynamic (CFD) frameworks, FLOW-3D HYDRO, and OpenFOAM to simulate waves generated by solid-block subaerial landslides in a 3D set-up. We calibrate and validate both numerical models using our physical experiments in a 3D wave tank and compare the performances of these models systematically. These two numerical models are selected among the existing CFD solvers because they have been reported to provide valuable insights into landslide-generated waves (Kim et al., 2020; Romano et al., 2020a, b ; Sabeti & Heidarzadeh, 2022a). As there is no study to compare the performances of these two models (FLOW-3D HYDRO and OpenFOAM) with each other in reproducing landslide-generated waves, this study is conducted to offer such a comparison, which can be helpful for model selection in future research studies or industrial projects. In the realm of tsunami generation by subaerial landslides, the solid-block approach serves as an effective representative for scenarios where the landslide mass is more cohesive and rigid, rather than granular. This methodology is particularly relevant in cases such as the 2018 Anak Krakatau or 1963 Vajont landslides, where the landslide’s nature aligns closely with the characteristics simulated by a solid-block model (Zaniboni & Tinti, 2014; Heidarzadeh et al., 2020a, 2020b).
The objectives of this research are: (i) To provide a detailed implementation and calibration for simulating solid-block subaerial landslide-generated waves using FLOW-3D HYDRO and OpenFOAM, and (ii) To compare the performance of these two numerical models based on three criteria: free surface elevation of the landslide-generated waves, capabilities of the models in simulating 3D features of the waves in the near-field, velocity fields, and velocity variations at different locations. The innovations of this study are twofold: firstly, it is a 3D study involving physical and numerical modelling and thus the data can be useful for other studies, and secondly, it compares the performance of two popular CFD models in modelling landslide-generated waves for the first time. The validated models such as those reported in this study and comparison of their performances can be useful for engineers and scientists addressing landslide tsunami hazards worldwide.
2 Data and Methods
2.1 Physical Modelling
To validate our numerical models, a series of three-dimensional physical experiments were carried out at the Hydraulic Laboratory of the Brunel University London (UK) in a 3D wave tank 2.40 m long, 2.60 m wide, and 0.60 m high (Figs. 1 and 2). To mitigate experimental errors and enhance the reliability of our results, each physical experiment was conducted three times. The reported data in the manuscript reflects the average of these three trials, assuming no anomalous outliers, thus ensuring an accurate reflection of the experimental tests. One experiment was used for validation of our numerical models. The slope angle (α) and water depth (h) were 45° and 0.246 m, respectively for this experiment. The movement of the sliding mass was recorded by a digital camera with a sampling frequency of 120 frames per second, which was used to calculate the slide impact velocity (vs). The travel distance (D), defined as the distance from the toe of the sliding mass to the water surface, was D=0.045 m. The material of the solid block used in our study was concrete with a density of 2600 kg/m3. Table 2 provides detailed information on the dimensions and kinematics of this solid block used in our physical experiments.
Figure 1. The geometrical and kinematic parameters of a subaerial landslide tsunami. Parameters are: h, water depth; aM, maximum wave amplitude; α, slope angle;vs, slide velocity; ls, length of landslide; bs, width of landslide; s, thickness of landslide; SWL, still water level; D, travel distance (the distance from the toe of the sliding mass to the water surface); L, length of the wave tank; and W, width of the wave tank and H, is the hight of the wave tank
Figure 2. a Wave tank setup of the physical experiments of this study. b Numerical simulation setup for the FLOW-3D HYDRO Model. c The numerical set-up for the OpenFOAM model. The location of the physical wave gauge (represented by numerical gauge WG-3 in the numerical simulations) is at X = 1.03 m, Y = 1.21 m, and Z = 0.046 m. d Top view showing the locations of numerical wave gauges (WG-1, WG-2, WG-3, WG-4, WG-5)
Parameter, unit
Value/type
Slide width (bs), m
0.26
Slide length (ls), m
0.20
Slide thickness (s), m
0.10
Slide volume (V), m3
2.60 × 10–3
Specific gravity, (γs)
2.60
Slide weight (ms), kg
6.86
Slide impact velocity (vs), m/s
1.84
Slide Froude number (Fr)
1.18
Material
Concrete
Table 2 Geometrical and kinematic information of the sliding mass used for physical experiments in this study
We took scale effects into account during physical experiments by considering the study by Heller et al. (2008) who proposed a criterion for avoiding scale effects. Heller et al. (2008) stated that the scale effects can be negligible as long as the Weber number (W=ρgh2/σ; where σ is surface tension coefficient) is greater than 5.0 × 103 and the Reynolds number (R=g0.5h1.5/ν; where ν is kinematic viscosity) is greater than 3.0 × 105 or water depth (h) is approximately above 0.20 m. Considering the water temperature of approximately 20 °C during our experiments, the kinematic viscosity (ν) and surface tension coefficient (σ) of water become 1.01 × 10–6 m2/s and 0.073 N/m, respectively. Therefore, the Reynolds and Weber numbers were as R= 3.8 × 105 and W= 8.1 × 105, indicating that the scale effect can be insignificant in our experiments. To record the waves, we used a twin wire wave gauge provided by HR Wallingford (https://equipit.hrwallingford.com). This wave gauge was placed at X = 1.03 m, Y = 1.21 m based on the coordinate system shown in Fig. 2a.
2.2 Numerical Simulations
The numerical simulations in this work were performed employing two CFD packages FLOW-3D HYDRO, and OpenFOAM which have been widely used in industry and academia (e.g., Bayon et al., 2016; Jasak, 2009; Rauter et al., 2021; Romano et al., 2020a, b; Yin et al., 2015).
2.2.1 Governing Equations and Turbulent Models
2.2.1.1 FLOW-3D HYDRO
The FLOW-3D HYDRO solver is based on the fundamental law of mass, momentum and energy conservation. To estimate the influence of turbulent fluctuations on the flow quantities, it is expressed by adding the diffusion terms in the following mass continuity and momentum transport equations:
quation (1) is the general mass continuity equation, where u is fluid velocity in the Cartesian coordinate directions (x), Ax is the fractional area open to flow in the x direction, VF is the fractional volume open to flow, ρ is the fluid density, R and ξ are coefficients that depend on the choice of the coordinate system. When Cartesian coordinates are used, R is set to unity and ξ is set to zero. RDIF and RSOR are the turbulent diffusion and density source terms, respectively. Uρ=Scμ∗/ρ, in which Sc is the turbulent Schmidt number, μ∗ is the dynamic viscosity, and ρ is fluid density. RSOR is applied to model mass injection through porous obstacle surfaces.
The 3D equations of motion are solved with the following Navier–Stokes equations with some additional terms:
where t is time, Gx is accelerations due to gravity, fx is viscous accelerations, and bx is the flow losses in porous media.
According to Flow Science (2022), FLOW-3D HYDRO’s turbulence models differ slightly from other formulations by generalizing the turbulence production with buoyancy forces at non-inertial accelerations and by including the influence of fractional areas/volumes of the FAVOR method (Fractional Area-Volume Obstacle Representation) method. Here we use k–ω model for turbulence modelling. The k–ω model demonstrates enhanced performance over the k-ε and Renormalization-Group (RNG) methods in simulating flows near wall boundaries. Also, for scenarios involving pressure changes that align with the flow direction, the k–ω model provides more accurate simulations, effectively capturing the effects of these pressure variations on the flow (Flow Science, 2022). The equations for turbulence kinetic energy are formulated as below based on Wilcox’s k–ω model (Flow Science, 2022):
where kT is turbulent kinetic energy, PT is the turbulent kinetic energy production, DiffKT is diffusion of turbulent kinetic energy, GT is buoyancy production, β∗=0.09 is closure coefficient, and ω is turbulent frequency.
2.2.1.2 OpenFOAM
For the simulations conducted in this study, OpenFOAM utilizes the Volume-Averaged RANS equations (VARANS) to enable the representation of flow within porous material, treated as a continuous medium. The momentum equation incorporates supplementary terms to accommodate frictional forces from the porous media. The mass and momentum conservation equations are linked to the VOF equation (Jesus et al., 2012) and are expressed as follows:
where the gravitational acceleration components are denoted bygj. The term u¯i=1Vf∫Vf0ujdV represents the volume averaged ensemble averaged velocity (or Darcy velocity) component, Vf is the fluid volume contained in the average volumeV,τ is the surface tension constant (assumed to be 1 for the water phase and 0 for the air phase), and fσi is surface tension, defined as fσi=σκ∂α∂xi, where σ (N/m) is the surface tension constant and κ (1/m) is the curvature (Brackbill et al., 1992). μeff is the effective dynamic viscosity that is defined as μeff=μ+ρνt and takes into account the dynamic molecular (μ) and the turbulent viscosity effects (ρνt). νt is eddy viscosity, which is provided by the turbulence closure model. n is the porosity, defined as the volume of voids over total volume, and P∗=1Vf∫∂Vf0P∗dS is the ensemble averaged pressure in excess of hydrostatic pressure. The coefficient A accounts for the frictional force induced by laminar Darcy-type flow, B considers the frictional force under turbulent flow conditions, and c accounts for the added mass. These coefficients (A,B, and c) are defined based on the work of Engelund (1953) and later modified by Van Gent (1995) as given below:
where D50 is the mean nominal diameter of the porous material, KC is the Keulegan–Carpenter number, a and b are empirical nondimensional coefficients (see Lara et al., 2011; Losada et al., 2016) and γ = 0.34 is a nondimensional parameter as proposed by Van Gent (1995). The k-ω Shear Stress Transport (SST) turbulence is employed to capture the effect of turbulent flow conditions (Zhang & Zhang, 2023) with the enhancement proposed by Larsen and Fuhrman (2018) for the over-production of turbulence beneath surface waves. Boundary layers are modelled with wall functions. The reader is referred to Larsen and Fuhrman (2018) for descriptions, validations, and discussions of the stabilized turbulence models.
2.2.2 FLOW-3D HYDRO Simulation Procedure
In our specific case in this study, FLOW-3D HYDRO utilizes the finite-volume method to numerically solve the equations described in the previous Sect. 2.2.1.1, ensuring a high level of accuracy in the computational modelling. The use of structured rectangular grids in FLOW-3D HYDRO offers the advantages of easier development of numerical methods, greater transparency in their relation to physical problems, and enhanced accuracy and stability of numerical solutions. (Flow Science, 2022). Curved obstacles, wall boundaries, or other geometric features are embedded in the mesh by defining the fractional face areas and fractional volumes of the cells that are open to flow (the FAVOR method). The VOF method is employed in FLOW-3D HYDRO for accurate capturing of the free-surface dynamics (Hirt and Nichols 1981). This approach then is upgraded to method of the TruVOF which is a split Lagrangian method that typically produces lower cumulative volume error than the alternative methods (Flow Science, 2022).
For numerical simulation using FLOW-3D HYDRO, the entire flow domain was 2.60 m wide, 0.60 m deep and 2.50 m long (Fig. 2b). The specific gravity (γs) for solid blocks was set to 2.60 in our model, aligning closely with the density of the actual sliding mass, which was approximately determined in our physical experiments. The fluid medium was modelled as water with a density of 1000 kg/m3 at 20 °C. A uniform grid comprising of one single mesh plane was applied with a grid size of 0.005 m. The top, front and back of the mesh areas were defined as symmetry, and the other surfaces were of wall type with no-slip conditions around the walls.
To simulate turbulent flows, k-ω model was used because of its accuracy in modelling turbulent flows (Menter 1992). Landslide movement was replicated in simulations using coupled motion objects, which implies that the movement of landslides is based on gravity and the friction between surfaces rather than a specified motion in which the model should be provided by force and torques. The time intervals of the numerical model outputs were set to 0.02 s to be consistent with the actual sampling rates of our wave gauges in the laboratory. In order to calibrate the FLOW-3D HYDRO model, the friction coefficient is set to 0.45, which is consistent with the Coulombic friction measurements in the laboratory. The Courant Number (C=UΔtΔx) is considered as the criterion for the stability of numerical simulations which gives the maximum time step (Δt) for a prespecified mesh size (Δx) and flow speed (U). The Courant number was always kept below one.
2.2.3 OpenFOAM Simulation Procedure
OpenFOAM is an open-source platform containing several C++ libraries which solves both 3D Reynolds-Averaged Navier–Stokes equations (RANS) and Volume-Averaged RANS equations (VARANS) for two-phase flows (https://www.openfoam.com/documentation/user-guide). Its implementation is based on a tensorial approach using object-oriented programming techniques and the Finite Volume Method (McDonald 1971). In order to simulate the subaerial landslide-generated waves, the IHFOAM solver based on interFoam (Higuera et al., 2013a, 2013b), and the overset mesh framework method are employed. The implementation of the overset mesh method for porous mediums in OpenFOAM is described in Romano et al. (2020a, b) for submerged rigid and impermeable landslides.
The overset mesh technique, as outlined by Romano et al. (2020a, b), uses two distinct domains: a moving domain that captures the dynamics of the rigid landslide and a static background domain to characterize the numerical wave tank. The overlapping of these domains results in a composite mesh that accurately depicts complex geometrical transformations while preserving mesh quality. A porous media with a very low permeability (n = 0.001) was used to simulate the impermeable sliding surfaces. RANS equations were solved within the porous media. The Multidimensional Universal Limiter with Explicit Solution (MULES) algorithm is employed for solving the (VOF) equation, ensuring precision in tracking fluid interfaces. Simultaneously, the PIMPLE algorithm is employed for the effective resolution of velocity–pressure coupling in the Eqs. 7 and 8. A background domain was created to reproduce the subaerial landslide waves with dimensions 2.50 m (x-direction) × 2.60 m (y-direction) × 0.6 m (z-direction) (Fig. 2c). The grid size is set to 0.005 m for the background mesh. A moving domain was applied in an area of 0.35 m (x-direction) × 0.46 m (y-direction) × 0.32 m (z-direction) with a grid spacing of 0.005 m and applying a body-fitted mesh approach, which contains the rigid and impermeable wedges. Wall condition with No-slip is defined as the boundary for the four side walls (left, right, front and back, in Fig. 1). Also, a non-slip boundary condition is specified to the bottom, whereas the top boundary is defined as open. The experimental slide movement time series is used to model the landslide motion in OpenFOAM. The applied equation is based on the analytical solution by Pelinovsky and Poplavsky (1996) which was later elaborated by Watts (1998). The motion of a sliding rigid body is governed by the following equation:
where, m represents the mass of the landslide, s is the displacement of the landslide down the slope, t is time elapsed, g stands for the acceleration due to gravity, θ is the slope angle, Cf is the Coulomb friction coefficient, Cm is the added mass coefficient, m0 denotes the mass of the water displaced by the moving landslide, A is the cross-sectional area of the landslide perpendicular to the direction of motion, ρ is the water density, and Cd is the drag coefficient.
2.2.4 Mesh Sensitivity Analysis
In order to find the most efficient mesh size, mesh sensitivity analyses were conducted for both numerical models (Fig. 3). We considered the influence of mesh density on simulated waveforms by considering three mesh sizes (Δx) of 0.0025 m, 0.005 m and 0.010 m. The results of FLOW-3D HYDRO revealed that the largest mesh deviates 9% (Fig. 3a, Δx = 0.0100 m) from two other finer meshes. Since the simulations by FLOW-3D HYDRO for the finest mesh (Δx = 0.0025 m) do not show any improvements in comparison with the 0.005 m mesh, therefore the mesh with the size of Δx = 0.0050 m is used for simulations (Fig. 3a). A similar approach was followed for mesh sensitivity of OpenFOAM mesh grids. The mesh with the grid spacing of Δx = 0.0050 m was selected for further simulations since a satisfactory independence was observed in comparison with the half size mesh (Δx = 0.0025 m). However, results showed that the mesh size with the double size of the selected mesh (Δx = 0.0100 m) was not sufficiently fine to minimize the errors (Fig. 3b).
Figure 3. a, b Sensitivity of numerical simulations to the sizes of the mesh (Δx) for FLOW-3D HYDRO, and OpenFOAM, respectively. The location of the wave gauge 3 (WG-3) is at X = 1.03 m, Y = 1.21 m, and Z = -0.55 m (see Fig. 2d)
In terms of computational cost, the time required for 2 s simulations by FLOW-3D HYDRO is approximately 4.0 h on a PC Intel® Core™ i7-8700 CPU with a frequency of 3.20 GHz equipped with a 32 GB RAM. OpenFOAM requires 20 h to run 2 s of numerical simulation on 2 processors on a PC Intel® Core™ i9-9900KF CPU with a frequency of 3.60 GHz equipped with a 364 GB RAM. Differences in computational time for simulations run with FLOW-3D HYDRO and OpenFOAM reflect the distinct characteristics of each numerical methods, and the specific hardware setups.
2.2.5 Validation
We validated both numerical models based on our laboratory experimental data (Fig. 4). The following criterion was used to assess the level of agreement between numerical simulations and laboratory observations:
where ε is the mismatch error, Obsi is the laboratory observation values, Simi is the simulation values, and the mathematical expression |X| represents the absolute value of X. The slope angle (α), water depth (h) and travel distance (D) were: α = 45°, h = 0.246 m and D = 0.045 m in both numerical models, consistent with the physical model. We find the percentage error between each simulated data point and its corresponding observed value, and subsequently average these errors to assess the overall accuracy of the simulation against the observed time series. Our results revealed that the mismatch errors between physical experiments and numerical models for the FLOW-3D HYDRO and OpenFOAM are 8% and 18%, respectively, indicating that our models reproduce the measured waveforms satisfactorily (Fig. 4). The simulated waveform by OpenFOAM shows a minor mismatch at t = 0.76 s which resulted from a droplet immediately after the slide hits the water surface in the splash zone. In term of the maximum negative amplitude, the simulated waves by OpenFOAM indicates a relatively better performance than FLOW-3D HYDRO, whereas the maximum positive amplitude (aM) simulated by FLOW-3D HYDRO is closer to the experimental value. The recorded maximum positive amplitude in physical experiment is 0.022 m, whereas it is 0.020 m for FLOW-3D HYDRO and 0.017 m for OpenFOAM simulations. In acknowledging the deviations observed, it is pertinent to highlight that while numerical models offer robust insights, the difference in meshing techniques and the distinct computational methods to resolve the governing equations in FLOW-3D HYDRO and OpenFOAM have contributed to the variance. Moreover, the intrinsic uncertainties associated with the physical experimentation process, including the precision of wave gauges and laboratory conditions, are non-negligible factors influencing the results.
Figure 4. Validation of the simulated waves (brown line for FLOW-3D HYDRO and green line for OpenFOAM) using the laboratory-measured waves (black solid diamonds). This physical experiment was conducted for wave gauge 3 (WG-3) located at X = 1.03 m, Y = 1.21 m, and Z = -0.55 m (see Fig. 2d). Here, ε shows the errors between simulations and actual physical measurements using Eq. (13)
3 Results
Following the validations of the two numerical models (FLOW-3D HYDRO and OpenFOAM), a series of simulations were performed to compare the performances of these two CFD solvers. The generation process of landslide waves, waveforms, and velocity fields are considered as the basis for comparing the performance of the two models (Figs. 5, 6, 7 and 8).
Figure 5.Comparison between the simulated waveforms by FLOW-3D HYDRO (black) and OpenFOAM (red) at four different locations in the near-field zone (WG-1,2,4 and 5). WG is the abbreviation for wave gauge. The mismatch (Δ) between the two models at each wave gauge is calculated using Eq. (14)
Figure 6. Comparison of water surface elevations produced by solid-block subaerial landslides for the two numerical models FLOW-3D-HYDRO (a–c) and OpenFOAM (e–g) at different times
Figure 7. Snapshots of the simulations at different times for FLOW-3D HYDRO (a–c) and OpenFOAM (e–g) showing velocity fields (colour maps and arrows). The colormaps indicate water particle velocity in m/s, and the lines indicate the velocities of water particles
Figure 8. Comparison of velocity variations at (WG-3) for FLOW-3D HYDRO (light blue) and OpenFOAM (brown)
3.1 Comparison of Waveforms
Five numerical wave gauges were placed in our numerical models to measure water surface oscillations in the near-field zone (Fig. 5). These gauges offer an azimuthal coverage of 60° (Fig. 2d). Figure 5 reveals that the simulated waveforms from two models (FLOW-3D HYDRO and OpenFOAM) are similar. The highest wave amplitude (aM) is recorded at WG-3 for both models, whereas the lowest amplitude is recorded at WG-5 and WG-1 which can be attributed to the longer distances of these gauges from the source region as well as their lateral offsets, resulting in higher wave energy dissipation at these gauges. The sharp peaks observed in the simulated waveforms, such as the red peak between 0.8–1.0 s in Fig. 5a from OpenFOAM, the red peak between 0.6–0.8 s in Fig. 5b also from OpenFOAM, and the black peak between 1.4–1.6 s in Fig. 5d from FLOW-3D HYDRO, are due to the models’ spatial and temporal discretization. They reflect the sensitivity of the models to capturing transient phenomena, where the chosen mesh and time-stepping intervals are key factors in the models’ ability to track rapid changes in the flow field. To quantify the deviations of the two models from one another, we apply the following equation for mismatch calculation:
where Δ is the mismatch error, Sim1 is the simulation values from FLOW-3D HYDRO, Sim2 is the simulation values from OpenFOAM, and the mathematical expression |X| implies the absolute value of X. We calculate the percentage difference for each corresponding pair of simulation results, then take the mean of these percentage differences to determine the average deviation between the two simulation time series. Using Eq. (14), we found a deviation range from 9 to 11% between the two models at various numerical gauges (Fig. 5), further confirming that the two models give similar simulation results.
3.2 Three-Dimensional Vision of Landslide Generation Process by Numerical Models
A sequence of four water surface elevation snapshots at different times is shown in Fig. 6 for both numerical modes. In both simulations, the sliding mass travels a constant distance of 0.045 m before hitting the water surface at t = 0.270 s which induces an initial change in water surface elevation (Figs. 6a and e). At t = 0.420 s, the mass is fully immersed for both simulations and an initial dipole wave is generated (Figs. 6b and f). Based on both numerical models, the maximum positive amplitude (0.020 m for FLOW-3D HYDRO, and 0.017 m for OpenFOAM) is observed at this stage (Fig. 6). The maximum propagation of landslide-simulated waves along with more droplets in the splash zone could be seen at t = 0.670 s for both models (Fig. 6c and g). The observed distinctions in water surface elevation simulations as illustrated in Fig. 6 are rooted in the unique computational methodologies intrinsic to each model. In the OpenFOAM simulations, a more diffused water surface elevation profile is evident. Such diffusion is an outcome of the simulation’s intrinsic treatment of turbulent kinetic energy dissipation, aligning with the solver’s numerical dissipation characteristics. These traits are influenced by the selected turbulence models and the numerical advection schemes, which prioritize computational stability, possibly at the expense of interface sharpness. The diffusion in the wave pattern as rendered by OpenFOAM reflects the application of a turbulence model with higher dissipative qualities, which serves to moderate the energy retained during wave propagation. This approach can provide insights into the potential overestimation of energy loss under specific simulation conditions. In contrast, the simulations from FLOW-3D HYDRO depict a more localized wave pattern, indicative of a different approach to turbulent dissipation. This coherence in wave fronts is a function of the model’s specific handling of the air–water interface and its targeted representation of the energy dynamics resulting from the landslide’s interaction with the water body. They each have specific attributes that cater to different aspects of wave simulation fidelity, thereby contributing to a more comprehensive understanding of the phenomena under study.
3.3 Wave Velocity Analysis
We show four velocity fields at different times during landslide motion in Fig. 7 and one time series of velocity (Fig. 8) for both numerical models. The velocity varies in the range of 0–1.9 m/s for both models, and the spatial distribution of water particle velocity appears to be similar in both. The models successfully reproduce the complex wavefield around the landslide generation area, which is responsible for splashing water and mixing with air around the source zone (Fig. 7). The first snapshot at t = 0.270 s (Fig. 7a and e) shows the initial contact of the sliding mass with water surface for both numerical models which generates a small elevation wave in front of the mass exhibiting a water velocity of approximately 1.2 m/s. The slide fully immerses for the first time at t = 0.420 s producing a water velocity of approximately 1.5 m/s at this time (Fig. 7b and f). The last snapshot (t = 0.670 s) shows 1.20 s after the slide hits the bottom of the wave tank. Both models show similar patterns for the propagation of the waves towards the right side of the wave tank. The differences in water surface profiles close to the slope and solid block at t = 0.67 s, observed in the FLOW-3D HYDRO and OpenFOAM simulations (Figs. 6 and 7), are due to the distinct turbulence models employed by each (RNG and k-ω SST, respectively) which handle the complex interactions of the landslide-induced waves with the structures differently. Additionally, the methods of simulating landslide movement further contribute to this discrepancy, with FLOW-3D HYDRO’s coupled motion objects possibly affecting the waves’ initiation and propagation unlike OpenFOAM’s prescribed motion from experimental data. In addition to the turbulence models, the variations in VOF methodologies between the two models also contribute to the observed discrepancies.
For the simulated time series of velocity, both models give similar patterns and close maximum velocities (Fig. 8). For both models the WG-3 located at X = 1.03 m, Y = 1.21 m, and Z = − 0.55 m (Fig. 2d) were used to record the time series. WG-3 is positioned 5 mm above the wave tank bottom, ensuring that the measurements taken reflect velocities very close to the bottom of the wave tank. The maximum velocity calculated by FLOW-3D HYDRO is 0.162 m/s while it is 0.132 m/s for OpenFOAM, implying a deviation of approximately 19% from one another. Some oscillations in velocity records are observed for both models, but these oscillations are clearer and sharper for OpenFOAM. Although it is hard to see velocity oscillations in the FLOW-3D HYDRO record, a close look may reveal some small oscillations (around t = 0.55 s and 0.9 s in Fig. 8). In fact, velocity oscillations are expected due to the variations in velocity of the sliding mass during the travel as well as due to the interferences of the initial waves with the reflected wave from the beach. In general, it appears that the velocity time series of the two models show similar patterns and similar maximum values although they have some differences in the amplitudes of the velocity oscillations. The differences between the two curves are attributed to factors such as difference in meshing between the two models, turbulence models, as well as the way that two models record the outputs.
4 Discussions
An important step for CFD modelling in academic or industrial projects is the selection of an appropriate numerical model that can deliver the task with satisfactory performance and at a reasonable computational cost. Obviously, the major drivers when choosing a CFD model are cost, capability, flexibility, and accessibility. In this sense, the existing options are of two types as follows:
Commercial models, such as FLOW-3D HYDRO, which are optimised to solve free-surface flow problems, with customer support and an intuitive Graphical User Interface (GUI) that significantly facilitates meshing, setup, simulation monitoring, visualization, and post-processing. They usually offer high-quality customer support. Although these models show high capabilities and flexibilities for numerical modelling, they are costly, and thus less accessible.
Open-source models, such as OpenFOAM, which come without a GUI but with coded tools for meshing, setup, parallel running, monitoring, post-processing, and visualization. Although these models offer no customer support, they have a big community support and online resources. Open-source models are free and widely accessible, but they may not be necessarily always flexible and capable.
OpenFOAM provides freedom for experimenting and diving through the code and formulating the problem for a user whereas FLOW-3D HYDRO comes with high-level customer supports, tutorial videos and access to an extensive set of example simulations (https://www.flow3d.com/). While FLOW-3D-HYDRO applies a semi-automatic meshing process where users only need to input the 3D model of the structure, OpenFOAM provides meshing options for simple cases, and in many advanced cases, users need to create the mesh in other software (e.g., ANSYS) (Ariza et al., 2018) and then convert it to OpenFOAM format. Auspiciously, there are numerous online resources (https://www.openfoam.com/trainings/about-trainings), and published examples for OpenFOAM (Rauter et al., 2021; Romano et al., 2020a, b; Zhang & Zhang, 2023).
The capabilities of both FLOW-3D HYDRO and OpenFOAM to simulate actual, complex landslide-generated wave events have been showcased in significant case studies. The study by Ersoy et al. (2022) applied FLOW-3D HYDRO to simulate impulse waves originating from landslides near an active fault at the Çetin Dam Reservoir, highlighting the model’s capacity for detailed, site-specific modelling. Concurrently, the work by Alexandre Paris (2021) applied OpenFOAM to model the 2017 Karrat Fjord landslide tsunami events, providing a robust validation of OpenFOAM’s utility in capturing the dynamics of real-world geophysical phenomena. Both instances exemplify the sophisticated computational approaches of these models in aiding the prediction and analysis of natural hazards from landslides.
As for limitations of this study, we acknowledge that our numerical models are validated by one real-world measured wave time series. However, it is believed that this one actual measurement was sufficient for validation of this study because it was out of the scope of this research to fully validate the FLOW-3D HYDRO and OpenFOAM models. These two models have been fully validated by more actual measurements by other researchers in the past (e.g., Sabeti & Heidarzadeh, 2022b). It is also noted that some of the comparisons made in this research were qualitative, such as the 3D wave propagation snapshots, as it was challenging to develop quantitative comparisons for snapshots. Another limitation of this study concerns the number of tests conducted here. We fixed properties such as water depth, slope angle, and travel distance throughout this study because it was out of the scope of this research to perform sensitivity analyses.
5 Conclusions
We configured, calibrated, validated and compared two numerical models, FLOW-3D HYDRO, and OpenFOAM, using physical experiments in a 3D wave tank. These validated models were used to simulate subaerial solid-block landslides in the near-field zone. Our results showed that both models are fully compatible with investigating waves generated by subaerial landslides, although they use different approaches to simulate the phenomenon. The properties of solid-block, water depth, slope angle, and travel distance were kept constant in this study as we focused on comparing the performance of the two models rather than conducting a full sensitivity analysis. The findings are as follows:
Different settings were used in the two models for modelling landslide-generated waves. In terms of turbulent flow modelling, we used the Renormalization Group (RNG) turbulence model in FLOW-3D HYDRO, and k-ω (SST) turbulence model in OpenFOAM. Regarding meshing techniques, the overset mesh method was used in OpenFOAM, whereas the structured cartesian mesh was applied in FLOW-3D HYDRO. As for simulation of landslide movement, the coupled motion objects method was used in FLOW-3D HYDRO, and the experimental slide movement time series were prescribed in OpenFOAM.
Our modelling revealed that both models successfully reproduced the physical experiments. The two models deviated 8% (FLOW-3D HYDRO) and 18% (OpenFOAM) from the physical experiments, indicating satisfactory performances. The maximum water particle velocity was approximately 1.9 m/s for both numerical models. When the simulated waveforms from the two numerical models are compared with each other, a deviation of 10% was achieved indicating that the two models perform approximately equally. Comparing the 3D snapshots of the two models showed that there are some minor differences in reproducing the details of the water splash in the near field.
Regarding computational costs, FLOW-3D HYDRO was able to complete the same simulations in 4 h as compared to nearly 20 h by OpenFOAM. However, the hardware that were used for modelling were not the same; the computer used for the OpenFOAM model was stronger than the one used for running FLOW-3D HYDRO. Therefore, it is challenging to provide a fair comparison for computational time costs.
Overall, we conclude that the two models give approximately similar performances, and they are both capable of accurately modelling landslide-generated waves. The choice of a model for research or industrial projects may depend on several factors such as availability of local knowledge of the models, computational costs, accessibility and flexibilities of the model, and the affordability of the cost of a license (either a commercial or an open-source model).
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In this study, experimental investigations were conducted on rectangular side weirs with different widths and heights. Corresponding simulations were also performed to analyze hydraulic characteristics including the water surface profile, flow velocity, and pressure. The relationship between the discharge coefficient and the Froude number, as well as the ratios of the side weir height and width to upstream water depth, was determined. A discharge formula was derived based on a dimensional analysis. The results demonstrated good agreement between simulated and experimental data, indicating the reliability of numerical simulations using FLOW-3D software (version 11.1). Notably, significant fluctuations in water surface profiles near the side weir were observed compared to those along the center line or away from the side weir in the main channel, suggesting that the entrance effect of the side weir did not propagate towards the center line of the main channel. The proposed discharge formula exhibited relative errors within 10%, thereby satisfying the flow measurement requirements for small channels and field inlets.
1. Introduction
Sharp crested weirs are used to obtain discharge in open channels by solely measuring the water head upstream of the water. Side weirs, as a kind of sharp-crested weir, are extensively used for flow measurement, flow diversion, and flow regulation in open channels. Side weirs can be placed directly in the channel direction or field inlet, without changing the original structure of the channel. Thus, side weirs have certain advantages in the promotion and application of flow measurement facilities in small channels and field inlets. The rectangular sharp-crested weir is the most commonly available, and many scholars have conducted research on it. Research on side weirs started in 1934. De Marchi studied the side weir in the rectangular channel and derived the theoretical formula based on the assumption that the specific energy of the main flow section of the rectangular channel in the side weir section was constant [1]. Ackers discussed the existing formulas for the prediction of the side weir discharge coefficient [2]. Chen concluded that the momentum theorem was more suitable for the analytical calculation of the side weir based on the experimental data [3]. Based on previous theoretical research, more and more scholars began to carry out experimental research on side weirs. Uyumaz and Muslu conducted experiments under subcritical and supercritical flow regimes and derived expressions for the side weir discharge and water surface profiles for these regimes by comparing them with experimental results [4]. Borghei et al. developed a discharge coefficient equation for rectangular side weirs in subcritical flow [5]. Ghodsian [6] and Durga and Pillai [7] developed a discharge coefficient equation of rectangular side weirs in supercritical flow. Mohamed proposed a new approach based on the video monitoring concept to measure the free surface of flow over rectangular side weirs [8]. Durga conducted experiments on rectangular side weirs of different lengths and sill heights and discussed the application of momentum and energy principles to the analysis of spatially varied flow under supercritical conditions. The results showed that the momentum principle was fitting better [7]. Omer et al. obtained sharp-crested rectangular side weirs discharge coefficients in the straight channel by using an artificial neural network model for a total of 843 experiments [9]. Emiroglu et al. studied water surface profile and surface velocity streamlines, and developed a discharge coefficient formula of the upstream Froude number, the ratios of weir length to channel width, weir length to flow depth, and weir height to flow depth [10]. Other investigators [11,12,13,14] have conducted experiments to study flow over rectangular side weirs in different flow conditions. Numerous studies have been conducted in laboratories to this day. Compared to experimental methods, the numerical simulation method has many attractive advantages. We can easily obtain a wide range of hydraulic parameters of side weirs using numerical simulation methods, without investing a lot of manpower and resources. In addition, we can conduct small changes in inlet condition, outlet condition, and geometric parameters, and study their impact on the flow characteristics of side weirs. Therefore, with the development and improvement of computational fluid dynamics, the numerical simulation method has begun to be widely applied on side weirs. Salimi et al. studied the free surface changes and the velocity field along a side weir located on a circular channel in the supercritical regime by numerical simulation [15]. Samadi et al. conducted a three-dimensional simulation on rectangular sharp-crested weirs with side contraction and without side contraction and verified the accuracy of numerical simulation compared with the experimental results [16]. Aydin investigated the effect of the sill on rectangular side weir flow by using a three-dimensional computational fluid dynamics model [17]. Azimi et al. studied the discharge coefficient of rectangular side weirs on circular channels in a supercritical flow regime using numerical simulation and experiments [18]. The discharge coefficient over the two compound side weirs (Rectangular and Semi-Circle) was modeled by using the FLOW-3D software to describe the flow characteristics in subcritical flow conditions [19]. Safarzadeh and Noroozi compared the hydraulics and 3D flow features of the ordinary rectangular and trapezoidal plan view piano key weirs (PKWs) using two-phase RANS numerical simulations [20]. Tarek et al. investigated the discharge performance, flow characteristics, and energy dissipation over PK and TL weirs under free-flow conditions using the FLOW-3D software [21]. As evident from the aforementioned, the majority of studies have primarily focused on determining the discharge coefficient, while comparatively less attention has been devoted to investigating the hydraulic characteristics of rectangular side weirs. Numerical simulations were conducted on different types of side weirs, including compound side weirs and piano key weirs, in different cross-section channels under different flow regimes. It is imperative to derive the discharge formula and investigate other crucial flow parameters such as depth, velocity, and pressure near side weirs for their effective implementation in water measurement. In this study, a combination of experimental and numerical simulation methods was employed to examine the relationship between the discharge coefficient and its influencing factors; furthermore, a dimensionless analysis was utilized to derive the discharge formula. Additionally, water surface profiles near side weirs and pressure distribution at the bottom of the side channel were analyzed to assess safety operation issues associated with installing side weirs.
2. Principle of Flow Measurement
Flow discharge over side weirs is a function of different dominant physical and geometrical quantities, which is defined as
where Q is flow discharge over the side weir, b is the side weir width, B is the channel width, P is the side weir height, v is the mean velocity, h1 is water depth upstream the side weir in the main channel, g is the gravitational acceleration, μ is the dynamic viscosity of fluid, ρ is fluid density, and i is the channel slope (Figure 1).
Figure 1. Definition sketch of parameters of rectangular side weir under subcritical flow. Note: h1 and h2 represent water depth upstream and downstream of the side weir in the main channel, respectively; y1 and y2 represent weir head upstream and downstream of the side weir in the main channel, respectively.
In experiments when the upstream weir head was over 30 mm, the effects of surface tension on discharge were found to be minor [22]. The viscosity effect was far less than the gravity effect in a turbulent flow. Hence μ and σ were excluded from the analysis [23,24]. In addition, the channel width, the channel slope, and the fluid density were all constant, so the discharge formula can be simplified as:
According to the Buckingham π theorem, the following relationship among the dimensionless parameters is established:
Selected h1 and g as basic fundamental quantities, and the remaining physical quantities were represented in terms of these fundamental quantities as follows:
In which
Based on dimensional analysis, the following equations were derived.
Namely
So the discharge formula can be simplified as:
In a sharp-crested weir, discharge over the weir is proportional to 𝐻1.51H11.5 (H1 is the upstream total head above the crest, namely H1 = y1 + v2/2 g), so Equation (6) can be transformed as follows:
Consequently, the discharge formula over rectangular side weirs is defined as follows, in which 𝑚=𝑓(𝑏ℎ1m=f(bh1,𝑃ℎ1,𝐹𝑟1)Ph1,Fr1). Parameter m represents the dimensionless discharge coefficient. Parameter Fr1 represents the Froude number at the upstream end of the side weir in the main channel.
3. Experiment Setup
The experimental setup contained a storage reservoir, a pumping station, an electromagnetic flow meter, a control valve, a stabilization pond, rectangular channels, a side weir, and a sluice gate. The layout of the experimental setup is shown in Figure 2. Water was supplied from the storage reservoir using a pump. The flow discharge was measured with an electromagnetic flow meter with precision of ±3‰. Water depth was measured with a point gauge with an accuracy of ±0.1 mm. The flow velocity was measured with a 3D Acoustic Doppler Velocimeter (Nortek Vectrino, manufactured by Nortek AS in Rud, Norway). In order to eliminate accidental and human error, multiple measurements of the water depth and flow velocity at the same point were performed and the average values were used as the actual water depth and flow velocity of the point. The main and side channels were both rectangular open channels measuring 47 cm in width and 60 cm in height. The geometrical parameters of rectangular side weirs are shown in Table 1.
Figure 2. Layout of the test system.
Table 1. The geometrical parameters of rectangular side weirs.
When water passes through a side weir, its quality point is affected not only by gravity but also by centrifugal inertia force, leading to an inclined water surface within that particular cross-section before reaching the weir. In order to examine water profiles adjacent to side weirs, cross-sectional measurements were conducted at regular intervals of 12 cm both upstream and downstream of each side weir, denoted as sections ① to ⑩, respectively. Measuring points were positioned near the side weir (referred to as “Side I”), along the center line of the main channel (referred to as “Side II”), and far away from the side weir (referred to as “Side III”) for each cross-section. The schematic diagram illustrating these measuring points is presented in Figure 3.
Figure 3. Schematic diagram of measurement points.
4. Numerical Simulation Settings
4.1. Mathematical Model
4.1.1. Governing Equations
Establishing the controlling equations is a prerequisite for solving any problem. For the flow analysis problem of water flowing over a side weir in a rectangular channel, assuming that no heat exchange occurs, the continuity equation (Equation (9)) and momentum equation (Equation (10)) can be used as the controlling equations as follows:
The continuity equation:
Momentum equation:
where: ρ is the fluid density, kg/m3; t is time, s; ui, uj are average flow velocities, u1, u2, u3 represent average flow velocity components in Cartesian coordinates x, y, and z, respectively, m/s; μ is dynamic viscosity of fluid, N·s/m2; p is the pressure, pa; Si is the body force, S1 = 0, S2 = 0, S3 = −ρg, N [24].
4.1.2. RNG k-ε Model
The water flow in the main channel is subcritical flow. When the water flows through the side weir, the flow line deviates sharply, the cross section suddenly decreases, and due to the blocking effect of the side weir, the water reflects and diffracts, resulting in strong changes in the water surface and obvious three-dimensional characteristics of the water flow [25]. Therefore the RNG k–ε model is selected. The model can better handle flows with greater streamline curvature, and its corresponding k and ε equation is, respectively, as follows:
where: k is the turbulent kinetic energy, m2/s2; μeff is the effective hydrodynamic viscous coefficient; Gk is the generation item of turbulent kinetic energy k due to gradient of the average flow velocity; C∗1εC1ε*, C2ε are empirical constants of 1.42 and 1.68, respectively; ε is turbulence dissipation rate, kg·m2/s2.
4.1.3. TruVOF Model
Because the shape of the free surface is very complex and the overall position is constantly changing, the fluid flow phenomenon with a free surface is a typical flow phenomenon that is difficult to simulate. The current methods used to simulate free surfaces mainly include elevation function method, the MAC method [26] and the VOF (Volume of Fluid) method [27]. The VOF method is a method proposed by Hirt and Nichols to deal with the complex motion of the free surface of a fluid, which can describe all the complexities of the free surface with only one function. The basic idea of the method is to define functions αw and αa, which represent the volume percentage of the calculation area occupied by water and air, respectively. In each unit cell, the sum of the volume fractions of water and air is equal to 1, i.e.,
The TruVOF calculation method can accurately track the change of free liquid level and accurately simulate the flow problems with free interface. Its equation is:
where: u_¯m is the average velocity of the mixture; t is the time; F is the volume fraction of the required fluid.
4.2. Parameter Setting and Boundary Conditions
To streamline the iterative calculation and minimize simulation time, we selected a main channel measuring 7.5 m in length and a side channel measuring 2.5 m in length for simulation. Three-dimensional geometrical models were developed using the software AutoCAD (version 2016-Simplified Chinese). The spatial domain was meshed using a constructed rectangular hexahedral mesh and each cell size was 2 cm. A volume flow rate was set in the channel inlet with an auto-adjusted fluid height. An outflow–outlet condition was positioned at the end of the side channel. A symmetry boundary condition was set in the air inlet at the top of the model, which represented that no fluid flows through the boundary. The lower Z (Zmin) and both of the side boundaries were treated as a rigid wall (W). No-slip conditions were applied at the wall boundaries. Figure 4 illustrates these boundary conditions.
Figure 4. Diagram of boundary conditions.
5. Results
5.1. Water Surface Profiles
Water surface profiles were crucial parameters for selecting water-measuring devices. Upon analyzing the consistent patterns observed in different conditions, one specific condition was chosen for further analysis. To validate the reliability of numerical simulation, measured and simulated water depths of rectangular side weirs with different widths and heights at a discharge rate of 25 L/s were extracted for comparison (Table 2 and Figure 5). The results in Table 2 and Figure 5 indicate a maximum absolute relative error value of 9.97% and all absolute relative error values within 10%, demonstrating satisfactory agreement between experimental and simulated results.
Figure 5. Comparison between measured and simulated flow depth.
P/cm
Section Position
b = 20 cm
b = 30 cm
b = 40 cm
b = 47 cm
hm/cm
hs/cm
R/%
hm/cm
hs/cm
R/%
hm/cm
hs/cm
R/%
hm/cm
hs/cm
R/%
7
④
21.49
19.4
9.73
17.74
16.9
4.74
16.07
14.51
9.71
13.79
12.50
9.35
④′
20.48
19.05
6.98
17.78
16.14
9.22
15.69
14.31
8.80
⑥
20.71
19.02
8.16
17.82
16.31
8.47
15.92
14.53
8.73
15.23
13.80
9.39
⑧′
22.00
20.22
8.09
18.27
16.74
8.37
16.59
14.96
9.83
⑧
22.37
20.17
9.83
17.73
16.80
5.25
16.27
15.08
7.31
15.36
14.36
6.51
10
④
24.15
22.6
6.42
19.96
18.84
5.61
19.03
18.58
2.36
16.83
15.85
5.82
④′
24.21
22.05
8.92
19.49
18.19
6.67
18.75
18.35
2.13
⑥
24.01
21.78
9.29
19.65
18.34
6.67
18.95
18.63
1.69
17.52
16.09
8.16
⑧′
24.88
22.4
9.97
20.65
19.21
6.97
20.12
19.29
4.13
⑧
24.03
22.96
4.45
21.16
19.34
8.60
19.71
19.43
1.42
18.39
17.36
5.60
15
④
28.85
27.56
4.47
25.86
24.09
6.84
24.05
21.89
8.98
22.73
20.80
8.49
④′
28.49
26.97
5.34
25.19
23.84
5.36
23.42
21.46
8.37
⑥
28.85
26.98
6.48
25.72
23.99
6.73
23.23
21.82
6.07
23.10
21.05
8.87
⑧′
28.96
27.30
5.73
26.38
24.19
8.30
24.18
22.27
7.90
⑧
29.18
27.96
4.18
26.57
24.54
7.64
24.57
22.33
9.12
23.20
21.10
9.05
20
④
33.29
32.34
2.85
30.63
29.02
5.26
28.49
26.87
5.69
26.99
25.81
4.37
④′
33.14
31.95
3.59
29.75
28.62
3.80
28.11
26.79
4.70
⑥
33.32
31.79
4.59
30.04
28.45
5.29
28.99
26.86
7.35
27.42
26.72
2.55
⑧′
34.02
32.39
4.79
30.69
28.95
5.67
29.59
27.25
7.91
⑧
34.62
32.84
5.14
31.44
29.29
6.84
29.51
27.31
7.46
28.21
27.00
4.29
Table 2. Comparison of measured and simulated water depths on Side I of each side weir at a discharge of 25 L/s
Due to the diversion caused by the side weir, there was a rapid variation in flow near the side weir in the main channel. In order to investigate the impact of the side weir on water flow in the main channel, water surface profiles on Side I, Side II, and Side III were plotted with a side weir width and height both set at 20 cm at a discharge rate of 25 L/s (Figure 6). As depicted in Figure 6, within a certain range of the upstream end of the main channel, water depths on Side I, Side II, and Side III were nearly equal with almost horizontal profiles. As the distance between the location of water flow and the upstream end of the weir crest decreased gradually, there was a gradual decrease in water depth on Side I along with an inclined trend in its corresponding profile; however, both Side II and Side III still maintained almost horizontal profiles. When approaching closer to the side weir area with flowing water, there was an evident reduction in water depth on Side I accompanied by a significant downward trend visible across an expanded decline range. The minimum point occurred near the upstream end of the weir crest before gradually increasing again towards downstream sections. At the crest section of the side weir, there is an upward trend observed in the water surface. The water surface tended to stabilize downstream of the main channel within a certain range from the downstream end of the weir crest. There was no significant change in the water surface profiles of Side Ⅱ and Side Ⅲ in the crest section. It can be inferred that the side weir entrance effect occurred only between Side Ⅰ and Side Ⅱ. M. Emin reported the same pattern [10].
Figure 6. Water surface profiles on Side I, Side II, and Side III with a side weir width of 20 cm and height of 15 cm at a discharge of 25 L/s.
For a more accurate study on the entrance effect of the side weir on the Water Surface Profile (WSP) for Side I; a comparative analysis conducted using different widths but the same height (15 cm) at a discharge rate of 25 L/s is presented through Figure 7, Figure 8, Figure 9 and Figure 10.
Figure 7. Water surface profile on Side Ⅰ with a side weir width of 20 cm and height of 15 cm at a discharge of 25 L/s.
Figure 8. Water surface profile on Side Ⅰ with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s.
Figure 9. Water surface profile on Side Ⅰ with a side weir width of 40 cm and height of 15 cm at a discharge of 25 L/s.
Figure 10. Water surface profile on Side Ⅰ with a side weir width of 47 cm and height of 15 cm at a discharge of 25 L/s.
According to Figure 7, Figure 8, Figure 9 and Figure 10, the water depth upstream of the main channel started to decrease as it approached the upstream end of the weir crest and then gradually increased at the weir crest section. In other words, the water surface profile exhibited a backwater curve along the length of the weir crest. The water depth remained relatively stable downstream of the main channel within a certain range from the downstream end of the weir crest. Additionally, there was a higher water depth downstream of the main channel compared to that upstream of the main channel. Furthermore, an increase in the width of the side weir led to a gradual reduction in fluctuations on its water surface.
5.2. Velocity Distribution
The law of flow velocity distribution near the side weir is the focus of research and analysis, so the simulated and measured values of flow velocity near the side weir were compared and analyzed. Take the discharge of 25 L/s, the height of 15 cm, and the width of 30 cm of the side weir as an example to illustrate. Figure 11 shows the measured and simulated velocity distribution in the x-direction of cross-section ④. As can be seen from Figure 11, the diagrams of the measured and simulated velocity distribution were relatively consistent, and the maximum absolute relative error between the measured and simulated values at the same measurement point was 9.37%, and the average absolute relative error was 3.97%, which indicated a satisfactory agreement between the experimental and simulated results.
Figure 11. Velocity distribution in the x-direction of section ④: when the discharge is 25 L/s, the height of the side weir is 15 cm and the width of the side weir is 30 cm. (a) Measured velocity distribution; (b) Simulated velocity distribution.
From Figure 11, it can be seen that the flow velocity gradually increased from the bottom of the channel towards the water surface in the Z-direction, and the flow velocity gradually increased from Side Ⅲ to Side Ⅰ in the Y-direction. The maximum flow velocity occurred near the weir crest.
Figure 12 shows the distribution of flow velocity at different depths (z/P = 0.3, z/P = 0.8, z/P = 1.6) with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s. The water flow line began to bend at a certain point upstream of the main channel, and the closer it was to the upstream end of the weir crest, the greater the curvature. The maximum curvature occurred at the downstream end of the weir crest. The flow patterns at the bottom, near the side weir crest, and above the side weir crest were significantly different. There was a reverse flow at the bottom of the main channel, where the forward and reverse flows intersect, resulting in a detention zone. The maximum flow velocity at the bottom layer occurred at the upstream end of the side weir crest. When the location of water flow approached the weir crest, the maximum flow velocity occurred at the upstream end of the weir crest. The maximum flow velocity on the water surface occurred at the downstream end of the weir crest. As the water depth decreased, the position of the maximum flow velocity gradually moved from the upstream end of the side weir to the downstream end of the side weir.
Figure 12. Distribution of flow velocity at different depths with a side weir width of 30 cm and height of 15 cm at a discharge of 25 L/s. (a) z/P = 0.3; (b) z/P = 0.8; (c) z/P = 1.6.
5.3. Side Channel Pressure Distribution
When water flowed through the side weir, an upstream water level was formed, resulting in a pressure zone at the junction with the side channel. This pressure zone led to increased water pressure on the floor of the side channel, which affected its stability and durability. In small channels or fields where erosion resistance is weak, excessive pressure can cause scour holes. Therefore, analyzing the pressure distribution in the side channel is necessary to select an appropriate height and width for the side weir that effectively reduces its impact on the bottom plate.
To investigate the impact of side weir width on hydraulic characteristics, pressure data was collected at a discharge rate of 25 L/s for side weirs with heights of 20 cm and widths ranging from 20 cm to 47 cm. The pressure distribution map was drawn, as shown in Figure 13.
Figure 13. Comparison of pressure distribution on the bottom plate of the side channel with different widths of side weirs when the discharge is 25 L/s and the height of side weirs is 20 cm. (a) P = 20 cm, b = 20 cm; (b) P = 20 cm, b = 30 cm; (c) P = 20 cm, b = 40 cm; (d) P = 20 cm, b = 47 cm.
As can be seen from Figure 13, the pressure at the bottom of the side channel decreased as the width of the side weir increased. This uneven distribution of water flow on the weir was caused by the sharp bending of water flow lines and the influence of centrifugal inertia force over a short period. After passing through the side weir, the water flow became symmetrically distributed with respect to the axis of the side channel, leaning towards the right bank at a certain distance. As we increased the width of the side weir, we noticed that its position gradually approached the side weir and maximum pressure decreased at this location where the water tongue formed due to flowing through it (Figure 13). For a constant height (20 cm) but varying widths (20 cm, 30 cm, 40 cm, and 47 cm), we measured maximum pressures at these positions as follows: 103,713 Pa, 103,558 Pa, 103,324 Pa, and 103,280 Pa, respectively. Consequently, increasing width reduced the impact on the side channel from water flowing through it while changing pressure distribution from concentration to dispersion in a vertical direction. In the practical application of side weirs, appropriate height should be selected based on the bottom plate’s capacity to withstand the pressure exerted by flowing water within channels.
To investigate how height affects the hydraulic characteristics of rectangular side weirs further (Figure 14), we extracted pressures on bottom plates when discharge was fixed at 25 L/s while varying heights were set as follows: 7 cm, 10 cm, 15 cm, and 20 cm, respectively.
Figure 14. Comparison of pressure distribution on the bottom plate of the side channel with different heights of side weirs when discharge is 25 L/s and the width of side weirs is 20 cm. (a) P = 7 cm, b = 20 cm; (b) P = 10 cm, b = 20 cm; (c) P = 15 cm, b = 20 cm; (d) P = 20 cm, b = 20 cm.
As shown in Figure 14, when the width of the side weir was constant, the pressure at the bottom of the side channel increased with the height of the side weir. As the height of the side weir increased, the water tongue formed by flow through the side weir gradually moved away from it in a downstream direction. In terms of vertical water flow, as the height of the side weir increased, the position of maximum pressure at which the water tongue falls shifted closer to the axis of the side channel from its right bank. Moreover, an increase in height resulted in higher maximum pressure at this falling point. For a constant width (20 cm) and varying heights (7 cm, 10 cm, 15 cm, and 20 cm), corresponding maximum pressures at this landing point were measured as 102,422 Pa, 102,700 Pa, 103,375 Pa, and 103,766 Pa, respectively. Consequently, increasing width led to a greater impact on both flow through and pressure distribution within the side channel; transforming it from scattered to concentrated along its lengthwise direction. Therefore, when applying such weirs practically one should select an appropriate width based on what pressure can be sustained by their respective channel bottom plates.
5.4. Discharge Coefficient
Based on dimensionless analysis, the influencing parameters of the discharge coefficient were obtained. To study the effect of parameters Fr1, b/h1, and P/h1, discharge coefficient values were plotted against Fr1, b/h1, and P/h1, shown in Figure 15, Figure 16 and Figure 17. The discharge coefficient decreased as parameters Fr1 and b/h1 increased. The discharge coefficient increased as parameter P/h1 increased. As Uyumaz and Muslu reported in a previous study, the variation of the discharge coefficient with respect to the Froude number showed a second-degree curve for a subcritical regime [4].
Figure 15. Variation of discharge coefficient values against Froude number.
Figure 16. Variation of discharge coefficient values against the percentage of the side weir width to the upstream flow depth over the side weir.
Figure 17. Variation of discharge coefficient values against the percentage of the side weir height to the upstream flow depth over the side weir.
Quantitative analysis between discharge coefficient values and parameters Fr1, b/h1, and P/h1 was conducted using data analysis software (IBM SPSS Statistics 19). The various coefficients obtained are shown in Table 3.
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig
B
Std. Error
Beta (β)
Constant
−1.294
0.155
−8.369
0.000
Fr1
3.430
0.286
3.401
12.013
0.000
b/h1
−0.004
0.004
−0.045
−0.944
0.348
P/h1
2.401
0.167
4.064
14.394
0.000
Table 3. Coefficient.
The value of t and Sig are the significance results of the independent variable, and the value of Sig corresponding to the value of t is less than 0.05, indicating that the independent variable has a significant impact on the dependent variable. Therefore, the values of Sig corresponding to the parameters Fr1 and P/h1 were less than 0.05, indicating that the parameters Fr1 and P/h1 have a significant impact on the discharge coefficient. On the contrary, the parameter b/h1 has less impact on the discharge coefficient. Therefore, quantitative analysis between discharge coefficient values and parameters Fr1, and P/h1 was conducted using data analysis software by removing factor b/h1. The model summary, ANOVA, and coefficient obtained are shown respectively in Table 4, Table 5 and Table 6. R and adjusted R square in Table 4 were approaching 1, which indicated the goodness of fit of the regression model was high. The value of Sig corresponding to the value of F in Table 5 was less than 0.05, which indicated that the regression equation was useful. The values of Sig corresponding to the parameters Fr1 and P/h1 in Table 6 were less than 0.05, indicating that the parameters Fr1 and P/h1 have a significant impact on the discharge coefficient.
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
0.913 a
0.833
0.829
0.03232
Table 4. Model Summary b. Note: a. Predictors:(Constant), Fr1, P/h1; b. Discharge coefficient.
Model
Sum of Squares
df
Mean Square
F
Sig
1
Regression
0.402
2
0.201
192.545
0.000 a
Residual
0.080
77
0.001
Total
0.483
79
Table 5. ANOVA b. Note: a. Predictors:(Constant), Fr1, P/h1; b. Discharge coefficient.
Model
Unstandardized Coefficients
Standardized Coefficients
t
Sig
B
Std. Error
Beta (β)
Constant
−1.326
0.151
−8.796
0.000
Fr1
3.479
0.281
3.449
12.396
0.000
P/h1
2.427
0.164
4.108
14.765
0.000
Table 6. Coefficient a. Note: a. Predictors:(Constant), Fr1, P/h1.
Based on the above analysis, the flow coefficient formula has been obtained, shown as follows:
Discharge formula were obtained by substituting Equation (15) into Equation (12), as shown in Equation (16).
where Q ∈ [0.006, 0.030], m3/s; b ∈ [0.20, 0.47], m; P ∈ [0.07, 0.20], m.
Figure 18 showed the measured discharge coefficient values with those calculated from discharge formulas in Table 3. The scatter of the data with respect to perfect line was limited to ±10%.
Figure 18. Comparison of the measured discharge coefficient values with those calculated from discharge formulas in Table 3.
6. Discussions
Determining water surface profile near the side weir in the main channel is one of the tasks of hydraulic calculation for side weirs. As the water flows through the side weir, discharge in the main channel is gradually decreasing, namely dQ/ds<0. According to the Equation (17) derived from Qimo Chen [3], it can be inferred that the value of 𝑑ℎ/𝑑𝑠 is greater than zero in subcritical flow (Fr < 1), that is, the water surface profile near the side weir in the main channel is a backwater curve. Due to the side weir entrance effect at the upstream end, water surface profiles drop slightly at the upstream end of the side weir crest, as EI-Khashab [28] and Emiroglu et al. [29] reported in previous experimental studies.
In this study, the water surface profile exhibited a backwater curve along the length of the weir crest. Therefore, during side weir application, it is crucial to ensure that downstream water levels do not exceed the highest water level of the channel.
The head on the weir is one of the important factors that flow over side weirs depends on. At the same time, the head depends on the water surface profile near the side weir in the main channel. Therefore, further research on the quantitative analysis of water surface profile needs to be conducted. Mohamed Khorchani proposed a new approach based on the video monitoring concept to measure the free surface of flow over side weirs. It points out a new direction for future research [8].
The maximum flow velocity, a key parameter in assessing the efficiency of a weir, occurs at the upstream end of the weir crest, typically near the crest. This is attributed to the convergence of the flow as it approaches the crest, resulting in a significant increase in velocity. It was found that in this study the minimum flow velocity occurred at the bottom of the main channel away from the side weir. Under such conditions, the accumulation of sediments could lead to siltation, which in turn can affect the accuracy of flow measurement through side weirs. This is because the presence of sediments can alter the flow patterns and cause errors in the measurement. Therefore, it becomes crucial to explore methods to optimize the selection of side weirs in order to minimize or eliminate the effects of sedimentation on flow measurement.
Pressure distribution plays a crucial role in ensuring structural safety for side weirs since small channels and field inlets have relatively limited pressure-bearing capacities. Therefore, it is important to select an appropriate geometrical parameter of rectangular side weirs based on their ability to withstand the pressure exerted on their bottom combined with pressure distribution data at the bottom of the side channel we have obtained in this study.
The discharge coefficient formula (Equation (15)), which incorporates Fr1 and P/h1, was derived based on dimensional analysis. However, it is worth noting that previous research has contradicted this formula by suggesting that the discharge coefficient solely depends on the Froude number. This conclusion can be observed in this study such as in Equations (18)–(23) in Table 7 of the manuscript [30,31,32,33,34,35], which clearly demonstrate the dependency of the discharge coefficient on the Froude number. In contrast, our derived discharge coefficient formula (Equation (15)) offers a more streamlined and simplified approach compared to Equation (25) [36] and Equation (29) [10]—making it easier to comprehend and apply—an advantageous feature particularly valuable in fluid dynamics where intricate calculations can be time-consuming. Furthermore, our derived discharge coefficient formula (Equation (15)) exhibits a broader application scope than that of Equation (24) [37] as shown in Table 8. Equation (26) [38] and Equation (27) [5] are specifically applicable under high flow discharge conditions. Conversely, our derived discharge coefficient formula (Equation (15)) is better suited for low-flow discharge conditions.
Table 7. Discharge coefficient formulas of rectangular side weirs presented in previous studies.
Discharge/(L·s−1)
Width of Side Weir/cm
Height of Side Weir/cm
Number of Formula
10~14
10~20
6~12
(24)
35–100
20~75
1~19
(26), (27)
6~30
20~47
7~20
(15)
Table 8. Application scope of discharge coefficient formulas.
In addition to the factors studied in the paper, factors such as the sediment content in the flow, the bottom slope, and the cross-section shape of the channel also have a certain impact on the hydraulic characteristics of the side weir. Further numerical simulation methods can be used to study the hydraulic characteristics and the influencing factors of the side weir. Water measurement facilities generally require high accuracy of water measurement, the flow of sharp-crested side weirs is complex, and the water surface fluctuates greatly. While conducting numerical simulations, experimental research on prototype channels is necessary to ensure the reliability of the results and provide reference for the body design and optimization of side weirs in small channels and field inlets.
7. Conclusions
This paper presents a comprehensive study that encompasses both experimental and numerical simulation research on rectangular side weirs of varying heights and widths within rectangular channels. A thorough analysis of the experimental and numerical simulation results has been conducted, leading to the derivation of several notable conclusions:
A comparative analysis was conducted on the measured and simulated values of water depth and flow velocity. Both of the maximum absolute relative errors were within 10%, which indicated that the numerical simulation of the side weir was feasible and effective.
