FLOW-3D POST 2023R2는 EXODUS II 형식을 기반으로 하는 완전히 새로운 결과 파일 형식을 도입하여 더 빠른 후처리를 가능하게 합니다. 이 새로운 파일 형식은 크고 복잡한 시뮬레이션의 후처리 작업에 소요되는 시간을 크게 줄이는 동시에(평균 최대 5배!) 다른 시각화 도구와의 연결성을 향상시킵니다.
FLOW-3D POST 2023R2 에서 사용자는 이제 선택한 데이터를 flsgrf , EXODUS II 또는 flsgrf 및 EXODUS II 파일 형식으로 쓸 수 있습니다 . 새로운 EXODUS II 파일 형식은 각 객체에 대해 유한 요소 메쉬를 활용하므로 사용자는 다른 호환 가능한 포스트 프로세서 및 FEA 코드를 사용 하여 FLOW-3D 결과를 열 수도 있습니다. 새로운 워크플로우를 통해 사용자는 크고 복잡한 사례를 신속하게 시각화하고 임의 슬라이싱, 볼륨 렌더링 및 통계를 사용하여 보조 정보를 추출할 수 있습니다.
새로운 결과 파일 형식은 솔버 엔진의 성능을 저하시키지 않으면서 flsgrf 에 비해 시각화 작업 흐름에서 놀라운 속도 향상을 자랑합니다. 이 흥미로운 새로운 개발은 결과 분석의 속도와 유연성이 향상되어 원활한 시뮬레이션 경험을 제공합니다.
FLOW-3D 2023R2는 2방정식(RANS) 난류 모델에 대한 동적 혼합 길이 계산을 크게 개선했습니다. 거의 층류 흐름 체계와 같은 특정 제한 사례에서는 이전 버전의 코드 계산 제한기가 때때로 과도하게 예측되어 사용자가 특정 혼합 길이를 수동으로 입력해야 할 수 있습니다.
새로운 동적 혼합 길이 계산은 이러한 상황에서 난류 길이와 시간 척도를 더 잘 설명합니다. 이제 사용자는 고정된(물리 기반) 혼합 길이를 설정하는 대신 더 넓은 범위의 흐름에 동적 모델을 적용할 수 있습니다.
사용자가 미리 정의된 유체 영역에서 정수압을 초기화해야 하는 경우가 많습니다. 이전에는 대규모의 복잡한 시뮬레이션에서 정수압 솔버의 수렴 속도가 느려지는 경우가 있었습니다. FLOW-3D 2023R2는 정수압 솔버의 성능을 크게 향상시켜 전처리 단계에서 최대 6배 빠르게 수렴할 수 있도록 해줍니다.
압축성 흐름 솔버 성능
FLOW-3D 2023R2는 최적화된 압력 솔버를 도입하여 압축성 흐름 문제에 대해 상당한 성능 향상을 제공합니다. 압축성 제트 흐름의 이 예에서 2023R2 솔버는 2023R1 버전보다 ~4배 빠릅니다.
FLOW-3D 2023R2 의 새로운 기능
FLOW-3D 소프트웨어 제품군의 모든 제품은 2023R2에서 IT 관련 개선 사항을 받았습니다. FLOW-3D 2023R2은 이제 Windows 11 및 RHEL 8을 지원합니다. Linux 설치 프로그램은 누락된 종속성을 보고하도록 개선되었으며 더 이상 루트 수준 권한이 필요하지 않으므로 설치가 더 쉽고 안전해집니다. 그리고 워크플로를 자동화한 분들을 위해 입력 파일 변환기에 명령줄 인터페이스를 추가하여 스크립트 환경에서도 워크플로가 업데이트된 입력 파일로 작동하는지 확인할 수 있습니다.
확장된 PQ 2 분석
제조에 사용되는 유압 시스템은 PQ 2 곡선을 사용하여 모델링할 수 있습니다. 장치의 세부 사항을 건너뛰고 흐름에 미치는 영향을 포함하기 위해 질량 운동량 소스 또는 속도 경계 조건을 사용하여 유압 시스템을 근사화하는 것이 편리한 단순화인 경우가 많습니다. 우리는 기존 PQ 2 분석 모델을 확장하여 이러한 유형의 기하학적 단순화를 허용하면서도 현실적인 결과를 제공했습니다. 이는 시뮬레이션 시간과 모델 복잡성의 감소를 의미합니다.
FLOW-3D 2022R2 의 새로운 기능
FLOW-3D 2022R2 제품군 출시로 Flow Science는 FLOW-3D 의 워크스테이션과 HPC 버전을 통합하여 단일 노드 CPU 구성에서 다중 노드에 이르기까지 모든 유형의 하드웨어 아키텍처를 활용할 수 있는 단일 솔버 엔진을 제공합니다. 노드 병렬 고성능 컴퓨팅 실행. 추가 개발에는 점탄성 흐름을 위한 새로운 로그 형태 텐서 방법, 지속적인 솔버 속도 성능 개선, 고급 냉각 채널 및 팬텀 구성요소 제어, 개선된 동반 공기 기능이 포함됩니다.
우리는 FLOW-3D 제품을 단일 통합 솔버로 마이그레이션하여 로컬 워크스테이션이나 고성능 컴퓨팅 하드웨어 환경에서 원활하게 실행했습니다.
많은 사용자가 노트북이나 로컬 워크스테이션에서 모델을 실행하지만, 고성능 컴퓨팅 클러스터에서도 더 큰 모델을 실행합니다. 2022R2 릴리스에서는 통합 솔버를 통해 사용자가 HPC 솔루션의 OpenMP/MPI 하이브리드 병렬화와 동일한 이점을 활용하여 워크스테이션과 노트북에서 실행할 수 있습니다.
솔버 성능 개선
멀티 소켓 워크스테이션
다중 소켓 워크스테이션은 이제 매우 일반적이며 대규모 시뮬레이션을 실행할 수 있습니다. 새로운 통합 솔버를 사용하면 이러한 유형의 하드웨어를 사용하는 사용자는 일반적으로 HPC 클러스터 구성에서만 사용할 수 있었던 OpenMP/MPI 하이브리드 병렬화를 활용하여 모델을 실행할 수 있어 성능이 향상되는 것을 확인할 수 있습니다.
낮은 수준의 루틴으로 향상된 벡터화 및 메모리 액세스
대부분의 테스트 사례에서 10~20% 정도의 성능 향상이 관찰되었으며 일부 사례에서는 20%를 초과하는 런타임 이점이 나타났습니다.
정제된 체적 대류 안정성 한계
시간 단계 안정성 제한은 모델 런타임의 주요 동인이며, 2022R2에서는 새로운 시간 단계 안정성 제한인 3D 대류 안정성 제한을 숫자 위젯에서 사용할 수 있습니다. 실행 중이고 대류가 제한된(cx, cy 또는 cz 제한) 모델의 경우 새 옵션은 일반적인 속도 향상을 30% 정도 보여줍니다.
압력 솔버 프리컨디셔너
경우에 따라 까다로운 흐름 구성의 경우 과도한 압력 솔버 반복으로 인해 실행 시간이 길어질 수 있습니다. 이러한 어려운 경우 2022R2에서는 모델이 너무 많이 반복되면 FLOW-3D가 자동으로 새로운 사전 조절기를 활성화하여 압력 수렴을 돕습니다. 테스트의 런타임이 1.9~335배 더 빨라졌습니다!
점탄성 유체에 대한 로그 형태 텐서 방법
점탄성 유체에 대한 새로운 솔버 옵션을 사용자가 사용할 수 있으며 특히 높은 Weissenberg 수에 효과적입니다.
활성 시뮬레이션 제어 확장
능동 시뮬레이션 제어 기능이 확장되어 연속 주조 및 적층 제조 응용 분야에 일반적으로 사용되는 팬텀 개체는 물론 주조 및 기타 여러 열 관리 응용 분야에 사용되는 냉각 채널에도 사용됩니다.
향상된 공기 동반 기능
디퓨저 및 이와 유사한 산업용 기포 흐름 응용 분야의 경우 이제 질량 공급원을 사용하여 물기둥에 공기를 유입할 수 있습니다. 또한, 동반 공기 및 용존 산소의 난류 확산에 대한 기본값이 업데이트되었으며 매우 낮은 공기 농도에 대한 모델 정확도가 향상되었습니다.
금속 적층 제조 중 고체 상 변형 예측: Inconel-738의 전자빔 분말층 융합에 대한 사례 연구
Nana Kwabena Adomako a, Nima Haghdadi a, James F.L. Dingle bc, Ernst Kozeschnik d, Xiaozhou Liao bc, Simon P. Ringer bc, Sophie Primig a
Metal additive manufacturing (AM) has now become the perhaps most desirable technique for producing complex shaped engineering parts. However, to truly take advantage of its capabilities, advanced control of AM microstructures and properties is required, and this is often enabled via modeling. The current work presents a computational modeling approach to studying the solid-state phase transformation kinetics and the microstructural evolution during AM. Our approach combines thermal and thermo-kinetic modelling. A semi-analytical heat transfer model is employed to simulate the thermal history throughout AM builds. Thermal profiles of individual layers are then used as input for the MatCalc thermo-kinetic software. The microstructural evolution (e.g., fractions, morphology, and composition of individual phases) for any region of interest throughout the build is predicted by MatCalc. The simulation is applied to an IN738 part produced by electron beam powder bed fusion to provide insights into how γ′ precipitates evolve during thermal cycling. Our simulations show qualitative agreement with our experimental results in predicting the size distribution of γ′ along the build height, its multimodal size character, as well as the volume fraction of MC carbides. Our findings indicate that our method is suitable for a range of AM processes and alloys, to predict and engineer their microstructures and properties.
Additive manufacturing (AM) is an advanced manufacturing method that enables engineering parts with intricate shapes to be fabricated with high efficiency and minimal materials waste. AM involves building up 3D components layer-by-layer from feedstocks such as powder . Various alloys, including steel, Ti, Al, and Ni-based superalloys, have been produced using different AM techniques. These techniques include directed energy deposition (DED), electron- and laser powder bed fusion (E-PBF and L-PBF), and have found applications in a variety of industries such as aerospace and power generation, , . Despite the growing interest, certain challenges limit broader applications of AM fabricated components in these industries and others. One of such limitations is obtaining a suitable and reproducible microstructure that offers the desired mechanical properties consistently. In fact, the AM as-built microstructure is highly complex and considerably distinctive from its conventionally processed counterparts owing to the complicated thermal cycles arising from the deposition of several layers upon each other , .
Several studies have reported that the solid-state phases and solidification microstructure of AM processed alloys such as CMSX-4, CoCr , , Ti-6Al-4V , , , IN738, 304L stainless steel, and IN718 ,  exhibit considerable variations along the build direction. For instance, references ,  have reported that there is a variation in the distribution of α and β phases along the build direction in Ti-alloys. Similarly, the microstructure of an L-PBF fabricated martensitic steel exhibits variations in the fraction of martensite . Furthermore, some of the present authors and others , , , , ,  have recently reviewed and reported that there is a difference in the morphology and fraction of nanoscale precipitates as a function of build height in Ni-based superalloys. These non-uniformities in the as-built microstructure result in an undesired heterogeneity in mechanical and other important properties such as corrosion and oxidation, , , . To obtain the desired microstructure and properties, additional processing treatments are utilized, but this incurs extra costs and may lead to precipitation of detrimental phases and grain coarsening. Therefore, a through-process understanding of the microstructure evolution under repeated heating and cooling is now needed to further advance 3D printed microstructure and property control.
It is now commonly understood that the microstructure evolution during printing is complex, and most AM studies concentrate on the microstructure and mechanical properties of the final build only. Post-printing studies of microstructure characteristics at room temperature miss crucial information on how they evolve. In-situ measurements and modelling approaches are required to better understand the complex microstructural evolution under repeated heating and cooling. Most in-situ measurements in AM focus on monitoring the microstructural changes, such as phase transformations and melt pool dynamics during fabrication using X-ray scattering and high-speed X-ray imaging , , , . For example, Zhao et al.  measured the rate of solidification and described the α/β phase transformation during L-PBF of Ti-6Al-4V in-situ. Also, Wahlmann et al.  recently used an L-PBF machine coupled with X-ray scattering to investigate the changes in CMSX-4 phase during successive melting processes. Although these techniques provide significant understanding of the basic principles of AM, they are not widely accessible. This is due to the great cost of the instrument, competitive application process, and complexities in terms of the experimental set-up, data collection, and analysis , .
Computational modeling techniques are promising and more widely accessible tools that enable advanced understanding, prediction, and engineering of microstructures and properties during AM. So far, the majority of computational studies have concentrated on physics based process models for metal AM, with the goal of predicting the temperature profile, heat transfer, powder dynamics, and defect formation (e.g., porosity) , . In recent times, there have been efforts in modeling of the AM microstructure evolution using approaches such as phase-field , Monte Carlo (MC) , and cellular automata (CA) , coupled with finite element simulations for temperature profiles. However, these techniques are often restricted to simulating the evolution of solidification microstructures (e.g., grain and dendrite structure) and defects (e.g., porosity). For example, Zinovieva et al.  predicted the grain structure of L-PBF Ti-6Al-4V using finite difference and cellular automata methods. However, studies on the computational modelling of the solid-state phase transformations, which largely determine the resulting properties, remain limited. This can be attributed to the multi-component and multi-phase nature of most engineering alloys in AM, along with the complex transformation kinetics during thermal cycling. This kind of research involves predictions of the thermal cycle in AM builds, and connecting it to essential thermodynamic and kinetic data as inputs for the model. Based on the information provided, the thermokinetic model predicts the history of solid-state phase microstructure evolution during deposition as output. For example, a multi-phase, multi-component mean-field model has been developed to simulate the intermetallic precipitation kinetics in IN718  and IN625  during AM. Also, Basoalto et al.  employed a computational framework to examine the contrasting distributions of process-induced microvoids and precipitates in two Ni-based superalloys, namely IN718 and CM247LC. Furthermore, McNamara et al.  established a computational model based on the Johnson-Mehl-Avrami model for non-isothermal conditions to predict solid-state phase transformation kinetics in L-PBF IN718 and DED Ti-6Al-4V. These models successfully predicted the size and volume fraction of individual phases and captured the repeated nucleation and dissolution of precipitates that occur during AM.
In the current study, we propose a modeling approach with appreciably short computational time to investigate the detailed microstructural evolution during metal AM. This may include obtaining more detailed information on the morphologies of phases, such as size distribution, phase fraction, dissolution and nucleation kinetics, as well as chemistry during thermal cycling and final cooling to room temperature. We utilize the combination of the MatCalc thermo-kinetic simulator and a semi-analytical heat conduction model. MatCalc is a software suite for simulation of phase transformations, microstructure evolution and certain mechanical properties in engineering alloys. It has successfully been employed to simulate solid-state phase transformations in Ni-based superalloys , , steels , and Al alloys during complex thermo-mechanical processes. MatCalc uses the classical nucleation theory as well as the so-called Svoboda-Fischer-Fratzl-Kozeschnik (SFFK) growth model as the basis for simulating precipitation kinetics . Although MatCalc was originally developed for conventional thermo-mechanical processes, we will show that it is also applicable for AM if the detailed time-temperature profile of the AM build is known. The semi-analytical heat transfer code developed by Stump and Plotkowski  is used to simulate these profile throughout the AM build.
1.1. Application to IN738
Inconel-738 (IN738) is a precipitation hardening Ni-based superalloy mainly employed in high-temperature components, e.g. in gas turbines and aero-engines owing to its exceptional mechanical properties at temperatures up to 980 °C, coupled with high resistance to oxidation and corrosion . Its superior high-temperature strength (∼1090 MPa tensile strength) is provided by the L12 ordered Ni3(Al,Ti) γ′ phase that precipitates in a face-centered cubic (FCC) γ matrix , . Despite offering great properties, IN738, like most superalloys with high γ′ fractions, is challenging to process owing to its propensity to hot cracking , . Further, machining of such alloys is challenging because of their high strength and work-hardening rates. It is therefore difficult to fabricate complex INC738 parts using traditional manufacturing techniques like casting, welding, and forging.
The emergence of AM has now made it possible to fabricate such parts from IN738 and other superalloys. Some of the current authors’ recent research successfully applied E-PBF to fabricate defect-free IN738 containing γ′ throughout the build , . The precipitated γ′ were heterogeneously distributed. In particular, Haghdadi et al.  studied the origin of the multimodal size distribution of γ′, while Lim et al.  investigated the gradient in γ′ character with build height and its correlation to mechanical properties. Based on these results, the present study aims to extend the understanding of the complex and site-specific microstructural evolution in E-PBF IN738 by using a computational modelling approach. New experimental evidence (e.g., micrographs not published previously) is presented here to support the computational results.
2. Materials and Methods
2.1. Materials preparation
IN738 Ni-based superalloy (59.61Ni-8.48Co-7.00Al-17.47Cr-3.96Ti-1.01Mo-0.81W-0.56Ta-0.49Nb-0.47C-0.09Zr-0.05B, at%) gas-atomized powder was used as feedstock. The powders, with average size of 60 ± 7 µm, were manufactured by Praxair and distributed by Astro Alloys Inc. An Arcam Q10 machine by GE Additive with an acceleration voltage of 60 kV was used to fabricate a 15 × 15 × 25 mm3 block (XYZ, Z: build direction) on a 316 stainless steel substrate. The block was 3D-printed using a ‘random’ spot melt pattern. The random spot melt pattern involves randomly selecting points in any given layer, with an equal chance of each point being melted. Each spot melt experienced a dwell time of 0.3 ms, and the layer thickness was 50 µm. Some of the current authors have previously characterized the microstructure of the very same and similar builds in more detail , . A preheat temperature of ∼1000 °C was set and kept during printing to reduce temperature gradients and, in turn, thermal stresses , , . Following printing, the build was separated from the substrate through electrical discharge machining. It should be noted that this sample was simultaneously printed with the one used in  during the same build process and on the same build plate, under identical conditions.
2.2. Microstructural characterization
The printed sample was longitudinally cut in the direction of the build using a Struers Accutom-50, ground, and then polished to 0.25 µm suspension via standard techniques. The polished x-z surface was electropolished and etched using Struers A2 solution (perchloric acid in ethanol). Specimens for image analysis were polished using a 0.06 µm colloidal silica. Microstructure analyses were carried out across the height of the build using optical microscopy (OM) and scanning electron microscopy (SEM) with focus on the microstructure evolution (γ′ precipitates) in individual layers. The position of each layer being analyzed was determined by multiplying the layer number by the layer thickness (50 µm). It should be noted that the position of the first layer starts where the thermal profile is tracked (in this case, 2 mm from the bottom). SEM images were acquired using a JEOL 7001 field emission microscope. The brightness and contrast settings, acceleration voltage of 15 kV, working distance of 10 mm, and other SEM imaging parameters were all held constant for analysis of the entire build. The ImageJ software was used for automated image analysis to determine the phase fraction and size of γ′ precipitates and carbides. A 2-pixel radius Gaussian blur, following a greyscale thresholding and watershed segmentation was used . Primary γ′ sizes (>50 nm), were measured using equivalent spherical diameters. The phase fractions were considered equal to the measured area fraction. Secondary γ′ particles (<50 nm) were not considered here. The γ′ size in the following refers to the diameter of a precipitate.
2.3. Hardness testing
A Struers DuraScan tester was utilized for Vickers hardness mapping on a polished x-z surface, from top to bottom under a maximum load of 100 mN and 10 s dwell time. 30 micro-indentations were performed per row. According to the ASTM standard , the indentations were sufficiently distant (∼500 µm) to assure that strain-hardened areas did not interfere with one another.
2.4. Computational simulation of E-PBF IN738 build
2.4.1. Thermal profile modeling
The thermal history was generated using the semi-analytical heat transfer code (also known as the 3DThesis code) developed by Stump and Plotkowski . This code is an open-source C++ program which provides a way to quickly simulate the conductive heat transfer found in welding and AM. The key use case for the code is the simulation of larger domains than is practicable with Computational Fluid Dynamics/Finite Element Analysis programs like FLOW-3D AM. Although simulating conductive heat transfer will not be an appropriate simplification for some investigations (for example the modelling of keyholding or pore formation), the 3DThesis code does provide fast estimates of temperature, thermal gradient, and solidification rate which can be useful for elucidating microstructure formation across entire layers of an AM build. The mathematics involved in the code is as follows:
In transient thermal conduction during welding and AM, with uniform and constant thermophysical properties and without considering fluid convection and latent heat effects, energy conservation can be expressed as:(1)��∂�∂�=�∇2�+�̇where � is density, � specific heat, � temperature, � time, � thermal conductivity, and �̇ a volumetric heat source. By assuming a semi-infinite domain, Eq. 1 can be analytically solved. The solution for temperature at a given time (t) using a volumetric Gaussian heat source is presented as:(2)��,�,�,�−�0=33�����32∫0�1������exp−3�′�′2��+�′�′2��+�′�′2����′(3)and��=12��−�′+��2for�=�,�,�(4)and�′�′=�−���′Where � is the vector �,�,� and �� is the location of the heat source.
The numerical integration scheme used is an adaptive Gaussian quadrature method based on the following nondimensionalization:(5)�=��xy2�,�′=��xy2�′,�=��xy,�=��xy,�=��xy,�=���xy
A more detailed explanation of the mathematics can be found in reference .
The main source of the thermal cycling present within a powder-bed fusion process is the fusion of subsequent layers. Therefore, regions near the top of a build are expected to undergo fewer thermal cycles than those closer to the bottom. For this purpose, data from the single scan’s thermal influence on multiple layers was spliced to represent the thermal cycles experienced at a single location caused by multiple subsequent layers being fused.
The cross-sectional area simulated by this model was kept constant at 1 × 1 mm2, and the depth was dependent on the build location modelled with MatCalc. For a build location 2 mm from the bottom, the maximum number of layers to simulate is 460. Fig. 1a shows a stitched overview OM image of the entire build indicating the region where this thermal cycle is simulated and tracked. To increase similarity with the conditions of the physical build, each thermal history was constructed from the results of two simulations generated with different versions of a random scan path. The parameters used for these thermal simulations can be found in Table 1. It should be noted that the main purpose of the thermal profile modelling was to demonstrate how the conditions at different locations of the build change relative to each other. Accurately predicting the absolute temperature during the build would require validation via a temperature sensor measurement during the build process which is beyond the scope of the study. Nonetheless, to establish the viability of the heat source as a suitable approximation for this study, an additional sensitivity analysis was conducted. This analysis focused on the influence of energy input on γ′ precipitation behavior, the central aim of this paper. This was achieved by employing varying beam absorption energies (0.76, 0.82 – the values utilized in the simulation, and 0.9). The direct impact of beam absorption efficiency on energy input into the material was investigated. Specifically, the initial 20 layers of the build were simulated and subsequently compared to experimental data derived from SEM. While phase fractions were found to be consistent across all conditions, disparities emerged in the mean size of γ′ precipitates. An absorption efficiency of 0.76 yielded a mean size of approximately 70 nm. Conversely, absorption efficiencies of 0.82 and 0.9 exhibited remarkably similar mean sizes of around 130 nm, aligning closely with the outcomes of the experiments.
The numerical analyses of the evolution of precipitates was performed using MatCalc version 6.04 (rel 0.011). The thermodynamic (‘mc_ni.tdb’, version 2.034) and diffusion (‘mc_ni.ddb’, version 2.007) databases were used. MatCalc’s basic principles are elaborated as follows:
The nucleation kinetics of precipitates are computed using a computational technique based on a classical nucleation theory that has been modified for systems with multiple components , . Accordingly, the transient nucleation rate (�), which expresses the rate at which nuclei are formed per unit volume and time, is calculated as:(6)�=�0��*∙�xp−�*�∙�∙exp−��where �0 denotes the number of active nucleation sites, �* the rate of atomic attachment, � the Boltzmann constant, � the temperature, �* the critical energy for nucleus formation, τ the incubation time, and t the time. � (Zeldovich factor) takes into consideration that thermal excitation destabilizes the nucleus as opposed to its inactive state . Z is defined as follows:(7)�=−12�kT∂2∆�∂�2�*12where ∆� is the overall change in free energy due to the formation of a nucleus and n is the nucleus’ number of atoms. ∆�’s derivative is evaluated at n* (critical nucleus size). �* accounts for the long-range diffusion of atoms required for nucleation, provided that the matrix’ and precipitates’ composition differ. Svoboda et al.  developed an appropriate multi-component equation for �*, which is given by:(8)�*=4��*2�4�∑�=1��ki−�0�2�0��0�−1where �* denotes the critical radius for nucleation, � represents atomic distance, and � is the molar volume. �ki and �0� represent the concentration of elements in the precipitate and matrix, respectively. The parameter �0� denotes the rate of diffusion of the ith element within the matrix. The expression for the incubation time � is expressed as :(9)�=12�*�2
and �*, which represents the critical energy for nucleation:(10)�*=16�3�3∆�vol2where � is the interfacial energy, and ∆Gvol the change in the volume free energy. The critical nucleus’ composition is similar to the γ′ phase’s equilibrium composition at the same temperature. � is computed based on the precipitate and matrix compositions, using a generalized nearest neighbor broken bond model, with the assumption of interfaces being planar, sharp, and coherent , , .
In Eq. 7, it is worth noting that �* represents the fundamental variable in the nucleation theory. It contains �3/∆�vol2 and is in the exponent of the nucleation rate. Therefore, even small variations in γ and/or ∆�vol can result in notable changes in �, especially if �* is in the order of �∙�. This is demonstrated in  for UDIMET 720 Li during continuous cooling, where these quantities change steadily during precipitation due to their dependence on matrix’ and precipitate’s temperature and composition. In the current work, these changes will be even more significant as the system is exposed to multiple cycles of rapid cooling and heating.
Once nucleated, the growth of a precipitate is assessed using the radius and composition evolution equations developed by Svoboda et al.  with a mean-field method that employs the thermodynamic extremal principle. The expression for the total Gibbs free energy of a thermodynamic system G, which consists of n components and m precipitates, is given as follows:(11)�=∑���0��0�+∑�=1�4���33��+∑�=1��ki�ki+∑�=1�4���2��.
The chemical potential of component � in the matrix is denoted as �0�(�=1,…,�), while the chemical potential of component � in the precipitate is represented by �ki(�=1,…,�,�=1,…,�). These chemical potentials are defined as functions of the concentrations �ki(�=1,…,�,�=1,…,�). The interface energy density is denoted as �, and �� incorporates the effects of elastic energy and plastic work resulting from the volume change of each precipitate.
Eq. (12) establishes that the total free energy of the system in its current state relies on the independent state variables: the sizes (radii) of the precipitates �� and the concentrations of each component �ki. The remaining variables can be determined by applying the law of mass conservation to each component �. This can be represented by the equation:(12)��=�0�+∑�=1�4���33�ki,
Furthermore, the global mass conservation can be expressed by equation:(13)�=∑�=1���When a thermodynamic system transitions to a more stable state, the energy difference between the initial and final stages is dissipated. This model considers three distinct forms of dissipation effects . These include dissipations caused by the movement of interfaces, diffusion within the precipitate and diffusion within the matrix.
Consequently, �̇� (growth rate) and �̇ki (chemical composition’s rate of change) of the precipitate with index � are derived from the linear system of equation system:(14)�ij��=��where �� symbolizes the rates �̇� and �̇ki . Index i contains variables for precipitate radius, chemical composition, and stoichiometric boundary conditions suggested by the precipitate’s crystal structure. Eq. (10) is computed separately for every precipitate �. For a more detailed description of the formulae for the coefficients �ij and �� employed in this work please refer to .
The MatCalc software was used to perform the numerical time integration of �̇� and �̇ki of precipitates based on the classical numerical method by Kampmann and Wagner . Detailed information on this method can be found in . Using this computational method, calculations for E-PBF thermal cycles (cyclic heating and cooling) were computed and compared to experimental data. The simulation took approximately 2–4 hrs to complete on a standard laptop.
Fig. 1 displays a stitched overview image and selected SEM micrographs of various γ′ morphologies and carbides after observations of the X-Z surface of the build from the top to 2 mm above the bottom. Fig. 2 depicts a graph that charts the average size and phase fraction of the primary γ′, as it changes with distance from the top to the bottom of the build. The SEM micrographs show widespread primary γ′ precipitation throughout the entire build, with the size increasing in the top to bottom direction. Particularly, at the topmost height, representing the 460th layer (Z = 22.95 mm), as seen in Fig. 1b, the average size of γ′ is 110 ± 4 nm, exhibiting spherical shapes. This is representative of the microstructure after it solidifies and cools to room temperature, without experiencing additional thermal cycles. The γ′ size slightly increases to 147 ± 6 nm below this layer and remains constant until 0.4 mm (∼453rd layer) from the top. At this position, the microstructure still closely resembles that of the 460th layer. After the 453rd layer, the γ′ size grows rapidly to ∼503 ± 19 nm until reaching the 437th layer (1.2 mm from top). The γ′ particles here have a cuboidal shape, and a small fraction is coarser than 600 nm. γ′ continue to grow steadily from this position to the bottom (23 mm from the top). A small fraction of γ′ is > 800 nm.
Besides primary γ′, secondary γ′ with sizes ranging from 5 to 50 nm were also found. These secondary γ′ precipitates, as seen in Fig. 1f, were present only in the bottom and middle regions. A detailed analysis of the multimodal size distribution of γ′ can be found in . There is no significant variation in the phase fraction of the γ′ along the build. The phase fraction is ∼ 52%, as displayed in Fig. 2. It is worth mentioning that the total phase fraction of γ′ was estimated based on the primary γ′ phase fraction because of the small size of secondary γ′. Spherical MC carbides with sizes ranging from 50 to 400 nm and a phase fraction of 0.8% were also observed throughout the build. The carbides are the light grey precipitates in Fig. 1g. The light grey shade of carbides in the SEM images is due to their composition and crystal structure . These carbides are not visible in Fig. 1b-e because they were dissolved during electro-etching carried out after electropolishing. In Fig. 1g, however, the sample was examined directly after electropolishing, without electro-etching.
Table 2 shows the nominal and measured composition of γ′ precipitates throughout the build by atom probe microscopy as determined in our previous study . No build height-dependent composition difference was observed in either of the γ′ precipitate populations. However, there was a slight disparity between the composition of primary and secondary γ′. Among the main γ′ forming elements, the primary γ′ has a high Ti concentration while secondary γ′ has a high Al concentration. A detailed description of the atom distribution maps and the proxigrams of the constituent elements of γ′ throughout the build can be found in .
Table 2. Bulk IN738 composition determined using inductively coupled plasma atomic emission spectroscopy (ICP-AES). Compositions of γ, primary γ′, and secondary γ′ at various locations in the build measured by APT. This information is reproduced from data in Ref.  with permission.
