그림 10. 수문이 고르지 않게 열리는 경우의 시뮬레이션 결과

홍수 시즌에 하수구를 운영할 때 흐름 회로를 제어하는 ​​기술, 푸토코무네 제방을 통해 제방에 적용

요약

대규모 홍수 구호 작업에 대한 일반적인 흐름 회로 현상의 영향은 많은 보고서에서 연구되었으며 비교적 자세하게 연구되었습니다. 그러나 유량 변동이 제방 암거 작동에 미치는 악영향에 대해서는 많이 언급되지 않았습니다.

실제 운영에 따르면 에너지 소산 탱크 또는 뒷마당의 국부적 불안정성, 운하 지붕의 부력 또는 붕괴, 하류 또는 제방 본체 일부 등 암거 구조 및 제방 시스템에 영향을 미치는 흐름 회로의 부정적인 영향이 많이 있음을 알 수 있습니다.

홍수기 운영 시 암거 및 제방의 안전성을 확보하기 위해서는 유동현상으로 인한 사업에 미치는 유해영향을 최소화할 필요가 있으며 본 연구에서는 이를 검증하고자 합니다.

제방을 통과하는 암거의 동적 회로 현상은 FLOW-3D 소프트웨어를 사용하여 수학적 모델로 시뮬레이션됩니다. 제방을 통한 암거 작동에 잠재적으로 영향을 미칠 수 있는 동맥 유형이 연구에서 구체적으로 논의됩니다.

동시에 암거와 제방 위치는 이 기사에서 지적한 흐름 회로 현상에 의해 부정적인 영향을 받을 가능성이 더 높습니다. 또한 Phuc Tho 마을의 제방을 가로지르는 암거 게이트 뒤의 회로 현상을 줄이기 위한 구조적, 비구조적 조치도 연구되고 평가됩니다.

이를 토대로 운영하수관로의 구조를 저해하지 않고 회로를 축소할 수 있는 방안을 제안한다.

The flow fluctuation has been studied in quite extensively for large-scale flood control works, however, this issue has been less addressed for culverts through levee. The operational experience has shown that there are many negative impacts of flow dynamics on the culvert structure and levee system such as the uplift instability, the local surface erosion of the stilling basin or the downstream channel, collapsing of part of the levee system, etc. According to the requirement of sluice and levee safety during flood season, the task of reducing fluctuation needs to be performed. The article not only pointed out the types of fluctuation that need to pay attention behind the operation gate, but also specified the locations where the sluice and levee could be destructively affected by the fluctuation. In addition, structural and non-structural countermeasures reducing negative impacts of fluctuation are also mentioned. Research has proposed measures to reduce flow dynamics for operating culverts without interfering with their structure.

키워드

Fluctuation, sluice, stilling basin, 흐름회로, 제방암거, 에너지소산조, Flow3D, 회로저감솔루션.

그림 10. 수문이 고르지 않게 열리는 경우의 시뮬레이션 결과
그림 10. 수문이 고르지 않게 열리는 경우의 시뮬레이션 결과

References

Hệ thống giám sát thiên tai Việt Nam (2023). http://vndms.dmc.gov.vn/ (Accessed: 28 February 2023).
Viện Thủy Công (2015) Nghiên cứu đánh giá các sự cố đê, cống dưới đê và đề xuất giải pháp
xử lý. HyCI.
Nguyễn Phương Dung (2017) Thí nghiệm xác định ảnh hưởng của áp lực thủy động tới độ dày bản
đáy bể tiêu năng và sân sau ở công trình tháo. TLU.
Nguyễn Chiến (2015) Tính toán thủy lực công trình tháo nước. NXB Xây dựng.
Nguyễn Ngọc Thắng (2019) Đề cương Đề tài cấp Bộ ‘Nghiên cứu đánh giá nguyên nhân, các biện
pháp đã áp dụng và đề xuất giải pháp xử lý sự cố cống dưới đê đảm bảo an toàn chống lũ’.
‘Flow 3D’ (2023). (User’s Manual).
Yong Peng (2018) ‘Experimental Optimization of Gate-Opening Modes to Minimize Near-Field
Vibrations in Hydropower Stations’, Water, 10(1435). Available at:
https://doi.org/doi:10.3390/w10101435.
Guibing HUANG (2021) ‘Pressure Fluctuations Characteristics of the Stilling Basin with a
Negative Step’, Hydraulic and Civil Engineering Technology VI [Preprint]. Available at:
https://doi.org/doi:10.3233/ATDE210196.
O. Fecarotta (2016) Experimental results on the physical model of an USBR type II stilling basin.
Taylor & Francis Group.

Schematic diagram of HP-LPBF melting process.

Modeling and numerical studies of high-precision laser powder bed fusion

Yi Wei ;Genyu Chen;Nengru Tao;Wei Zhou
https://doi.org/10.1063/5.0191504

In order to comprehensively reveal the evolutionary dynamics of the molten pool and the state of motion of the fluid during the high-precision laser powder bed fusion (HP-LPBF) process, this study aims to deeply investigate the specific manifestations of the multiphase flow, solidification phenomena, and heat transfer during the process by means of numerical simulation methods. Numerical simulation models of SS316L single-layer HP-LPBF formation with single and double tracks were constructed using the discrete element method and the computational fluid dynamics method. The effects of various factors such as Marangoni convection, surface tension, vapor recoil, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool have been paid attention to during the model construction process. The results show that the molten pool exhibits a “comet” shape, in which the temperature gradient at the front end of the pool is significantly larger than that at the tail end, with the highest temperature gradient up to 1.69 × 108 K/s. It is also found that the depth of the second track is larger than that of the first one, and the process parameter window has been determined preliminarily. In addition, the application of HP-LPBF technology helps to reduce the surface roughness and minimize the forming size.

Topics

Heat transferNonequilibrium thermodynamicsSolidification processComputer simulationDiscrete element methodLasersMass transferFluid mechanicsComputational fluid dynamicsMultiphase flows

I. INTRODUCTION

Laser powder bed fusion (LPBF) has become a research hotspot in the field of additive manufacturing of metals due to its advantages of high-dimensional accuracy, good surface quality, high density, and high material utilization.1,2 With the rapid development of electronics, medical, automotive, biotechnology, energy, communication, and optics, the demand for microfabrication technology is increasing day by day.3 High-precision laser powder bed fusion (HP-LPBF) is one of the key manufacturing technologies for tiny parts in the fields of electronics, medical, automotive, biotechnology, energy, communication, and optics because of its process characteristics such as small focal spot diameter, small powder particle size, and thin powder layup layer thickness.4–13 Compared with LPBF, HP-LPBF has the significant advantages of smaller focal spot diameter, smaller powder particle size, and thinner layer thickness. These advantages make HP-LPBF perform better in producing micro-fine parts, high surface quality, and parts with excellent mechanical properties.

HP-LPBF is in the exploratory stage, and researchers have already done some exploratory studies on the focal spot diameter, the amount of defocusing, and the powder particle size. In order to explore the influence of changing the laser focal spot diameter on the LPBF process characteristics of the law, Wildman et al.14 studied five groups of different focal spot diameter LPBF forming 316L stainless steel (SS316L) processing effect, the smallest focal spot diameter of 26 μm, and the results confirm that changing the focal spot diameter can be achieved to achieve the energy control, so as to control the quality of forming. Subsequently, Mclouth et al.15 proposed the laser out-of-focus amount (focal spot diameter) parameter, which characterizes the distance between the forming plane and the laser focal plane. The laser energy density was controlled by varying the defocusing amount while keeping the laser parameters constant. Sample preparation at different focal positions was investigated, and their microstructures were characterized. The results show that the samples at the focal plane have finer microstructure than those away from the focal plane, which is the effect of higher power density and smaller focal spot diameter. In order to explore the influence of changing the powder particle size on the characteristics of the LPBF process, Qian et al.16 carried out single-track scanning simulations on powder beds with average powder particle sizes of 70 and 40 μm, respectively, and the results showed that the melt tracks sizes were close to each other under the same process parameters for the two particle-size distributions and that the molten pool of powder beds with small particles was more elongated and the edges of the melt tracks were relatively flat. In order to explore the superiority of HP-LPBF technology, Xu et al.17 conducted a comparative analysis of HP-LPBF and conventional LPBF of SS316L. The results showed that the average surface roughness of the top surface after forming by HP-LPBF could reach 3.40 μm. Once again, it was verified that HP-LPBF had higher forming quality than conventional LPBF. On this basis, Wei et al.6 comparatively analyzed the effects of different laser focal spot diameters on different powder particle sizes formed by LPBF. The results showed that the smaller the laser focal spot diameter, the fewer the defects on the top and side surfaces. The above research results confirm that reducing the laser focal spot diameter can obtain higher energy density and thus better forming quality.