The water surface profile exhibited a backwater curve along the length of the weir crest. The side weir entrance effect occurred only between Side Ⅰ and Side Ⅱ. This indicates that flow patterns and associated hydraulic forces at the weir entrance play a crucial role in determining water level distribution along the weir crest.
The maximum flow velocity of the cross-section at the upstream end of the weir crest occurred near the weir crest, while the minimum flow velocity occurred at the bottom of the main channel away from the side weir. As the water depth decreased, the position of the maximum flow velocity gradually moved from the upstream end of the side weir to the downstream end of the side weir.
When the height of the side weir remains constant, an increase in the width of the side weir leads to a decrease in pressure at the bottom of the side channel. Conversely, when the width of the side weir is kept constant, an increase in its height results in an increase in pressure at the bottom of the side channel. Therefore, during practical applications involving side weirs, it is crucial to select an appropriate weir width based on the maximum pressure that can be sustained by the channel’s bottom plate.
The discharge coefficient was found to depend on the upstream Froude number Fr1 and the percentage of the side weir height to the upstream flow depth over the side weir P/h1. The relationship between the discharge coefficient and parameters Fr1 and P/h1 was obtained using multiple regression analysis, which was of linear form and provided an easy means to estimate the discharge coefficient. The discharge formula is of high accuracy with relative errors within 10%, which met the water measurement accuracy requirements of small channels in irrigation areas.
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Enhanced understanding of flow structure in braided rivers is essential for river regulation, flood control, and infrastructure safety across the river. It has been revealed that the basic morphological element of braided rivers is confluence-bifurcation units. However, flow structure in these units has so far remained poorly understood with previous studies having focused mainly on single confluences/bifurcations. Here, the flow structure in a laboratory-scale confluence-bifurcation unit is numerically investigated based on the FLOW–3D® software platform. Two discharges are considered, with the central bars submerged or exposed respectively when the discharge is high or low. The results show that flow convergence and divergence in the confluence-bifurcation unit are relatively weak when the central bars are submerged. Based on comparisons with a single confluence/bifurcation, it is found that the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar. Concurrently, the high-velocity zone in the confluence-bifurcation unit is less concentrated than that in a single confluence while being more concentrated than that observed in a single bifurcation. The present work unravels the flow structure in a confluence-bifurcation unit and provides a unique basis for further investigating morphodynamics in braided rivers.
1 Introduction
Confluences and bifurcations commonly exist in alluvial rivers and usually are important nodes of riverbed planform (Szupiany et al., 2012; Hackney et al., 2018). Flow convergence and divergence in these junctions result in highly three-dimensional (3D) flow characteristics, which greatly influence sediment transport, and hence riverbed evolution and channel formation (Le et al., 2019; Xie et al., 2020). Braided rivers, characterized by unstable networks of channels separated by central bars (Ashmore, 2013), have confluence-bifurcation units as their basic morphological elements (Ashmore, 1982; 1991; 2013; Federici & Paola, 2003; Jang & Shimizu, 2005). In particular, confluence-bifurcation units exhibit a distinct morphology from single confluences/bifurcations and bifurcation-confluence regions because two adjacent central bars are included. Within a confluence-bifurcation unit, two tributaries converge at the upstream bar tail and soon diverge to two anabranches again at the downstream bar head. Therefore, the flow structure in the unit may be significantly influenced by both the two central bars, and thus considerably different from that in single confluences, single bifurcations, and bifurcation-confluence regions, where the flow is affected by only one central bar. Enhanced understanding of flow structure in confluence-bifurcation units is urgently needed, which is essential for water resources management, river regulation, flood control, protection of river ecosystems and the safety of infrastructures across the rivers such as bridges, oil pipelines and communication cables (Redolfi et al., 2019; Ragno et al., 2021).
The flow dynamics, turbulent coherent structures, and turbulent characteristics in single confluences have been widely studied since the 1980s (Yuan et al., 2022). Flow dynamics at river channel confluences have been systematically and completely analyzed, which can be characterized by six major regions of flow stagnation, flow deflection, flow separation, maximum velocity, flow recovery and distinct shear layers (Best, 1987). For example, the field observation of Roy et al. (1988) and Roy and Bergeron (1990) highlighted the flow separation zones and recirculation at downstream natural confluence corners. Ashmore et al. (1992) measured the flow field in a natural confluence and found flow accelerates suddenly at the confluence junction with two separated high-velocity cores merging into one single core at the channel centre. De Serres et al. (1999) investigated the three-dimensional flow structure at a river confluence and identified the existence of the mixing layer, stagnation zones, separation zones and recovery zones. Sharifipour et al. (2015) numerically studied the flow structure in a 90° single confluence and found that the size of the separation zone decreases with the width ratio between the tributary and the main channel. Recently, three main classes of large-scale turbulent coherent structures (Duguay et al., 2022) have been presented, i.e. vertical-orientated vortices or Kelvin-Helmholtz instabilities (Rhoads & Sukhodolov, 2001; Constantinescu et al., 2011; 2016; Biron et al., 2019), channel-scale ‘back-to-back’ helical cells, (Mosley, 1976; Ashmore, 1982; Ashmore et al., 1992; Ashworth, 1996; Best, 1987; Rhoads & Kenworthy, 1995; Bradbrook et al., 1998; Lane et al., 2000), and smaller, strongly coherent streamwise-orientated vortices (Constantinescu et al., 2011; Sukhodolov & Sukhodolova, 2019; Duguay et al., 2022). However, no consensus on a universal turbulent coherent structure mode has been reached so far (Duguay et al., 2022). In addition, some studies (Ashworth, 1996; Constantinescu et al., 2011; Sukhodolov et al., 2017; Le et al., 2019; Yuan et al., 2023) have focused on turbulent characteristics, e.g. turbulent kinetic energy, turbulent dissipation rate and Reynolds stress, which can be critical parameters to further explaining the diversity of these turbulent coherent structure modes.
Investigations on the flow structure in single bifurcations have mainly focused on hydrodynamics in anabranches (Hua et al., 2009; van der Mark & Mosselman, 2013; Iwantoro et al., 2022) and around bifurcation bars (McLelland et al., 1999; Bertoldi & Tubino, 2005; 2007; Marra et al., 2014), whereas few studies have considered the effects of bifurcations on the upstream flow structure. Thomas et al. (2011) found that the velocity core upstream of the bifurcation is located near the water surface and towards the channel center in experimental investigations of a Y-shaped bifurcation. Miori et al. (2012) simulated flow in a Y-shaped bifurcation and found two circulation cells upstream of the bifurcation with flow converging at the water surface and diverging near the bed. Szupiany et al. (2012) reported velocity decreasing and back-to-back circulation cells upstream of the bifurcation junction in the field observation of a bifurcation of the Rio Parana River. These investigations provide insight into how bifurcations affect the flow patterns upstream, yet there is a need for further research on the dynamics of flow occurring immediately before the bifurcation junction.
Generally, the findings of studies on bifurcation-confluence regions are similar to those concerning single confluences and bifurcations. Hackney et al. (2018) measured the hydrodynamic characteristics in a bifurcation-confluence of the Mekong River and found the velocity cores located at the channel centre and strong secondary current occurring under low discharges. Le et al. (2019) reported a high-turbulent-kinetic-energy (high-TKE) zone located near the bed in their numerical simulation of flow in a natural bifurcation-confluence region. Moreover, a stagnation zone was found upstream of the confluence and back-to-back secondary current cells were detected at the confluence according to Xie et al. (2020) and Xu et al. (2022). Overall, these studies have further unraveled the flow patterns in river confluences and bifurcations.
Unfortunately, limited attention has been paid to the flow structure in confluence-bifurcation units. Parsons et al. (2007) investigated a large confluence-bifurcation unit in Rio Parana, Argentina, and no classical back-to-back secondary current cells were observed under a discharge of 12000 m3·s−1. To date, the differences in flow structure between confluence-bifurcation units and single confluences/bifurcations have remained far from clear. In addition, although the effects of discharge on flow structure have been investigated in several studies on single confluences/bifurcations, (Hua et al., 2009; Le et al., 2019; Luz et al., 2020; Xie et al., 2020; Xu et al., 2022), cases with fully submerged central bars were not considered, which is typical in braided rivers during floods. In-depth studies concerning these issues are urgently needed to gain better insight into the flow structure in confluence-bifurcation units of braided rivers.
This paper aims to (1) reveal the 3D flow structure in a confluence-bifurcation unit under different discharges and (2) elucidate the differences in the flow structure between confluence-bifurcation units and single confluence/bifurcation cases. Using the commercial computational fluid dynamics software FLOW-3D® (Version 11.2; https://www.flow3d.com; Flow Science, Inc.), fixed-bed simulations of a laboratory-scale confluence-bifurcation unit are conducted, and cases of a single confluence/bifurcation are also included for comparison. Two discharges are considered, with the central bars fully submerged or exposed respectively when the discharge is high or low. Based on the computational results, the 3D flow structure in the confluence-bifurcation unit conditions is analyzed from various aspects including free surface elevation, time-averaged flow velocity distribution, recirculation vortex structure, secondary current, and turbulent kinetic energy and dissipation rate. In particular, the flow structure in the confluence-bifurcation unit is compared with that in the single confluence/bifurcation cases to unravel the differences.h
2. Conceptual flume and computational cases
2.1. Conceptual flume
In this paper, a laboratory-scale conceptual flume is designed and used in numerical simulations. Figure 1(a–d) shows the morphological characteristics of the flume. To ensure that the conceptual flume reflects morphology features of natural braided channels, key parameters governing the flume morphology, e.g. unit length, width, and channel width-depth ratio, are determined according to studies on morphological characteristics of natural confluence-bifurcation units (Hundey & Ashmore, 2009; Ashworth, 1996; Orfeo et al., 2006; Parsons et al., 2007; Sambrook Smith et al., 2005; Kelly, 2006; Ashmore, 2013; Egozi & Ashmore, 2009; Redolfi et al., 2016; Ettema & Armstrong, 2019).
Figure 1. The sketch of the conceptual flume: (a) the original flume, (b) the central bar: (c) the sketch of cross-section C-C, (d) the sketch of cross-section D-D, (e) the modified part for the single confluence, (f) the modified part for the single bifurcation, (g) the position of different cross-sections. The red dashed boxes denote the regions of primary concern.
Figure 1. The sketch of the conceptual flume: (a) the original flume, (b) the central bar: (c) the sketch of cross-section C-C, (d) the sketch of cross-section D-D, (e) the modified part for the single confluence, (f) the modified part for the single bifurcation, (g) the position of different cross-sections. The red dashed boxes denote the regions of primary concern.
2.1.1. Length and width scales of the confluence-bifurcation unit
The length and width scales of the flume are first determined. The inner relation among the length LCB and average width B of a confluence-bifurcation unit and the average width Bi of a single branch was statistically studied by Hundey and Ashmore (2009), which indicates the following relations: 𝐿CB =(4∼5)𝐵 (1) 𝐵 =1.41𝐵𝑖 (2) In addition, Ashworth (1996) gave B = 2Bi in his experimental research on mid-bar formation downstream of a confluence, while the confluence-bifurcation unit of Rio Parana, Argentina has a relation of B≈1.71Bi (Orfeo et al., 2006; Parsons et al., 2007). Accordingly, the following relations are used in the present paper: 𝐿CB =4𝐵 (3) 𝐵 =1.88𝐵𝑖 (4) where LCB = 6 m, B = 1.5 m and Bi = 0.8 m.
2.1.2. Central bar morphology
The idealized plane pattern of central bars in braided rivers is a slightly fusiform leaf shape with a short upstream side and a long downstream side (Ashworth, 1996; Sambrook Smith et al., 2005; Kelly, 2006; Ashmore, 2013). To simplify the design, the bar is approximated as a combination of two different semi-ellipses (Figure 1(b)). The major axis Lb is two to ten times longer than the minor axis Bb according to the statistical data in Kelly’s study, and the regression equation is given as (Kelly, 2006): 𝐿𝑏=4.62𝐵0.96𝑏 (5) In this study, the bar width Bb is set as 0.8 m, whilst the lengths of downstream (LT1) and upstream sides (LT2) are 2 and 1.5 m, respectively (Figure 1(b)). Thus, the relation of Lb and Bb is given as: 𝐿𝑏=(𝐿𝑇1+𝐿𝑇2)=4.375𝐵𝑏 (6) The lengths of the inlet and outlet parts are determined as Lin = Lout = 8 m, which ensures negligible effects of boundary conditions without exceptional computational cost.
2.1.3. Width-depth ratio
Channel flow capacity can be significantly affected by cross-section shapes. For natural rivers, cross-section shapes can be generalized into three sorts based on the following width-depth curve (Redolfi et al., 2016): 𝐵=𝜓𝐻𝜑(7) Braided rivers usually have ψ = 5∼50 and φ>1, which indicates a rather wide and shallow cross-section. The central bar form should also be taken into account, so a parabolic cross-section shape is used here with ψ = 8 and φ>1 (Figure 1(c,d)).
2.1.4. Bed slope
In addition, natural braided rivers are usually located in mountainous areas and thus have a relatively large bed slope. According to flume experiments and field observations, the bed slope Sb is mostly in the range of 0.01∼0.02, and a few are below 0.01 (Ashworth, 1996; Egozi & Ashmore, 2009; Ashmore, 2013; Redolfi et al., 2016; Ettema & Armstrong, 2019). In this study, Sb takes 0.005.
2.1.5. Complete sketch of the conceptual flume
In summary, the flume is 29 m long, 2.4 m wide, and 0.6 m high. The plane coordinates (x-direction and y-direction) used in the calculation process are shown in Figure 1 (a). Note that the inlet corresponds to x = 0 m, and the centreline of the flume is located at y = 1.3 m. Besides, the thalweg elevation of the outlet is set as z = 0 m.
2.2. Computational cases
As stated before, the first aim of this paper is to reveal the flow structure in the confluence-bifurcation unit under different discharges. Therefore, two basic cases are set first: (1) case 1a under a low discharge (0.05 m3·s−1) with exposed central bars and (2) case 2a under a high discharge (0.30 m3·s−1) with fully submerged central bars. A total of 22 cross-sections are identified to examine the results (Figure 1(g)).
Further, cases of a single confluence/bifurcation are generated by splitting the original confluence-bifurcation unit into two parts. Part 1 only includes the upstream central bar and focuses on the flow convergence downstream of CS04 (Figure 1(e)), while Part 2 only includes the downstream central bar and focuses on the flow divergence upstream of CS19 (Figure 1(f)). Notably, the numbers of corresponding cross-sections in the original flume are reserved to facilitate comparison. The outlet section of the single confluence as well as the inlet section of the single bifurcation is extended to make the total length equivalent to the original flume (29 m). Also, two discharge conditions (0.05 and 0.30 m3·s−1), which correspond to exposed and fully submerged central bars, are considered for the single confluence/bifurcation. In total, six computational cases are conducted, as listed in Table 1. As the conceptual flume is designed to be symmetrical about the centreline, the momentum flux ratio (Mr) of the two branches should be 1 in all six cases. This is confirmed by further examining the computational results.
Case
Configuration
Qin (m3·s−1)
Zout (m)
Mr
Condition of bars
1a
CBU
0.05
0.15
1
Exposed
1b
SC
0.05
0.15
1
Exposed
1c
SB
0.05
0.15
1
Exposed
2a
CBU
0.30
0.34
1
Submerged
2b
SC
0.30
0.34
1
Submerged
2c
SB
0.30
0.34
1
Submerged
Table 1. Computational cases with inlet and outlet boundary conditions.
3. Numerical method
In this section, the 3D Large Eddy Simulation (LES) model integrated in the FLOW-3D® (Version 11.2; https://www.flow3d.com; Flow Science, Inc.) software platform is introduced, including governing equations and boundary conditions. Information on computational meshes with mesh independence test can be found in the Supplementary material.
3.1. Governing equations
The LES model was applied in the present paper to simulate flow in the laboratory-scale confluence-bifurcation unit. The LES model has been proven to be effective in simulating turbulent flow in river confluences and bifurcations (Constantinescu et al., 2011; Le et al., 2019). The basic idea of the LES model is that one should directly compute all turbulent flow structures that can be resolved by the computational meshes and only approximate those features that are too small to be resolved (Smagorinsky, 1963). Therefore, a filtering operation is applied to the original Navier-Stokes (NS) equations for incompressible fluids to distinguish the large-scale eddies and small-scale eddies (Liu et al., 2018). The filtered NS equations are then generated, which can be expressed in the form of a Cartesian tensor as (Liu, 2012):
(10) where ¯𝑢𝑖 is the resolved velocity component in the i – direction (i goes from 1 to 3, denoting the x-, y – and z-directions, respectively); t is the flow time; ρ is the density of the fluid; ¯𝑝 is the pressure; ν is the kinematic viscosity; τij is the sub-grid scale (SGS) stress; ¯𝐺𝑖 is the body acceleration. In FLOW–3D®, the full NS equations are discretized and solved using the finite-volume/finite-difference method (Bombardelli et al., 2011; Lu et al., 2023).
Due to the filtering process, the velocity can be divided into a resolved part (¯𝑢(𝑥,𝑡)) and an approximate part (𝑢′(𝑥,𝑡)) which is also known as the SGS part (Liu, 2012). To achieve model closure, the standard Smagorinsky SGS stress model is introduced here (Smagorinsky, 1963): 𝜏ij−13𝜏kk𝛿ij=−2𝜈SGS¯𝑆ij(11) where νSGS is the SGS turbulent viscosity, and ¯𝑆ij is the resolved rate-of-strain tensor for the resolved scale defined by (Smagorinsky, 1963): ¯𝑆ij=12(∂¯𝑢𝑖∂𝑥𝑗+∂¯𝑢𝑗∂𝑥𝑖)(12) In the standard Smagorinsky SGS stress model, the eddy viscosity is modelled by (Smagorinsky, 1963): 𝜈SGS=(𝐶𝑠¯𝛥)2∣¯𝑆∣,∣¯𝑆∣=√2¯𝑆ij¯𝑆ij(13) ¯𝛥=(ΔxΔyΔz)1/3(14) where Cs is the Smagorinsky constant, Δx, Δy, and Δz are mesh scales. In FLOW–3D®, Cs is between 0.1 to 0.2 (Smagorinsky, 1963). One of the key problems in simulating 3D open channel flow is the calculation of free surface. FLOW–3D® uses the Volume of Fluid (VOF) method (Hirt & Nichols, 1981) to track the change of free surface. The VOF method introduces a fluid phase fraction function f to characterize the proportion of a certain fluid in each mesh cell. In that case, the surface position can be precisely located if the mesh cell is fine enough. To monitor the change of f with time and space, the following convection equation is added:
For open channel flow, only two kinds of fluids are involved: water and air. If f is the fraction of water, the state of the fluid in each mesh cell can be defined as:
In FLOW–3D®, the interface between water and air is assumed to be shear-free, which means that the drag force on the water from the air is negligible. Moreover, in most cases, the details of the gas motion are not crucial for the heavier water motion so the computational processes will be more efficient.
3.2. Boundary conditions
Six boundary conditions need to be preset in the 3D numerical simulation process. Discharge boundary conditions are used for the inlet of the flume, where the free surface elevation is automatically calculated based on the free surface elevation boundary conditions set for the outlet. The specific information on the inlet and outlet boundary conditions for all computational cases is shown in Table 1. Moreover, because the free surface moves temporally, the free surface boundary conditions are just set as no shear stress and having a normal pressure, and the position of the free surface will be automatically adjusted over time by the VOF method in FLOW–3D®. Furthermore, the bed and two side walls are all set to be no-slip for fixed bed conditions, and a standard wall function is employed at the wall boundaries for wall treatment.
The inlet turbulent boundary conditions also need to be considered. They are set by default here. The turbulent velocity fluctuations V’ are assumed to be 10% of the mean flow velocity with the turbulent kinetic energy (TKE) (per unit mass) equaling 0.5V’2. The maximum turbulent mixing length is assumed to be 7% of the minimum computational domain scale, and the turbulent dissipation rate is evaluated automatically from the TKE.
4. Results and discussion
4.1. Flow structure in the confluence-bifurcation unit
4.1.1. Free surface elevation
Figure 2 shows the free surface elevation at five different longitudinal profiles (i.e. α = 0.2, 0.4, 0.5, 0.6, 0.8) for cases 1a and 2a. The parameter α was defined as follows:𝛼=𝑠𝐵(17) where s is the transverse distance between a certain profile and the left boundary of the flume. In general, the longitudinal change of free surface in the two cases is very similar despite different discharge levels. The free surface elevation decreases as the channel narrows from the upstream bifurcation to the front of the confluence-bifurcation unit. On the contrary, when the flow diverges again at the end of the confluence-bifurcation unit, the free surface elevation increases with channel widening. However, whether the fall or rise of free surface elevation in case 1a is much sharper than that in case 2a, especially at profiles with α = 0.2 and 0.8 (Figure 2(a)), which indicates there may be distinct flow states between the two cases. To further illustrate this finding, the Froude number Fr at different cross-sections (CS08∼CS15) is examined. In case 2a, the flow remains subcritical within the confluence-bifurcation unit. By contrast, in case 1a, a local supercritical flow is observed near the side banks of CS09 (i.e. α = 0.2 and 0.8), with Fr being about 1.2. This local supercritical flow can lead to a hydraulic drop followed by a hydraulic jump, which accounts for the sharp change of the free surface. The foregoing reveals that when central bars are exposed under relatively low discharge, supercritical flow is more likely to occur near the side banks of the confluence junction due to flow convergence.
Figure 2. Five time-averaged free surface elevation profiles in the confluence-bifurcation unit, in which α denotes the lateral position of the certain profile. Note that the black dashed line denotes the position of CS09, where Fr is about 1.2 near the side banks (α = 0.2 and 0.8) in case 1a. Z’ = z/h2, X’ = x/B, h2 is the maximum flow depth at the outlet boundary of cases 2a, 2b and 2c, h2 = 0.34 m.
Moreover, in both cases 1a and 2a, the free surface is higher at the channel centre than near the side banks, whether at the front or the end of the confluence-bifurcation unit. Thus, lateral free surface slopes from the centre to the side banks are generated. For example, the lateral free surface slopes at CS09 are 0.022 and 0.016 respectively for cases 1a and 2a. These lateral slopes can lead to lateral pressure gradient force whose direction is from the channel centreline to the side banks. Notably, the lateral surface slope in case 1a is steeper than that in case 2a, which may also result from the effect of the supercritical flow.
4.1.2. Time-averaged streamwise flow velocity
Figure 3. Time-averaged flow velocity distribution at three different slices over z-direction in the confluence-bifurcation unit: (a)∼(c) case 1a, (d)∼(f) case 2a. The flow direction is from the left to the right. StZ = Stagnation Zones, MiL = Mixing Layer. X’ = x/B, Y’ = y/B, Ui’ = Ui/Uti, Ui denotes the time-averaged streamwise flow velocity in case series i (i = 1,2), Uti denotes the cross-section-averaged streamwise flow velocity in case series i, Ut1 = 0.385 m/s, for case 2a Ut2 = 0.714 m/s.Figure 4. Time-averaged flow velocity contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a.
Besides the shared features described above, some differences between the two cases are also identified. First, flow stagnation zones at the upstream bar tail are found exclusively in case 1a as the central bars are exposed (Figure 3 (a–c)). Second, in case 1a the mixing layer is obvious in both the lower or upper flows (Figure 3 (a–c)), while in case 2a the mixing layer can be inconspicuous in the upper flow (Figure 3 (f)). Third, in case 1a, two high-velocity cores gradually transform into one single core downstream of the confluence [Figure 4 (a), CS08∼CS11] and are divided into two cores again at the downstream bar head [Figure 4 (a), CS15]. By contrast, in case 2a, the two cores merge much more rapidly [Figure 4 (a), CS08∼CS09], and no obvious reseparation of the merged core is found at the downstream bar head (Figure 3 (d–f)). The latter two differences between cases 1a and 2a indicate that the flow convergence and divergence are relatively weak when the central bars are fully submerged. It is noticed that when the central bars are exposed, the flow in branches needs to steer around the central bar, which can cause a large angle between the two flow directions at the confluence, and thus relatively strong flow convergence and divergence may occur. By contrast, when the central bars are fully submerged, the flow behavior resembles that of a straight channel, with flow predominantly moving straight along the main axis of the central bars. Therefore, a small angle between two tributary flow forms, and thus flow convergence and divergence are relatively mild.
4.1.3. Recirculation vortex
A recirculation vortex with a vertical axis is a typical structure usually found where flow steers sharply, and is generated from flow separation (Lu et al., 2023). This vortex structure is found in the confluence-bifurcation unit in the present study, marking several significant flow separation zones. Figure 5 shows the recirculation vortex structure at the bifurcation junction of the confluence-bifurcation unit. In both cases 1a and 2a, two recirculation vortices BV1 and BV2 are found at the bifurcation junction corner. Moreover, BV1 and BV2 seem well-established near the bed but tend to transform into premature ones in the upper flow, and there is also a tendency for the cores of BV1 and BV2 to shift downstream as they transition from the lower to the upper flow (Figure 5(a–c,d–f)). This finding indicates that flow separation zones exist at the bifurcation junction corner, and the vortex structure is similar in the separation zones under low and high discharges. These flow separation zones are generated due to the inertia effect as flow suddenly diverges and steers towards the curved side banks of the channel (Xie et al., 2020). Notably, two additional vortices BV3 and BV4 are found at both sides of the downstream bar in case 1a (Figure 5(a–c)), but no such vortices exist in case 2a. This difference shows that flow separation zones at both sides of the downstream bar are hard to form when the bars are completely submerged under the high discharge.
Figure 5. Recirculation vortices at the bifurcation junction (streamline view at three different slices over z-direction): (a)∼(c) case 1a, (d)∼(f) case 2a. The red solid line marked out the position of these vortices (BV1∼BV4).
Similarly, Figure 6 shows the recirculation vortex structure at the confluence junction of the confluence-bifurcation unit. No noteworthy similarities but a key difference between the two cases are observed at this site. Two vortices CV1 and CV2 are found downstream of the confluence junction corner in case 1a (Figure 6(c)), which mark two separation zones. Conversely, no such separation zones are found in case 2a. In fact, separation zones were reported at similar sites under relatively low discharges in some previous studies (Ashmore et al., 1992, Luz et al., 2020, Sukhodolov & Sukhodolova, 2019; Xie et al., 2020). Nevertheless, the flow separation zones at the confluence corner are very restricted in the present study (Figure 6(c)). Ashmore et al. (1992) also reported that no, or very restricted flow separation zones occur downstream of natural river confluence corners, primarily because of the relatively slow change in bank orientation compared with the sharp corners of laboratory confluences where separation is pronounced (Best & Reid, 1984; Best, 1988). In the present study, the bank orientation also changes slowly, which may explain why flow separation zones are inconspicuous at the confluence corner.
Figure 6. Recirculation vortices at the confluence junction (streamline view at three different slices over z-direction): (a)∼(c) case 1a, (d)∼(f) case 2a. The red solid line marked out the position of these vortices (CV1 & CV2).
The differences in the distribution of recirculation vortices discussed above may be mainly attributed to the difference in the angle between the tributary flows under different discharges. Some previous studies have reported that the confluence/bifurcation angle can significantly influence the flow structure at confluences/bifurcations (Best & Roy, 1991; Ashmore et al., 1992; Miori et al., 2012). Although the confluence/bifurcation angle is fixed due to the determined central bar shape in the present study, the angle between two tributary flows is affected by the varying discharge. When the central bars are exposed under the low discharge, the flow is characterized by a more pronounced curvature of the streamlines, and a large angle between the two tributary flows is noted (Figure 6(b)), causing strong flow convergence and divergence. By contrast, a small angle forms as the central bars are submerged, thereby leading to relatively weak flow convergence/divergence (Figure 6(e)). Overall, the differences mentioned above can be attributed to the differences in the intensity of flow convergence and divergence under different discharges.
It should be noted that some previous studies (Constantinescu et al., 2011; Sukhodolov & Sukhodolova, 2019) presented that there is a wake mode in the mixing layer of two streams at the confluence junction. The wake mode means that in the mixing layer, multiple streamwise coherent vortices moving downstream will form, which is similar to the flow structure around a bluffing body (Constantinescu et al., 2011). However, no such structure has been found within the confluence-bifurcation unit in this study. According to the numerical simulations of Constantinescu et al. (2011), a wake mode was found at a river confluence with a concordant bed and a momentum flux ratio of about 1. The confluence has a much larger angle (∼60°) between the two streams when compared to the confluence junction of the confluence-bifurcation unit in the present study where the angle is about 25°. As flow mechanics at river confluences may include several dominant mechanisms depending on confluence morphology, momentum ratio, the angle between the tributaries and the main channel, and other factors (Constantinescu et al., 2011), the relatively small confluence angle in the present study may explain why the wake mode is absent. The possible effects of the confluence/bifurcation angle are reserved for future study. Additionally, flow separation can lead to reduced local sediment transport capacity, thus causing considerable sediment deposition under natural conditions. Hence, the bank may migrate towards the inner side of the channel at the positions of CV1, CV2, BV1, and BV2, while the bar may expand laterally at the positions of BV3 and BV4.
4.1.4. Secondary current
Secondary current is the flow perpendicular to the mainstream axis (Thorne et al., 1985) and can be categorized into two primary types based on its origin: (1) Secondary current generated by the interaction between centrifugal force and pressure gradient force; (2) Secondary current resulting from turbulence heterogeneity and anisotropy (Lane et al., 2000). There are some widely recognized definitions of secondary current strength (SCS) (Lane et al., 2000). In this paper, the secondary current cells are identified by visible vortex with a streamwise axis, and the definition of SCS proposed by Shukry (1950) is used:
where ux, uy, and uz are flow velocities in x, y, and z directions and ux represents the mainstream flow velocity.
Figure 7 presents contour plots of SCS and the secondary current structure at key cross-sections of the study area. When the central bars are exposed, at the upstream bar tail (CS08), intense transverse flow occurs with flow converging to the centreline, but no secondary current cell is formed (Figure 7(a)). This is consistent with the findings of Hackney et al. (2018). At the confluence junction (CS09), transverse flow still plays a major role in the secondary current structure, with flow converging to the centreline at the surface and diverging to side banks near the bed (Figure 7(b)). Moreover, ‘back-to-back’ helical cells, which are two vortices rotating reversely, tend to generate at CS09 with their cores located near the side banks (Figure 7(b)) (Mosley, 1976; Ashmore, 1982; Ashmore et al., 1992), yet their forms are rather premature. As the flow goes downstream, the cores of the helical cells gradually rise to the upper flow and approach towards the centreline, and the helical cells become well-established (Figure 7(c–e)). When the flow diverges again at the downstream bar head (CS15), the helical cells attenuate rapidly, and the secondary current structure is once again characterized predominantly by transverse flow (Figure 7(f)).
Figure 7. Distribution of secondary current strength and secondary current cells at six different cross-sections: (a)∼(f) case 1a, (g)∼(l) case 2a. The secondary current cells are identified by visible lateral vortices (streamline view). The zero distance of each cross-section is located on the right bank.
When the central bars are fully submerged under the high discharge, the secondary current structure at the upstream bar tail and the confluence junction exhibits a resemblance to that under the low discharge (Figure 7(g,h)). However, at CS09, two pairs of cells with different scales tend to form under the high discharge (Figure 7(h)). The large and premature helical cells are similar to those under the low discharge, whereas the small helical cells are located near side banks possibly due to wall effects. As the flow moves downstream, the large helical cells tend to diminish rapidly and merge with the small ones near both side walls (Figure 7(i–k)). Moreover, the secondary current structure is once again characterized predominantly by transverse flow at CS14 under the high discharge, which occurs earlier than that under the low discharge (Figure 7(k)). At the downstream bar head, transverse flow still takes a dominant place, while the helical cells seem to become premature with increased scale (Figure 7(l)).
In general, in both cases 1a and 2a, the lateral distribution of SCS at all cross-sections is symmetrical about the channel centreline, where SCS is relatively small. A relatively high SCS is detected at both the upstream bar tail and the downstream bar head due to the effects of centrifugal force caused by flow steering. SCS decreases rapidly from the upstream bar tail (CS08) to the entrance of the downstream bifurcation junction (CS14), followed by a sudden increase at the downstream bar head (CS15) (Figure 7 (a–e, g–k)). However, the distribution of high-SCS zones is different between the two discharges. Under the low discharge, high-SCS zones appear along the bottom near the centerline and at the free surface on both sides of the centreline. Although similar high-SCS zones are found along the bottom near the centerline under the high discharge, the high-SCS zones are not found at the free surface. Furthermore, it is noticed that more obvious high-SCS zones appear under the low discharge compared with the high discharge, especially at CS09. This may be attributed to the differences in the intensity of flow convergence and divergence under different submerging conditions of the central bars. When the central bars are exposed, flow convergence and divergence are strong and sharp flow steering occurs, thereby causing large SCS. By contrast, when the central bars are fully submerged, flow convergence and divergence are relatively weak, and thus small SCS is observed.
4.1.5. Turbulent characteristics
Turbulent characteristics reflect the performance of energy and momentum transfer activities in flow (Sukhodolov et al., 2017). Comprehensive analysis of turbulent characteristics is crucial as they greatly impact the incipient motion, settling behavior, diffusion pattern, and transport process of sediment. Here, the TKE and turbulent dissipation rate (TDR) of flow in the confluence-bifurcation unit are analyzed.
Figure 8 shows the distribution of TKE on various cross-sections in cases 1a and 2a. In the same way, Figure 10 shows the distribution of TDR. The values of TKE and TDR are nondimensionalized with mid-values of TKE = 0.005 m2·s−2 and TDR = 0.007 m3·s−2. In both cases 1a and 2a, the distributions of TKE and TDR show symmetrical patterns concerning the channel centreline. High-TKE and high-TDR zones exhibit a belt distribution near the channel bottom (McLelland et al., 1999; Ashworth, 1996; Constantinescu et al., 2011), indicating that turbulence primarily originates at the channel bottom due to the influence of bed shear stress. A sudden increase of TKE (Weber et al., 2001) and TDR occurs near the channel bottom at the confluence junction [Figure 8 and 9, CS08∼CS09] and from the entrance of the bifurcation junction (CS14) to the downstream bar head (CS15) (Figures 8 and 9).
Figure 8. Turbulent kinetic energy contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a. TKE = turbulent kinetic energy. TKE’ = dimensionless value of TKE, with regard to a mid-value of TKE = 0.005 m2·s−2.Figure 9. Turbulent dissipation rate contours at eight different cross-sections in the confluence-bifurcation unit: (a) case 1a, (b) case 2a. TDR = turbulent dissipation rate. TDR’ = dimensionless value of TDR, with regard to a mid-value of TDR = 0.007 m3·s−2.Figure 10. Comparison of the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections between the confluence-bifurcation unit and the single confluence. (a)∼(f) 1a vs. 1b, (g)∼(l) 2a vs. 2b.
Despite the common turbulent characteristics between cases 1a and 2a, additional high-TKE zones are found in the upper flow at the upstream bar tail (CS08), the confluence junction (CS09) and the downstream bar head (CS15) (Figure 8) when the central bars are fully submerged. The formation mechanism of these high-TKE zones near the water surface is more complicated, which may result from interactions of velocity gradient, secondary current structure and wall shear stress (Engel & Rhoads, 2017; Lu et al., 2023).
4.2. Comparison with single confluence/bifurcation cases
In this section, the results of a single confluence (cases 1b and 2b) and a single bifurcation (cases 1c and 2c) are compared with those of the confluence-bifurcation unit (cases 1a and 2a) under two discharges. Flow structure at CS08∼CS15 is mainly concerned below.
4.2.1. Comparison with single confluence cases
First, the patterns of time-averaged streamwise velocity, TKE and TDR within the single confluence (presented by contour plots in the supplementary materials) are assessed and then compared with those within the confluence-bifurcation unit (Figures 4, 8, and 9). It is found that distributions of these parameters are similar in the confluence-bifurcation unit and the single confluence from the upstream bar tail (CS08) to the entrance of the bifurcation junction (CS14), despite varying discharges. As the existence of the downstream central bar is the main difference between the single confluence and the confluence-bifurcation unit, this finding indicates that the downstream bar may have limited influence on the flow structure in the confluence-bifurcation unit. In other words, the flow structure in the confluence-bifurcation unit appears to be mainly shaped by the presence of the upstream bar, with its impact potentially reaching as far as the entrance of the bifurcation (CS14). Moreover, under the low discharge, the two high-velocity cores seem to merge later (at CS11) in the single confluence than in the confluence-bifurcation unit (at CS10), which indicates the convergence of two tributary flows may achieve a steady state faster in the confluence-bifurcation unit. To further elucidate the differences, results on the distribution of time-averaged streamwise velocity and TKE along the flow depth are discussed below.
4.2.1.1. Time-averaged streamwise velocity
Figure 10 shows the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections. Note that α = 0.5 denotes the channel centreline and α = 0.7 denotes a position near the side banks. As only marginal differences are found at α = 0.3 and 0.7, only profiles at α = 0.7 are displayed for clarity.
Under the low discharge, no obvious difference in the distribution of time-averaged streamwise flow velocity is observed at the upstream bar tail (Figure 10(a)). At the confluence junction (Figure 10(b)), the velocities near the side banks (α = 0.7) are larger than those at the centre (α = 0.5) in both the confluence-bifurcation unit and the single confluence, which suggests that the two tributary flows have not sufficiently merged. The two tributary flows achieve convergence at CS11 in both the confluence-bifurcation unit and the single confluence (Figure 10(c)), with the velocity at the centre (α = 0.5) is larger than that near the side banks. Nevertheless, the velocities at the centre (α = 0.5) and near the side banks (α = 0.7) are closer to each other in the confluence-bifurcation unit than those in the single confluence, which represents less sufficient flow convergence in the confluence-bifurcation unit than in the single confluence. Therefore, it can be inferred that the convergence of two tributary flows may achieve a steady state faster in the confluence-bifurcation unit. After reaching the steady state, the velocity near the side banks (α = 0.7) is smaller in the single confluence than in the confluence-bifurcation unit despite close values at the centre (α = 0.5) (Figure 10(d,e)). This leads to a more pronounced disparity between velocities at the centre and near the side banks in the single confluence than that observed in the confluence-bifurcation unit. In other words, the high-velocity zone is more concentrated on the channel centreline in the single confluence, while the lateral distribution of flow velocity tends to be more uniform in the confluence-bifurcation unit. This may be attributed to the influence of the downstream central bar, which is further proved by comparing the velocity profiles at CS15 (Figure 10(e)).
As for the high discharge condition, from CS08 to CS14, the quantitative differences in velocity distribution between the confluence-bifurcation unit and the single confluence seem small. This indicates that the effect of morphology appears to be subdued when the central bars are fully submerged under the high discharge. It should be also noted that under both the low and high discharge, velocity profiles at the corresponding location exhibit the same shapes in the confluence-bifurcation unit and the single confluence, which indicates that the upstream confluence may dominate the flow structure in the confluence-bifurcation unit.
4.2.1.2. Secondary current
Figure 11 shows contour plots of SCS and the secondary current structure for single confluence cases. Compared with Figure 7, under both low and high discharge conditions, the distribution of SCS and the structure of helical cells in the confluence-bifurcation unit and the single confluence are very similar from CS08 to CS12 (Figure 7(a–d, g–j) and Figure 11(a–d, g–j)]. This indicates that the secondary current structure in the confluence-bifurcation unit exhibits certain consistent features when compared to those in the single confluence, thus proving that the effects of the upstream central bar may dominate the flow structure in the confluence-bifurcation unit. However, the secondary current structure at CS14 and CS15 is different between the confluence-bifurcation unit and the single confluence (Figure 7 and 11(e, f, k,l)). Under the low discharge, transverse flow is from the side banks to the centre and relatively high SCS is found near the side banks at CS14 in the single confluence, while the transverse flow is always from the centre to the side banks and SCS is relatively low at the corresponding sites in the confluence-bifurcation unit (Figure 11(e)). Under the high discharge, the helical cells near the side walls almost diminish in the single confluence, while they still exist in the confluence-bifurcation unit at CS14 (Figure 11(k)). Under both low and high discharges, the secondary current pattern at CS15 is similar to that at CS14 in the single confluence, while they are different in the confluence-bifurcation unit due to the existence of the downstream central bar. This comparison indicates that the existence of the downstream central bar can influence the upstream secondary current structure, nevertheless, the effects are fairly limited.
Figure 11. Secondary current at different cross-sections in the single confluence condition: (a)∼(f) case 1b, (g)∼(l) case 2b. The zero distance of each cross-section is located on the right bank.
4.2.1.3. Turbulent kinetic energy
Figure 12 shows TKE distribution along the flow depth at different cross-sections. Under the low discharge, in general, the maximum TKE tends to appear near the channel bottom in both the confluence-bifurcation unit and the single confluence. No obvious difference is observed at the upstream bar tail (CS08) (Figure 12(a)). Downstream this site (at CS09), the maximum TKE near the side banks (α = 0.7) is larger than that at the channel centre in the single confluence, while they are close to each other in the confluence-bifurcation unit (Figure 12(b)). This can also be attributed to the insufficient convergence of the two tributary flows. At CS11, flow convergence achieves a steady state in the confluence-bifurcation unit, while it remains insufficient in the single confluence. As flow convergence reaches a steady state at CS12, the maximum TKE in the single confluence exhibits a more concentrated distribution on the channel centre than that in the confluence-bifurcation unit (Figure 12(d)). This effect becomes more obvious downstream at CS14 (Figure 12(e)). The less-concentrated distribution of the maximum TKE in the confluence-bifurcation unit can be owing to the effects of the downstream central bar as well, which appears analogous to that mentioned in 4.2.1.1.
Figure 12. Comparison of the distribution of TKE along the flow depth at different cross-sections between the confluence-bifurcation unit and the single confluence. (a)∼(f) 1a vs. 1b, (g)∼(l) 2a vs. 2b.
Under the high discharge condition, two peaks of TKE appear in both the confluence-bifurcation unit and the single confluence (Figure 12(g–l)). Moreover, in both the confluence-bifurcation unit and the single confluence, from the upstream bar tail to the downstream bar head, the peak of TKE in the upper flow is larger at the channel centre (α = 0.5), while the peak of TKE in the lower flow is larger near the side banks (α = 0.7). However, the disparity between the TKE near the side banks and at the channel centre seems to be larger in the single confluence, while the TKE in the confluence-bifurcation unit takes a more uniform distribution. Even though, TKE profiles at the corresponding location exhibit highly similar shapes in the confluence-bifurcation unit and the single confluence, suggesting that the effects of channel morphology seem to be inhibited when the central bars are submerged under the high discharge.
4.2.2. Comparison with single bifurcation cases
Distributions of time-averaged streamwise velocity, TKE and TDR at corresponding cross-sections are also compared between the single bifurcation (see the Supplementary material) and the confluence-bifurcation unit (Figures 4, 8 and 9). Unlike the high similarity in flow characteristics exhibited between the confluence-bifurcation unit and the single confluence, significant differences are found between the confluence-bifurcation unit and the single bifurcation, especially at CS08∼CS14. On the one hand, the high-velocity zones are broader and asymmetrical concerning the channel centreline in the single bifurcation, with a belt-like and an approximately elliptic-like distribution respectively under the low and high discharges. By contrast, the high-velocity zone is a core that concentrates on the channel centre in the confluence-bifurcation unit. Moreover, the maximum velocity seems smaller in the single bifurcation than that in the confluence-bifurcation unit. On the other hand, the high-TKE belt near the channel bottom appears to be narrower in the single bifurcation than in the confluence-bifurcation unit, especially at CS08∼CS14 under the low discharge. Furthermore, additional high-TKE zones are found near the side walls at CS08∼CS11 in the single bifurcation, of which the scale is obviously smaller than those in the confluence-bifurcation unit. In addition, TKE at the channel centre is smaller near the free surface in the single bifurcation than that in the confluence-bifurcation unit. Nevertheless, the distributions of velocity, TKE and TDR seem to be similar in the confluence-bifurcation unit and the single bifurcation at CS15. As the existence of the upstream central bar is the main difference between the single confluence and the confluence-bifurcation unit, all the above findings indicate that the upstream central bar greatly influences the flow structure in the confluence-bifurcation unit. On the other hand, the downstream central bar may have a restricted influence on the flow structure in the confluence-bifurcation unit, whose impact may be limited to a range between the entrance of the bifurcation (CS14) and the downstream bar head (CS15). To further elucidate the differences, results on the distribution of time-averaged streamwise velocity and TKE along the flow depth are discussed below.
4.2.2.1. Time-averaged streamwise velocity
Figure 13 shows the distribution of time-averaged streamwise velocity along the flow depth at different cross-sections. Under the low discharge, distinct distribution patterns of flow velocity between the confluence-bifurcation unit and the single bifurcation are found at CS08, CS09 and CS11, which can be attributed to the effects of upstream flow convergence (Figure 13(a–c)). However, when the flow convergence reaches a steady state in the confluence-bifurcation unit (Figure 13(d–f)), the high-velocity zone is more concentrated in the confluence-bifurcation unit than in the single bifurcation due to to the significant influence of the upstream central bar on the flow structure. The velocity profiles at the downstream bar head can be a shred of evidence as well, with the maximum velocity larger at the channel centre but smaller near the side banks in the confluence-bifurcation unit than in the single bifurcation.
Figure 13. Comparison of the distribution of time-averaged streamwise flow velocity along the flow depth at different cross-sections between the confluence-bifurcation unit and the single bifurcation. (a)∼(f) 1a vs. 1c, (g)∼(l) 2a vs. 2c.
Under the high discharge, the distribution of velocity seems to exhibit limited differences between the two kinds of morphology, which indicates that the effects of channel morphology may be less noticeable when the central bars are fully submerged under the high discharge. Nevertheless, the velocity in the lower flow (below a relative depth of 0.45) shows a uniform lateral distribution in the single bifurcation, as the velocity profile at the channel centreline (α = 0.5) is in line with that near the side banks (α = 0.7) (Figure 13(g–l)). However, in the confluence-bifurcation unit, different velocity distributions in the lower flow can be observed at the channel centreline (α = 0.5) and near the side banks (α = 0.7). The foregoing results indicate that when the central bars are fully submerged, the high-velocity zones are more concentrated on the channel centreline in the confluence-bifurcation unit, while the lateral distribution of flow velocity within the single bifurcation tends to be more uniform, especially near the bifurcation junction (Figure 13(j,k)). This can also be attributed to the dominant influence of the upstream central bar over the downstream central bar.
It is also noted that the flow velocity distribution along the flow depth in the confluence-bifurcation unit is of a similar pattern despite varying discharges. As a critical point, the maximum velocity appears in the upper flow. The distribution above the critical point is approximately linear whereas it appears logarithmic below. By contrast, despite the similarity observed under the low discharge, the flow velocity distribution along the flow depth within the single bifurcation exhibits a distinct pattern under the high discharge, especially near the side banks (Figure 13(e–h)). On the one hand, the critical point in the upper flow no longer corresponds to the maximum velocity. On the other hand, the velocity distribution deviates from logarithmic below the critical point, with the maximum velocity appearing at a relative depth of 0.45. Succinctly, the distribution of streamwise velocity along the flow depth may retain the same pattern regardless of discharge levels in the confluence-bifurcation unit, while it may exhibit distinct patterns under different discharge levels in the single bifurcation.
4.2.2.2. Secondary current
Figure 14 shows contour plots of SCS and the distribution of secondary current for single bifurcation cases. In general, the value of SCS near the side banks at CS08∼CS14 (Figure 14(a–d, g–j)) in the single bifurcation seems smaller than that in the confluence-bifurcation unit (Figure 7(a–d, g–j)), especially under the low discharge. SCS distribution at CS14 is similar in the confluence-bifurcation unit and the single bifurcation under both low and high discharges. This difference in SCS distribution between the confluence-bifurcation unit and the single bifurcation indicates that the downstream bifurcation may have a restricted influence on the flow structure in the confluence-bifurcation unit. This influence is limited to a range between the entrance of the bifurcation (CS14) and the downstream bar head (CS15).
Figure 14. Secondary current at different cross-sections in the single bifurcation condition: (a)∼(f) case 1c, (g)∼(l) case 2c. The zero distance of each cross-section is located on the right bank.
In addition, the secondary current structure may also present different patterns in response to varying channel morphologies and discharge conditions. Under the low discharge condition, multiple unstable helical cells with asymmetrical distribution are formed from CS08 to CS12 in the single bifurcation (Figure 14(a–d)), while no obvious helical cells are found at CS14 and CS15 (Figure 14(d,e)). These findings are quite different from the stable and symmetrical helical cells at all cross-sections shown in the confluence-bifurcation unit (Figure 7). This difference may be attributed to the significant influence of the upstream central bar and the limited influence of the downstream central bar. Under the high discharge condition, only one pair of premature helical cells are found from CS08 to CS12 in the single bifurcation with their cores located near the side banks (Figure 14(e,f)). As the flow moves downstream, the helical cells gradually develop and become well-established (Figure 14(g,h)). These helical cells in the single bifurcation show symmetric cross-sectional distribution and a similar longitudinal development as in the confluence-bifurcation unit. However, in the confluence-bifurcation unit, two pairs of helical cells appear upstream of CS12 and CS14 and gradually fuse to one pair under the high discharge. As the ‘two-pairs’ structure in the confluence-bifurcation unit origins from the upstream confluence, the differences in the secondary current structure between the single bifurcation and the confluence-bifurcation unit under the high discharge can also be owing to the effects of the upstream central bar in excess of those of the downstream central bar.
4.2.2.3. Turbulent kinetic energy
Figure 15 shows the TKE distribution along the flow depth at different cross-sections. Under the low discharge, when the two tributary flows have not achieved sufficient convergence in the confluence-bifurcation unit, the maximum TKE is more concentrated in the single bifurcation (Figure 15(a–c)). As flow convergence achieves a steady state, more concentrated high-TKE zones appear at the channel centre within the confluence-bifurcation unit, confirming the finding that the effects of the upstream central bar reign over those of the downstream central bar in the confluence-bifurcation unit. However, things can be very complicated under the high discharge. For TKE distribution at the channel centreline, two peaks appear in the confluence-bifurcation unit with one close to the free surface and the other near the bed (Figure 15(g–l)). By contrast, only one peak near the bed is present in the single bifurcation. Therefore, a larger TKE can be found in the upper flow of the channel centreline in the confluence-bifurcation unit. For TKE distribution near the side banks, two peaks appear in both the confluence-bifurcation unit and the single bifurcation at CS09∼CS14 (Figure 15(h–l)). The upper peak is larger but the lower peak is smaller within the single bifurcation than those within the confluence-bifurcation unit. These significant discordances in TKE distribution under the high discharge further prove that the effects of the upstream bar on the flow structure in the confluence-bifurcation unit are more prominent than those of the downstream central bar.
Figure 15. Comparison of the distribution of TKE along the flow depth at different cross-sections between the confluence-bifurcation unit and the single bifurcation. (a)∼(f) 1a vs. 1c, (g)∼(l) 2a vs. 2c.
4.2.3. Further discussion of the comparisons
The above subsections have revealed significant differences in flow structure within the confluence-bifurcation unit and the single confluence and bifurcation, which directly result from the distinct channel morphologies and vary with the discharge conditions as well. These differences are summarized and further discussed below.
The distinctive morphology of a confluence-bifurcation unit plays a pivotal role in governing streamwise flow velocity distribution, secondary current structure, and turbulent kinetic energy distribution within the channel. Generally, from the upstream bar tail (CS08) to the entrance of the bifurcation (CS14), the flow structure in the confluence-bifurcation unit is highly similar to that in the single confluence, while it exhibits great differences (as shown in 4.2.2) between the confluence-bifurcation unit and the single bifurcation. This indicates that the upstream central bar greatly influences the flow structure in the confluence-bifurcation unit, with the effects spreading to the entrance of the bifurcation. At the downstream bar head (CS15), the flow structure (e.g. the transverse flow patterns) in the confluence-bifurcation unit exhibits high similarity to that in the single bifurcation. However, these similarities do not spread to upstream cross-sections, suggesting that the influence of the downstream central bar is limited at the bifurcation junction. In a word, the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit are in excess of those of the downstream central bar.
However, despite the influence of channel morphology, discharge may also have some important effects on the streamwise flow velocity distribution. On the one hand, when the central bars are exposed under the low discharge, the high-velocity zone is less concentrated in the confluence-bifurcation unit than in the single confluence, while it is more concentrated in the confluence-bifurcation unit than in the single bifurcation. On the other hand, it is noticed that when the central bars are fully submerged under the high discharge, reduced differences in flow structure between the confluence-bifurcation unit and the single confluence/bifurcation are witnessed, and thus the morphology effect seems to be subdued.
4.3. Implications
The present work unravels the flow structure in a laboratory-scale confluence-bifurcation unit and takes the first step to further investigating morphodynamics in such channel morphology. Based on the comparison with a single confluence/bifurcation, the findings provide insight into the complex 3D interactions between water flow and channel morphology. The distinct flow structure in the laboratory-scale confluence-bifurcation unit may appreciably alter sediment transport and morphological evolution, of which research is underway. As the basic morphological element of braided river planform is confluence-bifurcation units, the present work should have direct implications for flow structure in natural braided rivers. This is pivotal for the sustainable management of braided rivers which deals with water and land resources planning, eco-hydrological well-being, and infrastructure safety such as cross-river bridges and oil pipelines (Redolfi et al., 2019; Ragno et al., 2021).
Notably, braided rivers worldwide (e.g. in the Himalayas, North America, and New Zealand) have undergone increased pressures and will continue to evolve due to forces of global climate change and intensified anthropogenic activities (Caruso et al., 2017; Hicks et al., 2021; Lu et al., 2022). In particular, channel aggradation caused by increased sediment supply as well as exploitation of braidplain compromise space for flood conveyance, making the rivers prone to flooding. In this sense, an enhanced understanding of the flow structure under high discharge when central bars are fully submerged is essential for mitigating flooding hazards.
5. Conclusions
This study has numerically investigated the 3D flow structure in a laboratory-scale confluence-bifurcation unit based on the LES model integrated in the FLOW–3D® software platform. Two different discharges are considered with the central bars fully submerged or exposed respectively when the discharge is high or low. Cases of a single confluence/bifurcation are included for comparison. The key findings of this paper are as follows:
Several differences are highlighted in the comparison of the flow structure in the confluence-bifurcation unit between the two discharges. When the central bars are fully submerged under the high discharge, the mixing layer of two tributary flows is less obvious, and two high-velocity cores merge more rapidly as compared with those under the low discharge. Besides, flow separation zones are found neither at the confluence corner nor on both sides of the downstream bar when the central bars are fully submerged. Moreover, SCS seems to be smaller near the side banks under the high discharge than under the low discharge. Therefore, it is suggested that flow convergence/divergence is relatively weak in the confluence-bifurcation unit when central bars are fully submerged under the high discharge.
From the upstream bar tail to the entrance of the bifurcation, the flow structure in the confluence-bifurcation unit is highly similar to that in the single confluence, while it exhibits great differences from that in the single bifurcation. Only at the downstream bar head does the flow structure in the confluence-bifurcation unit exhibit high similarity to that in the single bifurcation. Consequently, the effects of the upstream central bar on the flow structure in the confluence-bifurcation unit reign over those of the downstream central bar.
Despite the influence of channel morphology, discharge may also have significant effects on the distribution of streamwise flow velocity. On the one hand, when the central bars are exposed under the low discharge, the high-velocity zone is less concentrated in the confluence-bifurcation unit than in the single confluence, while it is more concentrated in the confluence-bifurcation unit than in the single bifurcation. On the other hand, when the central bars are fully submerged under the high discharge, reduced differences in flow structure between the confluence-bifurcation unit and the single confluence/bifurcation are witnessed, and thus the morphology effect seems to be subdued.
It is noticed that the effects of other factors (e.g. confluence and bifurcation angles, bed discordance) on the flow structure in the confluence-bifurcation unit are not discussed here. Studies on these issues are warranted and reserved for future work.
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Turbidity currents are important carriers for transporting terrestrial sediment into the deep sea, facilitating the transfer of matter and energy between land and the deep sea. Previous studies have suggested that turbidity currents can exhibit high velocities during their movement in submarine canyons. However, the maximum vertical descent velocity of high-concentration turbid water simulating turbidity currents does not exceed 1 m/s, which does not support the understanding that turbidity currents can reach speeds of over twenty meters per second in submarine canyons. During their movement, turbidity currents can compress and push the water ahead, generating propagating waves. These waves, known as excitation waves, exert a force on the seafloor, resuspending bottom sediments and potentially leading to the generation of secondary turbidity currents downstream. Therefore, the propagation distance of excitation waves is not the same as the initial journey of the turbidity currents, and the velocity of excitation waves within this journey has been mistakenly regarded as the velocity of the turbidity currents. Research on the propagation velocity of excitation waves is of great significance for understanding the sediment supply patterns of turbidity currents and the transport patterns of deep-sea sediments. In this study, numerical simulations were conducted to investigate the velocity of excitation waves induced by turbidity currents and to explore the factors that can affect their propagation velocity and amplitude. The relationship between the velocity and amplitude of excitation waves and different influencing factors was determined. The results indicate that the propagation velocity of excitation waves induced by turbidity currents is primarily determined by the water depth, and an expression (v2 = 0.63gh) for the propagation velocity of excitation waves is provided.
Submarine turbidity currents, often referred to as underwater rivers, are important carriers that transport terrestrial sediments to the deep sea [1,2,3,4,5,6,7]. These turbidity currents, carrying a large amount of silt and sand, not only have strong erosive capabilities on the seabed [8,9,10], but also pose a threat to underwater communication cables, resulting in significant economic losses [11,12,13]. For example, the 2006 Pingdong earthquake in Taiwan caused the rupture of 11 submarine cables within the Kaoping Canyon, resulting in a slowdown in network speed in Southeast Asia for 49 days and requiring the deployment of 11 cable ships for repairs [13,14,15]. Investigating the velocity and patterns of turbidity currents in submarine canyons is of great significance for the protection of infrastructure such as pipelines and cables in these canyons. One of the main methods for quantitatively studying the velocity of turbidity currents in submarine canyons is to infer their speed through cable ruptures. The first confirmed occurrence of cable rupture caused by a turbidity current was in 1929, when the Grand Banks earthquake triggered the continuous rupture of 12 submarine cables. Inferred maximum turbidity current velocities reached 28 m/s [16,17,18]. Subsequently, multiple cable rupture incidents caused by turbidity currents have occurred worldwide. Table 1 summarizes the inferred maximum turbidity current velocities from these cable rupture incidents.
Event
Maximum Turbidity Velocity
References
18 November 1929 Grand Banks earthquake
28 m/s
[16,19,20,21]
1953 Suva earthquake in the Fiji Islands
5.1 m/s
[22]
The Orleansville earthquake of 9 September 1954, Algeria
20.6 m/s
[23]
Earthquake, Solomon Islands, Western Pacific, 23 December 1966
10.3 m/s
[24]
Incident at Nice airport, France, 16 October 1979
7 m/s
[25]
Taitung earthquake, 22 August 2002
9.8 m/s
[26]
21 May 2003 earthquake in Algeria
15.8 m/s
[27]
The Taitung earthquake of 10 December 2003
16.5 m/s
[26]
The Taitung earthquake of 18 December 2003
18.6 m/s
[26]
Pingtung earthquake on 26 December 2006
20 m/s
[28]
Typhoon Morakot on 7–9 August 2009
16.6 m/s
[29]
The 15 January 2022 eruption of Hunga volcano
33.9 m/s
[30]
Table 1. Cable breakage events caused by turbidity currents worldwide.
Previous studies have shown that the maximum vertical velocity of high-concentration turbidity currents in water does not exceed 1 m/s, and the maximum downward velocity of spherical particles in water does not exceed 10 m/s [31]. The maximum velocity of professional athlete Usain Bolt in the 100 m sprint on land is 9.58 m/s, while dolphins in the ocean can reach speeds of up to 20 m/s. Deep-sea turbidity currents, characterized by a small density difference compared to water, are primarily driven by the gravitational component along the direction of flow. However, factors such as bed friction also need to be considered. The driving force behind turbidity currents is primarily the density difference between the turbulent flow and the surrounding water, as well as the gravitational downslope component. Previous studies have detected a maximum sediment concentration of 12% in the basal layer of turbidity currents [32]. However, even high concentrations of suspended sediment, such as 1720 g/L, in seawater with a density of 1020 g/L, do not exceed a maximum vertical velocity of 1 m/s [33]. Similarly, spherical particles also have a maximum settling velocity in water of less than 10 m/s [33]. Turbidity currents, being density-driven flows, have relatively low density differences compared to water, and the gentle slope of submarine canyons also contributes to a smaller gravitational downslope force. Additionally, the influence of bed friction and other factors related to sediment deposition needs to be considered. It is incredible to think that turbidity currents can achieve flow velocities as high as 28 m/s [16,18,28,34,35]. When submarine landslides occur on continental slopes, the sliding mass entering the bottom of submarine canyons can cause the destruction of soft sediment beds. The mixing of sliding or flowing sediment with water forms turbidity currents. Turbidity currents exert pressure and propel the water ahead, forming an excitation wave. This aligns with Paull’s hypothesis that in the course of turbidity currents, a high-pressure zone is formed ahead, capable of causing an increase in pore water pressure in the sediment ahead [36]. Similar to surging waves, the excitation waves generated can propagate downstream along the submarine canyon, with a propagation velocity much greater than the velocity of turbidity currents [31]. The rapid propagation of excitation waves can exert a force on the seafloor of the submarine canyon, causing the resuspension of sediment in front of the head of the turbidity currents, which may lead to the formation of secondary turbidity currents at some downstream locations. The distance between the secondary and initial turbidity currents is actually the propagation distance of the excitation waves, rather than the journey of the initial turbidity currents. Therefore, the speed of the excitation waves within this distance is mistakenly considered as the velocity of the turbidity currents (see Figure 1). This may explain why the velocity of the turbidity currents as deduced from cable breakages is so high.
Figure 1. Diagram of excitation wave propagation due to turbidity current (v1 is the velocity of turbidity current. This refers to the ratio of distance to time experienced by a turbidity current mass moving underwater. v2 is the velocity of secondary turbidity current: the rapidly propagating excitation wave applies a force on the submarine canyon floor, leading to the destruction of the soft sediment floor and the secondary turbidity current. v is the propagation velocity of the excitation wave; this refers to the propagation velocity of the turbidity current excitation wave. This speed is not the velocity of the motion of the water mass. At time t0, the initial turbidity current moves underwater, pushing the stationary water in front to generate an excitation wave. At time t1, the excitation wave is propagating. At time t2, the rapidly propagating excitation wave exerts pressure on the soft bottom bed, resulting in the destruction of the bottom bed and secondary turbidity current).