Fig. 3a shows the Vickers hardness mapping performed along the entire X-Z surface, while Fig. 3b shows the plot of average hardness at different build heights. This hardness distribution is consistent with the γ′ precipitate size gradient across the build direction in Fig. 1, Fig. 2. The maximum hardness of ∼530 HV1 is found at ∼0.5 mm away from the top surface (Z = 22.5), where γ′ particles exhibit the smallest observed size in Fig. 2b. Further down the build (∼ 2 mm from the top), the hardness drops to the 440–490 HV1 range. This represents the region where γ′ begins to coarsen. The hardness drops further to 380–430 HV1 at the bottom of the build.
3.3. Modeling of the microstructural evolution during E-PBF
3.3.1. Thermal profile modeling
Fig. 4 shows the simulated thermal profile of the E-PBF build at a location of 23 mm from the top of the build, using a semi-analytical heat conduction model. This profile consists of the time taken to deposit 460 layers until final cooling, as shown in Fig. 4a. Fig. 4b-d show the magnified regions of Fig. 4a and reveal the first 20 layers from the top, a single layer (first layer from the top), and the time taken for the build to cool after the last layer deposition, respectively.
The peak temperatures experienced by previous layers decrease progressively as the number of layers increases but never fall below the build preheat temperature (1000 °C). Our simulated thermal cycle may not completely capture the complexity of the actual thermal cycle utilized in the E-PBF build. For instance, the top layer (Fig. 4c), also representing the first deposit’s thermal profile without additional cycles (from powder heating, melting, to solidification), recorded the highest peak temperature of 1390 °C. Although this temperature is above the melting range of the alloy (1230–1360 °C) , we believe a much higher temperature was produced by the electron beam to melt the powder. Nevertheless, the solidification temperature and dynamics are outside the scope of this study as our focus is on the solid-state phase transformations during deposition. It takes ∼25 s for each layer to be deposited and cooled to the build temperature. The interlayer dwell time is 125 s. The time taken for the build to cool to room temperature (RT) after final layer deposition is ∼4.7 hrs (17,000 s).
3.3.2. MatCalc simulation
During the MatCalc simulation, the matrix phase is defined as γ. γ′, and MC carbide are included as possible precipitates. The domain of these precipitates is set to be the matrix (γ), and nucleation is assumed to be homogenous. In homogeneous nucleation, all atoms of the unit volume are assumed to be potential nucleation sites. Table 3 shows the computational parameters used in the simulation. All other parameters were set at default values as recommended in the version 6.04.0011 of MatCalc. The values for the interfacial energies are automatically calculated according to the generalized nearest neighbor broken bond model and is one of the most outstanding features in MatCalc , , . It should be noted that the elastic misfit strain was not included in the calculation. The output of MatCalc includes phase fraction, size, nucleation rate, and composition of the precipitates. The phase fraction in MatCalc is the volume fraction. Although the experimental phase fraction is the measured area fraction, it is relatively similar to the volume fraction. This is because of the generally larger precipitate size and similar morphology at the various locations along the build . A reliable phase fraction comparison between experiment and simulation can therefore be made.
Table 3. Computational parameters used in the simulation.
γ′ = 0.080–0.140 J/m2 and MC carbide = 0.410–0.430 J/m2
126.96.36.199. Precipitate phase fraction
Fig. 5a shows the simulated phase fraction of γ′ and MC carbide during thermal cycling. Fig. 5b is a magnified view of 5a showing the simulated phase fraction at the center points of the top 70 layers, whereas Fig. 5c corresponds to the first two layers from the top. As mentioned earlier, the top layer (460th layer) represents the microstructure after solidification. The microstructure of the layers below is determined by the number of thermal cycles, which increases with distance to the top. For example, layers 459, 458, 457, up to layer 1 (region of interest) experience 1, 2, 3 and 459 thermal cycles, respectively. In the top layer in Fig. 5c, the volume fraction of γ′ and carbides increases with temperature. For γ′, it decreases to zero when the temperature is above the solvus temperature after a few seconds. Carbides, however, remain constant in their volume fraction reaching equilibrium (phase fraction ∼ 0.9%) in a short time. The topmost layer can be compared to the first deposit, and the peak in temperature symbolizes the stage where the electron beam heats the powder until melting. This means γ′ and carbide precipitation might have started in the powder particles during heating from the build temperature and electron beam until the onset of melting, where γ′ dissolves, but carbides remain stable .
During cooling after deposition, γ′ reprecipitates at a temperature of 1085 °C, which is below its solvus temperature. As cooling progresses, the phase fraction increases steadily to ∼27% and remains constant at 1000 °C (elevated build temperature). The calculated equilibrium fraction of phases by MatCalc is used to show the complex precipitation characteristics in this alloy. Fig. 6 shows that MC carbides form during solidification at 1320 °C, followed by γ′, which precipitate when the solidified layer cools to 1140 °C. This indicates that all deposited layers might contain a negligible amount of these precipitates before subsequent layer deposition, while being at the 1000 °C build temperature or during cooling to RT. The phase diagram also shows that the equilibrium fraction of the γ′ increases as temperature decreases. For instance, at 1000, 900, and 800 °C, the phase fractions are ∼30%, 38%, and 42%, respectively.
Deposition of subsequent layers causes previous layers to undergo phase transformations as they are exposed to several thermal cycles with different peak temperatures. In Fig. 5c, as the subsequent layer is being deposited, γ′ in the previous layer (459th layer) begins to dissolve as the temperature crosses the solvus temperature. This is witnessed by the reduction of the γ′ phase fraction. This graph also shows how this phase dissolves during heating. However, the phase fraction of MC carbide remains stable at high temperatures and no dissolution is seen during thermal cycling. Upon cooling, the γ′ that was dissolved during heating reprecipitates with a surge in the phase fraction until 1000 °C, after which it remains constant. This microstructure is similar to the solidification microstructure (layer 460), with a similar γ′ phase fraction (∼27%).
The complete dissolution and reprecipitation of γ′ continue for several cycles until the 50th layer from the top (layer 411), where the phase fraction does not reach zero during heating to the peak temperature (see Fig. 5d). This indicates the ‘partial’ dissolution of γ′, which continues progressively with additional layers. It should be noted that the peak temperatures for layers that underwent complete dissolution were much higher (1170–1300 °C) than the γ′ solvus.
The dissolution and reprecipitation of γ′ during thermal cycling are further confirmed in Fig. 7, which summarizes the nucleation rate, phase fraction, and concentration of major elements that form γ′ in the matrix. Fig. 7b magnifies a single layer (3rd layer from top) within the full dissolution region in Fig. 7a to help identify the nucleation and growth mechanisms. From Fig. 7b, γ′ nucleation begins during cooling whereby the nucleation rate increases to reach a maximum value of approximately 1 × 1020 m−3s−1. This fast kinetics implies that some rearrangement of atoms is required for γ′ precipitates to form in the matrix , . The matrix at this stage is in a non-equilibrium condition. Its composition is similar to the nominal composition and remains unchanged. The phase fraction remains insignificant at this stage although nucleation has started. The nucleation rate starts declining upon reaching the peak value. Simultaneously, diffusion-controlled growth of existing nuclei occurs, depleting the matrix of γ′ forming elements (Al and Ti). Thus, from (7), (11), ∆�vol continuously decreases until nucleation ceases. The growth of nuclei is witnessed by the increase in phase fraction until a constant level is reached at 27% upon cooling to and holding at build temperature. This nucleation event is repeated several times.
At the onset of partial dissolution, the nucleation rate jumps to 1 × 1021 m−3s−1, and then reduces sharply at the middle stage of partial dissolution. The nucleation rate reaches 0 at a later stage. Supplementary Fig. S1 shows a magnified view of the nucleation rate, phase fraction, and thermal profile, underpinning this trend. The jump in nucleation rate at the onset is followed by a progressive reduction in the solute content of the matrix. The peak temperatures (∼1130–1160 °C) are lower than those in complete dissolution regions but still above or close to the γ′ solvus. The maximum phase fraction (∼27%) is similar to that of the complete dissolution regions. At the middle stage, the reduction in nucleation rate is accompanied by a sharp drop in the matrix composition. The γ′ fraction drops to ∼24%, where the peak temperatures of the layers are just below or at γ′ solvus. The phase fraction then increases progressively through the later stage of partial dissolution to ∼30% towards the end of thermal cycling. The matrix solute content continues to drop although no nucleation event is seen. The peak temperatures are then far below the γ′ solvus. It should be noted that the matrix concentration after complete dissolution remains constant. Upon cooling to RT after final layer deposition, the nucleation rate increases again, indicating new nucleation events. The phase fraction reaches ∼40%, with a further depletion of the matrix in major γ′ forming elements.
188.8.131.52. γ′ size distribution
Fig. 8 shows histograms of the γ′ precipitate size distributions (PSD) along the build height during deposition. These PSDs are predicted at the end of each layer of interest just before final cooling to room temperature, to separate the role of thermal cycles from final cooling on the evolution of γ′. The PSD for the top layer (layer 460) is shown in Fig. 8a (last solidified region with solidification microstructure). The γ′ size ranges from 120 to 230 nm and is similar to the 44 layers below (2.2 mm from the top).
Further down the build, γ′ begins to coarsen after layer 417 (44th layer from top). Fig. 8c shows the PSD after the 44th layer, where the γ′ size exhibits two peaks at ∼120–230 and ∼300 nm, with most of the population being in the former range. This is the onset of partial dissolution where simultaneously with the reprecipitation and growth of fresh γ′, the undissolved γ′ grows rapidly through diffusive transport of atoms to the precipitates. This is shown in Fig. 8c, where the precipitate class sizes between 250 and 350 represent the growth of undissolved γ′. Although this continues in the 416th layer, the phase fractions plot indicates that the onset of partial dissolution begins after the 411th layer. This implies that partial dissolution started early, but the fraction of undissolved γ′ was too low to impact the phase fraction. The reprecipitated γ′ are mostly in the 100–220 nm class range and similar to those observed during full dissolution.
As the number of layers increases, coarsening intensifies with continued growth of more undissolved γ′, and reprecipitation and growth of partially dissolved ones. Fig. 8d, e, and f show this sequence. Further down the build, coarsening progresses rapidly, as shown in Figs. 8d, 8e, and 8f. The γ′ size ranges from 120 to 1100 nm, with the peaks at 160, 180, and 220 nm in Figs. 8d, 8e, and 8f, respectively. Coarsening continues until nucleation ends during dissolution, where only the already formed γ′ precipitates continue to grow during further thermal cycling. The γ′ size at this point is much larger, as observed in layers 361 and 261, and continues to increase steadily towards the bottom (layer 1). Two populations in the ranges of ∼380–700 and ∼750–1100 nm, respectively, can be seen. The steady growth of γ′ towards the bottom is confirmed by the gradual decrease in the concentration of solute elements in the matrix (Fig. 7a). It should be noted that for each layer, the γ′ class with the largest size originates from continuous growth of the earliest set of the undissolved precipitates.
Fig. 9, Fig. 10 and supplementary Figs. S2 and S3 show the γ′ size evolution during heating and cooling of a single layer in the full dissolution region, and early, middle stages, and later stages of partial dissolution, respectively. In all, the size of γ′ reduces during layer heating. Depending on the peak temperature of the layer which varies with build height, γ′ are either fully or partially dissolved as mentioned earlier. Upon cooling, the dissolved γ′ reprecipitate.
In Fig. 9, those layers that underwent complete dissolution (top layers) were held above γ′ solvus temperature for longer. In Fig. 10, layers at the early stage of partial dissolution spend less time in the γ′ solvus temperature region during heating, leading to incomplete dissolution. In such conditions, smaller precipitates are fully dissolved while larger ones shrink . Layers in the middle stages of partial dissolution have peak temperatures just below or at γ′ solvus, not sufficient to achieve significant γ′ dissolution. As seen in supplementary Fig. S2, only a few smaller γ′ are dissolved back into the matrix during heating, i.e., growth of precipitates is more significant than dissolution. This explains the sharp decrease in concentration of Al and Ti in the matrix in this layer.
The previous sections indicate various phenomena such as an increase in phase fraction, further depletion of matrix composition, and new nucleation bursts during cooling. Analysis of the PSD after the final cooling of the build to room temperature allows a direct comparison to post-printing microstructural characterization. Fig. 11 shows the γ′ size distribution of layer 1 (460th layer from the top) after final cooling to room temperature. Precipitation of secondary γ′ is observed, leading to the multimodal size distribution of secondary and primary γ′. The secondary γ′ size falls within the 10–80 nm range. As expected, a further growth of the existing primary γ′ is also observed during cooling.
184.108.40.206. γ′ chemistry after deposition
Fig. 12 shows the concentration of the major elements that form γ′ (Al, Ti, and Ni) in the primary and secondary γ′ at the bottom of the build, as calculated by MatCalc. The secondary γ′ has a higher Al content (13.5–14.5 at% Al), compared to 13 at% Al in the primary γ′. Additionally, within the secondary γ′, the smallest particles (∼10 nm) have higher Al contents than larger ones (∼70 nm). In contrast, for the primary γ′, there is no significant variation in the Al content as a function of their size. The Ni concentration in secondary γ′ (71.1–72 at%) is also higher in comparison to the primary γ′ (70 at%). The smallest secondary γ′ (∼10 nm) have higher Ni contents than larger ones (∼70 nm), whereas there is no substantial change in the Ni content of primary γ′, based on their size. As expected, Ti shows an opposite size-dependent variation. It ranges from ∼ 7.7–8.7 at% Ti in secondary γ′ to ∼9.2 at% in primary γ′. Similarly, within the secondary γ′, the smallest (∼10 nm) have lower Al contents than the larger ones (∼70 nm). No significant variation is observed for Ti content in primary γ′.
A combined modelling method is utilized to study the microstructural evolution during E-PBF of IN738. The presented results are discussed by examining the precipitation and dissolution mechanism of γ′ during thermal cycling. This is followed by a discussion on the phase fraction and size evolution of γ′ during thermal cycling and after final cooling. A brief discussion on carbide morphology is also made. Finally, a comparison is made between the simulation and experimental results to assess their agreement.
4.1. γ′ morphology as a function of build height
4.1.1. Nucleation of γ′
The fast precipitation kinetics of the γ′ phase enables formation of γ′ upon quenching from higher temperatures (above solvus) during thermal cycling . In Fig. 7b, for a single layer in the full dissolution region, during cooling, the initial increase in nucleation rate signifies the first formation of nuclei. The slight increase in nucleation rate during partial dissolution, despite a decrease in the concentration of γ′ forming elements, may be explained by the nucleation kinetics. During partial dissolution and as the precipitates shrink, it is assumed that the regions at the vicinity of partially dissolved precipitates are enriched in γ′ forming elements , . This differs from the full dissolution region, in which case the chemical composition is evenly distributed in the matrix. Several authors have attributed the solute supersaturation of the matrix around primary γ′ to partial dissolution during isothermal ageing , , , . The enhanced supersaturation in the regions close to the precipitates results in a much higher driving force for nucleation, leading to a higher nucleation rate upon cooling. This phenomenon can be closely related to the several nucleation bursts upon continuous cooling of Ni-based superalloys, where second nucleation bursts exhibit higher nucleation rates , , , .
At middle stages of partial dissolution, the reduction in the nucleation rate indicates that the existing composition and low supersaturation did not trigger nucleation as the matrix was closer to the equilibrium state. The end of a nucleation burst means that the supersaturation of Al and Ti has reached a low level, incapable of providing sufficient driving force during cooling to or holding at 1000 °C for further nucleation . Earlier studies on Ni-based superalloys have reported the same phenomenon during ageing or continuous cooling from the solvus temperature to RT , , .
4.1.2. Dissolution of γ′ during thermal cycling
γ′ dissolution kinetics during heating are fast when compared to nucleation due to exponential increase in phase transformation and diffusion activities with temperature . As shown in Fig. 9, Fig. 10, and supplementary Figs. S2 and S3, the reduction in γ′ phase fraction and size during heating indicates γ′ dissolution. This is also revealed in Fig. 5 where phase fraction decreases upon heating. The extent of γ′ dissolution mostly depends on the temperature, time spent above γ′ solvus, and precipitate size, , . Smaller γ′ precipitates are first to be dissolved , , . This is mainly because more solute elements need to be transported away from large γ′ precipitates than from smaller ones . Also, a high temperature above γ′ solvus temperature leads to a faster dissolution rate. The equilibrium solvus temperature of γ′ in IN738 in our MatCalc simulation (Fig. 6) and as reported by Ojo et al.  is 1140 °C and 1130–1180 °C, respectively. This means the peak temperature experienced by previous layers decreases progressively from γ′ supersolvus to subsolvus, near-solvus, and far from solvus as the number of subsequent layers increases. Based on the above, it can be inferred that the degree of dissolution of γ′ contributes to the gradient in precipitate distribution.
Although the peak temperatures during later stages of partial dissolution are much lower than the equilibrium γ′ solvus, γ′ dissolution still occurs but at a significantly lower rate (supplementary Fig. S3). Wahlmann et al.  also reported a similar case where they observed the rapid dissolution of γ′ in CMSX-4 during fast heating and cooling cycles at temperatures below the γ′ solvus. They attributed this to the γ′ phase transformation process taking place in conditions far from the equilibrium. While the same reasoning may be valid for our study, we further believe that the greater surface area to volume ratio of the small γ′ precipitates contributed to this. This ratio means a larger area is available for solute atoms to diffuse into the matrix even at temperatures much below the solvus .
4.2. γ′ phase fraction and size evolution
4.2.1. During thermal cycling
In the first layer, the steep increase in γ′ phase fraction during heating (Fig. 5), which also represents γ′ precipitation in the powder before melting, has qualitatively been validated in . The maximum phase fraction of 27% during the first few layers of thermal cycling indicates that IN738 theoretically could reach the equilibrium state (∼30%), but the short interlayer time at the build temperature counteracts this. The drop in phase fraction at middle stages of partial dissolution is due to the low number of γ′ nucleation sites . It has been reported that a reduction of γ′ nucleation sites leads to a delay in obtaining the final volume fraction as more time is required for γ′ precipitates to grow and reach equilibrium . This explains why even upon holding for 150 s before subsequent layer deposition, the phase fraction does not increase to those values that were observed in the previous full γ′ dissolution regions. Towards the end of deposition, the increase in phase fraction to the equilibrium value of 30% is as a result of the longer holding at build temperature or close to it .
During thermal cycling, γ′ particles begin to grow immediately after they first precipitate upon cooling. This is reflected in the rapid increase in phase fraction and size during cooling in Fig. 5 and supplementary Fig. S2, respectively. The rapid growth is due to the fast diffusion of solute elements at high temperatures . The similar size of γ′ for the first 44 layers from the top can be attributed to the fact that all layers underwent complete dissolution and hence, experienced the same nucleation event and growth during deposition. This corresponds with the findings by Balikci et al. , who reported that the degree of γ′ precipitation in IN738LC does not change when a solution heat treatment is conducted above a certain critical temperature.
The increase in coarsening rate (Fig. 8) during thermal cycling can first be ascribed to the high peak temperature of the layers . The coarsening rate of γ′ is known to increase rapidly with temperature due to the exponential growth of diffusion activity. Also, the simultaneous dissolution with coarsening could be another reason for the high coarsening rate, as γ′ coarsening is a diffusion-driven process where large particles grow by consuming smaller ones , , , . The steady growth of γ′ towards the bottom of the build is due to the much lower layer peak temperature, which is almost close to the build temperature, and reduced dissolution activity, as is seen in the much lower solute concentration in γ′ compared to those in the full and partial dissolution regions.
4.2.2. During cooling
The much higher phase fraction of ∼40% upon cooling signifies the tendency of γ′ to reach equilibrium at lower temperatures (Fig. 4). This is due to the precipitation of secondary γ′ and a further increase in the size of existing primary γ′, which leads to a multimodal size distribution of γ′ after cooling , , , , . The reason for secondary γ′ formation during cooling is as follows: As cooling progresses, it becomes increasingly challenging to redistribute solute elements in the matrix owing to their lower mobility , . A higher supersaturation level in regions away from or free of the existing γ′ precipitates is achieved, making them suitable sites for additional nucleation bursts. More cooling leads to the growth of these secondary γ′ precipitates, but as the temperature and in turn, the solute diffusivity is low, growth remains slow.
MC carbides in IN738 are known to have a significant impact on the high-temperature strength. They can also act as effective hardening particles and improve the creep resistance . Precipitation of MC carbides in IN738 and several other superalloys is known to occur during solidification or thermal treatments (e.g., hot isostatic pressing) . In our case, this means that the MC carbides within the E-PBF build formed because of the thermal exposure from the E-PBF thermal cycle in addition to initial solidification. Our simulation confirms this as MC carbides appear during layer heating (Fig. 5). The constant and stable phase fraction of MC carbides during thermal cycling can be attributed to their high melting point (∼1360 °C) and the short holding time at peak temperatures , , . The solvus temperature for most MC carbides exceeds most of the peak temperatures observed in our simulation, and carbide dissolution kinetics at temperatures above the solvus are known to be comparably slow . The stable phase fraction and random distribution of MC carbides signifies the slight influence on the gradient in hardness.
4.4. Comparison of simulations and experiments
4.4.1. Precipitate phase fraction and morphology as a function of build height
A qualitative agreement is observed for the phase fraction of carbides, i.e. ∼0.8% in the experiment and ∼0.9% in the simulation. The phase fraction of γ′ differs, with the experiment reporting a value of ∼51% and the simulation, 40%. Despite this, the size distribution of primary γ′ along the build shows remarkable consistency between experimental and computational analyses. It is worth noting that the primary γ′ morphology in the experimental analysis is observed in the as-fabricated state, whereas the simulation (Fig. 8) captures it during deposition process. The primary γ′ size in the experiment is expected to experience additional growth during the cooling phase. Regardless, both show similar trends in primary γ′ size increments from the top to the bottom of the build. The larger primary γ’ size in the simulation versus the experiment can be attributed to the fact that experimental and simulation results are based on 2D and 3D data, respectively. The absence of stereological considerations  in our analysis could have led to an underestimation of the precipitate sizes from SEM measurements. The early starts of coarsening (8th layer) in the experiment compared to the simulation (45th layer) can be attributed to a higher actual γ′ solvus temperature than considered in our simulation . The solvus temperature of γ′ in a Ni-based superalloy is mainly determined by the detailed composition. A high amount of Cr and Co are known to reduce the solvus temperature, whereas Ta and Mo will increase it , , . The elemental composition from our experimental work was used for the simulation except for Ta. It should be noted that Ta is not included in the thermodynamic database in MatCalc used, and this may have reduced the solvus temperature. This could also explain the relatively higher γ′ phase fraction in the experiment than in simulation, as a higher γ′ solvus temperature will cause more γ′ to precipitate and grow early during cooling , .
Another possible cause of this deviation can be attributed to the extent of γ′ dissolution, which is mainly determined by the peak temperature. It can be speculated that individual peak temperatures at different layers in the simulation may have been over-predicted. However, one needs to consider that the true thermal profile is likely more complicated in the actual E-PBF process . For example, the current model assumes that the thermophysical properties of the material are temperature-independent, which is not realistic. Many materials, including IN738, exhibit temperature-dependent properties such as thermal conductivity, specific heat capacity, and density . This means that heat transfer simulations may underestimate or overestimate the temperature gradients and cooling rates within the powder bed and the solidified part. Additionally, the model does not account for the reduced thermal diffusivity through unmelted powder, where gas separating the powder acts as insulation, impeding the heat flow . In E-PBF, the unmelted powder regions with trapped gas have lower thermal diffusivity compared to the fully melted regions, leading to localized temperature variations, and altered solidification behavior. These limitations can impact the predictions, particularly in relation to the carbide dissolution, as the peak temperatures may be underestimated.
While acknowledging these limitations, it is worth emphasizing that achieving a detailed and accurate representation of each layer’s heat source would impose tough computational challenges. Given the substantial layer count in E-PBF, our decision to employ a semi-analytical approximation strikes a balance between computational feasibility and the capture of essential trends in thermal profiles across diverse build layers. In future work, a dual-calibration strategy is proposed to further reduce simulation-experiment disparities. By refining temperature-independent thermophysical property approximations and absorptivity in the heat source model, and by optimizing interfacial energy descriptions in the kinetic model, the predictive precision could be enhanced. Further refining the simulation controls, such as adjusting the precipitate class size may enhance quantitative comparisons between modeling outcomes and experimental data in future work.
4.4.2. Multimodal size distribution of γ′ and concentration
Another interesting feature that sees qualitative agreement between the simulation and the experiment is the multimodal size distribution of γ′. The formation of secondary γ′ particles in the experiment and most E-PBF Ni-based superalloys is suggested to occur at low temperatures, during final cooling to RT , , . However, so far, this conclusion has been based on findings from various continuous cooling experiments, as the study of the evolution during AM would require an in-situ approach. Our simulation unambiguously confirms this in an AM context by providing evidence for secondary γ′ precipitation during slow cooling to RT. Additionally, it is possible to speculate that the chemical segregation occurring during solidification, due to the preferential partitioning of certain elements between the solid and liquid phases, can contribute to the multimodal size distribution during deposition . This is because chemical segregation can result in variations in the local composition of superalloys, which subsequently affects the nucleation and growth of γ′. Regions with higher concentrations of alloying elements will encourage the formation of larger γ′ particles, while regions with lower concentrations may favor the nucleation of smaller precipitates. However, it is important to acknowledge that the elevated temperature during the E-PBF process will largely homogenize these compositional differences , .
A good correlation is also shown in the composition of major γ′ forming elements (Al and Ti) in primary and secondary γ′. Both experiment and simulation show an increasing trend for Al content and a decreasing trend for Ti content from primary to secondary γ′. The slight composition differences between primary and secondary γ′ particles are due to the different diffusivity of γ′ stabilizers at different thermal conditions , . As the formation of multimodal γ′ particles with different sizes occurs over a broad temperature range, the phase chemistry of γ′ will be highly size dependent. The changes in the chemistry of various γ′ (primary, secondary, and tertiary) have received significant attention since they have a direct influence on the performance , , , , . Chen et al. , , reported a high Al content in the smallest γ′ precipitates compared to the largest, while Ti showed an opposite trend during continuous cooling in a RR1000 Ni-based superalloy. This was attributed to the temperature and cooling rate at which the γ′ precipitates were formed. The smallest precipitates formed last, at the lowest temperature and cooling rate. A comparable observation is evident in the present investigation, where the secondary γ′ forms at a low temperature and cooling rate in comparison to the primary. The temperature dependence of γ′ chemical composition is further evidenced in supplementary Fig. S4, which shows the equilibrium chemical composition of γ′ as a function of temperature.
A correlative modelling approach capable of predicting solid-state phase transformations kinetics in metal AM was developed. This approach involves computational simulations with a semi-analytical heat transfer model and the MatCalc thermo-kinetic software. The method was used to predict the phase transformation kinetics and detailed morphology and chemistry of γ′ and MC during E-PBF of IN738 Ni-based superalloy. The main conclusions are:
1.The computational simulations are in qualitative agreement with the experimental observations. This is particularly true for the γ′ size distribution along the build height, the multimodal size distribution of particles, and the phase fraction of MC carbides.
2.The deviations between simulation and experiment in terms of γ′ phase fraction and location in the build are most likely attributed to a higher γ′ solvus temperature during the experiment than in the simulation, which is argued to be related to the absence of Ta in the MatCalc database.
3.The dissolution and precipitation of γ′ occur fast and under non-equilibrium conditions. The level of γ′ dissolution determines the gradient in γ′ size distribution along the build. After thermal cycling, the final cooling to room temperature has further significant impacts on the final γ′ size, morphology, and distribution.
4.A negligible amount of γ′ forms in the first deposited layer before subsequent layer deposition, and a small amount of γ′ may also form in the powder induced by the 1000 °C elevated build temperature before melting.
Our findings confirm the suitability of MatCalc to predict the microstructural evolution at various positions throughout a build in a Ni-based superalloy during E-PBF. It also showcases the suitability of a tool which was originally developed for traditional thermo-mechanical processing of alloys to the new additive manufacturing context. Our simulation capabilities are likely extendable to other alloy systems that undergo solid-state phase transformations implemented in MatCalc (various steels, Ni-based superalloys, and Al-alloys amongst others) as well as other AM processes such as L-DED and L-PBF which have different thermal cycle characteristics. New tools to predict the microstructural evolution and properties during metal AM are important as they provide new insights into the complexities of AM. This will enable control and design of AM microstructures towards advanced materials properties and performances.
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.
This research was sponsored by the Department of Industry, Innovation, and Science under the auspices of the AUSMURI program – which is a part of the Commonwealth’s Next Generation Technologies Fund. The authors acknowledge the facilities and the scientific and technical assistance at the Electron Microscope Unit (EMU) within the Mark Wainwright Analytical Centre (MWAC) at UNSW Sydney and Microscopy Australia. Nana Adomako is supported by a UNSW Scientia PhD scholarship. Michael Haines’ (UNSW Sydney) contribution to the revised version of the original manuscript is thankfully acknowledged.
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Received: 10 June 2023 / Revised: 19 June 2023 / Accepted: 27 June 2023 / Published: 1 July 2023(This article belongs to the Section Ocean Engineering)
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Local scouring might result in the spanning of submarine cables, endangering their mechanical and electrical properties. In this contribution, a three-dimensional computational fluid dynamics simulation model is developed using FLOW-3D, and the scouring process of semi-exposed submarine cables is investigated. The effects of the sediment critical Shields number, sediment density, and ocean current velocity on local scouring are discussed, and variation rules for the submarine cables’ spanning time are provided. The results indicate that three scouring holes are formed around the submarine cables. The location of the bottom of the holes corresponds to that of the maximum shear velocity. The continuous development of scouring holes at the wake position leads to the spanning of the submarine cables. The increase in the sediment’s critical Shields number and sediment density, as well as the decrease in the ocean current velocity, will extend the time for maintaining the stability of the upstream scouring hole and retard the development velocity of the wake position and downstream scouring holes. The spanning time has a cubic relationship with the sediment’s critical Shields number, a linear relationship with the sediment density, and an exponential relationship with the ocean current velocity. In this paper, the local scouring process of semi-exposed submarine cables is studied, which provides a theoretical basis for the operation and maintenance of submarine cables.
As a key piece of equipment in cross-sea power grids, submarine cables are widely used to connect autonomous power grids, supply power to islands or offshore platforms, and transmit electric power generated by marine renewable energy installations to onshore substations . Once submarine cables break down due to natural disasters or human-made damage, the normal operation of other marine electric power equipment connected to them may be affected. These chain reactions will cause great economic losses and serious social impacts .