LPBF involves a variety of complex systems and mechanisms, and the final quality of the part is influenced by a large number of process parameters.18–24 Some research results have shown that there are more than 50 factors affecting the quality of the specimen. The influencing factors are mainly categorized into three main groups: (1) laser parameters, (2) powder parameters, and (3) equipment parameters, which interact with each other to determine the final specimen quality. With the continuous development of technologies such as computational materials science and computational fluid dynamics (CFD), the method of studying the influence of different factors on the forming quality of LPBF forming process has been shifted from time-consuming and laborious experimental characterization to the use of numerical simulation methods. As a result, more and more researchers are adopting this approach for their studies. Currently, numerical simulation studies on LPBF are mainly focused on the exploration of molten pool, temperature distribution, and residual stresses.

  1. Finite element simulation based on continuum mechanics and free surface fluid flow modeling based on fluid dynamics are two common approaches to study the behavior of LPBF molten pool.25–28 Finite element simulation focuses on the temperature and thermal stress fields, treats the powder bed as a continuum, and determines the molten pool size by plotting the elemental temperature above the melting point. In contrast, fluid dynamics modeling can simulate the 2D or 3D morphology of the metal powder pile and obtain the powder size and distribution by certain algorithms.29 The flow in the molten pool is mainly affected by recoil pressure and the Marangoni effect. By simulating the molten pool formation, it is possible to predict defects, molten pool shape, and flow characteristics, as well as the effect of process parameters on the molten pool geometry.30–34 In addition, other researchers have been conducted to optimize the laser processing parameters through different simulation methods and experimental data.35–46 Crystal growth during solidification is studied to further understand the effect of laser parameters on dendritic morphology and solute segregation.47–54 A multi-scale system has been developed to describe the fused deposition process during 3D printing, which is combined with the conductive heat transfer model and the dendritic solidification model.55,56
  2. Relevant scholars have adopted various different methods for simulation, such as sequential coupling theory,57 Lagrangian and Eulerian thermal models,58 birth–death element method,25 and finite element method,59 in order to reveal the physical phenomena of the laser melting process and optimize the process parameters. Luo et al.60 compared the LPBF temperature field and molten pool under double ellipsoidal and Gaussian heat sources by ANSYS APDL and found that the diffusion of the laser energy in the powder significantly affects the molten pool size and the temperature field.
  3. The thermal stresses obtained from the simulation correlate with the actual cracks,61 and local preheating can effectively reduce the residual stresses.62 A three-dimensional thermodynamic finite element model investigated the temperature and stress variations during laser-assisted fabrication and found that powder-to-solid conversion increases the temperature gradient, stresses, and warpage.63 Other scholars have predicted residual stresses and part deflection for LPBF specimens and investigated the effects of deposition pattern, heat, laser power, and scanning strategy on residual stresses, noting that high-temperature gradients lead to higher residual stresses.64–67 

In short, the process of LPBF forming SS316L is extremely complex and usually involves drastic multi-scale physicochemical changes that will only take place on a very small scale. Existing literature employs DEM-based mesoscopic-scale numerical simulations to investigate the effects of process parameters on the molten pool dynamics of LPBF-formed SS316L. However, a few studies have been reported on the key mechanisms of heating and solidification, spatter, and convective behavior of the molten pool of HP-LPBF-formed SS316L with small laser focal spot diameters. In this paper, the geometrical properties of coarse and fine powder particles under three-dimensional conditions were first calculated using DEM. Then, numerical simulation models for single-track and double-track cases in the single-layer HP-LPBF forming SS316L process were developed at mesoscopic scale using the CFD method. The flow genesis of the melt in the single-track and double-track molten pools is discussed, and their 3D morphology and dimensional characteristics are discussed. In addition, the effects of laser process parameters, powder particle size, and laser focal spot diameter on the temperature field, characterization information, and defects in the molten pool are discussed.

II. MODELING

A. 3D powder bed modeling

HP-LPBF is an advanced processing technique for preparing target parts layer by layer stacking, the process of which involves repetitive spreading and melting of powders. In this process, both the powder spreading and the morphology of the powder bed are closely related to the results of the subsequent melting process, while the melted surface also affects the uniform distribution of the next layer of powder. For this reason, this chapter focuses on the modeling of the physical action during the powder spreading process and the theory of DEM to establish the numerical model of the powder bed, so as to lay a solid foundation for the accuracy of volume of fluid (VOF) and CFD.

1. DEM

DEM is a numerical technique for calculating the interaction of a large number of particles, which calculates the forces and motions of the spheres by considering each powder sphere as an independent unit. The motion of the powder particles follows the laws of classical Newtonian mechanics, including translational and rotational,38,68–70 which are expressed as follows:����¨=���+∑��ij,

(1)����¨=∑�(�ij×�ij),

(2)

where �� is the mass of unit particle i in kg, ��¨ is the advective acceleration in m/s2, And g is the gravitational acceleration in m/s2. �ij is the force in contact with the neighboring particle � in N. �� is the rotational inertia of the unit particle � in kg · m2. ��¨ is the unit particle � angular acceleration in rad/s2. �ij is the vector pointing from unit particle � to the contact point of neighboring particle �⁠.

Equations (1) and (2) can be used to calculate the velocity and angular velocity variations of powder particles to determine their positions and velocities. A three-dimensional powder bed model of SS316L was developed using DEM. The powder particles are assumed to be perfect spheres, and the substrate and walls are assumed to be rigid. To describe the contact between the powder particles and between the particles and the substrate, a non-slip Hertz–Mindlin nonlinear spring-damping model71 was used with the following expression:�hz=��������+��[(�����ij−�eff����)−(�����+�eff����)],

(3)

where �hz is the force calculated using the Hertzian in M. �� and �� are the radius of unit particles � and � in m, respectively. �� is the overlap size of the two powder particles in m. ��⁠, �� are the elastic constants in the normal and tangential directions, respectively. �ij is the unit vector connecting the centerlines of the two powder particles. �eff is the effective mass of the two powder particles in kg. �� and �� are the viscoelastic damping constants in the normal and tangential directions, respectively. �� and �� are the components of the relative velocities of the two powder particles. ��� is the displacement vector between two spherical particles. The schematic diagram of overlapping powder particles is shown in Fig. 1.

FIG. 1.

VIEW LARGEDOWNLOAD SLIDE

Schematic diagram of overlapping powder particles.

Because the particle size of the powder used for HP-LPBF is much smaller than 100 μm, the effect of van der Waals forces must be considered. Therefore, the cohesive force �jkr of the Hertz–Mindlin model was used instead of van der Waals forces,72 with the following expression:�jkr=−4��0�*�1.5+4�*3�*�3,

(4)1�*=(1−��2)��+(1−��2)��,

(5)1�*=1��+1��,

(6)

where �* is the equivalent Young’s modulus in GPa; �* is the equivalent particle radius in m; �0 is the surface energy of the powder particles in J/m2; α is the contact radius in m; �� and �� are the Young’s modulus of the unit particles � and �⁠, respectively, in GPa; and �� and �� are the Poisson’s ratio of the unit particles � and �⁠, respectively.