Turbidity currents are mass movements composed of sediment particles, with a high concentration of the dense basal layer near the seabed. Depending on their density and granulometric composition, turbidity currents can move along submarine canyons through mechanisms such as diffusion, collapse, and flow [37], which differ from the downward movement as a single entity of landslide bodies after slope failure (this distinguishes them from surges). Additionally, during the long-distance movement of turbidity currents in canyons, the completion of subsequent water replenishment may generate multiple excitation waves. Furthermore, secondary excitation waves may also occur during the movement of secondary turbidity currents triggered by the initial turbidity current, which differs significantly from the surges caused by submarine landslides. Furthermore, previous studies [38,39,40,41] on sediment supply during turbidity current movements have mostly focused on the scouring action on the seabed, whereas the resuspension of sedimentary deposits in front of the initial turbidity current caused by excitation waves may serve as an effective mode of sediment supply during the long-distance transport of turbidity currents. In 2023, Ren et al. proposed that the cause of the long-distance high-speed motion of turbidity currents is due to the excitation waves caused by the primary turbidity currents. However, only preliminary research has been conducted on the comparison of excitation wave velocity and solitary wave velocity, and there has been no specific discussion on the reasons for the excitation wave velocity being much greater than that of the turbidity current. In an experiment conducted using an indoor flume, it was observed that the wavelength of the excitation waves was much larger than the water depth, similar to shallow water waves [33]. The amplitude of excitation waves in proportion to their wavelength was small, consistent with the theory of small-amplitude waves. Similar to the velocity model of shallow water waves, it is expected that the propagation speed of excitation waves is also influenced by the water depth. However, since excitation waves are triggered by sediment-laden turbidity currents, the velocity model may differ from that of surface waves induced by gravitational flows. The purpose of this study is to simulate and investigate the effects of different factors on the propagation velocity and amplitude of excitation waves through a validated numerical model based on laboratory experiments. The study aims to determine the maximum propagation velocity of excitation waves at a field scale and whether there is attenuation in the long-distance propagation after their formation. In recent studies, seafloor sediment flows have been collectively referred to as turbidity currents [42]. Therefore, we simulated the movement of turbidity currents by sediment flow. This study uses the CFD-based fluid computation software FLOW-3D to simulate the underwater movement process of turbidity currents. The numerical model is validated against indoor experimental results. During the simulation process, a velocity model for surging wave generation triggered by submarine landslides is used as a reference, and multiple factors that may affect the propagation velocity of the excitation wave are considered. By controlling a single variable, the main factors influencing the excitation wave propagation velocity are determined, and the corresponding expression for excitation wave propagation velocity is provided. The results indicate that the propagation velocity of the excitation wave induced by turbidity currents is primarily determined by the water depth. This research provides a new perspective for understanding the high-speed movement of turbidity currents in submarine canyons and enriches the understanding of the movement patterns of turbidity currents in submarine canyons. In addition, studying the propagation speed of excitation waves is highly significant for the resuspension of underwater sediments, as well as the re-circulation of carbon sequestration, nutrients, heavy metals, and microplastics.
2. Experimental Study on Excitation Waves Induced by Turbidity Currents
2.1. Experimental Design and Apparatus
The experimental apparatus used for the turbidity current-induced excitation wave tests is a straight water tank [33]. The water tank is 12.5 m long, 0.5 m wide, and 0.7 m high. A turbidity source area is located on the right side of the tank to generate turbidity currents. The tank is equipped with a terrain with a certain slope. Turbidity currents are generated underwater using a weir. The mass ratio of silt and clay used in the experimental turbid water solution was 8:2, with a density of 1600 kg/m3. Previous experiments have shown that this turbid mixture can reach a maximum flow velocity of 18.7 cm/s [31]. Three pressure sensors are placed along the straight section of the tank at intervals of 0.4 m. These sensors continuously monitor the bottom shear stress caused by the turbidity current-induced excitation wave, as well as the force exerted by the turbidity current itself on the bed. The monitoring frequency is set at 100 Hz.
2.2. Experimental Phenomenon and Results
In the laboratory water tank experiments, it was observed that as the turbidity current propagates, a wave is generated ahead of the turbidity front, moving in the same direction as the current and with a velocity greater than the turbidity current velocity [33]. By monitoring the pressure changes on the bed during the turbidity current motion [33], the propagation velocity of the excitation wave, the head movement velocity of the turbidity current, and the amplitude of the excitation wave (obtained from the measured surface elevation changes caused by the wave) can be estimated based on the distances between the sensors and the time when the pressure change peaks occur. The results of indoor experiments on turbidity currents indicate that they can compress and propel the water ahead of them, generating excitation waves similar to pulses. The propagation speed of these excitation waves caused by turbidity currents is found to be much greater than the velocity of the turbidity current movement at its head, as determined by pressure sensors installed on the seabed.
3. Numerical Simulation of Excitation Waves Induced by Turbidity Currents
FLOW-3D is a powerful computational fluid dynamics (CFD) software that excels in making accurate calculations of free surface and six-degrees-of-freedom motions of objects. Similar to other CFD software, FLOW-3D consists of three modules: pre-processing, solver, and post-processing. In recent years, there have been many simulations of turbidity currents using FLOW-3D due to its superior capabilities. For example, Heimsund (2007) simulated turbidity currents in the Monterey Canyon system using FLOW-3D based on high-resolution bathymetry and flow data [43]. Zhou et al. (2017) used FLOW-3D software to simulate turbidity currents in a flume with obstacles, analyzing the impact of the proportion between obstacle height and flume height on the movement of turbidity currents, including their velocity, flow state, and morphological evolution [44]. In this study, using the CFD software FLOW-3D, the underwater motion process of turbidity currents is simulated. The model is validated by comparing it with experimental results, and the motion of the waves induced by turbidity currents is simulated based on this validation.
3.1. Control Equations
FLOW-3D, a mature three-dimensional fluid simulation software, is used in this study. It employs the RNG turbulence model, which is capable of handling high strain rate flows and is suitable for simulating excitation waves. The research focus of this paper is on sediment gravity flows (turbulent flows), and the control equations used in the calculations include the basic continuity equation, the momentum equation, the turbulent kinetic energy k equation, and the turbulent kinetic energy dissipation rate ε equation.
The continuity equation:
The momentum equation:
The turbulence model:
k equation:
ε equation:
where u, v and w is the flow velocity component in x, y and z directions; Ax, Ay and Az represent the area fraction that can flow in x, y and z directions; Gx, Gy and Gz are the gravitational acceleration in x, y and z directions; fx, fy and fz are the viscous forces in the three directions; VF is the fraction of the volume that can flow; ρ is the fluid density; p is the pressure acting on the fluid element; k is the turbulence energy; ε is the turbulence kinetic energy dissipation rate; μ is turbulence viscosity coefficient
where u, v and w is the flow velocity component in x, y and z directions; Ax, Ay and Az represent the area fraction that can flow in x, y and z directions; Gx, Gy and Gz are the gravitational acceleration in x, y and z directions; fx, fy and fz are the viscous forces in the three directions; VF is the fraction of the volume that can flow; ρ is the fluid density; p is the pressure acting on the fluid element; k is the turbulence energy; ε is the turbulence kinetic energy dissipation rate;
μ is turbulence viscosity coefficient where Cμ = 0.0845;
Gk is the turbulent kinetic energy generation term, expressed as
and σk and σε are the Prandtl numbers corresponding to the turbulent kinetic energy and dissipation rate, respectively, both of which are 1.39.
In addition,where Cε1 and Cε2 are the empirical constants, 1.42 and 1.68, respectively.
Furthermore,
where , η0 = 4.377, β = 0.012.
The general mass continuity equation is as follows:
where VF is the fractional volume open to flow, ρ is the fluid density, RDIF is a turbulent diffusion term, and RSOR is the mass source.
3.2. Model Validation
To determine the factors affecting the velocity of the turbidity-induced excitation wave and its velocity expression, first, the indoor flume test was taken as the prototype. Then, a 1:1 geometric solid model was established, and the simulation parameters were set to be consistent with the flume test parameters [33]. Finally, the simulation results were compared with the laboratory test results.
The computational domain employs the method of unstructured grid and is entirely divided into structured orthogonal grids. Nested grids are used for local refinement at the interfaces of straight sections, resulting in a total of 800,000 grid cells after refinement.
The simulation results were compared with the indoor experimental results, with the velocity of the excitation wave and the turbidity current head being represented by changes in surface elevation and water density. The experimental and simulation results are shown in Table 2, and the calculation formula for the error is |Calculated value−Test value|Test value×100%Calculated value-Test valueTest value×100%.
Result
Propagation Velocity of Excitation Wave (m/s)
Velocity of Turbidity Current (m/s)
Excitation Wave Amplitude (m)
Sensor 1 to 2
Sensor 2 to 3
Sensor 1 to 2
Sensor 2 to 3
Sensor 1 to 2
Sensor 2 to 3
Test results
1.54
1.48
0.24
0.23
0.029
0.03
Computed results
1.55
1.52
0.25
0.23
0.03
0.03
Error range
0.6%
2.7%
4.2%
0%
3.4%
0%
Table 2. The test results of the propagation velocity of the excitation wave, the turbidity current velocity, and the excitation wave amplitude are compared with the simulation results.
From the above comparison, it can be observed that the simulated velocities of the excitation wave and the head of the turbidity current align well with the experimental results, indicating the rationality of using the numerical model established in this study for simulating the propagation velocity of the excitation wave induced by turbidity currents.
3.3. Analysis of Factors Affecting the Propagation Velocity of Excitation Waves
An analysis of the factors influencing the propagation velocity of excitation waves was conducted using numerical simulation. The reference model for wave velocity was based on the surge velocity model. The main factors affecting the propagation velocity of excitation waves were summarized, including the turbidity current density ρ, the thickness of the turbidity current source area d, the length of the turbidity current source area L, the depth at the initial flow of turbidity currents h, the canyon width l, and the initial velocity of the turbidity current v0 (as shown in Figure 2). The simulations were performed using a controlled variable approach for different parameters, and the velocity changes of the excitation wave were obtained, as shown in Table 3. The slope angle was fixed at 3°, and sensors were placed at intervals of 100 m starting from a distance of 500 m from the turbidity current source area (named Sensors 1, 2, 3). These sensors were used to extract surface elevation, density, and other relevant parameters at their respective locations. We can obtain the propagating velocity of excitation waves by measuring the time difference in surface elevation changes at the monitoring points. Similarly, we can determine the propagation velocity of turbidity currents by measuring the time difference in density changes.
Figure 2. Excitation wave velocity simulation model and parameters.
Group Order
Turbidity Current Density (kg/m3)
Length of Turbidity Source Area (m)
Canyon Width (m)
Thickness of Turbidity Source Area (m)
Depth (m)
Initial Velocity of Turbidity Current (m/s)
Propagation Velocity of Excitation Wave (m/s)
Excitation Wave Amplitude (m)
Velocity of Turbidity Current (m/s)
1
1600
1000
200
20
200
0
33.43
0.345
5.88
2
1500
1000
200
20
200
0
33.09
0.304
5.41
3
1400
1000
200
20
200
0
33.35
0.223
4.99
4
1300
1000
200
20
200
0
33.33
0.177
4.35
5
1200
1000
200
20
200
0
33.86
0.092
3.74
6
1600
1000
200
40
200
0
33.05
1.109
9.09
7
1600
1000
200
60
200
0
33.39
2.689
10.79
8
1600
1000
200
80
200
0
33.21
4.828
12.91
9
1600
1000
200
100
200
0
36.43
7.744
13.79
10
1600
200
200
20
200
0
32.93
0.181
5.58
11
1600
400
200
20
200
0
33.49
0.25
5.71
12
1600
600
200
20
200
0
33.06
0.278
5.79
13
1600
800
200
20
200
0
33.17
0.31
5.72
14
1600
1000
200
20
100
0
26.67
0.56
5.72
15
1600
1000
200
20
300
0
39.65
0.169
5.80
16
1600
1000
200
20
400
0
45.98
0.12
5.80
17
1600
1000
200
20
500
0
49.97
0.08
5.96
18
1600
1000
100
20
200
0
33.60
0.354
5.72
19
1600
1000
300
20
200
0
32.98
0.338
5.97
20
1600
1000
400
20
200
0
33.27
0.356
5.87
21
1600
1000
500
20
200
0
33.31
0.365
5.86
22
1600
1000
200
20
200
2
33.50
0.532
4.35
23
1600
1000
200
20
200
5
33.12
1.389
6.56
24
1600
1000
200
20
200
8
33.52
2.271
8.10
25
1600
1000
200
20
200
10
33.33
2.878
8.99
Table 3. Simulation results under different variables conditions.
The variations in surface elevation at three sensor locations in the simulated results of five different turbidity current density groups are presented in Figure 3.
Figure 3. Simulation of propagating velocity of excitation wave under the sole variable condition of turbulent current density. (Length of turbidity source area: 1000 m; canyon width: 200 m; thickness of turbidity source area: 20 m; depth: 200 m; initial velocity of turbidity current: 0 m/s).
Based on the simulation results described above, while keeping all other conditions constant, the impact of a single variable, namely, the turbidity current density, on the propagation velocity and amplitude of the excitation wave was analyzed. By fitting the data, the relationship between turbidity current density and the propagation velocity of turbidity currents as well as the amplitude of the excitation wave was obtained, as shown in Figure 4.
Figure 4. Relationship between turbidity current density and turbidity current velocity, as well as excitation wave amplitude.
The simulation results indicate that changes in turbidity current density, while keeping the other conditions constant, do not result in a change in the propagation velocity of the excitation waves. However, they do affect the amplitude of the excitation waves and the velocity of the turbidity current itself. The simulation reveals that within the selected density range, both the amplitude of the excitation waves and the velocity of the turbidity current increase with increasing turbidity current density. When the turbidity current density is equal to that of water (ρTurbidity current = ρWater), there is no turbidity current or excitation wave generation. Thus, the relationship between the turbidity current velocity (v) and density (ρ) is expressed as v = −34.80643 + 0.05082•ρ − 1.59286 × 10−5ρ2 (ρ > 1000, R2 = 0.994). Additionally, the relationship between the amplitude of the excitation waves (A) caused by turbidity currents and density (ρ) is expressed as A = −0.6021 + 5.9729 × 10−4ρ (ρ > 1000, R2 = 0.991).
3.3.2. The Influence of the Thickness of the Turbidity Source Area on the Propagation Velocity and Amplitude of Excitation Waves
The variations in surface elevation at three sensor locations in the simulated results of five different thickness of turbidity source area groups are presented in Figure 5.
Figure 5. Simulation of propagating velocity of excitation wave under the sole variable condition of thickness of turbidity source area. (Turbidity current density: 1600 kg/m3; length of turbidity source area: 1000 m; canyon width: 200 m; depth: 200 m; initial velocity of turbidity current: 0 m/s).
Based on the simulation results described above, while keeping all other conditions constant, the impact of a single variable, namely, the thickness of the turbidity source area, on the propagation velocity and amplitude of the excitation wave was analyzed. By fitting the data, the relationship between the thickness of the turbidity source area and the propagation velocity of the turbidity current as well as the amplitude of the excitation wave was obtained, as shown in Figure 6.
Figure 6. Relationship between thickness of turbidity source area and turbidity current velocity, as well as excitation wave amplitude.
Based on the simulated results mentioned above, it can be concluded that, while keeping the other conditions constant, changing only the thickness of the turbidity current source area does not affect the propagation velocity of the excitation waves. However, it does impact both the amplitude of the excitation waves and the velocity of the turbidity current itself. The simulation reveals that within the selected range of thickness values for the turbidity current source area, both the amplitude of the excitation waves and the velocity of the turbidity current increase with an increase in the thickness of the source area. Additionally, it is observed that when the length of the turbidity current source area is zero, neither the turbidity current nor the excitation waves are generated (i.e., no turbidity current is produced when hTurbidity current = 0). Therefore, the relationship between the velocity (v) of the turbidity current and its thickness (h) is expressed as v = 0.27983•h − 0.00146•h2 (h ≥ 0, R2 = 0.999). Similarly, the relationship between the amplitude (A) of the excitation waves caused by the turbidity current and its thickness (h) is A = −0.00375•h − 0.0008•h2 (h ≥ 0, R2 = 0.999).
3.3.3. The Influence of the Length of the Turbidity Source Area on the Propagation Velocity and Amplitude of Excitation Waves
The variations in surface elevation at three sensor locations in the simulated results of five different length of turbidity source area groups are presented in Figure 7.
Figure 7. Simulation of propagating velocity of excitation wave under the sole variable condition of length of turbidity source area. (Turbidity current density: 1600 kg/m3; canyon width: 200 m; thickness of turbidity source area: 20 m; depth: 200 m; initial velocity of turbidity current: 0 m/s).
Based on the simulation results described above, while keeping all other conditions constant, the impact of a single variable, namely, the length of the turbidity source area, on the propagation velocity and amplitude of the excitation wave was analyzed. By fitting the data, the relationship between the length of the turbidity source area and the amplitude of the excitation wave was obtained, as shown in Figure 8.
Figure 8. Relationship between length of turbidity source area and excitation wave amplitude.(Amplitude refers to the surface elevation change caused by the excitation wave).
Through simulations, it has been determined that within the chosen range of the length of the turbidity source area, the amplitude of the excitation waves increases with an increase in the length of the turbidity source area. When the length of the turbidity source area is zero, there is no turbidity current and no generation of excitation waves (i.e., when LTurbidity current = 0). Additionally, for large lengths of the turbidity source area, under the condition of sufficient sediment supply, the variations in surface elevation caused by the waves generated by turbidity currents are negligible. Therefore, the relationship between the amplitude of the excitation waves (A) generated by turbidity currents and the length of the turbidity source area (L) is expressed as follows: A = −0.3624 + 0.10305•ln(L − 6.15619) (L ≥ 0, R2 = 0.997).
3.3.4. The Influence of Depth on the Propagation Velocity and Amplitude of Excitation Waves
The variations in surface elevation at three sensor locations in the simulated results of five different depth groups are presented in Figure 9.
Figure 9. Simulation of propagation velocity of excitation wave under the sole variable condition of depth. (Turbidity current density: 1600 kg/m3; length of turbidity source area: 1000 m; canyon width: 200 m; thickness of turbidity source area: 20 m; initial velocity of turbidity current: 0 m/s).
Based on the simulation results described above, while keeping all other conditions constant, the impact of a single variable, namely, depth, on the propagation velocity and amplitude of the excitation wave was analyzed. By fitting the data, the relationship between depth and the propagation velocity of the excitation wave as well as the amplitude of the excitation wave was obtained, as shown in Figure 10.
Figure 10. Relationship between depth and propagating velocity of excitation wave, as well as excitation wave amplitude.
As the water depth approaches infinity, the excitation wave amplitude can only approach zero but cannot reach zero. Therefore, the characteristics of the excitation wave amplitude change with the water depth are similar to those of the velocity propagation of the excitation wave. The relationship between the velocity of the excitation wave induced by turbidity currents (vExcitation wave) and the water depth (H) can be described as vExcitation wave = −287.05446 + 48.59211•ln(H + 535.14863) (R2 = 0.998). The relationship between the excitation wave amplitude (A) and the water depth (H) can be expressed as A = 1.46573 − 0.22816•ln(H − 47.67563) (R2 = 0.985).
3.3.5. The Influence of the Canyon Width on the Propagation Velocity and Amplitude of Excitation Waves
The variations in surface elevation at three sensor locations in the simulated results of five different canyon width groups are presented in Figure 11.
Figure 11. Simulation of propagating velocity of excitation wave under the sole variable condition of canyon width. (Turbidity current density: 1600 kg/m3; length of turbidity source area: 1000 m; thickness of turbidity source area: 20 m; depth: 200 m; initial velocity of turbidity current: 0 m/s).
When the canyon width is taken as the single variable condition, changing the canyon width does not significantly affect the propagation velocity of excitation waves, the amplitude of excitation waves, and the velocity of turbidity currents. Therefore, it can be concluded that, without considering the impact of the differences in the terrain and sediment on the canyon width, the canyon width has no impact on the propagation of excitation waves and the movement of turbidity currents.
3.3.6. The Influence of the Initial Velocity of the Turbidity Current on the Propagation Velocity and Amplitude of Excitation Waves
The variations in surface elevation at three sensor locations in the simulated results of five different initial velocity of turbidity current groups are presented in Figure 12.
Figure 12. Simulation of propagating velocity of excitation wave under the sole variable condition of initial velocity of turbidity current. (Turbidity current density: 1600 kg/m3; length of turbidity source area: 1000 m; canyon width: 200 m; thickness of turbidity source area: 20 m; depth: 200 m).
Based on the simulation results described above, while keeping all other conditions constant, the impact of a single variable, namely, the initial velocity of the turbidity current, on the propagation velocity and amplitude of the excitation wave was analyzed. By fitting the data, the relationship between the initial velocity of the turbidity current and the amplitude of the excitation wave was obtained, as shown in Figure 13.
Figure 13. Relationship between initial velocity of turbidity current and excitation wave amplitude.
Based on the simulation, it is observed that within the selected range of the initial velocity of the turbidity current, the amplitude of the excitation wave increases linearly with the increase in the initial velocity of the turbidity current. Therefore, the relationship between the amplitude (A) of the excitation wave caused by the turbidity current and the initial velocity of the turbidity current (v0) can be expressed as A = 0.34 + 0.24084•v0 (A ≥ 0, R2 = 0.992).
Through controlling the simulation calculation of a single variable, it was found that there are several factors that can affect the amplitude of the excitation wave. These factors include the turbidity current density ρ, the thickness of the turbidity current source area d, the length of the turbidity current source area L, the water depth h, and the initial velocity of the turbidity current v0. In contrast, there are relatively few factors that influence the propagation velocity of the excitation wave. Within the selected parameter range, only the water depth can affect the propagation velocity of the excitation wave. The physical parameters of the turbidity current, including the turbidity current density ρ, the thickness of the turbidity current source area d, the length of the turbidity current source area L, the canyon width l, and the initial velocity of the turbidity current v0, have no direct influence on the propagation velocity of the excitation wave. Therefore, the turbidity current only serves as a triggering factor for the excitation wave and is not directly related to the propagation velocity of the excitation wave.
3.4. Analyze the Changes in Propagation Velocity of Excitation Waves along a Path
In order to further investigate the underlying truth behind the variation in the propagation velocity of the excitation wave, a discussion on whether there is velocity attenuation along the propagation path of the excitation wave is conducted. Since the seventh group of the excitation wave causes significant changes in surface elevation, the seventh group of the excitation wave is selected as the research object in order to study the variations in surface elevation along the propagation path of the excitation wave. The changes in surface elevation are extracted every 200 m along the sediment slope (with the first extraction point located 400 m away from the source area of the turbidity current). A total of six sets of surface elevation data are extracted (ranging from 400 m to 1400 m distance from the source area of the turbidity current), as shown in Figure 14.
Figure 14. Surface elevation changes during excitation wave propagation along sediment slopes.
The amplitudes and propagation velocities of the excitation wave at each point are shown in Table 4.
Distance from Turbidity Current Source Area (m)
Propagation Velocity of Excitation Wave (m/s)
Excitation Wave Amplitude (m)
400
33.34
2.524
600
36.79
2.596
800
37.13
2.589
1000
39.99
2.566
1200
40.04
2.542
1400
40.13
2.523
Table 4. Excitation wave velocity during the excitation wave propagation along the sediment slope.
From the table above, it can be observed that the amplitude of the excitation wave does not change while traveling along the slope. This indicates that the change in surface elevation caused by the propagation of the excitation wave does not attenuate. Furthermore, the propagation velocity of the excitation wave gradually increases, although the change is not very pronounced. This variation may be attributed to the change in the water depth caused by the sloping bed. To investigate this, a simulation was conducted in a straight channel with a length of 3000 m. Six sampling points were established from 400 m to 1400 m away from the turbidity current source area to extract the amplitude of the excitation wave. The results of the simulation are presented in Figure 15.
Figure 15. Surface elevation changes during wave propagation along a straight channel.
The amplitudes and propagation velocities of the excitation wave at each point are shown in Table 5.
Distance from Turbidity Current Source Area (m)
Propagation Velocity of Excitation Wave (m/s)
Excitation Wave Amplitude (m)
400
33.89
2.559
600
37.66
2.692
800
37.12
2.712
1000
36.92
2.717
1200
37.09
2.715
1400
37.48
2.718
Table 5. Excitation wave velocity during the propagation along the straight channel.
The data from the table above indicate that during the propagation of the excitation wave along a straight water channel, its velocity remains constant, except for a slight decrease at the initial point. This phenomenon may be attributed to the fact that in the starting phase, the excitation wave is not fully developed, and hence its velocity is relatively smaller. However, once it is fully developed, the propagation velocity of the excitation wave does not decrease in subsequent processes. Therefore, the propagation velocity of the excitation wave is only dependent on the real-time water depth of the wave. In future studies, we aim to explore the relationships between these influencing factors and other physical parameters, such as the speed of wave propagation, using the effective and accurate method of machine learning algorithms [45].
3.5. Expression of the Propagation Velocity of the Excitation Wave
The propagation of the excitation wave along a long distance does not experience an attenuation in velocity, as is the case with the propagation velocity of solitary waves. Referring to the estimated wave propagation velocity (the square of the propagation velocity is directly proportional to the water depth amplitude) [46], the wavelengths under different water depth conditions were extracted, as shown in Table 6.
Depth (m)
Propagation Velocity of Excitation Wave (m/s)
Excitation Wave Amplitude (m)
Excitation Wave Length (m)
100
26.67
0.56
2580
200
33.43
0.35
2850
300
39.65
0.17
3250
400
45.98
0.12
3600
500
49.97
0.08
4150
1000
66.67
0.04
6000
2000
90.91
0.02
9500
4000
165.84
0.3
17600
Table 6. Physical parameters of excitation wave under different water depth conditions.
From the simulation results of a single variable, the water depth, it could be seen that the wavelengths of the excitation waves were much larger than the water depth. Therefore, further simulations were conducted under water depth conditions ranging from 1000 m to 4000 m. Due to the minimal change in wave amplitude when the water depth reached 4000 m, it was not possible to observe a distinct waveform. However, through simulations with the thickness of the turbidity current source area as the single variable, it was found that an increase in the thickness of the source region led to a larger amplitude of the excitation waves, but it did not affect the wavelength of the excitation waves. Therefore, in order to better extract the wavelength of the excitation waves, the thickness of the source region in the simulation with a water depth of 4000 m was set to 200 m.
Through simulations at water depths of 1000 m and 4000 m, it is observed that the wavelengths of the excitation waves are much larger than the water depth, indicating that these waves belong to the category of shallow water waves. The amplitude of the excitation waves is relatively small compared to their wavelength, aligning with the small amplitude wave theory [47]. According to this theory, the wave velocity of shallow water waves is only dependent on the water depth (h) and gravity acceleration (g), regardless of the wave period. In the case of excitation waves induced by turbidity currents in deep water, the amplitudes of these waves are relatively small compared to the water depth. Referring to the expression for shallow water waves (when the relative water depth, which is the ratio of water depth to wavelength, is much smaller than 1/2), the wave velocity is denoted as 𝐶𝑠=√𝑔ℎ. This implies that the propagation velocity of the excitation waves is also solely related to the water depth. Therefore, a fitting of the square of the propagation velocity of the excitation waves (v2) and the water depth (h) was conducted (Figure 16).
Figure 16. The relationship between the propagation velocity of excitation wave and the depth.
Through fitting, the following can be obtained:
Through fitting, it can be discovered that the propagation model of the velocity of excitation waves is different from the shallow water wave theory. This is because turbidity currents, as granular materials, generate excitation waves by pushing the water in front of them with sediment particles underwater, which is different from the surges formed by solid blocks entering the ocean. Additionally, excitation waves formed by turbidity currents occur in an underwater environment, which may be the reason why the propagation velocity equation for the excitation waves behaves as if the velocity squared is equal to half the Earth’s gravity. This equation reveals the variation in the propagation velocity of the excitation wave with depth, explaining why the average velocity between the monitoring points in the field is greater than the instantaneous velocity measured at these points [41]. Further theoretical research on the propagation velocity of excitation waves requires subsequent field monitoring and the deployment of monitoring systems to more thoroughly investigate the fundamental causes.
4. Conclusions
This study aimed to investigate the velocity of turbidity current-induced excitation waves through numerical simulation. By fixing a single variable, different factors that could affect the propagation velocity and amplitude of the excitation waves were analyzed and discussed, leading to the following three conclusions:
Within the selected parameter range, there are several factors that can influence the amplitude of the excitation waves, including the turbidity current density ρ, the thickness of the turbidity current source area d, the length of the turbidity current source area L, the water depth h, and the initial velocity of the turbidity current v0.The amplitude of the excitation waves is positively correlated with the turbidity density, the thickness of the source area, the length of the source area, and the initial velocity, while it is negatively correlated with the water depth.
Within the selected parameter range, only the water depth can affect the propagation velocity of the excitation waves. As the water depth increases, the propagation velocity of the excitation waves also increases, and a relationship of v2 = 0.63gh (R2 = 0.967) is established between the square of the propagation velocity v2 and the water depth h.
During the propagation of the excitation waves, both the propagation velocity and the changes in surface elevation caused by the waves do not attenuate. Considering the relatively calm deep-sea environment, the high-speed propagation of the excitation waves and the resuspension of bottom sediments they cause not only complement the understanding of turbidity current motion patterns in canyons, but also provide new research directions for deep-sea sediment transport.
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Youkou Dong, Enjin Zhao, Lan Cui, Yizhe Li, Yang Wang
Abstract
Submarine pipelines are widely adopted around the world for transporting oil and gas from offshore fields. They tend to be severely ruined by the extreme waves induced by the natural disaster, such as hurricanes and tsunamis. To maintain the safety and function integrity of the pipelines, porous media have been used to wrap them from the external loads by the submarine environment. The functions of the porous wrappers under the hydrodynamic impact remain to be uncovered before they are widely accepted by the industry. In this study, a numerical wave tank is established with the immersed boundary method as one of the computational fluid dynamics. The submarine pipelines and their porous wrappers are two-way-coupled in terms of displacement and pressure at their interfaces. The impact from the solitary waves, which approximately represent the extreme waves in the reality, on the pipelines with different configurations of the porous wrapper is investigated. The results present significant protective functions of the wrappers on the internal pipelines, transferring the impact forces from the pipelines to the wrappers. The protective effects tend to be enhanced by the porosity and thickness of the wrappers. The influence of the pipeline configurations and the marine environment are then analysed. As for the front pipeline, an increase in the gap leads to a slight increase in the horizontal forces on both the wrapper and the pipeline, but a significant increase in the vertical forces. As for the rear pipeline, because of the shield function of the front pipeline, the velocity within the gap space and the forces on the pipes are decreased with the decrease in the gap size. The complex flow fields around the pipelines with wrappers are also illuminated, implying that the protection function of the wrapper is enhanced by the wave height reduction.
Pipelines that are laid on or below the seabed and continuously transport large amounts of oil (or gas) are collectively referred to as submarine pipelines. They constitute the main transporting structures and currently they are the most economical and reliable selections in the design of transportation tools. Pipelines are usually installed within the seabed sediments under the protection of rock berms [1]. However, the sediments around the pipelines may be scoured by contour currents and internal waves, which expose the pipelines to the threat of complex marine environments [2]. The scour mechanism and its evolution process around the in-position pipelines were investigated by many scholars, such as Reference [3]. Occasionally, segments of a pipeline may be suspended between high points through continental slopes due to an uneven seabed profile. For example, suspended pipelines were widely used in the Ormen Lange projects, with massive depressions and landslide blocks scattered along the 120-km-long route [4]. Natural disaster, such as hurricanes and tsunamis, may induce extreme waves that generate enormous impact loads on the pipelines and may cause serious ruins to the whole production and transportation system [5,6,7]. Tsunamis, one of the major marine disasters caused by earthquakes and submarine landslides [8,9], send surges of water with extremely long waves that are not especially steep [10]. The tsunami triggered by a 9.0-Mw earthquake in 2011 extensively destroyed 70% of the total 200,000 structures along the Miyagi coastline, including submarine pipelines, seawalls, and coastal bridges. A tsunami is typically composed of several transient waves with varying amplitudes, wave-lengths, and wave periods during propagation. Solitary waves were proposed to simulate the tsunami waves by decomposing them into N-waves through the Korteweg-de Vries equation [11,12,13,14]. Since then, the run-up process of the tsunami waves along the shoreline was investigated with the depth-averaged smooth particle hydrodynamics method [15,16]. References [17,18] quantified the impact loads over cylinders from a tsunami wave. To protect the marine structures from potential damages due to extreme marine conditions, engineers have developed outer protections in terms of wrappers made of porous media. A porous medium enhances the buffering performance of the structures and dissipates part of the incoming wave energy [19]. For example, the turbulent intensities on a permeable breakwater were significantly attenuated in the numerical analysis by References [20,21,22]. Naturally, porous media are expected to be protective to submarine pipelines under extreme marine conditions, although thermal insulation and erosion prevention were mainly considered in designing pipeline coatings in the industry [23,24]. Reference [25] quantified the wave forces on pipelines buried in an impermeable bed with coverings of porous media. References [26,27] evaluated the protective performance of a porous polymer coating on subsea pipelines under sudden impacts. The drag reduction function of the porous coatings over cylinders were then quantified by Reference [28]. Two factors were considered to influence the stabilization effect of the porous coatings on pipelines: the production of an entrainment layer through the coating and the triggering of turbulent transition of the detaching shear layers. In engineering practice, applications of porous coatings on submarine pipelines are limited. Concrete wrappers, mainly designed to counteract the buoyancy forces of pipelines, can be considered as one kind of porous wrapper with medium permeability. In addition, porous wrappers made with woven carbon-fiber materials or polyurethane foam may be designed in future for pipeline protection. The above literature review revealed that few studies were performed to examine the protective effect by the porous media on submarine pipelines, which is the main aim of this study. The porous wrapper and the submarine pipeline modules are simulated in a numerical wave tank (NWT) with the immersed boundary (IB) method. The numerical methods and equations will be provided in Section 2. Verification of the numerical model is provided in Section 3. The parametric simulations are in Section 4, in which the effects of different waves on various pipelines with porous wrappers are analysed. The conclusions are given in Section 5.
2. Numerical Methods
For simulating the interactions between pipelines and waves, the finite volume methods have been widely used. In this study, the commercial finite volume package FLOW-3D® (version 11.1.0; 2014; https://www.flow3d.com (accessed on 10 December 2022); Flow Science, Inc., Santa Fe, NM, USA). Flow-3D aims to solve the transient response of fluids under interactions with structures, internal and external loads and multi-physical processes. It features some advantages in terms of a high level of accuracy in solving the Navier-Stokes equation with the volume of fluid (VOF) method, efficient meshing techniques for complex geometries, and high efficiency level for large-scale problems. Also, Flow-3D provides the flexibility and utility for flowing through porous media. A two-dimensional numerical wave tank was constructed by using the immersed boundary (IB) method and an in-house subroutine termed as IFS_IB. A submarine pipeline and porous medium were two-way coupled at the interface described by the individual volume fractions [29]. The pipeline was wrapped with a layer of a porous medium. A solitary wave was generated at the inlet boundary of the tank to simulate an approaching tsunami. Non-slip wall conditions were assigned at the bottom of the tank and the pipe surface, which was also specified with a roughness coefficient. The top boundary was defined as a free boundary and configured with the atmospheric pressure. A Neumann-type absorbing boundary condition, a stable, local, and absorbing numerical boundary condition for discretized transport equations [30], was imposed on the outlet boundary to attenuate the reflections of the outgoing waves. A transition zone is set within a certain range from the boundary to reduce the horizontal gradient force of the elements near the boundary and suppress the calculation wave caused by this boundary condition. Through the relaxation coefficient, the predicted value on the inner boundary of the transition zone and the initial value on the outer boundary are continuously transitioned to achieve the purpose of reducing the reflection of propagating waves. The CUSTOMIZATION function of the software FLOW-3D was utilised to impose the Neumann-type absorbing boundary condition. The FLOW-3D distribution includes a variety of FORTRAN source subroutines that allow the user to customize FLOW-3D to meet their requirements. The FORTRAN subroutines provided allow the user to customize boundary conditions, include their own material property correlations, specify custom fluid forces (i.e., electromagnetic forces), add physical models to FLOW-3D, and have additional benefits. Several “dummy” variables have been provided in the input file namelists that users may use for custom options. A user definable namelist has also been provided for customization. Makefiles are provided for Linux and Windows distributions and Visual Studio solution files are provided for Windows distributions to allow users to recompile the FLOW-3D code with their customizations.
2.1. Governing Equations
The governing equations involved include the continuity equations and Reynolds-averaged Navier-Stokes equations. The mass and momentum are conserved in a two-dimensional zone [31]:
where U is the velocity vector, X is the Cartesian position vector, g denotes the gravitational acceleration vector, and ρ represents the weighted averaged density. The term μ is the viscosity. σκ∇α identifies the surface tension effects with σ as the surface tension and α as the fluid volume fraction. Each cell in the fluid domain has a water volume fraction (α) ranging between 0 and 1, where 1 represents cells that are fully occupied with water, while 0 represents cells that fully occupied with air. Values between 1 and 0 represent free surface between air and water. The free surface elevation is defined by using the volume of fluid (VOF) function:
where VF is the volume of fluid fraction, FSOR is the source function, FDIF is the diffusion function; Ax, Ay, and Az represent the fractional areas; and u, v, and w are the velocity components in the x, y, and z directions.
2.2. Porous Media Module
In FLOW-3D, the porous medium’s flow resistance is modelled by the inclusion of a drag term in the momentum equations (Equation (2)). Coarse granular material is used in most coastal engineering applications, in which case the Forchheimer model is suitable. Using this model, a drag term Fdui is added to the righthand-side of Equation (2):
where |U| is the norm of the velocity vector, n the porosity, and a and b are the factors.
2.3. Solitary Wave Boundary
The solitary wave is generated in terms of variations of the surface elevation η and velocities u and v by following McCowan’s theory [32]:
where h is the still water depth; Q is the reference value
where X = x − c0t; 𝑐0=√𝑔𝐻+ℎ; H is the wave height; and t is the elapsed time.
3. Validation
3.1. Propagation over a Porous Breakwater
An experimental test on the propagation process of a solitary wave over a permeable breakwater was performed by Reference [20], which was simulated in this study to validate the adopted two-way coupling model (Figure 1a). The length, width, and depth of the flume tank were 25, 0.5, and 0.6 m, respectively. A permeable breakwater was mounted at the bottom of the flume, which had dimensions of 13 cm and 6.5 cm in the length and height, respectively. The porous breakwater with an average porosity of 0.52 was configured by glass beads with a constant diameter of 1.5 cm. Two wave gauges were fixed before (WG1) and behind (WG2) the breakwater, respectively. The initial still water depth h was assumed to be 10.6 cm. Height of the solitary wave H was considered to be 4.77 cm. In the numerical model, the calculation zone had dimensions of 5 m in length and 0.25 m in height. The second order quadrilateral mesh elements were adopted. The grid around the breakwater was the finest of 0.001 m. The adopted time step size was 0.05 s. The numerical predictions of the water elevations at the locations WG1 and WG2 by the adopted numerical tool FLOW-3D are close to both the experimental measurements and the numerical predictions from another CFD FLUENT version 14.0.1 [33] (Figure 1). Figure 1b,c show the comparison of monitored water levels at the two water level monitoring points in Figure 1a. It can be seen that the experimental results of the two monitoring points are consistent with the numerical simulation results, indicating that the propagating solitary wave energy is basically completely dissipated and then flows out. If the propagating wave energy is not dissipated, the phenomenon of wave reflection will occur. The waves monitored at the two monitoring points will appear superposition of propagating waves and reflected waves. The numerical simulation results do not agree with the physical experiment results. The fluctuations of the water surface elevation after the bypass of the incoming wave are due to its residual reflection at the right absorbing boundary condition, which arrives at WG2 at an earlier time than WG1. Evolution of the wave surfaces was also compared between the experimental and the numerical models (Figure 2), which demonstrates that the numerical tool is sufficiently reliable. The velocity of the wave is reduced by the porous medium as it partially infiltrates into the breakwater, which is shown as in Figure 3 by comparing the horizontal velocity distributions between the experimental and numerical results at times of 1.5 s and 2 s. The numerical predictions of the flow velocities have slight discrepancies with the experimental measurements, which are attributed to the material assumptions made in the numerical model for the glass beads in the experimental setup.
Figure 1. The diagrammatic sketch of the numerical setup (non–scaled) (a) and the temporal evolution comparison of water surface between experimental and numerical results (b,c).
Figure 2. Water surface comparison between experimental and numerical results.
Figure 3. Comparison of horizontal velocity distribution between experimental and numerical results.
3.2. Forces on Pipeline
Another experimental test of a solitary wave impacting a pipeline was performed by Reference [34], which was also reproduced in this study for validation purposes. The calculation zone had dimensions of 40 m in length and 0.6 m in height. The solitary wave had a height of 0.0555 m with the initial water depth of 0.192 m. The pipe had a diameter of 0.048 m, which had a distance of 0.136 m over the bottom boundary of the model. A dense mesh consisting of 413,411 cells was employed with a mesh size of 0.1 mm around the pipe, which proved to be sufficiently fine through convergence studies. History of the horizontal and vertical forces, normalized by ρgL(πD2/4) with L as the unit length of 1 m, is compared between the experimental and numerical results (Figure 4). Both the peak values and the transient variations of the forces predicted by the numerical analysis converge to the measured values in the experimental test. The slight discrepancy between the numerical and experimental results at 2.5 s and 3.1 s, which may be induced by the error of the numerical model simulating the complicated turbulence behaviour, is acceptable in relation to the requirements of this study as our concern is mainly the peak values of forces.
Figure 4. Force comparison between the experimental and numerical results.
Therefore, the adopted numerical tool is sufficiently reliable to simulate the interactions between solitary waves and the permeable structure through the above validation cases.
4. Results and Discussion
Influence of the solitary waves on the performance of wrapped pipelines was investigated by considering different wave heights (H) and thicknesses (T) and wrapper porosities (n). The still water depth (h) was taken to be 6 m (Figure 5). The diameter of the porous medium was assumed to be 0.05 m. The pipeline diameter D was set at 1 m. In Figure 5 the variable G represents the gap between the permeable wrapper and the seabed. The scouring process had been completed before the simulation; therefore, the seabed boundary was taken as a rigid wall. The tandem pipelines had a distance of S between each other. The whole model had dimensions of 400 m in length and 12 m in height. The finest mesh around the pipeline was configured as 0.0025 m, which was verified to be sufficiently fine through trial calculations with finer meshes.
Figure 5. Layout for solitary wave impinging on the submarine pipeline encased in porous media.
4.1. Effect of Porous Wrapper
4.1.1. Wrapper Porosity
The pipeline was put on the seabed. The gap (G) between the wrapper and the seabed was considered to be zero. The height (H) of the solitary wave was considered to be 2 m. The porosity (n) was taken to be 0.0, 0.4, 0.6, and 1.0. Note that n = 0.0 indicates the impervious condition, while n = 1.0 corresponds to the non-wrapping condition. The thickness of the permeable wrapper remained at 0.5 m. In calculation, the wave approaches the pipe at around 6.3 s and departs from it at 10.2 s. When the wave approaches, the kinematic performance over the pipe is enhanced (Figure 6). Due to the wave disturbance, a number of small vortices are generated around the pipe (Figure 7). At the departure of the wave, the disturbance to the flow field seems to be more intense than that at its arrival, which further generates vortices around the pipeline. Without a wrapper, the pipe is fully exposed to the disturbance of the incoming wave, which maximises the velocity and vorticity values around the pipe. When the pipeline is wrapped by a porous medium, some water seeps into the wrapper, and the velocity in the wrapper is reduced to a very low value, which implies that the porous medium is capable of absorbing the dynamic energy of the flowing fluid. With an external coverage (n < 1.0), the disturbance is generated mainly at the outer surface of the wrapper. As the wrapper porosity increases, the domain of the low-speed flow underneath the pipeline expands.
Figure 6. The velocity contours of the flow fields under different porosities; (a) n = 0.0; (b) n = 0.4; (c) n = 0.6; (d) n = 1.0; left to right: arrival, departure.
Figure 7. The vorticity contours of the flow fields under different porosities; (a) n = 0.0; (b) n = 0.4; (c) n = 0.6; (d) n = 1.0; left to right: arrival, departure.
The peak velocity around the pipeline without a wrapper (1.9 m/s) is larger than that with a wrapper (1.6 m/s) (Figure 8). For pipes with wrappers, the peak velocities around them are similar to one another. In contrast, the velocity profiles at x = 23 m are quite different. When the pipeline has no wrapper (i.e., n = 1.0), the change in velocity is fairly moderate. When the pipeline has a wrapper, the porous wrapper causes a secondary fluctuation in the rear water body after the primary fluctuation due to the peak of the wave passing through the pipeline. This generates a series of velocity peaks. The secondary velocity peaks for a porosity coefficient of 0.4 are higher than those for a porosity coefficient of 0.6. Accordingly, the turbulent kinetic energy (TKE) also changes with the porosities, as shown in Figure 9. The TKE is expressed as
Figure 8. Comparison of horizontal and vertical velocities at front and rear of pipeline under different porosities.
Figure 9. Comparison of turbulent kinetic energy at front and rear of wrapper under different porosities.
With the propagation of the wave, the TKE increases gradually in front of the pipeline. The TKE value under the pipeline without a wrapper (n = 1.0) (0.0008 kJ) is nearly half of that with a wrapper (0.0015 kJ). In comparison, the TKE values for the wrapped pipelines (n < 1.0) are very close to each other. After the wave leaves the pipeline, the TKE in front of the pipeline decreases for around 50%. Then, the TKE in the rear of the pipeline with a porous wrapper increases intensively because the porous media perturb the flow field. Compared with the pipeline without the wrapper, the interaction between the wrapped pipeline with the flow field is more severe. Furthermore, the solid wrapper can cause a strong disturbance to the flow, but the interference of the solid wrapper (n = 0.0) in the rear flow is still weaker than the wrapper with the porosity of 0.4. The hydrodynamic forces (F), including the pressure and shear stress, are normalized by ρgL(πD2/4) (Figure 10). With a fully solid (i.e., n = 0.0) wrapper, the pipeline tends to be unaffected by the external flow. Hence, the hydrodynamic forces are zero while the forces on the wrapper reach their maximum. With porous wrappers, water seeps into the wrapper, buffering the impact of the incoming waves on the pipe. As the porosity coefficient increases, the induced forces on the pipeline increase while those on the wrapper decrease. When the porosity coefficient is 0.4, the forces on the external wrapper become higher than that on the internal pipeline. Therefore, the porous wrapper is capable of protecting the pipeline. The smaller the porosity coefficient the better protection the wrapper provides to the pipeline. The pressure gradient and shear stress forces are also shown in Figure 11.
Figure 10. Comparisons of the maximum hydrodynamic forces on the pipeline and wrapper.
Figure 11. Decomposed pressure gradient force (a) and shear stress (b) force on the pipeline.
4.1.2. Thickness of Wrapper
Seven wrapper thicknesses are considered: T = 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, and 0.5 m. The porosity coefficient is taken to be 0.6. At the moment that the wave goes through the pipe, the transient evolution of the vorticity contours around the pipeline with a wrapper thickness of 0.25 m is depicted in Figure 12. A couple of vortices emerge on the upper and lower vertices of the pipeline as the wave approaches the pipeline. As the wave propagates, many vortices flow along the wrapper and then shed off. Compared with the top vortices, the bottom vortices are shed off faster for two reasons. Firstly, as the friction at the seabed is small, the bottom flow velocity is higher than that on the top. Secondly, when the wave peak departs from the pipeline, a strong disturbance by the water body occurs behind the pipeline, followed by the irregular swing and fall off of the vortices. After the wave travels far away, the water flow near the pipeline becomes weak, and the vortices are scattered around the pipeline.
Figure 12. Temporal evolutions of vorticity contours around pipeline with wrapper thickness of 0.25 m (a) 6.0 s (b) 6.6 s (c) 7.2 s (d) 7.8 s (e) 8.1 s (f) 8.7 s (g) 9.0 s (h) 10.2 s (i) 12.6 s.
Figure 13 shows a comparison of flow field stream traces and the velocity contours. When the fluid penetrates the wrapper, the streamline starts to diverge, which indicates that the free flow is hindered. Therefore, the flow becomes slower and the flow direction becomes non-uniform. For the fluid flows out of the wrapper, the stream traces are quite complex and chaotic. The reason is that the seeping fluid mixes with the bypass flows and causes strong interference in the water body behind the pipeline. The streamlines passing through the wrapper indicates frequent water exchange at the wrapper surface. Along with the small-attached vortices on the wrapper surface, more fluid passes over the wrapper and causes a large vortex behind the wrapper.
Figure 13. Comparisons of flow field streamtraces and velocity contours under different wrapper thicknesses; (a) T = 0.2 m; (b) T = 0.3 m; (c) T = 0.4 m; (d) T = 0.5 m.
The highest free surface elevations and velocities at the front and at the rear of the pipeline with different wrapper thicknesses are depicted in Figure 14. As the wrapper thickness increases, the highest elevation at the front of the pipeline seems to be quite stable, although the peak velocity increases by around 6%. At the moment that the wave bypasses the pipeline, the maximum elevation reduces with an increase in the wrapper thickness. This is because the pipeline blocks the wave propagation. However, due to the strong mixing effect of the seepage and bypass water, the maximum velocity rises to be higher than that in front of the pipe. The maximum forces on the wrapper and the pipeline for different wrapper thicknesses are shown in Figure 15. With an increase in the wrapper thickness from 0.2 to 0.5 m, the normalized forces on the wrapper are doubled as a larger interaction area is involved. In contrast, the vertical forces on the pipeline decrease by 12.5%. Therefore, the larger the thickness of the wrapper the safer the pipeline.
Figure 14. Comparisons of the maximum elevations and velocities in front and rear of the pipelines with different wrapper thicknesses; (a) free surface elevation (note: original water depth is 6 m); (b) velocity.
Figure 15. Hydrodynamic forces on the pipeline and wrapper.
4.2. Effect of Pipeline Structure
The in-situ pipelines may be under various suspended conditions since the seabed topography is often uneven. Some pipelines are also laid in tandem for the sake of the transportation efficiency. In order to examine the effects of porous wrappers on pipelines under different conditions, a study was carried out considering two scenarios, namely, suspended pipelines and pipelines in tandem. In the numerical models, the porosity coefficient (n) remained at 0.6, the thickness (T) of the wrapper was kept at 0.5 m, and the wave height (H) was assumed to be 2.0 m.
4.2.1. Suspended Pipelines
Six gaps (G) between the wrapper and the seabed (0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 m) were considered [35,36,37]. The representative flow field at three points in time (6.3, 7.2, and 10.2 s) are shown in Figure 16. At the arrival of the wave at the pipeline (at 6.3 s), the flow is accelerated and the velocities over and beneath the pipe reach the maximum values due to the bypass effect of the fluid. At the moment that the wave peak is above the pipe (at 7.2 s), all the velocities around the pipe reach their highest values. After the wave passes over the pipe (at 10.2 s), the velocity decreases and several vortices are formed behind the pipeline. With a tiny wrapper-seabed gap, the velocity within the gap is very high while the flux is relatively small. An increase in the gap will result in an increase in the flux and a decrease in the velocity. A symmetric velocity distribution similar to a fisheye is observed behind the pipeline, which becomes more obvious when the gap increases (Figure 16c).
Figure 16. The velocity contours of the flow fields under different gaps; (a) G = 0.2 m; (b) G = 0.6 m; (c) G = 1.0 m. Left to right: 6.3 s, 7.2 s, and 10.2 s. Left to right: arrival, stay, departure.
With the bypass of the wave, the vortices generated around the pipeline become larger. The vorticity contours and the streamlines of the flow field are shown in Figure 17. As the solitary wave approaches, a pair of whirlpools shed off from the wrapper with a gap of 0.2 m. With an increase in the gap, the two whirlpools gradually disappear and are replaced with two smaller vortices. Due to the internal pores within the wrapper, the streamlines in the wrapper are dispersed, and it is hard for a vortex to be generated. With an increase in the gap, two anti-symmetric vortices shed off from the wrapper. Besides, some tiny vortices remain adhered to the wrapper due to the interaction by the seepage and the external flow. When the gap is very small, a few small vortices are generated between the wrapper and the seabed. In contrast to the interface of vortex from the flow around a solid cylinder, the vortex interface at the wrapper is not fully smooth. Because of the strong interactions of fluid over the wrapper surface, several small vortices mingle with the large shedding vortices. The flow direction also varies greatly according to the streamline mobilisation.
Figure 17. The vorticity contours of the flow fields under different gaps; (a) G = 0.2 m; (b) G = 0.6 m; (c) G = 1.0 m. Left to right: 6.3 s, 7.2 s and 10.2 s.
The gap is normalized by the pipeline diameter as β = G/D. With a small gap (β < 0.2), the horizontal forces on both the wrapper and the pipeline are slightly smaller than those on the wrapper and pipeline without a gap (Figure 18). With a further rise of the gap width, the horizontal forces are accordingly enlarged due to higher velocity around the pipeline as shown in Figure 16.
Figure 18. Comparisons of the maximum horizontal and vertical hydrodynamic forces on the pipeline and wrapper under different gaps.
In contrast, an increase in the gap width may inversely cause the reduction of vertical forces on both the wrapper and the pipeline. The vertical forces can be considered to consist of two parts. One is caused by the weight of the water body at the bypass of the wave from the pipeline, while the other can be caused by the velocity difference between the flow above and below the pipeline after the flow passes over. In summary, as the gap increases, the flow velocity within the gap initially increases when β < 0.2 and then decreases when β > 0.2. In contrast, the vertical forces caused by the wave’s weight always decrease with an increase in the gap.
4.2.2. Pipelines in Tandem
The hydrodynamic forces on pipelines in tandem are investigated considering five different distances (S) between the two pipeline centres (2.5, 3.0, 3.5, 4.0, and 4.5 m). The velocity and vorticity fields at 6.3, 7.2, and 10.2 s around the tandem pipelines with distances of 2.5, 3.5, and 4.5 m are depicted (Figure 19 and Figure 20). As the wave approaches the pipeline, the velocity within the pipeline gap is very small due to the blockage effect of the pipeline in front. As the distance increases, the velocity field within gap space is enhanced as more water flow is allowed. The velocity above the pipeline has its maximum value, and part of the high-speed fluid flows into the gap through the space underneath the pipeline. With a small distance, the vortices shedding off from the front pipeline impinge directly on the rear pipeline without any stretching. When the distance is increased, noticeable vortex shedding emerges in the middle space (Figure 20c). Similar vortex shedding behind the rear pipeline is observed for different distances. After the wave bypasses the pipeline, the increase in the distance between the pipelines will cause an increase in the velocity magnitudes in the space among the pipelines. As the distance increases, the flow becomes more chaotic due to the seepage from the wrapper and the limited flow space. In summary, influence of the distance between the pipelines over the whole kinematic field is not significant, although the local flow field around the pipelines is severely affected. When the wave bypasses the tandem pipelines, the largest forces on structures (i.e., the pipelines and wrappers) are shown in Figure 21, in which the distance ratio (θ) is calculated as θ = S/D.
Figure 19. The velocity contours of the flow fields under different spacings; (a) S = 2.5 m; (b) S = 3.5 m; (c) S = 4.5 m. Left to right: 6.3 s, 7.2 s and 10.2 s.
Figure 20. The vorticity contours of the flow fields under different porosities; (a) S = 2.5 m; (b) S = 3.5 m; (c) S = 4.5 m. Left to right: 6.3 s, 7.2 s and 10.2 s.
Figure 21. The maximum forces on the pipeline and wrapper under different distances.
As for the pipeline in front, as the distance ratio increases, the horizontal loads on the wrapper and pipeline increase slightly, while the vertical forces are almost doubled. As for the rear pipeline, as the distance reduces, the velocity in the gap becomes smaller and the forces on the pipelines and wrappers are also reduced, which is mainly attributed to the shield effect from the front pipeline. With an increase in the distance, the forces increase due to the increase in the turbulence energy in the gap. Different ratios of the forces on the front and rear pipelines are depicted in Figure 22. The difference ratio is defined as ΔFn = (ff,max−fr,max)/ff,max, where ff,max and fr,max are the maximum forces on the pipeline or wrapper. It is found that the horizontal loads on the rear pipe and wrapper tend to be always higher than their counterparts on the front pipe. This means that a turbulent flow in the horizontal direction on rear pipe is more intense than that on the front pipe. For different distances, deviations for the forces on the pipelines and wrappers are also different. The deviation is found to be maximized at a distance of 1 m and this indicates that the pipeline is not well protected and needs to be avoided in engineering practice.
Figure 22. The deviation of the forces on the front and rear pipelines and wrappers under different distances.
4.3. Effect of Wave Height
Six groups of wave heights (H), i.e., 1.6, 1.8, 2.0, 2.2, 2.4, and 2.6 m, are selected to consider different marine environment. After bypassing the pipeline, the height of the wave decreases because of the blockage effect of the pipeline and the dissipation of the flow energy (Figure 23a). The deviation ratio of the wave heights before and after the wave passes over the pipeline is shown in Figure 23b and is defined as δ = (Hf,max − Hr,max)/Hf,max. The wave height attenuation becomes more significant as the wave height increases. This means that waves with larger heights are more easily affected by the pipelines.
Figure 23. Waves with different wave heights; (a) temporal evolutions; (b) attenuation deviation.
At the bypass of the wave through the pipe, the loads are increased until they reach the maximum values at the moment that the wave peak appears above pipeline (Figure 24). The forces gradually decrease as the wave propagates. Because of some reflux after the wave bypasses the pipeline, the flow is in the opposite direction to that of the wave propagation, resulting in a negative force. The vibration of the water body by the wave propagation induces oscillations of the forces on the pipeline and wrapper. When the wave height is larger, the force oscillation becomes fiercer and the maximum loads on the pipeline and the wrapper increase (Figure 25). The vertical forces on the pipeline are the largest compared with other forces under the same conditions. Besides, as the wave height increases, the increased amplitude of vertical forces on the pipeline is the most significant change since the weight of the water above the pipeline increases. Therefore, given that the wave height is very high, the protective function of wrapper on the pipeline tends to be weakened compared with that of the wrapper for a low wave height.
Figure 24. The temporal evolutions of forces on the pipeline and wrapper; (a) Horizontal maximum force on pipeline; (b) Vertical maximum force on pipeline; (c) Horizontal maximum force on wrapper; (d) Vertical maximum force on wrapper.
Figure 25. Variation of hydrodynamic forces on the pipeline and wrapper under different distances.
5. Conclusions
The effect of porous media on the dynamic performance of submarine pipelines under solitary waves was investigated. The porosity of the wrapper, the seabed topography, the structure of the pipeline, and the marine environment were considered. The study had a limitation of the model sizes due to the limited computational resource and the simplification of the solitary wave due to its mathematical complication, which will be tackled in future works. The following main conclusions have been made.
(1) When a pipe is wrapped by a porous medium, the velocity in the wrapper is relatively small because the porous medium can consume the water energy and weaken the flow. With an increase in the porosity, the range of the low-speed flow at the bottom of the pipeline expands. This indicates that the porous wrapper can slow down the flow and affect a wider region of the surrounding water. After the bypass of the wave through the pipe, the number and volume of the vortices behind the porous wrapper are larger than those for a pipeline with a solid wrapper or without a wrapper. As the porosity coefficient increases, the impact forces on the pipe increase, while those on the wrapper decrease. This implies that the porous wrapper is capable of protecting the pipeline.
With an increase in the wrapper’s thickness, the hydrodynamic forces on the wrapper tend to increase. In particular, the horizontal forces on the pipeline decrease with an increase in the thickness due to the protection of the wrapper, while the vertical forces are increased because of variations in the fluid’s stagnation point.
(2) For a wave bypassing a pipe with different heights, a symmetric speed change similar to a fisheye appears behind the pipeline, along with two antisymmetric vortices shedding off from the wrapper.
As the internal seepage interacts with the external fluid flow, several small vortices are still attached to the wrapper. The hydrodynamic vertical forces on both the wrapper and the pipeline decrease with the pipeline distance. With an increase in the suspension of the pipe, the velocity and TKE within the gap space increase and both the vortex intensity and the number of vortices increase. Therefore, the flow pattern appears to be chaotic. As for the front pipeline, an increase in the gap leads to a slight increase in the horizontal forces on both the wrapper and the pipeline, but a significant increase in the vertical forces. As for the rear pipeline, because of the shield function of the front pipeline, the velocity within the gap space and the forces on the pipes decrease with a decrease in the gap size.
(3)When the waves with different heights pass over the pipeline, the height of the wave is reduced because of the blockage function from the pipeline and the dissipation characteristic of the flow energy. When the wave height is increased, the velocity around the pipeline increases, inducing an increase in the TKE. As the wave height increases, all the maximum forces on the pipeline and wrapper also increase. Note that an increase in the vertical forces on the pipeline is the most significant change because the weight of the water above the pipeline increases, which implies that the protection function of the wrapper is enhanced by the reduction in the wave height.
From the above investigation, the mechanism of load transfer from the pipeline to the external wrapper has been presented. This encourages industrial experts and academic scholars to arrange more investigations of the functions and cost-efficiency of porous wrappers, which could form a new branch of the pipeline design practice.
Author Contributions
Contributor Roles Taxonomy: E.Z.: Conceptualization, Methodology, Validation, Investigation and Writing—Original Draft; Y.D.: Data Curation, Formal analysis; Y.D.: Visualization, Project administration; Y.D., L.C., Y.W. and Y.L.: Writing—Review & Editing. All authors have read and agreed to the published version of the manuscript.
Funding
The paper was supported by the National Natural Science Foundations of China (Grants No. 52001286 and No. 42272328), GuangDong Basic and Applied Basic Research Foundation (Grant No. 2022A1515240002) and Comprehensive Survey of Natural Resources in Huizhou-Shanwei Coastal Zone (Grant No. DD20230415).
Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflict of interest.
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Md Tusher Mollah, Raphaël Comminal, Wilson Ricardo Leal da Silva, Berin Šeta, Jon Spangenberg
Abstract
A challenge for 3D Concrete Printing is to incorporate reinforcement bars without compromising the concrete-rebar bonding. In this paper, a Computational Fluid Dynamics (CFD) model is used to analyze the deposition of concrete around pre-installed rebars. The concrete is modelled with a yield-stress dependent elasto-viscoplastic constitutive model. The simulated cross-sections of the deposited layers are compared with experiments under different configurations and rebar sizes, and found capable of capturing the air void formation with high accuracy. This proves model robustness and provides a tool for running digital experiments prior to full-scale tests. Additionally, the model is employed to conduct a parametric study under three different rebar-configurations: i) no-rebar; ii) horizontal rebar; and iii) cross-shaped (horizontal and vertical) rebars. The results illustrate that air voids can be eliminated in all investigated cases by changing the toolpath, process parameters, and rebar joint geometry, which emphasizes the great potential of the digital model.