To protect submarine cables, they are usually buried 1 to 3 m below the seabed . However, submarine cables are still confronted with potential threats from the complex subsea environment. Under the influence of fishing, anchor damage, ocean current scouring, and other factors, the sediment above submarine cables will always inevitably migrate. When a submarine cable is partially exposed, the scouring at this position will be exacerbated; eventually, it will cause the submarine cable to span. According to a field investigation of the 500 kV oil-filled submarine cable that is part of the Hainan networking system, the total length of the span is 49 m . Under strong ocean currents, spanning submarine cables may experience vortex-induced vibrations. Fatigue stress caused by vortex-induced vibrations may lead to metal sheath rupture , which endangers the mechanical and electrical properties of submarine cables. Therefore, understanding the local scouring processes of partially exposed submarine cables is crucial for predicting scouring patterns. This is the basis for developing effective operation and maintenance strategies for submarine cables.
The mechanism and influencing factors of sediment erosion have been examined by researchers around the world. In 1988, Sumer  conducted experiments to show that the shedding vortex in the wake of a pipeline would increase the Shields parameter by 3–4 times, which would result in severe scouring. In 1991, Chiew  performed experiments to prove that the maximum scouring depth could be obtained when the pipeline was located on a flat bed and was scoured by a unidirectional water flow. Based on the test results, they provided a prediction formula for the maximum scouring depth. In 2003, Mastbergen  proposed a one-dimensional, steady-state numerical model of turbidity currents, which considered the negative pore pressures in the seabed. The calculated results of this model were basically consistent with the actual scouring of a submarine canyon. In 2007, Dey  presented a semitheoretical model for the computation of the maximum clear-water scour depth below underwater pipelines in uniform sediments under a steady flow, and the predicted scour depth in clear water satisfactorily agreed with the observed values. In 2008, Dey  conducted experiments on clear-water scour below underwater pipelines under a steady flow and obtained a variation pattern of the depth of the scouring hole. In 2008, Liang  used a two-dimensional numerical simulation to study the scouring process of a tube bundle under the action of currents and waves. They discovered that, compared with the scouring of a single tube, the scouring depth of the tube bundle was deeper, and the scouring time was longer. In 2012, Yang  found that placing rubber sheets under pipes can greatly accelerate their self-burial. The rubber sheets had the best performance when their length was about 1.5 times the size of the pipe. In 2020, Li  investigated the two-dimensional local scour beneath two submarine pipelines in tandem under wave-plus-current conditions via numerical simulation. They found that for conditions involving waves plus a low-strength current, the scour pattern beneath the two pipelines behaved like that in the pure-wave condition. Conversely, when the current had equal strength to the wave-induced flow, the scour pattern beneath the two pipelines resembled that in the pure-current condition. In 2020, Guan  studied and discussed the interactive coupling effects among a vibrating pipeline, flow field, and scour process through experiments, and the experimental data showed that the evolution of the scour hole had significant influences on the pipeline vibrations. In 2021, Liu  developed a two-dimensional finite element numerical model and researched the local scour around a vibrating pipeline. The numerical results showed that the maximum vibration amplitude of the pipeline could reach about 1.2 times diameter, and the maximum scour depth occurred on the wake side of the vibrating pipeline. In 2021, Huang  carried out two-dimensional numerical simulations to investigate the scour beneath a single pipeline and piggyback pipelines subjected to an oscillatory flow condition at a KC number of 11 and captured typical steady-streaming structures around the pipelines due to the oscillatory flow condition. In 2021, Cui  investigated the characteristics of the riverbed scour profile for a pipeline buried at different depths under the condition of riverbed sediments with different particle sizes. The results indicated that, in general, the equilibrium scour depth changed in a spoon shape with the gradual increase in the embedment ratio. In 2022, Li  used numerical simulation to study the influence of the burial depth of partially buried pipelines on the surrounding flow field, but they did not investigate the scour depth. In 2022, Zhu  performed experiments to prove that the scour hole propagation rate under a pipeline decreases with an increasing pipeline embedment ratio and rises with the KC number. In 2022, Najafzadeh  proposed equations for the prediction of the scouring propagation rate around pipelines due to currents based on a machine learning model, and the prediction results were consistent with the experimental data. In 2023, Ma  used the computational fluid dynamics coarse-grained discrete element method to simulate the scour process around a pipeline. The results showed that this method can effectively reduce the considerable need for computing resources and excessive computation time. In 2023, through numerical simulations, Hu  discovered that the water velocity and the pipeline diameter had a significant effect on the depth of scouring.
In the preceding works, the researchers investigated the mechanism of sediment scouring and the effect of various factors on the local scouring of submarine pipelines. However, submarine cables are buried beneath the seabed, while submarine pipelines are erected above the seabed. The difference in laying methods leads to a large discrepancy between their local scouring processes. Therefore, the conclusions of the above investigations are not applicable to the local scouring of submarine cables. Currently, there is no report on the research of the local scouring of partially exposed submarine cables.
In this paper, a three-dimensional computational fluid dynamics (CFD) finite element model, based on two-phase flow, is established using FLOW-3D. The local scouring process of semi-exposed submarine cables under steady-state ocean currents is studied, and the variation rules of the depth and the shape of the scouring holes, as well as the shear velocity with time, are obtained. By setting different critical Shields numbers of the sediment, different sediment densities, and different ocean current velocities, the change rule of the scouring holes’ development rate and the time required for the spanning of submarine cables are explored.
2. Sediment Scouring Model
In the sediment scouring model, the sediment is set as the dispersed particle, which is regarded as a kind of quasifluid. In this context, sediment scouring is considered as a two-phase flow process between the liquid phase and solid particle phase. The sediment in this process is further divided into two categories: one is suspended in the fluid, and the other is deposited on the bottom.When the local Shields number of sediment is greater than the critical Shields number, the deposited sediment will be transformed into the suspended sediment under the action of ocean currents. The calculation formulae of the local Shields numbers θ and the critical Shields numbers
𝑢�¯ is the mean velocity vector of the fluid and the sediment particle,
us is the velocity of the sediment particle,
fs is the volume fraction of the sediment particle, P is the pressure, F is the volumetric and viscous force, K is the drag force, and
ur is the relative velocity.
3. Numerical Setup and Modeling
In this paper, a three-dimensional submarine cable local scouring simulation model is established by FLOW-3D. Based on the numerical simulation, the process of the submarine cable, which gradually changes from semi-exposed to the spanning state under the steady-state ocean current, is studied. The geometric modeling, the mesh division, the physical field setup, and the grid independent test of CFD numerical model are as follows.
3.1. Geometric Modeling and Mesh Division
A three-dimensional (3D) numerical model of the local scouring of a semi-exposed submarine cable is established, which is shown in Figure 1. The dimensions of the model are marked in Figure 1. The inlet direction of the ocean current is defined as the upstream of the submarine cable (referred to as upstream), and the outlet direction of the ocean current is defined as the downstream of the submarine cable (referred to as downstream).
Figure 1. Three-dimensional finite element model of local scouring of semi-exposed submarine cable.
The submarine cable with a diameter of 0.2 m is positioned on sediment that is initially in a semi-exposed state. When the length of the span is short, the submarine cable will not show obvious deformation due to gravity or scouring from the ocean current. Therefore, the submarine cable surface is set as the fixed boundary. The model’s left boundary is set as the inlet, the right boundary is set as the outlet, the front and rear boundaries are set as symmetry, and the bottom boundary is set as the non-slip wall. Since the water depth above the submarine cable is more than 0.6 m in practice, the top boundary of the model is also set as symmetry. The sediment near the inlet and the outlet will be carried by ocean currents, which leads to the abnormal scouring terrain. At each end of the sediment, a baffle (thickness of 3 cm) is installed to ensure that the simulation results can reflect the real situation.
Due to the fact that the flow field around the semi-exposed submarine cable is not a simple two-dimensional symmetrical distribution, it should be solved by three-dimensional numerical simulation. Considering the accuracy and efficiency of the calculation, the size of mesh is set to 0.02 m. The total number of meshes after the dissection is 133,254.
3.2. Physical Field Setup
The CFD finite element model contains four physical field modules: sediment scouring module, gravity and non-inertial reference frame module, density evaluation module, and viscosity and turbulence module. In this paper, the renormalization group (RNG) k–ε turbulence model is used, which has high computational accuracy for turbulent vortices. Therefore, this turbulence model is suitable for calculating the sediment scouring process around the semi-exposed submarine cable . The key parameters of the numerical simulation are referring to the survey results of submarine sediments in the Korean Peninsula , as listed in Table 1.Table 1. Key parameters of numerical simulation.
3.3. Mesh Independent Test
In order to eliminate errors caused by the quantity of grids in the calculation process, two sizes of mesh are set on the validation model, and the scour profiles under different mesh sizes are compared. The validation model is shown in Figure 2, and the scouring terrain under different mesh size is given in Figure 3.
Figure 2. Validation model.
Figure 3. Scouring terrain under different mesh sizes.
It can be seen from Figure 3 that with the increase in the number of meshes, the scouring terrain of the verification model changes slightly, and the scouring depth is basically unchanged. Considering the accuracy of the numerical simulation and the calculation’s time cost, it is reasonable to consider setting the mesh size to 0.02 m.
4. Results and Analysis
4.1. Analysis of Local Scouring Process
Based on the CFD finite element numerical simulation, the local scouring process of the submarine cable under the steady-state ocean current is analyzed. The end time of the simulation is 9 h, the initial time step is 0.01 s, and the fluid velocity is 0.40 m/s. Simulation results are saved every minute. Figure 4 illustrates the scouring terrain around the semi-exposed submarine cable, which has been scoured by the steady-state current for 5 h.
Figure 4. Scouring terrain around semi-exposed submarine cable (scour for 5 h).
As can be seen from Figure 4, three scouring holes were separately formed in the upstream wake position and downstream of the semi-exposed submarine cable. The scouring holes are labeled according to their locations. The variation of the scouring terrain around the semi-exposed submarine cable over time is given in Figure 5. The red circle in the picture corresponds to the position of the submarine cable, and the red box in the legend marks the time when the submarine cable is spanning.
Figure 5. Variation of scouring terrain around semi-exposed submarine cable adapted to time.
From Figure 5, in the first hour of scouring, the upstream (−0.5 m to −0.1 m) and downstream (0.43 m to 1.5 m) scouring holes appeared. The upstream scouring hole was relatively flat with depth of 0.04 m. The depth of the downstream scouring hole increased with the increase in distance, and the maximum depth was 0.13 m. The scouring hole that developed at the wake position was very shallow, and its depth was only 0.007 m.
In the second hour of scouring, the upstream scouring hole’s depth remained nearly constant. The depth of the downstream scouring hole only increased by 0.002 m. The scouring hole at the wake position developed steadily, and its depth increased from 0.007 m to 0.014 m.
The upstream and downstream scouring holes did not continue to develop during the third to the sixth hour. Compared to the first two hours, the development of scouring holes at the wake position accelerated significantly, with an average growth rate of 0.028 m/h. The growth rate in the fifth hour of the scouring hole at the wake position was slightly faster than the other times. After 6 h of scouring, the sediment on the right side of the submarine cable had been hollowed out.
In the seventh and the eighth hour of scouring, the upstream scouring hole’s depth increased slightly, the downstream scouring hole still remained stable, and the depth of the scouring hole at wake position increased by 0.019 m. The sediment under the submarine cable was gradually eroded as well. By the end of the eighth hour, the lower right part of the submarine cable had been exposed to water as well.
At 8 h 21 min of the scouring, the submarine cable was completely spanned, and the scouring holes were connected to each other. Within the next 10 min, the development of the scouring holes sped up significantly, and the maximum depth of scouring holes increased greatly to 0.27 m.
In reference , researchers have studied the local scouring process of semi-buried pipelines in sandy riverbeds through experiments. The test results show that the scouring process can be divided into a start-up stage, micropore formation stage, extension stage, and equilibrium stage. In this paper, the first three stages are simulated, and the results are in good agreement with the experiment, which proves the accuracy of the present numerical model.
In this research, the velocity of ocean currents at the sediment surface is defined as the shear velocity, which plays an important role in the process of local scouring. Figure 6 provides visual data on how the shear velocity varies over time.
Figure 6. Shear velocity changes in the scouring process.
The semi-exposed submarine cable protrudes from the seabed, which makes the shear velocity of its surface much higher than other locations. After the submarine cable is spanned, the shear velocity of the scouring hole surface below it is taken. This is the reason for the sudden change of shear velocity at the submarine cable’s location in Figure 6.The shear velocity in the initial state of the upstream scouring hole is obviously greater than in subsequent times. After 1 h of scouring, the shear velocity in the upstream scouring hole rapidly decreased from 1.1 × 10
−2 m/s to 3.98 × 10
−3 m/s and remained stable until the end of the sixth hour. This phenomenon explains why the upstream scouring hole developed rapidly in the first hour but remained stable for the following 5 h.The shear velocity in the downstream scouring hole reduced at first and then increased; its initial value was 1.41 × 10
−2 m/s. It took approximately 5 h for the shear velocity to stabilize, and the stable shear velocity was 2.26 × 10
−3 m/s. Therefore, compared with the upstream scouring hole, the downstream scouring hole was deeper and required more time to reach stability.The initial shear velocity in the scouring hole at the wake position was only 7.1 × 10
−3 m/s, which almost does not change in the first hour. This leads to a very slow development of the scouring hole at the wake position in the early stages. The maximum shear velocity in this scouring hole gradually increased to 1.05 × 10
−2 m/s from the second to the fifth hour, and then decreased to 6.61 × 10
−3 m/s by the end of the eighth hour. This is why the scouring hole at the wake position grows fastest around the fifth hour. Consistent with the pattern of change in the scouring hole’s terrain, the location of the maximal shear velocity also shifted to the right with time.
The shear velocity of all three scouring holes rose dramatically in the last hour. Combined with the terrain in Figure 5, this can be attributed to the complete spanning of the submarine cable.
From Equations (3)–(5), one can see the movement of the sediment is related directly with the sediment’s critical Shields number, sediment density, and ocean current velocity. Based on the parameters in Table 1, the influence of the above parameters on the local scouring process of semi-exposed submarine cables will be discussed.
4.2. Influence Factors
4.2.1. Sediment’s Critical Shields Number
The sediment’s critical Shields number
θcr is set as 0.02, 0.03, 0.04, 0.05, 0.06, and 0.07, and the variations of scouring terrain over time under each
Figure 7. Influence of sediment’s critical Shields number
θcr on local scouring around semi-exposed submarine cable: (a
) θcr = 0.02; (b
) θcr = 0.03; (c
) θcr = 0.04; (d
) θcr = 0.05; (e
) θcr = 0.06; and (f
) θcr = 0.07.From Figure 7, one can see that a change in
θcr will affect the depth of the upstream scouring hole and the development speed of the scouring hole at the wake position, but it will have no significant impact on the expansion of the downstream scouring hole.Under conditions of different
θcr, the upstream scouring hole will reach a temporary plateau within 1 h, at which time the stable depth will be about 0.04 m. When
θcr ≤ 0.05, the upstream scouring hole will continue to expand after a few hours. The stable time is obviously affected by
θcr, which will gradually increase from 1 h to 11 h with the increase in
θcr. The terrain of the upstream scouring hole will gradually convert to deep on the left and to shallow on the right. Since the scouring hole at the wake position has not been stable, its state at the time of submarine cable spanning is studied emphatically. In the whole process of scouring, the scouring hole at the wake position continues to develop and does not reach a stable state. With the increase in
θcr, the development velocity of the scouring hole at the wake position will decrease considerably. Its average evolution velocity decreases from 3.88 cm/h to 1.62 cm/h, and its depth decreases from 21.9 cm to 18.8 cm. Under the condition of each
θcr, the downstream scouring hole will stabilize within 1 h, and the stable depth will be basically unchanged (all about 13.5 cm).As
θcr increases, so does the sediment’s ability to withstand shearing forces, which will cause it to become increasingly difficult to be eroded or carried away by ocean currents. This effect has been directly reflected in the depth of scouring holes (upstream and wake position). Due to the blocking effect of semi-exposed submarine cables, the wake is elongated, which is why the downstream scouring hole develops before the scouring hole at the wake position and quickly reaches a stable state. However, due to the high wake intensity, this process is not significantly affected by the change of
4.2.2. Sediment Density
The density of sediment
ρs is set as 1550 kg/m
3, 1600 kg/m
3, 1650 kg/m
3, 1700 kg/m
3, 1750 kg/m
3, and 1800 kg/m
3, and the variation of scouring terrain over time under each
ρs will also affect the depth of the upstream scouring hole and the development speed of the scouring hole at the wake position. In addition, it can even have an impact on the downstream scouring hole depth.Under different
ρs conditions, the upstream scouring hole will always reach a temporary stable state in 1 h, at which time the stable depth will be 0.04 m. When
ρs ≤ 1750 kg/m
3, the upstream scouring hole will continue to expand after a few hours. The stabilization time of upstream scouring hole is more clearly affected by
ρs, which will gradually increase from 3 h to 13 h with the increase in
ρs. The terrain of the upstream scouring hole will gradually change to deep on the left and to shallow on the right. Since the scouring hole at the wake position has not been stable, its state at the time of the submarine cable spanning is studied emphatically, too. In the whole process of scouring, the scouring hole at the wake position continues to develop and does not reach a stable state. When
ρs is large, the development rate of scouring hole obviously decreased with time. With the increase in
ρs, the development velocity of the scouring hole at the wake position reduces from 3.38 cm/h to 1.14 cm/h, and the depth of this scouring hole declines from 20 cm to 15 cm. As
ρs increases, the stabilization time of the downstream scouring hole increases from less than 1 h to about 2 h, but the stabilization depth of the downstream scouring hole remains essentially the same (all around 13.5 cm).As can be seen from Equation (1), the increase in
ρs will reduce the Shields number, thus weakening the shear action of the sediment by the ocean current, which explains the extension of the stability time of the upstream scouring hole. At the same time, with the increase in the depth of scouring hole at the wake position, its shear velocity will decreases. Therefore, under a larger
ρs value, the development speed of scouring hole at the wake position will decrease significantly with time. Possibly for the same reason,
ρs can affect the development rate of downstream scouring hole.
4.2.3. Ocean Current Velocity
The ocean current velocity v is set as 0.35 m/s, 0.40 m/s, 0.45 m/s, 0.50 m/s, 0.55 m/s, and 0.60 m/s. Figure 9 presents the variation in scouring terrain with time for each v.
Figure 9. Influence of ocean current velocity v on local scouring around semi-exposed submarine cable: (a) v = 0.35 m/s; (b) v = 0.40 m/s; (c) v = 0.45 m/s; (d) v = 0.50 m/s; (e) v = 0.55 m/s; and (f) v = 0.60 m/s.
Changes in v affect the depth of the upstream and downstream scouring holes, as well as the development velocity of the wake position and downstream scouring holes.
When v ≤ 0.45 m/s, the upstream scouring hole will reach a temporary stable state within 1 h, at which point the stable depth will be 0.04 m. The stabilization time of the upstream scouring hole is affected by v, which will gradually decrease from 15 h to 3 h with the increase in v. When v > 0.45 m/s, the upstream scouring hole is going to expand continuously. With the increase in v, its average development velocity increases from 6.68 cm/h to 8.66 cm/h, and its terrain changes to deep on the left and to shallow on the right. When the submarine cable is spanning, special attention should be paid to the depth of the scouring hole at the wake position. Throughout whole scouring process, the scouring hole at the wake position continues to develop and does not reach a stable state. With the increase in v, the depth of scouring hole at the wake position will increase from 14 cm to 20 cm, and the average development velocity will increase from 0.91 cm/h to 10.43 cm/h. As v increases, the time required to stabilize the downstream scouring hole is shortened from 1to 2 h to less than 1 h, but the stable depth is remains nearly constant at 13.5 cm.
An increase in v will increase the shear velocity. Therefore, when the depth of the scouring hole increases, the shear velocity in the hole will also increase, which can deepen both the upstream and downstream scouring hole. According to Equation (1), the Shields number is proportional to the square of the shear velocity. The increase in shear velocity significantly intensifies local scouring, which increases the development rate of scouring holes at the wake position and downstream.
4.3. Variation Rule of Spanning Time
In this paper, the spanning time is defined as the time taken for a semi-exposed submarine cable (initial state) to become a spanning submarine cable. Figure 10 illustrates the effect of the above parameters on the spanning time of the semi-exposed submarine cable.
Figure 10. Influence of different parameters on spanning time of the semi-exposed submarine cable: (a) Sediment critical Shields number; (b) Sediment density; and (c) Ocean current velocity.From Figure 10a, the spanning time monotonically increases with the increase in the critical Shields number of sediment. However, the slope of the curve decreases first and then increases, and the inflection point is at
θcr = 4.59 × 10
−2. The relationship between spanning time t and sediment’s critical Shields number
θcr can be formulated by a cubic function as shown in Equation (6):
𝑡=−2.98+6.76𝜃𝑐𝑟−1.45𝜃2𝑐𝑟+0.11𝜃3𝑐𝑟.�=−2.98+6.76���−1.45���2+0.11���3.(6)It can be seen from Figure 10b that with the increase in the sediment density, the spanning time increases monotonically and linearly. The relationship between the spanning time t and the sediment’s density
ρs can be formulated by the first order function as shown in Equation (7):
𝑡=−41.59+30.54𝜌𝑠.�=−41.59+30.54��.(7)Figure 10c shows that with the increase in the ocean current velocity, the spanning time decreases monotonically. The slope of the curve increases with the increase in the ocean current velocity, so it can be considered that there is saturation of the ocean current velocity effect. The relationship between the spanning time t and the ocean current velocity v can be formulated by the exponential function
In this paper, a three-dimensional CFD finite element numerical simulation model is established, which is used to research the local scouring process of the semi-exposed submarine cable under the steady-state ocean current. The relationship between shear velocity and scouring terrain is discussed, the influence of sediment critical Shields number, sediment density and ocean current velocity on the local scouring process is analyzed, and the variation rules of the spanning time of the semi-exposed submarine cable is given. The conclusions are as follows:
Under the steady-state ocean currents, scouring holes will be formed at the upstream, wake position and downstream of the semi-exposed submarine cable. The upstream and downstream scouring holes develop faster, which will reach a temporary stable state at about 1 h after the start of the scouring. The scouring hole at the wake position will continue to expand at a slower rate and eventually lead to the spanning of the submarine cable.
There is a close relationship between the distribution of shear velocity and the scouring terrain. As the local scouring process occurs, the location of the maximum shear velocity within the scouring hole shifts and causes the bottom of the hole to move as well.
When the sediment’s critical Shields number and density are significantly large and ocean current velocity is sufficiently low, the duration of the stable state of the upstream scouring hole will be prolonged, and the average development velocity of the scouring holes at the wake position and downstream will be reduced.
The relationship between the spanning time and the critical Shields number θcr can be formulated as a cubic function, in which the curve’s inflection point is θcr = 4.59 × 10−2. The relationship between spanning time and sediment density can be formulated as a linear function. The relationship between spanning time and ocean current velocity can be formulated by exponential function.
Based on the conclusions of this paper, even when it is too late to take measures or when the exposed position of the submarine cable cannot be located, the degree of burial depth development still can be predicted. This prediction is important for the operation and maintenance of the submarine cable. However, the study still leaves something to be desired. Only the local scouring process under the steady-state ocean current was studied, which is an extreme condition. In practice, exposed submarine cables are more likely to be scoured by reciprocating ocean currents. In the future, we will investigate the local scouring of submarine cables under the reciprocating ocean current.
Conceptualization, Y.H. and Q.L.; methodology, Q.L., P.Z. and H.T.; software, Q.L.; validation, Q.L., L.C. and W.T.; writing—original draft preparation, Q.L.; writing—review and editing, Y.H. and Q.L.; supervision, Y.H. and L.Y. All authors have read and agreed to the published version of the manuscript.
This research was funded by the [Smart Grid Joint Fund Key Project between National Natural Science Foundation of China and State Grid Corporation] grant number [U1766220].
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
The data supporting the reported results cannot be shared at this time, as they have been used in producing more publications on this research.
This work is supported by the Smart Grid Joint Fund Key Project of the National Natural Science Foundation of China and State Grid Corporation (Grant No. U1766220).
Conflicts of Interest
The authors declare no conflict of interest.
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액체-증기 상 변화 모델은 밀폐된 용기의 자체 가압 프로세스 시뮬레이션에 매우 큰 영향을 미칩니다. Hertz-Knudsen 관계, 에너지 점프 모델 및 그 파생물과 같은 널리 사용되는 액체-증기 상 변화 모델은 실온 유체를 기반으로 개발되었습니다. 액체-증기 전이를 통한 극저온 시뮬레이션에 널리 적용되었지만 각 모델의 성능은 극저온 조건에서 명시적으로 조사 및 비교되지 않았습니다. 본 연구에서는 171가지 일반적인 액체-증기 상 변화 모델을 통합한 통합 다상 솔버가 제안되었으며, 이를 통해 이러한 모델을 실험 데이터와 직접 비교할 수 있습니다. 증발 및 응축 모델의 예측 정확도와 계산 속도를 평가하기 위해 총 <>개의 자체 가압 시뮬레이션이 수행되었습니다. 압력 예측은 최적화 전략이 서로 다른 모델 계수에 크게 의존하는 것으로 나타났습니다. 에너지 점프 모델은 극저온 자체 가압 시뮬레이션에 적합하지 않은 것으로 나타났습니다. 평균 편차와 CPU 소비량에 따르면 Lee 모델과 Tanasawa 모델은 다른 모델보다 안정적이고 효율적인 것으로 입증되었습니다.
Liquid-vapor phase change models vitally influence the simulation of self-pressurization processes in closed containers. Popular liquid-vapor phase change models, such as the Hertz-Knudsen relation, energy jump model, and their derivations were developed based on room-temperature fluids. Although they had widely been applied in cryogenic simulations with liquid-vapor transitions, the performance of each model was not explicitly investigated and compared yet under cryogenic conditions. A unified multi-phase solver incorporating four typical liquid-vapor phase change models has been proposed in the present study, which enables direct comparison among those models against experimental data. A total number of 171 self-pressurization simulations were conducted to evaluate the evaporation and condensation models’ prediction accuracy and calculation speed. It was found that the pressure prediction highly depended on the model coefficients, whose optimization strategies differed from each other. The energy jump model was found inadequate for cryogenic self-pressurization simulations. According to the average deviation and CPU consumption, the Lee model and the Tanasawa model were proven to be more stable and more efficient than the others.
The liquid-vapor phase change of cryogenic fluids is widely involved in industrial applications, such as the hydrogen transport vehicles , shipborne liquid natural gas (LNG) containers  and on-orbit cryogenic propellant tanks . These applications require cryogenic fluids to be stored for weeks to months. Although high-performance insulation measures are adopted, heat inevitably enters the tank via radiation and conduction. The self-pressurization in the tank induced by the heat leakage eventually causes the venting loss of the cryogenic fluids and threatens the safety of the craft in long-term missions. To reduce the boil-off loss and extend the cryogenic storage duration, a more comprehensive understanding of the self-pressurization mechanism is needed.
Due to the difficulties and limitations in implementing cryogenic experiments, numerical modeling is a convenient and powerful way to study the self-pressurization process of cryogenic fluids. However, how the phase change models influence the mass and heat transfer under cryogenic conditions is still unsettled . As concluded by Persad and Ward , a seemingly slight variation in the liquid-vapor phase change models can lead to erroneous predictions.
Among the liquid-vapor phase change models, the kinetic theory gas (KTG) based models and the energy jump model are the most popular ones used in recent self-pressurization simulations . The KTG based models, also known as the Hertz-Knudsen relation models, were developed on the concept of the Maxwell-Boltzmann distribution of the gas molecular . The Hertz-Knudsen relation has evolved to several models, including the Schrage model , the Tanasawa model , the Lee model  and the statistical rate theory (SRT) , which will be described in Section 2.2. Since the Schrage model and the Lee model are embedded and configured as the default ones in the commercial CFD solvers Flow-3D® and Ansys Fluent® respectively, they have been widely used in self-pressurization simulations for liquid nitrogen ,  and liquid hydrogen , . The major drawback of the KTG models lies in the difficulty of selecting model coefficients, which were reported in a considerably wide range spanning three magnitudes even for the same working fluid , , , , , . Studies showed that the liquid level, pressure and mass transfer rate are directly influenced by the model coefficients , , , , . Wrong coefficients will lead to deviation or even divergence of the results. The energy jump model is also known as the thermal limitation model. It assumes that the evaporation and condensation at the liquid-vapor interface are induced only by heat conduction. The model is widely adopted in lumped node simulations due to its simplicity , , . To improve the accuracy of mass flux prediction, the energy jump model was modified by including the convection heat transfer , . However, the convection correlations are empirical and developed mainly for room-temperature fluids. Whether the correlation itself can be precisely applied in cryogenic simulations still needs further investigation.
Fig. 1 summarizes the cryogenic simulations involving the modeling of evaporation and condensation processes in recent years. The publication has been increasing rapidly. However, the characteristics of each evaporation and condensation model are not explicitly revealed when simulating self-pressurization. A comparative study of the phase change models is highly needed for cryogenic fluids for a better simulation of the self-pressurization processes.
In the present paper, a unified multi-phase solver incorporating four typical liquid-vapor phase change models, namely the Tanasawa model, the Lee model, the energy jump model, and the modified energy jump model has been proposed, which enables direct comparison among different models. The models are used to simulate the pressure and temperature evolutions in an experimental liquid nitrogen tank in normal gravity, which helps to evaluate themselves in the aspects of accuracy, calculation speed and robustness.
Governing equations for the self-pressurization tank
In the present study, both the fluid domain and the solid wall of the tank are modeled and discretized. The heat transportation at the solid boundaries is considered to be irrelevant with the nearby fluid velocity. Consequently, two sets of the solid and the fluid governing equations can be decoupled and solved separately. The pressures in the cryogenic container are usually from 100 kPa to 300 kPa. Under these conditions, the Knudsen number is far smaller than 0.01, and the fluids are
Self-pressurization results and phase change model comparison
This section compares the simulation results by different phase change models. Section 3.1 compares the pressure and temperature outputs from two KTG based models, namely the Lee model and the Tanasawa model. Section 3.2 presents the pressure predictions from the energy transport models, namely the energy jump model and the modified energy jump model, and compares pressure prediction performances between the KTG based models and the energy transport models. Section 3.3 evaluates the four models
A unified vapor-liquid-solid multi-phase numerical solver has been accomplished for the self pressurization simulation in cryogenic containers. Compared to the early fluid-only solver, the temperature prediction in the vicinity of the tank wall improves significantly. Four liquid-vapor phase change models were integrated into the solver, which enables fair and effective comparison for performances between each other. The pressure and temperature prediction accuracies, and the calculation speed
Authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Validity evaluation of popular liquid-vapor phase change models for cryogenic self-pressurization process”.