2. Model building

Figure 2 shows a 3D powder bed model generated using DEM with a coarse powder geometry of 1000 × 400 × 30 μm3. The powder layer thickness is 30 μm, and the powder bed porosity is 40%. The average particle size of this spherical powder is 31.7 μm and is normally distributed in the range of 15–53 μm. The geometry of the fine powder was 1000 × 400 × 20 μm3, with a layer thickness of 20 μm, and the powder bed porosity of 40%. The average particle size of this spherical powder is 11.5 μm and is normally distributed in the range of 5–25 μm. After the 3D powder bed model is generated, it needs to be imported into the CFD simulation software for calculation, and the imported geometric model is shown in Fig. 3. This geometric model is mainly composed of three parts: protective gas, powder bed, and substrate. Under the premise of ensuring the accuracy of the calculation, the mesh size is set to 3 μm, and the total number of coarse powder meshes is 1 704 940. The total number of fine powder meshes is 3 982 250.

FIG. 2.

VIEW LARGEDOWNLOAD SLIDE

Three-dimensional powder bed model: (a) coarse powder, (b) fine powder.

FIG. 3.

VIEW LARGEDOWNLOAD SLIDE

Geometric modeling of the powder bed computational domain: (a) coarse powder, (b) fine powder.

B. Modeling of fluid mechanics simulation

In order to solve the flow, melting, and solidification problems involved in HP-LPBF molten pool, the study must follow the three governing equations of conservation of mass, conservation of energy, and conservation of momentum.73 The VOF method, which is the most widely used in fluid dynamics, is used to solve the molten pool dynamics model.

1. VOF

VOF is a method for tracking the free interface between the gas and liquid phases on the molten pool surface. The core idea of the method is to define a volume fraction function F within each grid, indicating the proportion of the grid space occupied by the material, 0 ≤ F ≤ 1 in Fig. 4. Specifically, when F = 0, the grid is empty and belongs to the gas-phase region; when F = 1, the grid is completely filled with material and belongs to the liquid-phase region; and when 0 < F < 1, the grid contains free surfaces and belongs to the mixed region. The direction normal to the free surface is the direction of the fastest change in the volume fraction F (the direction of the gradient of the volume fraction), and the direction of the gradient of the volume fraction can be calculated from the values of the volume fractions in the neighboring grids.74 The equations controlling the VOF are expressed as follows:𝛻����+�⋅(��→)=0,

(7)

where t is the time in s and �→ is the liquid velocity in m/s.

FIG. 4.

VIEW LARGEDOWNLOAD SLIDE

Schematic diagram of VOF.

The material parameters of the mixing zone are altered due to the inclusion of both the gas and liquid phases. Therefore, in order to represent the density of the mixing zone, the average density �¯ is used, which is expressed as follows:72�¯=(1−�1)�gas+�1�metal,

(8)

where �1 is the proportion of liquid phase, �gas is the density of protective gas in kg/m3, and �metal is the density of metal in kg/m3.

2. Control equations and boundary conditions

Figure 5 is a schematic diagram of the HP-LPBF melting process. First, the laser light strikes a localized area of the material and rapidly heats up the area. Next, the energy absorbed in the region is diffused through a variety of pathways (heat conduction, heat convection, and surface radiation), and this process triggers complex phase transition phenomena (melting, evaporation, and solidification). In metals undergoing melting, the driving forces include surface tension and the Marangoni effect, recoil due to evaporation, and buoyancy due to gravity and uneven density. The above physical phenomena interact with each other and do not occur independently.

FIG. 5.

VIEW LARGEDOWNLOAD SLIDE

Schematic diagram of HP-LPBF melting process.

  1. Laser heat sourceThe Gaussian surface heat source model is used as the laser heat source model with the following expression:�=2�0����2exp(−2�12��2),(9)where � is the heat flow density in W/m2, �0 is the absorption rate of SS316L, �� is the radius of the laser focal spot in m, and �1 is the radial distance from the center of the laser focal spot in m. The laser focal spot can be used for a wide range of applications.
  2. Energy absorptionThe formula for calculating the laser absorption �0 of SS316L is as follows:�0=0.365(�0[1+�0(�−20)]/�)0.5,(10)where �0 is the direct current resistivity of SS316L at 20 °C in Ω m, �0 is the resistance temperature coefficient in ppm/°C, � is the temperature in °C, and � is the laser wavelength in m.
  3. Heat transferThe basic principle of heat transfer is conservation of energy, which is expressed as follows:𝛻𝛻𝛻�(��)��+�·(��→�)=�·(�0����)+��,(11)where � is the density of liquid phase SS316L in kg/m3, �� is the specific heat capacity of SS316L in J/(kg K), 𝛻� is the gradient operator, t is the time in s, T is the temperature in K, 𝛻�� is the temperature gradient, �→ is the velocity vector, �0 is the coefficient of thermal conduction of SS316L in W/(m K), and  �� is the thermal energy dissipation term in the molten pool.
  4. Molten pool flowThe following three conditions need to be satisfied for the molten pool to flow:
    • Conservation of mass with the following expression:𝛻�·(��→)=0.(12)
    • Conservation of momentum (Navier–Stokes equation) with the following expression:𝛻𝛻𝛻𝛻���→��+�(�→·�)�→=�·[−pI+�(��→+(��→)�)]+�,(13)where � is the pressure in Pa exerted on the liquid phase SS316L microelement, � is the unit matrix, � is the fluid viscosity in N s/m2, and � is the volumetric force (gravity, atmospheric pressure, surface tension, vapor recoil, and the Marangoni effect).
    • Conservation of energy, see Eq. (11)
  5. Surface tension and the Marangoni effectThe effect of temperature on the surface tension coefficient is considered and set as a linear relationship with the following expression:�=�0−��dT(�−��),(14)where � is the surface tension of the molten pool at temperature T in N/m, �� is the melting temperature of SS316L in K, �0 is the surface tension of the molten pool at temperature �� in Pa, and σdσ/ dT is the surface tension temperature coefficient in N/(m K).In general, surface tension decreases with increasing temperature. A temperature gradient causes a gradient in surface tension that drives the liquid to flow, known as the Marangoni effect.
  6. Metal vapor recoilAt higher input energy densities, the maximum temperature of the molten pool surface reaches the evaporation temperature of the material, and a gasification recoil pressure occurs vertically downward toward the molten pool surface, which will be the dominant driving force for the molten pool flow.75 The expression is as follows:��=0.54�� exp ���−���0���,(15)where �� is the gasification recoil pressure in Pa, �� is the ambient pressure in kPa, �� is the latent heat of evaporation in J/kg, �0 is the gas constant in J/(mol K), T is the surface temperature of the molten pool in K, and Te is the evaporation temperature in K.
  7. Solid–liquid–gas phase transitionWhen the laser hits the powder layer, the powder goes through three stages: heating, melting, and solidification. During the solidification phase, mutual transformations between solid, liquid, and gaseous states occur. At this point, the latent heat of phase transition absorbed or released during the phase transition needs to be considered.68 The phase transition is represented based on the relationship between energy and temperature with the following expression:�=�����,(�<��),�(��)+�−����−����,(��<�<��)�(��)+(�−��)����,(��<�),,(16)where �� and �� are solid and liquid phase density, respectively, of SS316L in kg/m3. �� and �� unit volume of solid and liquid phase-specific heat capacity, respectively, of SS316L in J/(kg K). �� and ��⁠, respectively, are the solidification temperature and melting temperature of SS316L in K. �� is the latent heat of the phase transition of SS316L melting in J/kg.