Keywords
3D Concrete Printing (3DCP); Reinforcement bars (rebars); Computational Fluid Dynamics (CFD); Multilayer deposition; Air voids
1. Introduction
3D Concrete Printing (3DCP) [1] is an extrusion-based automated construction process that belongs to Digital Fabrication with Concrete (DFC) [2,3]. The 3DCP offers high-quality built-structures with customizable structural design in a cost- and time-efficient manner [[4], [5], [6], [7]]. Structures in 3DCP are fabricated in a layer-by-layer approach, where a concrete extrusion nozzle is controlled by a robotic arm, cylindrical robot, gantry system, or delta system [[8], [9], [10], [11]]. Despite the enormous potential of 3DCP, one of its crucial limitations is the integration of reinforcement for the production of load-bearing structures. Most structural applications require the use of reinforcement to withstand tensile forces and introduce structural ductility [[12], [13], [14], [15]]. However, the introduction of reinforcement with 3DCP has never been an easy task, and difficulties were recognized at early stages of the technology [4] and various design solutions have been tested in practice to either circumvent the need for reinforcement or integrate reinforcement after the concrete is printed [[16], [17], [18], [19], [20]]. As a result, several reinforcement techniques have been proposed, such as bar reinforcement [21], micro-cable reinforcement [22,23], fiber reinforcement into the cementitious material [[24], [25], [26]], steel reinforcement using robotic arc welding [27,28], and in-process mesh reinforcement [29]. For comprehensive details on the reinforcement strategies, refer to [30]. Nevertheless, these reinforcement strategies are still rudimentary in many instances. This study focuses on bar reinforcement methods, where rebars are integrated with freshly deposited cementitious material. A few approaches can be found in the literature, for example, penetration of vertical bars through multiple printed layers [31,32], placement of horizontal bars into a printed layer along the printing direction and then covered by the next layer on top [33,34], and depositing around pre-installed bi-directional rebars [35]. However, in most approaches, the bonding between the rebar and concrete was compromised by the air void around the rebar [21,36]. To overcome this constraint, a large amount of trial and error is required, which is costly and time-consuming. An approach to mitigate extensive experimental campaigns is to apply numerical models. In the context of 3D printing technologies, like Fused Filament Fabrication (FFF), Robocasting, and 3DCP, CFD modelling has been found to be very beneficial [[37], [38], [39], [40], [41], [42], [43], [44]]. The morphology of the deposited strands in FFF was studied by Comminal et al. [45]. Furthermore, Serdeczny et al. [46] addressed how to reduce the porosities and enhance the bonding between subsequent layers. Mollah et al. [[47], [48], [49]] studied ways to minimize the deformation and thereby stabilize layers printed by Robocasting, while for 3DCP, the geometrical shapes of the single- and multiple-deposited layers have been investigated in detail in [[50], [51], [52]]. This paper uses the CFD model and extends the preliminary results recently published in [53]. The model uses elasto-viscoplastic constitutive equations to approximate the rheology of the concrete. The CFD model is validated by comparison with a number of experiments, and the model is subsequently exploited to make an in-depth analysis of air void formation between rebars and concrete using the cross-sections of the deposited part and the calculated volume fraction of air voids. Different material properties, such as yield stress and plastic viscosity, and processing parameters, like the rebar diameter, nozzle-rebar distance, a geometric ratio (i.e., the distance from nozzle to the substrate divided by the nozzle diameter), as well as a speed ratio (i.e., the printing speed divided by the extrusion speed) are varied. Section 2 describes the methodology of the study, along with the experimental and numerical details. Next, Section 3 presents and discusses the results. Finally, Section 4 summarizes the results with the conclusion.
2. Methodology
2.1. Materials’ properties and 3DCP experiments
A fresh cement-based mortar was used to perform the 3DCP experiment around the rebars. The mortar includes a binder system with white cement CEM I 52.5 R-SR 5 (EA), limestone filler with sand of maximum particle size 0.5 mm, admixtures, and water. The binder was prepared with a 75 L Eirich Intensive Mixer Type Ro8W. The water to cement ratio was 0.39. The admixtures dosage (by weight of cement) was set at 0.1 % high-range water-reducing agent, 0.1 % viscosity-modifying agent, and 0.5 % hydration retarder.
The rheological characterization of the mortar was done using an Anton Paar rheometer MCR 502, as used in [50,54]. The rotational and oscillatory tests were performed with a vane-in-cup measuring device. The obtained flow curve of the mortar from the rotational rheometric tests, with a ramp-down controlled shear rate (CSR), was fitted by a linear regression to determine the yield stress τ0= 630 Pa and plastic viscosity ηP= 7.5 Pa·s. The oscillatory test showed that the constitutive behavior of the unyielded mortar had a factorized relationship between the storage modulus G′ and loss modulus G′′ within the linear viscoelastic (LVE) region, where G′= 200 kPa was captured. Therefore, the mortar’s rheology was modelled as a yield stress limited elasto-viscoplastic material, where the storage modulus is used as the linear elastic shear modulus of the unyielded mortar. Furthermore, the rheological characterization showed that the mortar exhibited time-independent rheological characteristics within the actual printing process, see [50] for more details.
The setup for 3DCP experiments around ribbed rebars is presented in Fig. 1. It comprised a 6-axis industrial robot (Fanuc R-2000iC/165F) with a custom-designed nozzle ∅20 mm (i.e., nozzle diameter, Dn= 20 mm) made by fused filament fabrication of ABS thermoplastic, cf. Fig. 1-a. The robot also included a progressive cavity pump (NETZSCH) equipped with a hopper and a long steel-wire rubber hose (cf. Refs. [50, 52] for details). A 25 mm thick plywood plate was used as the built substrate as seen in Fig. 1-b. The 1000 mm long rebars of diameter Dr= 8 and 12 mm were placed horizontally on top of the substrate at a distance Hr= 14 mm. The horizontal rebars were held in place by two vertical rebars with a height of 37 mm. The setup was used to print a structure of four successive layers of parallel strands around the rebars. Details on the printing toolpath around the rebars are illustrated in the subsections below.
Fig. 1. 3DCP experiment around rebars: (a) 6-axis robotic arm [50]; (b) plywood built platform with integrated rebars; (c) example of printing (picture is taken during printing of the third layer).
The extrusion nozzle was placed above the substrate with a nozzle height Dn/2 for the first layer, whereas for subsequent number of layers (Nl), the nozzle height was set at Nl∗Dn/2. Thus, the nominal height of a layer was h=Dn/2. The print was done with a material extrusion rate 0.91 dm3/min and nozzle speed 35 mm/s. An example of a physical print is presented in Fig. 1-c. After the prints hardened, cross-sections were collected to investigate the rebar-concrete bonding. The cross-section slices were taken at specific positions to analyze the print around the horizontal rebar and cross-shaped rebar (i.e., horizontal and vertical rebars). To avoid destroying the specimens while cutting them, the printed part were impregnated with epoxy resin in a vacuum chamber.
2.2. Computational models and governing equations
Three different CFD models are built. The first model only simulated the mortar flow to understand the void formation pattern without rebars. The last two models simulated the 3DCP experiment around rebars: one model simulated the mortar flow around the horizontal rebar, while the other considered the cross-shaped rebar. This subdivision enabled the CFD models to consider a smaller computational domain than if the two scenarios were combined.
The CFD models comprised a cylindrical nozzle, a solid-substrate, and an artificial solid component (at the top) within the computational domain of size 8.5D×6D×2D+4h as shown in Fig. 2 (top), where Model 1 excluded rebars (left), Model 2 included the horizontal rebar (middle), and Model 3 considered the cross-shaped rebar (right). The printing toolpath of the models are illustrated in Fig. 2. The toolpath for Models 1 and 2 are presented in 3D (left bottom figure), where the only difference was the presence of the horizontal rebar. The toolpath in 2D presented at the bottom right is for Model 3. For all the models, the toolpath of the extrusion nozzle kept a distance of Dnr from the axis of the nearest rebar. The lengths of the horizontal and vertical rebars were 50 and 40 mm, respectively. The other printing parameters were similar to the ones used in the experiment, cf. Section 2.1. Finally, the models were used to simulate four successive layers with a length of 125 mm. Note that the rebars are modelled as cylindrical solid objects (i.e., smooth rebar).
Fig. 2. Model geometry with the extrusion nozzle, substrate, integrated rebars, and computational domain (top) and toolpath (bottom).
The computational domain was meshed by a uniform Cartesian grid. A mesh sensitivity test was performed for different meshes with cell sizes 0.9, 1.0, and 1.1 μm. Even if the change in absolute size of the cells were small, the total number of cells within the domain was 1.1, 1.5, and 2.0 million, respectively. A cell size of 1.0 μm was chosen as that was found to be time-efficient and had a negligible effect on the accuracy of the results. The top plane of the domain was an inlet boundary, where the artificial solid component was defined in order to prevent material flow outside the nozzle orifice, cf. Fig. 2 (top). On the bottom plane, a wall boundary was applied to represent the solid substrate. The other planes were assigned continuative boundary conditions, but had no influence on the results. Furthermore, no-slip boundary conditions were applied between fluids and solids.
Table 1 lists the printing parameters and their values for each of the investigated cases. All the models and cases are simulated for 4 successive layers.
Table 1. Description of case IDs with printing parameters and accompanying values. The reference values (corresponding to the experimental print) are written in bold, while the parameter change for each case is highlighted by underlining the value.
Empty Cell
Model/case ID
Parameters
Model 1 (no rebar), Model 2 (horizontal rebar), and Model 3 (cross-shaped rebar)
Case 1 (reference)
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Nozzle diameter Dn (mm)
20
20
20
20
20
20
20
20
20
Rebar diameter Dr (mm)
8
6
12
8
8
8
8
8
8
Nozzle-rebar distance Dnr (mm)
20
20
20
19
18
20
20
20
20
Layer height h (mm)
10
10
10
10
10
9
8
10
10
Geometric ratio Gr=h/Dn
0.5
0.5
0.5
0.5
0.5
0.45
0.4
0.5
0.5
Printing speed V (mm/s)
35
35
35
35
35
35
35
35
35
Extrusion speed U (mm/s)
48.42
48.42
48.42
48.42
48.42
48.42
48.42
51.47
53.84
Speed ratio Sr=V/U
0.72
0.72
0.72
0.72
0.72
0.72
0.72
0.68
0.65
The cementitious mortar flow was assumed transient and isothermal. Thus, the flow dynamics of the mortar are governed by the mass and momentum conservation equations of incompressible fluid:
where u is the velocity vector, ρ is the density, g=00−g is the gravitational acceleration vector, t is the time, p is the pressure, and σ is the deviatoric stress tensor.
The rheological behavior of the mortar was modelled by the following elasto-viscoplastic constitutive equation that represents σ as the sum of the deviatoric part of the viscous stress σV and elastic stress σE tensors; i.e.:
The deviatoric viscous stress tensor was predicted as:
is the deformation rate tensor, and T represents the transpose notation.
The deviatoric elastic stress tensor was modelled by the Hookean assumption of a small strain rate tensor E between each small time steps Δt=t−t0, to represent the elastic response of unyielded materials
Ewhere G is the shear modulus and Et=Et0+ΔtDT is the incremental strain rate tensor approximated by integrating the deformation rate tensor over Δt.
The incremental representation of Eq. (5) can be written as:
is the vorticity tensor. The first term of the left-hand side of Eq. (6) represents the change in stress at a fixed location in space. The change in stress due to advection and rotation of material particle is approximated by the second and third terms, respectively. The right-hand side takes into account the change in stress due to shearing.
The elastic stress tensor of the yielded material was approximated by imposing the yield stress τ0 limit as follows:
where σvM is the von Mises stress predicted as:
where IIσE∗=trσE∗2 is the second invariant of σE∗. The material was yielded when σvM exceeded the yield stress. The properties of the material used in the different models and cases are presented in Table 2. Note that the CFD model does not include the solidifications of the printed layers.
Table 2. Material properties.
Parameter with symbol
Unit
Value
Value for reference simulation
Density, ρ
kg·m−3
2112
2112
Shear modulus, G
kPa
20–100
20
Dynamic yield stress, τ0
Pa
400–800
630
Plastic viscosity, ηP
Pa·s
3.5–10
7.5
2.3. Numerical method
The computational model was developed in the commercial CFD tool FLOW-3D® (V12.0; Flow Science Inc.) [55]. It uses the FAVOR technique (Fractional Area/Volume Obstacle Representation) to embed solid objects (i.e., the nozzle, rebars, substrate, etc.) in the computational domain. The computational domain was meshed with a Cartesian grid and discretized with the Finite Volume Method. The governing equations of the mortar flow were solved by the implicit pressure-velocity solver GMRES (Generalized Minimum Residual) [[56], [57], [58]]. The predictions of pressure and forces near solid objects were modelled by the immersed boundary method [59]. The yield stress limited elasto-viscoplastic criterion was built in the software, and the elastic stress was calculated explicitly. An implicit technique, successive under-relaxation, was used to solve the viscous stress of the momentum equation (Eq. (2)). The free surface of the mortar was captured by the Volume-of-Fluid technique, see details in Ref. [60, 61]. The momentum advection was calculated explicitly by an upwind-difference technique and ensured first-order accuracy. The time step size was controlled dynamically with a stability limit in order to avoid numerical instabilities [55]. All the simulations were run with 20 cores on a high-performance computing cluster. The study was carried out with a first-order accuracy in both space and time in order to reduce the computational time, which was extensive due to the elasto-viscoplastic material model that computes both viscous and elastic stresses (e.g., the computational time was six days for model 1 case 1). In this regard, one should note that the model can simulate a multitude of scenarios simultaneous.
2.4. Results post-processing
The simulated results were processed in two steps. The first step was to show the cross-sectional shapes which were done in the post-processing tool FLOW-3D®POST, and the second step was to calculate the volume fraction of air voids inside the printed structure. The cross-sectional shapes were used to investigate the interior of the structure and the rebar-concrete bonding. The cross-sectional shapes were obtained in the plane parallel to the yz-plane at the middle of the layer’s length, as shown in Fig. 3-d. Fig. 3-c sketches the nominal positions of the air void creation in a four-layered structure. The positions of the air voids were defined as outer-bottom, bottom, mid, top, and outer-top in this study. The presence of the air void was quantified by calculating the volume fraction of air voids around the middle of the layer’s length. The calculation enabled to capture the presence of air voids for Model 3, i.e., around the vertical rebar, as seen in the dashed-box of Fig. 3-b.
Fig. 3. Post-processing of results; (a) introduction of volume sampling cuboid to calculate the volume fraction of air voids; (b) presence of air voids around the vertical rebar in the experiment; (c) schematic of air void creation; (d) cross-sectional shapes and void sampling area for the different models.
In order to calculate the volume fraction of air voids, a cuboid of size 20×25×3h mm3 was introduced to the CFD models as a volume sampling object, as seen in Fig. 3-a. Note that the size and position of the object were the same for all the models. The volume sampling was a three dimensional data collection tool built in the software that enabled calculating the amount of material as well as air void inside of it. Finally, the volume fraction of air voids VV was calculated as below:
3. Results and discussions
his section compares the simulated and experimental results of 3DCP around the rebars. Furthermore, it discusses the influence of different parameters on the air void formation in the cross-sectional shapes of printed parts and the volume fraction of air voids inside the structure. The parametric study includes the material properties (i.e., yield stress and plastic viscosity) and the printing properties (i.e., rebar diameter, rebar-nozzle distance, geometric ratio, and speed ratio).
3.1. Experiments and validation of the CFD model
The CFD models (Models 2 and 3) are compared and validated with the experiments. The results are presented in Fig. 4. Two rebar diameters, i.e., Dr= 8 and 12 mm, are taken into account, where the nozzle-rebar distances are 20 and 24 mm, respectively. The other printing and material parameters are kept constant in the experiments with different rebar diameters, see Table 1 (cases 1 and 3). In the case of the simulations, all parametric details are the same as implemented in the experiments except for the elastic shear modulus. The choice of shear modulus is subject to the analysis presented in Appendix A. The shear modulus is reduced to 100 kPa from the measured value (i.e., 200 kPa) to compare the simulated results with experiments. Furthermore, a shear modulus of 20 kPa is chosen for the parametric study in the later sections. These assumptions seem reasonable to avoid extensive computational time consumption since the differences found in void formation are limited (see Fig. A.1).
Fig. 4. 3DCP experiments (left column), simulations (middle column), and comparison (right column). (a) Horizontal rebar with Dr= 8 mm and Dnr= 20 mm; (b) horizontal rebar with Dr= 12 mm and Dnr= 24 mm; (c) cross-shaped rebar with Dr= 8 mm and Dnr= 20 mm; and (d) cross-shaped rebar with Dr= 12 mm and Dnr= 24 mm. The blue part in the experiments is epoxy resin. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The cross-sectional shapes of the 3DCP experiments around the horizontal rebar of 8 mm in Fig. 4-a illustrate the presence of a top air void as well as mid and bottom air voids positioned respectively above and below the horizontal rebar. The mid and bottom air voids are found to be smaller than the top one. This is due to the presence of the rebar that occupies the mortar’s flowable space as well as the deformation of the previously printed layers. For a detailed analysis of the deformation pattern, refer to [53]. The air voids at the top and bottom are significant when the rebar diameter is increased to 12 mm (Fig. 4-b). This is due to the fact that the nozzle-rebar distance was increased, which enhances the flowable space between the strands. In addition, the larger size of the rebar creates larger channels below and above itself, where the mortar of the second and third layers is forced to be squeezed into. The mid-air void is found to be absent as its area is fully occupied by the larger rebar. In the case of the cross-shaped rebars, the existence of the vertical rebar seems to restrict the merging of parallel strands, and therefore, the presence of air void content increases, cf. Fig. 4-c, d. This limitation is found to be pronounced for the larger rebar diameter with the larger nozzle-rebar distance. The cross-sectional shapes of the simulations (middle column in Fig. 4) illustrate high accuracy predictions of the position and size of the air voids when compared with the experiments. Particularly, the models capture small details around the vertical rebar for both diameters. This can clearly be seen in the comparison of experiments and simulation, cf. right column in Fig. 4. A discrepancy is found in the strand’s width of the bottom layer as well as the shape of the printed part, specifically in the shape of strands of all the layers next to the vertical rebar and the height of the part for the smaller rebar diameter. These could be due to a combined effect of the idealized rheological model, as well as slight differences in the processing parameters, e.g. nozzle height above the printing surface, nozzle-rebar distance, as well as printing- and extrusion-speed. Note that the height of the vertical rebar in the experiments is a bit shorter than the one in the simulations (40 mm), although it does not influence the results.
3.2. Influence of materials properties
The influence of the material properties, yield stress and plastic viscosity, on the air void formation is presented in Fig. 5, Fig. 6, Fig. 7. The process parameters of Case 1 (cf. Table 1) are utilized.
Fig. 5. Air void formation in the cross-sections of the printed parts for different yield stress.
Fig. 6. Air void formation in the cross-sections of the printed parts for different plastic viscosity.
Fig. 7. Volume fraction of air voids for different models as a function of (a) yield stress and (b) plastic viscosity.
Yield stress
Fig. 5 presents cross-sectional shapes for different yield stress, 400, 630, and 800 Pa. It can be seen that Models 1 and 2 predict a top air void, while for Model 3 the two topmost strands to the left are not in contact with the vertical rebar. The cross-sections illustrate that an increased yield stress causes less deformation of the printed layers and create stands with less round shape. This is due to the reduced effective gap (i.e., the distance between the nozzle and previous printed layer), which results in a reduced air void content for most models as seen quantitatively in Fig. 7-a. This behavior is converse to conventional concrete castings where a more fluid material (e.g. self-compacting concrete) can lead to a lower void content. An exception to the observed behavior that a higher yield stress leads to less voids formation is seen in case of Model 2 with 800 Pa, where the top air void is slightly larger as compared to the one for 630 Pa. Another exception is that an outer bottom air void appears for Models 1 and 2 when increasing the yield stress to 800 Pa. Both exceptions are a consequence of the yield stress now restricting the flow in confined spaces, which illustrates that it is a non-trivial task to fully eradicate air voids only by increasing the yield stress.
Plastic viscosity
As the plastic viscosity is varied, cross-sections for Model 1 show a slight change in air voids, see Fig. 6. A mid-air void is produced when the plastic viscosity is small, while the two largest plastic viscosities only produce the top air void. This is due to the increase in extrusion pressure that leads to larger deformation of the printed layers when the plastic viscosity is increased, cf. details in ref. [47]. When integrating a horizontal rebar (see Model 2), the air void formation increases at higher plastic viscosities. This could be due to the fact that the sideway flow of the depositing material (i.e., y-velocity) is limited by the flow resistance that comes from both the larger plastic viscosity and the presence of the solid rebar. No noticeable change can be seen in the cross-sections of Model 3 for different plastic viscosities. The same findings are quantitatively highlighted in Fig. 7-b, which illustrates that the volume fraction of air voids is not influenced much by the plastic viscosity except for Model 2.
3.3. Influence of processing conditions
The influence of processing conditions such as rebar diameter, nozzle-rebar distance, geometric ratio, and speed ratio on air void formation is presented in terms of cross-sectional shapes and volume fraction of air void, see Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12. Models 1 to 3 are simulated with the reference material properties, cf. Table 2.
Fig. 8. Air void formation in the cross-sections of the printed parts for different rebar diameters.
Fig. 9. Air void formation in the cross-sections of the printed parts for different nozzle-rebar distances.
Fig. 10. Air void formation in the cross-sections of the printed parts for different geometric ratios.
Fig. 11. Air void formation in the cross-sections of the printed parts for different speed ratios.
Fig. 12. Volume fraction of air voids for different models as a function of (a) rebar diameter, (b) nozzle-rebar distance, (c) geometric ratio, and (d) speed ratio.
Rebar diameter
The influence of the rebar diameter on the air void formation is presented in Fig. 8, Fig. 12-a. Models 2 and 3 are simulated (Model 1 does not contain a rebar) for Cases 1, 2, and 3, cf. Table 1. Fig. 8 illustrates that the top air void appears almost constant for Model 2, whereas the air void below the rebar increases with an enlarged rebar diameter. Two phenomena with opposite effects on the void formation play a role in this regard. On the one hand, increasing the rebar size reduces the space that the strands need to occupy to fully merge and thereby eliminate voids. On the other hand, the resistance towards flow and merging of the parallel strands next to the rebar increases proportionally with the size of the reinforcement. The latter effect is dominating in this case. For Model 3, the air void formation also increases when increasing the reinforcement. In addition to the previously mentioned argument, this is due to the left strands having to flow longer to reach the vertical rebar (i.e., Dnr+Dr/2). Conversely, additional air voids take place on the right-hand side of the vertical rebar for the smallest rebar diameter, because the nozzle-rebar distance Dnr= 20 mm is kept constant. Fig. 12-a underlines quantitatively that the volume fraction of air voids reduces when the rebar diameter is small. The trend is more pronounced for the cross-shaped rebar (Model 3), but in absolute values the air voids are substantially less for Model 2.
Nozzle-rebar distance
Fig. 9, Fig. 12-b show the effect of different nozzle-rebar distances on the formation of air voids. All the models are simulated for Cases 1, 4, and 5, cf. Table 1. Fig. 9 shows that the presence of air voids is reduced when the nozzle-rebar distance reduces. This is because the flowable space around the rebars shrinks. Interestingly, no significant air voids are present in Model 1 and 2 when Dnr= 18 mm, see Fig. 12-b. Fig. 12-b also depicts that the trend is more pronounced for the cross-shaped rebar (Model 3). One should be careful though about decreasing the nozzle-rebar distance too much, as a ridge is forming on the top layer since material from the left strand starts to flow on top of the right strand (Fig. 9), which potentially could affect the final shape of the structure. In case of the cross-shaped rebar model, air voids are formed for all investigated Dnr. One could potentially with benefit reduce the Dnr further, but not more than the sum of half of the nozzle diameter (10 mm), the nozzle wall thickness (2.5 mm), and half of the rebar diameter (4 mm), i.e., 16.5 mm, otherwise the nozzle will collide with the rebar.
Geometric ratio
The effect of the geometric ratio on the formation of air voids is presented in Fig. 10, Fig. 12-c. The considered simulations are Cases 1, 6, and 7 cf. Table 1. Fig. 10 illustrates that decreasing the geometric ratio can reduce the presence of air voids in the cross-sections. This is because a smaller geometric ratio results in wider strands, which then occupy more of the flowable space around the rebars. Note that when Gr= 0.50, 0.45, and 0.40 the layer height is 10, 9, and 8 mm, respectively. No air voids are formed for Model 1 and 2 when Gr= 0.45, and 0.40. However, for the smallest ratio ridges are obtained on either side of the strands as clearly seen for the top layer. These ridges can as previously mentioned have a negative effect on the final shape of the structure. Consequently, Gr= 0.45 is preferable for these two models. In the case of Model 3, air voids are still present next to the vertical rebar, even for the smallest investigated geometric ratio. The volume fraction of air voids is approximately 1.5 %, see Fig. 12-c. The geometric ratio could be reduced further in order to decrease the air voids even more, but the ridges already form at Gr= 0.40. Consequently, it is not possible to fully eliminate air voids while at the same time avoiding ridges when only varying the geometric ratio for Model 3.
Speed ratio
Fig. 11, Fig. 12-d illustrate the formation of air voids for different speed ratios. The considered simulations are Case 1, Case 8, and Case 9, cf. Table 1. The three speed ratios are obtained by applying an extrusion speed of 48.4 mm/s, 51.5 mm/s, and 53.8 mm/s. Fig. 11 show that less air voids are formed when decreasing the speed ratio (i.e., higher extrusion speed). Reducing the speed ratio increases the cross-sectional area of the strands proportionally, thereby decreasing the air voids. Model 1 obtains no air voids for the two smallest ratios, and the same is almost the case for Model 2; only a very small air void is formed when Sr= 0.68, see Fig. 12-d. Model 3 forms air voids for all speed ratios. For the lowest speed ratio, the third strand to the left is in contact with the vertical rebar, but air voids are still formed around the horizontal reinforcement, which underlines the fact that it is difficult to fully eliminate air voids for the cross-shaped rebars.
3.4. Cross-shaped reinforcement
Based on the above analysis, it is clear that the air voids around the horizontal rebar can be eliminated by changing some of the processing conditions, such as the nozzle-rebar distance, geometric ratio, and speed ratio. However, it remains unsolved to fully omit the presence of air voids around the cross-shaped rebar, although the processing conditions can reduce the volume fraction of air voids. A parametric study was conducted by varying some combinations of processing conditions; however, the same conclusion was achieved that the air voids could not be fully eliminated. In order to solve this predicament, a new stepped toolpath is investigated (see Fig. 13) along with three different rebar geometries: 1) cylindrical rebars as in the previous analysis, see Fig. 14-a; 2) a squared horizontal rebar, cf. Fig. 14-b; and 3) cylindrical rebars with a smooth transition between them, see Fig. 14-c. In all scenarios, the speed ratio is 0.665, the size (i.e., diameter or side of square) of the rebars are 6 mm, and the horizontal rebar is placed at a height of 8 mm from the substrate. The other processing parameters are the same as for Case 2 and reference material properties are applied. For scenarios one and two small air voids are formed, but for scenario three air voids are eliminated, see Fig. 14. This numerical analysis illustrates that although it is difficult to get rid of air voids for the cross-shaped rebars, one can do it when carefully selecting the material- and processing-parameters and remembering to have a smooth transition between the rebars.
Fig. 13. New toolpath planning around the cross-shaped rebar.
Fig. 14. Simulated structure with new toolpath and different rebar geometries. (a) Cylindrical horizontal- and vertical-rebar, (b) square horizontal rebar and cylindrical vertical rebar, (c) cylindrical horizontal- and vertical-rebar with smooth transition. Note that Dr= 6 mm, Sr= 0.665, and Hr= 8 mm.
4. Conclusions
A CFD model was used to predict the morphology of strands and the formation of air voids around reinforcement bars when integrated with 3DCP. The model used an elasto-viscoplastic constitutive model to mimic the cementitious mortar flow. The CFD model was compared with experiments that constituted a horizontal and a cross-shaped rebar configuration. The results illustrated that the model with high-accuracy could predict the air void formation in the structures. The simulations had though slightly less wide bottom strands as compared to the experimental counterpart, which was attributed to small differences in material behavior and processing parameters. The CFD model was exploited to investigate the effect of material properties on the air void formation. The results illustrated that by increasing the yield stress less air voids were formed due to the reduced effective gap. However, the air voids could not be eliminated as the increased yield stress also restricted the flow in confined spaces. In contrast to the effect of the yield stress, the void formation decreased somewhat when decreasing the plastic viscosity (although not enough to omit air voids fully). The process parameters were found to have a substantial effect on the air void formation. The air void formation increased when increasing the rebar diameter, because the resistance towards flow around the reinforcement and thereby merging of the strands increased proportional with the size of the rebars. The air voids could be reduced and in some of the horizontal cases fully avoided by reducing the nozzle-rebar distance, but it could come with the expense of ridges (which could affect the final geometry of the structure), since material from one strand would flow on top of a previously deposited stand. Similarly, decreasing the geometric ratio was found to reduce the presence of air voids, because a smaller geometric ratio resulted in wider stands that occupied more of the space around the rebars. However, the smallest ratios also resulted in ridges. It was also found that less air voids were formed when decreasing the speed ratio, since the cross-sectional area of the strands increased proportionally, thereby occupying more space around the rebars. By decreasing the nozzle-rebar distance, geometric ratio, and speed ratio, voids were omitted around the horizontal rebar, but air voids would still be introduced for the cross-shaped rebar. Those air voids could be eliminated by changing the toolpath and some processing parameters, as well as altering the geometry of the reinforcement joint to a smooth transition between the horizontal and vertical rebar. The results highlight that it is non-trivial to avoid air voids when integrating rebars in 3DCP, but that the CFD model is a very strong digital tool when it comes to securing a good bonding between the reinforcement and concrete. A limitation of the CFD model is that with the current computational power it is not possible to simulate a full 3DCP structure. Nevertheless, the CFD model is powerful when it comes to understanding and optimizing printing strategies for individual reinforcement details. In future research, it would be interesting to exploit the CFD model to systematically investigate various reinforcement setups and based on the model results generate response surfaces or lookup tables, which can be coupled with the 3DCP toolpath software. This approach would have several benefits: 1) the computational heavy CFD model could run all scenarios in parallel, thereby minimizing the physical time spent on calculations; 2) one could avoid repetition of individual reinforcement studies; and 3) the response surfaces or lookup tables could in a computational light manner come up with printing strategies for all reinforcement details in a full 3DCP structure.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to acknowledge the support of the Danish Council for Independent Research (DFF) | Technology and Production Sciences (FTP) (Contract No. 8022-00042B). Also, the authors would like to acknowledge the support of the Innovation Fund Denmark (Grant No. 8055-00030B: Next Generation of 3D-printed Concrete Structures and Grant no. 0223-00084B: ThermoForm – Robotic ThermoSetting Printing of Large-Scale Construction Formwork), Moreover, the support of FLOW-3D® regarding licenses is acknowledged.
Appendix A.
This analysis varies the shear modulus (i.e., 20, 50, and 100 kPa) in the case Dr= 8 mm, as seen in Fig. A.1, which presents the cross-sectional shapes (top) and the volume fraction of air voids (bottom). It can be seen that increasing the shear modulus slightly reduces the air void formation. This is because the larger shear modulus enhances the ability of the material to act against the shear deformation. However, an increase in shear modulus also extensively increases the computational time of solving the non-linear elastic response of the elasto-viscoplastic material. For example, the computational time of Model 3 is about 6, 12, and 18 days for a shear modulus of 20, 50, and 100 kPa, respectively. Therefore, the shear modulus is reduced to 100 kPa from the measured value (i.e., 200 kPa) to compare the simulated results with experiments. Furthermore, the shear modulus 20 kPa is chosen for the parametric study in 3.2 Influence of materials properties, 3.3 Influence of processing conditions, 3.4 Cross-shaped reinforcement. These assumptions seem reasonable to avoid extensive computational time consumption since the differences found in Fig. A.1 are not substantial.
Fig. A.1. Air void formation in the cross-sections of the printed parts (top) and the volume fraction of air voids (bottom) for different shear modulus.
References
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•Landslide travel distance is considered for the first time in a predictive equation.
•Predictive equation derived from databases using 3D physical and numerical modeling.
•The equation was successfully tested on the 2018 Anak Krakatau tsunami event.
•The developed equation using three-dimensional data exhibits a 91 % fitting quality.
Abstract
Landslide tsunamis, responsible for thousands of deaths and significant damage in recent years, necessitate the allocation of sufficient time and resources for studying these extreme natural hazards. This study offers a step change in the field by conducting a large number of three-dimensional numerical experiments, validated by physical tests, to develop a predictive equation for the maximum initial amplitude of tsunamis generated by subaerial landslides. We first conducted a few 3D physical experiments in a wave basin which were then applied for the validation of a 3D numerical model based on the Flow3D-HYDRO package. Consequently, we delivered 100 simulations using the validated model by varying parameters such as landslide volume, water depth, slope angle and travel distance. This large database was subsequently employed to develop a predictive equation for the maximum initial tsunami amplitude. For the first time, we considered travel distance as an independent parameter for developing the predictive equation, which can significantly improve the predication accuracy. The predictive equation was tested for the case of the 2018 Anak Krakatau subaerial landslide tsunami and produced satisfactory results.
The Anak Krakatau landslide tsunami on 22nd December 2018 was a stark reminder of the dangers posed by subaerial landslide tsunamis (Ren et al., 2020; Mulia et al. 2020a; Borrero et al., 2020; Heidarzadeh et al., 2020; Grilli et al., 2021). The collapse of the volcano’s southwest side into the ocean triggered a tsunami that struck the Sunda Strait, leading to approximately 450 fatalities (Syamsidik et al., 2020; Mulia et al., 2020b) (Fig. 1). As shown in Fig. 1, landslide tsunamis (both submarine and subaerial) have been responsible for thousands of deaths and significant damage to coastal communities worldwide. These incidents underscored the critical need for advanced research into landslide-generated waves to aid in hazard prediction and mitigation. This is further emphasized by recent events such as the 28th of November 2020 landslide tsunami in the southern coast mountains of British Columbia (Canada), where an 18 million m3 rockslide generated a massive tsunami, with over 100 m wave run-up, causing significant environmental and infrastructural damage (Geertsema et al., 2022).
Physical modelling and numerical simulation are crucial tools in the study of landslide-induced waves due to their ability to replicate and analyse the complex dynamics of landslide events (Kim et al., 2020). In two-dimensional (2D) modelling, the discrepancy between dimensions can lead to an artificial overestimation of wave amplification (e.g., Heller and Spinneken, 2015). This limitation is overcome with 3D modelling, which enables the scaled-down representation of landslide-generated waves while avoiding the simplifications inherent in 2D approaches (Erosi et al., 2019). Another advantage of 3D modelling in studying landslide-generated waves is its ability to accurately depict the complex dynamics of wave propagation, including lateral and radial spreading from the slide impact zone, a feature unattainable with 2D models (Heller and Spinneken, 2015).
Physical experiments in tsunami research, as presented by authors such as Romano et al. (2020), McFall and Fritz (2016), and Heller and Spinneken (2015), have supported 3D modelling works through validation and calibration of the numerical models to capture the complexities of wave generation and propagation. Numerical modelling has increasingly complemented experimental approach in tsunami research due to the latter’s time and resource-intensive nature, particularly for 3D models (Li et al., 2019; Kim et al., 2021). Various numerical approaches have been employed, from Eulerian and Lagrangian frameworks to depth-averaged and Navier–Stokes models, enhancing our understanding of tsunami dynamics (Si et al., 2018; Grilli et al., 2019; Heidarzadeh et al., 2017, 2020; Iorio et al., 2021; Zhang et al., 2021; Kirby et al., 2022; Wang et al., 2021, 2022; Hu et al., 2022). The sophisticated numerical techniques, including the Particle Finite Element Method and the Immersed Boundary Method, have also shown promising results in modelling highly dynamic landslide scenarios (Mulligan et al., 2020; Chen et al., 2020). Among these methods and techniques, FLOW-3D HYDRO stands out in simulating landslide-generated tsunami waves due to its sophisticated technical features such as offering Tru Volume of Fluid (VOF) method for precise free surface tracking (e.g., Sabeti and Heidarzadeh 2022a). TruVOF distinguishes itself through a split Lagrangian approach, adeptly reducing cumulative volume errors in wave simulations by dynamically updating cell volume fractions and areas with each time step. Its intelligent adaptation of time step size ensures precise capture of evolving free surfaces, offering unparalleled accuracy in modelling complex fluid interfaces and behaviour (Flow Science, 2023).
Predictive equations play a crucial role in assessing the potential hazards associated with landslide-generated tsunami waves due to their ability to provide risk assessment and warnings. These equations can offer swift and reasonable evaluations of potential tsunami impacts in the absence of detailed numerical simulations, which can be time-consuming and expensive to produce. Among multiple factors and parameters within a landslide tsunami generation, the initial maximum wave amplitude (Fig. 1) stands out due to its critical role. While it is most likely that the initial wave generated by a landslide will have the highest amplitude, it is crucial to clarify that the term “initial maximum wave amplitude” refers to the highest amplitude within the first set of impulse waves. This parameter is essential in determining the tsunami’s impact severity, with higher amplitudes signalling a greater destructive potential (Sabeti and Heidarzadeh 2022a). Additionally, it plays a significant role in tsunami modelling, aiding in the prediction of wave propagation and the assessment of potential impacts.
In this study, we initially validate the FLOW-3D HYDRO model through a series of physical experiments conducted in a 3D wave tank at University of Bath (UK). Upon confirmation of the model’s accuracy, we use it to systematically vary parameters namely landslide volume, water depth, slope angle, and travel distance, creating an extensive database. Alongside this, we perform a sensitivity analysis on these variables to discern their impacts on the initial maximum wave amplitude. The generated database was consequently applied to derive a non-dimensional predictive equation aimed at estimating the initial maximum wave amplitude in real-world landslide tsunami events.
Two innovations of this study are: (i) The predictive equation of this study is based on a large number of 3D experiments whereas most of the previous equations were based on 2D results, and (ii) For the first time, the travel distance is included in the predictive equation as an independent parameter. To evaluate the performance of our predictive equation, we applied it to a previous real-world subaerial landslide tsunami, i.e., the Anak Krakatau 2018 event. Furthermore, we compare the performance of our predictive equation with other existing equations.
2. Data and methods
The methodology applied in this research is a combination of physical and numerical modelling. Limited physical modelling was performed in a 3D wave basin at the University of Bath (UK) to provide data for calibration and validation of the numerical model. After calibration and validation, the numerical model was employed to model a large number of landslide tsunami scenarios which allowed us to develop a database for deriving a predictive equation.
2.1. Physical experiments
To validate our numerical model, we conducted a series of physical experiments including two sets in a 3D wave basin at University of Bath, measuring 2.50 m in length (WL), 2.60 m in width (WW), and 0.60 m in height (WH) (Fig. 2a). Conducting two distinct sets of experiments (Table 1), each with different setups (travel distance, location, and water depth), provided a robust framework for validation of the numerical model. For wave measurement, we employed a twin wire wave gauge from HR Wallingford (https://equipit.hrwallingford.com). In these experiments, we used a concrete prism solid block, the dimensions of which are outlined in Table 2. In our experiments, we employed a concrete prism solid block with a density of 2600 kg/m3, chosen for its similarity to the natural density of landslides, akin to those observed with the 2018 Anak Krakatau tsunami, where the landslide composition is predominantly solid rather than granular. The block’s form has also been endorsed in prior studies (Watts, 1998; Najafi-Jilani and Ataie-Ashtiani, 2008) as a suitable surrogate for modelling landslide-induced waves. A key aspect of our methodology was addressing scale effects, following the guidelines proposed by Heller et al. (2008) as it is described in Table 1. To enhance the reliability and accuracy of our experimental data, we conducted each physical experiment three times which revealed all three experimental waveforms were identical. This repetition was aimed at minimizing potential errors and inconsistencies in laboratory measurements.
Table 1. The locations and other information of the laboratory setups for making landslide-generated waves in the physical wave basin. This table details the specific parameters for each setup, including slope range (α), slide volume (V), kinematic viscosity (ν), water depth (h), travel distance (D), surface tension coefficient of water (σ), Reynolds number (R), Weber number (W), and the precise coordinates of the wave gauges (WG).
The acceptable ranges for avoiding scale effects are based on the study by Heller et al. (2008).⁎⁎
The Reynolds number (R) is given by g0.5h1.5/ν, with ν denoting the kinematic viscosity. The Weber number (W) is W = ρgh2/σ, where σ represents surface tension coefficient and ρ = 1000kg/m3 is the density of water. In our experiments, conducted at a water temperature of approximately 20 °C, the kinematic viscosity (ν) and the surface tension coefficient of water (σ) are 1.01 × 10−6 m²/s and 0.073 N/m, respectively (Kestin et al., 1978).
Table 2. Specifications of the solid block used in physical experiments for generating subaerial landslides in the laboratory.
Solid-block attributes
Property metrics
Geometric shape
Slide width (bs)
0.26 m
Slide length (ls)
0.20 m
Slide thickness (s)
0.10 m
Slide volume (V)
2.60 × 10−3 m3
Specific gravity, (γs)
2.60
Slide weight (ms)
6.86 kg
2.2. Numerical simulations applying FLOW-3D hydro
The detailed theoretical framework encompassing the governing equations, the computational methodologies employed, and the specific techniques used for tracking the water surface in these simulations are thoroughly detailed in the study by Sabeti et al. (2024). Here, we briefly explain some of the numerical details. We defined a uniform mesh for our flow domain, carefully crafted with a fine spatial resolution of 0.005 m (i.e., grid size). The dimensions of the numerical model directly matched those of our wave basin used in the physical experiment, being 2.60 m wide, 0.60 m deep, and 2.50 m long (Fig. 2). This design ensures comprehensive coverage of the study area. The output intervals of the numerical model are set at 0.02 s. This timing is consistent with the sampling rates of wave gauges used in laboratory settings. The friction coefficient in the FLOW-3D HYDRO is designated as 0.45. This value corresponds to the Coulombic friction measurements obtained in the laboratory, ensuring that the simulation accurately reflects real-world physical interactions.
In order to simulate the landslide motion, we applied coupled motion objects in FLOW-3D-HYDRO where the dynamics are predominantly driven by gravity and surface friction. This methodology stands in contrast to other models that necessitate explicit inputs of force and torque. This approach ensures that the simulation more accurately reflects the natural movement of landslides, which is heavily reliant on gravitational force and the interaction between sliding surfaces. The stability of the numerical simulations is governed by the Courant Number criterion (Courant et al., 1928), which dictates the maximum time step (Δt) for a given mesh size (Δx) and flow speed (U). According to Courant et al. (1928), this number is required to stay below one to ensure stability of numerical simulations. In our simulations, the Courant number is always maintained below one.
In alignment with the parameters of physical experiments, we set the fluid within the mesh to water, characterized by a density of 1000 kg/m³ at a temperature of 20 °C. Furthermore, we defined the top, front, and back surfaces of the mesh as symmetry planes. The remaining surfaces are designated as wall types, incorporating no-slip conditions to accurately simulate the interaction between the fluid and the boundaries. In terms of selection of an appropriate turbulence model, we selected the k–ω model that showed a better performance than other turbulence methods (e.g., Renormalization-Group) in a previous study (Sabeti et al., 2024). The simulations are conducted using a PC Intel® Core™ i7-10510U CPU with a frequency of 1.80 GHz, and a 16 GB RAM. On this PC, completion of a 3-s simulation required approximately 12.5 h.
2.3. Validation
The FLOW-3D HYDRO numerical model was validated using the two physical experiments (Fig. 3) outlined in Table 1. The level of agreement between observations (Oi) and simulations (Si) is examined using the following equation:(1)�=|��−����|×100where ε represents the mismatch error, Oi denotes the observed laboratory values, and Si represents the simulated values from the FLOW-3D HYDRO model. The results of this validation process revealed that our model could replicate the waves generated in the physical experiments with a reasonable degree of mismatch (ε): 14 % for Lab 1 and 8 % for Lab 2 experiments, respectively (Fig. 3). These values indicate that while the model is not perfect, it provides a sufficiently close approximation of the real-world phenomena.
In terms of mesh efficiency, we varied the mesh size to study sensitivity of the numerical results to mesh size. First, by halving the mesh size and then by doubling it, we repeated the modelling by keeping other parameters unchanged. This analysis guided that a mesh size of ∆x = 0.005 m is the most effective for the setup of this study. The total number of computational cells applying mesh size of 0.005 m is 9.269 × 106.
2.4. The dataset
The validated numerical model was employed to conduct 100 simulations, incorporating variations in four key landslide parameters namely water depth, slope angle, slide volume, and travel distance. This methodical approach was essential for a thorough sensitivity analysis of these variables, and for the creation of a detailed database to develop a predictive equation for maximum initial tsunami amplitude. Within the model, 15 distinct slide volumes were established, ranging from 0.10 × 10−3 m3 to 6.25 × 10−3 m3 (Table 3). The slope angle varied between 35° and 55°, and water depth ranged from 0.24 m to 0.27 m. The travel distance of the landslides was varied, spanning from 0.04 m to 0.07 m. Detailed configurations of each simulation, along with the maximum initial wave amplitudes and dominant wave periods are provided in Table 4.
Table 3. Geometrical information of the 15 solid blocks used in numerical modelling for generating landslide tsunamis. Parameters are: ls, slide length; bs, slide width; s, slide thickness; γs, specific gravity; and V, slide volume.
Solid block
ls (m)
bs (m)
s (m)
V (m3)
γs
Block-1
0.310
0.260
0.155
6.25 × 10−3
2.60
Block-2
0.300
0.260
0.150
5.85 × 10−3
2.60
Block-3
0.280
0.260
0.140
5.10 × 10−3
2.60
Block-4
0.260
0.260
0.130
4.39 × 10−3
2.60
Block-5
0.240
0.260
0.120
3.74 × 10−3
2.60
Block-6
0.220
0.260
0.110
3.15 × 10−3
2.60
Block-7
0.200
0.260
0.100
2.60 × 10−3
2.60
Block-8
0.180
0.260
0.090
2.11 × 10−3
2.60
Block-9
0.160
0.260
0.080
1.66 × 10−3
2.60
Block-10
0.140
0.260
0.070
1.27 × 10−3
2.60
Block-11
0.120
0.260
0.060
0.93 × 10−3
2.60
Block-12
0.100
0.260
0.050
0.65 × 10−3
2.60
Block-13
0.080
0.260
0.040
0.41 × 10−3
2.60
Block-14
0.060
0.260
0.030
0.23 × 10−3
2.60
Block-15
0.040
0.260
0.020
0.10 × 10−3
2.60
Table 4. The numerical simulation for the 100 tests performed in this study for subaerial solid-block landslide-generated waves. Parameters are aM, maximum wave amplitude; α, slope angle; h, water depth; D, travel distance; and T, dominant wave period. The location of the wave gauge is X=1.030 m, Y=1.210 m, and Z=0.050 m. The properties of various solid blocks are presented in Table 3.
Test-
Block No
α (°)
h (m)
D (m)
T(s)
aM (m)
1
Block-7
45
0.246
0.029
0.510
0.0153
2
Block-7
45
0.246
0.030
0.505
0.0154
3
Block-7
45
0.246
0.031
0.505
0.0156
4
Block-7
45
0.246
0.032
0.505
0.0158
5
Block-7
45
0.246
0.033
0.505
0.0159
6
Block-7
45
0.246
0.034
0.505
0.0160
7
Block-7
45
0.246
0.035
0.505
0.0162
8
Block-7
45
0.246
0.036
0.505
0.0166
9
Block-7
45
0.246
0.037
0.505
0.0167
10
Block-7
45
0.246
0.038
0.505
0.0172
11
Block-7
45
0.246
0.039
0.505
0.0178
12
Block-7
45
0.246
0.040
0.505
0.0179
13
Block-7
45
0.246
0.041
0.505
0.0181
14
Block-7
45
0.246
0.042
0.505
0.0183
15
Block-7
45
0.246
0.043
0.505
0.0190
16
Block-7
45
0.246
0.044
0.505
0.0197
17
Block-7
45
0.246
0.045
0.505
0.0199
18
Block-7
45
0.246
0.046
0.505
0.0201
19
Block-7
45
0.246
0.047
0.505
0.0191
20
Block-7
45
0.246
0.048
0.505
0.0217
21
Block-7
45
0.246
0.049
0.505
0.0220
22
Block-7
45
0.246
0.050
0.505
0.0226
23
Block-7
45
0.246
0.051
0.505
0.0236
24
Block-7
45
0.246
0.052
0.505
0.0239
25
Block-7
45
0.246
0.053
0.510
0.0240
26
Block-7
45
0.246
0.054
0.505
0.0241
27
Block-7
45
0.246
0.055
0.505
0.0246
28
Block-7
45
0.246
0.056
0.505
0.0247
29
Block-7
45
0.246
0.057
0.505
0.0248
30
Block-7
45
0.246
0.058
0.505
0.0249
31
Block-7
45
0.246
0.059
0.505
0.0251
32
Block-7
45
0.246
0.060
0.505
0.0257
33
Block-1
45
0.246
0.045
0.505
0.0319
34
Block-2
45
0.246
0.045
0.505
0.0294
35
Block-3
45
0.246
0.045
0.505
0.0282
36
Block-4
45
0.246
0.045
0.505
0.0262
37
Block-5
45
0.246
0.045
0.505
0.0243
38
Block-6
45
0.246
0.045
0.505
0.0223
39
Block-7
45
0.246
0.045
0.505
0.0196
40
Block-8
45
0.246
0.045
0.505
0.0197
41
Block-9
45
0.246
0.045
0.505
0.0198
42
Block-10
45
0.246
0.045
0.505
0.0184
43
Block-11
45
0.246
0.045
0.505
0.0173
44
Block-12
45
0.246
0.045
0.505
0.0165
45
Block-13
45
0.246
0.045
0.404
0.0153
46
Block-14
45
0.246
0.045
0.404
0.0124
47
Block-15
45
0.246
0.045
0.505
0.0066
48
Block-7
45
0.202
0.045
0.404
0.0220
49
Block-7
45
0.204
0.045
0.404
0.0219
50
Block-7
45
0.206
0.045
0.404
0.0218
51
Block-7
45
0.208
0.045
0.404
0.0217
52
Block-7
45
0.210
0.045
0.404
0.0216
53
Block-7
45
0.212
0.045
0.404
0.0215
54
Block-7
45
0.214
0.045
0.505
0.0214
55
Block-7
45
0.216
0.045
0.505
0.0214
56
Block-7
45
0.218
0.045
0.505
0.0213
57
Block-7
45
0.220
0.045
0.505
0.0212
58
Block-7
45
0.222
0.045
0.505
0.0211
59
Block-7
45
0.224
0.045
0.505
0.0208
60
Block-7
45
0.226
0.045
0.505
0.0203
61
Block-7
45
0.228
0.045
0.505
0.0202
62
Block-7
45
0.230
0.045
0.505
0.0201
63
Block-7
45
0.232
0.045
0.505
0.0201
64
Block-7
45
0.234
0.045
0.505
0.0200
65
Block-7
45
0.236
0.045
0.505
0.0199
66
Block-7
45
0.238
0.045
0.404
0.0196
67
Block-7
45
0.240
0.045
0.404
0.0194
68
Block-7
45
0.242
0.045
0.404
0.0193
69
Block-7
45
0.244
0.045
0.404
0.0192
70
Block-7
45
0.246
0.045
0.505
0.0190
71
Block-7
45
0.248
0.045
0.505
0.0189
72
Block-7
45
0.250
0.045
0.505
0.0187
73
Block-7
45
0.252
0.045
0.505
0.0187
74
Block-7
45
0.254
0.045
0.505
0.0186
75
Block-7
45
0.256
0.045
0.505
0.0184
76
Block-7
45
0.258
0.045
0.505
0.0182
77
Block-7
45
0.259
0.045
0.505
0.0183
78
Block-7
45
0.260
0.045
0.505
0.0191
79
Block-7
45
0.261
0.045
0.505
0.0192
80
Block-7
45
0.262
0.045
0.505
0.0194
81
Block-7
45
0.263
0.045
0.505
0.0195
82
Block-7
45
0.264
0.045
0.505
0.0195
83
Block-7
45
0.265
0.045
0.505
0.0197
84
Block-7
45
0.266
0.045
0.505
0.0197
85
Block-7
45
0.267
0.045
0.505
0.0198
86
Block-7
45
0.270
0.045
0.505
0.0199
87
Block-7
30
0.246
0.045
0.505
0.0101
88
Block-7
35
0.246
0.045
0.505
0.0107
89
Block-7
36
0.246
0.045
0.505
0.0111
90
Block-7
37
0.246
0.045
0.505
0.0116
91
Block-7
38
0.246
0.045
0.505
0.0117
92
Block-7
39
0.246
0.045
0.505
0.0119
93
Block-7
40
0.246
0.045
0.505
0.0121
94
Block-7
41
0.246
0.045
0.505
0.0127
95
Block-7
42
0.246
0.045
0.404
0.0154
96
Block-7
43
0.246
0.045
0.404
0.0157
97
Block-7
44
0.246
0.045
0.404
0.0162
98
Block-7
45
0.246
0.045
0.505
0.0197
99
Block-7
50
0.246
0.045
0.505
0.0221
100
Block-7
55
0.246
0.045
0.505
0.0233
In all these 100 simulations, the wave gauge was consistently positioned at coordinates X=1.09 m, Y=1.21 m, and Z=0.05 m. The dominant wave period for each simulation was determined using the Fast Fourier Transform (FFT) function in MATLAB (MathWorks, 2023). Furthermore, the classification of wave types was carried out using a wave categorization graph according to Sorensen (2010), as shown in Fig. 4a. The results indicate that the majority of the simulated waves are on the border between intermediate and deep-water waves, and they are categorized as Stokes waves (Fig. 4a). Four sample waveforms from our 100 numerical experiments are provided in Fig. 4b.
The dataset in Table 4 was used to derive a new predictive equation that incorporates travel distance for the first time to estimate the initial maximum tsunami amplitude. In developing this equation, a genetic algorithm optimization technique was implemented using MATLAB (MathWorks 2023). This advanced approach entailed the use of genetic algorithms (GAs), an evolutionary algorithm type inspired by natural selection processes (MathWorks, 2023). This technique is iterative, involving selection, crossover, and mutation processes to evolve solutions over several generations. The goal was to identify the optimal coefficients and powers for each landslide parameter in the predictive equation, ensuring a robust and reliable model for estimating maximum wave amplitudes. Genetic Algorithms excel at optimizing complex models by navigating through extensive combinations of coefficients and exponents. GAs effectively identify highly suitable solutions for the non-linear and complex relationships between inputs (e.g., slide volume, slope angle, travel distance, water depth) and the output (i.e., maximum initial wave amplitude, aM). MATLAB’s computational environment enhances this process, providing robust tools for GA to adapt and evolve solutions iteratively, ensuring the precision of the predictive model (Onnen et al., 1997). This approach leverages MATLAB’s capabilities to fine-tune parameters dynamically, achieving an optimal equation that accurately estimates aM. It is important to highlight that the nondimensionalized version of this dataset is employed to develop a predictive equation which enables the equation to reproduce the maximum initial wave amplitude (aM) for various subaerial landslide cases, independent of their dimensional differences (e.g., Heler and Hager 2014; Heller and Spinneken 2015; Sabeti and Heidarzadeh 2022b). For this nondimensionalization, we employed the water depth (h) to nondimensionalize the slide volume (V/h3) and travel distance (D/h). The slide thickness (s) was applied to nondimensionalize the water depth (h/s).
2.5. Landslide velocity
In discussing the critical role of landslide velocity for simulating landslide-generated waves, we focus on the mechanisms of landslide motion and the techniques used to record landslide velocity in our simulations (Fig. 5). Also, we examine how these methods were applied in two distinct scenarios: Lab 1 and Lab 2 (see Table 1 for their details). Regarding the process of landslide movement, a slide starts from a stationary state, gaining momentum under the influence of gravity and this acceleration continues until the landslide collides with water, leading to a significant reduction in its speed before eventually coming to a stop (Fig. 5) (e.g., Panizzo et al. 2005).
To measure the landslide’s velocity in our simulations, we attached a probe at the centre of the slide, which supplied a time series of the velocity data. The slide’s velocity (vs) peaks at the moment it enters the water (Fig. 5), a point referred to as the impact time (tImp). Following this initial impact, the slides continue their underwater movement, eventually coming to a complete halt (tStop). Given the results in Fig. 5, it can be seen that Lab 1, with its longer travel distance (0.070 m), exhibits a higher peak velocity of 1.89 m/s. This increase in velocity is attributed to the extended travel distance allowing more time for the slide to accelerate under gravity. Whereas Lab 2, featuring a shorter travel distance (0.045 m), records a lower peak velocity of 1.78 m/s. This difference underscores how travel distance significantly influences the dynamics of landslide motion. After reaching the peak, both profiles show a sharp decrease in velocity, marking the transition to submarine motion until the slides come to a complete stop (tStop). There are noticeable differences observable in Fig. 5 between the Lab-1 and Lab-2 simulations, including the peaks at 0.3 s . These variations might stem from the placement of the wave gauge, which differs slightly in each scenario, as well as the water depth’s minor discrepancies and, the travel distance.
2.6. Effect of air entrainment
In this section we examine whether it is required to consider air entrainment for our modelling or not as the FLOW-3D HYDRO package is capable of modelling air entrainment. The process of air entrainment in water during a landslide tsunami and its subsequent transport involve two key components: the quantification of air entrainment at the water surface, and the simulation of the air’s transport within the fluid (Hirt, 2003). FLOW-3D HYDRO employs the air entrainment model to compute the volume of air entrained at the water’s surface utilizing three approaches: a constant density model, a variable density model accounting for bulking, and a buoyancy model that adds the Drift-FLUX mechanism to variable density conditions (Flow Science, 2023). The calculation of the entrainment rate is based on the following equation:(2)�������=������[2(��−�����−2�/���)]1/2where parameters are: Vair, volume of air; Cair, entrainment rate coefficient; As, surface area of fluid; ρ, fluid density; k, turbulent kinetic energy; gn, gravity normal to surface; Lt, turbulent length scale; and σ, surface tension coefficient. The value of k is directly computed from the Reynolds-averaged Navier-Stokes (RANS) (k–w) calculations in our model.
In this study, we selected the variable density + Drift-FLUX model, which effectively captures the dynamics of phase separation and automatically activates the constant density and variable density models. This method simplifies the air-water mixture, treating it as a single, homogeneous fluid within each computational cell. For the phase volume fractions f1and f2, the velocities are expressed in terms of the mixture and relative velocities, denoted as u and ur, respectively, as follows:(3)��1��+�.(�1�)=��1��+�.(�1�)−�.(�1�2��)=0(4)��2��+�.(�2�)=��2��+�.(�2�)−�.(�1�2��)=0
The outcomes from this simulation are displayed in Fig. 6, which indicates that the influence of air entrainment on the generated wave amplitude is approximately 2 %. A value of 0.02 for the entrained air volume fraction means that, in the simulated fluid, approximately 2 % of the volume is composed of entrained air. In other words, for every unit volume of the fluid-air mixture at that location, 2 % is air and the remaining 98 % is water. The configuration of Test-17 (Table 4) was employed for this simulation. While the effect of air entrainment is anticipated to be more significant in models of granular landslide-generated waves (Fritz, 2002), in our simulations we opted not to incorporate this module due to its negligible impact on the results.
3. Results
In this section, we begin by presenting a sequence of our 3D simulations capturing different time steps to illustrate the generation process of landslide-generated waves. Subsequently, we derive a new predictive equation to estimate the maximum initial wave amplitude of landslide-generated waves and assess its performance.
3.1. Wave generation and propagation
To demonstrate the wave generation process in our simulation, we reference Test-17 from Table 4, where we employed Block-7 (Tables 3, 4). In this configuration, the slope angle was set to 45°, with a water depth of 0.246 m and a travel distance at 0.045 m (Fig. 7). At 0.220 s, the initial impact of the moving slide on the water is depicted, marking the onset of the wave generation process (Fig. 7a). Disturbances are localized to the immediate area of impact, with the rest of the water surface remaining undisturbed. At this time, a maximum water particle velocity of 1.0 m/s – 1.2 m/s is seen around the impact zone (Fig. 7d). Moving to 0.320 s, the development of the wave becomes apparent as energy transfer from the landslide to the water creates outwardly radiating waves with maximum water particle velocity of up to around 1.6 m/s – 1.8 m/s (Fig. 7b, e). By the time 0.670 s, the wave has fully developed and is propagating away from the impact point exhibiting maximum water particle velocity of up to 2.0 m/s – 2.1 m/s. Concentric wave fronts are visible, moving outwards in all directions, with a colour gradient signifying the highest wave amplitude near the point of landslide entry, diminishing with distance (Fig. 7c, f).
3.2. Influence of landslide parameters on tsunami amplitude
In this section, we investigate the effects of various landslide parameters namely slide volume (V), water depth (h), slipe angle (α) and travel distance (D) on the maximum initial wave amplitude (aM). Fig. 8 presents the outcome of these analyses. According to Fig. 8, the slide volume, slope angle, and travel distance exhibit a direct relationship with the wave amplitude, meaning that as these parameters increase, so does the amplitude. Conversely, water depth is inversely related to the maximum initial wave amplitude, suggesting that the deeper the water depth, the smaller the maximum wave amplitude will be (Fig. 8b).
Fig. 8a highlights the pronounced impact of slide volume on the aM, demonstrating a direct correlation between the two variables. For instance, in the range of slide volumes we modelled (Fig. 8a), The smallest slide volume tested, measuring 0.10 × 10−3 m3, generated a low initial wave amplitude (aM= 0.0066 m) (Table 4). In contrast, the largest volume tested, 6.25 × 10−3 m3, resulted in a significantly higher initial wave amplitude (aM= 0.0319 m) (Table 4). The extremities of these results emphasize the slide volume’s paramount impact on wave amplitude, further elucidated by their positions as the smallest and largest aM values across all conducted tests (Table 4). This is corroborated by findings from the literature (e.g., Murty, 2003), which align with the observed trend in our simulations.
The slope angle’s influence on aM was smooth. A steady increase of wave amplitude was observed as the slope angle increased (Fig. 8c). In examining travel distance, an anomaly was identified. At a travel distance of 0.047 m, there was an unexpected dip in aM, which deviates from the general increasing trend associated with longer travel distances. This singular instance could potentially be attributed to a numerical error. Beyond this point, the expected pattern of increasing aM with longer travel distances resumes, suggesting that the anomaly at 0.047 m is an outlier in an otherwise consistent trend, and thus this single data point was overlooked while deriving the predictive equation. Regarding the inverse relationship between water depth and wave amplitude, our result (Fig. 8b) is consistent with previous reports by Fritz et al. (2003), (2004), and Watts et al. (2005).