This project is supported by the National Natural Science Foundation of China (No. 51936006).
Progress in physical modelling and numerical simulation of phase transitions in cryogenic pool boiling and cavitation2023, Applied Mathematical ModellingCitation Excerpt :We will not delve into cryogenic evaporation phenomena, that predominantly drive the phase-change in well-insulated storage facilities, and thus are less relevant to spill scenarios. We instead refer the reader to the works of Zuo et al. [31–33]. If the static pressure at any location in a turbomachine drops below a fluid’s saturation pressure, localized evaporation events may occur, followed by rapid collapse of the vapour cavities in a process termed “cavitation” .Show abstract
Interfacial mass and energy transport during steady-state evaporation in liquid oxygen storage tanks2022, Applied EnergyCitation Excerpt :However, most of them simply used the Lee model for mass transport as did for regular fluids, and seldom focus on the evaporation itself related to the interfacial temperature distribution or were unable to validate their results against credible experimental data. A recent study proposed an optimized evaporation model for the cryogenic self-pressurization with a thorough comparison between popular phase change models , but still lacked of experimental data to validate the results. A series of experiments have been conducted on the heat and mass transport in a thin liquid layer in the vicinity of the liquid–vapor interface of room-temperature fluids [9–14].Show abstract
Thermal destratification of cryogenic liquid storage tanks by continuous bubbling of gases2022, International Journal of Hydrogen EnergyCitation Excerpt :It was concluded that a single injector with a larger diameter configuration showed a higher chance of developing a vertical temperature gradient. Zuo et al.  carried out a numerical analysis to investigate the temperature distribution within the LH2 storage tank with a self-pinning spraying bar. They used the SST turbulence model coupled with the 6-DOF model.Show abstract
A numerical study was performed on the embankment weir overflows with various surface roughness and tailwater submergence, to better understand the effects of weir roughness on discharge performances under the free and submerged conditions. The variation of flow regime is captured, from the free overflow, submerged hydraulic jump, to surface flow with increasing tailwater depth. A roughness factor is introduced to reflect the reduction in discharge caused by weir roughness. The roughness factor decreases with the roughness height, and it also depends on the tailwater depth, highlighting various relations of the roughness factor with the roughness height between different flow regimes, which is linear for the free overflow and submerged hydraulic jump while exponential for the surface flow. Accordingly, the effects of weir roughness on overflow discharge appear nonnegligible for the significant roughness height and the surface flow regime occurring under considerable tailwater submergence. The established empirical expressions of discharge coefficient and submergence and roughness factors make it possible to predict the discharge over embankment weirs considering both tailwater submergence and surface roughness.
자유 및 침수 조건에서 방류 성능에 대한 둑 거칠기의 영향을 더 잘 이해하기 위해 다양한 표면 거칠기와 테일워터 침수를 갖는 제방 둑 범람에 대한 수치 연구가 수행되었습니다.
자유 범람, 수중 수압 점프, 테일워터 깊이가 증가하는 표면 유동에 이르기까지 유동 체제의 변화가 캡처됩니다. 위어 거칠기로 인한 배출 감소를 반영하기 위해 거칠기 계수가 도입되었습니다.
조도 계수는 조도 높이와 함께 감소하고, 또한 테일워터 깊이에 따라 달라지며, 서로 다른 흐름 영역 사이의 조도 높이와 조도 계수의 다양한 관계를 강조합니다.
이는 자유 범람 및 수중 수압 점프에 대해 선형인 반면 표면에 대해 지수적입니다. 흐름. 따라서 월류 방류에 대한 웨어 조도의 영향은 상당한 조도 높이와 상당한 방수 침수 하에서 발생하는 표면 흐름 체제에 대해 무시할 수 없는 것으로 보입니다.
방류계수와 침수 및 조도계수의 확립된 실증식은 방류수 침수와 지표조도를 모두 고려한 제방보 위의 방류량을 예측할 수 있게 합니다.
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A series of numerical simulation were conducted to study the local scour around umbrella suction anchor foundation (USAF) under random waves. In this study, the validation was carried out firstly to verify the accuracy of the present model. Furthermore, the scour evolution and scour mechanism were analyzed respectively. In addition, two revised models were proposed to predict the equilibrium scour depth Seq around USAF. At last, a parametric study was carried out to study the effects of the Froude number Fr and Euler number Eu for the Seq. The results indicate that the present numerical model is accurate and reasonable for depicting the scour morphology under random waves. The revised Raaijmakers’s model shows good agreement with the simulating results of the present study when KCs,p < 8. The predicting results of the revised stochastic model are the most favorable for n = 10 when KCrms,a < 4. The higher Fr and Eu both lead to the more intensive horseshoe vortex and larger Seq.
The rapid expansion of cities tends to cause social and economic problems, such as environmental pollution and traffic jam. As a kind of clean energy, offshore wind power has developed rapidly in recent years. The foundation of offshore wind turbine (OWT) supports the upper tower, and suffers the cyclic loading induced by waves, tides and winds, which exerts a vital influence on the OWT system. The types of OWT foundation include the fixed and floating foundation, and the fixed foundation was used usually for nearshore wind turbine. After the construction of fixed foundation, the hydrodynamic field changes in the vicinity of the foundation, leading to the horseshoe vortex formation and streamline compression at the upside and sides of foundation respectively [1,2,3,4]. As a result, the neighboring soil would be carried away by the shear stress induced by vortex, and the scour hole would emerge in the vicinity of foundation. The scour holes increase the cantilever length, and weaken the lateral bearing capacity of foundation [5,6,7,8,9]. Moreover, the natural frequency of OWT system increases with the increase of cantilever length, causing the resonance occurs when the system natural frequency equals the wave or wind frequency [10,11,12]. Given that, an innovative foundation called umbrella suction anchor foundation (USAF) has been designed for nearshore wind power. The previous studies indicated the USAF was characterized by the favorable lateral bearing capacity with the low cost [6,13,14]. The close-up of USAF is shown in Figure 1, and it includes six parts: 1-interal buckets, 2-external skirt, 3-anchor ring, 4-anchor branch, 5-supporting rod, 6-telescopic hook. The detailed description and application method of USAF can be found in reference .
Figure 1. The close-up of umbrella suction anchor foundation (USAF).
Numerical and experimental investigations of scour around OWT foundation under steady currents and waves have been extensively studied by many researchers [1,2,15,16,17,18,19,20,21,22,23,24]. The seabed scour can be classified as two types according to Shields parameter θ, i.e., clear bed scour (θ < θcr) or live bed scour (θ > θcr). Due to the set of foundation, the adverse hydraulic pressure gradient exists at upstream foundation edges, resulting in the streamline separation between boundary layer flow and seabed. The separating boundary layer ascended at upstream anchor edges and developed into the horseshoe vortex. Then, the horseshoe vortex moved downstream gradually along the periphery of the anchor, and the vortex shed off continually at the lee-side of the anchor, i.e., wake vortex. The core of wake vortex is a negative pressure center, liking a vacuum cleaner. Hence, the soil particles were swirled into the negative pressure core and carried away by wake vortexes. At the same time, the onset of scour at rear side occurred. Finally, the wake vortex became downflow when the turbulence energy could not support the survival of wake vortex. According to Tavouktsoglou et al. , the scale of pile wall boundary layer is proportional to 1/ln(Rd) (Rd is pile Reynolds), which means the turbulence intensity induced by the flow-structure interaction would decrease with Rd increases, but the effects of Rd can be neglected only if the flow around the foundation is fully turbulent . According to previous studies [1,15,27,28,29,30,31,32], the scour development around pile foundation under waves was significantly influenced by Shields parameter θ and KC number simultaneously (calculated by Equation (1)). Sand ripples widely existed around pile under waves in the case of live bed scour, and the scour morphology is related with θ and KC. Compared with θ, KC has a greater influence on the scour morphology [21,27,28]. The influence mechanism of KC on the scour around the pile is reflected in two aspects: the horseshoe vortex at upstream and wake vortex shedding at downstream.
where, Uwm is the maximum velocity of the undisturbed wave-induced oscillatory flow at the sea bottom above the wave boundary layer, T is wave period, and D is pile diameter.
There are two prerequisites to satisfy the formation of horseshoe vortex at upstream pile edges: (1) the incoming flow boundary layer with sufficient thickness and (2) the magnitude of upstream adverse pressure gradient making the boundary layer separating [1,15,16,18,20]. The smaller KC results the lower adverse pressure gradient, and the boundary layer cannot separate, herein, there is almost no horseshoe vortex emerging at upside of pile. Sumer et al. [1,15] carried out several sets of wave flume experiments under regular and irregular waves respectively, and the experiment results show that there is no horseshoe vortex when KC is less than 6. While the scale and lifespan of horseshoe vortex increase evidently with the increase of KC when KC is larger than 6. Moreover, the wake vortex contributes to the scour at lee-side of pile. Similar with the case of horseshoe vortex, there is no wake vortex when KC is less than 6. The wake vortex is mainly responsible for scour around pile when KC is greater than 6 and less than O(100), while horseshoe vortex controls scour nearly when KC is greater than O(100).
Sumer et al.  found that the equilibrium scour depth was nil around pile when KC was less than 6 under regular waves for live bed scour, while the equilibrium scour depth increased with the increase of KC. Based on that, Sumer proposed an equilibrium scour depth predicting equation (Equation (2)). Carreiras et al.  revised Sumer’s equation with m = 0.06 for nonlinear waves. Different with the findings of Sumer et al.  and Carreiras et al. , Corvaro et al.  found the scour still occurred for KC ≈ 4, and proposed the revised equilibrium scour depth predicting equation (Equation (3)) for KC > 4.
Rudolph and Bos  conducted a series of wave flume experiments to investigate the scour depth around monopile under waves only, waves and currents combined respectively, indicting KC was one of key parameters in influencing equilibrium scour depth, and proposed the equilibrium scour depth predicting equation (Equation (4)) for low KC (1 < KC < 10). Through analyzing the extensive data from published literatures, Raaijmakers and Rudolph  developed the equilibrium scour depth predicting equation (Equation (5)) for low KC, which was suitable for waves only, waves and currents combined. Khalfin  carried out several sets of wave flume experiments to study scour development around monopile, and proposed the equilibrium scour depth predicting equation (Equation (6)) for low KC (0.1 < KC < 3.5). Different with above equations, the Khalfin’s equation considers the Shields parameter θ and KC number simultaneously in predicting equilibrium scour depth. The flow reversal occurred under through in one wave period, so sand particles would be carried away from lee-side of pile to upside, resulting in sand particles backfilled into the upstream scour hole [20,29]. Considering the backfilling effects, Zanke et al.  proposed the equilibrium scour depth predicting equation (Equation (7)) around pile by theoretical analysis, and the equation is suitable for the whole range of KC number under regular waves and currents combined.
where, n is the 1/n’th highest wave for random waves
For predicting equilibrium scour depth under irregular waves, i.e., random waves, Sumer and Fredsøe  found it’s suitable to take Equation (2) to predict equilibrium scour depth around pile under random waves with the root-mean-square (RMS) value of near-bed orbital velocity amplitude Um and peak wave period TP to calculate KC. Khalfin  recommended the RMS wave height Hrms and peak wave period TP were used to calculate KC for Equation (6). References [37,38,39,40] developed a series of stochastic theoretical models to predict equilibrium scour depth around pile under random waves, nonlinear random waves plus currents respectively. The stochastic approach thought the 1/n’th highest wave were responsible for scour in vicinity of pile under random waves, and the KC was calculated in Equation (8) with Um and mean zero-crossing wave period Tz. The results calculated by Equation (8) agree well with experimental values of Sumer and Fredsøe  if the 1/10′th highest wave was used. To author’s knowledge, the stochastic approach proposed by Myrhaug and Rue  is the only theoretical model to predict equilibrium scour depth around pile under random waves for the whole range of KC number in published documents. Other methods of predicting scour depth under random waves are mainly originated from the equation for regular waves-only, waves and currents combined, which are limited to the large KC number, such as KC > 6 for Equation (2) and KC > 4 for Equation (3) respectively. However, situations with relatively low KC number (KC < 4) often occur in reality, for example, monopile or suction anchor for OWT foundations in ocean environment. Moreover, local scour around OWT foundations under random waves has not yet been investigated fully. Therefore, further study are still needed in the aspect of scour around OWT foundations with low KC number under random waves. Given that, this study presents the scour sediment model around umbrella suction anchor foundation (USAF) under random waves. In this study, a comparison of equilibrium scour depth around USAF between this present numerical models and the previous theoretical models and experimental results was presented firstly. Then, this study gave a comprehensive analysis for the scour mechanisms around USAF. After that, two revised models were proposed according to the model of Raaijmakers and Rudolph  and the stochastic model developed by Myrhaug and Rue  respectively to predict the equilibrium scour depth. Finally, a parametric study was conducted to study the effects of the Froude number (Fr) and Euler number (Eu) to equilibrium scour depth respectively.
2. Numerical Method
2.1. Governing Equations of Flow
The following equations adopted in present model are already available in Flow 3D software. The authors used these theoretical equations to simulate scour in random waves without modification. The incompressible viscous fluid motion satisfies the Reynolds-averaged Navier-Stokes (RANS) equation, so the present numerical model solves RANS equations:
where, VF is the volume fraction; u, v, and w are the velocity components in x, y, z direction respectively with Cartesian coordinates; Ai is the area fraction; ρf is the fluid density, fi is the viscous fluid acceleration, Gi is the fluid body acceleration (i = x, y, z).
2.2. Turbulent Model
The turbulence closure is available by the turbulent model, such as one-equation, the one-equation k-ε model, the standard k-ε model, RNG k-ε turbulent model and large eddy simulation (LES) model. The LES model requires very fine mesh grid, so the computational time is large, which hinders the LES model application in engineering. The RNG k-ε model can reduce computational time greatly with high accuracy in the near-wall region. Furthermore, the RNG k-ε model computes the maximum turbulent mixing length dynamically in simulating sediment scour model. Therefore, the RNG k-ε model was adopted to study the scour around anchor under random waves [41,42].
where, kT is specific kinetic energy involved with turbulent velocity, GT is the turbulent energy generated by buoyancy; εT is the turbulent energy dissipating rate, PT is the turbulent energy, Diffε and DiffkT are diffusion terms associated with VF, Ai; CDIS1, CDIS2 and CDIS3 are dimensionless parameters, and CDIS1, CDIS3 have default values of 1.42, 0.2 respectively. CDIS2 can be obtained from PT and kT.
2.3. Sediment Scour Model
The sand particles may suffer four processes under waves, i.e., entrainment, bed load transport, suspended load transport, and deposition, so the sediment scour model should depict the above processes efficiently. In present numerical simulation, the sediment scour model includes the following aspects:
2.3.1. Entrainment and Deposition
The combination of entrainment and deposition determines the net scour rate of seabed in present sediment scour model. The entrainment lift velocity of sand particles was calculated as :
where, αi is the entrainment parameter, ns is the outward point perpendicular to the seabed, d* is the dimensionless diameter of sand particles, which was calculated by Equation (15), θcr is the critical Shields parameter, g is the gravity acceleration, di is the diameter of sand particles, ρi is the density of seabed species.
In Equation (14), the entrainment parameter αi confirms the rate at which sediment erodes when the given shear stress is larger than the critical shear stress, and the recommended value 0.018 was adopted according to the experimental data of Mastbergen and Von den Berg . ns is the outward pointing normal to the seabed interface, and ns = (0,0,1) according to the Cartesian coordinates used in present numerical model.
The shields parameter was obtained from the following equation:
where, Uf,m is the maximum value of the near-bed friction velocity; d50 is the median diameter of sand particles. The detailed calculation procedure of θ was available in Soulsby .
The critical shields parameter θcr was obtained from the Equation (17) 
The sand particles begin to deposit on seabed when the turbulence energy weaken and cann’t support the particles suspending. The setting velocity of the particles was calculated from the following equation :
where, qb,i is the bed load transport rate, which was obtained from Equation (20), δi is the bed load thickness, which was calculated by Equation (21), cb,i is the volume fraction of sand i in the multiple species, fb is the critical packing fraction of the seabed.
where, u¯�¯ is the velocity of mixed fluid-particles, which can be calculated by the RANS equation with turbulence model, cs,i is the suspended sand particles volume concentration, which was computed from Equation (24).
3. Model Setup
The seabed-USAF-wave three-dimensional scour numerical model was built using Flow-3D software. As shown in Figure 2, the model includes sandy seabed, USAF model, sea water, two baffles and porous media. The dimensions of USAF are shown in Table 1. The sandy bed (210 m in length, 30 m in width and 11 m in height) is made up of uniform fine sand with median diameter d50 = 0.041 cm. The USAF model includes upper steel tube with the length of 20 m, which was installed in the middle of seabed. The location of USAF is positioned at 140 m from the upstream inflow boundary and 70 m from the downstream outflow boundary. Two baffles were installed at two ends of seabed. In order to eliminate the wave reflection basically, the porous media was set at the outflow side on the seabed.
Figure 2. (a) The sketch of seabed-USAF-wave three-dimensional model; (b) boundary condation:Wv-wave boundary, S-symmetric boundary, O-outflow boundary; (c) USAF model.
Table 1. Numerical simulating cases.
3.1. Mesh Geometric Dimensions
In the simulation of the scour under the random waves, the model includes the umbrella suction anchor foundation, seabed and fluid. As shown in Figure 3, the model mesh includes global mesh grid and nested mesh grid, and the total number of grids is 1,812,000. The basic procedure for building mesh grid consists of two steps. Step 1: Divide the global mesh using regular hexahedron with size of 0.6 × 0.6. The global mesh area is cubic box, embracing the seabed and whole fluid volume, and the dimensions are 210 m in length, 30 m in width and 32 m in height. The details of determining the grid size can see the following mesh sensitivity section. Step 2: Set nested fine mesh grid in vicinity of the USAF with size of 0.3 × 0.3 so as to shorten the computation cost and improve the calculation accuracy. The encryption range is −15 m to 15 m in x direction, −15 m to 15 m in y direction and 0 m to 32 m in z direction, respectively. In order to accurately capture the free-surface dynamics, such as the fluid-air interface, the volume of fluid (VOF) method was adopted for tracking the free water surface. One specific algorithm called FAVORTM (Fractional Area/Volume Obstacle Representation) was used to define the fractional face areas and fractional volumes of the cells which are open to fluid flow.
Figure 3. The sketch of mesh grid.
3.2. Boundary Conditions
As shown in Figure 2, the initial fluid length is 210 m as long as seabed. A wave boundary was specified at the upstream offshore end. The details of determining the random wave spectrum can see the following wave parameters section. The outflow boundary was set at the downstream onshore end. The symmetry boundary was used at the top and two sides of the model. The symmetric boundaries were the better strategy to improve the computation efficiency and save the calculation cost . At the seabed bottom, the wall boundary was adopted, which means the u = v = w= 0. Besides, the upper steel tube of USAF was set as no-slip condition.
3.3. Wave Parameters
The random waves with JONSWAP wave spectrum were used for all simulations as realistic representation of offshore conditions. The unidirectional JONSWAP frequency spectrum was described as :
where, α is wave energy scale parameter, which is calculated by Equation (26), ω is frequency, ωp is wave spectrum peak frequency, which can be obtained from Equation (27). γ is wave spectrum peak enhancement factor, in this study γ = 3.3. σ is spectral width factor, σ equals 0.07 for ω ≤ ωp and 0.09 for ω > ωp respectively.
where, X is fetch length, U is average wind velocity at 10 m height from mean sea level.
In present numerical model, the input key parameters include X and U for wave boundary with JONSWAP wave spectrum. The objective wave height and period are available by different combinations of X and U. In this study, we designed 9 cases with different wave heights, periods and water depths for simulating scour around USAF under random waves (see Table 2). For random waves, the wave steepness ε and Ursell number Ur were acquired form Equations (28) and (29) respectively
where, Hs is significant wave height, Ta is average wave period, k is wave number, hw is water depth. The Shield parameter θ satisfies θ>θcr for all simulations in current study, indicating the live bed scour prevails.
Table 2. Numerical simulating cases.
3.4. Mesh Sensitivity
In this section, a mesh sensitivity analysis was conducted to investigate the influence of mesh grid size to results and make sure the calculation is mesh size independent and converged. Three mesh grid size were chosen: Mesh 1—global mesh grid size of 0.75 × 0.75, nested fine mesh grid size of 0.4 × 0.4, and total number of grids 1,724,000, Mesh 2—global mesh grid size of 0.6 × 0.6, nested fine mesh grid size of 0.3 × 0.3, and total number of grids 1,812,000, Mesh 3—global mesh grid size of 0.4 × 0.4, nested fine mesh grid size of 0.2 × 0.2, and total number of grids 1,932,000. The near-bed shear velocity U* is an important factor for influencing scour process [1,15], so U* at the position of (4,0,11.12) was evaluated under three mesh sizes. As the Figure 4 shown, the maximum error of shear velocity ∆U*1,2 is about 39.8% between the mesh 1 and mesh 2, and 4.8% between the mesh 2 and mesh 3. According to the mesh sensitivity criterion adopted by Pang et al. , it’s reasonable to think the results are mesh size independent and converged with mesh 2. Additionally, the present model was built according to prototype size, and the mesh size used in present model is larger than the mesh size adopted by Higueira et al.  and Corvaro et al. . If we choose the smallest cell size, it will take too much time. For example, the simulation with Mesh3 required about 260 h by using a computer with Intel Xeon Scalable Gold 4214 CPU @24 Cores, 2.2 GHz and 64.00 GB RAM. Therefore, in this case, considering calculation accuracy and computation efficiency, the mesh 2 was chosen for all the simulation in this study.
Figure 4. Comparison of near-bed shear velocity U* with different mesh grid size.
The nested mesh block was adopted for seabed in vicinity of the USAF, which was overlapped with the global mesh block. When two mesh blocks overlap each other, the governing equations are by default solved on the mesh block with smaller average cell size (i.e., higher grid resolution). It is should be noted that the Flow 3D software used the moving mesh captures the scour evolution and automatically adjusts the time step size to be as large as possible without exceeding any of the stability limits, affecting accuracy, or unduly increasing the effort required to enforce the continuity condition .
3.5. Model Validation
In order to verify the reliability of the present model, the results of present study were compared with the experimental data of Khosronejad et al. . The experiment was conducted in an open channel with a slender vertical pile under unidirectional currents. The comparison of scour development between the present results and the experimental results is shown in Figure 5. The Figure 5 reveals that the present results agree well with the experimental data of Khosronejad et al. . In the first stage, the scour depth increases rapidly. After that, the scour depth achieves a maximum value gradually. The equilibrium scour depth calculated by the present model is basically corresponding with the experimental results of Khosronejad et al. , although scour depth in the present model is slightly larger than the experimental results at initial stage.
Figure 5. Comparison of time evolution of scour between the present study and Khosronejad et al. , Petersen et al. .
Secondly, another comparison was further conducted between the results of present study and the experimental data of Petersen et al. . The experiment was carried out in a flume with a circular vertical pile in combined waves and current. Figure 4 shows a comparison of time evolution of scour depth between the simulating and the experimental results. As Figure 5 indicates, the scour depth in this study has good overall agreement with the experimental results proposed in Petersen et al. . The equilibrium scour depth calculated by the present model is 0.399 m, which equals to the experimental value basically. Overall, the above verifications prove the present model is accurate and capable in dealing with sediment scour under waves.
In addition, in order to calibrate and validate the present model for hydrodynamic parameters, the comparison of water surface elevation was carried out with laboratory experiments conducted by Stahlmann  for wave gauge No. 3. The Figure 6 depicts the surface wave profiles between experiments and numerical model results. The comparison indicates that there is a good agreement between the model results and experimental values, especially the locations of wave crest and trough. Comparison of the surface elevation instructs the present model has an acceptable relative error, and the model is a calibrated in terms of the hydrodynamic parameters.
Figure 6. Comparison of surface elevation between the present study and Stahlmann .
Finally, another comparison was conducted for equilibrium scour depth or maximum scour depth under random waves with the experimental data of Sumer and Fredsøe  and Schendel et al. . The Figure 7 shows the comparison between the numerical results and experimental data of Run01, Run05, Run21 and Run22 in Sumer and Fredsøe  and test A05 and A09 in Schendel et al. . As shown in Figure 7, the equilibrium scour depth or maximum scour depth distributed within the ±30 error lines basically, meaning the reliability and accuracy of present model for predicting equilibrium scour depth around foundation in random waves. However, compared with the experimental values, the present model overestimated the equilibrium scour depth generally. Given that, a calibration for scour depth was carried out by multiplying the mean reduced coefficient 0.85 in following section.
Figure 7. Comparison of equilibrium (or maximum) scour depth between the present study and Sumer and Fredsøe , Schendel et al. .
Through the various examination for hydrodynamic and morphology parameters, it can be concluded that the present model is a validated and calibrated model for scour under random waves. Thus, the present numerical model would be utilized for scour simulation around foundation under random waves.
4. Numerical Results and Discussions
4.1. Scour Evolution
Figure 8 displays the scour evolution for case 1–9. As shown in Figure 8a, the scour depth increased rapidly at the initial stage, and then slowed down at the transition stage, which attributes to the backfilling occurred in scour holes under live bed scour condition, resulting in the net scour decreasing. Finally, the scour reached the equilibrium state when the amount of sediment backfilling equaled to that of scouring in the scour holes, i.e., the net scour transport rate was nil. Sumer and Fredsøe  proposed the following formula for the scour development under waves
where Tc is time scale of scour process.
Figure 8. Time evolution of scour for case 1–9: (a) Case 1–5; (b) Case 6–9.
The computing time is 3600 s and the scour development curves in Figure 8 kept fluctuating, meaning it’s still not in equilibrium scour stage in these cases. According to Sumer and Fredsøe , the equilibrium scour depth can be acquired by fitting the data with Equation (30). From Figure 8, it can be seen that the scour evolution obtained from Equation (30) is consistent with the present study basically at initial stage, but the scour depth predicted by Equation (30) developed slightly faster than the simulating results and the Equation (30) overestimated the scour depth to some extent. Overall, the whole tendency of the results calculated by Equation (30) agrees well with the simulating results of the present study, which means the Equation (30) is applicable to depict the scour evolution around USAF under random waves.
4.2. Scour Mechanism under Random Waves
The scour morphology and scour evolution around USAF are similar under random waves in case 1~9. Taking case 7 as an example, the scour morphology is shown in Figure 9.
Figure 9. Scour morphology under different times for case 7.
From Figure 9, at the initial stage (t < 1200 s), the scour occurred at upstream foundation edges between neighboring anchor branches. The maximum scour depth appeared at the lee-side of the USAF. Correspondingly, the sediments deposited at the periphery of the USAF, and the location of the maximum accretion depth was positioned at an angle of about 45° symmetrically with respect to the wave propagating direction in the lee-side of the USAF. After that, when t > 2400 s, the location of the maximum scour depth shifted to the upside of the USAF at an angle of about 45° with respect to the wave propagating direction.
According to previous studies [1,15,16,19,30,31], the horseshoe vortex, streamline compression and wake vortex shedding were responsible for scour around foundation. The Figure 10 displays the distribution of flow velocity in vicinity of foundation, which reflects the evolving processes of horseshoe vertex.
Figure 10. Velocity profile around USAF: (a) Flow runup and down stream at upstream anchor edges; (b) Horseshoe vortex at upstream anchor edges; (c) Flow reversal during wave through stage at lee side.
As shown in Figure 10, the inflow tripped to the upstream edges of the USAF and it was blocked by the upper tube of USAF. Then, the downflow formed the horizontal axis clockwise vortex and rolled on the seabed bypassing the tube, that is, the horseshoe vortex (Figure 11). The Figure 12 displays the turbulence intensity around the tube on the seabed. From Figure 12, it can be seen that the turbulence intensity was high-intensity with respect to the region of horseshoe vortex. This phenomenon occurred because of drastic water flow momentum exchanging in the horseshoe vortex. As a result, it created the prominent shear stress on the seabed, causing the local scour at the upstream edges of USAF. Besides, the horseshoe vortex moved downstream gradually along the periphery of the tube and the wake vortex shed off continually at the lee-side of the USAF, i.e., wake vortex.
Figure 11. Sketch of scour mechanism around USAF under random waves.
Figure 12. Turbulence intensity: (a) Turbulence intensity of horseshoe vortex; (b) Turbulence intensity of wake vortex; (c) Turbulence intensity of accretion area.
The core of wake vortex is a negative pressure center, liking a vacuum cleaner [11,42]. Hence, the soil particles were swirled into the negative pressure core and carried away by wake vortex. At the same time, the onset of scour at rear side occurred. Finally, the wake vortex became downflow at the downside of USAF. As is shown in Figure 12, the turbulence intensity was low where the downflow occurred at lee-side, which means the turbulence energy may not be able to support the survival of wake vortex, leading to accretion happening. As mentioned in previous section, the formation of horseshoe vortex was dependent with adverse pressure gradient at upside of foundation. As shown in Figure 13, the evaluated range of pressure distribution is −15 m to 15 m in x direction. The t = 450 s and t = 1800 s indicate that the wave crest and trough arrived at the upside and lee-side of the foundation respectively, and the t = 350 s was neither the wave crest nor trough. The adverse gradient pressure reached the maximum value at t = 450 s corresponding to the wave crest phase. In this case, it’s helpful for the wave boundary separating fully from seabed, which leads to the formation of horseshoe vortex with high turbulence intensity. Therefore, the horseshoe vortex is responsible for the local scour between neighboring anchor branches at upside of USAF. What’s more, due to the combination of the horseshoe vortex and streamline compression, the maximum scour depth occurred at the upside of the USAF with an angle of about 45° corresponding to the wave propagating direction. This is consistent with the findings of Pang et al.  and Sumer et al. [1,15] in case of regular waves. At the wave trough phase (t = 1800 s), the pressure gradient became positive at upstream USAF edges, which hindered the separating of wave boundary from seabed. In the meantime, the flow reversal occurred (Figure 10) and the adverse gradient pressure appeared at downstream USAF edges, but the magnitude of adverse gradient pressure at lee-side was lower than the upstream gradient pressure under wave crest. In this way, the intensity of horseshoe vortex behind the USAF under wave trough was low, which explains the difference of scour depth at upstream and downstream, i.e., the scour asymmetry. In other words, the scour asymmetry at upside and downside of USAF was attributed to wave asymmetry for random waves, and the phenomenon became more evident for nonlinear waves . Briefly speaking, the vortex system at wave crest phase was mainly related to the scour process around USAF under random waves.