3. Assumptions

The CFD model was computed using the commercial software package FLOW-3D.76 In order to simplify the calculation and solution process while ensuring the accuracy of the results, the model makes the following assumptions:

  1. It is assumed that the effects of thermal stress and material solid-phase thermal expansion on the calculation results are negligible.
  2. The molten pool flow is assumed to be a Newtonian incompressible laminar flow, while the effects of liquid thermal expansion and density on the results are neglected.
  3. It is assumed that the surface tension can be simplified to an equivalent pressure acting on the free surface of the molten pool, and the effect of chemical composition on the results is negligible.
  4. Neglecting the effect of the gas flow field on the molten pool.
  5. The mass loss due to evaporation of the liquid metal is not considered.
  6. The influence of the plasma effect of the molten metal on the calculation results is neglected.

It is worth noting that the formulation of assumptions requires a trade-off between accuracy and computational efficiency. In the above models, some physical phenomena that have a small effect or high difficulty on the calculation results are simplified or ignored. Such simplifications make numerical simulations more efficient and computationally tractable, while still yielding accurate results.

4. Initial conditions

The preheating temperature of the substrate was set to 393 K, at which time all materials were in the solid state and the flow rate was zero.

5. Material parameters

The material used is SS316L and the relevant parameters required for numerical simulations are shown in Table I.46,77,78

TABLE I.

SS316L-related parameters.

PropertySymbolValue
Density of solid metal (kg/m3�metal 7980 
Solid phase line temperature (K) �� 1658 
Liquid phase line temperature (K) �� 1723 
Vaporization temperature (K) �� 3090 
Latent heat of melting (⁠ J/kg⁠) �� 2.60×105 
Latent heat of evaporation (⁠ J/kg⁠) �� 7.45×106 
Surface tension of liquid phase (N /m⁠) � 1.60 
Liquid metal viscosity (kg/m s) �� 6×10−3 
Gaseous metal viscosity (kg/m s) �gas 1.85×10−5 
Temperature coefficient of surface tension (N/m K) ��/�T 0.80×10−3 
Molar mass (⁠ kg/mol⁠) 0.05 593 
Emissivity � 0.26 
Laser absorption �0 0.35 
Ambient pressure (kPa) �� 101 325 
Ambient temperature (K) �0 300 
Stefan–Boltzmann constant (W/m2 K4� 5.67×10−8 
Thermal conductivity of metals (⁠ W/m K⁠) � 24.55 
Density of protective gas (kg/m3�gas 1.25 
Coefficient of thermal expansion (/K) �� 16×10−6 
Generalized gas constant (⁠ J/mol K⁠) 8.314 

III. RESULTS AND DISCUSSION

With the objective of studying in depth the evolutionary patterns of single-track and double-track molten pool development, detailed observations were made for certain specific locations in the model, as shown in Fig. 6. In this figure, P1 and P2 represent the longitudinal tangents to the centers of the two melt tracks in the XZ plane, while L1 is the transverse profile in the YZ plane. The scanning direction is positive and negative along the X axis. Points A and B are the locations of the centers of the molten pool of the first and second melt tracks, respectively (x = 1.995 × 10−4, y = 5 × 10−7, and z = −4.85 × 10−5).

FIG. 6.

VIEW LARGEDOWNLOAD SLIDE

Schematic diagram of observation position.

A. Single-track simulation

A series of single-track molten pool simulation experiments were carried out in order to investigate the influence law of laser power as well as scanning speed on the HP-LPBF process. Figure 7 demonstrates the evolution of the 3D morphology and temperature field of the single-track molten pool in the time period of 50–500 μs under a laser power of 100 W and a scanning speed of 800 mm/s. The powder bed is in the natural cooling state. When t = 50 μs, the powder is heated by the laser heat and rapidly melts and settles to form the initial molten pool. This process is accompanied by partial melting of the substrate and solidification together with the melted powder. The molten pool rapidly expands with increasing width, depth, length, and temperature, as shown in Fig. 7(a). When t = 150 μs, the molten pool expands more obviously, and the temperature starts to transfer to the surrounding area, forming a heat-affected zone. At this point, the width of the molten pool tends to stabilize, and the temperature in the center of the molten pool has reached its peak and remains largely stable. However, the phenomenon of molten pool spatter was also observed in this process, as shown in Fig. 7(b). As time advances, when t = 300 μs, solidification begins to occur at the tail of the molten pool, and tiny ripples are produced on the solidified surface. This is due to the fact that the melt flows toward the region with large temperature gradient under the influence of Marangoni convection and solidifies together with the melt at the end of the bath. At this point, the temperature gradient at the front of the bath is significantly larger than at the end. While the width of the molten pool was gradually reduced, the shape of the molten pool was gradually changed to a “comet” shape. In addition, a slight depression was observed at the top of the bath because the peak temperature at the surface of the bath reached the evaporation temperature, which resulted in a recoil pressure perpendicular to the surface of the bath downward, creating a depressed region. As the laser focal spot moves and is paired with the Marangoni convection of the melt, these recessed areas will be filled in as shown in Fig. 7(c). It has been shown that the depressed regions are the result of the coupled effect of Marangoni convection, recoil pressure, and surface tension.79 By t = 500 μs, the width and height of the molten pool stabilize and show a “comet” shape in Fig. 7(d).

FIG. 7.

VIEW LARGEDOWNLOAD SLIDE

Single-track molten pool process: (a) t = 50  ��⁠, (b) t = 150  ��⁠, (c) t = 300  ��⁠, (d) t = 500  ��⁠.

Figure 8 depicts the velocity vector diagram of the P1 profile in a single-track molten pool, the length of the arrows represents the magnitude of the velocity, and the maximum velocity is about 2.36 m/s. When t = 50 μs, the molten pool takes shape, and the velocities at the two ends of the pool are the largest. The variation of the velocities at the front end is especially more significant in Fig. 8(a). As the time advances to t = 150 μs, the molten pool expands rapidly, in which the velocity at the tail increases and changes more significantly, while the velocity at the front is relatively small. At this stage, the melt moves backward from the center of the molten pool, which in turn expands the molten pool area. The melt at the back end of the molten pool center flows backward along the edge of the molten pool surface and then converges along the edge of the molten pool to the bottom center, rising to form a closed loop. Similarly, a similar closed loop is formed at the front end of the center of the bath, but with a shorter path. However, a large portion of the melt in the center of the closed loop formed at the front end of the bath is in a nearly stationary state. The main cause of this melt flow phenomenon is the effect of temperature gradient and surface tension (the Marangoni effect), as shown in Figs. 8(b) and 8(e). This dynamic behavior of the melt tends to form an “elliptical” pool. At t = 300 μs, the tendency of the above two melt flows to close the loop is more prominent and faster in Fig. 8(c). When t = 500 μs, the velocity vector of the molten pool shows a stable trend, and the closed loop of melt flow also remains stable. With the gradual laser focal spot movement, the melt is gradually solidified at its tail, and finally, a continuous and stable single track is formed in Fig. 8(d).

FIG. 8.

VIEW LARGEDOWNLOAD SLIDE

Vector plot of single-track molten pool velocity in XZ longitudinal section: (a) t = 50  ��⁠, (b) t = 150  ��⁠, (c) t = 300  ��⁠, (d) t = 500  ��⁠, (e) molten pool flow.