The insights from Fig. 8 informed the architecture of the predictive equation in the next Section, with slide volume, travel distance, and slope angle being multiplicatively linked to wave amplitude underscoring their direct correlations with wave amplitude. Conversely, water depth is incorporated as a divisor, representing its inverse relationship with wave amplitude. This structure encapsulates the dynamics between the landslide parameters and their influence on the maximum initial wave amplitude as discussed in more detail in the next Section.
3.3. Predictive equation
Building on our sensitivity analysis of landslide parameters, as detailed in Section 3.2, and utilizing our nondimensional dataset, we have derived a new predictive equation as follows:(5)��/ℎ=0.015(tan�)0.10(�ℎ3)0.90(�ℎ)0.10(ℎ�)−0.11where, V is sliding volume, h is water depth, α is slope angle, and s is landslide thickness. It is important to note that this equation is valid only for subaerial solid-block landslide tsunamis as all our experiments were for this type of waves. The performance of this equation in predicting simulation data is demonstrated by the satisfactory alignment of data points around a 45° line, indicating its accuracy and reliability with regard to the experimental dataset (Fig. 9). The quality of fit between the dataset and Eq. (5) is 91 % indicating that Eq. (5) represents the dataset very well. Table 5 presents Eq. (5) alongside four other similar equations previously published. Two significant distinctions between our Eq. (5) and these others are: (i) Eq. (5) is derived from 3D experiments, whereas the other four equations are based on 2D experiments. (ii) Unlike the other equations, our Eq. (5) incorporates travel distance as an independent parameter.
Table 5. Performance comparison among our newly-developed equation and existing equations for estimating the maximum initial amplitude (aM) of the 2018 Anak Krakatau subaerial landslide tsunami. Parameters: aM, initial maximum wave amplitude; h, water depth; vs, landslide velocity; V, slide volume; bs, slide width; ls, slide length; s, slide thickness; α, slope angle; and ����, volume of the final immersed landslide. We considered ����= V as the slide volume.
Geometrical and kinematic parameters of the 2018 Anak Krakatau subaerial landslide based on Heidarzadeh et al. (2020), Grilli et al. (2019) and Grilli et al. (2021): V=2.11 × 107 m3, h= 50 m; s= 114 m; α= 45°; ls=1250 m; bs= 2700 m; vs=44.9 m/s; D= 2500 m; aM= 100 m −150 m.⁎⁎
aM= An average value of aM = 134 m is considered in this study.⁎⁎⁎
The equation of Bolin et al. (2014) is based on the reformatted one reported by Lindstrøm (2016).⁎⁎⁎⁎
Error is calculated using Eq. (1), where the calculated aM is assumed as the simulated value.
Additionally, we evaluated the performance of this equation using the real-world data from the 2018 Anak Krakatau subaerial landslide tsunami. Based on previous studies (Heidarzadeh et al., 2020; Grilli et al., 2019, 2021), we were able to provide a list of parameters for the subaerial landslide and associated tsunami for the 2018 Anak Krakatau event (see footnote of Table 5). We note that the data of the 2018 Anak Krakatau event was not used while deriving Eq. (5). The results indicate that Eq. (5) predicts the initial amplitude of the 2018 Anak Krakatau tsunami as being 130 m indicating an error of 2.9 % compared to the reported average amplitude of 134 m for this event. This performance indicates an improvement compared to the previous equation reported by Sabeti and Heidarzadeh (2022a) (Table 5). In contrast, the equations from Robbe-Saule et al. (2021) and Bolin et al. (2014) demonstrate higher discrepancies of 4200 % and 77 %, respectively (Table 5). Although Noda’s (1970) equation reproduces the tsunami amplitude of 134 m accurately (Table 5), it is crucial to consider its limitations, notably not accounting for parameters such as slope angle and travel distance.
It is essential to recognize that both travel distance and slope angle significantly affect wave amplitude. In our model, captured in Eq. (5), we integrate the slope angle (α) through the tangent function, i.e., tan α. This choice diverges from traditional physical interpretations that often employ the cosine or sine function (e.g., Heller and Hager, 2014; Watts et al., 2003). We opted for the tangent function because it more effectively reflects the direct impact of slope steepness on wave generation, yielding superior estimations compared to conventional methods.
The significance of this study lies in its application of both physical and numerical 3D experiments and the derivation of a predictive equation based on 3D results. Prior research, e.g. Heller et al. (2016), has reported notable discrepancies between 2D and 3D wave amplitudes, highlighting the important role of 3D experiments. It is worth noting that the suitability of applying an equation derived from either 2D or 3D data depends on the specific geometry and characteristics inherent in the problem being addressed. For instance, in the case of a long, narrow dam reservoir, an equation derived from 2D data would likely be more suitable. In such contexts, the primary dynamics of interest such as flow patterns and potential wave propagation are predominantly two-dimensional, occurring along the length and depth of the reservoir. This simplification to 2D for narrow dam reservoirs allows for more accurate modelling of these dynamics.
This study specifically investigates waves initiated by landslides, focusing on those characterized as solid blocks instead of granular flows, with slope angles confined to a range of 25° to 60°. We acknowledge the additional complexities encountered in real-world scenarios, such as dynamic density and velocity of landslides, which could affect the estimations. The developed equation in this study is specifically designed to predict the maximum initial amplitude of tsunamis for the aforementioned specified ranges and types of landslides.
4. Conclusions
Both physical and numerical experiments were undertaken in a 3D wave basin to study solid-block landslide-generated waves and to formulate a predictive equation for their maximum initial wave amplitude. At the beginning, two physical experiments were performed to validate and calibrate a 3D numerical model, which was subsequently utilized to generate 100 experiments by varying different landslide parameters. The generated database was then used to derive a predictive equation for the maximum initial wave amplitude of landslide tsunamis. The main features and outcomes are:
•The predictive equation of this study is exclusively derived from 3D data and exhibits a fitting quality of 91 % when applied to the database.
•For the first time, landslide travel distance was considered in the predictive equation. This inclusion provides more accuracy and flexibility for applying the equation.
•To further evaluate the performance of the predictive equation, it was applied to a real-world subaerial landslide tsunami (i.e., the 2018 Anak Krakatau event) and delivered satisfactory performance.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
RS is supported by the Leverhulme Trust Grant No. RPG-2022-306. MH is funded by open funding of State Key Lab of Hydraulics and Mountain River Engineering, Sichuan University, grant number SKHL2101. We acknowledge University of Bath Institutional Open Access Fund. MH is also funded by the Great Britain Sasakawa Foundation grant no. 6217 (awarded in 2023).
Acknowledgements
Authors are sincerely grateful to the laboratory technician team, particularly Mr William Bazeley, at the Faculty of Engineering, University of Bath for their support during the laboratory physical modelling of this research. We appreciate the valuable insights provided by Mr. Brian Fox (Senior CFD Engineer at Flow Science, Inc.) regarding air entrainment modelling in FLOW-3D HYDRO. We acknowledge University of Bath Institutional Open Access Fund.
Data availability
All data used in this study are given in the body of the article.
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Gonzalo Duró, Mariano De Dios, Alfredo López, Sergio O. Liscia
ABSTRACT
This study presents comparisons between the results of a commercial CFD code and physical model measurements. The case study is a hydro-combined power station operating in spillway mode for a given scenario. Two turbulence models and two scales are implemented to identify the capabilities and limitations of each approach and to determine the selection criteria for CFD modeling for this kind of structure. The main flow characteristics are considered for analysis, but the focus is on a fluctuating frequency phenomenon for accurate quantitative comparisons. Acceptable representations of the general hydraulic functioning are found in all approaches, according to physical modeling. The k-ε RNG, and LES models give good representation of the discharge flow, mean water depths, and mean pressures for engineering purposes. The k-ε RNG is not able to characterize fluctuating phenomena at a model scale but does at a prototype scale. The LES is capable of identifying the dominant frequency at both prototype and model scales. A prototype-scale approach is recommended for the numerical modeling to obtain a better representation of fluctuating pressures for both turbulence models, with the complement of physical modeling for the ultimate design of the hydraulic structures.
본 연구에서는 상용 CFD 코드 결과와 물리적 모델 측정 결과를 비교합니다. 사례 연구는 주어진 시나리오에 대해 배수로 모드에서 작동하는 수력 복합 발전소입니다.
각 접근 방식의 기능과 한계를 식별하고 이러한 종류의 구조에 대한 CFD 모델링의 선택 기준을 결정하기 위해 두 개의 난류 모델과 두 개의 스케일이 구현되었습니다. 주요 흐름 특성을 고려하여 분석하지만 정확한 정량적 비교를 위해 변동하는 주파수 현상에 중점을 둡니다.
일반적인 수리학적 기능에 대한 허용 가능한 표현은 물리적 모델링에 따라 모든 접근 방식에서 발견됩니다. k-ε RNG 및 LES 모델은 엔지니어링 목적을 위한 배출 유량, 평균 수심 및 평균 압력을 잘 표현합니다.
k-ε RNG는 모델 규모에서는 변동 현상을 특성화할 수 없지만 프로토타입 규모에서는 특성을 파악합니다. LES는 프로토타입과 모델 규모 모두에서 주요 주파수를 식별할 수 있습니다.
수력학적 구조의 궁극적인 설계를 위한 물리적 모델링을 보완하여 두 난류 모델에 대한 변동하는 압력을 더 잘 표현하기 위해 수치 모델링에 프로토타입 규모 접근 방식이 권장됩니다.
Figure 1 – Physical scale model (left). Upstream flume and point gauge (right)
Figure 4 – Water levels: physical model (maximum values) and CFD results (mean values)Figure 5 – Instantaneous pressures [Pa] and velocities [m/s] at model scale (bay center)
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Alireza Khoshkonesh1, Blaise Nsom2, Saeid Okhravi3*, Fariba Ahmadi Dehrashid4, Payam Heidarian5, Silvia DiFrancesco6 1 Department of Geography, School of Social Sciences, History, and Philosophy, Birkbeck University of London, London, UK. 2 Université de Bretagne Occidentale. IRDL/UBO UMR CNRS 6027. Rue de Kergoat, 29285 Brest, France. 3 Institute of Hydrology, Slovak Academy of Sciences, Dúbravská cesta 9, 84104, Bratislava, Slovak Republic. 4Department of Water Science and Engineering, Faculty of Agriculture, Bu-Ali Sina University, 65178-38695, Hamedan, Iran. 5 Department of Civil, Environmental, Architectural Engineering and Mathematics, University of Brescia, 25123 Brescia, Italy. 6Niccol`o Cusano University, via Don C. Gnocchi 3, 00166 Rome, Italy. * Corresponding author. Tel.: +421-944624921. E-mail: saeid.okhravi@savba.sk
Abstract
This study aimed to comprehensively investigate the influence of substrate level difference and material composition on dam break wave evolution over two different erodible beds. Utilizing the Volume of Fluid (VOF) method, we tracked free surface advection and reproduced wave evolution using experimental data from the literature. For model validation, a comprehensive sensitivity analysis encompassed mesh resolution, turbulence simulation methods, and bed load transport equations. The implementation of Large Eddy Simulation (LES), non-equilibrium sediment flux, and van Rijn’s (1984) bed load formula yielded higher accuracy compared to alternative approaches. The findings emphasize the significant effect of substrate level difference and material composition on dam break morphodynamic characteristics. Decreasing substrate level disparity led to reduced flow velocity, wavefront progression, free surface height, substrate erosion, and other pertinent parameters. Initial air entrapment proved substantial at the wavefront, illustrating pronounced air-water interaction along the bottom interface. The Shields parameter experienced a one-third reduction as substrate level difference quadrupled, with the highest near-bed concentration observed at the wavefront. This research provides fresh insights into the complex interplay of factors governing dam break wave propagation and morphological changes, advancing our comprehension of this intricate phenomenon.
이 연구는 두 개의 서로 다른 침식층에 대한 댐 파괴파 진화에 대한 기질 수준 차이와 재료 구성의 영향을 종합적으로 조사하는 것을 목표로 했습니다. VOF(유체량) 방법을 활용하여 자유 표면 이류를 추적하고 문헌의 실험 데이터를 사용하여 파동 진화를 재현했습니다.
모델 검증을 위해 메쉬 해상도, 난류 시뮬레이션 방법 및 침대 하중 전달 방정식을 포함하는 포괄적인 민감도 분석을 수행했습니다. LES(Large Eddy Simulation), 비평형 퇴적물 플럭스 및 van Rijn(1984)의 하상 부하 공식의 구현은 대체 접근 방식에 비해 더 높은 정확도를 산출했습니다.
연구 결과는 댐 붕괴 형태역학적 특성에 대한 기질 수준 차이와 재료 구성의 중요한 영향을 강조합니다. 기판 수준 차이가 감소하면 유속, 파면 진행, 자유 표면 높이, 기판 침식 및 기타 관련 매개변수가 감소했습니다.
초기 공기 포집은 파면에서 상당한 것으로 입증되었으며, 이는 바닥 경계면을 따라 뚜렷한 공기-물 상호 작용을 보여줍니다. 기판 레벨 차이가 4배로 증가함에 따라 Shields 매개변수는 1/3로 감소했으며, 파면에서 가장 높은 베드 근처 농도가 관찰되었습니다.
이 연구는 댐 파괴파 전파와 형태학적 변화를 지배하는 요인들의 복잡한 상호 작용에 대한 새로운 통찰력을 제공하여 이 복잡한 현상에 대한 이해를 향상시킵니다.
Fig. 3. Free surface and substrate profiles in all Sp and Ls cases at t = 1 s, t = 3 s, and t = 5 s, arranged left to right (note: the colour contours
correspond to the horizontal component of the flow velocity (u), expressed in m/s).
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Unintended end-of-process depression (EOPD) commonly occurs in laser powder bed fusion (LPBF), leading to poor surface quality and lower fatigue strength, especially for many implants. In this study, a high-fidelity multi-physics meso-scale simulation model is developed to uncover the forming mechanism of this defect. A defect-process map of the EOPD phenomenon is obtained using this simulation model. It is found that the EOPD formation mechanisms are different under distinct regions of process parameters. At low scanning speeds in keyhole mode, the long-lasting recoil pressure and the large temperature gradient easily induce EOPD. While at high scanning speeds in keyhole mode, the shallow molten pool morphology and the large solidification rate allow the keyhole to evolve into an EOPD quickly. Nevertheless, in the conduction mode, the Marangoni effects along with a faster solidification rate induce EOPD. Finally, a ‘step’ variable power strategy is proposed to optimise the EOPD defects for the case with high volumetric energy density at low scanning speeds. This work provides a profound understanding and valuable insights into the quality control of LPBF fabrication.
의도하지 않은 공정 종료 후 함몰(EOPD)은 LPBF(레이저 분말층 융합)에서 흔히 발생하며, 특히 많은 임플란트의 경우 표면 품질이 떨어지고 피로 강도가 낮아집니다. 본 연구에서는 이 결함의 형성 메커니즘을 밝히기 위해 충실도가 높은 다중 물리학 메조 규모 시뮬레이션 모델을 개발했습니다.
이 시뮬레이션 모델을 사용하여 EOPD 현상의 결함 프로세스 맵을 얻습니다. EOPD 형성 메커니즘은 공정 매개변수의 별개 영역에서 서로 다른 것으로 밝혀졌습니다.
키홀 모드의 낮은 스캔 속도에서는 오래 지속되는 반동 압력과 큰 온도 구배로 인해 EOPD가 쉽게 유발됩니다. 키홀 모드에서 높은 스캐닝 속도를 유지하는 동안 얕은 용융 풀 형태와 큰 응고 속도로 인해 키홀이 EOPD로 빠르게 진화할 수 있습니다.
그럼에도 불구하고 전도 모드에서는 더 빠른 응고 속도와 함께 마랑고니 효과가 EOPD를 유발합니다. 마지막으로, 낮은 스캐닝 속도에서 높은 체적 에너지 밀도를 갖는 경우에 대해 EOPD 결함을 최적화하기 위한 ‘단계’ 가변 전력 전략이 제안되었습니다.
이 작업은 LPBF 제조의 품질 관리에 대한 심오한 이해와 귀중한 통찰력을 제공합니다.
Figure 5. Simulation of the molten pool under low-speed scanning (1.06 m/s). (a) Sequential solidification of the molten pool at the
end of the melt track for laser powers of 190 and 340 W, respectively. (b) Recoil pressure on the molten pool at the keyhole for laser
powers of 190 and 340 W, respectively. (c) The force diagram of the melt at the back of the keyhole at t = 750 μs in case B. (d) Temperature gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case A. (e) Temperature
gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case B.
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Yi Wei ;Genyu Chen;Nengru Tao;Wei Zhou https://doi.org/10.1063/5.0191504
In order to comprehensively reveal the evolutionary dynamics of the molten pool and the state of motion of the fluid during the high-precision laser powder bed fusion (HP-LPBF) process, this study aims to deeply investigate the specific manifestations of the multiphase flow, solidification phenomena, and heat transfer during the process by means of numerical simulation methods. Numerical simulation models of SS316L single-layer HP-LPBF formation with single and double tracks were constructed using the discrete element method and the computational fluid dynamics method. The effects of various factors such as Marangoni convection, surface tension, vapor recoil, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool have been paid attention to during the model construction process. The results show that the molten pool exhibits a “comet” shape, in which the temperature gradient at the front end of the pool is significantly larger than that at the tail end, with the highest temperature gradient up to 1.69 × 108 K/s. It is also found that the depth of the second track is larger than that of the first one, and the process parameter window has been determined preliminarily. In addition, the application of HP-LPBF technology helps to reduce the surface roughness and minimize the forming size.
Laser powder bed fusion (LPBF) has become a research hotspot in the field of additive manufacturing of metals due to its advantages of high-dimensional accuracy, good surface quality, high density, and high material utilization.1,2 With the rapid development of electronics, medical, automotive, biotechnology, energy, communication, and optics, the demand for microfabrication technology is increasing day by day.3 High-precision laser powder bed fusion (HP-LPBF) is one of the key manufacturing technologies for tiny parts in the fields of electronics, medical, automotive, biotechnology, energy, communication, and optics because of its process characteristics such as small focal spot diameter, small powder particle size, and thin powder layup layer thickness.4–13 Compared with LPBF, HP-LPBF has the significant advantages of smaller focal spot diameter, smaller powder particle size, and thinner layer thickness. These advantages make HP-LPBF perform better in producing micro-fine parts, high surface quality, and parts with excellent mechanical properties.
HP-LPBF is in the exploratory stage, and researchers have already done some exploratory studies on the focal spot diameter, the amount of defocusing, and the powder particle size. In order to explore the influence of changing the laser focal spot diameter on the LPBF process characteristics of the law, Wildman et al.14 studied five groups of different focal spot diameter LPBF forming 316L stainless steel (SS316L) processing effect, the smallest focal spot diameter of 26 μm, and the results confirm that changing the focal spot diameter can be achieved to achieve the energy control, so as to control the quality of forming. Subsequently, Mclouth et al.15 proposed the laser out-of-focus amount (focal spot diameter) parameter, which characterizes the distance between the forming plane and the laser focal plane. The laser energy density was controlled by varying the defocusing amount while keeping the laser parameters constant. Sample preparation at different focal positions was investigated, and their microstructures were characterized. The results show that the samples at the focal plane have finer microstructure than those away from the focal plane, which is the effect of higher power density and smaller focal spot diameter. In order to explore the influence of changing the powder particle size on the characteristics of the LPBF process, Qian et al.16 carried out single-track scanning simulations on powder beds with average powder particle sizes of 70 and 40 μm, respectively, and the results showed that the melt tracks sizes were close to each other under the same process parameters for the two particle-size distributions and that the molten pool of powder beds with small particles was more elongated and the edges of the melt tracks were relatively flat. In order to explore the superiority of HP-LPBF technology, Xu et al.17 conducted a comparative analysis of HP-LPBF and conventional LPBF of SS316L. The results showed that the average surface roughness of the top surface after forming by HP-LPBF could reach 3.40 μm. Once again, it was verified that HP-LPBF had higher forming quality than conventional LPBF. On this basis, Wei et al.6 comparatively analyzed the effects of different laser focal spot diameters on different powder particle sizes formed by LPBF. The results showed that the smaller the laser focal spot diameter, the fewer the defects on the top and side surfaces. The above research results confirm that reducing the laser focal spot diameter can obtain higher energy density and thus better forming quality.
LPBF involves a variety of complex systems and mechanisms, and the final quality of the part is influenced by a large number of process parameters.18–24 Some research results have shown that there are more than 50 factors affecting the quality of the specimen. The influencing factors are mainly categorized into three main groups: (1) laser parameters, (2) powder parameters, and (3) equipment parameters, which interact with each other to determine the final specimen quality. With the continuous development of technologies such as computational materials science and computational fluid dynamics (CFD), the method of studying the influence of different factors on the forming quality of LPBF forming process has been shifted from time-consuming and laborious experimental characterization to the use of numerical simulation methods. As a result, more and more researchers are adopting this approach for their studies. Currently, numerical simulation studies on LPBF are mainly focused on the exploration of molten pool, temperature distribution, and residual stresses.
Finite element simulation based on continuum mechanics and free surface fluid flow modeling based on fluid dynamics are two common approaches to study the behavior of LPBF molten pool.25–28 Finite element simulation focuses on the temperature and thermal stress fields, treats the powder bed as a continuum, and determines the molten pool size by plotting the elemental temperature above the melting point. In contrast, fluid dynamics modeling can simulate the 2D or 3D morphology of the metal powder pile and obtain the powder size and distribution by certain algorithms.29 The flow in the molten pool is mainly affected by recoil pressure and the Marangoni effect. By simulating the molten pool formation, it is possible to predict defects, molten pool shape, and flow characteristics, as well as the effect of process parameters on the molten pool geometry.30–34 In addition, other researchers have been conducted to optimize the laser processing parameters through different simulation methods and experimental data.35–46 Crystal growth during solidification is studied to further understand the effect of laser parameters on dendritic morphology and solute segregation.47–54 A multi-scale system has been developed to describe the fused deposition process during 3D printing, which is combined with the conductive heat transfer model and the dendritic solidification model.55,56
Relevant scholars have adopted various different methods for simulation, such as sequential coupling theory,57 Lagrangian and Eulerian thermal models,58 birth–death element method,25 and finite element method,59 in order to reveal the physical phenomena of the laser melting process and optimize the process parameters. Luo et al.60 compared the LPBF temperature field and molten pool under double ellipsoidal and Gaussian heat sources by ANSYS APDL and found that the diffusion of the laser energy in the powder significantly affects the molten pool size and the temperature field.
The thermal stresses obtained from the simulation correlate with the actual cracks,61 and local preheating can effectively reduce the residual stresses.62 A three-dimensional thermodynamic finite element model investigated the temperature and stress variations during laser-assisted fabrication and found that powder-to-solid conversion increases the temperature gradient, stresses, and warpage.63 Other scholars have predicted residual stresses and part deflection for LPBF specimens and investigated the effects of deposition pattern, heat, laser power, and scanning strategy on residual stresses, noting that high-temperature gradients lead to higher residual stresses.64–67
In short, the process of LPBF forming SS316L is extremely complex and usually involves drastic multi-scale physicochemical changes that will only take place on a very small scale. Existing literature employs DEM-based mesoscopic-scale numerical simulations to investigate the effects of process parameters on the molten pool dynamics of LPBF-formed SS316L. However, a few studies have been reported on the key mechanisms of heating and solidification, spatter, and convective behavior of the molten pool of HP-LPBF-formed SS316L with small laser focal spot diameters. In this paper, the geometrical properties of coarse and fine powder particles under three-dimensional conditions were first calculated using DEM. Then, numerical simulation models for single-track and double-track cases in the single-layer HP-LPBF forming SS316L process were developed at mesoscopic scale using the CFD method. The flow genesis of the melt in the single-track and double-track molten pools is discussed, and their 3D morphology and dimensional characteristics are discussed. In addition, the effects of laser process parameters, powder particle size, and laser focal spot diameter on the temperature field, characterization information, and defects in the molten pool are discussed.
II. MODELING
A. 3D powder bed modeling
HP-LPBF is an advanced processing technique for preparing target parts layer by layer stacking, the process of which involves repetitive spreading and melting of powders. In this process, both the powder spreading and the morphology of the powder bed are closely related to the results of the subsequent melting process, while the melted surface also affects the uniform distribution of the next layer of powder. For this reason, this chapter focuses on the modeling of the physical action during the powder spreading process and the theory of DEM to establish the numerical model of the powder bed, so as to lay a solid foundation for the accuracy of volume of fluid (VOF) and CFD.
1. DEM
DEM is a numerical technique for calculating the interaction of a large number of particles, which calculates the forces and motions of the spheres by considering each powder sphere as an independent unit. The motion of the powder particles follows the laws of classical Newtonian mechanics, including translational and rotational,38,68–70 which are expressed as follows:����¨=���+∑��ij,
(1)����¨=∑�(�ij×�ij),
(2)
where �� is the mass of unit particle i in kg, ��¨ is the advective acceleration in m/s2, And g is the gravitational acceleration in m/s2. �ij is the force in contact with the neighboring particle � in N. �� is the rotational inertia of the unit particle � in kg · m2. ��¨ is the unit particle � angular acceleration in rad/s2. �ij is the vector pointing from unit particle � to the contact point of neighboring particle �.
Equations (1) and (2) can be used to calculate the velocity and angular velocity variations of powder particles to determine their positions and velocities. A three-dimensional powder bed model of SS316L was developed using DEM. The powder particles are assumed to be perfect spheres, and the substrate and walls are assumed to be rigid. To describe the contact between the powder particles and between the particles and the substrate, a non-slip Hertz–Mindlin nonlinear spring-damping model71 was used with the following expression:�hz=��������+��[(�����ij−�eff����)−(�����+�eff����)],
(3)
where �hz is the force calculated using the Hertzian in M. �� and �� are the radius of unit particles � and � in m, respectively. �� is the overlap size of the two powder particles in m. ��, �� are the elastic constants in the normal and tangential directions, respectively. �ij is the unit vector connecting the centerlines of the two powder particles. �eff is the effective mass of the two powder particles in kg. �� and �� are the viscoelastic damping constants in the normal and tangential directions, respectively. �� and �� are the components of the relative velocities of the two powder particles. ��� is the displacement vector between two spherical particles. The schematic diagram of overlapping powder particles is shown in Fig. 1.
Schematic diagram of overlapping powder particles.
Because the particle size of the powder used for HP-LPBF is much smaller than 100 μm, the effect of van der Waals forces must be considered. Therefore, the cohesive force �jkr of the Hertz–Mindlin model was used instead of van der Waals forces,72 with the following expression:�jkr=−4��0�*�1.5+4�*3�*�3,
(4)1�*=(1−��2)��+(1−��2)��,
(5)1�*=1��+1��,
(6)
where �* is the equivalent Young’s modulus in GPa; �* is the equivalent particle radius in m; �0 is the surface energy of the powder particles in J/m2; α is the contact radius in m; �� and �� are the Young’s modulus of the unit particles � and �, respectively, in GPa; and �� and �� are the Poisson’s ratio of the unit particles � and �, respectively.
2. Model building
Figure 2 shows a 3D powder bed model generated using DEM with a coarse powder geometry of 1000 × 400 × 30 μm3. The powder layer thickness is 30 μm, and the powder bed porosity is 40%. The average particle size of this spherical powder is 31.7 μm and is normally distributed in the range of 15–53 μm. The geometry of the fine powder was 1000 × 400 × 20 μm3, with a layer thickness of 20 μm, and the powder bed porosity of 40%. The average particle size of this spherical powder is 11.5 μm and is normally distributed in the range of 5–25 μm. After the 3D powder bed model is generated, it needs to be imported into the CFD simulation software for calculation, and the imported geometric model is shown in Fig. 3. This geometric model is mainly composed of three parts: protective gas, powder bed, and substrate. Under the premise of ensuring the accuracy of the calculation, the mesh size is set to 3 μm, and the total number of coarse powder meshes is 1 704 940. The total number of fine powder meshes is 3 982 250.
Geometric modeling of the powder bed computational domain: (a) coarse powder, (b) fine powder.
B. Modeling of fluid mechanics simulation
In order to solve the flow, melting, and solidification problems involved in HP-LPBF molten pool, the study must follow the three governing equations of conservation of mass, conservation of energy, and conservation of momentum.73 The VOF method, which is the most widely used in fluid dynamics, is used to solve the molten pool dynamics model.
1. VOF
VOF is a method for tracking the free interface between the gas and liquid phases on the molten pool surface. The core idea of the method is to define a volume fraction function F within each grid, indicating the proportion of the grid space occupied by the material, 0 ≤ F ≤ 1 in Fig. 4. Specifically, when F = 0, the grid is empty and belongs to the gas-phase region; when F = 1, the grid is completely filled with material and belongs to the liquid-phase region; and when 0 < F < 1, the grid contains free surfaces and belongs to the mixed region. The direction normal to the free surface is the direction of the fastest change in the volume fraction F (the direction of the gradient of the volume fraction), and the direction of the gradient of the volume fraction can be calculated from the values of the volume fractions in the neighboring grids.74 The equations controlling the VOF are expressed as follows:𝛻����+�⋅(��→)=0,
(7)
where t is the time in s and �→ is the liquid velocity in m/s.
The material parameters of the mixing zone are altered due to the inclusion of both the gas and liquid phases. Therefore, in order to represent the density of the mixing zone, the average density �¯ is used, which is expressed as follows:72�¯=(1−�1)�gas+�1�metal,
(8)
where �1 is the proportion of liquid phase, �gas is the density of protective gas in kg/m3, and �metal is the density of metal in kg/m3.
2. Control equations and boundary conditions
Figure 5 is a schematic diagram of the HP-LPBF melting process. First, the laser light strikes a localized area of the material and rapidly heats up the area. Next, the energy absorbed in the region is diffused through a variety of pathways (heat conduction, heat convection, and surface radiation), and this process triggers complex phase transition phenomena (melting, evaporation, and solidification). In metals undergoing melting, the driving forces include surface tension and the Marangoni effect, recoil due to evaporation, and buoyancy due to gravity and uneven density. The above physical phenomena interact with each other and do not occur independently.
Laser heat sourceThe Gaussian surface heat source model is used as the laser heat source model with the following expression:�=2�0����2exp(−2�12��2),(9)where � is the heat flow density in W/m2, �0 is the absorption rate of SS316L, �� is the radius of the laser focal spot in m, and �1 is the radial distance from the center of the laser focal spot in m. The laser focal spot can be used for a wide range of applications.
Energy absorptionThe formula for calculating the laser absorption �0 of SS316L is as follows:�0=0.365(�0[1+�0(�−20)]/�)0.5,(10)where �0 is the direct current resistivity of SS316L at 20 °C in Ω m, �0 is the resistance temperature coefficient in ppm/°C, � is the temperature in °C, and � is the laser wavelength in m.
Heat transferThe basic principle of heat transfer is conservation of energy, which is expressed as follows:𝛻𝛻𝛻�(��)��+�·(��→�)=�·(�0����)+��,(11)where � is the density of liquid phase SS316L in kg/m3, �� is the specific heat capacity of SS316L in J/(kg K), 𝛻� is the gradient operator, t is the time in s, T is the temperature in K, 𝛻�� is the temperature gradient, �→ is the velocity vector, �0 is the coefficient of thermal conduction of SS316L in W/(m K), and �� is the thermal energy dissipation term in the molten pool.
Molten pool flowThe following three conditions need to be satisfied for the molten pool to flow:
Conservation of mass with the following expression:𝛻�·(��→)=0.(12)
Conservation of momentum (Navier–Stokes equation) with the following expression:𝛻𝛻𝛻𝛻���→��+�(�→·�)�→=�·[−pI+�(��→+(��→)�)]+�,(13)where � is the pressure in Pa exerted on the liquid phase SS316L microelement, � is the unit matrix, � is the fluid viscosity in N s/m2, and � is the volumetric force (gravity, atmospheric pressure, surface tension, vapor recoil, and the Marangoni effect).
Surface tension and the Marangoni effectThe effect of temperature on the surface tension coefficient is considered and set as a linear relationship with the following expression:�=�0−��dT(�−��),(14)where � is the surface tension of the molten pool at temperature T in N/m, �� is the melting temperature of SS316L in K, �0 is the surface tension of the molten pool at temperature �� in Pa, and σdσ/ dT is the surface tension temperature coefficient in N/(m K).In general, surface tension decreases with increasing temperature. A temperature gradient causes a gradient in surface tension that drives the liquid to flow, known as the Marangoni effect.
Metal vapor recoilAt higher input energy densities, the maximum temperature of the molten pool surface reaches the evaporation temperature of the material, and a gasification recoil pressure occurs vertically downward toward the molten pool surface, which will be the dominant driving force for the molten pool flow.75 The expression is as follows:��=0.54�� exp ���−���0���,(15)where �� is the gasification recoil pressure in Pa, �� is the ambient pressure in kPa, �� is the latent heat of evaporation in J/kg, �0 is the gas constant in J/(mol K), T is the surface temperature of the molten pool in K, and Te is the evaporation temperature in K.
Solid–liquid–gas phase transitionWhen the laser hits the powder layer, the powder goes through three stages: heating, melting, and solidification. During the solidification phase, mutual transformations between solid, liquid, and gaseous states occur. At this point, the latent heat of phase transition absorbed or released during the phase transition needs to be considered.68 The phase transition is represented based on the relationship between energy and temperature with the following expression:�=�����,(�<��),�(��)+�−����−����,(��<�<��)�(��)+(�−��)����,(��<�),,(16)where �� and �� are solid and liquid phase density, respectively, of SS316L in kg/m3. �� and �� unit volume of solid and liquid phase-specific heat capacity, respectively, of SS316L in J/(kg K). �� and ��, respectively, are the solidification temperature and melting temperature of SS316L in K. �� is the latent heat of the phase transition of SS316L melting in J/kg.
3. Assumptions
The CFD model was computed using the commercial software package FLOW-3D.76 In order to simplify the calculation and solution process while ensuring the accuracy of the results, the model makes the following assumptions:
It is assumed that the effects of thermal stress and material solid-phase thermal expansion on the calculation results are negligible.
The molten pool flow is assumed to be a Newtonian incompressible laminar flow, while the effects of liquid thermal expansion and density on the results are neglected.
It is assumed that the surface tension can be simplified to an equivalent pressure acting on the free surface of the molten pool, and the effect of chemical composition on the results is negligible.
Neglecting the effect of the gas flow field on the molten pool.
The mass loss due to evaporation of the liquid metal is not considered.
The influence of the plasma effect of the molten metal on the calculation results is neglected.
It is worth noting that the formulation of assumptions requires a trade-off between accuracy and computational efficiency. In the above models, some physical phenomena that have a small effect or high difficulty on the calculation results are simplified or ignored. Such simplifications make numerical simulations more efficient and computationally tractable, while still yielding accurate results.
4. Initial conditions
The preheating temperature of the substrate was set to 393 K, at which time all materials were in the solid state and the flow rate was zero.
5. Material parameters
The material used is SS316L and the relevant parameters required for numerical simulations are shown in Table I.46,77,78
TABLE I.
SS316L-related parameters.
Property
Symbol
Value
Density of solid metal (kg/m3)
�metal
7980
Solid phase line temperature (K)
��
1658
Liquid phase line temperature (K)
��
1723
Vaporization temperature (K)
��
3090
Latent heat of melting ( J/kg)
��
2.60×105
Latent heat of evaporation ( J/kg)
��
7.45×106
Surface tension of liquid phase (N /m)
�
1.60
Liquid metal viscosity (kg/m s)
��
6×10−3
Gaseous metal viscosity (kg/m s)
�gas
1.85×10−5
Temperature coefficient of surface tension (N/m K)
��/�T
0.80×10−3
Molar mass ( kg/mol)
M
0.05 593
Emissivity
�
0.26
Laser absorption
�0
0.35
Ambient pressure (kPa)
��
101 325
Ambient temperature (K)
�0
300
Stefan–Boltzmann constant (W/m2 K4)
�
5.67×10−8
Thermal conductivity of metals ( W/m K)
�
24.55
Density of protective gas (kg/m3)
�gas
1.25
Coefficient of thermal expansion (/K)
��
16×10−6
Generalized gas constant ( J/mol K)
R
8.314
III. RESULTS AND DISCUSSION
With the objective of studying in depth the evolutionary patterns of single-track and double-track molten pool development, detailed observations were made for certain specific locations in the model, as shown in Fig. 6. In this figure, P1 and P2 represent the longitudinal tangents to the centers of the two melt tracks in the XZ plane, while L1 is the transverse profile in the YZ plane. The scanning direction is positive and negative along the X axis. Points A and B are the locations of the centers of the molten pool of the first and second melt tracks, respectively (x = 1.995 × 10−4, y = 5 × 10−7, and z = −4.85 × 10−5).
A series of single-track molten pool simulation experiments were carried out in order to investigate the influence law of laser power as well as scanning speed on the HP-LPBF process. Figure 7 demonstrates the evolution of the 3D morphology and temperature field of the single-track molten pool in the time period of 50–500 μs under a laser power of 100 W and a scanning speed of 800 mm/s. The powder bed is in the natural cooling state. When t = 50 μs, the powder is heated by the laser heat and rapidly melts and settles to form the initial molten pool. This process is accompanied by partial melting of the substrate and solidification together with the melted powder. The molten pool rapidly expands with increasing width, depth, length, and temperature, as shown in Fig. 7(a). When t = 150 μs, the molten pool expands more obviously, and the temperature starts to transfer to the surrounding area, forming a heat-affected zone. At this point, the width of the molten pool tends to stabilize, and the temperature in the center of the molten pool has reached its peak and remains largely stable. However, the phenomenon of molten pool spatter was also observed in this process, as shown in Fig. 7(b). As time advances, when t = 300 μs, solidification begins to occur at the tail of the molten pool, and tiny ripples are produced on the solidified surface. This is due to the fact that the melt flows toward the region with large temperature gradient under the influence of Marangoni convection and solidifies together with the melt at the end of the bath. At this point, the temperature gradient at the front of the bath is significantly larger than at the end. While the width of the molten pool was gradually reduced, the shape of the molten pool was gradually changed to a “comet” shape. In addition, a slight depression was observed at the top of the bath because the peak temperature at the surface of the bath reached the evaporation temperature, which resulted in a recoil pressure perpendicular to the surface of the bath downward, creating a depressed region. As the laser focal spot moves and is paired with the Marangoni convection of the melt, these recessed areas will be filled in as shown in Fig. 7(c). It has been shown that the depressed regions are the result of the coupled effect of Marangoni convection, recoil pressure, and surface tension.79 By t = 500 μs, the width and height of the molten pool stabilize and show a “comet” shape in Fig. 7(d).
Single-track molten pool process: (a) t = 50 ��, (b) t = 150 ��, (c) t = 300 ��, (d) t = 500 ��.
Figure 8 depicts the velocity vector diagram of the P1 profile in a single-track molten pool, the length of the arrows represents the magnitude of the velocity, and the maximum velocity is about 2.36 m/s. When t = 50 μs, the molten pool takes shape, and the velocities at the two ends of the pool are the largest. The variation of the velocities at the front end is especially more significant in Fig. 8(a). As the time advances to t = 150 μs, the molten pool expands rapidly, in which the velocity at the tail increases and changes more significantly, while the velocity at the front is relatively small. At this stage, the melt moves backward from the center of the molten pool, which in turn expands the molten pool area. The melt at the back end of the molten pool center flows backward along the edge of the molten pool surface and then converges along the edge of the molten pool to the bottom center, rising to form a closed loop. Similarly, a similar closed loop is formed at the front end of the center of the bath, but with a shorter path. However, a large portion of the melt in the center of the closed loop formed at the front end of the bath is in a nearly stationary state. The main cause of this melt flow phenomenon is the effect of temperature gradient and surface tension (the Marangoni effect), as shown in Figs. 8(b) and 8(e). This dynamic behavior of the melt tends to form an “elliptical” pool. At t = 300 μs, the tendency of the above two melt flows to close the loop is more prominent and faster in Fig. 8(c). When t = 500 μs, the velocity vector of the molten pool shows a stable trend, and the closed loop of melt flow also remains stable. With the gradual laser focal spot movement, the melt is gradually solidified at its tail, and finally, a continuous and stable single track is formed in Fig. 8(d).
Vector plot of single-track molten pool velocity in XZ longitudinal section: (a) t = 50 ��, (b) t = 150 ��, (c) t = 300 ��, (d) t = 500 ��, (e) molten pool flow.
In order to explore in depth the transient evolution of the molten pool, the evolution of the single-track temperature field and the melt flow was monitored in the YZ cross section. Figure 9(a) shows the state of the powder bed at the initial moment. When t = 250 μs, the laser focal spot acts on the powder bed and the powder starts to melt and gradually collects in the molten pool. At this time, the substrate will also start to melt, and the melt flow mainly moves in the downward and outward directions and the velocity is maximum at the edges in Fig. 9(b). When t = 300 μs, the width and depth of the molten pool increase due to the recoil pressure. At this time, the melt flows more slowly at the center, but the direction of motion is still downward in Fig. 9(c). When t = 350 μs, the width and depth of the molten pool further increase, at which time the intensity of the melt flow reaches its peak and the direction of motion remains the same in Fig. 9(d). When t = 400 μs, the melt starts to move upward, and the surrounding powder or molten material gradually fills up, causing the surface of the molten pool to begin to flatten. At this time, the maximum velocity of the melt is at the center of the bath, while the velocity at the edge is close to zero, and the edge of the melt starts to solidify in Fig. 9(e). When t = 450 μs, the melt continues to move upward, forming a convex surface of the melt track. However, the melt movement slows down, as shown in Fig. 9(f). When t = 500 μs, the melt further moves upward and its speed gradually becomes smaller. At the same time, the melt solidifies further, as shown in Fig. 9(g). When t = 550 μs, the melt track is basically formed into a single track with a similar “mountain” shape. At this stage, the velocity is close to zero only at the center of the molten pool, and the flow behavior of the melt is poor in Fig. 9(h). At t = 600 μs, the melt stops moving and solidification is rapidly completed. Up to this point, a single track is formed in Fig. 9(i). During the laser action on the powder bed, the substrate melts and combines with the molten state powder. The powder-to-powder fusion is like the convergence of water droplets, which are rapidly fused by surface tension. However, the fusion between the molten state powder and the substrate occurs driven by surface tension, and the molten powder around the molten pool is pulled toward the substrate (a wetting effect occurs), which ultimately results in the formation of a monolithic whole.38,80,81
Evolution of single-track molten pool temperature and melt flow in the YZ cross section: (a) t = 0 ��, (b) t = 250 ��, (c) t = 300 ��, (d) t = 350 ��, (e) t = 400 ��, (f) t = 450 ��, (g) t = 500 ��, (h) t = 550 ��, (i) t = 600 ��.
The wetting ability between the liquid metal and the solid substrate in the molten pool directly affects the degree of balling of the melt,82,83 and the wetting ability can be measured by the contact angle of a single track in Fig. 10. A smaller value of contact angle represents better wettability. The contact angle α can be calculated by�=�1−�22,
(17)
where �1 and �2 are the contact angles of the left and right regions, respectively.
Relevant studies have confirmed that the wettability is better at a contact angle α around or below 40°.84 After measurement, a single-track contact angle α of about 33° was obtained under this process parameter, which further confirms the good wettability.
B. Double-track simulation
In order to deeply investigate the influence of hatch spacing on the characteristics of the HP-LPBF process, a series of double-track molten pool simulation experiments were systematically carried out. Figure 11 shows in detail the dynamic changes of the 3D morphology and temperature field of the double-track molten pool in the time period of 2050–2500 μs under the conditions of laser power of 100 W, scanning speed of 800 mm/s, and hatch spacing of 0.06 mm. By comparing the study with Fig. 7, it is observed that the basic characteristics of the 3D morphology and temperature field of the second track are similar to those of the first track. However, there are subtle differences between them. The first track exhibits a basically symmetric shape, but the second track morphology shows a slight deviation influenced by the difference in thermal diffusion rate between the solidified metal and the powder. Otherwise, the other characteristic information is almost the same as that of the first track. Figure 12 shows the velocity vector plot of the P2 profile in the double-track molten pool, with a maximum velocity of about 2.63 m/s. The melt dynamics at both ends of the pool are more stable at t = 2050 μs, where the maximum rate of the second track is only 1/3 of that of the first one. Other than that, the rest of the information is almost no significant difference from the characteristic information of the first track. Figure 13 demonstrates a detailed observation of the double-track temperature field and melts flow in the YZ cross section, and a comparative study with Fig. 9 reveals that the width of the second track is slightly wider. In addition, after the melt direction shifts from bottom to top, the first track undergoes four time periods (50 μs) to reach full solidification, while the second track takes five time periods. This is due to the presence of significant heat buildup in the powder bed after the forming of the first track, resulting in a longer dynamic time of the melt and an increased molten pool lifetime. In conclusion, the level of specimen forming can be significantly optimized by adjusting the laser power and hatch spacing.
Evolution of double-track molten pool temperature and melt flow in the YZ cross section: (a) t = 2250 ��, (b) t = 2300 ��, (c) t = 2350 ��, (d) t = 2400 ��, (e) t = 2450 ��, (f) t = 2500 ��, (g) t = 2550 ��, (h) t = 2600 ��, (i) t = 2650 ��.
In order to quantitatively detect the molten pool dimensions as well as the remolten region dimensions, the molten pool characterization information in Fig. 14 is constructed by drawing the boundary on the YZ cross section based on the isothermal surface of the liquid phase line. It can be observed that the heights of the first track and second track are basically the same, but the depth of the second track increases relative to the first track. The molten pool width is mainly positively correlated with the laser power as well as the scanning speed (the laser line energy density �). However, the remelted zone width is negatively correlated with the hatch spacing (the overlapping ratio). Overall, the forming quality of the specimens can be directly influenced by adjusting the laser power, scanning speed, and hatch spacing.
Double-track molten pool characterization information on YZ cross section.
In order to study the variation rule of the temperature in the center of the molten pool with time, Fig. 15 demonstrates the temperature variation curves with time for two reference points, A and B. Among them, the red dotted line indicates the liquid phase line temperature of SS316L. From the figure, it can be seen that the maximum temperature at the center of the molten pool in the first track is lower than that in the second track, which is mainly due to the heat accumulation generated after passing through the first track. The maximum temperature gradient was calculated to be 1.69 × 108 K/s. When the laser scanned the first track, the temperature in the center of the molten pool of the second track increased slightly. Similarly, when the laser scanned the second track, a similar situation existed in the first track. Since the temperature gradient in the second track is larger than that in the first track, the residence time of the liquid phase in the molten pool of the first track is longer than that of the second track.
Temperature profiles as a function of time for two reference points A and B.
C. Simulation analysis of molten pool under different process parameters
In order to deeply investigate the effects of various process parameters on the mesoscopic-scale temperature field, molten pool characteristic information and defects of HP-LPBF, numerical simulation experiments on mesoscopic-scale laser power, scanning speed, and hatch spacing of double-track molten pools were carried out.
1. Laser power
Figure 16 shows the effects of different laser power on the morphology and temperature field of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. When P = 50 W, a smaller molten pool is formed due to the lower heat generated by the Gaussian light source per unit time. This leads to a smaller track width, which results in adjacent track not lapping properly and the presence of a large number of unmelted powder particles, resulting in an increase in the number of defects, such as pores in the specimen. The surface of the track is relatively flat, and the depth is small. In addition, the temperature gradient before and after the molten pool was large, and the depression location appeared at the biased front end in Fig. 16(a). When P = 100 W, the surface of the track is flat and smooth with excellent lap. Due to the Marangoni effect, the velocity field of the molten pool is in the form of “vortex,” and the melt has good fluidity, and the maximum velocity reaches 2.15 m/s in Fig. 16(b). When P = 200 W, the heat generated by the Gaussian light source per unit time is too large, resulting in the melt rapidly reaching the evaporation temperature, generating a huge recoil pressure, forming a large molten pool, and the surface of the track is obviously raised. The melt movement is intense, especially the closed loop at the center end of the molten pool. At this time, the depth and width of the molten pool are large, leading to the expansion of the remolten region and the increased chance of the appearance of porosity defects in Fig. 16(c). The results show that at low laser power, the surface tension in the molten pool is dominant. At high laser power, recoil pressure is its main role.
Simulation results of double-track molten pool under different laser powers: (a) P = 50 W, (b) P = 100 W, (c) P = 200 W.
Table II shows the effect of different laser powers on the characteristic information of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. The negative overlapping ratio in the table indicates that the melt tracks are not lapped, and 26/29 indicates the melt depth of the first track/second track. It can be seen that with the increase in laser power, the melt depth, melt width, melt height, and remelted zone show a gradual increase. At the same time, the overlapping ratio also increases. Especially in the process of laser power from 50 to 200 W, the melting depth and melting width increased the most, which increased nearly 2 and 1.5 times, respectively. Meanwhile, the overlapping ratio also increases with the increase in laser power, which indicates that the melting and fusion of materials are better at high laser power. On the other hand, the dimensions of the molten pool did not change uniformly with the change of laser power. Specifically, the depth-to-width ratio of the molten pool increased from about 0.30 to 0.39 during the increase from 50 to 120 W, which further indicates that the effective heat transfer in the vertical direction is greater than that in the horizontal direction with the increase in laser power. This dimensional response to laser power is mainly affected by the recoil pressure and also by the difference in the densification degree between the powder layer and the metal substrate. In addition, according to the experimental results, the contact angle shows a tendency to increase and then decrease during the process of laser power increase, and always stays within the range of less than 33°. Therefore, in practical applications, it is necessary to select the appropriate laser power according to the specific needs in order to achieve the best processing results.
TABLE II.
Double-track molten pool characterization information at different laser powers.
Laser power (W)
Depth (μm)
Width (μm)
Height (μm)
Remolten region (μm)
Overlapping ratio (%)
Contact angle (°)
50
16
54
11
/
−10
23
100
26/29
74
14
18
23.33
33
200
37/45
116
21
52
93.33
28
2. Scanning speed
Figure 17 demonstrates the effect of different scanning speeds on the morphology and temperature field of the double-track molten pool at a laser power of 100 W and a hatch spacing of 0.06 mm. With the gradual increase in scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. When � = 200 mm/s, the slow scanning speed causes the material to absorb too much heat, which is very easy to trigger the overburning phenomenon. At this point, the molten pool is larger and the surface morphology is uneven. This situation is consistent with the previously discussed scenario with high laser power in Fig. 17(a). However, when � = 1600 mm/s, the scanning speed is too fast, resulting in the material not being able to absorb sufficient heat, which triggers the powder particles that fail to melt completely to have a direct effect on the bonding of the melt to the substrate. At this time, the molten pool volume is relatively small and the neighboring melt track cannot lap properly. This result is consistent with the previously discussed case of low laser power in Fig. 17(b). Overall, the ratio of the laser power to the scanning speed (the line energy density �) has a direct effect on the temperature field and surface morphology of the molten pool.
Simulation results of double-track molten pool under different scanning speed: (a) � = 200 mm/s, (b) � = 1600 mm/s.
Table III shows the effects of different scanning speed on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and hatch spacing of 0.06 mm. It can be seen that the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. With the increase in scanning speed, the melt depth, melt width, melt height, remelted zone, and overlapping ratio show a gradual decreasing trend. Among them, the melt depth and melt width decreased faster, while the melt height and remolten region decreased relatively slowly. In addition, when the scanning speed was increased from 200 to 800 mm/s, the decreasing speeds of melt depth and melt width were significantly accelerated, while the decreasing speeds of overlapping ratio were relatively slow. When the scanning speed was further increased to 1600 mm/s, the decreasing speeds of melt depth and melt width were further accelerated, and the un-lapped condition of the melt channel also appeared. In addition, the contact angle increases and then decreases with the scanning speed, and both are lower than 33°. Therefore, when selecting the scanning speed, it is necessary to make reasonable trade-offs according to the specific situation, and take into account the factors of melt depth, melt width, melt height, remolten region, and overlapping ratio, in order to achieve the best processing results.
TABLE III.
Double-track molten pool characterization information at different scanning speeds.
Scanning speed (mm/s)
Depth (μm)
Width (μm)
Height (μm)
Remolten region (μm)
Overlapping ratio (%)
Contact angle (°)
200
55/68
182
19/32
124
203.33
22
1600
13
50
11
/
−16.67
31
3. Hatch spacing
Figure 18 shows the effect of different hatch spacing on the morphology and temperature field of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. The surface morphology and temperature field of the first track and second track are basically the same, but slightly different. The first track shows a basically symmetric morphology along the scanning direction, while the second track shows a slight offset due to the difference in the heat transfer rate between the solidified material and the powder particles. When the hatch spacing is too small, the overlapping ratio increases and the probability of defects caused by remelting phenomenon grows. When the hatch spacing is too large, the neighboring melt track cannot overlap properly, and the powder particles are not completely melted, leading to an increase in the number of holes. In conclusion, the ratio of the line energy density � to the hatch spacing (the volume energy density E) has a significant effect on the temperature field and surface morphology of the molten pool.
Simulation results of double-track molten pool under different hatch spacings: (a) H = 0.03 mm, (b) H = 0.12 mm.
Table IV shows the effects of different hatch spacing on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. It can be seen that the hatch spacing has little effect on the melt depth, melt width, and melt height, but has some effect on the remolten region. With the gradual expansion of hatch spacing, the remolten region shows a gradual decrease. At the same time, the overlapping ratio also decreased with the increase in hatch spacing. In addition, it is observed that the contact angle shows a tendency to increase and then remain stable when the hatch spacing increases, which has a more limited effect on it. Therefore, trade-offs and decisions need to be made on a case-by-case basis when selecting the hatch spacing.
TABLE IV.
Double-track molten pool characterization information at different hatch spacings.
Hatch spacing (mm)
Depth (μm)
Width (μm)
Height (μm)
Remolten region (μm)
Overlapping ratio (%)
Contact angle (°)
0.03
25/27
82
14
59
173.33
30
0.12
26
78
14
/
−35
33
In summary, the laser power, scanning speed, and hatch spacing have a significant effect on the formation of the molten pool, and the correct selection of these three process parameters is crucial to ensure the forming quality. In addition, the melt depth of the second track is slightly larger than that of the first track at higher line energy density � and volume energy density E. This is mainly due to the fact that a large amount of heat accumulation is generated after the first track, forming a larger molten pool volume, which leads to an increase in the melt depth.
D. Simulation analysis of molten pool with powder particle size and laser focal spot diameter
Figure 19 demonstrates the effect of different powder particle sizes and laser focal spot diameters on the morphology and temperature field of the double-track molten pool under a laser power of 100 W, a scanning speed of 800 mm/s, and a hatch spacing of 0.06 mm. In the process of melting coarse powder with small laser focal spot diameter, the laser energy cannot completely melt the larger powder particles, resulting in their partial melting and further generating excessive pore defects. The larger powder particles tend to generate zigzag molten pool edges, which cause an increase in the roughness of the melt track surface. In addition, the molten pool is also prone to generate the present spatter phenomenon, which can directly affect the quality of forming. The volume of the formed molten pool is relatively small, while the melt depth, melt width, and melt height are all smaller relative to the fine powder in Fig. 19(a). In the process of melting fine powders with a large laser focal spot diameter, the laser energy is able to melt the fine powder particles sufficiently, even to the point of overmelting. This results in a large number of fine spatters being generated at the edge of the molten pool, which causes porosity defects in the melt track in Fig. 19(b). In addition, the maximum velocity of the molten pool is larger for large powder particle sizes compared to small powder particle sizes, which indicates that the temperature gradient in the molten pool is larger for large powder particle sizes and the melt motion is more intense. However, the size of the laser focal spot diameter has a relatively small effect on the melt motion. However, a larger focal spot diameter induces a larger melt volume with greater depth, width, and height. In conclusion, a small powder size helps to reduce the surface roughness of the specimen, and a small laser spot diameter reduces the minimum forming size of a single track.
Simulation results of double-track molten pool with different powder particle size and laser focal spot diameter: (a) focal spot = 25 μm, coarse powder, (b) focal spot = 80 μm, fine powder.
Table V shows the maximum temperature gradient at the reference point for different powder sizes and laser focal spot diameters. As can be seen from the table, the maximum temperature gradient is lower than that of HP-LPBF for both coarse powders with a small laser spot diameter and fine powders with a large spot diameter, a phenomenon that leads to an increase in the heat transfer rate of HP-LPBF, which in turn leads to a corresponding increase in the cooling rate and, ultimately, to the formation of finer microstructures.
TABLE V.
Maximum temperature gradient at the reference point for different powder particle sizes and laser focal spot diameters.
Laser power (W)
Scanning speed (mm/s)
Hatch spacing (mm)
Average powder size (μm)
Laser focal spot diameter (μm)
Maximum temperature gradient (×107 K/s)
100
800
0.06
31.7
25
7.89
11.5
80
7.11
IV. CONCLUSIONS
In this study, the geometrical characteristics of 3D coarse and fine powder particles were first calculated using DEM and then numerical simulations of single track and double track in the process of forming SS316L from monolayer HP-LPBF at mesoscopic scale were developed using CFD method. The effects of Marangoni convection, surface tension, recoil pressure, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool were considered in this model. The effects of laser power, scanning speed, and hatch spacing on the dynamics of the single-track and double-track molten pools, as well as on other characteristic information, were investigated. The effects of the powder particle size on the molten pool were investigated comparatively with the laser focal spot diameter. The main conclusions are as follows:
The results show that the temperature gradient at the front of the molten pool is significantly larger than that at the tail, and the molten pool exhibits a “comet” morphology. At the top of the molten pool, there is a slightly concave region, which is the result of the coupling of Marangoni convection, recoil pressure, and surface tension. The melt flow forms two closed loops, which are mainly influenced by temperature gradients and surface tension. This special dynamic behavior of the melt tends to form an “elliptical” molten pool and an almost “mountain” shape in single-track forming.
The basic characteristics of the three-dimensional morphology and temperature field of the second track are similar to those of the first track, but there are subtle differences. The first track exhibits a basically symmetrical shape; however, due to the difference in thermal diffusion rates between the solidified metal and the powder, a slight asymmetry in the molten pool morphology of the second track occurs. After forming through the first track, there is a significant heat buildup in the powder bed, resulting in a longer dynamic time of the melt, which increases the life of the molten pool. The heights of the first track and second track remained essentially the same, but the depth of the second track was greater relative to the first track. In addition, the maximum temperature gradient was 1.69 × 108 K/s during HP-LPBF forming.
At low laser power, the surface tension in the molten pool plays a dominant role. At high laser power, recoil pressure becomes the main influencing factor. With the increase of laser power, the effective heat transfer in the vertical direction is superior to that in the horizontal direction. With the gradual increase of scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. In addition, the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. Too large or too small hatch spacing will lead to remelting or non-lap phenomenon, which in turn causes the formation of defects.
When using a small laser focal spot diameter, it is difficult to completely melt large powder particle sizes, resulting in partial melting and excessive porosity generation. At the same time, large powder particles produce curved edges of the molten pool, resulting in increased surface roughness of the melt track. In addition, spatter occurs, which directly affects the forming quality. At small focal spot diameters, the molten pool volume is relatively small, and the melt depth, the melt width, and the melt height are correspondingly small. Taken together, the small powder particle size helps to reduce surface roughness, while the small spot diameter reduces the forming size.
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Farhoud Kalateh a,*, Ehsan Aminvash a and Rasoul Daneshfaraz b a Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran b Faculty of Engineering, University of Maragheh, Maragheh, Iran *Corresponding author. E-mail: f.kalateh@gmail.com
ABSTRACT
The main goal of the present study is to investigate the effects of macro-roughnesses downstream of the inclined drop through numerical models. Due to the vital importance of geometrical properties of the macro-roughnesses in the hydraulic performance and efficient energy dissipation downstream of inclined drops, two different geometries of macro-roughnesses, i.e., semi-circular and triangular geometries, have been investigated using the Flow-3D model. Numerical simulation showed that with the flow rate increase and relative critical depth, the flow energy consumption has decreased. Also, relative energy dissipation increases with the increase in height and slope angle, so that this amount of increase in energy loss compared to the smooth bed in semi-circular and triangular elements is 86.39 and 76.80%, respectively, in the inclined drop with a height of 15 cm and 86.99 and 65.78% in the drop with a height of 20 cm. The Froude number downstream on the uneven bed has been dramatically reduced, so this amount of reduction has been approximately 47 and 54% compared to the control condition. The relative depth of the downstream has also increased due to the turbulence of the flow on the uneven bed with the increase in the flow rate.
본 연구의 주요 목표는 수치 모델을 통해 경사 낙하 하류의 거시 거칠기 효과를 조사하는 것입니다. 수력학적 성능과 경사 낙하 하류의 효율적인 에너지 소산에서 거시 거칠기의 기하학적 특성이 매우 중요하기 때문에 두 가지 서로 다른 거시 거칠기 형상, 즉 반원형 및 삼각형 형상이 Flow를 사용하여 조사되었습니다.
3D 모델 수치 시뮬레이션을 통해 유량이 증가하고 상대 임계 깊이가 증가함에 따라 유동 에너지 소비가 감소하는 것으로 나타났습니다. 또한, 높이와 경사각이 증가함에 따라 상대적인 에너지 소산도 증가하는데, 반원형 요소와 삼각형 요소에서 평활층에 비해 에너지 손실의 증가량은 경사낙하에서 각각 86.39%와 76.80%입니다.
높이 15cm, 높이 20cm의 드롭에서 86.99%, 65.78%입니다. 고르지 못한 베드 하류의 프루드 수가 극적으로 감소하여 이 감소량은 대조 조건에 비해 약 47%와 54%였습니다. 유속이 증가함에 따라 고르지 못한 층에서의 흐름의 난류로 인해 하류의 상대적 깊이도 증가했습니다.
Key words
flow energy dissipation, Froude number, inclined drop, numerical simulation
Figure 1 | Schematic of the present research model with dimensions and macro-roughnesses installed.Figure 2 | Meshing, boundary condition, and solution field network
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험프 웨어는 수위 제어 및 배출 측정을 위한 기존의 수력 구조물 중 하나입니다. 상류 및 하류 경사로의 경사는 자유 및 침수 흐름 조건 모두에서 험프 웨어의 성능에 영향을 미치는 설계 매개변수입니다.
침수된 험프보의 유출 특성 및 수위 변화에 대한 램프 경사 및 유출의 영향을 조사하기 위해 일련의 수치 시뮬레이션이 수행되었습니다. 1V:1H에서 1V:5H까지의 5개 램프 경사를 다양한 업스트림 방전에서 테스트했습니다.
수치모델의 검증을 위해 수치결과를 실험실 데이터와 비교하였다. 수면수위 예측과 유출계수의 시뮬레이션 불일치는 각각 전체 범위의 ±10%와 ±5% 이내였습니다.
모듈 한계 및 방전 감소 계수의 변화에 대한 램프 경사의 영향을 연구했습니다. 험프보의 경사로 경사가 증가함에 따라 상대적으로 높은 침수율에서 모듈러 한계가 발생함을 알 수 있었다.