Figure 13. Pressure distribution around USAF.
4.3. Equilibrium Scour Depth
The KC number is a key parameter for horseshoe vortex emerging and evolving under waves. According to Equation (1), when pile diameter D is fixed, the KC depends on the maximum near-bed velocity Uwm and wave period T. For random waves, the Uwm can be denoted by the root-mean-square (RMS) value of near-bed velocity amplitude Uwm,rms or the significant value of near-bed velocity amplitude Uwm,s. The Uwm,rms and Uwm,s for all simulating cases of the present study are listed in Table 3 and Table 4. The T can be denoted by the mean up zero-crossing wave period Ta, peak wave period Tp, significant wave period Ts, the maximum wave period Tm, 1/10′th highest wave period Tn = 1/10 and 1/5′th highest wave period Tn = 1/5 for random waves, so the different combinations of Uwm and T will acquire different KC. The Table 3 and Table 4 list 12 types of KC, for example, the KCrms,s was calculated by Uwm,rms and Ts. Sumer and Fredsøe  conducted a series of wave flume experiments to investigate the scour depth around monopile under random waves, and found the equilibrium scour depth predicting equation (Equation (2)) for regular waves was applicable for random waves with KCrms,p. It should be noted that the Equation (2) is only suitable for KC > 6 under regular waves or KCrms,p > 6 under random waves.
Table 3.Uwm,rms and KC for case 1~9.
Table 4.Uwm,s and KC for case 1~9.
Raaijmakers and Rudolph  proposed the equilibrium scour depth predicting model (Equation (5)) around pile under waves, which is suitable for low KC. The format of Equation (5) is similar with the formula proposed by Breusers , which can predict the equilibrium scour depth around pile at different scour stages. In order to verify the applicability of Raaijmakers’s model for predicting the equilibrium scour depth around USAF under random waves, a validation of the equilibrium scour depth Seq between the present study and Raaijmakers’s equation was conducted. The position where the scour depth Seq was evaluated is the location of the maximum scour depth, and it was depicted in Figure 14. The Figure 15 displays the comparison of Seq with different KC between the present study and Raaijmakers’s model.
Figure 14. Sketch of the position where the Seq was evaluated.
Figure 15. Comparison of the equilibrium scour depth between the present model and the model of Raaijmakers and Rudolph : (a) KCrms,s, KCrms,a; (b) KCrms,p, KCrms,m; (c) KCrms,n = 1/10, KCrms,n = 1/5; (d) KCs,s, KCs,a; (e) KCs,p, KCs,m; (f) KCs,n = 1/10, KCs,n = 1/5.
As shown in Figure 15, there is an error in predicting Seq between the present study and Raaijmakers’s model, and Raaijmakers’s model underestimates the results generally. Although the error exists, the varying trend of Seq with KC obtained from Raaijmakers’s model is consistent with the present study basically. What’s more, the error is minimum and the Raaijmakers’s model is of relatively high accuracy for predicting scour around USAF under random waves by using KCs,p. Based on this, a further revision was made to eliminate the error as much as possible, i.e., add the deviation value ∆S/D in the Raaijmakers’s model. The revised equilibrium scour depth predicting equation based on Raaijmakers’s model can be written as
As the Figure 16 shown, through trial-calculation, when ∆S/D = 0.05, the results calculated by Equation (31) show good agreement with the simulating results of the present study. The maximum error is about 18.2% and the engineering requirements have been met basically. In order to further verify the accuracy of the revised model for large KC (KCs,p > 4) under random waves, a validation between the revised model and the previous experimental results . The experiment was conducted in a flume (50 m in length, 1.0 m in width and 1.3 m in height) with a slender vertical pile (D = 0.1 m) under random waves. The seabed is composed of 0.13 m deep layer of sand with d50 = 0.6 mm and the water depth is 0.5 m for all tests. The significant wave height is 0.12~0.21 m and the KCs,p is 5.52~11.38. The comparison between the predicting results by Equation (31) and the experimental results of Corvaro et al.  is shown in Figure 17. From Figure 17, the experimental data evenly distributes around the predicted results and the prediction accuracy is favorable when KCs,p < 8. However, the gap between the predicting results and experimental data becomes large and the Equation (31) overestimates the equilibrium scour depth to some extent when KCs,p > 8.
Figure 16. Comparison of Seq between the simulating results and the predicting values by Equation (31).
Figure 17. Comparison of Seq/D between the Experimental results of Corvaro et al.  and the predicting values by Equation (31).
In ocean environment, the waves are composed of a train of sinusoidal waves with different frequencies and amplitudes. The energy of constituent waves with very large and very small frequencies is relatively low, and the energy of waves is mainly concentrated in a certain range of moderate frequencies. Myrhaug and Rue  thought the 1/n’th highest wave was responsible for scour and proposed the stochastic model to predict the equilibrium scour depth around pile under random waves for full range of KC. Noteworthy is that the KC was denoted by KCrms,a in the stochastic model. To verify the application of the stochastic model for predicting scour depth around USAF, a validation between the simulating results of present study and predicting results by the stochastic model with n = 2,3,5,10,20,500 was carried out respectively.
As shown in Figure 18, compared with the simulating results, the stochastic model underestimates the equilibrium scour depth around USAF generally. Although the error exists, the varying trend of Seq with KCrms,a obtained from the stochastic model is consistent with the present study basically. What’s more, the gap between the predicting values by stochastic model and the simulating results decreases with the increase of n, but for large n, for example n = 500, the varying trend diverges between the predicting values and simulating results, meaning it’s not feasible only by increasing n in stochastic model to predict the equilibrium scour depth around USAF.
Figure 18. Comparison of Seq between the simulating results and the predicting values by Equation (8).
The Figure 19 lists the deviation value ∆Seq/D′ between the predicting values and simulating results with different KCrms,a and n. Then, fitted the relationship between the ∆S′and n under different KCrms,a, and the fitting curve can be written by Equation (32). The revised stochastic model (Equation (33)) can be acquired by adding ∆Seq/D′ to Equation (8).
The comparison between the predicting results by Equation (33) and the simulating results of present study is shown in Figure 20. According to the Figure 20, the varying trend of Seq with KCrms,a obtained from the stochastic model is consistent with the present study basically. Compared with predicting results by the stochastic model, the results calculated by Equation (33) is favorable. Moreover, comparison with simulating results indicates that the predicting results are the most favorable for n = 10, which is consistent with the findings of Myrhaug and Rue  for equilibrium scour depth predicting around slender pile in case of random waves.
Figure 20. Comparison of Seq between the simulating results and the predicting values by Equation (33).
In order to further verify the accuracy of the Equation (33) for large KC (KCrms,a > 4) under random waves, a validation was conducted between the Equation (33) and the previous experimental results of Sumer and Fredsøe  and Corvaro et al. . The details of experiments conducted by Corvaro et al.  were described in above section. Sumer and Fredsøe  investigated the local scour around pile under random waves. The experiments were conducted in a wave basin with a slender vertical pile (D = 0.032, 0.055 m). The seabed is composed of 0.14 m deep layer of sand with d50 = 0.2 mm and the water depth was maintained at 0.5 m. The JONSWAP wave spectrum was used and the KCrms,a was 5.29~16.95. The comparison between the predicting results by Equation (33) and the experimental results of Sumer and Fredsøe  and Corvaro et al.  are shown in Figure 21. From Figure 21, contrary to the case of low KCrms,a (KCrms,a < 4), the error between the predicting values and experimental results increases with decreasing of n for KCrms,a > 4. Therefore, the predicting results are the most favorable for n = 2 when KCrms,a > 4.
Figure 21. Comparison of Seq between the experimental results of Sumer and Fredsøe  and Corvaro et al.  and the predicting values by Equation (33).
Noteworthy is that the present model was built according to prototype size, so the errors between the numerical results and experimental data of References [16,21] may be attribute to the scale effects. In laboratory experiments on scouring process, it is typically impossible to ensure a rigorous similarity of all physical parameters between the model and prototype structure, leading to the scale effects in the laboratory experiments. To avoid a cohesive behaviour, the bed material was not scaled geometrically according to model scale. As a consequence, the relatively large-scaled sediments sizes may result in the overestimation of bed load transport and underestimation of suspended load transport compared with field conditions. What’s more, the disproportional scaled sediment presumably lead to the difference of bed roughness between the model and prototype, and thus large influences for wave boundary layer on the seabed and scour process. Besides, according to Corvaro et al.  and Schendel et al. , the pile Reynolds numbers and Froude numbers both affect the scour depth for the condition of non fully developed turbulent flow in laboratory experiments.
4.4. Parametric Study
4.4.1. Influence of Froude Number
As described above, the set of foundation leads to the adverse pressure gradient appearing at upstream, leading to the wave boundary layer separating from seabed, then horseshoe vortex formatting and the horseshoe vortex are mainly responsible for scour around foundation (see Figure 22). The Froude number Fr is the key parameter to influence the scale and intensity of horseshoe vortex. The Fr under waves can be calculated by the following formula 
where Uw is the mean water particle velocity during 1/4 cycle of wave oscillation, obtained from the following formula. Noteworthy is that the root-mean-square (RMS) value of near-bed velocity amplitude Uwm,rms is used for calculating Uwm.
Figure 22. Sketch of flow field at upstream USAF edges.
Tavouktsoglou et al.  proposed the following formula between Fr and the vertical location of the stagnation y
where e is constant.
The Figure 23 displays the relationship between Seq/D and Fr of the present study. In order to compare with the simulating results, the experimental data of Corvaro et al.  was also depicted in Figure 23. As shown in Figure 23, the equilibrium scour depth appears a logarithmic increase as Fr increases and approaches the mathematical asymptotic value, which is also consistent with the experimental results of Corvaro et al. . According to Figure 24, the adverse pressure gradient pressure at upstream USAF edges increases with the increase of Fr, which is benefit for the wave boundary layer separating from seabed, resulting in the high-intensity horseshoe vortex, hence, causing intensive scour around USAF. Based on the previous study of Tavouktsoglou et al.  for scour around pile under currents, the high Fr leads to the stagnation point is closer to the mean sea level for shallow water, causing the stronger downflow kinetic energy. As mentioned in previous section, the energy of downflow at upstream makes up the energy of the subsequent horseshoe vortex, so the stronger downflow kinetic energy results in the more intensive horseshoe vortex. Therefore, the higher Fr leads to the more intensive horseshoe vortex by influencing the position of stagnation point y presumably. Qi and Gao  carried out a series of flume tests to investigate the scour around pile under regular waves, and proposed the fitting formula between Seq/D and Fr as following
where A, B and C are constant.
Figure 23. The fitting curve between Seq/D and Fr.
Figure 24. Sketch of adverse pressure gradient at upstream USAF edges.
Took the Equation (37) to fit the simulating results with A = −0.002, B = 0.686 and C = −0.808, and the results are shown in Figure 23. From Figure 23, the simulating results evenly distribute around the Equation (37) and the varying trend of Seq/D and Fr in present study is consistent with Equation (37) basically, meaning the Equation (37) is applicable to express the relationship of Seq/D with Fr around USAF under random waves.
4.4.2. Influence of Euler Number
The Euler number Eu is the influencing factor for the hydrodynamic field around foundation. The Eu under waves can be calculated by the following formula. The Eu can be represented by the Equation (38) for uniform cylinders . The root-mean-square (RMS) value of near-bed velocity amplitude Um,rms is used for calculating Um.
where Um is depth-averaged flow velocity.
The Figure 25 displays the relationship between Seq/D and Eu of the present study. In order to compare with the simulating results, the experimental data of Sumer and Fredsøe  and Corvaro et al.  were also plotted in Figure 25. As shown in Figure 25, similar with the varying trend of Seq/D and Fr, the equilibrium scour depth appears a logarithmic increase as Eu increases and approaches the mathematical asymptotic value, which is also consistent with the experimental results of Sumer and Fredsøe  and Corvaro et al. . According to Figure 24, the adverse pressure gradient pressure at upstream USAF edges increases with the increasing of Eu, which is benefit for the wave boundary layer separating from seabed, inducing the high-intensity horseshoe vortex, hence, causing intensive scour around USAF.
Figure 25. The fitting curve between Seq/D and Eu.
Therefore, the variation of Fr and Eu reflect the magnitude of adverse pressure gradient pressure at upstream. Given that, the Equation (37) also was used to fit the simulating results with A = 8.875, B = 0.078 and C = −9.601, and the results are shown in Figure 25. From Figure 25, the simulating results evenly distribute around the Equation (37) and the varying trend of Seq/D and Eu in present study is consistent with Equation (37) basically, meaning the Equation (37) is also applicable to express the relationship of Seq/D with Eu around USAF under random waves. Additionally, according to the above description of Fr, it can be inferred that the higher Fr and Eu both lead to the more intensive horseshoe vortex by influencing the position of stagnation point y presumably.
A series of numerical models were established to investigate the local scour around umbrella suction anchor foundation (USAF) under random waves. The numerical model was validated for hydrodynamic and morphology parameters by comparing with the experimental data of Khosronejad et al. , Petersen et al. , Sumer and Fredsøe  and Schendel et al. . Based on the simulating results, the scour evolution and scour mechanisms around USAF under random waves were analyzed respectively. Two revised models were proposed according to the model of Raaijmakers and Rudolph  and the stochastic model developed by Myrhaug and Rue  to predict the equilibrium scour depth around USAF under random waves. Finally, a parametric study was carried out with the present model to study the effects of the Froude number Fr and Euler number Eu to the equilibrium scour depth around USAF under random waves. The main conclusions can be described as follows.(1)
The packed sediment scour model and the RNG k−ε turbulence model were used to simulate the sand particles transport processes and the flow field around UASF respectively. The scour evolution obtained by the present model agrees well with the experimental results of Khosronejad et al. , Petersen et al. , Sumer and Fredsøe  and Schendel et al. , which indicates that the present model is accurate and reasonable for depicting the scour morphology around UASF under random waves.(2)
The vortex system at wave crest phase is mainly related to the scour process around USAF under random waves. The maximum scour depth appeared at the lee-side of the USAF at the initial stage (t < 1200 s). Subsequently, when t > 2400 s, the location of the maximum scour depth shifted to the upside of the USAF at an angle of about 45° with respect to the wave propagating direction.(3)
The error is negligible and the Raaijmakers’s model is of relatively high accuracy for predicting scour around USAF under random waves when KC is calculated by KCs,p. Given that, a further revision model (Equation (31)) was proposed according to Raaijmakers’s model to predict the equilibrium scour depth around USAF under random waves and it shows good agreement with the simulating results of the present study when KCs,p < 8.(4)
Another further revision model (Equation (33)) was proposed according to the stochastic model established by Myrhaug and Rue  to predict the equilibrium scour depth around USAF under random waves, and the predicting results are the most favorable for n = 10 when KCrms,a < 4. However, contrary to the case of low KCrms,a, the predicting results are the most favorable for n = 2 when KCrms,a > 4 by the comparison with experimental results of Sumer and Fredsøe  and Corvaro et al. .(5)
The same formula (Equation (37)) is applicable to express the relationship of Seq/D with Eu or Fr, and it can be inferred that the higher Fr and Eu both lead to the more intensive horseshoe vortex and larger Seq.
Conceptualization, H.L. (Hongjun Liu); Data curation, R.H. and P.Y.; Formal analysis, X.W. and H.L. (Hao Leng); Funding acquisition, X.W.; Writing—original draft, R.H. and P.Y.; Writing—review & editing, X.W. and H.L. (Hao Leng); The final manuscript has been approved by all the authors. All authors have read and agreed to the published version of the manuscript.
This research was funded by the Fundamental Research Funds for the Central Universities (grant number 202061027) and the National Natural Science Foundation of China (grant number 41572247).
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
The data presented in this study are available on request from the corresponding author.
Conflicts of Interest
The authors declare no conflict of interest.
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Erick Mattos-Villarroel a, Jorge Flores-Velázquez b, Waldo Ojeda-Bustamante c, Carlos Díaz-Delgado d, Humberto Salinas-Tapia dShow moreAdd to MendeleyShareCite
aMexican Institute of Water Technology, Mexico bPostgraduate College, Hydrosciences, Carr. Mex-Tex Km 36.5, Texcoco, Mexico State, 56230, Mexico cAgricultural Engineering Graduate Program, University of Chapingo, Mexicod Inter-American Institute of Water Science and Technology, Mexico
•Optimizing the geometric design of weirs can improve hydraulic performance.
•Labyrinth type weirs allow the discharge capacity to be increased compared to linear weirs.
•Hydraulic heads with ratio HT/P > 0.5 generated sub-atmospheric pressures on the side walls of the weir.
•Numerical simulation it is a strong tool to analyze and get optimized the weir function.
Labyrinth type weirs are structures that, due to their geometry, allow the discharge capacity to be increased compared to linear weirs. They are a favorable option for dam rehabilitation and upstream level control. There are various geometries of labyrinth type weirs such as trapezoidal, triangular or piano key as well as different types of crest profiles. Geometric changes are directly related to hydraulic efficiency. The objective of this work was to analyze the hydraulic performance of a labyrinth type weir, by simulating several geometries of the apex and of the crest using Computational Fluid Dynamics (CFD). For model validation, experimental studies reported in the literature were used. Tests were carried out with trapezoidal and circular apexes and four types of crest profiles: sharp-crest, half-round, quarter-round and Waterways Experiment Station (WES). The results revealed a determination coefficient of R2 = 0.984 between experimental and simulated data with CFD, which provides statistical agreement. Simulations showed that circular-apex weirs are more efficient than those with trapezoidal apex, because they have a higher discharge coefficient (4.7% higher). Of the four types of crest profiles analyzed, the half-round and the WES crest profiles had similar discharge coefficients and were generally greater than those of the sharp-crest and the quarter-round (5.26% y 8.5% higher) profiles. Nevertheless, to facilitate a practical construction process, it is recommended to use a half-round profile. For hydraulic heads with HT/P > 0.5 ratio, all profiles generated sub-atmospheric pressures on the side walls of the weir. However, when HT/P ≈ 0.8 ratio the half-round crest generated a higher negative pressure (−1500 Pa), while the sharp-crest profile managed to increase the pressure by 76% (−350 Pa), but with a greater area of negative pressure. On the other hand, the WES profile reduced the negative-pressure area by 50%.
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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
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.
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Fatemehsadat Mirshafiee1, Emad Shahbazi 2, Mohadeseh Safi 3, Rituraj Rituraj 4,* 1Department of Electrical and Computer Engineering, K.N. Toosi University of Technology, Tehran 1999143344 , Iran 2Department of Mechatronic, Amirkabir University of Technology, Tehran 158754413, Iran 3Department of Mechatronic, Electrical and Computer Engineering, University of Tehran, Tehran 1416634793, Iran 4 Faculty of Informatics, Obuda University, 1023, Budapest, Hungary
본 연구는 지속가능한 에너지 변환기의 전력 및 수소 발생 모델링을 위한 데이터 기반 방법론을 제안합니다. 파고와 풍속을 달리하여 파고와 수소생산을 예측합니다.
또한 이 연구는 파도에서 수소를 추출할 수 있는 가능성을 강조하고 장려합니다. FLOW-3D 소프트웨어 시뮬레이션에서 추출한 데이터와 해양 특수 테스트의 실험 데이터를 사용하여 두 가지 데이터 기반 학습 방법의 비교 분석을 수행합니다.
결과는 수소 생산의 양은 생성된 전력의 양에 비례한다는 것을 보여줍니다. 제안된 재생 에너지 변환기의 신뢰성은 지속 가능한 스마트 그리드 애플리케이션으로 추가로 논의됩니다.
This study proposes a data-driven methodology for modeling power and hydrogen generation of a sustainable energy converter. The wave and hydrogen production at different wave heights and wind speeds are predicted. Furthermore, this research emphasizes and encourages the possibility of extracting hydrogen from ocean waves. By using the extracted data from FLOW-3D software simulation and the experimental data from the special test in the ocean, the comparison analysis of two data-driven learning methods is conducted. The results show that the amount of hydrogen production is proportional to the amount of generated electrical power. The reliability of the proposed renewable energy converter is further discussed as a sustainable smart grid application.
Cavity, Combustion efficiency, hydrogen fuel, Computational Fluent and Gambit.
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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. 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 . 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◦
 M. Michaelis, Kapillarinduzierte Schwingungen freier Fl¨ussigkeitsoberfl¨achen, Dissertation Universit¨at Bremen, Fortschritt-Berichte Nr. 454 (VDI Verlag, D¨usseldorf, 2003).
결합된 Bi-level 메타휴리스틱 접근법을 사용한 해양 재생 에너지 변환기의 설계 최적화
Erfan Amini a1, Mahdieh Nasiri b1, Navid Salami Pargoo a, Zahra Mozhgani c, Danial Golbaz d, Mehrdad Baniesmaeil e, Meysam Majidi Nezhad f, Mehdi Neshat gj, Davide Astiaso Garcia h, Georgios Sylaios i
In recent years, there has been an increasing interest in renewable energies in view of the fact that fossil fuels are the leading cause of catastrophic environmental consequences. Ocean wave energy is a renewable energy source that is particularly prevalent in coastal areas. Since many countries have tremendous potential to extract this type of energy, a number of researchers have sought to determine certain effective factors on wave converters’ performance, with a primary emphasis on ambient factors. In this study, we used metaheuristic optimization methods to investigate the effects of geometric factors on the performance of an Oscillating Surge Wave Energy Converter (OSWEC), in addition to the effects of hydrodynamic parameters. To do so, we used CATIA software to model different geometries which were then inserted into a numerical model developed in Flow3D software. A Ribed-surface design of the converter’s flap is also introduced in this study to maximize wave-converter interaction. Besides, a Bi-level Hill Climbing Multi-Verse Optimization (HCMVO) method was also developed for this application. The results showed that the converter performs better with greater wave heights, flap freeboard heights, and shorter wave periods. Additionally, the added ribs led to more wave-converter interaction and better performance, while the distance between the flap and flume bed negatively impacted the performance. Finally, tracking the changes in the five-dimensional objective function revealed the optimum value for each parameter in all scenarios. This is achieved by the newly developed optimization algorithm, which is much faster than other existing cutting-edge metaheuristic approaches.
Wave Energy Converter
The increase in energy demand, the limitations of fossil fuels, as well as environmental crises, such as air pollution and global warming, are the leading causes of calling more attention to harvesting renewable energy recently , , . While still in its infancy, ocean wave energy has neither reached commercial maturity nor technological convergence. In recent decades, remarkable progress has been made in the marine energy domain, which is still in the early stage of development, to improve the technology performance level (TPL) , and technology readiness level (TRL) of wave energy converters (WECs). This has been achieved using novel modeling techniques , , , , , , , ,  to gain the following advantages : (i) As a source of sustainable energy, it contributes to the mix of energy resources that leads to greater diversity and attractiveness for coastal cities and suppliers.  (ii) Since wave energy can be exploited offshore and does not require any land, in-land site selection would be less expensive and undesirable visual effects would be reduced.  (iii) When the best layout and location of offshore site are taken into account, permanent generation of energy will be feasible (as opposed to using solar energy, for example, which is time-dependent) .
In general, the energy conversion process can be divided into three stages in a WEC device, including primary, secondary, and tertiary stages , . In the first stage of energy conversion, which is the subject of this study, the wave power is converted to mechanical power by wave-structure interaction (WSI) between ocean waves and structures. Moreover, the mechanical power is transferred into electricity in the second stage, in which mechanical structures are coupled with power take-off systems (PTO). At this stage, optimal control strategies are useful to tune the system dynamics to maximize power output , , . Furthermore, the tertiary energy conversion stage revolves around transferring the non-standard AC power into direct current (DC) power for energy storage or standard AC power for grid integration , . We discuss only the first stage regardless of the secondary and tertiary stages. While Page 1 of 16 WECs include several categories and technologies such as terminators, point absorbers, and attenuators , , we focus on oscillating surge wave energy converters (OSWECs) in this paper due to its high capacity for industrialization .
Over the past two decades, a number of studies have been conducted to understand how OSWECs’ structures and interactions between ocean waves and flaps affect converters performance. Henry et al.’s experiment on oscillating surge wave energy converters is considered as one of the most influential pieces of research , which demonstrated how the performance of oscillating surge wave energy converters (OSWECs) is affected by seven different factors, including wave period, wave power, flap’s relative density, water depth, free-board of the flap, the gap between the tubes, gap underneath the flap, and flap width. These parameters were assessed in their two models in order to estimate the absorbed energy from incoming waves , . In addition, Folly et al. investigated the impact of water depth on the OSWECs performance analytically, numerically, and experimentally. According to this and further similar studies, the average annual incident wave power is significantly reduced by water depth. Based on the experimental results, both the surge wave force and the power capture of OSWECs increase in shallow water , . Following this, Sarkar et al. found that under such circumstances, the device that is located near the coast performs much better than those in the open ocean . On the other hand, other studies are showing that the size of the converter, including height and width, is relatively independent of the location (within similar depth) . Subsequently, Schmitt et al. studied OSWECs numerically and experimentally. In fact, for the simulation of OSWEC, OpenFOAM was used to test the applicability of Reynolds-averaged Navier-Stokes (RANS) solvers. Then, the experimental model reproduced the numerical results with satisfying accuracy . In another influential study, Wang et al. numerically assessed the effect of OSWEC’s width on their performance. According to their findings, as converter width increases, its efficiency decreases in short wave periods while increases in long wave periods . One of the main challenges in the analysis of the OSWEC is the coupled effect of hydrodynamic and geometric variables. As a result, numerous cutting-edge geometry studies have been performed in recent years in order to find the optimal structure that maximizes power output and minimizes costs. Garcia et al. reviewed hull geometry optimization studies in the literature in . In addition, Guo and Ringwood surveyed geometric optimization methods to improve the hydrodynamic performance of OSWECs at the primary stage . Besides, they classified the hull geometry of OSWECs based on Figure 1. Subsequently, Whittaker et al. proposed a different design of OSWEC called Oyster2. There have been three examples of different geometries of oysters with different water depths. Based on its water depth, they determined the width and height of the converter. They also found that in the constant wave period the less the converter’s width, the less power captures the converter has . Afterward, O’Boyle et al. investigated a type of OSWEC called Oyster 800. They compared the experimental and numerical models with the prototype model. In order to precisely reproduce the shape, mass distribution, and buoyancy properties of the prototype, a 40th-scale experimental model has been designed. Overall, all the models were fairly accurate according to the results .
Inclusive analysis of recent research avenues in the area of flap geometry has revealed that the interaction-based designs of such converters are emerging as a novel approach. An initiative workflow is designed in the current study to maximizing the wave energy extrication by such systems. To begin with, a sensitivity analysis plays its role of determining the best hydrodynamic values for installing the converter’s flap. Then, all flap dimensions and characteristics come into play to finalize the primary model. Following, interactive designs is proposed to increase the influence of incident waves on the body by adding ribs on both sides of the flap as a novel design. Finally, a new bi-level metaheuristic method is proposed to consider the effects of simultaneous changes in ribs properties and other design parameters. We hope this novel approach will be utilized to make big-scale projects less costly and justifiable. The efficiency of the method is also compared with four well known metaheuristic algorithms and out weight them for this application.
This paper is organized as follows. First, the research methodology is introduced by providing details about the numerical model implementation. To that end, we first introduced the primary model’s geometry and software details. That primary model is later verified with a benchmark study with regard to the flap angle of rotation and water surface elevation. Then, governing equations and performance criteria are presented. In the third part of the paper, we discuss the model’s sensitivity to lower and upper parts width (we proposed a two cross-sectional design for the flap), bottom elevation, and freeboard. Finally, the novel optimization approach is introduced in the final part and compared with four recent metaheuristic algorithms.
2. Numerical Methods
In this section, after a brief introduction of the numerical software, Flow3D, boundary conditions are defined. Afterwards, the numerical model implementation, along with primary model properties are described. Finally, governing equations, as part of numerical process, are discussed.
2.1. Model Setup
FLOW-3D is a powerful and comprehensive CFD simulation platform for studying fluid dynamics. This software has several modules to solve many complex engineering problems. In addition, modeling complex flows is simple and effective using FLOW-3D’s robust meshing capabilities . Interaction between fluid and moving objects might alter the computational range. Dynamic meshes are used in our modeling to take these changes into account. At each time step, the computational node positions change in order to adapt the meshing area to the moving object. In addition, to choose mesh dimensions, some factors are taken into account such as computational accuracy, computational time, and stability. The final grid size is selected based on the detailed procedure provided in . To that end, we performed grid-independence testing on a CFD model using three different mesh grid sizes of 0.01, 0.015, and 0.02 meters. The problem geometry and boundary conditions were defined the same, and simulations were run on all three grids under the same conditions. The predicted values of the relevant variable, such as velocity, was compared between the grids. The convergence behavior of the numerical solution was analyzed by calculating the relative L2 norm error between two consecutive grids. Based on the results obtained, it was found that the grid size of 0.02 meters showed the least error, indicating that it provided the most accurate and reliable solution among the three grids. Therefore, the grid size of 0.02 meters was selected as the optimal spatial resolution for the mesh grid.
In this work, the flume dimensions are 10 meters long, 0.1 meters wide, and 2.2 meters high, which are shown in figure2. In addition, input waves with linear characteristics have a height of 0.1 meters and a period of 1.4 seconds. Among the linear wave methods included in this software, RNGk-ε and k- ε are appropriate for turbulence model. The research of Lopez et al. shows that RNGk- ε provides the most accurate simulation of turbulence in OSWECs . We use CATIA software to create the flap primary model and other innovative designs for this project. The flap measures 0.1 m x 0.65 m x 0.360 m in x, y and z directions, respectively. In Figure 3, the primary model of flap and its dimensions are shown. In this simulation, five boundaries have been defined, including 1. Inlet, 2. Outlet, 3. Converter flap, 4. Bed flume, and 5. Water surface, which are shown in figure 2. Besides, to avoid wave reflection in inlet and outlet zones, Flow3D is capable of defining some areas as damping zones, the length of which has to be one to one and a half times the wavelength. Therefore, in the model, this length is considered equal to 2 meters. Furthermore, there is no slip in all the boundaries. In other words, at every single time step, the fluid velocity is zero on the bed flume, while it is equal to the flap velocity on the converter flap. According to the wave theory defined in the software, at the inlet boundary, the water velocity is called from the wave speed to be fed into the model.