In order to explore in depth the transient evolution of the molten pool, the evolution of the single-track temperature field and the melt flow was monitored in the YZ cross section. Figure 9(a) shows the state of the powder bed at the initial moment. When t = 250 μs, the laser focal spot acts on the powder bed and the powder starts to melt and gradually collects in the molten pool. At this time, the substrate will also start to melt, and the melt flow mainly moves in the downward and outward directions and the velocity is maximum at the edges in Fig. 9(b). When t = 300 μs, the width and depth of the molten pool increase due to the recoil pressure. At this time, the melt flows more slowly at the center, but the direction of motion is still downward in Fig. 9(c). When t = 350 μs, the width and depth of the molten pool further increase, at which time the intensity of the melt flow reaches its peak and the direction of motion remains the same in Fig. 9(d). When t = 400 μs, the melt starts to move upward, and the surrounding powder or molten material gradually fills up, causing the surface of the molten pool to begin to flatten. At this time, the maximum velocity of the melt is at the center of the bath, while the velocity at the edge is close to zero, and the edge of the melt starts to solidify in Fig. 9(e). When t = 450 μs, the melt continues to move upward, forming a convex surface of the melt track. However, the melt movement slows down, as shown in Fig. 9(f). When t = 500 μs, the melt further moves upward and its speed gradually becomes smaller. At the same time, the melt solidifies further, as shown in Fig. 9(g). When t = 550 μs, the melt track is basically formed into a single track with a similar “mountain” shape. At this stage, the velocity is close to zero only at the center of the molten pool, and the flow behavior of the melt is poor in Fig. 9(h). At t = 600 μs, the melt stops moving and solidification is rapidly completed. Up to this point, a single track is formed in Fig. 9(i). During the laser action on the powder bed, the substrate melts and combines with the molten state powder. The powder-to-powder fusion is like the convergence of water droplets, which are rapidly fused by surface tension. However, the fusion between the molten state powder and the substrate occurs driven by surface tension, and the molten powder around the molten pool is pulled toward the substrate (a wetting effect occurs), which ultimately results in the formation of a monolithic whole.38,80,81

FIG. 9.

VIEW LARGEDOWNLOAD SLIDE

Evolution of single-track molten pool temperature and melt flow in the YZ cross section: (a) t = 0  ��⁠, (b) t = 250  ��⁠, (c) t = 300  ��⁠, (d) t = 350  ��⁠, (e) t = 400  ��⁠, (f) t = 450  ��⁠, (g) t = 500  ��⁠, (h) t = 550  ��⁠, (i) t = 600  ��⁠.

The wetting ability between the liquid metal and the solid substrate in the molten pool directly affects the degree of balling of the melt,82,83 and the wetting ability can be measured by the contact angle of a single track in Fig. 10. A smaller value of contact angle represents better wettability. The contact angle α can be calculated by�=�1−�22,

(17)

where �1 and �2 are the contact angles of the left and right regions, respectively.

FIG. 10.

VIEW LARGEDOWNLOAD SLIDE

Schematic of contact angle.

Relevant studies have confirmed that the wettability is better at a contact angle α around or below 40°.84 After measurement, a single-track contact angle α of about 33° was obtained under this process parameter, which further confirms the good wettability.

B. Double-track simulation

In order to deeply investigate the influence of hatch spacing on the characteristics of the HP-LPBF process, a series of double-track molten pool simulation experiments were systematically carried out. Figure 11 shows in detail the dynamic changes of the 3D morphology and temperature field of the double-track molten pool in the time period of 2050–2500 μs under the conditions of laser power of 100 W, scanning speed of 800 mm/s, and hatch spacing of 0.06 mm. By comparing the study with Fig. 7, it is observed that the basic characteristics of the 3D morphology and temperature field of the second track are similar to those of the first track. However, there are subtle differences between them. The first track exhibits a basically symmetric shape, but the second track morphology shows a slight deviation influenced by the difference in thermal diffusion rate between the solidified metal and the powder. Otherwise, the other characteristic information is almost the same as that of the first track. Figure 12 shows the velocity vector plot of the P2 profile in the double-track molten pool, with a maximum velocity of about 2.63 m/s. The melt dynamics at both ends of the pool are more stable at t = 2050 μs, where the maximum rate of the second track is only 1/3 of that of the first one. Other than that, the rest of the information is almost no significant difference from the characteristic information of the first track. Figure 13 demonstrates a detailed observation of the double-track temperature field and melts flow in the YZ cross section, and a comparative study with Fig. 9 reveals that the width of the second track is slightly wider. In addition, after the melt direction shifts from bottom to top, the first track undergoes four time periods (50 μs) to reach full solidification, while the second track takes five time periods. This is due to the presence of significant heat buildup in the powder bed after the forming of the first track, resulting in a longer dynamic time of the melt and an increased molten pool lifetime. In conclusion, the level of specimen forming can be significantly optimized by adjusting the laser power and hatch spacing.

FIG. 11.

VIEW LARGEDOWNLOAD SLIDE

Double-track molten pool process: (a) t = 2050  ��⁠, (b) t = 2150  ��⁠, (c) t = 2300  ��⁠, (d) t = 2500  ��⁠.

FIG. 12.

VIEW LARGEDOWNLOAD SLIDE

Vector plot of double-track molten pool velocity in XZ longitudinal section: (a) t = 2050  ��⁠, (b) t = 2150  ��⁠, (c) t = 2300  ��⁠, (d) t = 2500  ��⁠.

FIG. 13.

VIEW LARGEDOWNLOAD SLIDE

Evolution of double-track molten pool temperature and melt flow in the YZ cross section: (a) t = 2250  ��⁠, (b) t = 2300  ��⁠, (c) t = 2350  ��⁠, (d) t = 2400  ��⁠, (e) t = 2450  ��⁠, (f) t = 2500  ��⁠, (g) t = 2550  ��⁠, (h) t = 2600  ��⁠, (i) t = 2650  ��⁠.

In order to quantitatively detect the molten pool dimensions as well as the remolten region dimensions, the molten pool characterization information in Fig. 14 is constructed by drawing the boundary on the YZ cross section based on the isothermal surface of the liquid phase line. It can be observed that the heights of the first track and second track are basically the same, but the depth of the second track increases relative to the first track. The molten pool width is mainly positively correlated with the laser power as well as the scanning speed (the laser line energy density �⁠). However, the remelted zone width is negatively correlated with the hatch spacing (the overlapping ratio). Overall, the forming quality of the specimens can be directly influenced by adjusting the laser power, scanning speed, and hatch spacing.

FIG. 14.

VIEW LARGEDOWNLOAD SLIDE

Double-track molten pool characterization information on YZ cross section.

In order to study the variation rule of the temperature in the center of the molten pool with time, Fig. 15 demonstrates the temperature variation curves with time for two reference points, A and B. Among them, the red dotted line indicates the liquid phase line temperature of SS316L. From the figure, it can be seen that the maximum temperature at the center of the molten pool in the first track is lower than that in the second track, which is mainly due to the heat accumulation generated after passing through the first track. The maximum temperature gradient was calculated to be 1.69 × 108 K/s. When the laser scanned the first track, the temperature in the center of the molten pool of the second track increased slightly. Similarly, when the laser scanned the second track, a similar situation existed in the first track. Since the temperature gradient in the second track is larger than that in the first track, the residence time of the liquid phase in the molten pool of the first track is longer than that of the second track.

FIG. 15.

VIEW LARGEDOWNLOAD SLIDE

Temperature profiles as a function of time for two reference points A and B.

C. Simulation analysis of molten pool under different process parameters

In order to deeply investigate the effects of various process parameters on the mesoscopic-scale temperature field, molten pool characteristic information and defects of HP-LPBF, numerical simulation experiments on mesoscopic-scale laser power, scanning speed, and hatch spacing of double-track molten pools were carried out.

1. Laser power

Figure 16 shows the effects of different laser power on the morphology and temperature field of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. When P = 50 W, a smaller molten pool is formed due to the lower heat generated by the Gaussian light source per unit time. This leads to a smaller track width, which results in adjacent track not lapping properly and the presence of a large number of unmelted powder particles, resulting in an increase in the number of defects, such as pores in the specimen. The surface of the track is relatively flat, and the depth is small. In addition, the temperature gradient before and after the molten pool was large, and the depression location appeared at the biased front end in Fig. 16(a). When P = 100 W, the surface of the track is flat and smooth with excellent lap. Due to the Marangoni effect, the velocity field of the molten pool is in the form of “vortex,” and the melt has good fluidity, and the maximum velocity reaches 2.15 m/s in Fig. 16(b). When P = 200 W, the heat generated by the Gaussian light source per unit time is too large, resulting in the melt rapidly reaching the evaporation temperature, generating a huge recoil pressure, forming a large molten pool, and the surface of the track is obviously raised. The melt movement is intense, especially the closed loop at the center end of the molten pool. At this time, the depth and width of the molten pool are large, leading to the expansion of the remolten region and the increased chance of the appearance of porosity defects in Fig. 16(c). The results show that at low laser power, the surface tension in the molten pool is dominant. At high laser power, recoil pressure is its main role.