침수 시작은 방류 수위를 작은 증분으로 조심스럽게 증가시켜 모델링되었으며 그 결과는 모듈 한계의 고전적인 정의와 비교되었습니다. 램프 경사와 방전이 증가함에 따라 모듈러 한계가 증가하는 것으로 밝혀졌지만, 모듈러 한계의 고전적인 정의는 모듈러 한계가 방전과 무관하다는 것을 나타냅니다.
Hump weir 하류의 속도와 와류장은 램프 경사에 의해 제어되는 와류 구조 형성을 나타냅니다. 에너지 손실은 수치 출력으로부터 계산되었으며 정규화된 에너지 손실은 침수에 따라 선형적으로 감소하는 것으로 나타났습니다.
Hump weirs are amongst conventional hydraulic structures for water level control and discharge measurement. The slope in the upstream and downstream ramps is a design parameter that affects the performance of Hump weirs in both free and submerged flow conditions. A series of numerical simulations was performed to investigate the effects of ramp slope and discharge on discharge characteristics and water level variations of submerged Hump weirs. Five ramp slopes ranging from 1V:1H to 1V:5H were tested at different upstream discharges. The numerical results were compared with the laboratory data for verifications of the numerical model. The simulation discrepancies in prediction of water surface level and discharge coefficient were within ±10 % and ±5 % of the full range, respectively. The effects of ramp slope on variations of modular limit and discharge reduction factor were studied. It was found that the modular limit occurred at relatively higher submergence ratios as the ramp slope in Hump weirs increased. The onset of submergence was modeled by carefully increasing tailwater level with small increments and the results were compared with the classic definition of modular limit. It was found that the modular limit increases with increasing the ramp slope and discharge while the classic definition of modular limit indicated that the modular limit is independent of the discharge. The velocity and vortex fields in the downstream of Hump weirs indicated the formation vortex structure, which is controlled by the ramp slope. The energy losses were calculated from the numerical outputs, and it was found that the normalized energy losses decreased linearly with submergence.
Weirs have been utilized predominantly for discharge measurement, flow diversion, and water level control in open channels, irrigation canal, and natural streams due to their simplicity of operation and accuracy. Several research studies have been conducted to determine the head-discharge relationship in weirs as one of the most common hydraulic structures for flow measurement (Rajaratnam and Muralidhar, 1969 [[1], [2], [3]]; Vatankhah, 2010, [[4], [5], [6]]; b [[7], [8], [9]]; Azimi and Seyed Hakim, 2019; Salehi et al., 2019; Salehi and Azimi, 2019, [10]. Weirs in general are classified into two major categories named as sharp-crested weirs and weirs of finite-crest length (Rajaratnam and Muralidhar, 1969; [11]. Sharp-crested weirs are typically used for flow measurement in small irrigation canals and laboratory flumes. In contrast, weirs of finite crest length are more suitable for water level control and flow diversion in rivers and natural streams [7,[12], [13], [14]].
The head-discharge relationship in sharp-crested weirs is developed by employing energy equation between two sections in the upstream and downstream of the weir and integration of the velocity profile at the crest of the weir as:
where Qf is the free flow discharge, B is the channel width, g is the acceleration due to gravity, ho is the water head in free-flow condition, and Cd is the discharge coefficient. Rehbock [15] proposed a linear correlation between discharge coefficient and the ratio of water head, ho, and the weir height, P as Cd = 0.605 + 0.08 (ho/P).
Upstream and/or downstream ramp(s) can be added to sharp-crested weirs to enhance the structural stability of the weir. A sharp-crested weir with upstream and/or downstream ramp(s) are known as triangular weirs in the literature. Triangular weirs with both upstream and downstream ramps are also known as Hump weirs and are first introduced in the experimental study of Bazin [16]. The ramps are constructed upstream and downstream of sharp-crested weirs to enhance the weir’s structural integrity and improve the hydraulic performance of the weir. In free-flow condition, the discharge coefficient of Hump weirs increases with increasing downstream ramp slope but decreases as upstream ramp slope increases (Azimi et al., 2013).
The hydraulic performance of weirs is evaluated in both free and submerged flow conditions. In free flow condition, water freely flows over weirs since the downstream water level is lower than that of the crest level of the weir. Channel blockage or flood in the downstream of weirs can raise the tailwater level, t. As tailwater passes the crest elevation in sharp-crested weirs, the upstream flow decelerates due to the excess pressure force in the downstream and the upstream water level increases. The onset of water level raise due to tailwater raise is called the modular limit. Once the tailwater level passes the modular limit, the weir is submerged. In sharp-crested weirs, the submerged flow regime may occur even before the tailwater reaches the crest elevation [8,14], whereas, in weirs of finite crest length, the upstream water level remains unchanged even if the tailwater raises above the crest elevation and it normally causes submergence once the tailwater level passes the critical depth at the crest of the weir [7,17]. The degree of submergence can be estimated by careful observation of the water surface profile. Observations of water surface at different submergence levels indicated two distinct flow patterns in submerged sharp-crested weirs that was initially classified as impinging jet and surface flow regimes [14]. [8] analyzed the variations of water surface profiles over submerged sharp-crested weirs with different submergence ratios and defined four distinct regimes of impinging jet, surface jump, surface wave, and surface jet.
[18] characterized the onset of submergence by defining the modular limit as a stage when the free flow head increases by +1 mm due to tailwater rise. The definition of modular limit is somewhat arbitrary, and it is difficult to identify for large discharges because the upstream water surface begins to fluctuate. This definition did not consider the effects of channel and weir geometries. The experimental data in triangular weirs and weirs finite-crest length with upstream and downstream ramp(s) revealed that the modular limit varied with the ratio of the free-flow head to the total streamwise length of the weir [17]. Weirs of finite crest length with upstream and downstream ramps are known as embankment weirs in literature [1,19,20] and Azimi et al., 2013) [19]. conducted two series of laboratory experiments to study the hydraulics of submerged embankment weirs with the upstream and downstream ramps of 1V:1H and 1V:2H. Empirical correlations were proposed to directly estimate the flow discharge in submerged embankment weirs for t/h > 0.7 where h is the water head in submerged flow condition. He found that the free flow discharge is a function of upstream water head, but the submerged discharge is a function of submergence level, t/h [21]. studied the hydraulics of four embankment weirs with different weir heights ranging from 0.09 m to 0.36 m. It was found that submerged embankments with a higher ho/P, where P is the height of the weir, have a smaller discharge reduction due to submergence. Effects of crest length in embankment weirs with both upstream and downstream ramps of 1V:2H was studied in both free and submerged flow conditions [1]. It was found that the modular limit in submerged embankment weirs decreased linearly with the relative crest length, Ho/(Ho + L), where Ho is the total head and L is the crest length.
In submerged flow condition, the performance of weirs is quantified by the discharge reduction factor, ψ, which is a ratio of the submerged discharge, Qs, to the corresponding free-flow discharge, Qf, based on the upstream head, h [12]. In submerged-flow conditions, flow discharge can be estimated as:��=���
[1] proposed a formula to predict ψ that could be used for embankment weirs with different crest lengths ranging from 0 to 0.3 m as:�=(1−��)�where n is an exponent varying from 4 to 7 and Yt is the normalized submergence defined as:��=�ℎ−[0.85−(0.5��+�)]1−[0.85−(0.5��+�)]where H is the total upstream head in submerged-flow conditions [7]. proposed a simpler formula to predict ψ for weirs of finite-crest length as:�=[1−(�ℎ)�]�where m and n are exponents varying for different types of weirs. Hakim and Azimi (2017) employed regression analysis to propose values of n = 0.25 and m = 0.28 (ho/L)−2.425 for triangular weirs.
The discharge capacity of weirs decreases in submerged flow condition and the onset of submergence occurs at the modular limit. Therefore, the determination of modular limit in weirs with different geometries is critical to understanding the sensitivity of a particular weir model with tailwater level variations. The available definition of modular limit as when head water raises by +1 mm due to tailwater rise does not consider the effects of channel and weir geometries. Therefore, a new and more accurate definition of modular limit is proposed in this study to consider the effect of other geometry and approaching flow parameters. The second objective of this study is to evaluate the effects of upstream and downstream ramps and ramps slopes on the hydraulic performance of submerged Hump weirs. The flow patterns, velocity distributions, and energy dissipation rates were extracted from validated numerical data to better understand the discharge reduction mechanism in Hump weirs in both free and submerged flow conditions.
Section snippets
Governing equations
Numerical simulation has been employed as an efficient and effective method to analyze free surface flow problems and in particular investigating on the hydraulics of flow over weirs [22]. The weir models were developed in numerical domain and the water pressure and velocity field were simulated by employing the FLOW-3D solver (Flow Science, Inc., Santa Fe, USA). The numerical results were validated with the laboratory measurements and the effects of ramps slopes on the performance of Hump
Verification of numerical model
The experimental observations of Bazin [16,17] were used for model validation in free and submerged flow conditions, respectively. The weir height in the study of Bazin was P = 0.5 m and two ramp slopes of 1V:1H and 1V:2H were tested. The bed and sides of the channel were made of glass, and the roughness distribution of the bed and walls were uniform. The Hump weir models in the study of Seyed Hakim and Azimi (2017) had a weir height of 0.076 m and ramp slopes of 1V:2H in both upstream and
Conclusions
A series of numerical simulations was performed to study the hydraulics and velocity pattern downstream of a Hump weir with symmetrical ramp slopes. Effects of ramp slope and discharge on formation of modular limit and in submerged flow condition were tested by conducting a series of numerical simulations on Hump weirs with ramp slopes varying from 1V:1H to 1V:5H. A comparison between numerical results and experimental data indicated that the proposed numerical model is accurate with a mean
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Review on Blood Flow Dynamics in Lab-on-a-Chip Systems: An Engineering Perspective
Bin-Jie Lai
,
Li-Tao Zhu
,
Zhe Chen*
,
Bo Ouyang*
, and
Zheng-Hong Luo*
Abstract
다양한 수송 메커니즘 하에서, “LOC(lab-on-a-chip)” 시스템에서 유동 전단 속도 조건과 밀접한 관련이 있는 혈류 역학은 다양한 수송 현상을 초래하는 것으로 밝혀졌습니다.
본 연구는 적혈구의 동적 혈액 점도 및 탄성 거동과 같은 점탄성 특성의 역할을 통해 LOC 시스템의 혈류 패턴을 조사합니다. 모세관 및 전기삼투압의 주요 매개변수를 통해 LOC 시스템의 혈액 수송 현상에 대한 연구는 실험적, 이론적 및 수많은 수치적 접근 방식을 통해 제공됩니다.
전기 삼투압 점탄성 흐름에 의해 유발되는 교란은 특히 향후 연구 기회를 위해 혈액 및 기타 점탄성 유체를 취급하는 LOC 장치의 혼합 및 분리 기능 향상에 논의되고 적용됩니다. 또한, 본 연구는 보다 정확하고 단순화된 혈류 모델에 대한 요구와 전기역학 효과 하에서 점탄성 유체 흐름에 대한 수치 연구에 대한 강조와 같은 LOC 시스템 하에서 혈류 역학의 수치 모델링의 문제를 식별합니다.
전기역학 현상을 연구하는 동안 제타 전위 조건에 대한 보다 실용적인 가정도 강조됩니다. 본 연구는 모세관 및 전기삼투압에 의해 구동되는 미세유체 시스템의 혈류 역학에 대한 포괄적이고 학제적인 관점을 제공하는 것을 목표로 한다.
1.1. Microfluidic Flow in Lab-on-a-Chip (LOC) Systems
Over the past several decades, the ability to control and utilize fluid flow patterns at microscales has gained considerable interest across a myriad of scientific and engineering disciplines, leading to growing interest in scientific research of microfluidics.
(1) Microfluidics, an interdisciplinary field that straddles physics, engineering, and biotechnology, is dedicated to the behavior, precise control, and manipulation of fluids geometrically constrained to a small, typically submillimeter, scale.
(2) The engineering community has increasingly focused on microfluidics, exploring different driving forces to enhance working fluid transport, with the aim of accurately and efficiently describing, controlling, designing, and applying microfluidic flow principles and transport phenomena, particularly for miniaturized applications.
(3) This attention has chiefly been fueled by the potential to revolutionize diagnostic and therapeutic techniques in the biomedical and pharmaceutical sectorsUnder various driving forces in microfluidic flows, intriguing transport phenomena have bolstered confidence in sustainable and efficient applications in fields such as pharmaceutical, biochemical, and environmental science. The “lab-on-a-chip” (LOC) system harnesses microfluidic flow to enable fluid processing and the execution of laboratory tasks on a chip-sized scale. LOC systems have played a vital role in the miniaturization of laboratory operations such as mixing, chemical reaction, separation, flow control, and detection on small devices, where a wide variety of fluids is adapted. Biological fluid flow like blood and other viscoelastic fluids are notably studied among the many working fluids commonly utilized by LOC systems, owing to the optimization in small fluid sample volumed, rapid response times, precise control, and easy manipulation of flow patterns offered by the system under various driving forces.
(4)The driving forces in blood flow can be categorized as passive or active transport mechanisms and, in some cases, both. Under various transport mechanisms, the unique design of microchannels enables different functionalities in driving, mixing, separating, and diagnosing blood and drug delivery in the blood.
(5) Understanding and manipulating these driving forces are crucial for optimizing the performance of a LOC system. Such knowledge presents the opportunity to achieve higher efficiency and reliability in addressing cellular level challenges in medical diagnostics, forensic studies, cancer detection, and other fundamental research areas, for applications of point-of-care (POC) devices.
1.2. Engineering Approach of Microfluidic Transport Phenomena in LOC Systems
Different transport mechanisms exhibit unique properties at submillimeter length scales in microfluidic devices, leading to significant transport phenomena that differ from those of macroscale flows. An in-depth understanding of these unique transport phenomena under microfluidic systems is often required in fluidic mechanics to fully harness the potential functionality of a LOC system to obtain systematically designed and precisely controlled transport of microfluids under their respective driving force. Fluid mechanics is considered a vital component in chemical engineering, enabling the analysis of fluid behaviors in various unit designs, ranging from large-scale reactors to separation units. Transport phenomena in fluid mechanics provide a conceptual framework for analytically and descriptively explaining why and how experimental results and physiological phenomena occur. The Navier–Stokes (N–S) equation, along with other governing equations, is often adapted to accurately describe fluid dynamics by accounting for pressure, surface properties, velocity, and temperature variations over space and time. In addition, limiting factors and nonidealities for these governing equations should be considered to impose corrections for empirical consistency before physical models are assembled for more accurate controls and efficiency. Microfluidic flow systems often deviate from ideal conditions, requiring adjustments to the standard governing equations. These deviations could arise from factors such as viscous effects, surface interactions, and non-Newtonian fluid properties from different microfluid types and geometrical layouts of microchannels. Addressing these nonidealities supports the refining of theoretical models and prediction accuracy for microfluidic flow behaviors.
The analytical calculation of coupled nonlinear governing equations, which describes the material and energy balances of systems under ideal conditions, often requires considerable computational efforts. However, advancements in computation capabilities, cost reduction, and improved accuracy have made numerical simulations using different numerical and modeling methods a powerful tool for effectively solving these complex coupled equations and modeling various transport phenomena. Computational fluid dynamics (CFD) is a numerical technique used to investigate the spatial and temporal distribution of various flow parameters. It serves as a critical approach to provide insights and reasoning for decision-making regarding the optimal designs involving fluid dynamics, even prior to complex physical model prototyping and experimental procedures. The integration of experimental data, theoretical analysis, and reliable numerical simulations from CFD enables systematic variation of analytical parameters through quantitative analysis, where adjustment to delivery of blood flow and other working fluids in LOC systems can be achieved.
Numerical methods such as the Finite-Difference Method (FDM), Finite-Element-Method (FEM), and Finite-Volume Method (FVM) are heavily employed in CFD and offer diverse approaches to achieve discretization of Eulerian flow equations through filling a mesh of the flow domain. A more in-depth review of numerical methods in CFD and its application for blood flow simulation is provided in Section 2.2.2.
1.3. Scope of the Review
In this Review, we explore and characterize the blood flow phenomena within the LOC systems, utilizing both physiological and engineering modeling approaches. Similar approaches will be taken to discuss capillary-driven flow and electric-osmotic flow (EOF) under electrokinetic phenomena as a passive and active transport scheme, respectively, for blood transport in LOC systems. Such an analysis aims to bridge the gap between physical (experimental) and engineering (analytical) perspectives in studying and manipulating blood flow delivery by different driving forces in LOC systems. Moreover, the Review hopes to benefit the interests of not only blood flow control in LOC devices but also the transport of viscoelastic fluids, which are less studied in the literature compared to that of Newtonian fluids, in LOC systems.
Section 2 examines the complex interplay between viscoelastic properties of blood and blood flow patterns under shear flow in LOC systems, while engineering numerical modeling approaches for blood flow are presented for assistance. Sections 3 and 4 look into the theoretical principles, numerical governing equations, and modeling methodologies for capillary driven flow and EOF in LOC systems as well as their impact on blood flow dynamics through the quantification of key parameters of the two driving forces. Section 5 concludes the characterized blood flow transport processes in LOC systems under these two forces. Additionally, prospective areas of research in improving the functionality of LOC devices employing blood and other viscoelastic fluids and potentially justifying mechanisms underlying microfluidic flow patterns outside of LOC systems are presented. Finally, the challenges encountered in the numerical studies of blood flow under LOC systems are acknowledged, paving the way for further research.
Blood, an essential physiological fluid in the human body, serves the vital role of transporting oxygen and nutrients throughout the body. Additionally, blood is responsible for suspending various blood cells including erythrocytes (red blood cells or RBCs), leukocytes (white blood cells), and thrombocytes (blood platelets) in a plasma medium.Among the cells mentioned above, red blood cells (RBCs) comprise approximately 40–45% of the volume of healthy blood.
(7) An RBC possesses an inherent elastic property with a biconcave shape of an average diameter of 8 μm and a thickness of 2 μm. This biconcave shape maximizes the surface-to-volume ratio, allowing RBCs to endure significant distortion while maintaining their functionality.
(8,9) Additionally, the biconcave shape optimizes gas exchange, facilitating efficient uptake of oxygen due to the increased surface area. The inherent elasticity of RBCs allows them to undergo substantial distortion from their original biconcave shape and exhibits high flexibility, particularly in narrow channels.RBC deformability enables the cell to deform from a biconcave shape to a parachute-like configuration, despite minor differences in RBC shape dynamics under shear flow between initial cell locations. As shown in Figure 1(a), RBCs initiating with different resting shapes and orientations displaying display a similar deformation pattern
(10) in terms of its shape. Shear flow induces an inward bending of the cell at the rear position of the rim to the final bending position,
(11) resulting in an alignment toward the same position of the flow direction.
Figure 1. Images of varying deformation of RBCs and different dynamic blood flow behaviors. (a) The deforming shape behavior of RBCs at four different initiating positions under the same experimental conditions of a flow from left to right, (10) (b) RBC aggregation, (13) (c) CFL region. (18) Reproduced with permission from ref (10). Copyright 2011 Elsevier. Reproduced with permission from ref (13). Copyright 2022 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/. Reproduced with permission from ref (18). Copyright 2019 Elsevier.
The flexible property of RBCs enables them to navigate through narrow capillaries and traverse a complex network of blood vessels. The deformability of RBCs depends on various factors, including the channel geometry, RBC concentration, and the elastic properties of the RBC membrane.
(12) Both flexibility and deformability are vital in the process of oxygen exchange among blood and tissues throughout the body, allowing cells to flow in vessels even smaller than the original cell size prior to deforming.As RBCs serve as major components in blood, their collective dynamics also hugely affect blood rheology. RBCs exhibit an aggregation phenomenon due to cell to cell interactions, such as adhesion forces, among populated cells, inducing unique blood flow patterns and rheological behaviors in microfluidic systems. For blood flow in large vessels between a diameter of 1 and 3 cm, where shear rates are not high, a constant viscosity and Newtonian behavior for blood can be assumed. However, under low shear rate conditions (0.1 s
–1) in smaller vessels such as the arteries and venules, which are within a diameter of 0.2 mm to 1 cm, blood exhibits non-Newtonian properties, such as shear-thinning viscosity and viscoelasticity due to RBC aggregation and deformability. The nonlinear viscoelastic property of blood gives rise to a complex relationship between viscosity and shear rate, primarily influenced by the highly elastic behavior of RBCs. A wide range of research on the transient behavior of the RBC shape and aggregation characteristics under varied flow circumstances has been conducted, aiming to obtain a better understanding of the interaction between blood flow shear forces from confined flows.
For a better understanding of the unique blood flow structures and rheological behaviors in microfluidic systems, some blood flow patterns are introduced in the following section.
2.1.1. RBC Aggregation
RBC aggregation is a vital phenomenon to be considered when designing LOC devices due to its impact on the viscosity of the bulk flow. Under conditions of low shear rate, such as in stagnant or low flow rate regions, RBCs tend to aggregate, forming structures known as rouleaux, resembling stacks of coins as shown in Figure 1(b).
(13) The aggregation of RBCs increases the viscosity at the aggregated region,
(14) hence slowing down the overall blood flow. However, when exposed to high shear rates, RBC aggregates disaggregate. As shear rates continue to increase, RBCs tend to deform, elongating and aligning themselves with the direction of the flow.
(15) Such a dynamic shift in behavior from the cells in response to the shear rate forms the basis of the viscoelastic properties observed in whole blood. In essence, the viscosity of the blood varies according to the shear rate conditions, which are related to the velocity gradient of the system. It is significant to take the intricate relationship between shear rate conditions and the change of blood viscosity due to RBC aggregation into account since various flow driving conditions may induce varied effects on the degree of aggregation.
2.1.2. Fåhræus-Lindqvist Effect
The Fåhræus–Lindqvist (FL) effect describes the gradual decrease in the apparent viscosity of blood as the channel diameter decreases.
(16) This effect is attributed to the migration of RBCs toward the central region in the microchannel, where the flow rate is higher, due to the presence of higher pressure and asymmetric distribution of shear forces. This migration of RBCs, typically observed at blood vessels less than 0.3 mm, toward the higher flow rate region contributes to the change in blood viscosity, which becomes dependent on the channel size. Simultaneously, the increase of the RBC concentration in the central region of the microchannel results in the formation of a less viscous region close to the microchannel wall. This region called the Cell-Free Layer (CFL), is primarily composed of plasma.
(17) The combination of the FL effect and the following CFL formation provides a unique phenomenon that is often utilized in passive and active plasma separation mechanisms, involving branched and constriction channels for various applications in plasma separation using microfluidic systems.
2.1.3. Cell-Free Layer Formation
In microfluidic blood flow, RBCs form aggregates at the microchannel core and result in a region that is mostly devoid of RBCs near the microchannel walls, as shown in Figure 1(c).
(18) The region is known as the cell-free layer (CFL). The CFL region is often known to possess a lower viscosity compared to other regions within the blood flow due to the lower viscosity value of plasma when compared to that of the aggregated RBCs. Therefore, a thicker CFL region composed of plasma correlates to a reduced apparent whole blood viscosity.
(19) A thicker CFL region is often established following the RBC aggregation at the microchannel core under conditions of decreasing the tube diameter. Apart from the dependence on the RBC concentration in the microchannel core, the CFL thickness is also affected by the volume concentration of RBCs, or hematocrit, in whole blood, as well as the deformability of RBCs. Given the influence CFL thickness has on blood flow rheological parameters such as blood flow rate, which is strongly dependent on whole blood viscosity, investigating CFL thickness under shear flow is crucial for LOC systems accounting for blood flow.
2.1.4. Plasma Skimming in Bifurcation Networks
The uneven arrangement of RBCs in bifurcating microchannels, commonly termed skimming bifurcation, arises from the axial migration of RBCs within flowing streams. This uneven distribution contributes to variations in viscosity across differing sizes of bifurcating channels but offers a stabilizing effect. Notably, higher flow rates in microchannels are associated with increased hematocrit levels, resulting in higher viscosity compared with those with lower flow rates. Parametric investigations on bifurcation angle,
(21) and RBC dynamics, including aggregation and deformation,
(22) may alter the varying viscosity of blood and its flow behavior within microchannels.
2.2. Modeling on Blood Flow Dynamics
2.2.1. Blood Properties and Mathematical Models of Blood Rheology
Under different shear rate conditions in blood flow, the elastic characteristics and dynamic changes of the RBC induce a complex velocity and stress relationship, resulting in the incompatibility of blood flow characterization through standard presumptions of constant viscosity used for Newtonian fluid flow. Blood flow is categorized as a viscoelastic non-Newtonian fluid flow where constitutive equations governing this type of flow take into consideration the nonlinear viscometric properties of blood. To mathematically characterize the evolving blood viscosity and the relationship between the elasticity of RBC and the shear blood flow, respectively, across space and time of the system, a stress tensor (τ) defined by constitutive models is often coupled in the Navier–Stokes equation to account for the collective impact of the constant dynamic viscosity (η) and the elasticity from RBCs on blood flow.The dynamic viscosity of blood is heavily dependent on the shear stress applied to the cell and various parameters from the blood such as hematocrit value, plasma viscosity, mechanical properties of the RBC membrane, and red blood cell aggregation rate. The apparent blood viscosity is considered convenient for the characterization of the relationship between the evolving blood viscosity and shear rate, which can be defined by Casson’s law, as shown in eq 1.
𝜇=𝜏0𝛾˙+2𝜂𝜏0𝛾˙⎯⎯⎯⎯⎯⎯⎯√+𝜂�=�0�˙+2��0�˙+�
(1)where τ
0 is the yield stress–stress required to initiate blood flow motion, η is the Casson rheological constant, and γ̇ is the shear rate. The value of Casson’s law parameters under blood with normal hematocrit level can be defined as τ
0 = 0.0056 Pa and η = 0.0035 Pa·s.
(23) With the known property of blood and Casson’s law parameters, an approximation can be made to the dynamic viscosity under various flow condition domains. The Power Law model is often employed to characterize the dynamic viscosity in relation to the shear rate, since precise solutions exist for specific geometries and flow circumstances, acting as a fundamental standard for definition. The Carreau and Carreau–Yasuda models can be advantageous over the Power Law model due to their ability to evaluate the dynamic viscosity at low to zero shear rate conditions. However, none of the above-mentioned models consider the memory or other elastic behavior of blood and its RBCs. Some other commonly used mathematical models and their constants for the non-Newtonian viscosity property characterization of blood are listed in Table 1 below.
(24−26)Table 1. Comparison of Various Non-Newtonian Models for Blood Viscosity
The blood rheology is commonly known to be influenced by two key physiological factors, namely, the hematocrit value (H
t) and the fibrinogen concentration (c
f), with an average value of 42% and 0.252 gd·L
–1, respectively. Particularly in low shear conditions, the presence of varying fibrinogen concentrations affects the tendency for aggregation and rouleaux formation, while the occurrence of aggregation is contingent upon specific levels of hematocrit.
(28) modifies the Casson model through emphasizing its reliance on hematocrit and fibrinogen concentration parameter values, owing to the extensive knowledge of the two physiological blood parameters.The viscoelastic response of blood is heavily dependent on the elasticity of the RBC, which is defined by the relationship between the deformation and stress relaxation from RBCs under a specific location of shear flow as a function of the velocity field. The stress tensor is usually characterized by constitutive equations such as the Upper-Convected Maxwell Model
(30) to track the molecule effects under shear from different driving forces. The prominent non-Newtonian features, such as shear thinning and yield stress, have played a vital role in the characterization of blood rheology, particularly with respect to the evaluation of yield stress under low shear conditions. The nature of stress measurement in blood, typically on the order of 1 mPa, is challenging due to its low magnitude. The occurrence of the CFL complicates the measurement further due to the significant decrease in apparent viscosity near the wall over time and a consequential disparity in viscosity compared to the bulk region.In addition to shear thinning viscosity and yield stress, the formation of aggregation (rouleaux) from RBCs under low shear rates also contributes to the viscoelasticity under transient flow
(32) of whole blood. Given the difficulty in evaluating viscoelastic behavior of blood under low strain magnitudes and limitations in generalized Newtonian models, the utilization of viscoelastic models is advocated to encompass elasticity and delineate non-shear components within the stress tensor. Extending from the Oldroyd-B model, Anand et al.
(33) developed a viscoelastic model framework for adapting elasticity within blood samples and predicting non-shear stress components. However, to also address the thixotropic effects, the model developed by Horner et al.
(34) serves as a more comprehensive approach than the viscoelastic model from Anand et al. Thixotropy
(32) typically occurs from the structural change of the rouleaux, where low shear rate conditions induce rouleaux formation. Correspondingly, elasticity increases, while elasticity is more representative of the isolated RBCs, under high shear rate conditions. The model of Horner et al.
(34) considers the contribution of rouleaux to shear stress, taking into account factors such as the characteristic time for Brownian aggregation, shear-induced aggregation, and shear-induced breakage. Subsequent advancements in the model from Horner et al. often revolve around refining the three aforementioned key terms for a more substantial characterization of rouleaux dynamics. Notably, this has led to the recently developed mHAWB model
(35) and other model iterations to enhance the accuracy of elastic and viscoelastic contributions to blood rheology, including the recently improved model suggested by Armstrong et al.
Numerical simulation has become increasingly more significant in analyzing the geometry, boundary layers of flow, and nonlinearity of hyperbolic viscoelastic flow constitutive equations. CFD is a powerful and efficient tool utilizing numerical methods to solve the governing hydrodynamic equations, such as the Navier–Stokes (N–S) equation, continuity equation, and energy conservation equation, for qualitative evaluation of fluid motion dynamics under different parameters. CFD overcomes the challenge of analytically solving nonlinear forms of differential equations by employing numerical methods such as the Finite-Difference Method (FDM), Finite-Element Method (FEM), and Finite-Volume Method (FVM) to discretize and solve the partial differential equations (PDEs), allowing for qualitative reproduction of transport phenomena and experimental observations. Different numerical methods are chosen to cope with various transport systems for optimization of the accuracy of the result and control of error during the discretization process.FDM is a straightforward approach to discretizing PDEs, replacing the continuum representation of equations with a set of finite-difference equations, which is typically applied to structured grids for efficient implementation in CFD programs.
(37) However, FDM is often limited to simple geometries such as rectangular or block-shaped geometries and struggles with curved boundaries. In contrast, FEM divides the fluid domain into small finite grids or elements, approximating PDEs through a local description of physics.
(38) All elements contribute to a large, sparse matrix solver. However, FEM may not always provide accurate results for systems involving significant deformation and aggregation of particles like RBCs due to large distortion of grids.
(39) FVM evaluates PDEs following the conservation laws and discretizes the selected flow domain into small but finite size control volumes, with each grid at the center of a finite volume.
(40) The divergence theorem allows the conversion of volume integrals of PDEs with divergence terms into surface integrals of surface fluxes across cell boundaries. Due to its conservation property, FVM offers efficient outcomes when dealing with PDEs that embody mass, momentum, and energy conservation principles. Furthermore, widely accessible software packages like the OpenFOAM toolbox
(41) include a viscoelastic solver, making it an attractive option for viscoelastic fluid flow modeling.
The complexity in the blood flow simulation arises from deformability and aggregation that RBCs exhibit during their interaction with neighboring cells under different shear rate conditions induced by blood flow. Numerical models coupled with simulation programs have been applied as a groundbreaking method to predict such unique rheological behavior exhibited by RBCs and whole blood. The conventional approach of a single-phase flow simulation is often applied to blood flow simulations within large vessels possessing a moderate shear rate. However, such a method assumes the properties of plasma, RBCs and other cellular components to be evenly distributed as average density and viscosity in blood, resulting in the inability to simulate the mechanical dynamics, such as RBC aggregation under high-shear flow field, inherent in RBCs. To accurately describe the asymmetric distribution of RBC and blood flow, multiphase flow simulation, where numerical simulations of blood flows are often modeled as two immiscible phases, RBCs and blood plasma, is proposed. A common assumption is that RBCs exhibit non-Newtonian behavior while the plasma is treated as a continuous Newtonian phase.Numerous multiphase numerical models have been proposed to simulate the influence of RBCs on blood flow dynamics by different assumptions. In large-scale simulations (above the millimeter range), continuum-based methods are wildly used due to their lower computational demands.
(43) Eulerian multiphase flow simulations offer the solution of a set of conservation equations for each separate phase and couple the phases through common pressure and interphase exchange coefficients. Xu et al.
(44) utilized the combined finite-discrete element method (FDEM) to replicate the dynamic behavior and distortion of RBCs subjected to fluidic forces, utilizing the Johnson–Kendall–Roberts model
(45) to define the adhesive forces of cell-to-cell interactions. The iterative direct-forcing immersed boundary method (IBM) is commonly employed in simulations of the fluid–cell interface of blood. This method effectively captures the intricacies of the thin and flexible RBC membranes within various external flow fields.
(44) also adopts this approach to bridge the fluid dynamics and RBC deformation through IBM. Yoon and You utilized the Maxwell model to define the viscosity of the RBC membrane.
(47) It was discovered that the Maxwell model could represent the stress relaxation and unloading processes of the cell. Furthermore, the reduced flexibility of an RBC under particular situations such as infection is specified, which was unattainable by the Kelvin–Voigt model
(48) when compared to the Maxwell model in the literature. The Yeoh hyperplastic material model was also adapted to predict the nonlinear elasticity property of RBCs with FEM employed to discretize the RBC membrane using shell-type elements. Gracka et al.
(49) developed a numerical CFD model with a finite-volume parallel solver for multiphase blood flow simulation, where an updated Maxwell viscoelasticity model and a Discrete Phase Model are adopted. In the study, the adapted IBM, based on unstructured grids, simulates the flow behavior and shape change of the RBCs through fluid-structure coupling. It was found that the hybrid Euler–Lagrange (E–L) approach
(50) for the development of the multiphase model offered better results in the simulated CFL region in the microchannels.To study the dynamics of individual behaviors of RBCs and the consequent non-Newtonian blood flow, cell-shape-resolved computational models are often adapted. The use of the boundary integral method has become prevalent in minimizing computational expenses, particularly in the exclusive determination of fluid velocity on the surfaces of RBCs, incorporating the option of employing IBM or particle-based techniques. The cell-shaped-resolved method has enabled an examination of cell to cell interactions within complex ambient or pulsatile flow conditions
(51) surrounding RBC membranes. Recently, Rydquist et al.
(52) have looked to integrate statistical information from macroscale simulations to obtain a comprehensive overview of RBC behavior within the immediate proximity of the flow through introduction of respective models characterizing membrane shape definition, tension, bending stresses of RBC membranes.At a macroscopic scale, continuum models have conventionally been adapted for assessing blood flow dynamics through the application of elasticity theory and fluid dynamics. However, particle-based methods are known for their simplicity and adaptability in modeling complex multiscale fluid structures. Meshless methods, such as the boundary element method (BEM), smoothed particle hydrodynamics (SPH), and dissipative particle dynamics (DPD), are often used in particle-based characterization of RBCs and the surrounding fluid. By representing the fluid as discrete particles, meshless methods provide insights into the status and movement of the multiphase fluid. These methods allow for the investigation of cellular structures and microscopic interactions that affect blood rheology. Non-confronting mesh methods like IBM can also be used to couple a fluid solver such as FEM, FVM, or the Lattice Boltzmann Method (LBM) through membrane representation of RBCs. In comparison to conventional CFD methods, LBM has been viewed as a favorable numerical approach for solving the N–S equations and the simulation of multiphase flows. LBM exhibits the notable advantage of being amenable to high-performance parallel computing environments due to its inherently local dynamics. In contrast to DPD and SPH where RBC membranes are modeled as physically interconnected particles, LBM employs the IBM to account for the deformation dynamics of RBCs
(53,54) under shear flows in complex channel geometries.
(54,55) However, it is essential to acknowledge that the utilization of LBM in simulating RBC flows often entails a significant computational overhead, being a primary challenge in this context. Krüger et al.
(56) proposed utilizing LBM as a fluid solver, IBM to couple the fluid and FEM to compute the response of membranes to deformation under immersed fluids. This approach decouples the fluid and membranes but necessitates significant computational effort due to the requirements of both meshes and particles.Despite the accuracy of current blood flow models, simulating complex conditions remains challenging because of the high computational load and cost. Balachandran Nair et al.
(57) suggested a reduced order model of RBC under the framework of DEM, where the RBC is represented by overlapping constituent rigid spheres. The Morse potential force is adapted to account for the RBC aggregation exhibited by cell to cell interactions among RBCs at different distances. Based upon the IBM, the reduced-order RBC model is adapted to simulate blood flow transport for validation under both single and multiple RBCs with a resolved CFD-DEM solver.
(58) In the resolved CFD-DEM model, particle sizes are larger than the grid size for a more accurate computation of the surrounding flow field. A continuous forcing approach is taken to describe the momentum source of the governing equation prior to discretization, which is different from a Direct Forcing Method (DFM).
(59) As no body-conforming moving mesh is required, the continuous forcing approach offers lower complexity and reduced cost when compared to the DFM. Piquet et al.
(60) highlighted the high complexity of the DFM due to its reliance on calculating an additional immersed boundary flux for the velocity field to ensure its divergence-free condition.The fluid–structure interaction (FSI) method has been advocated to connect the dynamic interplay of RBC membranes and fluid plasma within blood flow such as the coupling of continuum–particle interactions. However, such methodology is generally adapted for anatomical configurations such as arteries
(63) where both the structural components and the fluid domain undergo substantial deformation due to the moving boundaries. Due to the scope of the Review being blood flow simulation within microchannels of LOC devices without deformable boundaries, the Review of the FSI method will not be further carried out.In general, three numerical methods are broadly used: mesh-based, particle-based, and hybrid mesh–particle techniques, based on the spatial scale and the fundamental numerical approach, mesh-based methods tend to neglect the effects of individual particles, assuming a continuum and being efficient in terms of time and cost. However, the particle-based approach highlights more of the microscopic and mesoscopic level, where the influence of individual RBCs is considered. A review from Freund et al.
(64) addressed the three numerical methodologies and their respective modeling approaches of RBC dynamics. Given the complex mechanics and the diverse levels of study concerning numerical simulations of blood and cellular flow, a broad spectrum of numerical methods for blood has been subjected to extensive review.
(65) offered an extensive review of the application of the DPD, SPH, and LBM for numerical simulations of RBC, while Rathnayaka et al.
(67) conducted a review of the particle-based numerical modeling for liquid marbles through drawing parallels to the transport of RBCs in microchannels. A comparative analysis between conventional CFD methods and particle-based approaches for cellular and blood flow dynamic simulation can be found under the review by Arabghahestani et al.
(69) offer an overview of both continuum-based models at micro/macroscales and multiscale particle-based models encompassing various length and temporal dimensions. Furthermore, these reviews deliberate upon the potential of coupling continuum-particle methods for blood plasma and RBC modeling. Arciero et al.
(70) investigated various modeling approaches encompassing cellular interactions, such as cell to cell or plasma interactions and the individual cellular phases. A concise overview of the reviews is provided in Table 2 for reference.
Table 2. List of Reviews for Numerical Approaches Employed in Blood Flow Simulation
Capillary driven (CD) flow is a pivotal mechanism in passive microfluidic flow systems
(9) such as the blood circulation system and LOC systems.
(71) CD flow is essentially the movement of a liquid to flow against drag forces, where the capillary effect exerts a force on the liquid at the borders, causing a liquid–air meniscus to flow despite gravity or other drag forces. A capillary pressure drops across the liquid–air interface with surface tension in the capillary radius and contact angle. The capillary effect depends heavily on the interaction between the different properties of surface materials. Different values of contact angles can be manipulated and obtained under varying levels of surface wettability treatments to manipulate the surface properties, resulting in different CD blood delivery rates for medical diagnostic device microchannels. CD flow techniques are appealing for many LOC devices, because they require no external energy. However, due to the passive property of liquid propulsion by capillary forces and the long-term instability of surface treatments on channel walls, the adaptability of CD flow in geometrically complex LOC devices may be limited.
3.2. Theoretical and Numerical Modeling of Capillary Driven Blood Flow
3.2.1. Theoretical Basis and Assumptions of Microfluidic Flow
The study of transport phenomena regarding either blood flow driven by capillary forces or externally applied forces under microfluid systems all demands a comprehensive recognition of the significant differences in flow dynamics between microscale and macroscale. The fundamental assumptions and principles behind fluid transport at the microscale are discussed in this section. Such a comprehension will lay the groundwork for the following analysis of the theoretical basis of capillary forces and their role in blood transport in LOC systems.
At the macroscale, fluid dynamics are often strongly influenced by gravity due to considerable fluid mass. However, the high surface to volume ratio at the microscale shifts the balance toward surface forces (e.g., surface tension and viscous forces), much larger than the inertial force. This difference gives rise to transport phenomena unique to microscale fluid transport, such as the prevalence of laminar flow due to a very low Reynolds number (generally lower than 1). Moreover, the fluid in a microfluidic system is often assumed to be incompressible due to the small flow velocity, indicating constant fluid density in both space and time.Microfluidic flow behaviors are governed by the fundamental principles of mass and momentum conservation, which are encapsulated in the continuity equation and the Navier–Stokes (N–S) equation. The continuity equation describes the conservation of mass, while the N–S equation captures the spatial and temporal variations in velocity, pressure, and other physical parameters. Under the assumption of the negligible influence of gravity in microfluidic systems, the continuity equation and the Eulerian representation of the incompressible N–S equation can be expressed as follows:
∇·𝐮⇀=0∇·�⇀=0
(7)
−∇𝑝+𝜇∇2𝐮⇀+∇·𝝉⇀−𝐅⇀=0−∇�+�∇2�⇀+∇·�⇀−�⇀=0
(8)Here, p is the pressure, u is the fluid viscosity,
𝝉⇀�⇀ represents the stress tensor, and F is the body force exerted by external forces if present.
3.2.2. Theoretical Basis and Modeling of Capillary Force in LOC Systems
The capillary force is often the major driving force to manipulate and transport blood without an externally applied force in LOC systems. Forces induced by the capillary effect impact the free surface of fluids and are represented not directly in the Navier–Stokes equations but through the pressure boundary conditions of the pressure term p. For hydrophilic surfaces, the liquid generally induces a contact angle between 0° and 30°, encouraging the spread and attraction of fluid under a positive cos θ condition. For this condition, the pressure drop becomes positive and generates a spontaneous flow forward. A hydrophobic solid surface repels the fluid, inducing minimal contact. Generally, hydrophobic solids exhibit a contact angle larger than 90°, inducing a negative value of cos θ. Such a value will result in a negative pressure drop and a flow in the opposite direction. The induced contact angle is often utilized to measure the wall exposure of various surface treatments on channel walls where different wettability gradients and surface tension effects for CD flows are established. Contact angles between different interfaces are obtainable through standard values or experimental methods for reference.
(72)For the characterization of the induced force by the capillary effect, the Young–Laplace (Y–L) equation
(73) is widely employed. In the equation, the capillary is considered a pressure boundary condition between the two interphases. Through the Y–L equation, the capillary pressure force can be determined, and subsequently, the continuity and momentum balance equations can be solved to obtain the blood filling rate. Kim et al.
(74) studied the effects of concentration and exposure time of a nonionic surfactant, Silwet L-77, on the performance of a polydimethylsiloxane (PDMS) microchannel in terms of plasma and blood self-separation. The study characterized the capillary pressure force by incorporating the Y–L equation and further evaluated the effects of the changing contact angle due to different levels of applied channel wall surface treatments. The expression of the Y–L equation utilized by Kim et al.
(9)where σ is the surface tension of the liquid and θ
b, θ
t, θ
l, and θ
r are the contact angle values between the liquid and the bottom, top, left, and right walls, respectively. A numerical simulation through Coventor software is performed to evaluate the dynamic changes in the filling rate within the microchannel. The simulation results for the blood filling rate in the microchannel are expressed at a specific time stamp, shown in Figure 2. The results portray an increasing instantaneous filling rate of blood in the microchannel following the decrease in contact angle induced by a higher concentration of the nonionic surfactant treated to the microchannel wall.
Figure 2. Numerical simulation of filling rate of capillary driven blood flow under various contact angle conditions at a specific timestamp. (74) Reproduced with permission from ref (74). Copyright 2010 Elsevier.
When in contact with hydrophilic or hydrophobic surfaces, blood forms a meniscus with a contact angle due to surface tension. The Lucas–Washburn (L–W) equation
(75) is one of the pioneering theoretical definitions for the position of the meniscus over time. In addition, the L–W equation provides the possibility for research to obtain the velocity of the blood formed meniscus through the derivation of the meniscus position. The L–W equation
(10)Here L(t) represents the distance of the liquid driven by the capillary forces. However, the generalized L–W equation solely assumes the constant physical properties from a Newtonian fluid rather than considering the non-Newtonian fluid behavior of blood. Cito et al.
(76) constructed an enhanced version of the L–W equation incorporating the power law to consider the RBC aggregation and the FL effect. The non-Newtonian fluid apparent viscosity under the Power Law model is defined as
𝜇=𝑘·(𝛾˙)𝑛−1�=�·(�˙)�−1
(11)where γ̇ is the strain rate tensor defined as
𝛾˙=12𝛾˙𝑖𝑗𝛾˙𝑗𝑖⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√�˙=12�˙���˙��. The stress tensor term τ is computed as τ = μγ̇
(12)where k is the flow consistency index and n is the power law index, respectively. The power law index, from the Power Law model, characterizes the extent of the non-Newtonian behavior of blood. Both the consistency and power law index rely on blood properties such as hematocrit, the appearance of the FL effect, the formation of RBC aggregates, etc. The updated L–W equation computes the location and velocity of blood flow caused by capillary forces at specified time points within the LOC devices, taking into account the effects of blood flow characteristics such as RBC aggregation and the FL effect on dynamic blood viscosity.Apart from the blood flow behaviors triggered by inherent blood properties, unique flow conditions driven by capillary forces that are portrayed under different microchannel geometries also hold crucial implications for CD blood delivery. Berthier et al.
(77) studied the spontaneous Concus–Finn condition, the condition to initiate the spontaneous capillary flow within a V-groove microchannel, as shown in Figure 3(a) both experimentally and numerically. Through experimental studies, the spontaneous Concus–Finn filament development of capillary driven blood flow is observed, as shown in Figure 3(b), while the dynamic development of blood flow is numerically simulated through CFD simulation.
Figure 3. (a) Sketch of the cross-section of Berthier’s V-groove microchannel, (b) experimental view of blood in the V-groove microchannel, (78) (c) illustration of the dynamic change of the extension of filament from FLOW 3D under capillary flow at three increasing time intervals. (78) Reproduced with permission from ref (78). Copyright 2014 Elsevier.
Berthier et al.
(77) characterized the contact angle needed for the initiation of the capillary driving force at a zero-inlet pressure, through the half-angle (α) of the V-groove geometry layout, and its relation to the Concus–Finn filament as shown below:
(13)Three possible regimes were concluded based on the contact angle value for the initiation of flow and development of Concus–Finn filament:
𝜃>𝜃1𝜃1>𝜃>𝜃0𝜃0no SCFSCF without a Concus−Finn filamentSCF without a Concus−Finn filament{�>�1no SCF�1>�>�0SCF without a Concus−Finn filament�0SCF without a Concus−Finn filament
(14)Under Newton’s Law, the force balance with low Reynolds and Capillary numbers results in the neglect of inertial terms. The force balance between the capillary forces and the viscous force induced by the channel wall is proposed to derive the analytical fluid velocity. This relation between the two forces offers insights into the average flow velocity and the penetration distance function dependent on time. The apparent blood viscosity is defined by Berthier et al.
(23) given in eq 1. The research used the FLOW-3D program from Flow Science Inc. software, which solves transient, free-surface problems using the FDM in multiple dimensions. The Volume of Fluid (VOF) method
(79) is utilized to locate and track the dynamic extension of filament throughout the advancing interface within the channel ahead of the main flow at three progressing time stamps, as depicted in Figure 3(c).
The utilization of external forces, such as electric fields, has significantly broadened the possibility of manipulating microfluidic flow in LOC systems.
(80) Externally applied electric field forces induce a fluid flow from the movement of ions in fluid terms as the “electro-osmotic flow” (EOF).Unique transport phenomena, such as enhanced flow velocity and flow instability, induced by non-Newtonian fluids, particularly viscoelastic fluids, under EOF, have sparked considerable interest in microfluidic devices with simple or complicated geometries within channels.
(81) However, compared to the study of Newtonian fluids and even other electro-osmotic viscoelastic fluid flows, the literature focusing on the theoretical and numerical modeling of electro-osmotic blood flow is limited due to the complexity of blood properties. Consequently, to obtain a more comprehensive understanding of the complex blood flow behavior under EOF, theoretical and numerical studies of the transport phenomena in the EOF section will be based on the studies of different viscoelastic fluids under EOF rather than that of blood specifically. Despite this limitation, we believe these studies offer valuable insights that can help understand the complex behavior of blood flow under EOF.
4.1. EOF Phenomena
Electro-osmotic flow occurs at the interface between the microchannel wall and bulk phase solution. When in contact with the bulk phase, solution ions are absorbed or dissociated at the solid–liquid interface, resulting in the formation of a charge layer, as shown in Figure 4. This charged channel surface wall interacts with both negative and positive ions in the bulk sample, causing repulsion and attraction forces to create a thin layer of immobilized counterions, known as the Stern layer. The induced electric potential from the wall gradually decreases with an increase in the distance from the wall. The Stern layer potential, commonly termed the zeta potential, controls the intensity of the electrostatic interactions between mobile counterions and, consequently, the drag force from the applied electric field. Next to the Stern layer is the diffuse mobile layer, mainly composed of a mobile counterion. These two layers constitute the “electrical double layer” (EDL), the thickness of which is directly proportional to the ionic strength (concentration) of the bulk fluid. The relationship between the two parameters is characterized by a Debye length (λ
D), expressed as
𝜆𝐷=𝜖𝑘B𝑇2(𝑍𝑒)2𝑐0⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√��=��B�2(��)2�0
(15)where ϵ is the permittivity of the electrolyte solution, k
B is the Boltzmann constant, T is the electron temperature, Z is the integer valence number, e is the elementary charge, and c
0 is the ionic density.
Figure 4. Schematic diagram of an electro-osmotic flow in a microchannel with negative surface charge. (82) Reproduced with permission from ref (82). Copyright 2012 Woodhead Publishing.
When an electric field is applied perpendicular to the EDL, viscous drag is generated due to the movement of excess ions in the EDL. Electro-osmotic forces can be attributed to the externally applied electric potential (ϕ) and the zeta potential, the system wall induced potential by charged walls (ψ). As illustrated in Figure 4, the majority of ions in the bulk phase have a uniform velocity profile, except for a shear rate condition confined within an extremely thin Stern layer. Therefore, EOF displays a unique characteristic of a “near flat” or plug flow velocity profile, different from the parabolic flow typically induced by pressure-driven microfluidic flow (Hagen–Poiseuille flow). The plug-shaped velocity profile of the EOF possesses a high shear rate above the Stern layer.Overall, the EOF velocity magnitude is typically proportional to the Debye Length (λ
D), zeta potential, and magnitude of the externally applied electric field, while a more viscous liquid reduces the EOF velocity.
4.2. Modeling on Electro-osmotic Viscoelastic Fluid Flow
4.2.1. Theoretical Basis of EOF Mechanisms
The EOF of an incompressible viscoelastic fluid is commonly governed by the continuity and incompressible N–S equations, as shown in eqs 7 and 8, where the stress tensor and the electrostatic force term are coupled. The electro-osmotic body force term F, representing the body force exerted by the externally applied electric force, is defined as
𝐹⇀=𝑝𝐸𝐸⇀�⇀=���⇀, where ρ
E and
𝐸⇀�⇀ are the net electric charge density and the applied external electric field, respectively.Numerous models are established to theoretically study the externally applied electric potential and the system wall induced potential by charged walls. The following Laplace equation, expressed as eq 16, is generally adapted and solved to calculate the externally applied potential (ϕ).
∇2𝜙=0∇2�=0
(16)Ion diffusion under applied electric fields, together with mass transport resulting from convection and diffusion, transports ionic solutions in bulk flow under electrokinetic processes. The Nernst–Planck equation can describe these transport methods, including convection, diffusion, and electro-diffusion. Therefore, the Nernst–Planck equation is used to determine the distribution of the ions within the electrolyte. The electric potential induced by the charged channel walls follows the Poisson–Nernst–Plank (PNP) equation, which can be written as eq 17.
i are the diffusion coefficient, ionic concentration, and ionic valence of the ionic species I, respectively. However, due to the high nonlinearity and numerical stiffness introduced by different lengths and time scales from the PNP equations, the Poisson–Boltzmann (PB) model is often considered the major simplified method of the PNP equation to characterize the potential distribution of the EDL region in microchannels. In the PB model, it is assumed that the ionic species in the fluid follow the Boltzmann distribution. This model is typically valid for steady-state problems where charge transport can be considered negligible, the EDLs do not overlap with each other, and the intrinsic potentials are low. It provides a simplified representation of the potential distribution in the EDL region. The PB equation governing the EDL electric potential distribution is described as
0 is the ion bulk concentration, z is the ionic valence, and ε
0 is the electric permittivity in the vacuum. Under low electric potential conditions, an even further simplified model to illustrate the EOF phenomena is the Debye–Hückel (DH) model. The DH model is derived by obtaining a charge density term by expanding the exponential term of the Boltzmann equation in a Taylor series.
4.2.2. EOF Modeling for Viscoelastic Fluids
Many studies through numerical modeling were performed to obtain a deeper understanding of the effect exhibited by externally applied electric fields on viscoelastic flow in microchannels under various geometrical designs. Bello et al.
(83) found that methylcellulose solution, a non-Newtonian polymer solution, resulted in stronger electro-osmotic mobility in experiments when compared to the predictions by the Helmholtz–Smoluchowski equation, which is commonly used to define the velocity of EOF of a Newtonian fluid. Being one of the pioneers to identify the discrepancies between the EOF of Newtonian and non-Newtonian fluids, Bello et al. attributed such discrepancies to the presence of a very high shear rate in the EDL, resulting in a change in the orientation of the polymer molecules. Park and Lee
(84) utilized the FVM to solve the PB equation for the characterization of the electric field induced force. In the study, the concept of fractional calculus for the Oldroyd-B model was adapted to illustrate the elastic and memory effects of viscoelastic fluids in a straight microchannel They observed that fluid elasticity and increased ratio of viscoelastic fluid contribution to overall fluid viscosity had a significant impact on the volumetric flow rate and sensitivity of velocity to electric field strength compared to Newtonian fluids. Afonso et al.
(85) derived an analytical expression for EOF of viscoelastic fluid between parallel plates using the DH model to account for a zeta potential condition below 25 mV. The study established the understanding of the electro-osmotic viscoelastic fluid flow under low zeta potential conditions. Apart from the electrokinetic forces, pressure forces can also be coupled with EOF to generate a unique fluid flow behavior within the microchannel. Sousa et al.
(86) analytically studied the flow of a standard viscoelastic solution by combining the pressure gradient force with an externally applied electric force. It was found that, at a near wall skimming layer and the outer layer away from the wall, macromolecules migrating away from surface walls in viscoelastic fluids are observed. In the study, the Phan-Thien Tanner (PTT) constitutive model is utilized to characterize the viscoelastic properties of the solution. The approach is found to be valid when the EDL is much thinner than the skimming layer under an enhanced flow rate. Zhao and Yang
(87) solved the PB equation and Carreau model for the characterization of the EOF mechanism and non-Newtonian fluid respectively through the FEM. The numerical results depict that, different from the EOF of Newtonian fluids, non-Newtonian fluids led to an increase of electro-osmotic mobility for shear thinning fluids but the opposite for shear thickening fluids.Like other fluid transport driving forces, EOF within unique geometrical layouts also portrays unique transport phenomena. Pimenta and Alves
(88) utilized the FVM to perform numerical simulations of the EOF of viscoelastic fluids considering the PB equation and the Oldroyd-B model, in a cross-slot and flow-focusing microdevices. It was found that electroelastic instabilities are formed due to the development of large stresses inside the EDL with streamlined curvature at geometry corners. Bezerra et al.
(89) used the FDM to numerically analyze the vortex formation and flow instability from an electro-osmotic non-Newtonian fluid flow in a microchannel with a nozzle geometry and parallel wall geometry setting. The PNP equation is utilized to characterize the charge motion in the EOF and the PTT model for non-Newtonian flow characterization. A constriction geometry is commonly utilized in blood flow adapted in LOC systems due to the change in blood flow behavior under narrow dimensions in a microchannel. Ji et al.
(90) recently studied the EOF of viscoelastic fluid in a constriction microchannel connected by two relatively big reservoirs on both ends (as seen in Figure 5) filled with the polyacrylamide polymer solution, a viscoelastic fluid, and an incompressible monovalent binary electrolyte solution KCl.
Figure 5. Schematic diagram of a negatively charged constriction microchannel connected to two reservoirs at both ends. An electro-osmotic flow is induced in the system by the induced potential difference between the anode and cathode. (90) Reproduced with permission from ref (90). Copyright 2021 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
In studying the EOF of viscoelastic fluids, the Oldroyd-B model is often utilized to characterize the polymeric stress tensor and the deformation rate of the fluid. The Oldroyd-B model is expressed as follows:
𝜏=𝜂p𝜆(𝐜−𝐈)�=�p�(�−�)
(19)where η
p, λ, c, and I represent the polymer dynamic viscosity, polymer relaxation time, symmetric conformation tensor of the polymer molecules, and the identity matrix, respectively.A log-conformation tensor approach is taken to prevent convergence difficulty induced by the viscoelastic properties. The conformation tensor (c) in the polymeric stress tensor term is redefined by a new tensor (Θ) based on the natural logarithm of the c. The new tensor is defined as
Θ=ln(𝐜)=𝐑ln(𝚲)𝐑Θ=ln(�)=�ln(�)�
(20)in which Λ is the diagonal matrix and R is the orthogonal matrix.Under the new conformation tensor, the induced EOF of a viscoelastic fluid is governed by the continuity and N–S equations adapting the Oldroyd-B model, which is expressed as
(21)where Ω and B represent the anti-symmetric matrix and the symmetric traceless matrix of the decomposition of the velocity gradient tensor ∇u, respectively. The conformation tensor can be recovered by c = exp(Θ). The PB model and Laplace equation are utilized to characterize the charged channel wall induced potential and the externally applied potential.The governing equations are numerically solved through the FVM by RheoTool,
(42) an open-source viscoelastic EOF solver on the OpenFOAM platform. A SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm was applied to solve the velocity-pressure coupling. The pressure field and velocity field were computed by the PCG (Preconditioned Conjugate Gradient) solver and the PBiCG (Preconditioned Biconjugate Gradient) solver, respectively.Ranging magnitudes of an applied electric field or fluid concentration induce both different streamlines and velocity magnitudes at various locations and times of the microchannel. In the study performed by Ji et al.,
(90) notable fluctuation of streamlines and vortex formation is formed at the upper stream entrance of the constriction as shown in Figure 6(a) and (b), respectively, due to the increase of electrokinetic effect, which is seen as a result of the increase in polymeric stress (τ
xx).
(90) The contraction geometry enhances the EOF velocity within the constriction channel under high E
app condition (600 V/cm). Such phenomena can be attributed to the dependence of electro-osmotic viscoelastic fluid flow on the system wall surface and bulk fluid properties.
Figure 6. Schematic diagram of vortex formation and streamlines of EOF depicting flow instability at (a) 1.71 s and (b) 1.75 s. Spatial distribution of the elastic normal stress at (c) high Eapp condition. Streamline of an electro-osmotic flow under Eapp of 600 V/cm (90) for (d) non-Newtonian and (e) Newtonian fluid through a constriction geometry. Reproduced with permission from ref (90). Copyright 2021 The Authors, under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
As elastic normal stress exceeds the local shear stress, flow instability and vortex formation occur. The induced elastic stress under EOF not only enhances the instability of the flow but often generates an irregular secondary flow leading to strong disturbance.
(92) It is also vital to consider the effect of the constriction layout of microchannels on the alteration of the field strength within the system. The contraction geometry enhances a larger electric field strength compared with other locations of the channel outside the constriction region, resulting in a higher velocity gradient and stronger extension on the polymer within the viscoelastic solution. Following the high shear flow condition, a higher magnitude of stretch for polymer molecules in viscoelastic fluids exhibits larger elastic stresses and enhancement of vortex formation at the region.
(93)As shown in Figure 6(c), significant elastic normal stress occurs at the inlet of the constriction microchannel. Such occurrence of a polymeric flow can be attributed to the dominating elongational flow, giving rise to high deformation of the polymers within the viscoelastic fluid flow, resulting in higher elastic stress from the polymers. Such phenomena at the entrance result in the difference in velocity streamline as circled in Figure 6(d) compared to that of the Newtonian fluid at the constriction entrance in Figure 6(e).
(90) The difference between the Newtonian and polymer solution at the exit, as circled in Figure 6(d) and (e), can be attributed to the extrudate swell effect of polymers
(94) within the viscoelastic fluid flow. The extrudate swell effect illustrates that, as polymers emerge from the constriction exit, they tend to contract in the flow direction and grow in the normal direction, resulting in an extrudate diameter greater than the channel size. The deformation of polymers within the polymeric flow at both the entrance and exit of the contraction channel facilitates the change in shear stress conditions of the flow, leading to the alteration in streamlines of flows for each region.
4.3. EOF Applications in LOC Systems
4.3.1. Mixing in LOC Systems
Rather than relying on the micromixing controlled by molecular diffusion under low Reynolds number conditions, active mixers actively leverage convective instability and vortex formation induced by electro-osmotic flows from alternating current (AC) or direct current (DC) electric fields. Such adaptation is recognized as significant breakthroughs for promotion of fluid mixing in chemical and biological applications such as drug delivery, medical diagnostics, chemical synthesis, and so on.
(95)Many researchers proposed novel designs of electro-osmosis micromixers coupled with numerical simulations in conjunction with experimental findings to increase their understanding of the role of flow instability and vortex formation in the mixing process under electrokinetic phenomena. Matsubara and Narumi
(96) numerically modeled the mixing process in a microchannel with four electrodes on each side of the microchannel wall, which generated a disruption through unstable electro-osmotic vortices. It was found that particle mixing was sensitive to both the convection effect induced by the main and secondary vortex within the micromixer and the change in oscillation frequency caused by the supplied AC voltage when the Reynolds number was varied. Qaderi et al.
(97) adapted the PNP equation to numerically study the effect of the geometry and zeta potential configuration of the microchannel on the mixing process with a combined electro-osmotic pressure driven flow. It was reported that the application of heterogeneous zeta potential configuration enhances the mixing efficiency by around 23% while the height of the hurdles increases the mixing efficiency at most 48.1%. Cho et al.