In the current study, we utilize the Schmitt experimental model as a benchmark for verification, which was developed at the Queen’s University of Belfast. The experiments were conducted on the flap of the converter, its rotation, and its interaction with the water surface. Thus, the details of the experiments are presented below based up on the experimental setup’s description . In the experiment, the laboratory flume has a length of 20m and a width of 4.58m. Besides, in order to avoid incident wave reflection, a wave absorption source is devised at the end of the left flume. The flume bed, also, includes two parts with different slops. The flap position and dimensions of the flume can be seen in Figure4. In addition, a wave-maker with 6 paddles is installed at one end. At the opposite end, there is a beach with wire meshes. Additionally, there are 6 indicators to extract the water level elevation. In the flap model, there are three components: the fixed support structure, the hinge, and the flap. The flap measures 0.1m x 0.65m x 0.341m in x, y and z directions, respectively. In Figure5, the details are given . The support structure consists of a 15 mm thick stainless steel base plate measuring 1m by 1.4m, which is screwed onto the bottom of the tank. The hinge is supported by three bearing blocks. There is a foam centerpiece on the front and back of the flap which is sandwiched between two PVC plates. Enabling changes of the flap, three metal fittings link the flap to the hinge. Moreover, in this experiment, the selected wave is generated based on sea wave data at scale 1:40. The wave height and the wave period are equal to 0.038 (m) and 2.0625 (s), respectively, which are tantamount to a wave with a period of 13 (s) and a height of 1.5 (m).
Two distinct graphs illustrate the numerical and experi-mental study results. Figure6 and Figure7 are denoting the angle of rotation of flap and surface elevation in computational and experimental models, respectively. The two figures roughly represent that the numerical and experimental models are a good match. However, for the purpose of verifying the match, we calculated the correlation coefficient (C) and root mean square error (RMSE). According to Figure6, correlation coefficient and RMSE are 0.998 and 0.003, respectively, and in Figure7 correlation coefficient and RMSE are respectively 0.999 and 0.001. Accordingly, there is a good match between the numerical and empirical models. It is worth mentioning that the small differences between the numerical and experimental outputs may be due to the error of the measuring devices and the calibration of the data collection devices.
Including continuity equation and momentum conserva- tion for incompressible fluid are given as , :(1)
where P represents the pressure, g denotes gravitational acceleration, u represents fluid velocity, and Di is damping coefficient. Likewise, the model uses the same equation. to calculate the fluid velocity in other directions as well. Considering the turbulence, we use the two-equation model of RNGK- ε. These equations are:
(3)��t(��)+����(����)=����[�eff�������]+��-��and(4)���(��)+����(����)=����[�eff�������]+�1�∗����-��2��2�Where �2� and �1� are constants. In addition, �� and �� represent the turbulent Prandtl number of � and k, respectively.
�� also denote the production of turbulent kinetic energy of k under the effect of velocity gradient, which is calculated as follows:(5)��=�eff[�����+�����]�����(6)�eff=�+��(7)�eff=�+��where � is molecular viscosity,�� represents turbulence viscosity, k denotes kinetic energy, and ∊∊ is energy dissipation rate. The values of constant coefficients in the two-equation RNGK ∊-∊ model is as shown in the Table 1.Table 2.
Table 1. Constant coefficients in RNGK-∊ model
Table 2. Flap properties
Joint height (m)
Height of the center of mass (m)
It is worth mentioning that the volume of fluid method is used to separate water and air phases in this software . Below is the equation of this method .(8)����+����(���)=0where α and 1 − α are portion of water phase and air phase, respectively. As a weighting factor, each fluid phase portion is used to determine the mixture properties. Finally, using the following equations, we calculate the efficiency of converters , , :(9)�=14|�|2�+�2+(�+�a)2(�n2-�2)2where �� represents natural frequency, I denotes the inertia of OSWEC, Ia is the added inertia, F is the complex wave force, and B denotes the hydrodynamic damping coefficient. Afterward, the capture factor of the converter is calculated by :(10)��=�1/2��2����gw where �� represents the capture factor, which is the total efficiency of device per unit length of the wave crest at each time step , �� represent the dimensional amplitude of the incident wave, w is the flap’s width, and Cg is the group velocity of the incident wave, as below:(11)��=��0·121+2�0ℎsinh2�0ℎwhere �0 denotes the wave number, h is water depth, and H is the height of incident waves.
According to previous sections ∊,����-∊ modeling is used for all models simulated in this section. For this purpose, the empty boundary condition is used for flume walls. In order to preventing wave reflection at the inlet and outlet of the flume, the length of wave absorption is set to be at least one incident wavelength. In addition, the structured mesh is chosen, and the mesh dimensions are selected in two distinct directions. In each model, all grids have a length of 2 (cm) and a height of 1 (cm). Afterwards, as an input of the software for all of the models, we define the time step as 0.001 (s). Moreover, the run time of every simulation is 30 (s). As mentioned before, our primary model is Schmitt model, and the flap properties is given in table2. For all simulations, the flume measures 15 meters in length and 0.65 meters in width, and water depth is equal to 0.335 (m). The flap is also located 7 meters from the flume’s inlet.
Finally, in order to compare the results, the capture factor is calculated for each simulation and compared to the primary model. It is worth mentioning that capture factor refers to the ratio of absorbed wave energy to the input wave energy.
According to primary model simulation and due to the decreasing horizontal velocity with depth, the wave crest has the highest velocity. Considering the fact that the wave’s orbital velocity causes the flap to move, the contact between the upper edge of the flap and the incident wave can enhance its performance. Additionally, the numerical model shows that the dynamic pressure decreases as depth increases, and the hydrostatic pressure increases as depth increases.
To determine the OSWEC design, it is imperative to understand the correlation between the capture factor, wave period, and wave height. Therefore, as it is shown in Figure8, we plot the change in capture factor over the variations in wave period and wave height in 3D and 2D. In this diagram, the first axis features changes in wave period, the second axis displays changes in wave height, and the third axis depicts changes in capture factor. According to our wave properties in the numerical model, the wave period and wave height range from 2 to 14 seconds and 2 to 8 meters, respectively. This is due to the fact that the flap does not oscillate if the wave height is less than 2 (m), and it does not reverse if the wave height is more than 8 (m). In addition, with wave periods more than 14 (s), the wavelength would be so long that it would violate the deep-water conditions, and with wave periods less than 2 (s), the flap would not oscillate properly due to the shortness of wavelength. The results of simulation are shown in Figure 8. As it can be perceived from Figure 8, in a constant wave period, the capture factor is in direct proportion to the wave height. It is because of the fact that waves with more height have more energy to rotate the flap. Besides, in a constant wave height, the capture factor increases when the wave period increases, until a given wave period value. However, the capture factor falls after this point. These results are expected since the flap’s angular displacement is not high in lower wave periods, while the oscillating motion of that is not fast enough to activate the power take-off system in very high wave periods.
As is shown in Figure 9, we plot the change in capture factor over the variations in wave period (s) and water depth (m) in 3D. As it can be seen in this diagram, the first axis features changes in water depth (m), the second axis depicts the wave period (s), and the third axis displays OSWEC’s capture factor. The wave period ranges from 0 to 10 seconds based on our wave properties, which have been adopted from Schmitt’s model, while water depth ranges from 0 to 0.5 meters according to the flume and flap dimensions and laboratory limitations. According to Figure9, for any specific water depth, the capture factor increases in a varying rate when the wave period increases, until a given wave period value. However, the capture factor falls steadily after this point. In fact, the maximum capture factor occurs when the wave period is around 6 seconds. This trend is expected since, in a specific water depth, the flap cannot oscillate properly when the wavelength is too short. As the wave period increases, the flap can oscillate more easily, and consequently its capture factor increases. However, the capture factor drops in higher wave periods because the wavelength is too large to move the flap. Furthermore, in a constant wave period, by changing the water depth, the capture factor does not alter. In other words, the capture factor does not depend on the water depth when it is around its maximum value.
3. Sensitivity Analysis
Based on previous studies, in addition to the flap design, the location of the flap relative to the water surface (freeboard) and its elevation relative to the flume bed (flap bottom elevation) play a significant role in extracting energy from the wave energy converter. This study measures the sensitivity of the model to various parameters related to the flap design including upper part width of the flap, lower part width of the flap, the freeboard, and the flap bottom elevation. Moreover, as a novel idea, we propose that the flap widths differ in the lower and upper parts. In Figure10, as an example, a flap with an upper thickness of 100 (mm) and a lower thickness of 50 (mm) and a flap with an upper thickness of 50 (mm) and a lower thickness of 100 (mm) are shown. The influence of such discrepancy between the widths of the upper and lower parts on the interaction between the wave and the flap, or in other words on the capture factor, is evaluated. To do so, other parameters are remained constant, such as the freeboard, the distance between the flap and the flume bed, and the wave properties.
In Figure11, models are simulated with distinct upper and lower widths. As it is clear in this figure, the first axis depicts the lower part width of the flap, the second axis indicates the upper part width of the flap, and the colors represent the capture factor values. Additionally, in order to consider a sufficient range of change, the flap thickness varies from half to double the value of the primary model for each part.
According to this study, the greater the discrepancy in these two parts, the lower the capture factor. It is on account of the fact that when the lower part of the flap is thicker than the upper part, and this thickness difference in these two parts is extremely conspicuous, the inertia against the motion is significant at zero degrees of rotation. Consequently, it is difficult to move the flap, which results in a low capture factor. Similarly, when the upper part of the flap is thicker than the lower part, and this thickness difference in these two parts is exceedingly noticeable, the inertia is so great that the flap can not reverse at the maximum degree of rotation. As the results indicate, the discrepancy can enhance the performance of the converter if the difference between these two parts is around 20%. As it is depicted in the Figure11, the capture factor reaches its own maximum amount, when the lower part thickness is from 5 to 6 (cm), and the upper part thickness is between 6 and 7 (cm). Consequently, as a result of this discrepancy, less material will be used, and therefore there will be less cost.
As illustrated in Figure12, this study examines the effects of freeboard (level difference between the flap top and water surface) and the flap bottom elevation (the distance between the flume bed and flap bottom) on the converter performance. In this diagram, the first axis demonstrates the freeboard and the second axis on the left side displays the flap bottom elevation, while the colors indicate the capture factor. In addition, the feasible range of freeboard is between -15 to 15 (cm) due to the limitation of the numerical model, so that we can take the wave slamming and the overtopping into consideration. Additionally, based on the Schmitt model and its scaled model of 1:40 of the base height, the flap bottom should be at least 9 (cm) high. Since the effect of surface waves is distributed over the depth of the flume, it is imperative to maintain a reasonable flap height exposed to incoming waves. Thus, the maximum flap bottom elevation is limited to 19 (cm). As the Figure12 pictures, at constant negative values of the freeboard, the capture factor is in inverse proportion with the flap bottom elevation, although slightly.
Furthermore, at constant positive values of the freeboard, the capture factor fluctuates as the flap bottom elevation decreases while it maintains an overall increasing trend. This is on account of the fact that increasing the flap bottom elevation creates turbulence flow behind the flap, which encumbers its rotation, as well as the fact that the flap surface has less interaction with the incoming waves. Furthermore, while keeping the flap bottom elevation constant, the capture factor increases by raising the freeboard. This is due to the fact that there is overtopping with adverse impacts on the converter performance when the freeboard is negative and the flap is under the water surface. Besides, increasing the freeboard makes the wave slam more vigorously, which improves the converter performance.
Adding ribs to the flap surface, as shown in Figure13, is a novel idea that is investigated in the next section. To achieve an optimized design for the proposed geometry of the flap, we determine the optimal number and dimensions of ribs based on the flap properties as our decision variables in the optimization process. As an example, Figure13 illustrates a flap with 3 ribs on each side with specific dimensions.
Figure14 shows the flow velocity field around the flap jointed to the flume bed. During the oscillation of the flap, the pressure on the upper and lower surfaces of the flap changes dynamically due to the changing angle of attack and the resulting change in the direction of fluid flow. As the flap moves upwards, the pressure on the upper surface decreases, and the pressure on the lower surface increases. Conversely, as the flap moves downwards, the pressure on the upper surface increases, and the pressure on the lower surface decreases. This results in a cyclic pressure variation around the flap. Under certain conditions, the pressure field around the flap can exhibit significant variations in magnitude and direction, forming vortices and other flow structures. These flow structures can affect the performance of the OSWEC by altering the lift and drag forces acting on the flap.
4. Design Optimization
We consider optimizing the design parameters of the flap of converter using a nature-based swarm optimization method, that fall in the category of metaheuristic algorithms . Accordingly, we choose four state-of-the-art algorithms to perform an optimization study. Then, based on their performances to achieve the highest capture factor, one of them will be chosen to be combined with the Hill Climb algorithm to carry out a local search. Therefore, in the remainder of this section, we discuss the search process of each algorithm and visualize their performance and convergence curve as they try to find the best values for decision variables.
4.1. Metaheuristic Approaches
As the first considered algorithm, the Gray Wolf Optimizer (GWO) algorithm simulates the natural leadership and hunting performance of gray wolves which tend to live in colonies. Hunters must obey the alpha wolf, the leader, who is responsible for hunting. Then, the beta wolf is at the second level of the gray wolf hierarchy. A subordinate of alpha wolf, beta stands under the command of the alpha. At the next level in this hierarchy, there are the delta wolves. They are subordinate to the alpha and beta wolves. This category of wolves includes scouts, sentinels, elders, hunters, and caretakers. In this ranking, omega wolves are at the bottom, having the lowest level and obeying all other wolves. They are also allowed to eat the prey just after others have eaten. Despite the fact that they seem less important than others, they are really central to the pack survival. Since, it has been shown that without omega wolves, the entire pack would experience some problems like fighting, violence, and frustration. In this simulation, there are three primary steps of hunting including searching, surrounding, and finally attacking the prey. Mathematically model of gray wolves’ hunting technique and their social hierarchy are applied in determined by optimization. this study. As mentioned before, gray wolves can locate their prey and surround them. The alpha wolf also leads the hunt. Assuming that the alpha, beta, and delta have more knowledge about prey locations, we can mathematically simulate gray wolf hunting behavior. Hence, in addition to saving the top three best solutions obtained so far, we compel the rest of the search agents (also the omegas) to adjust their positions based on the best search agent. Encircling behavior can be mathematically modeled by the following equations: .(12)�→=|�→·��→(�)-�→(�)|(13)�→(�+1)=��→(�)-�→·�→(14)�→=2.�2→(15)�→=2�→·�1→-�→Where �→indicates the position vector of gray wolf, ��→ defines the vector of prey, t indicates the current iteration, and �→and �→are coefficient vectors. To force the search agent to diverge from the prey, we use �→ with random values greater than 1 or less than -1. In addition, C→ contains random values in the range [0,2], and �→ 1 and �2→ are random vectors in [0,1]. The second considered technique is the Moth Flame Optimizer (MFO) algorithm. This method revolves around the moths’ navigation mechanism, which is realized by positioning themselves and maintaining a fixed angle relative to the moon while flying. This effective mechanism helps moths to fly in a straight path. However, when the source of light is artificial, maintaining an angle with the light leads to a spiral flying path towards the source that causes the moth’s death . In MFO algorithm, moths and flames are both solutions. The moths are actual search agents that fly in hyper-dimensional space by changing their position vectors, and the flames are considered pins that moths drop when searching the search space . The problem’s variables are the position of moths in the space. Each moth searches around a flame and updates it in case of finding a better solution. The fitness value is the return value of each moth’s fitness (objective) function. The position vector of each moth is passed to the fitness function, and the output of the fitness function is assigned to the corresponding moth. With this mechanism, a moth never loses its best solution . Some attributes of this algorithm are as follows:
•It takes different values to converge moth in any point around the flame.
•Distance to the flame is lowered to be eventually minimized.
•When the position gets closer to the flame, the updated positions around the flame become more frequent.
As another method, the Multi-Verse Optimizer is based on a multiverse theory which proposes there are other universes besides the one in which we all live. According to this theory, there are more than one big bang in the universe, and each big bang leads to the birth of a new universe . Multi-Verse Optimizer (MVO) is mainly inspired by three phenomena in cosmology: white holes, black holes, and wormholes. A white hole has never been observed in our universe, but physicists believe the big bang could be considered a white hole . Black holes, which behave completely in contrast to white holes, attract everything including light beams with their extremely high gravitational force . In the multiverse theory, wormholes are time and space tunnels that allow objects to move instantly between any two corners of a universe (or even simultaneously from one universe to another) . Based on these three concepts, mathematical models are designed to perform exploration, exploitation, and local search, respectively. The concept of white and black holes is implied as an exploration phase, while the concept of wormholes is considered as an exploitation phase by MVO. Additionally, each solution is analogous to a universe, and each variable in the solution represents an object in that universe. Furthermore, each solution is assigned an inflation rate, and the time is used instead of iterations. Following are the universe rules in MVO:
•The possibility of having white hole increases with the inflation rate.
•The possibility of having black hole decreases with the inflation rate.
•Objects tend to pass through black holes more frequently in universes with lower inflation rates.
•Regardless of inflation rate, wormholes may cause objects in universes to move randomly towards the best universe. 
Modeling the white/black hole tunnels and exchanging objects of universes mathematically was accomplished by using the roulette wheel mechanism. With every iteration, the universes are sorted according to their inflation rates, then, based on the roulette wheel, the one with the white hole is selected as the local extremum solution. This is accomplished through the following steps:
Where ��� represents the jth parameter of the ith universe, Ui indicates the ith universe, NI(Ui) is normalized inflation rate of the ith universe, r1 is a random number in [0,1], and j xk shows the jth parameter of the kth universe selected by a roulette wheel selection mechanism . It is assumed that wormhole tunnels always exist between a universe and the best universe formed so far. This mechanism is as follows:(17)���=if�2<���:��+���×((���-���)×�4+���)�3<0.5��-���×((���-���)×�4+���)�3≥0.5����:���where Xj indicates the jth parameter of the best universe formed so far, TDR and WEP are coefficients, where Xj indicates the jth parameter of the best universelbjshows the lower bound of the jth variable, ubj is the upper bound of the jth variable, and r2, r3, and r4 are random numbers in , .
Finally, one of the newest optimization algorithms is WOA. The WOA algorithm simulates the movement of prey and the whale’s discipline when looking for their prey. Among several species, Humpback whales have a specific method of hunting . Humpback whales can recognize the location of prey and encircle it before hunting. The optimal design position in the search space is not known a priori, and the WOA algorithm assumes that the best candidate solution is either the target prey or close to the optimum. This foraging behavior is called the bubble-net feeding method. Two maneuvers are associated with bubbles: upward spirals and double loops. A unique behavior exhibited only by humpback whales is bubble-net feeding. In fact, The WOA algorithm starts with a set of random solutions. At each iteration, search agents update their positions for either a randomly chosen search agent or the best solution obtained so far , . When the best search agent is determined, the other search agents will attempt to update their positions toward that agent. It is important to note that humpback whales swim around their prey simultaneously in a circular, shrinking circle and along a spiral-shaped path. By using a mathematical model, the spiral bubble-net feeding maneuver is optimized. The following equation represents this behavior:(18)�→(�+1)=�′→·�bl·cos(2��)+�∗→(�)
X→(t+ 1) indicates the distance of the it h whale to the prey (best solution obtained so far),� is a constant for defining the shape of the logarithmic spiral, l is a random number in [−1,1], and dot (.) is an element-by-element multiplication .
Comparing the four above-mentioned methods, simulations are run with 10 search agents for 400 iterations. In Figure 15, there are 20 plots the optimal values of different parameters in optimization algorithms. The five parameters of this study are freeboard, bottom elevations, number of ribs on the converter, rib thickness, and rib Height. The optimal value for each was found by optimization algorithms, naming WOA, MVO, MFO, and GWO. By looking through the first row, the freeboard parameter converges to its maximum possible value in the optimization process of GWO after 300 iterations. Similarly, MFO finds the same result as GWO. In contrast, the freeboard converges to its minimum possible value in MVO optimizing process, which indicates positioning the converter under the water. Furthermore, WOA found the optimal value of freeboard as around 0.02 after almost 200 iterations. In the second row, the bottom elevation is found at almost 0.11 (m) in all algorithms; however, the curves follow different trends in each algorithm. The third row shows the number of ribs, where results immediately reveal that it should be over 4. All algorithms coincide at 5 ribs as the optimal number in this process. The fourth row displays the trends of algorithms to find optimal rib thickness. MFO finds the optimal value early and sets it to around 0.022, while others find the same value in higher iterations. Finally, regarding the rib height, MVO, MFO, and GWO state that the optimal value is 0.06 meters, but WOA did not find a higher value than 0.039.
4.2. HCMVO Bi-level Approach
Despite several strong search characteristics of MVO and its high performance in various optimization problems, it suffers from a few deficiencies in local and global search mechanisms. For instance, it is trapped in the local optimum when wormholes stochastically generate many solutions near the best universe achieved throughout iterations, especially in solving complex multimodal problems with high dimensions . Furthermore, MVO needs to be modified by an escaping strategy from the local optima to enhance the global search abilities. To address these shortages, we propose a fast and effective meta-algorithm (HCMVO) to combine MVO with a Random-restart hill-climbing local search. This meta-algorithm uses MVO on the upper level to develop global tracking and provide a range of feasible and proper solutions. The hill-climbing algorithm is designed to develop a comprehensive neighborhood search around the best-found solution proposed by the upper-level (MVO) when MVO is faced with a stagnation issue or falling into a local optimum. The performance threshold is formulated as follows.(20)Δ����THD=∑�=1�����TH��-����TH��-1�where BestTHDis the best-found solution per generation, andM is related to the domain of iterations to compute the average performance of MVO. If the proposed best solution by the local search is better than the initial one, the global best of MVO will be updated. HCMVO iteratively runs hill climbing when the performance of MVO goes down, each time with an initial condition to prepare for escaping such undesirable situations. In order to get a better balance between exploration and exploitation, the search step size linearly decreases as follows:(21)��=��-����Ma�iter��+1where iter and Maxiter are the current iteration and maximum number of evaluation, respectively. �� stands for the step size of the neighborhood search. Meanwhile, this strategy can improve the convergence rate of MVO compared with other algorithms.
Algorithm 1 shows the technical details of the proposed optimization method (HCMVO). The initial solution includes freeboard (�), bottom elevation (�), number of ribs (Nr), rib thickness (�), and rib height(�).
The high trend of diminishing worldwide energy resources has entailed a great crisis upon vulnerable societies. To withstand this effect, developing renewable energy technologies can open doors to a more reliable means, among which the wave energy converters will help the coastal residents and infrastructure. This paper set out to determine the optimized design for such devices that leads to the highest possible power output. The main goal of this research was to demonstrate the best design for an oscillating surge wave energy converter using a novel metaheuristic optimization algorithm. In this regard, the methodology was devised such that it argued the effects of influential parameters, including wave characteristics, WEC design, and interaction criteria.
To begin with, a numerical model was developed in Flow 3D software to simulate the response of the flap of a wave energy converter to incoming waves, followed by a validation study based upon a well-reputed experimental study to verify the accuracy of the model. Secondly, the hydrodynamics of the flap was investigated by incorporating the turbulence. The effect of depth, wave height, and wave period are also investigated in this part. The influence of two novel ideas on increasing the wave-converter interaction was then assessed: i) designing a flap with different widths in the upper and lower part, and ii) adding ribs on the surface of the flap. Finally, four trending single-objective metaheuristic optimization methods
Algorithm 1:Hill Climb Multiverse Optimization
Initialize parameters�ER,�DR,�EP,Best�,���ite��▹Wormhole existence probability (WEP)
��=Normalize the inflation rate��
for iter in[1,⋯,���iter]do
�(Black HoleIndex,�)=��(White HoleIndex,�)
ifΔBestTHD<��then▹Perform hill climbing local search
The implementation details of the hill-climbing algorithm applied in HCMPA can be seen in Algorithm 2. One of the critical parameters isg, which denotes the resolution of the neighborhood search around the proposed global best by MVO. If we set a small step size for hill-climbing, the convergence speed will be decreased. On the other hand, a large step size reinforces the exploration ability. Still, it may reduce the exploitation ability and in return increase the act of jumping from a global optimum or surfaces with high-potential solutions. Per each decision variable, the neighborhood search evaluates two different direct searches, incremental or decremental. After assessing the generated solutions, the best candidate will be selected to iterate the search algorithm. It is noted that the hill-climbing algorithm should not be applied in the initial iteration of the optimization process due to the immense tendency for converging to local optima. Meanwhile, for optimizing largescale problems, hill-climbing is not an appropriate selection. In order to improve understanding of the proposed hybrid optimization algorithm’s steps, the flowchart of HCMVO is designed and can be seen in Figure 16.
Figure 17 shows the observed capture factor (which is the absorbed energy with respect to the available energy) by each optimization algorithm from iterations 1 to 400. The algorithms use ten search agents in their modified codes to find the optimal solutions. While GWO and MFO remain roughly constant after iterations 54 and 40, the other three algorithms keep improving the capture factor. In this case, HCMVO and MVO worked very well in the optimizing process with a capture factor obtained by the former as 0.594 and by the latter as 0.593. MFO almost found its highest value before the iteration 50, which means the exploration part of the algorithm works out well. Similarly, HCMVO does the same. However, it keeps finding the better solution during the optimization process until the last iteration, indicating the strong exploitation part of the algorithm. GWO reveals a weakness in exploration and exploitation because not only does it evoke the least capture factor value, but also the curve remains almost unchanged throughout 350 iterations.
Figure 18 illustrates complex interactions between the five optimization parameters and the capture factor for HCMVO (a), MPA (b), and MFO (c) algorithms. The first interesting observation is that there is a high level of nonlinear relationships among the setting parameters that can make a multi-modal search space. The dark blue lines represent the best-found configuration throughout the optimisation process. Based on both HCMVO (a) and MVO (b), we can infer that the dark blue lines concentrate in a specific range, showing the high convergence ability of both HCMVO and MVO. However, MFO (c) could not find the exact optimal range of the decision variables, and the best-found solutions per generation distribute mostly all around the search space.
Algorithm 1:Hill Climb Multiverse Optimization
Initialize the constraints��1�,��1�
�1�=Mi�1�+���1�/�▹Compute the step size,�is search resolution
were utilized to illuminate the optimum values of the design parameters, and the best method was chosen to develop a new algorithm that performs both local and global search methods.
The correlation between hydrodynamic parameters and the capture factor of the converter was supported by the results. For any given water depth, the capture factor increases as the wave period increases, until a certain wave period value (6 seconds) is reached, after which the capture factor gradually decreases. It is expected since the flap cannot oscillate effectively when the wavelength is too short for a certain water depth. Conversely, when the wavelength is too long, the capture factor decreases. Furthermore, under a constant wave period, increasing the water depth does not affect the capture factor. Regarding the sensitivity analysis, the study found that increasing the flap bottom elevation causes turbulence flow behind the flap and limitation of rotation, which leads to less interaction with the incoming waves. Furthermore, while keeping the flap bottom elevation constant, increasing the freeboard improves the capture factor. Overtopping happens when the freeboard is negative and the flap is below the water surface, which has a detrimental influence on converter performance. Furthermore, raising the freeboard causes the wave impact to become more violent, which increases converter performance.
In the last part, we discussed the search process of each algorithm and visualized their performance and convergence curves as they try to find the best values for decision variables. Among the four selected metaheuristic algorithms, the Multi-verse Optimizer proved to be the most effective in achieving the best answer in terms of the WEC capture factor. However, the MVO needed modifications regarding its escape approach from the local optima in order to improve its global search capabilities. To overcome these constraints, we presented a fast and efficient meta-algorithm (HCMVO) that combines MVO with a Random-restart hill-climbing local search. On a higher level, this meta-algorithm employed MVO to generate global tracking and present a range of possible and appropriate solutions. Taken together, the results demonstrated that there is a significant degree of nonlinearity among the setup parameters that might result in a multimodal search space. Since MVO was faced with a stagnation issue or fell into a local optimum, we constructed a complete neighborhood search around the best-found solution offered by the upper level. In sum, the newly-developed algorithm proved to be highly effective for the problem compared to other similar optimization methods. The strength of the current findings may encourage future investigation on design optimization of wave energy converters using developed geometry as well as the novel approach.
CRediT authorship contribution statement
Erfan Amini: Conceptualization, Methodology, Validation, Data curation, Writing – original draft, Writing – review & editing, Visualization. Mahdieh Nasiri: Conceptualization, Methodology, Validation, Data curation, Writing – original draft, Writing – review & editing, Visualization. Navid Salami Pargoo: Writing – original draft, Writing – review & editing. Zahra Mozhgani: Conceptualization, Methodology. Danial Golbaz: Writing – original draft. Mehrdad Baniesmaeil: Writing – original draft. Meysam Majidi Nezhad: . Mehdi Neshat: Supervision, Conceptualization, Writing – original draft, Writing – review & editing, Visualization. Davide Astiaso Garcia: Supervision. Georgios Sylaios: Supervision.
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.
This research has been carried out within ILIAD (Inte-grated Digital Framework for Comprehensive Maritime Data and Information Services) project that received funding from the European Union’s H2020 programme.
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Analyses of Cryogenic Propellant Tank Pressurization based upon Experiments and Numerical Simulations Carina Ludwig? and Michael Dreyer** *DLR – German Aerospace Center, Space Launcher Systems Analysis (SART), Institute of Space Systems, 28359 Bremen, Germany, Carina.Ludwig@dlr.de **ZARM – Center for Applied Space Technology and Microgravity, University of Bremen, 28359 Bremen, Germany
본 연구에서는 발사대 적용을 위한 극저온 추진제 탱크의 능동 가압을 분석하였다. 따라서 지상 실험, 수치 시뮬레이션 및 분석 연구를 수행하여 다음과 같은 중요한 결과를 얻었습니다.
필요한 가압 기체 질량을 최소화하기 위해 더 높은 가압 기체 온도가 유리하거나 헬륨을 가압 기체로 적용하는 것이 좋습니다.
Flow-3D를 사용한 가압 가스 질량의 수치 시뮬레이션은 실험 결과와 잘 일치함을 보여줍니다. 가압 중 지배적인 열 전달은 주입된 가압 가스에서 축방향 탱크 벽으로 나타나고 능동 가압 단계 동안 상 변화의 주된 방식은 가압 가스의 유형에 따라 다릅니다.
가압 단계가 끝나면 상당한 압력 강하가 발생합니다. 이 압력 강하의 분석적 결정을 위해 이론적 모델이 제공됩니다.