FIG. 16.

VIEW LARGEDOWNLOAD SLIDE

Simulation results of double-track molten pool under different laser powers: (a) P = 50 W, (b) P = 100 W, (c) P = 200 W.

Table II shows the effect of different laser powers on the characteristic information of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. The negative overlapping ratio in the table indicates that the melt tracks are not lapped, and 26/29 indicates the melt depth of the first track/second track. It can be seen that with the increase in laser power, the melt depth, melt width, melt height, and remelted zone show a gradual increase. At the same time, the overlapping ratio also increases. Especially in the process of laser power from 50 to 200 W, the melting depth and melting width increased the most, which increased nearly 2 and 1.5 times, respectively. Meanwhile, the overlapping ratio also increases with the increase in laser power, which indicates that the melting and fusion of materials are better at high laser power. On the other hand, the dimensions of the molten pool did not change uniformly with the change of laser power. Specifically, the depth-to-width ratio of the molten pool increased from about 0.30 to 0.39 during the increase from 50 to 120 W, which further indicates that the effective heat transfer in the vertical direction is greater than that in the horizontal direction with the increase in laser power. This dimensional response to laser power is mainly affected by the recoil pressure and also by the difference in the densification degree between the powder layer and the metal substrate. In addition, according to the experimental results, the contact angle shows a tendency to increase and then decrease during the process of laser power increase, and always stays within the range of less than 33°. Therefore, in practical applications, it is necessary to select the appropriate laser power according to the specific needs in order to achieve the best processing results.

TABLE II.

Double-track molten pool characterization information at different laser powers.

Laser power (W)Depth (μm)Width (μm)Height (μm)Remolten region (μm)Overlapping ratio (%)Contact angle (°)
50 16 54 11 −10 23 
100 26/29 74 14 18 23.33 33 
200 37/45 116 21 52 93.33 28 

2. Scanning speed

Figure 17 demonstrates the effect of different scanning speeds on the morphology and temperature field of the double-track molten pool at a laser power of 100 W and a hatch spacing of 0.06 mm. With the gradual increase in scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. When � = 200 mm/s, the slow scanning speed causes the material to absorb too much heat, which is very easy to trigger the overburning phenomenon. At this point, the molten pool is larger and the surface morphology is uneven. This situation is consistent with the previously discussed scenario with high laser power in Fig. 17(a). However, when � = 1600 mm/s, the scanning speed is too fast, resulting in the material not being able to absorb sufficient heat, which triggers the powder particles that fail to melt completely to have a direct effect on the bonding of the melt to the substrate. At this time, the molten pool volume is relatively small and the neighboring melt track cannot lap properly. This result is consistent with the previously discussed case of low laser power in Fig. 17(b). Overall, the ratio of the laser power to the scanning speed (the line energy density �⁠) has a direct effect on the temperature field and surface morphology of the molten pool.

FIG. 17.

VIEW LARGEDOWNLOAD SLIDE

Simulation results of double-track molten pool under different scanning speed: (a)  � = 200 mm/s, (b)  � = 1600 mm/s.

Table III shows the effects of different scanning speed on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and hatch spacing of 0.06 mm. It can be seen that the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. With the increase in scanning speed, the melt depth, melt width, melt height, remelted zone, and overlapping ratio show a gradual decreasing trend. Among them, the melt depth and melt width decreased faster, while the melt height and remolten region decreased relatively slowly. In addition, when the scanning speed was increased from 200 to 800 mm/s, the decreasing speeds of melt depth and melt width were significantly accelerated, while the decreasing speeds of overlapping ratio were relatively slow. When the scanning speed was further increased to 1600 mm/s, the decreasing speeds of melt depth and melt width were further accelerated, and the un-lapped condition of the melt channel also appeared. In addition, the contact angle increases and then decreases with the scanning speed, and both are lower than 33°. Therefore, when selecting the scanning speed, it is necessary to make reasonable trade-offs according to the specific situation, and take into account the factors of melt depth, melt width, melt height, remolten region, and overlapping ratio, in order to achieve the best processing results.

TABLE III.

Double-track molten pool characterization information at different scanning speeds.

Scanning speed (mm/s)Depth (μm)Width (μm)Height (μm)Remolten region (μm)Overlapping ratio (%)Contact angle (°)
200 55/68 182 19/32 124 203.33 22 
1600 13 50 11 −16.67 31 

3. Hatch spacing

Figure 18 shows the effect of different hatch spacing on the morphology and temperature field of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. The surface morphology and temperature field of the first track and second track are basically the same, but slightly different. The first track shows a basically symmetric morphology along the scanning direction, while the second track shows a slight offset due to the difference in the heat transfer rate between the solidified material and the powder particles. When the hatch spacing is too small, the overlapping ratio increases and the probability of defects caused by remelting phenomenon grows. When the hatch spacing is too large, the neighboring melt track cannot overlap properly, and the powder particles are not completely melted, leading to an increase in the number of holes. In conclusion, the ratio of the line energy density � to the hatch spacing (the volume energy density E) has a significant effect on the temperature field and surface morphology of the molten pool.

FIG. 18.

VIEW LARGEDOWNLOAD SLIDE

Simulation results of double-track molten pool under different hatch spacings: (a) H = 0.03 mm, (b) H = 0.12 mm.

Table IV shows the effects of different hatch spacing on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. It can be seen that the hatch spacing has little effect on the melt depth, melt width, and melt height, but has some effect on the remolten region. With the gradual expansion of hatch spacing, the remolten region shows a gradual decrease. At the same time, the overlapping ratio also decreased with the increase in hatch spacing. In addition, it is observed that the contact angle shows a tendency to increase and then remain stable when the hatch spacing increases, which has a more limited effect on it. Therefore, trade-offs and decisions need to be made on a case-by-case basis when selecting the hatch spacing.

TABLE IV.

Double-track molten pool characterization information at different hatch spacings.

Hatch spacing (mm)Depth (μm)Width (μm)Height (μm)Remolten region (μm)Overlapping ratio (%)Contact angle (°)
0.03 25/27 82 14 59 173.33 30 
0.12 26 78 14 −35 33 

In summary, the laser power, scanning speed, and hatch spacing have a significant effect on the formation of the molten pool, and the correct selection of these three process parameters is crucial to ensure the forming quality. In addition, the melt depth of the second track is slightly larger than that of the first track at higher line energy density � and volume energy density E. This is mainly due to the fact that a large amount of heat accumulation is generated after the first track, forming a larger molten pool volume, which leads to an increase in the melt depth.

D. Simulation analysis of molten pool with powder particle size and laser focal spot diameter

Figure 19 demonstrates the effect of different powder particle sizes and laser focal spot diameters on the morphology and temperature field of the double-track molten pool under a laser power of 100 W, a scanning speed of 800 mm/s, and a hatch spacing of 0.06 mm. In the process of melting coarse powder with small laser focal spot diameter, the laser energy cannot completely melt the larger powder particles, resulting in their partial melting and further generating excessive pore defects. The larger powder particles tend to generate zigzag molten pool edges, which cause an increase in the roughness of the melt track surface. In addition, the molten pool is also prone to generate the present spatter phenomenon, which can directly affect the quality of forming. The volume of the formed molten pool is relatively small, while the melt depth, melt width, and melt height are all smaller relative to the fine powder in Fig. 19(a). In the process of melting fine powders with a large laser focal spot diameter, the laser energy is able to melt the fine powder particles sufficiently, even to the point of overmelting. This results in a large number of fine spatters being generated at the edge of the molten pool, which causes porosity defects in the melt track in Fig. 19(b). In addition, the maximum velocity of the molten pool is larger for large powder particle sizes compared to small powder particle sizes, which indicates that the temperature gradient in the molten pool is larger for large powder particle sizes and the melt motion is more intense. However, the size of the laser focal spot diameter has a relatively small effect on the melt motion. However, a larger focal spot diameter induces a larger melt volume with greater depth, width, and height. In conclusion, a small powder size helps to reduce the surface roughness of the specimen, and a small laser spot diameter reduces the minimum forming size of a single track.