(98) utilized the PB model and Laplace equation to numerically simulate the electro-osmotic non-Newtonian fluid mixing process within a wavy and block layout of microchannel walls. The Power Law model is adapted to describe the fluid rheological characteristic. It was found that shear-thinning fluids possess a higher volumetric flow rate, which could result in poorer mixing efficiency compared to that of Newtonian fluids. Numerous studies have revealed that flow instability and vortex generation, in particular secondary vortices produced by barriers or greater magnitudes of heterogeneous zeta potential distribution, enhance mixing by increasing bulk flow velocity and reducing flow distance.To better understand the mechanism of disturbance formed in the system due to externally applied forces, known as electrokinetic instability, literature often utilize the Rayleigh (Ra) number,
(22)where γ is the conductivity ratio of the two streams and can be written as
𝛾=𝜎el,H𝜎el,L�=�el,H�el,L. The Ra number characterizes the ratio between electroviscous and electro-osmotic flow. A high Ra
v value often results in good mixing. It is evident that fluid properties such as the conductivity (σ) of the two streams play a key role in the formation of disturbances to enhance mixing in microsystems. At the same time, electrokinetic parameters like the zeta potential (ζ) in the Ra number is critical in the characterization of electro-osmotic velocity and a slip boundary condition at the microchannel wall.To understand the mixing result along the channel, the concentration field can be defined and simulated under the assumption of steady state conditions and constant diffusion coefficient for each of the working fluid within the system through the convection–diffusion equation as below:
∂𝑐𝒊∂𝑡+∇⇀(𝑐𝑖𝑢⇀−𝐷𝑖∇⇀𝑐𝒊)=0∂��∂�+∇⇀(���⇀−��∇⇀��)=0
(23)where c
i is the species concentration of species i and D
i is the diffusion coefficient of the corresponding species.The standard deviation of concentration (σ
sd) can be adapted to evaluate the mixing quality of the system.
(97) The standard deviation for concentration at a specific portion of the channel may be calculated using the equation below:
m are the non-dimensional concentration profile and the mean concentration at the portion, respectively. C* is the non-dimensional concentration and can be calculated as
𝐶∗=𝐶𝐶ref�*=��ref, where C
ref is the reference concentration defined as the bulk solution concentration. The mean concentration profile can be calculated as
𝐶m=∫10(𝐶∗(𝑦∗)d𝑦∗∫10d𝑦∗�m=∫01(�*(�*)d�*∫01d�*. With the standard deviation of concentration, the mixing efficiency
sd,0 is the standard derivation of the case of no mixing. The value of the mixing efficiency is typically utilized in conjunction with the simulated flow field and concentration field to explore the effect of geometrical and electrokinetic parameters on the optimization of the mixing results.
Viscoelastic fluids such as blood flow in LOC systems are an essential topic to proceed with diagnostic analysis and research through microdevices in the biomedical and pharmaceutical industries. The complex blood flow behavior is tightly controlled by the viscoelastic characteristics of blood such as the dynamic viscosity and the elastic property of RBCs under various shear rate conditions. Furthermore, the flow behaviors under varied driving forces promote an array of microfluidic transport phenomena that are critical to the management of blood flow and other adapted viscoelastic fluids in LOC systems. This review addressed the blood flow phenomena, the complicated interplay between shear rate and blood flow behaviors, and their numerical modeling under LOC systems through the lens of the viscoelasticity characteristic. Furthermore, a theoretical understanding of capillary forces and externally applied electric forces leads to an in-depth investigation of the relationship between blood flow patterns and the key parameters of the two driving forces, the latter of which is introduced through the lens of viscoelastic fluids, coupling numerical modeling to improve the knowledge of blood flow manipulation in LOC systems. The flow disturbances triggered by the EOF of viscoelastic fluids and their impact on blood flow patterns have been deeply investigated due to their important role and applications in LOC devices. Continuous advancements of various numerical modeling methods with experimental findings through more efficient and less computationally heavy methods have served as an encouraging sign of establishing more accurate illustrations of the mechanisms for multiphase blood and other viscoelastic fluid flow transport phenomena driven by various forces. Such progress is fundamental for the manipulation of unique transport phenomena, such as the generated disturbances, to optimize functionalities offered by microdevices in LOC systems.
The following section will provide further insights into the employment of studied blood transport phenomena to improve the functionality of micro devices adapting LOC technology. A discussion of the novel roles that external driving forces play in microfluidic flow behaviors is also provided. Limitations in the computational modeling of blood flow and electrokinetic phenomena in LOC systems will also be emphasized, which may provide valuable insights for future research endeavors. These discussions aim to provide guidance and opportunities for new paths in the ongoing development of LOC devices that adapt blood flow.
5.2. Future Directions
5.2.1. Electro-osmosis Mixing in LOC Systems
Despite substantial research, mixing results through flow instability and vortex formation phenomena induced by electro-osmotic mixing still deviate from the effective mixing results offered by chaotic mixing results such as those seen in turbulent flows. However, recent discoveries of a mixing phenomenon that is generally observed under turbulent flows are found within electro-osmosis micromixers under low Reynolds number conditions. Zhao
(99) experimentally discovered a rapid mixing process in an AC applied micromixer, where the power spectrum of concentration under an applied voltage of 20 V
p-p induces a −5/3 slope within a frequency range. This value of the slope is considered as the O–C spectrum in macroflows, which is often visible under relatively high Re conditions, such as the Taylor microscale Reynolds number Re > 500 in turbulent flows.
(100) However, the Re value in the studied system is less than 1 at the specific location and applied voltage. A secondary flow is also suggested to occur close to microchannel walls, being attributed to the increase of convective instability within the system.Despite the experimental phenomenon proposed by Zhao et al.,
(99) the range of effects induced by vital parameters of an EOF mixing system on the enhanced mixing results and mechanisms of disturbance generated by the turbulent-like flow instability is not further characterized. Such a gap in knowledge may hinder the adaptability and commercialization of the discovery of micromixers. One of the parameters for further evaluation is the conductivity gradient of the fluid flow. A relatively strong conductivity gradient (5000:1) was adopted in the system due to the conductive properties of the two fluids. The high conductivity gradients may contribute to the relatively large Rayleigh number and differences in EDL layer thickness, resulting in an unusual disturbance in laminar flow conditions and enhanced mixing results. However, high conductivity gradients are not always achievable by the working fluids due to diverse fluid properties. The reliance on turbulent-like phenomena and rapid mixing results in a large conductivity gradient should be established to prevent the limited application of fluids for the mixing system. In addition, the proposed system utilizes distinct zeta potential distributions at the top and bottom walls due to their difference in material choices, which may be attributed to the flow instability phenomena. Further studies should be made on varying zeta potential magnitude and distribution to evaluate their effect on the slip boundary conditions of the flow and the large shear rate condition close to the channel wall of EOF. Such a study can potentially offer an optimized condition in zeta potential magnitude through material choices and geometrical layout of the zeta potential for better mixing results and manipulation of mixing fluid dynamics. The two vital parameters mentioned above can be varied with the aid of numerical simulation to understand the effect of parameters on the interaction between electro-osmotic forces and electroviscous forces. At the same time, the relationship of developed streamlines of the simulated velocity and concentration field, following their relationship with the mixing results, under the impact of these key parameters can foster more insight into the range of impact that the two parameters have on the proposed phenomena and the microfluidic dynamic principles of disturbances.
In addition, many of the current investigations of electrokinetic mixers commonly emphasize the fluid dynamics of mixing for Newtonian fluids, while the utilization of biofluids, primarily viscoelastic fluids such as blood, and their distinctive response under shear forces in these novel mixing processes of LOC systems are significantly less studied. To develop more compatible microdevice designs and efficient mixing outcomes for the biomedical industry, it is necessary to fill the knowledge gaps in the literature on electro-osmotic mixing for biofluids, where properties of elasticity, dynamic viscosity, and intricate relationship with shear flow from the fluid are further considered.
5.2.2. Electro-osmosis Separation in LOC Systems
Particle separation in LOC devices, particularly in biological research and diagnostics, is another area where disturbances may play a significant role in optimization.
(101) Plasma analysis in LOC systems under precise control of blood flow phenomena and blood/plasma separation procedures can detect vital information about infectious diseases from particular antibodies and foreign nucleic acids for medical treatments, diagnostics, and research,
(102) offering more efficient results and simple operating procedures compared to that of the traditional centrifugation method for blood and plasma separation. However, the adaptability of LOC devices for blood and plasma separation is often hindered by microchannel clogging, where flow velocity and plasma yield from LOC devices is reduced due to occasional RBC migration and aggregation at the filtration entrance of microdevices.
(103)It is important to note that the EOF induces flow instability close to microchannel walls, which may provide further solutions to clogging for the separation process of the LOC systems. Mohammadi et al.
(104) offered an anti-clogging effect of RBCs at the blood and plasma separating device filtration entry, adjacent to the surface wall, through RBC disaggregation under high shear rate conditions generated by a forward and reverse EOF direction.
Further theoretical and numerical research can be conducted to characterize the effect of high shear rate conditions near microchannel walls toward the detachment of binding blood cells on surfaces and the reversibility of aggregation. Through numerical modeling with varying electrokinetic parameters to induce different degrees of disturbances or shear conditions at channel walls, it may be possible to optimize and better understand the process of disrupting the forces that bind cells to surface walls and aggregated cells at filtration pores. RBCs that migrate close to microchannel walls are often attracted by the adhesion force between the RBC and the solid surface originating from the van der Waals forces. Following RBC migration and attachment by adhesive forces adjacent to the microchannel walls as shown in Figure 7, the increase in viscosity at the region causes a lower shear condition and encourages RBC aggregation (cell–cell interaction), which clogs filtering pores or microchannels and reduces flow velocity at filtration region. Both the impact that shear forces and disturbances may induce on cell binding forces with surface walls and other cells leading to aggregation may suggest further characterization. Kinetic parameters such as activation energy and the rate-determining step for cell binding composition attachment and detachment should be considered for modeling the dynamics of RBCs and blood flows under external forces in LOC separation devices.
Figure 7. Schematic representations of clogging at a microchannel pore following the sequence of RBC migration, cell attachment to channel walls, and aggregation. (105) Reproduced with permission from ref (105). Copyright 2018 The Authors under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
5.2.3. Relationship between External Forces and Microfluidic Systems
In blood flow, a thicker CFL suggests a lower blood viscosity, suggesting a complex relationship between shear stress and shear rate, affecting the blood viscosity and blood flow. Despite some experimental and numerical studies on electro-osmotic non-Newtonian fluid flow, limited literature has performed an in-depth investigation of the role that applied electric forces and other external forces could play in the process of CFL formation. Additional studies on how shear rates from external forces affect CFL formation and microfluidic flow dynamics can shed light on the mechanism of the contribution induced by external driving forces to the development of a separate phase of layer, similar to CFL, close to the microchannel walls and distinct from the surrounding fluid within the system, then influencing microfluidic flow dynamics.One of the mechanisms of phenomena to be explored is the formation of the Exclusion Zone (EZ) region following a “Self-Induced Flow” (SIF) phenomenon discovered by Li and Pollack,
(106) as shown in Figure 8(a) and (b), respectively. A spontaneous sustained axial flow is observed when hydrophilic materials are immersed in water, resulting in the buildup of a negative layer of charges, defined as the EZ, after water molecules absorb infrared radiation (IR) energy and break down into H and OH
+–.
Figure 8. Schematic representations of (a) the Exclusion Zone region and (b) the Self Induced Flow through visualization of microsphere movement within a microchannel. (106) Reproduced with permission from ref (106). Copyright 2020 The Authors under the terms of the Creative Commons (CC BY 4.0) License https://creativecommons.org/licenses/by/4.0/.
Despite the finding of such a phenomenon, the specific mechanism and role of IR energy have yet to be defined for the process of EZ development. To further develop an understanding of the role of IR energy in such phenomena, a feasible study may be seen through the lens of the relationships between external forces and microfluidic flow. In the phenomena, the increase of SIF velocity under a rise of IR radiation resonant characteristics is shown in the participation of the external electric field near the microchannel walls under electro-osmotic viscoelastic fluid flow systems. The buildup of negative charges at the hydrophilic surfaces in EZ is analogous to the mechanism of electrical double layer formation. Indeed, research has initiated the exploration of the core mechanisms for EZ formation through the lens of the electrokinetic phenomena.
(107) Such a similarity of the role of IR energy and the transport phenomena of SIF with electrokinetic phenomena paves the way for the definition of the unknown SIF phenomena and EZ formation. Furthermore, Li and Pollack
(106) suggest whether CFL formation might contribute to a SIF of blood using solely IR radiation, a commonly available source of energy in nature, as an external driving force. The proposition may be proven feasible with the presence of the CFL region next to the negatively charged hydrophilic endothelial glycocalyx layer, coating the luminal side of blood vessels.
(108) Further research can dive into the resonating characteristics between the formation of the CFL region next to the hydrophilic endothelial glycocalyx layer and that of the EZ formation close to hydrophilic microchannel walls. Indeed, an increase in IR energy is known to rapidly accelerate EZ formation and SIF velocity, depicting similarity to the increase in the magnitude of electric field forces and greater shear rates at microchannel walls affecting CFL formation and EOF velocity. Such correlation depicts a future direction in whether SIF blood flow can be observed and characterized theoretically further through the lens of the relationship between blood flow and shear forces exhibited by external energy.
The intricate link between the CFL and external forces, more specifically the externally applied electric field, can receive further attention to provide a more complete framework for the mechanisms between IR radiation and EZ formation. Such characterization may also contribute to a greater comprehension of the role IR can play in CFL formation next to the endothelial glycocalyx layer as well as its role as a driving force to propel blood flow, similar to the SIF, but without the commonly assumed pressure force from heart contraction as a source of driving force.
5.3. Challenges
Although there have been significant improvements in blood flow modeling under LOC systems over the past decade, there are still notable constraints that may require special attention for numerical simulation applications to benefit the adaptability of the designs and functionalities of LOC devices. Several points that require special attention are mentioned below:
1.
The majority of CFD models operate under the relationship between the viscoelasticity of blood and the shear rate conditions of flow. The relative effect exhibited by the presence of highly populated RBCs in whole blood and their forces amongst the cells themselves under complex flows often remains unclearly defined. Furthermore, the full range of cell populations in whole blood requires a much more computational load for numerical modeling. Therefore, a vital goal for future research is to evaluate a reduced modeling method where the impact of cell–cell interaction on the viscoelastic property of blood is considered.
2.
Current computational methods on hemodynamics rely on continuum models based upon non-Newtonian rheology at the macroscale rather than at molecular and cellular levels. Careful considerations should be made for the development of a constructive framework for the physical and temporal scales of micro/nanoscale systems to evaluate the intricate relationship between fluid driving forces, dynamic viscosity, and elasticity.
3.
Viscoelastic fluids under the impact of externally applied electric forces often deviate from the assumptions of no-slip boundary conditions due to the unique flow conditions induced by externally applied forces. Furthermore, the mechanism of vortex formation and viscoelastic flow instability at laminar flow conditions should be better defined through the lens of the microfluidic flow phenomenon to optimize the prediction of viscoelastic flow across different geometrical layouts. Mathematical models and numerical methods are needed to better predict such disturbance caused by external forces and the viscoelasticity of fluids at such a small scale.
4.
Under practical situations, zeta potential distribution at channel walls frequently deviates from the common assumption of a constant distribution because of manufacturing faults or inherent surface charges prior to the introduction of electrokinetic influence. These discrepancies frequently lead to inconsistent surface potential distribution, such as excess positive ions at relatively more negatively charged walls. Accordingly, unpredicted vortex formation and flow instability may occur. Therefore, careful consideration should be given to these discrepancies and how they could trigger the transport process and unexpected results of a microdevice.
Zhe Chen – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; Email: zaccooky@sjtu.edu.cn
Bo Ouyang – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; Email: bouy93@sjtu.edu.cn
Zheng-Hong Luo – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0000-0001-9011-6020; Email: luozh@sjtu.edu.cn
Authors
Bin-Jie Lai – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0009-0002-8133-5381
Li-Tao Zhu – Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China; https://orcid.org/0000-0001-6514-8864
NotesThe authors declare no competing financial interest.
This work was supported by the National Natural Science Foundation of China (No. 22238005) and the Postdoctoral Research Foundation of China (No. GZC20231576).
the field of technological and scientific study that investigates fluid flow in channels with dimensions between 1 and 1000 μm
Lab-on-a-Chip Technology
the field of research and technological development aimed at integrating the micro/nanofluidic characteristics to conduct laboratory processes on handheld devices
Computational Fluid Dynamics (CFD)
the method utilizing computational abilities to predict physical fluid flow behaviors mathematically through solving the governing equations of corresponding fluid flows
Shear Rate
the rate of change in velocity where one layer of fluid moves past the adjacent layer
Viscoelasticity
the property holding both elasticity and viscosity characteristics relying on the magnitude of applied shear stress and time-dependent strain
Electro-osmosis
the flow of fluid under an applied electric field when charged solid surface is in contact with the bulk fluid
Vortex
the rotating motion of a fluid revolving an axis line
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웨어의 두 가지 서로 다른 배열(즉, 직선형 웨어와 직사각형 미로 웨어)을 사용하여 웨어 모양, 웨어 간격, 웨어의 오리피스 존재, 흐름 영역에 대한 바닥 경사와 같은 기하학적 매개변수의 영향을 평가했습니다.
유량과 수심의 관계, 수심 평균 속도의 변화와 분포, 난류 특성, 어도에서의 에너지 소산. 흐름 조건에 미치는 영향을 조사하기 위해 FLOW-3D® 소프트웨어를 사용하여 전산 유체 역학 시뮬레이션을 수행했습니다.
수치 모델은 계산된 표면 프로파일과 속도를 문헌의 실험적으로 측정된 값과 비교하여 검증되었습니다. 수치 모델과 실험 데이터의 결과, 급락유동의 표면 프로파일과 표준화된 속도 프로파일에 대한 평균 제곱근 오차와 평균 절대 백분율 오차가 각각 0.014m와 3.11%로 나타나 수치 모델의 능력을 확인했습니다.
수영장과 둑의 흐름 특성을 예측합니다. 각 모델에 대해 L/B = 1.83(L: 웨어 거리, B: 수로 폭) 값에서 급락 흐름이 발생할 수 있고 L/B = 0.61에서 스트리밍 흐름이 발생할 수 있습니다. 직사각형 미로보 모델은 기존 모델보다 무차원 방류량(Q+)이 더 큽니다.
수중 흐름의 기존 보와 직사각형 미로 보의 경우 Q는 각각 1.56과 1.47h에 비례합니다(h: 보 위 수심). 기존 웨어의 풀 내 평균 깊이 속도는 직사각형 미로 웨어의 평균 깊이 속도보다 높습니다.
그러나 주어진 방류량, 바닥 경사 및 웨어 간격에 대해 난류 운동 에너지(TKE) 및 난류 강도(TI) 값은 기존 웨어에 비해 직사각형 미로 웨어에서 더 높습니다. 기존의 웨어는 직사각형 미로 웨어보다 에너지 소산이 더 낮습니다.
더 낮은 TKE 및 TI 값은 미로 웨어 상단, 웨어 하류 벽 모서리, 웨어 측벽과 채널 벽 사이에서 관찰되었습니다. 보와 바닥 경사면 사이의 거리가 증가함에 따라 평균 깊이 속도, 난류 운동 에너지의 평균값 및 난류 강도가 증가하고 수영장의 체적 에너지 소산이 감소했습니다.
둑에 개구부가 있으면 평균 깊이 속도와 TI 값이 증가하고 풀 내에서 가장 높은 TKE 범위가 감소하여 두 모델 모두에서 물고기를 위한 휴식 공간이 더 넓어지고(TKE가 낮아짐) 에너지 소산율이 감소했습니다.
Two different arrangements of the weir (i.e., straight weir and rectangular labyrinth weir) were used to evaluate the effects of geometric parameters such as weir shape, weir spacing, presence of an orifice at the weir, and bed slope on the flow regime and the relationship between discharge and depth, variation and distribution of depth-averaged velocity, turbulence characteristics, and energy dissipation at the fishway. Computational fluid dynamics simulations were performed using FLOW-3D® software to examine the effects on flow conditions. The numerical model was validated by comparing the calculated surface profiles and velocities with experimentally measured values from the literature. The results of the numerical model and experimental data showed that the root-mean-square error and mean absolute percentage error for the surface profiles and normalized velocity profiles of plunging flows were 0.014 m and 3.11%, respectively, confirming the ability of the numerical model to predict the flow characteristics of the pool and weir. A plunging flow can occur at values of L/B = 1.83 (L: distance of the weir, B: width of the channel) and streaming flow at L/B = 0.61 for each model. The rectangular labyrinth weir model has larger dimensionless discharge values (Q+) than the conventional model. For the conventional weir and the rectangular labyrinth weir at submerged flow, Q is proportional to 1.56 and 1.47h, respectively (h: the water depth above the weir). The average depth velocity in the pool of a conventional weir is higher than that of a rectangular labyrinth weir. However, for a given discharge, bed slope, and weir spacing, the turbulent kinetic energy (TKE) and turbulence intensity (TI) values are higher for a rectangular labyrinth weir compared to conventional weir. The conventional weir has lower energy dissipation than the rectangular labyrinth weir. Lower TKE and TI values were observed at the top of the labyrinth weir, at the corner of the wall downstream of the weir, and between the side walls of the weir and the channel wall. As the distance between the weirs and the bottom slope increased, the average depth velocity, the average value of turbulent kinetic energy and the turbulence intensity increased, and the volumetric energy dissipation in the pool decreased. The presence of an opening in the weir increased the average depth velocity and TI values and decreased the range of highest TKE within the pool, resulted in larger resting areas for fish (lower TKE), and decreased the energy dissipation rates in both models.
1 Introduction
Artificial barriers such as detour dams, weirs, and culverts in lakes and rivers prevent fish from migrating and completing the upstream and downstream movement cycle. This chain is related to the life stage of the fish, its location, and the type of migration. Several riverine fish species instinctively migrate upstream for spawning and other needs. Conversely, downstream migration is a characteristic of early life stages [1]. A fish ladder is a waterway that allows one or more fish species to cross a specific obstacle. These structures are constructed near detour dams and other transverse structures that have prevented such migration by allowing fish to overcome obstacles [2]. The flow pattern in fish ladders influences safe and comfortable passage for ascending fish. The flow’s strong turbulence can reduce the fish’s speed, injure them, and delay or prevent them from exiting the fish ladder. In adult fish, spawning migrations are usually complex, and delays are critical to reproductive success [3].
Various fish ladders/fishways include vertical slots, denil, rock ramps, and pool weirs [1]. The choice of fish ladder usually depends on many factors, including water elevation, space available for construction, and fish species. Pool and weir structures are among the most important fish ladders that help fish overcome obstacles in streams or rivers and swim upstream [1]. Because they are easy to construct and maintain, this type of fish ladder has received considerable attention from researchers and practitioners. Such a fish ladder consists of a sloping-floor channel with series of pools directly separated by a series of weirs [4]. These fish ladders, with or without underwater openings, are generally well-suited for slopes of 10% or less [1, 2]. Within these pools, flow velocities are low and provide resting areas for fish after they enter the fish ladder. After resting in the pools, fish overcome these weirs by blasting or jumping over them [2]. There may also be an opening in the flooded portion of the weir through which the fish can swim instead of jumping over the weir. Design parameters such as the length of the pool, the height of the weir, the slope of the bottom, and the water discharge are the most important factors in determining the hydraulic structure of this type of fish ladder [3]. The flow over the weir depends on the flow depth at a given slope S0 and the pool length, either “plunging” or “streaming.” In plunging flow, the water column h over each weir creates a water jet that releases energy through turbulent mixing and diffusion mechanisms [5]. The dimensionless discharges for plunging (Q+) and streaming (Q*) flows are shown in Fig. 1, where Q is the total discharge, B is the width of the channel, w is the weir height, S0 is the slope of the bottom, h is the water depth above the weir, d is the flow depth, and g is the acceleration due to gravity. The maximum velocity occurs near the top of the weir for plunging flow. At the water’s surface, it drops to about half [6].
Fig. 1
Extensive experimental studies have been conducted to investigate flow patterns for various physical geometries (i.e., bed slope, pool length, and weir height) [2]. Guiny et al. [7] modified the standard design by adding vertical slots, orifices, and weirs in fishways. The efficiency of the orifices and vertical slots was related to the velocities at their entrances. In the laboratory experiments of Yagci [8], the three-dimensional (3D) mean flow and turbulence structure of a pool weir fishway combined with an orifice and a slot is investigated. It is shown that the energy dissipation per unit volume and the discharge have a linear relationship.
Considering the beneficial characteristics reported in the limited studies of researchers on the labyrinth weir in the pool-weir-type fishway, and knowing that the characteristics of flow in pool-weir-type fishways are highly dependent on the geometry of the weir, an alternative design of the rectangular labyrinth weir instead of the straight weirs in the pool-weir-type fishway is investigated in this study [7, 9]. Kim [10] conducted experiments to compare the hydraulic characteristics of three different weir types in a pool-weir-type fishway. The results show that a straight, rectangular weir with a notch is preferable to a zigzag or trapezoidal weir. Studies on natural fish passes show that pass ability can be improved by lengthening the weir’s crest [7]. Zhong et al. [11] investigated the semi-rigid weir’s hydraulic performance in the fishway’s flow field with a pool weir. The results showed that this type of fishway performed better with a lower invert slope and a smaller radius ratio but with a larger pool spacing.
Considering that an alternative method to study the flow characteristics in a fishway with a pool weir is based on numerical methods and modeling from computational fluid dynamics (CFD), which can easily change the geometry of the fishway for different flow fields, this study uses the powerful package CFD and the software FLOW-3D to evaluate the proposed weir design and compare it with the conventional one to extend the application of the fishway. The main objective of this study was to evaluate the hydraulic performance of the rectangular labyrinth pool and the weir with submerged openings in different hydraulic configurations. The primary objective of creating a new weir configuration for suitable flow patterns is evaluated based on the swimming capabilities of different fish species. Specifically, the following questions will be answered: (a) How do the various hydraulic and geometric parameters relate to the effects of water velocity and turbulence, expressed as turbulent kinetic energy (TKE) and turbulence intensity (TI) within the fishway, i.e., are conventional weirs more affected by hydraulics than rectangular labyrinth weirs? (b) Which weir configurations have the greatest effect on fish performance in the fishway? (c) In the presence of an orifice plate, does the performance of each weir configuration differ with different weir spacing, bed gradients, and flow regimes from that without an orifice plate?
2 Materials and Methods
2.1 Physical Model Configuration
This paper focuses on Ead et al. [6]’s laboratory experiments as a reference, testing ten pool weirs (Fig. 2). The experimental flume was 6 m long, 0.56 m wide, and 0.6 m high, with a bottom slope of 10%. Field measurements were made at steady flow with a maximum flow rate of 0.165 m3/s. Discharge was measured with magnetic flow meters in the inlets and water level with point meters (see Ead et al. [6]. for more details). Table 1 summarizes the experimental conditions considered for model calibration in this study.
Fig. 2
Table 1 Experimental conditions considered for calibration
Computational fluid dynamics (CFD) simulations were performed using FLOW-3D® v11.2 to validate a series of experimental liner pool weirs by Ead et al. [6] and to investigate the effects of the rectangular labyrinth pool weir with an orifice. The dimensions of the channel and data collection areas in the numerical models are the same as those of the laboratory model. Two types of pool weirs were considered: conventional and labyrinth. The proposed rectangular labyrinth pool weirs have a symmetrical cross section and are sized to fit within the experimental channel. The conventional pool weir model had a pool length of l = 0.685 and 0.342 m, a weir height of w = 0.141 m, a weir width of B = 0.56 m, and a channel slope of S0 = 5 and 10%. The rectangular labyrinth weirs have the same front width as the offset, i.e., a = b = c = 0.186 m. A square underwater opening with a width of 0.05 m and a depth of 0.05 m was created in the middle of the weir. The weir configuration considered in the present study is shown in Fig. 3.
Fig. 3
2.3 Governing Equations
FLOW-3D® software solves the Navier–Stokes–Reynolds equations for three-dimensional analysis of incompressible flows using the fluid-volume method on a gridded domain. FLOW -3D® uses an advanced free surface flow tracking algorithm (TruVOF) developed by Hirt and Nichols [12], where fluid configurations are defined in terms of a VOF function F (x, y, z, t). In this case, F (fluid fraction) represents the volume fraction occupied by the fluid: F = 1 in cells filled with fluid and F = 0 in cells without fluid (empty areas) [4, 13]. The free surface area is at an intermediate value of F. (Typically, F = 0.5, but the user can specify a different intermediate value.) The equations in Cartesian coordinates (x, y, z) applicable to the model are as follows:
�f∂�∂�+∂(���x)∂�+∂(���y)∂�+∂(���z)∂�=�SOR
(1)
∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�x+�x
(2)
∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�y+�y
(3)
∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�z+�z
(4)
where (u, v, w) are the velocity components, (Ax, Ay, Az) are the flow area components, (Gx, Gy, Gz) are the mass accelerations, and (fx, fy, fz) are the viscous accelerations in the directions (x, y, z), ρ is the fluid density, RSOR is the spring term, Vf is the volume fraction associated with the flow, and P is the pressure. The k–ε turbulence model (RNG) was used in this study to solve the turbulence of the flow field. This model is a modified version of the standard k–ε model that improves performance. The model is a two-equation model; the first equation (Eq. 5) expresses the turbulence’s energy, called turbulent kinetic energy (k) [14]. The second equation (Eq. 6) is the turbulent dissipation rate (ε), which determines the rate of dissipation of kinetic energy [15]. These equations are expressed as follows Dasineh et al. [4]:
In these equations, k is the turbulent kinetic energy, ε is the turbulent energy consumption rate, Gk is the generation of turbulent kinetic energy by the average velocity gradient, with empirical constants αε = αk = 1.39, C1ε = 1.42, and C2ε = 1.68, eff is the effective viscosity, μeff = μ + μt [15]. Here, μ is the hydrodynamic density coefficient, and μt is the turbulent density of the fluid.
2.4 Meshing and the Boundary Conditions in the Model Setup
The numerical area is divided into three mesh blocks in the X-direction. The meshes are divided into different sizes, a containing mesh block for the entire spatial domain and a nested block with refined cells for the domain of interest. Three different sizes were selected for each of the grid blocks. By comparing the accuracy of their results based on the experimental data, the reasonable mesh for the solution domain was finally selected. The convergence index method (GCI) evaluated the mesh sensitivity analysis. Based on this method, many researchers, such as Ahmadi et al. [16] and Ahmadi et al. [15], have studied the independence of numerical results from mesh size. Three different mesh sizes with a refinement ratio (r) of 1.33 were used to perform the convergence index method. The refinement ratio is the ratio between the larger and smaller mesh sizes (r = Gcoarse/Gfine). According to the recommendation of Celik et al. [17], the recommended number for the refinement ratio is 1.3, which gives acceptable results. Table 2 shows the characteristics of the three mesh sizes selected for mesh sensitivity analysis.Table 2 Characteristics of the meshes tested in the convergence analysis
The results of u1 = umax (u1 = velocity component along the x1 axis and umax = maximum velocity of u1 in a section perpendicular to the invert of the fishway) at Q = 0.035 m3/s, × 1/l = 0.66, and Y1/b = 0 in the pool of conventional weir No. 4, obtained from the output results of the software, were used to evaluate the accuracy of the calculation range. As shown in Fig. 4, x1 = the distance from a given weir in the x-direction, Y1 = the water depth measured in the y-direction, Y0 = the vertical distance in the Cartesian coordinate system, h = the water column at the crest, b = the distance between the two points of maximum velocity umax and zero velocity, and l = the pool length.
Fig. 4
The apparent index of convergence (p) in the GCI method is calculated as follows:
�=ln(�3−�2)(�2−�1)/ln(�)
(7)
f1, f2, and f3 are the hydraulic parameters obtained from the numerical simulation (f1 corresponds to the small mesh), and r is the refinement ratio. The following equation defines the convergence index of the fine mesh:
GCIfine=1.25|ε|��−1
(8)
Here, ε = (f2 − f1)/f1 is the relative error, and f2 and f3 are the values of hydraulic parameters considered for medium and small grids, respectively. GCI12 and GCI23 dimensionless indices can be calculated as:
GCI12=1.25|�2−�1�1|��−1
(9)
Then, the independence of the network is preserved. The convergence index of the network parameters obtained by Eqs. (7)–(9) for all three network variables is shown in Table 3. Since the GCI values for the smaller grid (GCI12) are lower compared to coarse grid (GCI23), it can be concluded that the independence of the grid is almost achieved. No further change in the grid size of the solution domain is required. The calculated values (GCI23/rpGCI12) are close to 1, which shows that the numerical results obtained are within the convergence range. As a result, the meshing of the solution domain consisting of a block mesh with a mesh size of 0.012 m and a block mesh within a larger block mesh with a mesh size of 0.009 m was selected as the optimal mesh (Fig. 5).Table 3 GCI calculation
The boundary conditions applied to the area are shown in Fig. 6. The boundary condition of specific flow rate (volume flow rate-Q) was used for the inlet of the flow. For the downstream boundary, the flow output (outflow-O) condition did not affect the flow in the solution area. For the Zmax boundary, the specified pressure boundary condition was used along with the fluid fraction = 0 (P). This type of boundary condition considers free surface or atmospheric pressure conditions (Ghaderi et al. [19]). The wall boundary condition is defined for the bottom of the channel, which acts like a virtual wall without friction (W). The boundary between mesh blocks and walls were considered a symmetrical condition (S).
Fig. 6
The convergence of the steady-state solutions was controlled during the simulations by monitoring the changes in discharge at the inlet boundary conditions. Figure 7 shows the time series plots of the discharge obtained from the Model A for the three main discharges from the numerical results. The 8 s to reach the flow equilibrium is suitable for the case of the fish ladder with pool and weir. Almost all discharge fluctuations in the models are insignificant in time, and the flow has reached relative stability. The computation time for the simulations was between 6 and 8 h using a personal computer with eight cores of a CPU (Intel Core i7-7700K @ 4.20 GHz and 16 GB RAM).
Fig. 7
3 Results
3.1 Verification of Numerical Results
Quantitative outcomes, including free surface and normalized velocity profiles obtained using FLOW-3D software, were reviewed and compared with the results of Ead et al. [6]. The fourth pool was selected to present the results and compare the experiment and simulation. For each quantity, the percentage of mean absolute error (MAPE (%)) and root-mean-square error (RMSE) are calculated. Equations (10) and (11) show the method used to calculate the errors.
MAPE(%)100×1�∑1�|�exp−�num�exp|
(10)
RMSE(−)1�∑1�(�exp−�num)2
(11)
Here, Xexp is the value of the laboratory data, Xnum is the numerical data value, and n is the amount of data. As shown in Fig. 8, let x1 = distance from a given weir in the x-direction and Y1 = water depth in the y-direction from the bottom. The trend of the surface profiles for each of the numerical results is the same as that of the laboratory results. The surface profiles of the plunging flows drop after the flow enters and then rises to approach the next weir. The RMSE and MAPE error values for Model A are 0.014 m and 3.11%, respectively, indicating acceptable agreement between numerical and laboratory results. Figure 9 shows the velocity vectors and plunging flow from the numerical results, where x and y are horizontal and vertical to the flow direction, respectively. It can be seen that the jet in the fish ladder pool has a relatively high velocity. The two vortices, i.e., the enclosed vortex rotating clockwise behind the weir and the surface vortex rotating counterclockwise above the jet, are observed for the regime of incident flow. The point where the jet meets the fish passage bed is shown in the figure. The normalized velocity profiles upstream and downstream of the impact points are shown in Fig. 10. The figure shows that the numerical results agree well with the experimental data of Ead et al. [6].
Fig. 8Fig. 9Fig. 10
3.2 Flow Regime and Discharge-Depth Relationship
Depending on the geometric shape of the fishway, including the distance of the weir, the slope of the bottom, the height of the weir, and the flow conditions, the flow regime in the fishway is divided into three categories: dipping, transitional, and flow regimes [4]. In the plunging flow regime, the flow enters the pool through the weir, impacts the bottom of the fishway, and forms a hydraulic jump causing two eddies [2, 20]. In the streamwise flow regime, the surface of the flow passing over the weir is almost parallel to the bottom of the channel. The transitional regime has intermediate flow characteristics between the submerged and flow regimes. To predict the flow regime created in the fishway, Ead et al. [6] proposed two dimensionless parameters, Qt* and L/w, where Qt* is the dimensionless discharge, L is the distance between weirs, and w is the height of the weir:
��∗=���0���
(12)
Q is the total discharge, B is the width of the channel, S0 is the slope of the bed, and g is the gravity acceleration. Figure 11 shows different ranges for each flow regime based on the slope of the bed and the distance between the pools in this study. The results of Baki et al. [21], Ead et al. [6] and Dizabadi et al. [22] were used for this comparison. The distance between the pools affects the changes in the regime of the fish ladder. So, if you decrease the distance between weirs, the flow regime more likely becomes. This study determined all three flow regimes in a fish ladder. When the corresponding range of Qt* is less than 0.6, the flow regime can dip at values of L/B = 1.83. If the corresponding range of Qt* is greater than 0.5, transitional flow may occur at L/B = 1.22. On the other hand, when Qt* is greater than 1, streamwise flow can occur at values of L/B = 0.61. These observations agree well with the results of Baki et al. [21], Ead et al. [6] and Dizabadi et al. [22].
Fig. 11
For plunging flows, another dimensionless discharge (Q+) versus h/w given by Ead et al. [6] was used for further evaluation:
�+=��ℎ�ℎ=23�d�
(13)
where h is the water depth above the weir, and Cd is the discharge coefficient. Figure 12a compares the numerical and experimental results of Ead et al. [6]. In this figure, Rehbock’s empirical equation is used to estimate the discharge coefficient of Ead et al. [6].
�d=0.57+0.075ℎ�
(14)
Fig. 12
The numerical results for the conventional weir (Model A) and the rectangular labyrinth weir (Model B) of this study agree well with the laboratory results of Ead et al. [6]. When comparing models A and B, it is also found that a rectangular labyrinth weir has larger Q + values than the conventional weir as the length of the weir crest increases for a given channel width and fixed headwater elevation. In Fig. 12b, Models A and B’s flow depth plot shows the plunging flow regime. The power trend lines drawn through the data are the best-fit lines. The data shown in Fig. 12b are for different bed slopes and weir geometries. For the conventional weir and the rectangular labyrinth weir at submerged flow, Q can be assumed to be proportional to 1.56 and 1.47h, respectively. In the results of Ead et al. [6], Q is proportional to 1.5h. If we assume that the flow through the orifice is Qo and the total outflow is Q, the change in the ratio of Qo/Q to total outflow for models A and B can be shown in Fig. 13. For both models, the flow through the orifice decreases as the total flow increases. A logarithmic trend line was also found between the total outflow and the dimensionless ratio Qo/Q.
Fig. 13
3.3 Depth-Averaged Velocity Distributions
To ensure that the target fish species can pass the fish ladder with maximum efficiency, the average velocity in the fish ladder should be low enough [4]. Therefore, the average velocity in depth should be as much as possible below the critical swimming velocities of the target fishes at a constant flow depth in the pool [20]. The contour plot of depth-averaged velocity was used instead of another direction, such as longitudinal velocity because fish are more sensitive to depth-averaged flow velocity than to its direction under different hydraulic conditions. Figure 14 shows the distribution of depth-averaged velocity in the pool for Models A and B in two cases with and without orifice plates. Model A’s velocity within the pool differs slightly in the spanwise direction. However, no significant variation in velocity was observed. The flow is gradually directed to the sides as it passes through the rectangular labyrinth weir. This increases the velocity at the sides of the channel. Therefore, the high-velocity zone is located at the sides. The low velocity is in the downstream apex of the weir. This area may be suitable for swimming target fish. The presence of an opening in the weir increases the flow velocity at the opening and in the pool’s center, especially in Model A. The flow velocity increase caused by the models’ opening varied from 7.7 to 12.48%. Figure 15 illustrates the effect of the inverted slope on the averaged depth velocity distribution in the pool at low and high discharge. At constant discharge, flow velocity increases with increasing bed slope. In general, high flow velocity was found in the weir toe sidewall and the weir and channel sidewalls.
Fig. 14Fig. 15
On the other hand, for a constant bed slope, the high-velocity area of the pool increases due to the increase in runoff. For both bed slopes and different discharges, the most appropriate path for fish to travel from upstream to downstream is through the middle of the cross section and along the top of the rectangular labyrinth weirs. The maximum dominant velocities for Model B at S0 = 5% were 0.83 and 1.01 m/s; at S0 = 10%, they were 1.12 and 1.61 m/s at low and high flows, respectively. The low mean velocities for the same distance and S0 = 5 and 10% were 0.17 and 0.26 m/s, respectively.
Figure 16 shows the contour of the averaged depth velocity for various distances from the weir at low and high discharge. The contour plot shows a large variation in velocity within short distances from the weir. At L/B = 0.61, velocities are low upstream and downstream of the top of the weir. The high velocities occur in the side walls of the weir and the channel. At L/B = 1.22, the low-velocity zone displaces the higher velocity in most of the pool. Higher velocities were found only on the sides of the channel. As the discharge increases, the velocity zone in the pool becomes wider. At L/B = 1.83, there is an area of higher velocities only upstream of the crest and on the sides of the weir. At high discharge, the prevailing maximum velocities for L/B = 0.61, 1.22, and 1.83 were 1.46, 1.65, and 1.84 m/s, respectively. As the distance between weirs increases, the range of maximum velocity increases.
Fig. 16
On the other hand, the low mean velocity for these distances was 0.27, 0.44, and 0.72 m/s, respectively. Thus, the low-velocity zone decreases with increasing distance between weirs. Figure 17 shows the pattern distribution of streamlines along with the velocity contour at various distances from the weir for Q = 0.05 m3/s. A stream-like flow is generally formed in the pool at a small distance between weirs (L/B = 0.61). The rotation cell under the jet forms clockwise between the two weirs. At the distances between the spillways (L/B = 1.22), the transition regime of the flow is formed. The transition regime occurs when or shortly after the weir is flooded. The rotation cell under the jet is clockwise smaller than the flow regime and larger than the submergence regime. At a distance L/B = 1.83, a plunging flow is formed so that the plunging jet dips into the pool and extends downstream to the center of the pool. The clockwise rotation of the cell is bounded by the dipping jet of the weir and is located between the bottom and the side walls of the weir and the channel.
Fig. 17
Figure 18 shows the average depth velocity bar graph for each weir at different bed slopes and with and without orifice plates. As the distance between weirs increases, all models’ average depth velocity increases. As the slope of the bottom increases and an orifice plate is present, the average depth velocity in the pool increases. In addition, the average pool depth velocity increases as the discharge increases. Among the models, Model A’s average depth velocity is higher than Model B’s. The variation in velocity ranged from 8.11 to 12.24% for the models without an orifice plate and from 10.26 to 16.87% for the models with an orifice plate.
Fig. 18
3.4 Turbulence Characteristics
The turbulent kinetic energy is one of the important parameters reflecting the turbulent properties of the flow field [23]. When the k value is high, more energy and a longer transit time are required to migrate the target species. The turbulent kinetic energy is defined as follows:
�=12(�x′2+�y′2+�z′2)
(15)
where ux, uy, and uz are fluctuating velocities in the x, y, and z directions, respectively. An illustration of the TKE and the effects of the geometric arrangement of the weir and the presence of an opening in the weir is shown in Fig. 19. For a given bed slope, in Model A, the highest TKE values are uniformly distributed in the weir’s upstream portion in the channel’s cross section. In contrast, for the rectangular labyrinth weir (Model B), the highest TKE values are concentrated on the sides of the pool between the crest of the weir and the channel wall. The highest TKE value in Models A and B is 0.224 and 0.278 J/kg, respectively, at the highest bottom slope (S0 = 10%). In the downstream portion of the conventional weir and within the crest of the weir and the walls of the rectangular labyrinth, there was a much lower TKE value that provided the best conditions for fish to recover in the pool between the weirs. The average of the lowest TKE for bottom slopes of 5 and 10% in Model A is 0.041 and 0.056 J/kg, and for Model B, is 0.047 and 0.064 J/kg. The presence of an opening in the weirs reduces the area of the highest TKE within the pool. It also increases the resting areas for fish (lower TKE). The highest TKE at the highest bottom slope in Models A and B with an orifice is 0.208 and 0.191 J/kg, respectively.
Fig. 19
Figure 20 shows the effect of slope on the longitudinal distribution of TKE in the pools. TKE values significantly increase for a given discharge with an increasing bottom slope. Thus, for a low bed slope (S0 = 5%), a large pool area has expanded with average values of 0.131 and 0.168 J/kg for low and high discharge, respectively. For a bed slope of S0 = 10%, the average TKE values are 0.176 and 0.234 J/kg. Furthermore, as the discharge increases, the area with high TKE values within the pool increases. Lower TKE values are observed at the apex of the labyrinth weir, at the corner of the wall downstream of the weir, and between the side walls of the weir and the channel wall for both bottom slopes. The effect of distance between weirs on TKE is shown in Fig. 21. Low TKE values were observed at low discharge and short distances between weirs. Low TKE values are located at the top of the rectangular labyrinth weir and the downstream corner of the weir wall. There is a maximum value of TKE at the large distances between weirs, L/B = 1.83, along the center line of the pool, where the dip jet meets the bottom of the bed. At high discharge, the maximum TKE value for the distance L/B = 0.61, 1.22, and 1.83 was 0.246, 0.322, and 0.417 J/kg, respectively. In addition, the maximum TKE range increases with the distance between weirs.
Fig. 20Fig. 21
For TKE size, the average value (TKEave) is plotted against q in Fig. 22. For all models, the TKE values increase with increasing q. For example, in models A and B with L/B = 0.61 and a slope of 10%, the TKE value increases by 41.66 and 86.95%, respectively, as q increases from 0.1 to 0.27 m2/s. The TKE values in Model B are higher than Model A for a given discharge, bed slope, and weir distance. The TKEave in Model B is higher compared to Model A, ranging from 31.46 to 57.94%. The presence of an orifice in the weir reduces the TKE values in both weirs. The intensity of the reduction is greater in Model B. For example, in Models A and B with L/B = 0.61 and q = 0.1 m2/s, an orifice reduces TKEave values by 60.35 and 19.04%, respectively. For each model, increasing the bed slope increases the TKEave values in the pool. For example, for Model B with q = 0.18 m2/s, increasing the bed slope from 5 to 10% increases the TKEave value by 14.34%. Increasing the distance between weirs increases the TKEave values in the pool. For example, in Model B with S0 = 10% and q = 0.3 m2/s, the TKEave in the pool increases by 34.22% if you increase the distance between weirs from L/B = 0.61 to L/B = 0.183.
Fig. 22
Cotel et al. [24] suggested that turbulence intensity (TI) is a suitable parameter for studying fish swimming performance. Figure 23 shows the plot of TI and the effects of the geometric arrangement of the weir and the presence of an orifice. In Model A, the highest TI values are found upstream of the weirs and are evenly distributed across the cross section of the channel. The TI values increase as you move upstream to downstream in the pool. For the rectangular labyrinth weir, the highest TI values were concentrated on the sides of the pool, between the top of the weir and the side wall of the channel, and along the top of the weir. Downstream of the conventional weir, within the apex of the weir, and at the corners of the walls of the rectangular labyrinth weir, the percentage of TI was low. At the highest discharge, the average range of TI in Models A and B was 24–45% and 15–62%, respectively. The diversity of TI is greater in the rectangular labyrinth weir than the conventional weir. Fish swimming performance is reduced due to higher turbulence intensity. However, fish species may prefer different disturbance intensities depending on their swimming abilities; for example, Salmo trutta prefers a disturbance intensity of 18–53% [25]. Kupferschmidt and Zhu [26] found a higher range of TI for fishways, such as natural rock weirs, of 40–60%. The presence of an orifice in the weir increases TI values within the pool, especially along the middle portion of the cross section of the fishway. With an orifice in the weir, the average range of TI in Models A and B was 28–59% and 22–73%, respectively.
Fig. 23
The effect of bed slope on TI variation is shown in Fig. 24. TI increases in different pool areas as the bed slope increases for a given discharge. For a low bed slope (S0 = 5%), a large pool area has increased from 38 to 63% and from 56 to 71% for low and high discharge, respectively. For a bed slope of S0 = 10%, the average values of TI are 45–67% and 61–73% for low and high discharge, respectively. Therefore, as runoff increases, the area with high TI values within the pool increases. A lower TI is observed for both bottom slopes in the corner of the wall, downstream of the crest walls, and between the side walls in the weir and channel. Figure 25 compares weir spacing with the distribution of TI values within the pool. The TI values are low at low flows and short distances between weirs. A maximum value of TI occurs at long spacing and where the plunging stream impinges on the bed and the area around the bed. TI ranges from 36 to 57%, 58–72%, and 47–76% for the highest flow in a wide pool area for L/B = 0.61, 1.22, and 1.83, respectively.
Fig. 24Fig. 25
The average value of turbulence intensity (TIave) is plotted against q in Fig. 26. The increase in TI values with the increase in q values is seen in all models. For example, the average values of TI for Models A and B at L/B = 0.61 and slope of 10% increased from 23.9 to 33.5% and from 42 to 51.8%, respectively, with the increase in q from 0.1 to 0.27 m2/s. For a given discharge, a given gradient, and a given spacing of weirs, the TIave is higher in Model B than Model A. The presence of an orifice in the weirs increases the TI values in both types. For example, in Models A and B with L/B = 0.61 and q = 0.1 m2/s, the presence of an orifice increases TIave from 23.9 to 37.1% and from 42 to 48.8%, respectively. For each model, TIave in the pool increases with increasing bed slope. For Model B with q = 0.18 m2/s, TIave increases from 37.5 to 45.8% when you increase the invert slope from 5 to 10%. Increasing the distance between weirs increases the TIave in the pool. In Model B with S0 = 10% and q = 0.3 m2/s, the TIave in the pool increases from 51.8 to 63.7% as the distance between weirs increases from L/B = 0.61 to L/B = 0.183.
Fig. 26
3.5 Energy Dissipation
To facilitate the passage of various target species through the pool of fishways, it is necessary to pay attention to the energy dissipation of the flow and to keep the flow velocity in the pool slow. The average volumetric energy dissipation (k) in the pool is calculated using the following basic formula:
�=����0��
(16)
where ρ is the water density, and H is the average water depth of the pool. The change in k versus Q for all models at two bottom slopes, S0 = 5%, and S0 = 10%, is shown in Fig. 27. Like the results of Yagci [8] and Kupferschmidt and Zhu [26], at a constant bottom slope, the energy dissipation in the pool increases with increasing discharge. The trend of change in k as a function of Q from the present study at a bottom gradient of S0 = 5% is also consistent with the results of Kupferschmidt and Zhu [26] for the fishway with rock weir. The only difference between the results is the geometry of the fishway and the combination of boulders instead of a solid wall. Comparison of the models shows that the conventional model has lower energy dissipation than the rectangular labyrinth for a given discharge. Also, increasing the distance between weirs decreases the volumetric energy dissipation for each model with the same bed slope. Increasing the slope of the bottom leads to an increase in volumetric energy dissipation, and an opening in the weir leads to a decrease in volumetric energy dissipation for both models. Therefore, as a guideline for volumetric energy dissipation, if the value within the pool is too high, the increased distance of the weir, the decreased slope of the bed, or the creation of an opening in the weir would decrease the volumetric dissipation rate.
Fig. 27
To evaluate the energy dissipation inside the pool, the general method of energy difference in two sections can use:
ε=�1−�2�1
(17)
where ε is the energy dissipation rate, and E1 and E2 are the specific energies in Sects. 1 and 2, respectively. The distance between Sects. 1 and 2 is the same. (L is the distance between two upstream and downstream weirs.) Figure 28 shows the changes in ε relative to q (flow per unit width). The rectangular labyrinth weir (Model B) has a higher energy dissipation rate than the conventional weir (Model A) at a constant bottom gradient. For example, at S0 = 5%, L/B = 0.61, and q = 0.08 m3/s.m, the energy dissipation rate in Model A (conventional weir) was 0.261. In Model B (rectangular labyrinth weir), however, it was 0.338 (22.75% increase). For each model, the energy dissipation rate within the pool increases as the slope of the bottom increases. For Model B with L/B = 1.83 and q = 0.178 m3/s.m, the energy dissipation rate at S0 = 5% and 10% is 0.305 and 0.358, respectively (14.8% increase). Figure 29 shows an orifice’s effect on the pools’ energy dissipation rate. With an orifice in the weir, both models’ energy dissipation rates decreased. Thus, the reduction in energy dissipation rate varied from 7.32 to 9.48% for Model A and from 8.46 to 10.57 for Model B.
Fig. 28Fig. 29
4 Discussion
This study consisted of entirely of numerical analysis. Although this study was limited to two weirs, the hydraulic performance and flow characteristics in a pooled fishway are highlighted by the rectangular labyrinth weir and its comparison with the conventional straight weir. The study compared the numerical simulations with laboratory experiments in terms of surface profiles, velocity vectors, and flow characteristics in a fish ladder pool. The results indicate agreement between the numerical and laboratory data, supporting the reliability of the numerical model in capturing the observed phenomena.
When the configuration of the weir changes to a rectangular labyrinth weir, the flow characteristics, the maximum and minimum area, and even the location of each hydraulic parameter change compared to a conventional weir. In the rectangular labyrinth weir, the flow is gradually directed to the sides as it passes the weir. This increases the velocity at the sides of the channel [21]. Therefore, the high-velocity area is located on the sides. In the downstream apex of the weir, the flow velocity is low, and this area may be suitable for swimming target fish. However, no significant change in velocity was observed at the conventional weir within the fish ladder. This resulted in an average increase in TKE of 32% and an average increase in TI of about 17% compared to conventional weirs.
In addition, there is a slight difference in the flow regime for both weir configurations. In addition, the rectangular labyrinth weir has a higher energy dissipation rate for a given discharge and constant bottom slope than the conventional weir. By reducing the distance between the weirs, this becomes even more intense. Finally, the presence of an orifice in both configurations of the weir increased the flow velocity at the orifice and in the middle of the pool, reducing the highest TKE value and increasing the values of TI within the pool of the fish ladder. This resulted in a reduction in volumetric energy dissipation for both weir configurations.
The results of this study will help the reader understand the direct effects of the governing geometric parameters on the hydraulic characteristics of a fishway with a pool and weir. However, due to the limited configurations of the study, further investigation is needed to evaluate the position of the weir’s crest on the flow direction and the difference in flow characteristics when combining boulders instead of a solid wall for this type of labyrinth weir [26]. In addition, hydraulic engineers and biologists must work together to design an effective fishway with rectangular labyrinth configurations. The migration habits of the target species should be considered when designing the most appropriate design [27]. Parametric studies and field observations are recommended to determine the perfect design criteria.
The current study focused on comparing a rectangular labyrinth weir with a conventional straight weir. Further research can explore other weir configurations, such as variations in crest position, different shapes of labyrinth weirs, or the use of boulders instead of solid walls. This would help understand the influence of different geometric parameters on hydraulic characteristics.
5 Conclusions
A new layout of the weir was evaluated, namely a rectangular labyrinth weir compared to a straight weir in a pool and weir system. The differences between the weirs were highlighted, particularly how variations in the geometry of the structures, such as the shape of the weir, the spacing of the weir, the presence of an opening at the weir, and the slope of the bottom, affect the hydraulics within the structures. The main findings of this study are as follows:
The calculated dimensionless discharge (Qt*) confirmed three different flow regimes: when the corresponding range of Qt* is smaller than 0.6, the regime of plunging flow occurs for values of L/B = 1.83. (L: distance of the weir; B: channel width). When the corresponding range of Qt* is greater than 0.5, transitional flow occurs at L/B = 1.22. On the other hand, if Qt* is greater than 1, the streaming flow is at values of L/B = 0.61.
For the conventional weir and the rectangular labyrinth weir with the plunging flow, it can be assumed that the discharge (Q) is proportional to 1.56 and 1.47h, respectively (h: water depth above the weir). This information is useful for estimating the discharge based on water depth in practical applications.
In the rectangular labyrinth weir, the high-velocity zone is located on the side walls between the top of the weir and the channel wall. A high-velocity variation within short distances of the weir. Low velocity occurs within the downstream apex of the weir. This area may be suitable for swimming target fish.
As the distance between weirs increased, the zone of maximum velocity increased. However, the zone of low speed decreased. The prevailing maximum velocity for a rectangular labyrinth weir at L/B = 0.61, 1.22, and 1.83 was 1.46, 1.65, and 1.84 m/s, respectively. The low mean velocities for these distances were 0.27, 0.44, and 0.72 m/s, respectively. This finding highlights the importance of weir spacing in determining the flow characteristics within the fishway.
The presence of an orifice in the weir increased the flow velocity at the orifice and in the middle of the pool, especially in a conventional weir. The increase ranged from 7.7 to 12.48%.
For a given bottom slope, in a conventional weir, the highest values of turbulent kinetic energy (TKE) are uniformly distributed in the upstream part of the weir in the cross section of the channel. In contrast, for the rectangular labyrinth weir, the highest TKE values were concentrated on the sides of the pool between the crest of the weir and the channel wall. The highest TKE value for the conventional and the rectangular labyrinth weir was 0.224 and 0.278 J/kg, respectively, at the highest bottom slope (S0 = 10%).
For a given discharge, bottom slope, and weir spacing, the average values of TI are higher for the rectangular labyrinth weir than for the conventional weir. At the highest discharge, the average range of turbulence intensity (TI) for the conventional and rectangular labyrinth weirs was between 24 and 45% and 15% and 62%, respectively. This reveals that the rectangular labyrinth weir may generate more turbulent flow conditions within the fishway.
For a given discharge and constant bottom slope, the rectangular labyrinth weir has a higher energy dissipation rate than the conventional weir (22.75 and 34.86%).
Increasing the distance between weirs decreased volumetric energy dissipation. However, increasing the gradient increased volumetric energy dissipation. The presence of an opening in the weir resulted in a decrease in volumetric energy dissipation for both model types.
Availability of data and materials
Data is contained within the article.
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Yupeng Ren abc, Huiguang Zhou cd, Houjie Wang ab, Xiao Wu ab, Guohui Xu cd, Qingsheng Meng cd
Abstract
해저 퇴적물 흐름은 퇴적물을 심해로 운반하는 주요 수단 중 하나이며, 종종 장거리를 이동하고 수십 또는 수백 킬로미터에 걸쳐 상당한 양의 퇴적물을 운반합니다. 그것의 강력한 파괴력은 종종 이동 과정에서 잠수함 유틸리티에 심각한 손상을 초래합니다.
퇴적물 흐름의 퇴적물 농도는 주변 해수와의 밀도차를 결정하며, 이 밀도 차이는 퇴적물 흐름의 흐름 능력을 결정하여 이송된 퇴적물의 최종 퇴적 위치에 영향을 미칩니다. 본 논문에서는 다양한 미사 및 점토 중량비(미사/점토 비율이라고 함)를 갖는 다양한 퇴적물 농도의 퇴적물 흐름을 수로 테스트를 통해 연구합니다.
우리의 테스트 결과는 특정 퇴적물 구성에 대해 퇴적물 흐름이 가장 빠르게 이동하는 임계 퇴적물 농도가 있음을 나타냅니다. 4가지 미사/점토 비율 각각에 대한 임계 퇴적물 농도와 이에 상응하는 최대 속도가 구해집니다. 결과는 점토 함량이 임계 퇴적물 농도와 선형적으로 음의 상관 관계가 있음을 나타냅니다.
퇴적물 농도가 증가함에 따라 퇴적물의 흐름 거동은 흐름 상태에서 붕괴된 상태로 변환되고 흐름 거동이 변화하는 두 탁한 현탁액의 유체 특성은 모두 Bingham 유체입니다.
또한 본 논문에서는 퇴적물 흐름 내 입자 배열을 분석하여 위에서 언급한 결과에 대한 미시적 설명도 제공합니다.
Submarine sediment flows is one of the main means for transporting sediment to the deep sea, often traveling long-distance and transporting significant volumes of sediment for tens or even hundreds of kilometers. Its strong destructive force often causes serious damage to submarine utilities on its course of movement. The sediment concentration of the sediment flow determines its density difference with the ambient seawater, and this density difference determines the flow ability of the sediment flow, and thus affects the final deposition locations of the transported sediment. In this paper, sediment flows of different sediment concentration with various silt and clay weight ratios (referred to as silt/clay ratio) are studied using flume tests. Our test results indicate that there is a critical sediment concentration at which sediment flows travel the fastest for a specific sediment composition. The critical sediment concentrations and their corresponding maximum velocities for each of the four silt/clay ratios are obtained. The results further indicate that the clay content is linearly negatively correlated with the critical sediment concentration. As the sediment concentration increases, the flow behaviors of sediment flows transform from the flow state to the collapsed state, and the fluid properties of the two turbid suspensions with changing flow behaviors are both Bingham fluids. Additionally, this paper also provides a microscopic explanation of the above-mentioned results by analyzing the arrangement of particles within the sediment flow.
Introduction
Submarine sediment flows are important carriers for sea floor sediment movement and may carry and transport significant volumes of sediment for tens or even hundreds of kilometers (Prior et al., 1987; Pirmez and Imran, 2003; Zhang et al., 2018). Earthquakes, storms, and floods may all trigger submarine sediment flow events (Hsu et al., 2008; Piper and Normark, 2009; Pope et al., 2017b; Gavey et al., 2017). Sediment flows have strong forces during the movement, which will cause great harm to submarine structures such as cables and pipelines (Pope et al., 2017a). It was first confirmed that the cable breaking event caused by the sediment flow occurred in 1929. The sediment flow triggered by the Grand Banks earthquake damaged 12 cables. According to the time sequence of the cable breaking, the maximum velocity of the sediment flow is as high as 28 m/s (Heezen and Ewing, 1952; Kuenen, 1952; Heezen et al., 1954). Subsequent research shows that the lowest turbidity velocity that can break the cable also needs to reach 19 m/s (Piper et al., 1988). Since then, there have been many damage events of submarine cables and oil and gas pipelines caused by sediment flows in the world (Hsu et al., 2008; Carter et al., 2012; Cattaneo et al., 2012; Carter et al., 2014). During its movement, the sediment flow will gradually deposit a large amount of sediment carried by it along the way, that is, the deposition process of the sediment flow. On the one hand, this process brings a large amount of terrestrial nutrients and other materials to the ocean, while on the other hand, it causes damage and burial to benthic organisms, thus forming the largest sedimentary accumulation on Earth – submarine fans, which are highly likely to become good reservoirs for oil and gas resources (Daly, 1936; Yuan et al., 2010; Wu et al., 2022). The study on sediment flows (such as, the study of flow velocity and the forces acting on seabed structures) can provide important references for the safe design of seabed structures, the protection of submarine ecosystems, and exploration of turbidity sediments related oil and gas deposits. Therefore, it is of great significance to study the movement of sediment flows.
The sediment flow, as a highly sediment-concentrated fluid flowing on the sea floor, has a dense bottom layer and a dilute turbulent cloud. Observations at the Monterey Canyon indicated that the sediment flow can maintain its movement over long distances if its bottom has a relatively high sediment concentration. This dense bottom layer can be very destructive along its movement path to any facilities on the sea floor (Paull et al., 2018; Heerema et al., 2020; Wang et al., 2020). The sediment flow mentioned in this research paper is the general term of sediment density flow.