The active-pressurization of cryogenic propellant tanks for the launcher application was analyzed in this study. Therefore, ground experiments, numerical simulations and analytical studies were performed with the following important results: In order to minimize the required pressurant gas mass, a higher pressurant gas temperature is advantageous or the application of helium as pressurant gas. Numerical simulations of the pressurant gas mass using Flow-3D show good agreement to the experimental results. The dominating heat transfer during pressurization appears from the injected pressurant gas to the axial tank walls and the predominant way of phase change during the active-pressurization phase depends on the type of the pressurant gas. After the end of the pressurization phase, a significant pressure drop occurs. A theoretical model is presented for the analytical determination of this pressure drop.
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Pan Lu1 , Zhang Cheng-Lin2,6,Wang Liang3, Liu Tong4 and Liu Jiang-lin5 1 Aviation and Materials College, Anhui Technical College of Mechanical and Electrical Engineering, Wuhu Anhui 241000, People’s Republic of China 2 School of Engineering Science, University of Science and Technology of China, Hefei Anhui 230026, People’s Republic of China 3 Anhui Top Additive Manufacturing Technology Co., Ltd., Wuhu Anhui 241300, People’s Republic of China 4 Anhui Chungu 3D Printing Institute of Intelligent Equipment and Industrial Technology, Anhui 241300, People’s Republic of China 5 School of Mechanical and Transportation Engineering, Taiyuan University of Technology, Taiyuan Shanxi 030024, People’s Republic of China 6 Author to whom any correspondence should be addressed. E-mail: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org,email@example.com and firstname.lastname@example.org
선택적 레이저 용융(SLM)은 열 전달, 용융, 상전이, 기화 및 물질 전달을 포함하는 복잡한 동적 비평형 프로세스인 금속 적층 제조(MAM)에서 가장 유망한 기술 중 하나가 되었습니다. 용융 풀의 특성(구조, 온도 흐름 및 속도 흐름)은 SLM의 최종 성형 품질에 결정적인 영향을 미칩니다. 이 연구에서는 선택적 레이저 용융 AlCu5MnCdVA 합금의 용융 풀 구조, 온도 흐름 및 속도장을 연구하기 위해 수치 시뮬레이션과 실험을 모두 사용했습니다.
그 결과 용융풀의 구조는 다양한 형태(깊은 오목 구조, 이중 오목 구조, 평면 구조, 돌출 구조 및 이상적인 평면 구조)를 나타냈으며, 용융 풀의 크기는 약 132 μm × 107 μm × 50 μm였습니다. : 용융풀은 초기에는 여러 구동력에 의해 깊이 15μm의 깊은 오목형상이었으나, 성형 후기에는 장력구배에 의해 높이 10μm의 돌출형상이 되었다. 용융 풀 내부의 금속 흐름은 주로 레이저 충격력, 금속 액체 중력, 표면 장력 및 반동 압력에 의해 구동되었습니다.
AlCu5MnCdVA 합금의 경우, 금속 액체 응고 속도가 매우 빠르며(3.5 × 10-4 S), 가열 속도 및 냉각 속도는 각각 6.5 × 107 K S-1 및 1.6 × 106 K S-1 에 도달했습니다. 시각적 표준으로 표면 거칠기를 선택하고, 낮은 레이저 에너지 AlCu5MnCdVA 합금 최적 공정 매개변수 창을 수치 시뮬레이션으로 얻었습니다: 레이저 출력 250W, 부화 공간 0.11mm, 층 두께 0.03mm, 레이저 스캔 속도 1.5m s-1 .
또한, 실험 프린팅과 수치 시뮬레이션과 비교할 때, 용융 풀의 폭은 각각 약 205um 및 약 210um이었고, 인접한 두 용융 트랙 사이의 중첩은 모두 약 65um이었다. 결과는 수치 시뮬레이션 결과가 실험 인쇄 결과와 기본적으로 일치함을 보여 수치 시뮬레이션 모델의 정확성을 입증했습니다.
Selective Laser Melting (SLM) has become one of the most promising technologies in Metal Additive Manufacturing (MAM), which is a complex dynamic non-equilibrium process involving heat transfer, melting, phase transition, vaporization and mass transfer. The characteristics of the molten pool (structure, temperature flow and velocity flow) have a decisive influence on the final forming quality of SLM. In this study, both numerical simulation and experiments were employed to study molten pool structure, temperature flow and velocity field in Selective Laser Melting AlCu5MnCdVA alloy. The results showed the structure of molten pool showed different forms(deep-concave structure, double-concave structure, plane structure, protruding structure and ideal planar structure), and the size of the molten pool was approximately 132 μm × 107 μm × 50 μm: in the early stage, molten pool was in a state of deep-concave shape with a depth of 15 μm due to multiple driving forces, while a protruding shape with a height of 10 μm duo to tension gradient in the later stages of forming. The metal flow inside the molten pool was mainly driven by laser impact force, metal liquid gravity, surface tension and recoil pressure. For AlCu5MnCdVA alloy, metal liquid solidification speed was extremely fast(3.5 × 10−4 S), the heating rate and cooling rate reached 6.5 × 107 K S−1 and 1.6 × 106 K S−1 , respectively. Choosing surface roughness as a visual standard, low-laser energy AlCu5MnCdVA alloy optimum process parameters window was obtained by numerical simulation: laser power 250 W, hatching space 0.11 mm, layer thickness 0.03 mm, laser scanning velocity 1.5 m s−1 . In addition, compared with experimental printing and numerical simulation, the width of the molten pool was about 205 um and about 210 um, respectively, and overlapping between two adjacent molten tracks was all about 65 um. The results showed that the numerical simulation results were basically consistent with the experimental print results, which proved the correctness of the numerical simulation model.
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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 . This criterion is described by the following relationship:
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 .
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:
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 :
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:
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.
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.
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 .
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.
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 .
Characteristic of the DPM model.
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 :
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.
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.
where Pa is the atmospheric pressure, and h is the the height of metal in the reactor.
Themelis and Goyal  developed a model for calculating mixing power delivered by gas injection.
where Q is the gas flow, and m is the mass of liquid metal.
Zhang  proposed a model taking into account the temperature difference between gas and alloy (metal).
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.
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.
Mixing power calculated from mathematical models.
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.
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 .
where D is the diffusion coefficient, and dB is the bubble diameter.
After substituting appropriate values, we get
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.
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 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.
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 (220.127.116.114 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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Publication Date:2013-07-24 Research Org.: Los Alamos National Lab. (LANL), Los Alamos, NM (United States) Sponsoring Org.: DOE/LANL OSTI Identifier: 1088904 Report Number(s): LA-UR-13-25537 DOE Contract Number: AC52-06NA25396 Resource Type: Technical Report Country of Publication: United States Language: English Subject: Engineering(42); Materials Science(36); Radiation Chemistry, Radiochemistry, & Nuclear Chemistry(38)
The plutonium foundry at Los Alamos National Laboratory casts products for various special nuclear applications. However, plutonium’s radioactivity, material properties, and security constraints complicate the ability to perform experimental analysis of mold behavior. The Manufacturing Engineering and Technologies (MET-2) group previously developed a graphite mold to vacuum cast small plutonium disks to be used by the Department of Homeland Security as point sources for radiation sensor testing.
A two-stage pouring basin consisting of a funnel and an angled cavity directs the liquid into a vertical runner. A stack of ten disk castings connect to the runner by horizontal gates. Volumetric flow rates were implemented to limit overflow into the funnel and minimize foundry returns. Models using Flow-3D computational fluid dynamics software are employed here to determine liquid Pu flow paths, optimal pour regimes, temperature changes, and pressure variations.
Hardcopy drawings provided necessary information to create 3D .stl models for import into Flow-3D (Figs. 1 and 2). The mesh was refined over several iterations to isolate the disk cavities, runner, angled cavity, funnel, and input pour. The final flow and mold-filling simulation utilizes a fine mesh with ~5.5 million total cells. For the temperature study, the mesh contained 1/8 as many cells to reduce computational time and set temperatures to 850 °C for the molten plutonium and 500 °C for the solid graphite mold components (Fig. 3).
Flow-3D solves mass continuity and Navier-Stokes momentum equations over the structured rectangular grid model using finite difference and finite volume numerical algorithms. The solver includes terms in the momentum equation for body and viscous accelerations and uses convective heat transfer.
Simulation settings enabled Flow-3D physics calculations for gravity at 980.665 cm/s 2 in the negative Z direction (top of mold to bottom); viscous, turbulent, incompressible flow using dynamically-computed Renormalized Group Model turbulence calculations and no-slip/partial slip wall shear, and; first order, full energy equation heat transfer.
Mesh boundaries were all set to symmetric boundary conditions except for the Zmin boundary set to outflow and the Zmax boundary set to a volume flow. Vacuum casting conditions and the high reactivity of remaining air molecules with Pu validate the assumption of an initially fluidless void.
The flow follows a unique three-dimensional path. The mold fills upwards with two to three disks receiving fluid in a staggered sequence. Figures 5-9 show how the fluid fills the cavity, and Figure 7 includes the color scale for pressure levels in these four figures. The narrow gate causes a high pressure region which forces the fluid to flow down the cavity centerline.
It proceeds to splash against the far wall and then wrap around the circumference back to the gate (Figs. 5 and 6). Flow in the angled region of the pouring basin cascades over the bottom ledge and attaches to the far wall of the runner, as seen in Figure 7.
This channeling becomes less pronounced as fluid volume levels increase. Finally, two similar but non-uniform depressed regions form about the centerline. These regions fill from their perimeter and bottom until completion (Fig. 8). Such a pattern is counter, for example, to a steady scenario in which a circle of molten Pu encompassing the entire bottom surface rises as a growing cylinder.
Cavity pressure becomes uniform when the cavity is full. Pressure levels build in the rising well section of the runner, where impurities were found to settle in actual casting. Early test simulations optimized the flow as three pours so that the fluid would never overflow to the funnel, the cavities would all fill completely, and small amounts of fluid would remain as foundry returns in the angled cavity.
These rates and durations were translated to the single 2.7s pour at 100 cm 3 per second used here. Figure 9 shows anomalous pressure fluctuations which occurred as the cavities became completely filled. Multiple simulations exhibited a rapid change in pressure from positive to negative and back within the newly-full disk and surrounding, already-full disks.
The time required to completely fill each cavity is plotted in Figure 10. Results show negligible temperature change within the molten Pu during mold filling and, as seen in Figure 11, at fill completion.
Non-uniform cavity filling could cause crystal microstructure irregularities during solidification. However, the small temperature changes seen – due to large differences in specific heat between Pu and graphite – over a relatively short time make such problems unlikely in this case.
In the actual casting, cooling required approximately ten minutes. This large difference in time scales further reduces the chance for temperature effects in such a superheated scenario. Pouring basin emptying decreases pressure at the gate which extends fill time of the top two cavities.
The bottom cavity takes longer to fill because fluid must first enter the runner and fill the well. Fill times continue linearly until the top two cavities. The anomalous pressure fluctuations may be due to physical attempts by the system to reach equilibrium, but they are more likely due to numerical errors in the Flow3D solver.
Unsuccessful tests were performed to remove them by halving fluid viscosity. The fine mesh reduced, but did not eliminate, the extent of the fluctuations. Future work is planned to study induction and heat transfer in the full Pu furnace system, including quantifying temporal lag of the cavity void temperature to the mold wall temperature during pre-heat and comparing heat flux levels between furnace components during cool-down.
Thanks to Doug Kautz for the opportunity to work with MET-2 and for assigning an interesting unclassified project. Additional thanks to Mike Bange for CFD guidance, insight of the project’s history, and draft review.
The elimination of internal macro-defects is a key issue in Ti–6Al–4V alloys fabricated via powder bed fusion using electron beams (PBF-EB), wherein internal macro-defects mainly originate from the virgin powder and inappropriate printing parameters. This study compares different types powders by combining support vector machine techniques to determine the most suitable powder for PBF-EB and to predict the processing window for the printing parameters without internal macro-defects. The results show that powders fabricated via plasma rotating electrode process have the best sphericity, flowability, and minimal porosity and are most suitable for printing. Surface roughness criterion was also applied to determine the quality of the even surfaces, and support vector machine was used to construct processing maps capable of predicting a wide range of four-dimensional printing parameters to obtain macro-defect-free samples, offering the possibility of subsequent development of Ti–6Al–4V alloys with excellent properties. The macro-defect-free samples exhibited good elongation, with the best overall mechanical properties being the ultimate tensile strength and elongation of 934.7 MPa and 24.3%, respectively. The elongation of the three macro-defect-free samples was much higher than that previously reported for additively manufactured Ti–6Al–4V alloys. The high elongation of the samples in this work is mainly attributed to the elimination of internal macro-defects.
Additive manufacturing (AM) technologies can rapidly manufacture complex or custom parts, reducing process steps and saving manufacturing time [, , , ], and are widely used in the aerospace, automotive, and other precision industries [5,6]. Powder bed fusion using an electron beam (PBF-EB) is an additive manufacturing method that uses a high-energy electron beam to melt metal powders layer by layer to produce parts. In contrast to selective laser melting, PBF-EB involves the preparation of samples in a high vacuum environment, which effectively prevents the introduction of impurities such as O and N. It also involves a preheating process for the print substrate and powder, which reduces residual thermal stress on the sample and subsequent heat treatment processes [, , ,7]. Due to these features and advantages, PBF-EB technology is a very important AM technology with great potential in metallic materials. Moreover, PBF-EB is the ideal AM technology for the manufacture of complex components made of many alloys, such as titanium alloys, nickel-based superalloys, aluminum alloys and stainless steels [, , ,8].
Ti–6Al–4V alloy is one of the prevalent commercial titanium alloys possessing high specific strength, excellent mechanical properties, excellent corrosion resistance, and good biocompatibility [9,10]. It is widely used in applications requiring low density and excellent corrosion resistance, such as the aerospace industry and biomechanical applications [11,12]. The mechanical properties of PBF-EB-processed Ti–6Al–4V alloys are superior to those fabricated by casting or forging, because the rapid cooling rate in PBF-EB results in finer grains [, , , , , , ]. However, the PBF-EB-fabricated parts often include internal macro-defects, which compromises their mechanical properties [, , , ]. This study focused on the elimination of macro-defects, such as porosity, lack of fusion, incomplete penetration and unmelted powders, which distinguishes them from micro-defects such as vacancies, dislocations, grain boundaries and secondary phases, etc. Large-sized fusion defects cause a severe reduction in mechanical strength. Smaller defects, such as pores and cracks, lead to the initiation of fatigue cracking and rapidly accelerate the cracking process . The issue of internal macro-defects must be addressed to expand the application of the PBF-EB technology. The main studies for controlling internal macro-defects are online monitoring of defects, remelting and hot isostatic pressing (HIP). The literatures [24,25] report the use of infrared imaging or other imaging techniques to identify defects, but the monitoring of smaller sized defects is still not adequate. And in some cases remelting does not reduce the internal macro-defects of the part, but instead causes coarsening of the macrostructure and volatilization of some metal elements . The HIP treatment does not completely eliminate the internal macro-defects, the original defect location may still act as a point of origin of the crack, and the subsequent treatment will consume more time and economic costs . Therefore, optimizing suitable printing parameters to avoid internal macro-defects in printed parts at source is of great industrial value and research significance, and is an urgent issue in PBF-EB related technology.
There are two causes of internal macro-defects in the AM process: gas pores trapped in the virgin powder and the inappropriate printing parameters [7,23]. Gui et al.  classify internal macro-defects during PBF-EB process according to their shape, such as spherical defects, elongated shape defects, flat shape defects and other irregular shape defects. Of these, spherical defects mainly originate from raw material powders. Other shape defects mainly originate from lack of fusion or unmelted powders caused by unsuitable printing parameters, etc. The PBF-EB process requires powders with good flowability, and spherical powders are typically chosen as raw materials. The prevalent techniques for the fabrication of pre-alloyed powders are gas atomization (GA), plasma atomization (PA), and the plasma rotating electrode process (PREP) [27,28]. These methods yield powders with different characteristics that affect the subsequent fabrication. The selection of a suitable powder for PBF-EB is particularly important to produce Ti–6Al–4V alloys without internal macro-defects. The need to optimize several printing parameters such as beam current, scan speed, line offset, and focus offset make it difficult to eliminate internal macro-defects that occur during printing . Most of the studies [11,12,22,, , , , ] on the optimization of AM processes for Ti–6Al–4V alloys have focused on samples with a limited set of parameters (e.g., power–scan speed) and do not allow for the guidance and development of unknown process windows for macro-defect-free samples. In addition, process optimization remains a time-consuming problem, with the traditional ‘trial and error’ method demanding considerable time and economic costs. The development of a simple and efficient method to predict the processing window for alloys without internal macro-defects is a key issue. In recent years, machine learning techniques have increasingly been used in the field of additive manufacturing and materials development [, , , ]. Aoyagi et al.  recently proposed a novel and efficient method based on a support vector machine (SVM) to optimize the two-dimensional process parameters (current and scan speed) and obtain PBF-EB-processed CoCr alloys without internal macro-defects. The method is one of the potential approaches toward effective optimization of more than two process parameters and makes it possible for the machine learning techniques to accelerate the development of alloys without internal macro-defects.
Herein, we focus on the elimination of internal macro-defects, such as pores, lack of fusion, etc., caused by raw powders and printing parameters. The Ti–6Al–4V powders produced by three different methods were compared, and the powder with the best sphericity, flowability, and minimal porosity was selected as the feedstock for subsequent printing. The relationship between the surface roughness and internal macro-defects in the Ti–6Al–4V components was also investigated. The combination of SVM and surface roughness indices (Sdr) predicted a wider four-dimensional processing window for obtaining Ti–6Al–4V alloys without internal macro-defects. Finally, we investigated the tensile properties of Ti–6Al–4V alloys at room temperature with different printing parameters, as well as the corresponding microstructures and fracture types.
Three types of Ti–6Al–4V alloy powders, produced by GA, PA, and PREP, were compared. The particle size distribution of the powders was determined using a laser particle size analyzer (LS230, Beckman Coulter, USA), and the flowability was measured using a Hall flowmeter (JIS-Z2502, Tsutsui Scientific Instruments Co., Ltd., Japan), according to the ASTM B213 standard. The powder morphology and internal macro-defects were determined using scanning electron microscopy (SEM, JEOL JCM-6000) and X-ray
Comparison of the characteristics of GA, PA, and PREP Ti–6Al–4V powders
The particle size distributions (PSDs) and flowability of the three types of Ti–6Al–4V alloy powders produced by GA, PA, and PREP are shown in Fig. 2. Although the average particle sizes are similar (89.4 μm for GA, 82.5 μm for PA, and 86.1μm for PREP), the particle size range is different for the three types of powder (6.2–174.8 μm for GA, 27.3–139.2 μm for PA, and 39.4–133.9 μm for PREP). The flowability of the GA, PA, and PREP powders was 30.25 ± 0.98, 26.54 ± 0.37, and 25.03 ± 0.22 (s/50
The characteristics of the three types of Ti–6Al–4V alloy powders produced via GA, PA, and PREP were compared. The PREP powder with the best sphericity, flowability, and low porosity was found to be the most favorable powder for subsequent printing of Ti–6Al–4V alloys without internal macro-defects. The quantitative criterion of Sdr <0.015 for even surfaces was also found to be applicable to Ti–6Al–4V alloys. The process maps of Ti–6Al–4V alloys include two regions, high beam current/scan speed
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.
This study was based on the results obtained from project JPNP19007, commissioned by the New Energy and Industrial Technology Development Organization (NEDO). This work was also supported by JSPS KAKENHI (Proposal No. 21K03801) and the Inter-University Cooperative Research Program (Proposal nos. 18G0418, 19G0411, and 20G0418) of the Cooperative Research and Development Center for Advanced Materials, Institute for Materials Research, Tohoku University. It was also supported by the Council for
이 프로젝트의 주요 목표는 FLOW-3D를 사용하여 계단식 여수로에서 스키밍 흐름의 수치 모델링을 개발하는 것입니다. 이러한 구조의 설계는 물리적 모델링에서 얻은 경험적 표현과 CFD 코드를 지원하는 계단식 여수로를 통한 흐름의 수치 모델링에서 보완 연구를 기반으로 합니다. 수치 모델은 균일한 영역의 유속과 계단 여수로의 마찰 계수를 추정하는 데 사용됩니다(ϴ = 45º, Hd=4.61m). 흐름에 대한 자동 통기의 표현은 복잡하므로 프로그램은 공기 연행 모델을 사용하여 특정 제한이 있는 솔루션에 근접합니다.
The main objective of this project is to develop the numerical modeling of the skimming flow in a stepped spillway using FLOW-3D. The design of these structures is based on the use of empirical expressions obtained from physical modeling and complementary studies in the numerical modeling of flow over the stepped spillway with support of CFD code. The numerical model is used to estimate the flow velocity in the uniform region and the friction coefficient of the stepped spillway (ϴ = 45º, Hd=4.61m). The representation of auto aeration a flow is complex, so the program approximates the solution with certain limitations, using an air entrainment model; drift flux model and turbulence model k-ԑ RNG. The results obtained with numerical modeling and physical modeling at the beginning of natural auto aeration of flow and depth of the biphasic flow in the uniform region presents deviations above to 10% perhaps the flow is highly turbulent.
ARAGUA. (2013). “Modelación numérica y experimental de flujos aire-agua en caídas en colectores.”, Laboratório Nacional de Engenharia Civil, I. P. Av do Brasil 101 • 1700-066 Lisboa. Bombardelli, F.A., Meireles, I. and Matos, J., (2010), “Laboratory measurement and multi-block numerical simulations of the mean flow and turbulence in the non-aerated skimming flow region of steep stepped spillways”, Environ Fluid Mechanics. Castro M. (2015) “Análisis Dimensional y Modelación física en Hidráulica”. Escuela Politécnica Nacional. Quito Ecuador. 50 p. Chanson H., D. B. Bung., J. Matos (2015). “Stepped spillways and cascades”. IAHR Monograph. School of Civil Engineering, University of Queensland, Brisbane, Australia. Chanson H. (1993). “Stepped Spillway Flows and Air Entrainment.” Can. Jl of Civil Eng., Vol. 20, No. 3, June, pp. 422-435 (ISSN 0315-1468). CIERHI, EPN TECH, (2016). “Estudio experimental en modelo físico de las rápidas con perfil escalonado y liso de la quebrada el Batán Fase I y Fase II”, Escuela Politécnica Nacional, Quito Ecuador. Fernández Oro J. M. (2012)., “Técnicas Numéricas en Ingeniería de Fluidos: Introducción a la Dinámica de Fluidos Computacional (CFD) por el Método de Volúmenes Finitos”. Barcelona: Reverté. Flow Science, Inc. (2012). “FLOW 3D 10.1.0 Documentation Release. Manual de Usuario”, Los Alamos National Laboratory. Santa Fe, New México Khatsuria, R.M., (2005)., “Hydraulics of Spillways and Energy Dissipators”. Department of Civil and Environmental Engineering Georgia Institute of Technology Atlanta. Lucio I., Matos J., Meireles I. (2015). “Stepped spillway flow over small embankment dams: some computational experiments”. 15th FLOW-3D European users conference. Mohammad S., Jalal A. and Michael P., (2012). “Numerical Computation of Inception Point Location for Steeply Sloping Stepped Spillways” 9th International Congress on Civil Engineering. Isfahan University of Technology (IUT), Isfahan, Iran Pfister M., Chanson H., (2013), “Scale Effects in Modelling Two-phase Airwater Flows”, Proceedings of 2013 IAHR World Congress. Sarfaraz, M. and Attari, J. (2011), “Numerical Simulation of Uniform Flow Region over a Steeply Sloping Stepped Spillway”, 6th National Congress on Civil Engineering, Semnan University, Semnan, Iran. Valero, D., Bung, D., (2015), “Hybrid investigation of air transport processes in moderately sloped stepped spillway flows”, E-proceedings of the 36th IAHR World Congress 28 June – 3 July, 2015, The Hague, the Netherlands.
Modeling of Mesh Screen for Use in Surface TensionTankUsing Flow-3d Software
Hyuntak Kim․ Sang Hyuk Lim․Hosung Yoon․Jeong-Bae Park*․Sejin Kwon†
Mesh screen modeling and liquid propellant discharge simulation of surface tension tank wereperformed using commercial CFD software Flow-3d. 350 × 2600, 400 × 3000 and 510 × 3600 DTW mesh screen were modeled using macroscopic porous media model. Porosity, capillary pressure, and drag coefficient were assigned for each mesh screen model, and bubble point simulations were performed. The mesh screen model was validated with the experimental data. Based on the screen modeling, liquidpropellant discharge simulation from PMD tank was performed. NTO was assigned as the liquidpropellant, and void was set to flow into the tank inlet to achieve an initial volume flowrate of liquid propellant in 3 × 10-3 g acceleration condition. The intial flow pressure drop through the meshscreen was approximately 270 Pa, and the pressure drop increased with time. Liquid propellant discharge was sustained until the flow pressure drop reached approximately 630 Pa, which was near the estimated bubble point value of the screen model.
상용 CFD 프로그램 Flow-3d를 활용하여, 표면 장력 탱크 적용을 위한 메시 스크린의 모델링 및 추진제 배출 해석을 수행하였다. Flow-3d 내 거시적 다공성 매체 모델을 사용하였으며, 350 × 2600, 400× 3000, 510 × 3600 DTW 메시 스크린에 대한 공극률, 모세관압, 항력계수를 스크린 모델에 대입 후, 기포점 측정 시뮬레이션을 수행하였다.
시뮬레이션 결과를 실험 데이터와 비교하였으며, 메시 스크린 모델링의 적절성을 검증하였다. 이를 기반으로 스크린 모델을 포함한 PMD 구조체에 대한 추진제 배출 해석을 수행하였다. 추진제는 액상의 NTO를 가정하였으며, 3 × 10-3 g 가속 조건에서 초기 유량을만족하도록 void를 유입시켰다. 메시 스크린을 통한 차압은 초기 약 270 Pa에서 시간에 따라 증가하였으며, 스크린 모델의 예상 기포점과 유사한 630 Pa에 이르기까지 액상 추진제 배출을 지속하였다.
Surface Tension Tank(표면장력 탱크), Propellant Management Device(추진제 관리 장치), Mesh Screen(메시 스크린), Porous Media Model(다공성 매체 모델), Bubble Point(기포점)
우주비행체를 미소 중력 조건 내에서 운용하 는 경우, 가압 기체가 액상의 추진제와 혼합되어 엔진으로 공급될 우려가 있으므로 이를 방지하 기 위한 탱크의 설계가 필요하다.
다이어프램 (Diaphragm), 피스톤(Piston) 등 다양한 장치들 이 활용되고 있으며, 이 중 표면 장력 탱크는 내 부의 메시 스크린(Mesh screen), 베인(Vane) 등 의 구조체에서 추진제의 표면장력을 활용함으로 써 액상 추진제의 이송 및 배출을 유도하는 방 식이다.
표면 장력 탱크는 구동부가 없는 구조로 신뢰성이 높고, 전 부분을 티타늄 등의 금속 재 질로 구성함으로써 부식성 추진제의 사용 조건 에서도 장기 운용이 가능한 장점이 있다. 위에서 언급한 메시 스크린(Mesh screen)은 수 십 마이크로미터 두께의 금속 와이어를 직조한 다공성 재질로 표면 장력 탱크의 핵심 구성 요소 중 하나이다.
미세 공극 상 추진제의 표면장력에 의해 기체와 액체 간 계면을 일정 차압 내에서 유지시킬 수 있다. 이러한 성질로 인해 일정 조 건에서 가압 기체가 메시 스크린을 통과하지 못 하게 되고, 스크린을 탱크 유로에 설치함으로써 액상의 추진제 배출을 유도할 수 있다.
메시 스크린이 가압 기체를 통과시키기 직전 의 기체-액체 계면에 형성되는 최대 차압을 기포 점 (Bubble point) 이라 칭하며, 메시 스크린의 주 요 성능 지표 중 하나이다. IPA, 물, LH2, LCH4 등 다양한 기준 유체 및 추진제, 다양한 메시 스 크린 사양에 대해 기포점 측정 관련 실험적 연 구가 이루어져 왔다 [1-3].
위 메시 스크린을 포함하여 표면 장력 탱크 내 액상의 추진제 배출을 유도하는 구조물 일체 를 PMD(Propellant management device)라 칭하 며, 갤러리(Gallery), 베인(Vane), 스펀지(Sponge), 트랩(Trap) 등 여러 종류의 구조물에 대해 각종 형상 변수를 내포한다[4, 5].
따라서 다양한 파라미터를 고려한 실험적 연구는 제약이 따를 수 있으며, 베인 등 상대적으로 작은 미소 중력 조건에서 개방형 유로를 활용하는 경우 지상 추진제 배출 실험이 불가능하다. 그러므로 CFD를 통한 표면장력 탱크 추진제 배출 해석은 다양한 작동 조건 및 PMD 형상 변수에 따른 추진제 거동을 이해하고, 탱크를 설계하는 데 유용하게 활용될 수 있다.
상기 추진제 배출 해석을 수행하기 위해서는 핵심 요소 중 하나인 메시 스크린에 대한 모델링이 필수적이다. Chato, McQuillen 등은 상용 CFD 프로그램인 Fluent를 통해, 갤러리 내 유동 시뮬레이션을 수행하였으며, 이 때 메시 스크린에 ‘porous jump’ 경계 조건을 적용함으로써 액상의 추진제가 스크린을 통과할 때 생기는 압력 강하를 모델링하였다[7, 8].
그러나 앞서 언급한 메시 스크린의 기포점 특성을 모델링한 사례는 찾아보기 힘들다. 이는 스크린을 활용하는 표면 장력 탱크 내 액상 추진제 배출 현상을 해석적으로 구현하기 위해 반드시 필요한 부분이다. 본 연구에서는 자유표면 해석에 상대적으로 강점을 지닌 상용 CFD 프로그램 Flow-3d를 사용하여, 메시 스크린을 모델링하였다.
거시적 다공성 매체 모델(Macroscopic porous mediamodel)을 활용하여 메시 스크린 모델 영역에 공극률(Porosity), 모세관압(Capillary pressure), 항력 계수(Drag coefficient)를 지정하고, 이를 기반으로 기포점 측정 시뮬레이션을 수행, 해석 결과와 실험 데이터 간 비교 및 검증을 수행하였다.
이를 기반으로 메시 스크린 및 PMD구조체를 포함한 탱크의 추진제 배출 해석을 수행하고, 기포점 특성의 반영 여부를 확인하였다.