FIG. 19.

VIEW LARGEDOWNLOAD SLIDE

Simulation results of double-track molten pool with different powder particle size and laser focal spot diameter: (a) focal spot = 25 μm, coarse powder, (b) focal spot = 80 μm, fine powder.

Table V shows the maximum temperature gradient at the reference point for different powder sizes and laser focal spot diameters. As can be seen from the table, the maximum temperature gradient is lower than that of HP-LPBF for both coarse powders with a small laser spot diameter and fine powders with a large spot diameter, a phenomenon that leads to an increase in the heat transfer rate of HP-LPBF, which in turn leads to a corresponding increase in the cooling rate and, ultimately, to the formation of finer microstructures.

TABLE V.

Maximum temperature gradient at the reference point for different powder particle sizes and laser focal spot diameters.

Laser power (W)Scanning speed (mm/s)Hatch spacing (mm)Average powder size (μm)Laser focal spot diameter (μm)Maximum temperature gradient (×107 K/s)
100 800 0.06 31.7 25 7.89 
11.5 80 7.11 

IV. CONCLUSIONS

In this study, the geometrical characteristics of 3D coarse and fine powder particles were first calculated using DEM and then numerical simulations of single track and double track in the process of forming SS316L from monolayer HP-LPBF at mesoscopic scale were developed using CFD method. The effects of Marangoni convection, surface tension, recoil pressure, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool were considered in this model. The effects of laser power, scanning speed, and hatch spacing on the dynamics of the single-track and double-track molten pools, as well as on other characteristic information, were investigated. The effects of the powder particle size on the molten pool were investigated comparatively with the laser focal spot diameter. The main conclusions are as follows:

  1. The results show that the temperature gradient at the front of the molten pool is significantly larger than that at the tail, and the molten pool exhibits a “comet” morphology. At the top of the molten pool, there is a slightly concave region, which is the result of the coupling of Marangoni convection, recoil pressure, and surface tension. The melt flow forms two closed loops, which are mainly influenced by temperature gradients and surface tension. This special dynamic behavior of the melt tends to form an “elliptical” molten pool and an almost “mountain” shape in single-track forming.
  2. The basic characteristics of the three-dimensional morphology and temperature field of the second track are similar to those of the first track, but there are subtle differences. The first track exhibits a basically symmetrical shape; however, due to the difference in thermal diffusion rates between the solidified metal and the powder, a slight asymmetry in the molten pool morphology of the second track occurs. After forming through the first track, there is a significant heat buildup in the powder bed, resulting in a longer dynamic time of the melt, which increases the life of the molten pool. The heights of the first track and second track remained essentially the same, but the depth of the second track was greater relative to the first track. In addition, the maximum temperature gradient was 1.69 × 108 K/s during HP-LPBF forming.
  3. At low laser power, the surface tension in the molten pool plays a dominant role. At high laser power, recoil pressure becomes the main influencing factor. With the increase of laser power, the effective heat transfer in the vertical direction is superior to that in the horizontal direction. With the gradual increase of scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. In addition, the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. Too large or too small hatch spacing will lead to remelting or non-lap phenomenon, which in turn causes the formation of defects.
  4. When using a small laser focal spot diameter, it is difficult to completely melt large powder particle sizes, resulting in partial melting and excessive porosity generation. At the same time, large powder particles produce curved edges of the molten pool, resulting in increased surface roughness of the melt track. In addition, spatter occurs, which directly affects the forming quality. At small focal spot diameters, the molten pool volume is relatively small, and the melt depth, the melt width, and the melt height are correspondingly small. Taken together, the small powder particle size helps to reduce surface roughness, while the small spot diameter reduces the forming size.

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에스티아이씨앤디가 대한민국 최대 생산제조기술 전회(SIMTOS 2024)에 참여합니다.

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Figura 1 – Mapa de localização da PCH Salto Paraopeba

하천 저수지 물리적 모델의 침적 과정에 대한 전산 유체 역학 모델링(CFD) 기준

Natália Melo da Silva1 1; Jorge Luis Zegarra Tarqui2,Edna Maria de Faria Viana 3

Abstract

저수지 침전은 수력 발전의 지속 가능한 발전을 위한 주요 문제 중 하나이며 브라질에 매우 중요합니다. 브라질의 주요 에너지원은 수력발전소에서 나옵니다. 소규모 수력 발전소(SHP)는 재생 에너지의 보완적 발전을 위한 중요한 대안입니다.

이들의 설계, 건설, 운영 및 재동력을 최적화하기 위해 저수지 내 퇴적물의 유체 역학 및 이동을 연구하는 것이 매우 중요합니다.

3차원 전산유체역학 – CFD 3D 모델링은 복잡한 흐름 문제에 가장 적합한 방법입니다. 제안된 방법은 MG Jeceaba 자치구에 위치한 PCH Salto Paraopeba의 유체 역학 및 퇴적물 이동 현상을 재현하고 평가하는 것을 목표로 하며, 취수구의 완전한 침전으로 인해 작동이 중단되었습니다.

모델의 검증은 미나스제라이스 연방대학교의 수력학 연구 센터(CPH)에 구축된 축소된 물리적 모델의 실험 데이터를 사용하여 수행됩니다.

Abstract: The reservoir silting is one of the main problems for sustainable development in the
generation of hydroelectric energy and it is of great significance for Brazil. The main source of energy
in Brazil comes from hydroelectric power plant. The Small Hydroelectric Power Plant (SHP) are an
important alternative for complementary generation of renewable energy.
Seeking to optimize the design, construction, operation, and repowering of these, it is extremely
important to study the hydrodynamics and transport of sediments in their reservoirs. Threedimensional Computational Fluid Dynamics – CFD 3D modeling is the most appropriate method for
complex flow problems. The proposed method aims to reproduce and evaluate the hydrodynamic and
sediment transport phenomena of the PCH Salto Paraopeba, located in the municipality of Jeceaba,
MG, which stopped working due to the complete silting up of its water intake. The validation of the
model will be done using experimental data from the reduced physical model, built at the Hydraulic
Research Center (CPH) at the Federal University of Minas Gerais.

Keywords

퇴적물 수송, 물리적 모델, 소규모 수력 발전소, Sediment transport, physical model, Small Hydroelectric Power Plant.

Figura 1 – Mapa de localização da PCH Salto Paraopeba
Figura 1 – Mapa de localização da PCH Salto Paraopeba
Figura 2 – PCH Salto Paraopeba e modelo reduzido.
Figura 2 – PCH Salto Paraopeba e modelo reduzido.

REFERÊNCIAS

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비선형 파력의 영향에 따른 잔해 언덕 방파제 형상의 효과에 대한 수치 분석

비선형 파력의 영향에 따른 잔해 언덕 방파제 형상의 효과에 대한 수치 분석

Numerical Analysis of the Effects of Rubble Mound Breakwater Geometry Under the Effect of Nonlinear Wave Force

Arabian Journal for Science and EngineeringAims and scopeSubmit manuscript

Cite this article

Abstract

Assessing the interaction of waves and porous offshore structures such as rubble mound breakwaters plays a critical role in designing such structures optimally. This study focused on the effect of the geometric parameters of a sloped rubble mound breakwater, including the shape of the armour, method of its arrangement, and the breakwater slope. Thus, three main design criteria, including the wave reflection coefficient (Kr), transmission coefficient (Kt), and depreciation wave energy coefficient (Kd), are discussed. Based on the results, a decrease in wavelength reduced the Kr and increased the Kt and Kd. The rubble mound breakwater with the Coreloc armour layer could exhibit the lowest Kr compared to other armour geometries. In addition, a decrease in the breakwater slope reduced the Kr and Kd by 3.4 and 1.25%, respectively. In addition, a decrease in the breakwater slope from 33 to 25° increased the wave breaking height by 6.1% on average. Further, a decrease in the breakwater slope reduced the intensity of turbulence depreciation. Finally, the armour geometry and arrangement of armour layers on the breakwater with its different slopes affect the wave behaviour and interaction between the wave and breakwater. Thus, layering on the breakwater and the correct use of the geometric shapes of the armour should be considered when designing such structures.