The sediment flow, which occurs on the seafloor, has the potential to cause erosion along its path. In this process, the suspended sediment is replenished, allowing the sediment flow to maintain its continuous flow capacity (Zhao et al., 2018). The dynamic force of sediment flow movement stem from its own gravity and density difference with surrounding water. In cases that the gravity drive of the slope is absent (on a flat sea floor), the flow velocity and distance of sediment flows are essentially determined by the sediment composition and concentration of the sediment flows as previous studies have demonstrated. Ilstad et al. (2004) conducted underwater flow tests in a sloped tank and employed high speed video camera to perform particle tracking. The results indicated that the premixed sand-rich and clay-rich slurries demonstrated different flow velocity and flow behavior. Using mixed kaolinite(d50 = 6 μm) and silica flour(d50 = 9 μm) in three compositions with total volumetric concentration ranged 22% or 28%, Felix and Peakall (2006) carried out underwater flow tests in a 5° slope Perspex channel and found that the flow ability of sediment flows is different depending on sediment compositions and concentrations. Sumner et al. (2009) used annular flume experiments to investigate the depositional dynamics and deposits of waning sediment-laden flows, finding that decelerating fast flows with fixed sand content and variable mud content resulted in four different deposit types. Chowdhury and Testik (2011) used lock-exchange tank, and experimented the kaolin clay sediment flows in the concentration range of 25–350 g/L, and predicted the fluid mud sediment flows propagation characteristics, but this study focused on giving sediment flows propagate phase transition time parameters, and is limited to clay. Lv et al. (2017) found through experiments that the rheological properties and flow behavior of kaolin clay (d50 = 3.7 μm) sediment flows were correlated to clay concentrations. In the field monitoring conducted by Liu et al. (2023) at the Manila Trench in the South China Sea in 2021, significant differences in the velocity, movement distance, and flow morphology of turbidity currents were observed. These differences may be attributed to variations in the particle composition of the turbidity currents.
On low and gentle slopes, although sediment flow with sand as the main sediment composition moves faster, it is difficult to propagate over long distances because sand has greater settling velocity and subaqueous angle of repose. Whereas the sediment flows with silt and clay as main composition may maintain relatively stable currents. Although its movement speed is slow, it has the ability to propagate over long distances because of the low settling rate of the fine particles (Ilstad et al., 2004; Liu et al., 2023). In a field observation at the Gaoping submarine canyon, the sediments collected from the sediment flows exhibited grain size gradation and the sediment was mostly composed of silt and clay (Liu et al., 2012). At the largest deltas in the world, for instance, the Mississippi River Delta, the sediments are mainly composed of silt and clay, which generally distributed along the coast in a wide range and provided the sediment sources for further distribution. The sediment flows originated and transported sediment from the coast to the deep sea are therefore share the same sediment compositions as delta sediments. To study the sediment flows composed of silt and clay is of great importance.
The sediment concentration of the sediment flows determines the density difference between the sediment flows and the ambient water and plays a key role in its flow ability. For the sediment flow with sediment composed of silt and clay, low sediment concentration means low density and therefore leads to low flow ability; however, although high sediment concentration results in high density, since there is cohesion between fine particles, it changes fluid properties and leads to low flow ability as well. Therefore, there should be a critical sediment concentration with mixed composition of silt and clay, at which the sediment flow maintains its strongest flow capacity and have the highest movement speed. In other words, the two characteristics of particle diameter and concentration of the sediment flow determine its own motion ability, which, if occurs, may become the most destructive force to submarine structures.
The objectives of this work was to study how the sediment composition (measured in relative weight of silt and clay, and referred as silt/clay ratio) and sediment concentration affect flow ability and behavior of the sediment flows, and to quantify the critical sediment concentration at which the sediment flows reached the greatest flow velocity under the experiment setting. We used straight flume without slope and conducted a series of flume tests with varying sediment compositions (silt-rich or clay-rich) and concentrations (96 to 1212 g/L). Each sediment flow sample was tested and analyzed for rheological properties using a rheometer, in order to characterize the relationship between flow behavior and rheological properties. Combined with the particle diameter, density and viscosity characteristics of the sediment flows measured in the experiment, a numerical modeling study is conducted, which are mutually validated with the experimental results.
The sediment concentration determines the arrangements of the sediment particles in the turbid suspension, and the arrangement impacts the fluid properties of the turbid suspension. The microscopic mode of particle arrangement in the turbid suspension can be constructed to further analyze the relationship between the fluid properties of turbid suspension and the flow behaviors of the sediment flow, and then characterize the critical sediment concentration at which the sediment flow runs the fastest. A simplified microscopic model of particle arrangement in turbid suspension was constructed to analyze the microscopic arrangement characteristics of sediment particles in turbid suspension with the fastest velocity.
Section snippets
Equipment and materials
The sediment flows flow experiments were performed in a Perspex channel with smooth transparent walls. The layout and dimensions of the experimental set-up were shown in Fig. 1. The bottom of the channel was flat and straight, and a gate was arranged to separate the two tanks. In order to study the flow capacity of turbidity currents from the perspective of their own composition (particle size distribution and concentration), we used a straight channel instead of an inclined one, to avoid any
Relationship between sediment flow flow velocity and sediment concentration
After the sediment flow is generated, its movement in the first half (50 cm) of the channel is relatively stable, and there is obvious shock diffusion in the second half. The reason is that the excitation wave (similar to the surge) will be formed during the sediment flow movement, and its speed is much faster than the speed of the sediment flow head. When the excitation wave reaches the tail of the channel, it will be reflected, thus affecting the subsequent flow of the sediment flow.
Sediment flows motion simulation based on FLOW-3D
As a relatively mature 3D fluid simulation software, FLOW-3D can accurately predict the free surface flow, and has been used to simulate the movement process of sediment flows for many times (Heimsund, 2007). The model adopted in this paper is RNG turbulence model, which can better deal with the flow with high strain rate and is suitable for the simulation of sediment flows with variable shape during movement. The governing equations of the numerical model involved include continuity equation,
Conclusions
In this study, we conducted a series of sediment flow flume tests with mixed silt and clay sediment samples in four silt/clay ratios on a flat slope. Rheological measurements were carried out on turbid suspension samples and microstructure analysis of the sediment particle arrangements was conducted, we concluded that:
(1)The flow velocity of the sediment flow is controlled by the sediment concentration and its own particle diameter composition, the flow velocity increased with the increase of the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China [Grant no. 42206055]; the National Natural Science Foundation of China [Grant no. 41976049]; and the National Natural Science Foundation of China [Grant no. 42272327].
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Sous la direction de : Marc Jolin, directeur de recherche Benoit Bissonnette, codirecteur de recherche
Modélisation de l’écoulement du béton frais
Abstract
현재의 기후 비상 사태와 기후 변화에 관한 다양한 과학적 보고서를 고려할 때 인간이 만든 오염을 대폭 줄이는 것은 필수적이며 심지어 중요합니다. 최신 IPCC(기후변화에 관한 정부 간 패널) 보고서(2022)는 2030년까지 배출량을 절반으로 줄여야 함을 나타내며, 지구 보존을 위해 즉각적인 조치를 취해야 한다고 강력히 강조합니다.
이러한 의미에서 콘크리트 생산 산업은 전체 인간 이산화탄소 배출량의 4~8%를 담당하고 있으므로 환경에 미치는 영향을 줄이기 위한 진화가 시급히 필요합니다.
본 연구의 주요 목적은 이미 사용 가능한 기술적 품질 관리 도구를 사용하여 생산을 최적화하고 혼합 시간을 단축하며 콘크리트 폐기물을 줄이기 위한 신뢰할 수 있고 활용 가능한 수치 모델을 개발함으로써 이러한 산업 전환에 참여하는 것입니다.
실제로, 혼합 트럭 내부의 신선한 콘크리트의 거동과 흐름 프로파일을 더 잘 이해할 수 있는 수치 시뮬레이션을 개발하면 혼합 시간과 비용을 더욱 최적화할 수 있으므로 매우 유망합니다. 이러한 복잡한 수치 도구를 활용할 수 있으려면 수치 시뮬레이션을 검증, 특성화 및 보정하기 위해 기본 신 콘크리트 흐름 모델의 구현이 필수적입니다.
이 논문에서는 세 가지 단순 유동 모델의 개발이 논의되고 얻은 결과는 신선한 콘크리트 유동의 수치적 거동을 검증하는 데 사용됩니다. 이러한 각 모델은 강점과 약점을 갖고 있으며, 신선한 콘크리트의 유변학과 유동 거동을 훨씬 더 잘 이해할 수 있는 수치 작업 환경을 만드는 데 기여합니다.
따라서 이 연구 프로젝트는 새로운 콘크리트 생산의 완전한 모델링을 위한 진정한 관문입니다.
In view of the current climate emergency and the various scientific reports on climate change, it is essential and even vital to drastically reduce man-made pollution. The latest IPCC (Intergovernmental Panel on Climate Change) report (2022) indicates that emissions must be halved by 2030 and strongly emphasizes the need to act immediately to preserve the planet. In this sense, the concrete production industry is responsible for 4-8% of total human carbon dioxide emissions and therefore urgently needs to evolve to reduce its environmental impact. The main objective of this study is to participate in this industrial transition by developing a reliable and exploitable numerical model to optimize the production, reduce mixing time and also reduce concrete waste by using technological quality control tools already available. Indeed, developing a numerical simulation allowing to better understand the behavior and flow profiles of fresh concrete inside a mixing-truck is extremely promising as it allows for further optimization of mixing times and costs. In order to be able to exploit such a complex numerical tool, the implementation of elementary fresh concrete flow models is essential to validate, characterize and calibrate the numerical simulations. In this thesis, the development of three simple flow models is discussed and the results obtained are used to validate the numerical behavior of fresh concrete flow. Each of these models has strengths and weaknesses and contributes to the creation of a numerical working environment that provides a much better understanding of the rheology and flow behavior of fresh concrete. This research project is therefore a real gateway to a full modelling of fresh concrete production.
Figure 2-15: Système expérimental du plan inclinéFigure 2-19: Essai d’affaissement au cône d’Abrams
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Field observations provide valuable data regarding nearshore tsunami impact, yet only in inundation areas where tsunami waves have already flooded. Therefore, tsunami modeling is essential to understand tsunami behavior and prepare for tsunami inundation. It is necessary that all numerical models used in tsunami emergency planning be subject to benchmark tests for validation and verification. This study focuses on two numerical codes, NAMI DANCE and FLOW-3D®, for validation and performance comparison. NAMI DANCE is an in-house tsunami numerical model developed by the Ocean Engineering Research Center of Middle East Technical University, Turkey and Laboratory of Special Research Bureau for Automation of Marine Research, Russia. FLOW-3D® is a general purpose computational fluid dynamics software, which was developed by scientists who pioneered in the design of the Volume-of-Fluid technique. The codes are validated and their performances are compared via analytical, experimental and field benchmark problems, which are documented in the ‘‘Proceedings and Results of the 2011 National Tsunami Hazard Mitigation Program (NTHMP) Model Benchmarking Workshop’’ and the ‘‘Proceedings and Results of the NTHMP 2015 Tsunami Current Modeling Workshop”. The variations between the numerical solutions of these two models are evaluated through statistical error analysis.
현장 관찰은 연안 쓰나미 영향에 관한 귀중한 데이터를 제공하지만 쓰나미 파도가 이미 범람한 침수 지역에서만 가능합니다. 따라서 쓰나미 모델링은 쓰나미 행동을 이해하고 쓰나미 범람에 대비하는 데 필수적입니다.
쓰나미 비상 계획에 사용되는 모든 수치 모델은 검증 및 검증을 위한 벤치마크 테스트를 받아야 합니다. 이 연구는 검증 및 성능 비교를 위해 NAMI DANCE 및 FLOW-3D®의 두 가지 숫자 코드에 중점을 둡니다.
NAMI DANCE는 터키 중동 기술 대학의 해양 공학 연구 센터와 러시아 해양 연구 자동화를 위한 특별 조사국 연구소에서 개발한 사내 쓰나미 수치 모델입니다. FLOW-3D®는 Volume-of-Fluid 기술의 설계를 개척한 과학자들이 개발한 범용 전산 유체 역학 소프트웨어입니다.
코드의 유효성이 검증되고 분석, 실험 및 현장 벤치마크 문제를 통해 코드의 성능이 비교되며, 이는 ‘2011년 NTHMP(National Tsunami Hazard Mitigation Program) 모델 벤치마킹 워크숍의 절차 및 결과’와 ”절차 및 NTHMP 2015 쓰나미 현재 모델링 워크숍 결과”. 이 두 모델의 수치 해 사이의 변동은 통계적 오류 분석을 통해 평가됩니다.
The distribution of the computed maximum current speed during the entire duration of the NAMI DANCE and FLOW-3D simulations. The
resolution of computational domain is 10 m
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The authors wish to thank Dr. Andrey Zaytsev due to his undeniable contributions to the development of in-house numerical model, NAMI DANCE. The Turkish branch of Flow Science, Inc. is also acknowledged. Finally, the National Tsunami Hazard Mitigation Program (NTHMP), who provided most of the benchmark data, is appreciated. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author information
Author notes
Deniz Velioglu SogutPresent address: 1212 Computer Science, Department of Civil Engineering, Stony Brook University, Stony Brook, NY, 11794, USA
Authors and Affiliations
Middle East Technical University, 06800, Ankara, TurkeyDeniz Velioglu Sogut & Ahmet Cevdet Yalciner
Velioglu Sogut, D., Yalciner, A.C. Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems. Pure Appl. Geophys.176, 3115–3153 (2019). https://doi.org/10.1007/s00024-018-1907-9
Journal of Hydrodynamicsvolume 33, pages833–843 (2021)
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Abstract
이 백서는 Flow-3D를 적용하여 다양한 흐름 배출 및 식생 시나리오가 흐름 속도(세로, 가로 및 수직 속도 포함)에 미치는 영향을 조사합니다.
실험적 측정을 통한 검증 후 식생직경, 식생높이, 유량방류량에 대한 민감도 분석을 수행하였다. 종방향 속도의 경우 흐름 구조에 가장 큰 영향을 미치는 것은 배출보다는 식생 직경에서 비롯됩니다.
그러나 식생 높이는 수직 분포의 변곡점을 결정합니다. 식생지 내 두 지점, 즉 상류와 하류의 횡속도를 비교하면 수심에 따른 대칭적인 패턴을 확인할 수 있다. 식생 지역의 가로 및 세로 유체 순환 패턴을 포함하여 흐름 또는 식생 시나리오와 관계없이 수직 속도에 대해서도 동일한 패턴이 관찰됩니다.
또한 식생의 직경이 클수록 이러한 패턴이 더 분명해집니다. 상부 순환은 초목 캐노피 근처에서 발생합니다. 식생지역의 가로방향과 세로방향의 순환에 관한 이러한 발견은 침수식생을 통한 3차원 유동구조를 밝혀준다.
This paper applies the Flow-3D to investigate the impacts of different flow discharge and vegetation scenarios on the flow velocity (including the longitudinal, transverse and vertical velocities). After the verification by using experimental measurements, a sensitivity analysis is conducted for the vegetation diameter, the vegetation height and the flow discharge. For the longitudinal velocity, the greatest impact on the flow structure originates from the vegetation diameter, rather than the discharge. The vegetation height, however, determines the inflection point of the vertical distribution. Comparing the transverse velocities at two positions in the vegetated area, i.e., the upstream and the downstream, a symmetric pattern is identified along the water depth. The same pattern is also observed for the vertical velocity regardless of the flow or vegetation scenario, including both transverse and vertical fluid circulation patterns in the vegetated area. Moreover, the larger the vegetation diameter is, the more evident these patterns become. The upper circulation occurs near the vegetation canopy. These findings regarding the circulations along the transverse and vertical directions in the vegetated region shed light on the 3-D flow structure through the submerged vegetation.
Key words
Submerged rigid vegetation
longitudinal velocity
transverse velocity
vertical velocity
open channel
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Melquisedec Cortés Zambrano*, Helmer Edgardo Monroy González, Wilson Enrique Amaya Tequia Faculty of Civil Engineering, Santo Tomas Tunja University. Address Av. Universitaria No. 45-202. Tunja – Boyacá – Colombia
Abstract
홍수는 지반이동 및 이동의 원인 중 하나이며, 급속한 도시화 및 도시화로 인해 이전보다 빈번하게 발생할 수 있다. 도시 배수 시스템의 특성은 집수 요소가 결정적인 역할을 하는 범람의 발생 및 범위를 정의할 수 있습니다. 이 문서는 7가지 유형의 화격자 유입구의 수력 유입 효율 및 배출 계수에 대한 수치 조사를 제시합니다. FLOW-3D® 시뮬레이터는 Q = 24, 34.1, 44, 100, 200 및 300 L/s의 유속에서 풀 스케일로 격자를 테스트하는 데 사용되며 종방향 기울기가 1.0인 실험 프로토타입의 구성을 유지합니다. %, 1.5% 및 2.0% 및 고정 횡단 경사, 총 126개 모델. 그 결과를 바탕으로 종류별 및 종단경사 조건에 따른 수력유입구 효율곡선과 토출계수를 구성하였다. 결과는 다른 조사에서 제안된 경험적 공식으로 조정되어 프로토타입의 물리적 테스트 결과를 검증하는 역할을 합니다.
Floods are one of the causes of ground movement and displacement, and due to rapid urbanization and urban growth may occur more frequently than before. The characteristics of an urban drainage system can define the occurrence and extent of flooding, where catchment elements have a determining role. This document presents the numerical investigation of the hydraulic inlet efficiency and the discharge coefficient of seven types of grate inlets. The FLOW-3D® simulator is used to test the gratings at a full scale, under flow rates of Q = 24, 34.1, 44, 100, 200 and 300 L/s, preserving the configuration of the experimental prototype with longitudinal slopes of 1.0%, 1.5% and 2.0% and a fixed cross slope, for a total of 126 models. Based on the results, hydraulic inlet efficiency curves and discharge coefficients are constructed for each type and a longitudinal slope condition. The results are adjusted with empirical formulations proposed in other investigations, serving to verify the results of physical testing of prototypes.
Fig. 1. Physical model of the experimental campaign (source: Chaparro Andrade and Abaunza Tabares, 2021)Fig. 2. Design of the grate inlet types studied: (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, (f) R6, (g) R7 (source: based on geometries of Chaparro Andrade
and Abaunza Tabares, 2021)Fig. 4. Comparison between the results obtained during physical experimentation in prototype 7 and simulation results with FLOW-3D® (source:
made with FlowSight® and photographic record by Chaparro Andrade and Abaunza Tabares, 2021)Fig. 6. Example of the results of flow depth and velocity vectors in the xy plane, for a stable flow condition in a grate inlet type and free surface
configuration and flow regime, of some grating types (source: produced with FlowSight®)
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Assistant Professor, Faculty of Civil Engineering, K. N. Toosi Univ. of Technology, Tehran 1996715433, Iran; formerly, Postdoctoral Research Fellow, Dept. of Civil and Environmental Engineering, Amirkabir Univ. of Technology (Tehran Polytechnic), Tehran 1591634311, Iran. ORCID: https://orcid.org/0000-0002-5162-6332. Email: mohammad.tehrani@kntu.ac.ir
https://doi.org/10.1061/JHEND8.HYENG-12914
Received: May 15, 2021
Accepted: September 30, 2022
Published online: December 21, 2022Journal of Hydraulic Engineering
chute 여수로에서는 난류 경계층 가장자리가 충분히 길면 자유 표면에 접근하는 시작점의 하류에서 자체 통기가 발생합니다. 시작 지점의 하류에서 공기-물 혼합물을 포함하는 층이 팽창 효과와 함께 흐름을 통해 점진적으로 확장됩니다.
유동 벌킹은 측벽 건현 설계 측면에서 필수적입니다. 또한 고체 경계 근처에 충분한 양의 공기를 도입하면 캐비테이션 손상을 방지할 수 있습니다. 현재 연구에서, 매끄러운 chute 을 따라 유동 벌킹과 함께 깊이와 자유 표면 위치에 걸쳐 자체 폭기 및 공기 농도 프로파일을 예측하기 위해 2D 수치 모델이 개발되었습니다.
개발된 모델은 혼합물 연속성, 기단 및 공기-물 혼합물 운동량 보존의 일방향 포물선 방정식의 해를 다룹니다. 이러한 방정식은 행진 기법과 Prandtl의 혼합 길이 난류 모델을 활용하여 자유 표면에 대한 동적 방정식과 함께 해결됩니다.
프로토타입 측정 및 실험실 테스트를 통해 얻은 실험 데이터를 사용하여 수치 모델의 정확도를 평가했습니다. 관련 결과는 경계층 발달의 유도된 시작점, 자체 유입 흐름 내의 공기 농도 프로파일 및 그에 따른 흐름의 벌킹 측면에서 비교되었습니다.
실용적인 목적을 위한 수치 모델의 기능은 상당히 정확한 결과에 따라 의미가 있으며 추가 연구를 위한 새로운 지평을 밝힙니다.
In chute spillways, self-aeration occurs downstream of the inception point, where the turbulent boundary layer edge approaches the free surface, if they are long enough. Downstream of the inception point, a layer containing an air–water mixture extends gradually through the flow with the bulking effect. Flow bulking is essential in terms of sidewall freeboard design. In addition, the introduction of enough air quantity near the solid boundaries prevents cavitation damage. In the present work, a 2D numerical model was developed for the prediction of self-aeration and air concentration profiles across the depth and the free-surface location, together with flow bulking along the smooth chutes. The developed model deals with the solution of the one-way direction parabolic equations of mixture continuity, air mass, and air–water mixture momentum conservation. These equations are solved accompanied by the dynamic equation for the free surface, utilizing the marching technique and Prandtl’s mixing length turbulent model. The experimental data obtained by prototype measurements and laboratory tests were used to assess the accuracy of the numerical model. The relevant results were compared in terms of the induced inception point of the boundary layer development, air concentration profiles within self-entrained flows, and the consequent bulking of the flow. The capability of the numerical model for practical purposes is signified in accordance with the fairly accurate obtained results, shedding light on new horizons for further research.
선박의 동력 요구 사항을 설계할 때 고려해야 할 가장 중요한 요소는 선박 저항 또는 선박에 작용하는 항력입니다. 항력을 극복하는 데 필요한 동력이 추진 시스템의 ‘손실’에 기여하기 때문에 추진 시스템을 설계하는 동안 선박 저항을 추정하는 것이 중요합니다. 선박 저항을 계산하는 세 가지 주요 방법이 있습니다:
Holtrop-Mennen(HM) 방법과 같은 통계적 방법, 수치 분석 또는 CFD(전산 유체 역학) 시뮬레이션 및 모델 테스트, 즉 예인 탱크에서 축소된 모델 테스트. 설계 단계 초기에는 기본 선박 매개변수만 사용할 수 있을 때 HM 방법과 같은 통계 모델만 사용할 수 있습니다.
수치 해석/CFD 시뮬레이션 및 모델 테스트는 선박의 완전한 3D 설계가 완료된 경우에만 수행할 수 있습니다. 본 논문은 Flow-3D 소프트웨어 패키지를 사용하여 CFD 시뮬레이션을 사용하여 잔잔한 수상 선박 저항을 예측하는 것을 목표로 합니다.
롤온/롤오프 승객(RoPax) 페리에 대한 사례 연구를 조사했습니다. 선박 저항은 다양한 선박 속도에서 계산되었습니다. 메쉬는 모든 CFD 시뮬레이션의 결과에 영향을 미치기 때문에 메쉬 민감도를 확인하기 위해 여러 개의 메쉬가 사용되었습니다. 시뮬레이션의 결과를 HM 방법의 추정치와 비교했습니다.
시뮬레이션 결과는 낮은 선박 속도에 대한 HM 방법과 잘 일치했습니다. 더 높은 선속을 위한 HM 방법에 비해 결과의 차이가 상당히 컸다. 선박 저항 분석을 수행하는 Flow-3D의 기능이 시연되었습니다.
While designing the power requirements of a ship, the most important factor to be considered is the ship resistance, or the sea drag forces acting on the ship. It is important to have an estimate of the ship resistance while designing the propulsion system since the power required to overcome the sea drag forces contribute to ‘losses’ in the propulsion system. There are three main methods to calculate ship resistance: Statistical methods like the Holtrop-Mennen (HM) method, numerical analysis or CFD (Computational Fluid Dynamics) simulations, and model testing, i.e. scaled model tests in towing tanks. At the start of the design stage, when only basic ship parameters are available, only statistical models like the HM method can be used. Numerical analysis/ CFD simulations and model tests can be performed only when the complete 3D design of the ship is completed. The present paper aims at predicting the calm water ship resistance using CFD simulations, using the Flow-3D software package. A case study of a roll-on/roll-off passenger (RoPax) ferry was investigated. Ship resistance was calculated at various ship speeds. Since the mesh affects the results in any CFD simulation, multiple meshes were used to check the mesh sensitivity. The results from the simulations were compared with the estimate from the HM method. The results from simulations agreed well with the HM method for low ship speeds. The difference in the results was considerably high compared to the HM method for higher ship speeds. The capability of Flow-3D to perform ship resistance analysis was demonstrated.
Figure 1: Simplified ship geometryFigure 3: Wave pattern at sea surface at 20 knots (10.29 m/s) for mesh 1Figure 4: Ship Resistance (kN) vs Ship Speed (knots)
Publisher
International Society of Multiphysics
Citation
Deshpande SR, Sundsbø P, Das S. Ship resistance analysis using CFD simulations in Flow-3D. The International Journal of Multiphysics. 2020;14(3):227-236
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[1] K. Min and S. Kang, “Study on the form factor and full-scale ship resistance prediction method,” Journal of Marine Science and Technology, vol. 15, pp. 108-118, June 2010. [2] A. Molland, S. Turnock and D. Hudson, “Ship Resistance and Propulsion” Second Edition. In Ship Resistance and Propulsion: Practical Estimation of Ship Propulsive Power (pp. 12-69), August 2017, Cambridge University Press. [3] K. Niklas and H. Pruszko, “Full-scale CFD simulations for the determination of ship resistance as a rational, alternative method to towing tank experiments,” Ocean Engineering, vol. 190, October 2019. [4] A. Elkafas, M. Elgohary and A. Zeid, “Numerical study on the hydrodynamic drag force of a container ship model,” Alexandria Engineering Journal, vol. 58, no. 3, pp. 849-859, September 2019. [5] J. Holtrop and G. Mennen, “An approximate power prediction method,” International Shipbuilding Progress, vol. 29, no. 335, pp. 166-170, July 1982. [6] E. Bøckmann and S. Steen, “Model test and simulation of a ship with wavefoils,” Applied Ocean research, vol. 57, pp. 8-18, April 2016. [7] K. Atreyapurapu, B. Tallapragada and K. Voonna, “Simulation of a Free Surface Flow over a Container Vessel Using CFD,” International Journal of Engineering Trends and Technology (IJETT), vol. 18, no. 7, pp. 334-339, December 2014. [8] J. Petersen, D. Jacobsen and O. Winther, “Statistical modelling for ship propulsion efficiency,” Journal of Marine Science and Technology, vol. 17, pp. 30-39, December 2011. [9] H. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics: the finite volume method (second edition), Harlow, England: Pearson Education Ltd, 2007. [10]C. Hirth and B. Nichols, “Volume of fluid (VOF) method for the dynamics of free boundaries,” Journal of Computational Physics, vol. 39, no. 1, pp. 201-225, January 1981. [11] A. Nordli and H. Khawaja, “Comparison of Explicit Method of Solution for CFD Euler Problems using MATLAB® and FORTRAN 77,” International Journal of Multiphysics, vol. 13, no. 2, 2019. [12] FLOW-3D® Version 12.0 User’s Manual (2018). FLOW-3D [Computer software]. Santa Fe, NM: Flow Science, Inc. https://www.flow3d.com. [13] D. McCluskey and A. Holdø, “Optimizing the hydrocyclone for ballast water treatment using computational fluid dynamics,” International Journal of Multiphysics, vol. 3, no. 3, 2009. [14]M. Breuer, D. Lakehal and W. Rodi, “Flow around a Surface Mounted Cubical Obstacle: Comparison of Les and Rans-Results,” Computation of Three-Dimensional Complex Flows. Notes on Numerical Fluid Mechanics, vol. 49, p. 1996. [15] G. Wei, “A Fixed-Mesh Method for General Moving Objects in Fluid Flow”, Modern Physics Letters B, vol. 19, no. 28, pp. 1719-1722, 2005. [16]J. Michell, “The wave-resistance of a ship,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vols. 45, 1898, no. 272, pp. 106-123, May 2009.
Malte Stief∗, Jens Gerstmann∗∗, and Michael E. Dreyer∗∗∗ ZARM, Center of Applied Space Technology and Microgravity, University of Bremen, Am Fallturm, D-28359 Bremen Experiments to observe the surface oscillation of cryogenic liquids have been performed with liquid nitrogen inside a 50 mm diameter right circular cylinder. The surface oscillation is driven by the capillary force that becomes dominant after a sudden reduction of the gravity acceleration acting on the liquid. The experiments show differences from the speculated behavior and enables one to observe new features.
Introduction and motivation
최근 몇 년 동안 Bremen의 낙하탑에서 중력의 단계적 감소 시 방향 재지향 거동과 표면 진동을 조사하기 위해 수많은 실험이 수행되었습니다[1]. 이 실험의 원리는 그림 1에 나와 있습니다.
그림 1의 왼쪽에 표시된 것처럼 오른쪽 원형 원통형 용기에 테스트 액체를 레벨 h0까지 채웁니다. 처음에 액체는 정지 상태이며 중앙에서 평평한 인터페이스를 형성합니다.
초기 중력 가속도 kzi ≈ 9.81 [m/s2]와 결과적으로 높은 BOND 수(Bo = ρkziR2/σ)로 인해 실린더의 대칭축에서. 낙하탑에서 실험 캡슐의 방출에 의해 확립된 μ-중력 환경 kz ≈ 0 [m/s2]로의 갑작스러운 전환과 함께 자유 표면은 진동 운동으로 새로운 평형 구성을 찾기 시작합니다(그림의 오른쪽) 1). 이러한 움직임은 그림 1의 중앙에 스케치되어 있습니다.
표면 진동의 구동력은 접착력과 결합된 표면 장력이며, 댐핑은 액체의 점도에 의해 제어됩니다. 위치가 zw인 벽에서 접촉선의 이동은 접촉각 γ에 의해 제어됩니다. 접촉각이 작은 액체용 γ ≈ 0◦
In recent years numerous experiments have been carried out to investigate the reorientation behavior and surface oscillations upon step reduction of gravity at the drop tower in Bremen [1]. The principals of these experiments are shown in figure 1. A right circular cylindrical container is filled up to the level h0 with the test liquid, as shown on the left of figure 1. Initially the liquid is quiescent and forms a flat interface at the center, in the symmetry axis of the cylinder, due to the initial gravity acceleration kzi ≈ 9.81 [m/s2] and the resulting high BOND number (Bo = ρkziR2/σ). With the sudden transition to the µ-gravity environment kz ≈ 0 [m/s2], which is established by the release of the experiment capsular in the drop tower, the free surface is initiated to search its new equilibrium configuration (right side of figure 1) with an oscillatory motion. These movements are sketched in the center of figure 1. The driving force for the surface oscillation is the surface tension in combination with the adhesion force where the damping is controlled by the viscosity of the liquid. The movement of the contact line at the wall, with its position zw, is governed by the contact angle γ. For liquids with small contact angle γ ≈ 0◦
Fig. 1 Oscillation of a free surface due to the step reduction of gravity acceleration from kzi ≈ 9.81 to kz ≈ 0Fig. 2 Experiment picture-series showing the oscillation of the free surface at different times for a 50 mm diameter cylinder.
References
[1] M. Michaelis, Kapillarinduzierte Schwingungen freier Fl¨ussigkeitsoberfl¨achen, Dissertation Universit¨at Bremen, Fortschritt-Berichte Nr. 454 (VDI Verlag, D¨usseldorf, 2003).
This paper presents the results of tests on the suitability of designed heads (impellers) for aluminum refining. The research was carried out on a physical model of the URO-200, followed by numerical simulations in the FLOW 3D program. Four design variants of impellers were used in the study. The degree of dispersion of the gas phase in the model liquid was used as a criterion for evaluating the performance of each solution using different process parameters, i.e., gas flow rate and impeller speed. Afterward, numerical simulations in Flow 3D software were conducted for the best solution. These simulations confirmed the results obtained with the water model and verified them.
Constantly increasing requirements concerning metallurgical purity in terms of hydrogen content and nonmetallic inclusions make casting manufacturers use effective refining techniques. The answer to this demand is the implementation of the aluminum refining technique making use of a rotor with an original design guaranteeing efficient refining [1,2,3,4]. The main task of the impeller (rotor) is to reduce the contamination of liquid metal (primary and recycled aluminum) with hydrogen and nonmetallic inclusions. An inert gas, mainly argon or a mixture of gases, is introduced through the rotor into the liquid metal to bring both hydrogen and nonmetallic inclusions to the metal surface through the flotation process. Appropriately and uniformly distributed gas bubbles in the liquid metal guarantee achieving the assumed level of contaminant removal economically. A very important factor in deciding about the obtained degassing effect is the optimal rotor design [5,6,7,8]. Thanks to the appropriate geometry of the rotor, gas bubbles introduced into the liquid metal are split into smaller ones, and the spinning movement of the rotor distributes them throughout the volume of the liquid metal bath. In this solution impurities in the liquid metal are removed both in the volume and from the upper surface of the metal. With a well-designed impeller, the costs of refining aluminum and its alloys can be lowered thanks to the reduced inert gas and energy consumption (optimal selection of rotor rotational speed). Shorter processing time and a high degree of dehydrogenation decrease the formation of dross on the metal surface (waste). A bigger produced dross leads to bigger process losses. Consequently, this means that the choice of rotor geometry has an indirect impact on the degree to which the generated waste is reduced [9,10].
Another equally important factor is the selection of process parameters such as gas flow rate and rotor speed [11,12]. A well-designed gas injection system for liquid metal meets two key requirements; it causes rapid mixing of the liquid metal to maintain a uniform temperature throughout the volume and during the entire process, to produce a chemically homogeneous metal composition. This solution ensures effective degassing of the metal bath. Therefore, the shape of the rotor, the arrangement of the nozzles, and their number are significant design parameters that guarantee the optimum course of the refining process. It is equally important to complete the mixing of the metal bath in a relatively short time, as this considerably shortens the refining process and, consequently, reduces the process costs. Another important criterion conditioning the implementation of the developed rotor is the generation of fine diffused gas bubbles which are distributed throughout the metal volume, and whose residence time will be sufficient for the bubbles to collide and adsorb the contaminants. The process of bubble formation by the spinning rotors differs from that in the nozzles or porous molders. In the case of a spinning rotor, the shear force generated by the rotor motion splits the bubbles into smaller ones. Here, the rotational speed, mixing force, surface tension, and fluid density have a key effect on the bubble size. The velocity of the bubbles, which depends mainly on their size and shape, determines their residence time in the reactor and is, therefore, very important for the refining process, especially since gas bubbles in liquid aluminum may remain steady only below a certain size [13,14,15].
The impeller designs presented in the article were developed to improve the efficiency of the process and reduce its costs. The impellers used so far have a complicated structure and are very pricey. The success of the conducted research will allow small companies to become independent of external supplies through the possibility of making simple and effective impellers on their own. The developed structures were tested on the water model. The results of this study can be considered as pilot.
Rotors were realized with the SolidWorks computer design technique and a 3D printer. The developed designs were tested on a water model. Afterward, the solution with the most advantageous refining parameters was selected and subjected to calculations with the Flow3D package. As a result, an impeller was designed for aluminum refining. Its principal lies in an even distribution of gas bubbles in the entire volume of liquid metal, with the largest possible participation of the bubble surface, without disturbing the metal surface. This procedure guarantees the removal of gaseous, as well as metallic and nonmetallic, impurities.
2.1. Rotor Designs
The developed impeller constructions, shown in Figure 1, Figure 2, Figure 3 and Figure 4, were printed on a 3D printer using the PLA (polylactide) material. The impeller design models differ in their shape and the number of holes through which the inert gas flows. Figure 1, Figure 2 and Figure 3 show the same impeller model but with a different number of gas outlets. The arrangement of four, eight, and 12 outlet holes was adopted in the developed design. A triangle-shaped structure equipped with three gas outlet holes is presented in Figure 4.
A schematic of the water model of reactor URO 200.
The URO 200 reactor can be classified as a cyclic reactor. The main element of the device is a rotor, which ends the impeller. The whole system is attached to a shaft via which the refining gas is supplied. Then, the shaft with the rotor is immersed in the liquid metal in the melting pot or the furnace chamber. In URO 200 reactors, the refining process lasts 600 s (10 min), the gas flow rate that can be obtained ranges from 5 to 20 dm3·min−1, and the speed at which the rotor can move is 0 to 400 rpm. The permissible quantity of liquid metal for barbotage refining is 300 kg or 700 kg [8,16,17]. The URO 200 has several design solutions which improve operation and can be adapted to the existing equipment in the foundry. These solutions include the following [8,16]:
URO-200XR—used for small crucible furnaces, the capacity of which does not exceed 250 kg, with no control system and no control of the refining process.
URO-200SA—used to service several crucible furnaces of capacity from 250 kg to 700 kg, fully automated and equipped with a mechanical rotor lift.
URO-200KA—used for refining processes in crucible furnaces and allows refining in a ladle. The process is fully automated, with a hydraulic rotor lift.
URO-200KX—a combination of the XR and KA models, designed for the ladle refining process. Additionally, refining in heated crucibles is possible. The unit is equipped with a manual hydraulic rotor lift.
URO-200PA—designed to cooperate with induction or crucible furnaces or intermediate chambers, the capacity of which does not exceed one ton. This unit is an integral part of the furnace. The rotor lift is equipped with a screw drive.
Studies making use of a physical model can be associated with the observation of the flow and circulation of gas bubbles. They require meeting several criteria regarding the similarity of the process and the object characteristics. The similarity conditions mainly include geometric, mechanical, chemical, thermal, and kinetic parameters. During simulation of aluminum refining with inert gas, it is necessary to maintain the geometric similarity between the model and the real object, as well as the similarity related to the flow of liquid metal and gas (hydrodynamic similarity). These quantities are characterized by the Reynolds, Weber, and Froude numbers. The Froude number is the most important parameter characterizing the process, its magnitude is the same for the physical model and the real object. Water was used as the medium in the physical modeling. The factors influencing the choice of water are its availability, relatively low cost, and kinematic viscosity at room temperature, which is very close to that of liquid aluminum.
The physical model studies focused on the flow of inert gas in the form of gas bubbles with varying degrees of dispersion, particularly with respect to some flow patterns such as flow in columns and geysers, as well as disturbance of the metal surface. The most important refining parameters are gas flow rate and rotor speed. The barbotage refining studies for the developed impeller (variants B4, B8, B12, and RT3) designs were conducted for the following process parameters:
Rotor speed: 200, 300, 400, and 500 rpm,
Ideal gas flow: 10, 20, and 30 dm3·min−1,
Temperature: 293 K (20 °C).
These studies were aimed at determining the most favorable variants of impellers, which were then verified using the numerical modeling methods in the Flow-3D program.
2.3. Numerical Simulations with Flow-3D Program
Testing different rotor impellers using a physical model allows for observing the phenomena taking place while refining. This is a very important step when testing new design solutions without using expensive industrial trials. Another solution is modeling by means of commercial simulation programs such as ANSYS Fluent or Flow-3D [18,19]. Unlike studies on a physical model, in a computer program, the parameters of the refining process and the object itself, including the impeller design, can be easily modified. The simulations were performed with the Flow-3D program version 12.03.02. A three-dimensional system with the same dimensions as in the physical modeling was used in the calculations. The isothermal flow of liquid–gas bubbles was analyzed. As in the physical model, three speeds were adopted in the numerical tests: 200, 300, and 500 rpm. During the initial phase of the simulations, the velocity field around the rotor generated an appropriate direction of motion for the newly produced bubbles. When the required speed was reached, the generation of randomly distributed bubbles around the rotor was started at a rate of 2000 per second. Table 1 lists the most important simulation parameters.
In the case of the CFD analysis, the numerical solutions require great care when generating the computational mesh. Therefore, computational mesh tests were performed prior to the CFD calculations. The effect of mesh density was evaluated by taking into account the velocity of water in the tested object on the measurement line A (height of 0.065 m from the bottom) in a characteristic cross-section passing through the object axis (see Figure 6). The mesh contained 3,207,600, 6,311,981, 7,889,512, 11,569,230, and 14,115,049 cells.
The velocity of the water depending on the size of the computational grid.
The quality of the generated computational meshes was checked using the criterion skewness angle QEAS [18]. This criterion is described by the following relationship:
QEAS=max{βmax−βeq180−βeq,βeq−βminβeq},
(1)
where βmax, βmin are the maximal and minimal angles (in degrees) between the edges of the cell, and βeq is the angle corresponding to an ideal cell, which for cubic cells is 90°.
Normalized in the interval [0;1], the value of QEAS should not exceed 0.75, which identifies the permissible skewness angle of the generated mesh. For the computed meshes, this value was equal to 0.55–0.65.
Moreover, when generating the computational grids in the studied facility, they were compacted in the areas of the highest gradients of the calculated values, where higher turbulence is to be expected (near the impeller). The obtained results of water velocity in the studied object at constant gas flow rate are shown in Figure 6.
The analysis of the obtained water velocity distributions (see Figure 6) along the line inside the object revealed that, with the density of the grid of nodal points, the velocity changed and its changes for the test cases of 7,889,512, 11,569,230, and 14,115,049 were insignificant. Therefore, it was assumed that a grid containing not less than 7,900,000 (7,889,512) cells would not affect the result of CFD calculations.
A single-block mesh of regular cells with a size of 0.0034 m was used in the numerical calculations. The total number of cells was approximately 7,900,000 (7,889,512). This grid resolution (see Figure 7) allowed the geometry of the system to be properly represented, maintaining acceptable computation time (about 3 days on a workstation with 2× CPU and 12 computing cores).
Structured equidistant mesh used in numerical calculations: (a) mesh with smoothed, surface cells (the so-called FAVOR method) used in Flow-3D; (b) visualization of the applied mesh resolution.
The calculations were conducted with an explicit scheme. The timestep was selected by the program automatically and controlled by stability and convergence. From the moment of the initial velocity field generation (start of particle generation), it was 0.0001 s.
When modeling the degassing process, three fluids are present in the system: water, gas supplied through the rotor head (impeller), and the surrounding air. Modeling such a multiphase flow is a numerically very complex issue. The necessity to overcome the liquid backpressure by the gas flowing out from the impeller leads to the formation of numerical instabilities in the volume of fluid (VOF)-based approach used by Flow-3D software. Therefore, a mixed description of the analyzed flow was used here. In this case, water was treated as a continuous medium, while, in the case of gas bubbles, the discrete phase model (DPM) model was applied. The way in which the air surrounding the system was taken into account is later described in detail.
The following additional assumptions were made in the modeling:
—The liquid phase was considered as an incompressible Newtonian fluid.
—The effect of chemical reactions during the refining process was neglected.
—The composition of each phase (gas and liquid) was considered homogeneous; therefore, the viscosity and surface tension were set as constants.
—Only full turbulence existed in the liquid, and the effect of molecular viscosity was neglected.
—The gas bubbles were shaped as perfect spheres.
—The mutual interaction between gas bubbles (particles) was neglected.
2.3.1. Modeling of Liquid Flow
The motion of the real fluid (continuous medium) is described by the Navier–Stokes Equation [20].
dudt=−1ρ∇p+ν∇2u+13ν∇(∇⋅ u)+F,
(2)
where du/dt is the time derivative, u is the velocity vector, t is the time, and F is the term accounting for external forces including gravity (unit components denoted by X, Y, Z).
In the simulations, the fluid flow was assumed to be incompressible, in which case the following equation is applicable:
∂u∂t+(u⋅∇)u=−1ρ∇p+ν∇2u+F.
(3)
Due to the large range of liquid velocities during flows, the turbulence formation process was included in the modeling. For this purpose, the k–ε model turbulence kinetic energy k and turbulence dissipation ε were the target parameters, as expressed by the following equations [21]:
where ρ is the gas density, σκ and σε are the Prandtl turbulence numbers, k and ε are constants of 1.0 and 1.3, and Gk and Gb are the kinetic energy of turbulence generated by the average velocity and buoyancy, respectively.
As mentioned earlier, there are two gas phases in the considered problem. In addition to the gas bubbles, which are treated here as particles, there is also air, which surrounds the system. The boundary of phase separation is in this case the free surface of the water. The shape of the free surface can change as a result of the forming velocity field in the liquid. Therefore, it is necessary to use an appropriate approach to free surface tracking. The most commonly used concept in liquid–gas flow modeling is the volume of fluid (VOF) method [22,23], and Flow-3D uses a modified version of this method called TrueVOF. It introduces the concept of the volume fraction of the liquid phase fl. This parameter can be used for classifying the cells of a discrete grid into areas filled with liquid phase (fl = 1), gaseous phase, or empty cells (fl = 0) and those through which the phase separation boundary (fl ∈ (0, 1)) passes (free surface). To determine the local variations of the liquid phase fraction, it is necessary to solve the following continuity equation:
dfldt=0.
(6)
Then, the fluid parameters in the region of coexistence of the two phases (the so-called interface) depend on the volume fraction of each phase.
ρ=flρl+(1−fl)ρg,
(7)
ν=flνl+(1−fl)νg,
(8)
where indices l and g refer to the liquid and gaseous phases, respectively.
The parameter of fluid velocity in cells containing both phases is also determined in the same way.
u=flul+(1−fl)ug.
(9)
Since the processes taking place in the surrounding air can be omitted, to speed up the calculations, a single-phase, free-surface model was used. This means that no calculations were performed in the gas cells (they were treated as empty cells). The liquid could fill them freely, and the air surrounding the system was considered by the atmospheric pressure exerted on the free surface. This approach is often used in modeling foundry and metallurgical processes [24].
2.3.2. Modeling of Gas Bubble Flow
As stated, a particle model was used to model bubble flow. Spherical particles (gas bubbles) of a given size were randomly generated in the area marked with green in Figure 7b. In the simulations, the gas bubbles were assumed to have diameters of 0.016 and 0.02 m corresponding to the gas flow rates of 10 and 30 dm3·min−1, respectively.
Experimental studies have shown that, as a result of turbulent fluid motion, some of the bubbles may burst, leading to the formation of smaller bubbles, although merging of bubbles into larger groupings may also occur. Therefore, to be able to observe the behavior of bubbles of different sizes (diameter), the calculations generated two additional particle types with diameters twice smaller and twice larger, respectively. The proportion of each species in the system was set to 33.33% (Table 2).
The velocity of the particle results from the generated velocity field (calculated from Equation (3) in the liquid ul around it and its velocity resulting from the buoyancy force ub. The effect of particle radius r on the terminal velocity associated with buoyancy force can be determined according to Stokes’ law.
ub=29 (ρg−ρl)μlgr2,
(10)
where g is the acceleration (9.81).
The DPM model was used for modeling the two-phase (water–air) flow. In this model, the fluid (water) is treated as a continuous phase and described by the Navier–Stokes equation, while gas bubbles are particles flowing in the model fluid (discrete phase). The trajectories of each bubble in the DPM system are calculated at each timestep taking into account the mass forces acting on it. Table 3 characterizes the DPM model used in our own research [18].
Table 3
Characteristic of the DPM model.
Method
Equations
Euler–Lagrange
Balance equation: dugdt=FD(u−ug)+g(ϱg−ϱ)ϱg+F. FD (u − up) denotes the drag forces per mass unit of a bubble, and the expression for the drag coefficient FD is of the form FD=18μCDReϱ⋅gd2g24. The relative Reynolds number has the form Re≡ρdg|ug−u|μ. On the other hand, the force resulting from the additional acceleration of the model fluid has the form F=12dρdtρg(u−ug), where ug is the gas bubble velocity, u is the liquid velocity, dg is the bubble diameter, and CD is the drag coefficient.
3.1. Calculations of Power and Mixing Time by the Flowing Gas Bubbles
One of the most important parameters of refining with a rotor is the mixing power induced by the spinning rotor and the outflowing gas bubbles (via impeller). The mixing power of liquid metal in a ladle of height (h) by gas injection can be determined from the following relation [15]:
pgVm=ρ⋅g⋅uB,
(11)
where pg is the mixing power, Vm is the volume of liquid metal in the reactor, ρ is the density of liquid aluminum, and uB is the average speed of bubbles, given below.
uB=n⋅R⋅TAc⋅Pm⋅t,
(12)
where n is the number of gas moles, R is the gas constant (8.314), Ac is the cross-sectional area of the reactor vessel, T is the temperature of liquid aluminum in the reactor, and Pm is the pressure at the middle tank level. The pressure at the middle level of the tank is calculated by a function of the mean logarithmic difference.
Pm=(Pa+ρ⋅g⋅h)−Paln(Pa+ρ⋅g⋅h)Pa,
(13)
where Pa is the atmospheric pressure, and h is the the height of metal in the reactor.
Themelis and Goyal [25] developed a model for calculating mixing power delivered by gas injection.
pg=2Q⋅R⋅T⋅ln(1+m⋅ρ⋅g⋅hP),
(14)
where Q is the gas flow, and m is the mass of liquid metal.
Zhang [26] proposed a model taking into account the temperature difference between gas and alloy (metal).
pg=QRTgVm[ln(1+ρ⋅g⋅hPa)+(1−TTg)],
(15)
where Tg is the gas temperature at the entry point.
Data for calculating the mixing power resulting from inert gas injection into liquid aluminum are given below in Table 4. The design parameters were adopted for the model, the parameters of which are shown in Figure 5.
Table 4
Data for calculating mixing power introduced by an inert gas.
Table 5 presents the results of mixing power calculations according to the models of Themelis and Goyal and of Zhang for inert gas flows of 10, 20, and 30 dm3·min−1. The obtained calculation results significantly differed from each other. The difference was an order of magnitude, which indicates that the model is highly inaccurate without considering the temperature of the injected gas. Moreover, the calculations apply to the case when the mixing was performed only by the flowing gas bubbles, without using a rotor, which is a great simplification of the phenomenon.
Table 5
Mixing power calculated from mathematical models.
Mathematical Model
Mixing Power (W·t−1) for a Given Inert Gas Flow (dm3·min−1)
The mixing time is defined as the time required to achieve 95% complete mixing of liquid metal in the ladle [27,28,29,30]. Table 6 groups together equations for the mixing time according to the models.
Figure 8 and Figure 9 show the mixing time as a function of gas flow rate for various heights of the liquid column in the ladle and mixing power values.
Mixing time as a function of mixing power (Szekly model).
3.2. Determining the Bubble Size
The mechanisms controlling bubble size and mass transfer in an alloy undergoing refining are complex. Strong mixing conditions in the reactor promote impurity mass transfer. In the case of a spinning rotor, the shear force generated by the rotor motion separates the bubbles into smaller bubbles. Rotational speed, mixing force, surface tension, and liquid density have a strong influence on the bubble size. To characterize the kinetic state of the refining process, parameters k and A were introduced. Parameters k, A, and uB can be calculated using the below equations [33].
k=2D⋅uBdB⋅π−−−−−−√,
(16)
A=6Q⋅hdB⋅uB,
(17)
uB=1.02g⋅dB,−−−−−√
(18)
where D is the diffusion coefficient, and dB is the bubble diameter.
After substituting appropriate values, we get
dB=3.03×104(πD)−2/5g−1/5h4/5Q0.344N−1.48.
(19)
According to the last equation, the size of the gas bubble decreases with the increasing rotational speed (see Figure 10).
Effect of rotational speed on the bubble diameter.
In a flow of given turbulence intensity, the diameter of the bubble does not exceed the maximum size dmax, which is inversely proportional to the rate of kinetic energy dissipation in a viscous flow ε. The size of the gas bubble diameter as a function of the mixing energy, also considering the Weber number and the mixing energy in the negative power, can be determined from the following equations [31,34]:
The first stage of experiments (using the URO-200 water model) included conducting experiments with impellers equipped with four, eight, and 12 gas outlets (variants B4, B8, B12). The tests were carried out for different process parameters. Selected results for these experiments are presented in Figure 11, Figure 12, Figure 13 and Figure 14.
Impeller variant B4—gas bubbles dispersion registered for a gas flow rate of 10 dm3·min−1 and rotor speed of (a) 200, (b) 300, (c) 400, and (d) 500 rpm.
Impeller variant B8—gas bubbles dispersion registered for a gas flow rate of 10 dm3·min−1 and rotor speed of (a) 200, (b) 300, (c) 400, and (d) 500 rpm.
Gas bubble dispersion registered for different processing parameters (impeller variant RT3).
The analysis of the refining variants presented in Figure 11, Figure 12, Figure 13 and Figure 14 reveals that the proposed impellers design model is not useful for the aluminum refining process. The number of gas outlet orifices, rotational speed, and flow did not affect the refining efficiency. In all the variants shown in the figures, very poor dispersion of gas bubbles was observed in the object. The gas bubble flow had a columnar character, and so-called dead zones, i.e., areas where no inert gas bubbles are present, were visible in the analyzed object. Such dead zones were located in the bottom and side zones of the ladle, while the flow of bubbles occurred near the turning rotor. Another negative phenomenon observed was a significant agitation of the water surface due to excessive (rotational) rotor speed and gas flow (see Figure 13, cases 20; 400, 30; 300, 30; 400, and 30; 500).
Research results for a ‘red triangle’ impeller equipped with three gas supply orifices (variant RT3) are presented in Figure 14.
In this impeller design, a uniform degree of bubble dispersion in the entire volume of the modeling fluid was achieved for most cases presented (see Figure 14). In all tested variants, single bubbles were observed in the area of the water surface in the vessel. For variants 20; 200, 30; 200, and 20; 300 shown in Figure 14, the bubble dispersion results were the worst as the so-called dead zones were identified in the area near the bottom and sidewalls of the vessel, which disqualifies these work parameters for further applications. Interestingly, areas where swirls and gas bubble chains formed were identified only for the inert gas flows of 20 and 30 dm3·min−1 and 200 rpm in the analyzed model. This means that the presented model had the best performance in terms of dispersion of gas bubbles in the model liquid. Its design with sharp edges also differed from previously analyzed models, which is beneficial for gas bubble dispersion, but may interfere with its suitability in industrial conditions due to possible premature wear.
3.4. Qualitative Comparison of Research Results (CFD and Physical Model)
The analysis (physical modeling) revealed that the best mixing efficiency results were obtained with the RT3 impeller variant. Therefore, numerical calculations were carried out for the impeller model with three outlet orifices (variant RT3). The CFD results are presented in Figure 15 and Figure 16.
Simulation results of the impeller RT3, for given flows and rotational speeds after a time of 1 s: simulation variants (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F.
Simulation results of the impeller RT3, for given flows and rotational speeds after a time of 5.4 s.: simulation variants (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F.
CFD results are presented for all analyzed variants (impeller RT3) at two selected calculation timesteps of 1 and 5.40 s. They show the velocity field of the medium (water) and the dispersion of gas bubbles.
Figure 15 shows the initial refining phase after 1 s of the process. In this case, the gas bubble formation and flow were observed in an area close to contact with the rotor. Figure 16 shows the phase when the dispersion and flow of gas bubbles were advanced in the reactor area of the URO-200 model.
The quantitative evaluation of the obtained results of physical and numerical model tests was based on the comparison of the degree of gas dispersion in the model liquid. The degree of gas bubble dispersion in the volume of the model liquid and the areas of strong turbulent zones formation were evaluated during the analysis of the results of visualization and numerical simulations. These two effects sufficiently characterize the required course of the process from the physical point of view. The known scheme of the below description was adopted as a basic criterion for the evaluation of the degree of dispersion of gas bubbles in the model liquid.
Minimal dispersion—single bubbles ascending in the region of their formation along the ladle axis; lack of mixing in the whole bath volume.
Accurate dispersion—single and well-mixed bubbles ascending toward the bath mirror in the region of the ladle axis; no dispersion near the walls and in the lower part of the ladle.
Uniform dispersion—most desirable; very good mixing of fine bubbles with model liquid.
Excessive dispersion—bubbles join together to form chains; large turbulence zones; uneven flow of gas.
The numerical simulation results give a good agreement with the experiments performed with the physical model. For all studied variants (used process parameters), the single bubbles were observed in the area of water surface in the vessel. For variants presented in Figure 13 (200 rpm, gas flow 20 and dm3·min−1) and relevant examples in numerical simulation Figure 16, the worst bubble dispersion results were obtained because the dead zones were identified in the area near the bottom and sidewalls of the vessel, which disqualifies these work parameters for further use. The areas where swirls and gas bubble chains formed were identified only for the inert gas flows of 20 and 30 dm3·min−1 and 200 rpm in the analyzed model (physical model). This means that the presented impeller model had the best performance in terms of dispersion of gas bubbles in the model liquid. The worst bubble dispersion results were obtained because the dead zones were identified in the area near the bottom and side walls of the vessel, which disqualifies these work parameters for further use.
Figure 17 presents exemplary results of model tests (CFD and physical model) with marked gas bubble dispersion zones. All variants of tests were analogously compared, and this comparison allowed validating the numerical model.
Compilations of model research results (CFD and physical): A—single gas bubbles formed on the surface of the modeling liquid, B—excessive formation of gas chains and swirls, C—uniform distribution of gas bubbles in the entire volume of the tank, and D—dead zones without gas bubbles, no dispersion. (a) Variant B; (b) variant F.
It should be mentioned here that, in numerical simulations, it is necessary to make certain assumptions and simplifications. The calculations assumed three particle size classes (Table 2), which represent the different gas bubbles that form due to different gas flow rates. The maximum number of particles/bubbles (Table 1) generated was assumed in advance and related to the computational capabilities of the computer. Too many particles can also make it difficult to visualize and analyze the results. The size of the particles, of course, affects their behavior during simulation, while, in the figures provided in the article, the bubbles are represented by spheres (visualization of the results) of the same size. Please note that, due to the adopted Lagrangian–Eulerian approach, the simulation did not take into account phenomena such as bubble collapse or fusion. However, the obtained results allow a comprehensive analysis of the behavior of gas bubbles in the system under consideration.
The comparative analysis of the visualization (quantitative) results obtained with the water model and CFD simulations (see Figure 17) generated a sufficient agreement from the point of view of the trends. A precise quantitative evaluation is difficult to perform because of the lack of a refraction compensating system in the water model. Furthermore, in numerical simulations, it is not possible to determine the geometry of the forming gas bubbles and their interaction with each other as opposed to the visualization in the water model. The use of both research methods is complementary. Thus, a direct comparison of images obtained by the two methods requires appropriate interpretation. However, such an assessment gives the possibility to qualitatively determine the types of the present gas bubble dispersion, thus ultimately validating the CFD results with the water model.
A summary of the visualization results for impellers RT3, i.e., analysis of the occurring gas bubble dispersion types, is presented in Table 8.
Table 8
Summary of visualization results (impeller RT3)—different types of gas bubble dispersion.
Tests carried out for impeller RT3 confirmed the high efficiency of gas bubble distribution in the volume of the tested object at a low inert gas flow rate of 10 dm3·min−1. The most optimal variant was variant B (300 rpm, 10 dm3·min−1). However, the other variants A and C (gas flow rate 10 dm3·min−1) seemed to be favorable for this type of impeller and are recommended for further testing. The above process parameters will be analyzed in detail in a quantitative analysis to be performed on the basis of the obtained efficiency curves of the degassing process (oxygen removal). This analysis will give an unambiguous answer as to which process parameters are the most optimal for this type of impeller; the results are planned for publication in the next article.
It should also be noted here that the high agreement between the results of numerical calculations and physical modelling prompts a conclusion that the proposed approach to the simulation of a degassing process which consists of a single-phase flow model with a free surface and a particle flow model is appropriate. The simulation results enable us to understand how the velocity field in the fluid is formed and to analyze the distribution of gas bubbles in the system. The simulations in Flow-3D software can, therefore, be useful for both the design of the impeller geometry and the selection of process parameters.
The results of experiments carried out on the physical model of the device for the simulation of barbotage refining of aluminum revealed that the worst results in terms of distribution and dispersion of gas bubbles in the studied object were obtained for the black impellers variants B4, B8, and B12 (multi-orifice impellers—four, eight, and 12 outlet holes, respectively).
In this case, the control of flow, speed, and number of gas exit orifices did not improve the process efficiency, and the developed design did not meet the criteria for industrial tests. In the case of the ‘red triangle’ impeller (variant RT3), uniform gas bubble dispersion was achieved throughout the volume of the modeling fluid for most of the tested variants. The worst bubble dispersion results due to the occurrence of the so-called dead zones in the area near the bottom and sidewalls of the vessel were obtained for the flow variants of 20 dm3·min−1 and 200 rpm and 30 dm3·min−1 and 200 rpm. For the analyzed model, areas where swirls and gas bubble chains were formed were found only for the inert gas flow of 20 and 30 dm3·min−1 and 200 rpm. The model impeller (variant RT3) had the best performance compared to the previously presented impellers in terms of dispersion of gas bubbles in the model liquid. Moreover, its design differed from previously presented models because of its sharp edges. This can be advantageous for gas bubble dispersion, but may negatively affect its suitability in industrial conditions due to premature wearing.
The CFD simulation results confirmed the results obtained from the experiments performed on the physical model. The numerical simulation of the operation of the ‘red triangle’ impeller model (using Flow-3D software) gave good agreement with the experiments performed on the physical model. This means that the presented model impeller, as compared to other (analyzed) designs, had the best performance in terms of gas bubble dispersion in the model liquid.
In further work, the developed numerical model is planned to be used for CFD simulations of the gas bubble distribution process taking into account physicochemical parameters of liquid aluminum based on industrial tests. Consequently, the obtained results may be implemented in production practice.
This paper was created with the financial support grants from the AGH-UST, Faculty of Foundry Engineering, Poland (16.16.170.654 and 11/990/BK_22/0083) for the Faculty of Materials Engineering, Silesian University of Technology, Poland.
Conceptualization, K.K. and D.K.; methodology, J.P. and T.M.; validation, M.S. and S.G.; formal analysis, D.K. and T.M.; investigation, J.P., K.K. and S.G.; resources, M.S., J.P. and K.K.; writing—original draft preparation, D.K. and T.M.; writing—review and editing, D.K. and T.M.; visualization, J.P., K.K. and S.G.; supervision, D.K.; funding acquisition, D.K. and T.M. All authors have read and agreed to the published version of the manuscript.
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![Figure 5 – Instantaneous pressures [Pa] and velocities [m/s] at model scale (bay center)](https://flow3d.co.kr/wp-content/uploads/s-at-model-scale-bay-center.png)
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