참 고 문 헌
David J. C and Maureen T. K, ScreenChannel Liquid Aquisition Devices for Cryogenic Propellants” NASA-TM-2005- 213638, 2005
Hartwig, J., Mann, J. A. Jr., Darr, S. R., “Parametric Analysis of the LiquidHydrogen and Nitrogen Bubble Point Pressure for Cryogenic Liquid AcquisitionDevices”, Cryogenics, Vol. 63, 2014, pp. 25-36
Jurns, J. M., McQuillen, J. B.,BubblePoint Measurement with Liquid Methane of a Screen Capillary Liquid AcquisitionDevice”, NASA-TM-2009-215496, 2009
Jaekle, D. E. Jr., “Propellant Management Device: Conceptual Design and Analysis: Galleries”, AIAA 29th Joint PropulsionConference, AIAA-97-2811, 1997
Jaekle, D. E. Jr., “Propellant Management Device: Conceptual Design and Analysis: Traps and Troughs”, AIAA 31th Joint Propulsion Conference, AIAA-95-2531, 1995
Yu, A., Ji, B., Zhuang, B. T., Hu, Q., Luo, X. W., Xu, H. Y., “Flow Analysis inaVane-type Surface Tension Propellant Tank”, IOP Conference Series: MaterialsScience and Engineering, Vol. 52, No. 7, – 990 – 2013, Article number: 072018
Chato, D. J., McQuillen, J. B., Motil, B. J., Chao, D. F., Zhang, N., CFD simulation of Pressure Drops in Liquid Acquisition Device Channel with Sub-Cooled Oxygen”, World Academy of Science, Engineering and Technology, Vol. 3, 2009, pp. 144-149
McQuillen, J. B., Chao, D. F., Hall, N. R., Motil, B. J., Zhang, N., CFD simulation of Flow in Capillary Flow Liquid Acquisition Device Channel”, World Academy of Science, Engineering and Technology, Vol. 6, 2012, pp. 640-646
Hartwig, J., Chato, D., McQuillen, J., Screen Channel LAD Bubble Point Tests in Liquid Hydrogen”, International Journal of Hydrogen Energy, Vol. 39, No. 2, 2014, pp. 853-861
Fischer, A., Gerstmann, J., “Flow Resistance of Metallic Screens in Liquid, Gaseous and Cryogenic Flow”, 5th European Conferencefor Aeronautics and Space Sciences, Munich, Germany, 2013
Fries, N., Odic, K., Dreyer, M., Wickingof Perfectly Wetting Liquids into a MetallicMesh”, 2nd International Conference onPorous Media and its Applications inScience and Engineering, 2007
Seo, M, K., Kim, D, H., Seo, C, W., Lee, S, Y., Jang, S, P., Koo, J., “Experimental Study of Pressure Drop in CompressibleFluid through Porous Media”, Transactionsof the Korean Society of Mechanical Engineers – B, Vol. 37, No. 8, pp. 759-765, 2013.
Hartwig, J., Mann, J. A., “Bubble Point Pressures of Binary Methanol/Water Mixtures in Fine-Mesh Screens”, AlChEJournal, Vol. 60, No. 2, 2014, pp. 730-739
Hydraulic model test was used to analyze the rapidly varied flow on the spillway. But, it has some shortcomings such as error of scale effect and expensive costs. Recently, through the development of three dimensional computational fluid dynamics (CFD), rapidly varied flow and turbulence can be simulated. In this study, the applicability of CFD model to simulate flow on the spillway was reviewed. The Karian dam in Indonesia was selected as the study area. The FLOW-3d model, which is well known to simulate a flow having a free surface, was used to analyze flow. The flow stability in approach channel was investigated with the initial plan design, and the results showed that the flow in approach channel is unstable in the initial plan design. To improve flow stability in the spillway, therefore, the revised plan design was formulated. The appropriateness of the revised design was examined by a numerical modeling. The results showed that the flow in spillway is stable in the revised design.
여수로의 급격하게 변화하는 흐름을 분석하기 위해 수리학적 모델 테스트를 사용했습니다. 그러나 스케일 효과의 오차와 고가의 비용 등의 단점이 있다. 최근에는 3차원 전산유체역학(CFD)의 발달로 급변하는 유동과 난류를 모사할 수 있다. 본 연구에서는 여수로의 흐름을 시뮬레이션하기 위한 CFD 모델의 적용 가능성을 검토했습니다. 인도네시아의 Karian 댐이 연구 지역으로 선정되었습니다. 자유표면을 갖는 유동을 모의하는 것으로 잘 알려진 FLOW-3d 모델을 유동해석에 사용하였다. 접근수로의 흐름 안정성은 초기 계획설계와 함께 조사한 결과 초기 계획설계에서 접근수로의 흐름이 불안정한 것으로 나타났다. 따라서 방수로의 흐름 안정성을 향상시키기 위해 수정된 계획 설계가 공식화되었습니다. 수정된 설계의 적합성을 수치모델링을 통해 검토하였다. 결과는 수정된 설계에서 여수로의 흐름이 안정적이라는 것을 보여주었습니다.
Betts PL (1979). A variation principle in terms of stream function for free surface flows and its application to finite element method. Comp. Fluids, 7(2): 145-153. Cassidy JJ (1965). Irrotational flow over spillways of finite height. J. Eng. Mech. Div. ASCE., 91(6): 155-173. Flow Science (2002). FLOW-3D -Theory manual. Los Alamos, NM. Guo Y, Wen X, Wu C, Fang D (1998). Numerical modeling of spillway flow with free drop and initially unknown discharge. J. Hydraulic Res. IAHR, 36(5): 785-801. Ho DKH, Donohoo SM (2001). Investigation of spillway behavior under increased maximum flood by computational fluid dynamics technique. Proceeding 14 th Australasian Fluid Mech. Conference, Adelaide University, Adelaide, Australia, pp. 10-14. Ikegawa M, Washizu K (1973). Finite element method applied to analysis of flow over a spillway crest. Int. J. Numerical Methods Eng., 6: 179-189. Kim DG, Park JH (2005). Analysis of flow structure over ogee-spillway in consideration of scale and roughness effects by using CFD model. J. Civil Eng. KSCE., pp. 161-169. KRA, KWATER (2006). Feasibility study and detail design of the Karian dam project. Indonesia. Li W, Xie Q, Chen CJ (1989). Finite analytic solution of flow over spillways, J. Eng. Mech. ASCE, 115(2): 2645-2648. Olsen NR, Kjellesvig HM (1998).Three-dimensional numerical flow modeling for estimation of spillway capacity. J. Hydraulic Res. IAHR., 36(5): 775-784. Savage BM, Johnson MC (2001). Flow over ogee spillway: Physical and numerical model case study. J. Hydraulic Eng. ASCE., 127(8): 640- 649. Tabbara M, Chatial J, Awwad R (2005). Computational simulation of flow over stepped spillways. Comput. Structure, 83: 2215-2224.
Numerical simulation of ship waves in the presence of a uniform current
CongfangAiYuxiangMaLeiSunGuohaiDongState Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China
• Ship waves in the presence of a uniform current are studied by a non-hydrostatic model.
• Effects of a following current on characteristic wave parameters are investigated.
• Effects of an opposing current on characteristic wave parameters are investigated.
• The response of the maximum water level elevation to the ship draft is discussed.
이 논문은 균일한 해류가 존재할 때 선박파의 생성 및 전파를 시뮬레이션하기 위한 비정역학적 모델을 제시합니다. 선박 선체의 움직임을 표현하기 위해 움직이는 압력장 방법이 모델에 통합되었습니다.
뒤따르거나 반대 방향의 균일한 흐름이 있는 경우의 선박 파도의 수치 결과를 흐름이 없는 선박 파도의 수치 결과와 비교합니다. 추종 또는 반대 균일 전류가 존재할 때 계산된 첨단선 각도는 분석 솔루션과 잘 일치합니다. 추종 균일 전류와 반대 균일 전류가 특성파 매개변수에 미치는 영향을 제시하고 논의합니다.
선박 흘수에 대한 최대 수위 상승의 응답은 추종 또는 반대의 균일한 흐름이 있는 경우에도 표시되며 흐름이 없는 선박 파도의 응답과 비교됩니다. 선박 선체 측면의 최대 수위 상승은 Froude 수 Fr’=Us/gh의 특정 범위에 대해 다음과 같은 균일한 흐름의 존재에 의해 증가될 수 있음이 밝혀졌습니다.
여기서 Us는 선박 속도이고 h는 물입니다. 깊이. 균일한 해류를 무시하면 추종류나 반대류가 존재할 때 선박 흘수에 대한 최대 수위 상승의 응답이 과소평가될 수 있습니다.
본 연구는 선박파의 해석에 있어 균일한 해류의 영향을 고려해야 함을 시사합니다.
This paper presents a non-hydrostatic model to simulate the generation and propagation of ship waves in the presence of a uniform current. A moving pressure field method is incorporated into the model to represent the movement of a ship hull. Numerical results of ship waves in the presence of a following or an opposing uniform current are compared with those of ship waves without current. The calculated cusp-line angles in the presence of a following or opposing uniform current agree well with analytical solutions. The effects of a following uniform current and an opposing uniform current on the characteristic wave parameters are presented and discussed. The response of the maximum water level elevation to the ship draft is also presented in the presence of a following or an opposing uniform current and is compared with that for ship waves without current. It is found that the maximum water level elevation lateral to the ship hull can be increased by the presence of a following uniform current for a certain range of Froude numbers Fr′=Us/gh, where Us is the ship speed and h is the water depth. If the uniform current is neglected, the response of the maximum water level elevation to the ship draft in the presence of a following or an opposing current can be underestimated. The present study indicates that the effect of a uniform current should be considered in the analysis of ship waves.
Ship waves, Non-hydrostatic model, Following current, Opposing current, Wave parameters
Similar to wind waves, ships sailing across the sea can also create free-surface undulations ranging from ripples to waves of large size (Grue, 2017, 2020). Ship waves can cause sediment suspension and engineering structures damage and even pose a threat to flora and fauna living near the embankments of waterways (Dempwolff et al., 2022). It is quite important to understand ship waves in various environments. The study of ship waves has been conducted over a century. A large amount of research (Almström et al., 2021; Bayraktar and Beji, 2013; David et al., 2017; Ertekin et al., 1986; Gourlay, 2001; Havelock, 1908; Lee and Lee, 2019; Samaras and Karambas, 2021; Shi et al., 2018) focused on the generation and propagation of ship waves without current. When a ship navigates in the sea or in a river where tidal flows or river flows always exist, the effect of currents should be taken into account. However, the effect of currents on the characteristic parameters of ship waves is still unclear, because very few publications have been presented on this topic.
In addition to Boussinesq-type models, numerical models based on the Navier-Stokes equations (NSE) or Euler equations are also capable of resolving ship waves. Lee and Lee (2019, 2021) employed the FLOW-3D model to simulate ship waves without current and ship waves in the presence of a uniform current to confirm their equations for ship wave crests. FLOW-3D is a computational fluid dynamics (CFD) software based on the NSE, and the volume of fluid (VOF) method is used to capture the moving free surface. However, VOF-based NSE models are computationally expensive due to the treatment of the free surface. To efficiently track the free surface, non-hydrostatic models employ the so-called free surface equation and can be solved efficiently. One pioneering application for the simulation of ship waves by the non-hydrostatic model was initiated by Ma (2012) and named XBeach. Recently, Almström et al. (2021) validated XBeach with improved dispersive behavior by comparison with field measurements. XBeach employed in Almström et al. (2021) is a 2-layer non-hydrostatic model and is accurate up to Kh=4 for the linear dispersion relation (de Ridder et al., 2020), where K=2π/L is the wavenumber. L is the wavelength, and h is the still water depth. However, no applications of non-hydrostatic models on the simulation of ship waves in the presence of a uniform current have been published. For more advances in the numerical modelling of ship waves, the reader is referred to Dempwolff et al. (2022).
This paper investigates ship waves in the presence of a uniform current by using a non-hydrostatic model (Ai et al., 2019), in which a moving pressure field method is incorporated to represent the movement of a ship hull. The model solves the incompressible Euler equations by using a semi-implicit algorithm and is associated with iterating to solve the Poisson equation. The model with two, three and five layers is accurate up to Kh= 7, 15 and 40, respectively (Ai et al., 2019) in resolving the linear dispersion relation. To the best of our knowledge, ship waves in the presence of currents have been studied theoretically (Benjamin et al., 2017; Ellingsen, 2014; Li and Ellingsen, 2016; Li et al., 2019.) and numerically (Dam et al., 2008; Lee and Lee, 2019, 2021). However, no publications have presented the effects of a uniform current on characteristic wave parameters except for Dam et al. (2008), who investigated only the effect of currents on the maximum wave height in a narrow channel for the narrow relative Froude number Fr=(Us−Uc)/gh ranging from 0.47 to 0.76, where Us is the ship speed and Uc is the current velocity. To reveal the effect of currents on the characteristic parameters of ship waves, the main objectives of this paper are (1) to validate the capability of the proposed model to resolve ship waves in the presence of a uniform current, (2) to investigate the effects of a following or an opposing current on characteristic wave parameters including the maximum water level elevation and the leading wave period in the ship wave train, (3) to show the differences in characteristic wave parameters between ship waves in the presence of a uniform current and those without current when the same relative Froude number Fr is specified, and (4) to examine the response of the maximum water level elevation to the ship draft in the presence of a uniform current.
The remainder of this paper is organized as follows. The non-hydrostatic model for ship waves is described in Section 2. Section 3 presents numerical validations for ship waves. Numerical results and discussions about the effects of a uniform current on characteristic wave parameters are provided in Section 4, and a conclusion is presented in Section 5.
2. Non-hydrostatic model for ship waves
2.1. Governing equations
The 3D incompressible Euler equations are expressed in the following form:(1)∂u∂x+∂v∂y+∂w∂z=0(2)∂u∂t+∂u2∂x+∂uv∂y+∂uw∂z=−∂p∂x(3)∂v∂t+∂uv∂x+∂v2∂y+∂vw∂z=−∂p∂y(4)∂w∂t+∂uw∂x+∂vw∂y+∂w2∂z=−∂p∂z−gwhere t is the time; u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) are the velocity components in the horizontal x, y and vertical z directions, respectively; p(x,y,z,t) is the pressure divided by a constant reference density; and g is the gravitational acceleration.
The pressure p(x,y,z,t) can be expressed as(5)p=ps+g(η−z)+qwhere ps(x,y,t) is the pressure at the free surface, η(x,y,t) is the free surface elevation, and q(x,y,z,t) is the non-hydrostatic pressure.
η(x,y,t) is calculated by the following free-surface equation:(6)∂η∂t+∂∂x∫−hηudz+∂∂y∫−hηvdz=0where z=−h(x,y) is the bottom surface.
For −L/2≤x’≤L/2,−B/2≤y’≤B/2(7)ps(x,y,t)|t=0=pm[1−cL(x′/L)4][1−cB(y′/B)2]exp[−a(y′/B)2]where x′=x−x0 and y′=y−y0. (x0,y0) is the center of the pressure field, pm is the peak pressure defined at (x0,y0), and L and B are the lengthwise and breadthwise parameters, respectively. cL, cB and a are set to 16, 2 and 16, respectively.
2.2. Numerical algorithms
In this study, the generation of ship waves is incorporated into the semi-implicit non-hydrostatic model developed by Ai et al. (2019). The 3D grid system used in the model is built from horizontal rectangular grids by adding horizontal layers. The horizontal layers are distributed uniformly along the water depth, which means the layer thickness is defined by Δz=(η+h)/Nz, where Nz is the number of horizontal layers.
In the solution procedure, the first step is to generate ship waves by implementing Eq. (7) together with the prescribed ship track. In the second step, Eqs. (1), (2), (3), (4) are solved by the pressure correction method, which can be subdivided into three stages. The first stage is to compute intermediate velocities un+1/2, vn+1/2, and wn+1/2 by solving Eqs. (2), (3), (4), which contain the non-hydrostatic pressure at the preceding time level. In the second stage, the Poisson equation for the non-hydrostatic pressure correction term is solved on the graphics processing unit (GPU) in conjunction with the conjugate gradient method. The third stage is to compute the new velocities un+1, vn+1, and wn+1 by correcting the intermediate values after including the non-hydrostatic pressure correction term. In the discretization of Eqs. (2), (3), the gradient terms of the water surface ∂η/∂x and ∂η/∂y are discretized by means of the semi-implicit method (Vitousek and Fringer, 2013), in which the implicitness factor θ=0.5 is used. The model is second-order accurate in time for free-surface flows. More details about the model can be found in Ai et al. (2019).
3. Model validation
In this section, we validate the proposed model in resolving ship waves. The numerical experimental conditions are provided in Table 1 and Table 2. In Table 2, Case A with the current velocity of Uc = 0.0 m/s represents ship waves without current. Both Case B and Case C correspond to the cases in the presence of a following current, while Case D and Case E represent the cases in the presence of an opposing current. The current velocities are chosen based on the observed currents at 40.886° N, 121.812° E, which is in the Liaohe Estuary. The measured data were collected from 14:00 on September 18 (GMT + 08:00) to 19:00 on September 19 in 2021. The maximum flood velocity is 1.457 m/s, and the maximum ebb velocity is −1.478 m/s. The chosen current velocities are between the maximum flood velocity and the maximum ebb velocity.
Table 1. Summary of ship speeds.
Water depth h (m)
Ship speed Us (m/s)
Froude number Fr′=Us/gh
Table 2. Summary of current velocities.
Current velocity Uc (m/s)
Notably, the Froude number Fr′=Us/gh presented in Table 1 is defined by the ship speed Us only and is different from the relative Froude number Fr when a uniform current is presented. According to the theory of Lee and Lee (2021), with the same relative Froude number, the cusp-line angles in the presence of a following or an opposing uniform current are identical to those without current. As a result, for the test cases presented in Table 1, Table 2, all calculated cusp-line angles follow the analytical solution of Havelock (1908), when the relative Froude number Fr is introduced.
As shown in Fig. 1, the dimensions of the computational domain are −420≤x≤420 m and −200≤y≤200 m, which are similar to those of David et al. (2017). The ship track follows the x axis and ranges from −384 m to 384 m. The ship hull is represented by Eq. (7), in which the length L and the beam B are set to 14.0 m and 7.0 m, respectively, and the peak pressure value is pm= 5000 Pa. In the numerical simulations, grid convergence tests reveal that the horizontal grid spacing of Δx=Δy= 1.0 m and two horizontal layers are adequate. The numerical results with different numbers of horizontal layers are shown in the Appendix.
Fig. 2, Fig. 3 compare the calculated cusp-line angles θc with the analytical solutions of Havelock (1908) for ship waves in the presence of a following uniform current and an opposing uniform current, respectively. The calculated cusp-line angles without current are also depicted in Fig. 2, Fig. 3. All calculated cusp-line angles are in good agreement with the analytical solutions, except that the model tends to underpredict the cusp-line angle for 0.9<Fr<1.0. Notably, a similar underprediction of the cusp-line angle can also be found in David et al. (2017).
4. Results and discussions
This section presents the effects of a following current and opposing current on the maximum water level elevation and the leading wave period in the wave train based on the test cases presented in Table 1, Table 2. Moreover, the response of the maximum water level elevation to the ship draft in the presence of a uniform current is examined.
4.1. Effects of a following current on characteristic wave parameters
To present the effect of a following current on the maximum wave height, the variations of the maximum water level elevation ηmax with the Froude number Fr′ at gauge points G1 and G2 are depicted in Fig. 4. The positions of gauge points G1 and G2 are shown in Fig. 1. The maximum water level elevation is an analogue to the maximum wave height and is presented in this study, because maximum wave heights at different positions away from the ship track vary throughout the wave train (David et al., 2017). In general, the variations of ηmax with the Froude number Fr′ in the three cases show a similar behavior, in which with the increase in Fr′, ηmax increases and then decreases. The presence of the following currents decreases ηmax for Fr′≤0.8 and Fr′≥1.2. Specifically, the following currents have a significant effect on ηmax for Fr′≤0.8. Notably, ηmax can be increased by the presence of the following currents for 0.9≤Fr′≤1.1. Compared with Case A, at location G1 ηmax is amplified 1.25 times at Fr′=0.925 in Case B and 1.31 times at Fr′=1.025 in Case C. Similarly, at location G2 ηmax is amplified 1.15 times at Fr′=1.025 in Case B and 1.11 times at Fr′=1.075 in Case C. The fact that ηmax can be increased by the presence of a following current for 0.9≤Fr′≤1.1 implies that if a following uniform current is neglected, then ηmax may be underestimated.
To show the effect of a following current on the wave period, Fig. 5 depicts the variation of the leading wave period Tp in the wave train at gauge point G2 with the Froude number Fr′. Similar to David et al. (2017), Tp is defined by the wave period of the first wave with a leading trough in the wave train. The leading wave periods for Fr′= 0.6 and 0.7 were not given in Case B and Case C, because the leading wave heights for Fr′= 0.6 and 0.7 are too small to discern the leading wave periods. Compared with Case A, the presence of a following current leads to a larger Tp for 0.925≤Fr′≤1.1 and a smaller Tp for Fr′≥1.3. For Fr′= 0.8 and 0.9, Tp in Case B is larger than that in Case A and Tp in Case C is smaller than that in Case A. In all three cases, Tp decreases with increasing Fr′ for Fr′>1.0. However, this decreasing trend becomes very gentle after Fr′≥1.4. Notably, as shown in Fig. 5, Fr′=1.2 tends to be a transition point at which the following currents have a very limited effect on Tp. Moreover, before the transition point, Tp in Case B and Case C are larger than that in Case A (only for 0.925≤Fr′≤1.2), but after the transition point the reverse is true.
As mentioned previously, the cusp-line angles for ship waves in the presence of a following or an opposing current are identical to those for ship waves only with the same relative Froude number Fr. However, with the same Fr, the characteristic parameters of ship waves in the presence of a following or an opposing current are quite different from those of ship waves without current. Fig. 6 shows the variations of the maximum water level elevation ηmax with Fr at gauge points G1 and G2 for ship waves in the presence of a following uniform current. Overall, the relationship curves between ηmax and Fr in Case B and Case C are lower than those in Case A. It is inferred that with the same Fr, ηmax in the presence of a following current is smaller than that without current. Fig. 7 shows the variation of the leading wave period Tp in the wave train at gauge point G2 with Fr for ship waves in the presence of a following uniform current. The overall relationship curves between Tp and Fr in Case B and Case C are also lower than those in Case A for 0.9≤Fr≤2.0. It can be inferred that with the same Fr, Tp in the presence of a following current is smaller than that without current for Fr≥0.9.
To compare the numerical results between the case of ship waves only and the case of ship waves in the presence of a following current with the same Fr, Fig. 8 shows the wave patterns for Fr=1.2. To obtain the case of ship waves in the presence of a following current with Fr=1.2, the ship speed Us=9.7 m/s and the current velocity Uc=0.5 m/s are adopted. Fig. 8 indicates that both the calculated cusp-line angles for the case of Us=9.2 m/s and Uc=0.0 m/s and the case of Us=9.7 m/s and Uc=0.5 m/s are equal to 56.5°, which follows the theory of Lee and Lee (2021). Fig. 9 depicts the comparison of the time histories of the free surface elevation at gauge point G2 for Fr=1.2 between the case of ship waves only and the case of ship waves in the presence of a following current. The time when the ship wave just arrived at gauge point G2 is defined as t′=0. Both the maximum water level elevation and the leading wave period in the case of Us=9.2 m/s and Uc=0.0 m/s are larger than those in the case of Us=9.7 m/s and Uc=0.5 m/s, which is consistent with the inferences based on Fig. 6, Fig. 7.
Fig. 10 shows the response of the maximum water level elevation ηmax to the ship draft at gauge point G2 for Fr′= 1.2 in the presence of a following uniform current. pm ranges from 2500 Pa to 40,000 Pa with an interval of Δp= 2500 Pa pm0= 2500 Pa represents a reference case. ηmax0 denotes the maximum water level elevation corresponding to the case of pm0= 2500 Pa. The best-fit linear trend lines obtained by linear regression analysis for the three responses are also depicted in Fig. 10. In general, all responses of ηmax to the ship draft show a linear relationship. The coefficients of determination for the three linear trend lines are R2= 0.9901, 0.9941 and 0.9991 for Case A, Case B and Case C, respectively. R2 is used to measure how close the numerical results are to the linear trend lines. The closer R2 is to 1.0, the more linear the numerical results tend to be. As a result, the relationship curve between ηmax and the ship draft in the presence of a following uniform current tends to be more linear than that without current. Notably, with the increase in pmpm0, ηmax increases faster in Case B and Case C than Case A. This implies that neglecting the following currents can lead to the underestimation of the response of ηmax to the ship draft.
4.2. Effects of an opposing current on characteristic wave parameters
Fig. 11 shows the variations of the maximum water level elevation ηmax with the Froude number Fr′ at gauge points G1 and G2 for ship waves in the presence of an opposing uniform current. The presence of opposing uniform currents leads to a significant reduction in ηmax at the two gauge points for 0.6≤Fr′≤2.0. Especially for Fr′=0.6, the decrease in ηmax is up to 73.8% in Case D and 78.4% in Case E at location G1 and up to 93.8% in Case D and 95.3% in Case E at location G2 when compared with Case A. Fig. 12 shows the variations of the leading wave period Tp at gauge point G2 with the Froude number Fr′ for ship waves in the presence of an opposing uniform current. The leading wave periods for Fr′= 0.6 and 0.7 were also not provided in Case D and Case E due to the small leading wave heights. In general, Tp decreases with increasing Fr′ in Case D and Case E for 0.8≤Fr′≤2.0. Tp in Case D and Case E are larger than that in Case A for Fr′≥1.0.
Fig. 13 depicts the variations of the maximum water level elevation ηmax with the relative Froude number Fr at gauge points G1 and G2 for ship waves in the presence of an opposing uniform current. Similar to Case B and Case C shown in Fig. 6, the overall relationship curves between ηmax and Fr in Case D and Case E are lower than those in Case A. This implies that with the same Fr, ηmax in the presence of an opposing current is also smaller than that without current. Fig. 14 depicts the variations of the leading wave period Tp in the wave train at gauge point G2 with Fr for ship waves in the presence of an opposing uniform current. Similar to Case B and Case C shown in Fig. 7, the overall relationship curves between Tp and Fr in Case D and Case E are lower than those in Case A for 0.9≤Fr≤2.0. This also implies that with the same Fr, Tp in the presence of an opposing current is smaller than that without current.
Fig. 15 shows a comparison of the wave pattern for Fr=1.2 between the case of ship waves only and the case of ship waves in the presence of an opposing current. The case of the ship wave in the presence of an opposing current with Fr=1.2 is obtained by setting the ship speed Us=8.7 m/s and the current velocity Uc=−0.5 m/s. As expected (Lee and Lee, 2021), both calculated cusp-line angles are identical. Fig. 16 depicts the comparison of the time histories of the free surface elevation at gauge point G2 for Fr=1.2 between the case of ship waves only and the case of ship waves in the presence of an opposing current. The maximum water level elevation in the case of Us=9.2 m/s and Uc=0.0 m/s is larger than that in the case of Us=8.7 m/s and Uc=−0.5 m/s, while the reverse is true for the leading wave period. Fig. 16 is consistent with the inferences based on Fig. 13, Fig. 14.
Fig. 17 depicts the response of the maximum water level elevation ηmax to the ship draft at gauge point G2 for Fr′= 1.2 in the presence of an opposing uniform current. Similarly, the response of ηmax to the ship draft in the presence of an opposing uniform current shows a linear relationship. The coefficients of determination for the three linear trend lines are R2= 0.9901, 0.9955 and 0.9987 for Case A, Case D and Case E, respectively. This indicates that the relationship curve between ηmax and the ship draft in the presence of an opposing uniform current also tends to be more linear than that without current. In addition, ηmax increases faster with increasing pmpm0 in Case D and Case E than Case A, implying that the response of ηmax to the ship draft can also be underestimated by neglecting opposing currents.
A non-hydrostatic model incorporating a moving pressure field method was used to investigate characteristic wave parameters for ship waves in the presence of a uniform current. The calculated cusp-line angles for ship waves in the presence of a following or an opposing uniform current were in good agreement with analytical solutions, demonstrating that the proposed model can accurately resolve ship waves in the presence of a uniform current.
The model results showed that the presence of a following current can result in an increase in the maximum water level elevation ηmax for 0.9≤Fr′≤1.1, while the presence of an opposing current leads to a significant reduction in ηmax for 0.6≤Fr′≤2.0. The leading wave period Tp can be increased for 0.925≤Fr′≤1.2 and reduced for Fr′≥1.3 due to the presence of a following current. However, the presence of an opposing current leads to an increase in Tp for Fr′≥1.0.
Although with the same relative Froude number Fr, the cusp-line angles for ship waves in the presence of a following or an opposing current are identical to those for ship waves without current, the maximum water level elevation ηmax and leading wave period Tp in the presence of a following or an opposing current are quite different from those without current. The present model results imply that with the same Fr, ηmax in the presence of a following or an opposing current is smaller than that without current for Fr≥0.6, and Tp in the presence of a following or an opposing current is smaller than that without current for Fr≥0.9.
The response of ηmax to the ship draft in the presence of a following current or an opposing current is similar to that without current and shows a linear relationship. However, the presence of a following or an opposing uniform current results in more linear responses of ηmax to the ship draft. Moreover, more rapid responses of ηmax to the ship draft are obtained when a following current or an opposing current is presented. This implies that the response of ηmax to the ship draft in the presence of a following current or an opposing current can be underestimated if the uniform current is neglected.
The present results have implications for ships sailing across estuarine and coastal environments, where river flows or tidal flows are significant. In these environments, ship waves can be larger than expected and the response of the maximum water level elevation to the ship draft may be more remarkable. The effect of a uniform current should be considered in the analysis of ship waves.
The present study considered only slender-body type ships. For different hull shapes, the effects of a uniform current on characteristic wave parameters need to be further investigated. Moreover, the effects of an oblique uniform current on ship waves need to be examined in future work.
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.
This research is financially supported by the National Natural Science Foundation of China (Grant No. 52171248, 51720105010, 51979029), LiaoNing Revitalization Talents Program (Grant No. XLYC1807010) and the Fundamental Research Funds for the Central Universities (Grant No. DUT21LK01).
Appendix. Numerical results with different numbers of horizontal layers
Fig. 18 shows comparisons of the time histories of the free surface elevation at gauge point G1 for Case B and Fr′= 1.2 between the three sets of numerical results with different numbers of horizontal layers. The maximum water level elevations ηmax obtained by Nz= 3 and 4 are 0.24% and 0.35% larger than ηmax with Nz= 2, respectively. Correspondingly, the leading wave periods Tp obtained by Nz= 3 and 4 are 0.45% and 0.55% larger than Tp with Nz= 2, respectively. In general, the three sets of numerical results are very close. To reduce the computational cost, two horizontal layers Nz= 2 were chosen for this study.