파도와 잔해 더미 방파제와 같은 다공성 해양 구조물의 상호 작용을 평가하는 것은 이러한 구조물을 최적으로 설계하는 데 중요한 역할을 합니다. 본 연구는 경사진 잔해 둔덕 방파제의 기하학적 매개변수의 효과에 초점을 맞추었는데, 여기에는 갑옷의 형태, 배치 방법, 방파제 경사 등이 포함된다. 따라서 파동 반사 계수(Kr), 투과 계수(Kt) 및 감가상각파 에너지 계수(Kd)에 대해 논의합니다. 결과에 따르면 파장이 감소하면 K가 감소합니다.r그리고 K를 증가시켰습니다t 및 Kd. Coreloc 장갑 층이 있는 잔해 언덕 방파제는 가장 낮은 K를 나타낼 수 있습니다.r 다른 갑옷 형상과 비교했습니다. 또한 방파제 경사가 감소하여 K가 감소했습니다.r 및 Kd 각각 3.4%, 1.25% 증가했다. 또한 방파제 경사가 33°에서 25°로 감소하여 파도 파쇄 높이가 평균 6.1% 증가했습니다. 또한, 방파제 경사의 감소는 난류 감가상각의 강도를 감소시켰다. 마지막으로, 경사가 다른 방파제의 장갑 형상과 장갑 층의 배열은 파도 거동과 파도와 방파제 사이의 상호 작용에 영향을 미칩니다. 따라서 이러한 구조를 설계 할 때 방파제에 층을 쌓고 갑옷의 기하학적 모양을 올바르게 사용하는 것을 고려해야합니다.

Keywords

  • Rubble mound breakwater
  • Computational fluid dynamics
  • Armour layer
  • Wave reflection coefficient
  • Wave transmission coefficient
  • Wave energy dissipation coefficient

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통합 솔버

FLOW-3D 제품을 단일 통합 솔버로 마이그레이션하여 로컬 워크스테이션이나 고성능 컴퓨팅 하드웨어 환경에서 원활하게 실행할 수 있습니다.

많은 사용자가 노트북이나 로컬 워크스테이션에서 모델을 실행하지만, 고성능 컴퓨팅 클러스터에서 더 큰 모델을 실행합니다. 2022R2 릴리스에서는 통합 솔버를 통해 사용자가 HPC 솔루션의 Open MP/MPI 하이브리드 병렬화와 동일한 이점을 활용하여 워크스테이션과 노트북에서 실행할 수 있습니다.

성능 확장의 예
CPU 코어 수 증가에 따른 성능 확장의 예
메쉬 분해의 예
Open MP/MPI 하이브리드 병렬화를 위한 메시 분해의 예

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

Time step 안정성 한계는 모델 런타임의 주요 요인이며, 2022R2에서는 새로운 time step 안정성 한계인 3D 대류 안정성 한계를 Numerics 탭에서 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 유동 해석의 경우 과도한 압력 솔버 반복으로 인해 실행 시간이 길어질 수 있습니다. 이러한 어려운 경우 2022R2에서는 모델이 너무 많이 반복되면 FLOW-3D가 자동으로 새로운 프리컨디셔너 기능을 활성화하여 압력 수렴을 돕습니다. 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.

점탄성 흐름을 위한 개선된 솔루션
로그 구조 텐서 솔루션을 사용하여 점탄성 흐름에 대한 높은 Weissenberg 수의 개선된 솔루션의 예입니다. 제공: MF Tome 외, J. Non-Newton. Fluid. Mech. 175-176 (2012) 44–54

활성 시뮬레이션 제어 확장

Active simulation 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.

팬텀 물체 속도 제어의 예
연속 주조 응용 분야에 대한 가상 물체 속도 제어의 예
동적 열 제어의 예
융합 증착 모델링 애플리케이션을 위한 동적 열 제어의 예
동적 냉각 채널 제어의 예
산업용 탱크 적용을 위한 동적 냉각 채널 제어의 예

향상된 공기 동반 기능

디퓨저 및 이와 유사한 산업용 기포 흐름 응용 분야의 경우 이제 질량 공급원을 사용하여 물기둥에 공기를 유입할 수 있습니다. 또한, 동반된 공기 및 용존 산소의 난류 확산에 대한 기본값이 업데이트되었으며 매우 낮은 공기 농도에 대한 모델 정확도가 향상되었습니다.

디퓨저 모델의 예
디퓨저 모델의 예: 이제 질량 소스를 사용하여 물기둥에 공기를 유입할 수 있습니다.

FLOW-3D 아카이브 의 새로운 기능

FLOW-3D 2022R1 의 새로운 기능

FLOW-3D v12.0 의 새로운 기능

Figure 11. Sketch of scour mechanism around USAF under random waves.

Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves

by Ruigeng Hu 1,Hongjun Liu 2,Hao Leng 1,Peng Yu 3 andXiuhai Wang 1,2,*

1College of Environmental Science and Engineering, Ocean University of China, Qingdao 266000, China

2Key Lab of Marine Environment and Ecology (Ocean University of China), Ministry of Education, Qingdao 266000, China

3Qingdao Geo-Engineering Survering Institute, Qingdao 266100, China

*Author to whom correspondence should be addressed.

J. Mar. Sci. Eng. 20219(8), 886; https://doi.org/10.3390/jmse9080886

Received: 6 July 2021 / Revised: 8 August 2021 / Accepted: 13 August 2021 / Published: 17 August 2021

(This article belongs to the Section Ocean Engineering)

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Abstract

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.

Keywords: 

scournumerical investigationrandom wavesequilibrium scour depthKC number

1. Introduction

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 [13].

Jmse 09 00886 g001 550

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. [25], 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 [26]. 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 θKC 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.

KC=UwmTD��=�wm��(1)

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. [1] 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. [33] revised Sumer’s equation with m = 0.06 for nonlinear waves. Different with the findings of Sumer et al. [1] and Carreiras et al. [33], Corvaro et al. [21] 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 [2] 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 [34] developed the equilibrium scour depth predicting equation (Equation (5)) for low KC, which was suitable for waves only, waves and currents combined. Khalfin [35] 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. [36] 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.

S/D=1.3(1−exp([−m(KC−6)])�/�=1.3(1−exp(−�(��−6))(2)

where, m = 0.03 for linear waves.

S/D=1.3(1−exp([−0.02(KC−4)])�/�=1.3(1−exp(−0.02(��−4))(3)

S/D=1.3γKwaveKhw�/�=1.3��wave�ℎw(4)

where, γ is safety factor, depending on design process, typically γ = 1.5, Kwave is correction factor considering wave action, Khw is correction factor considering water depth.

S/D=1.5[tanh(hwD)]KwaveKhw�/�=1.5tanh(ℎw�)�wave�ℎw(5)

where, hw is water depth.

S/D=0.0753(θθcr−−−√−0.5)0.69KC0.68�/�=0.0753(��cr−0.5)0.69��0.68(6)

where, θ is shields parameter, θcr is critical shields parameter.

S/D=2.5(1−0.5u/uc)xrelxrel=xeff/(1+xeff)xeff=0.03(1−0.35ucr/u)(KC−6)⎫⎭⎬⎪⎪�/�=2.5(1−0.5�/��)��������=����/(1+����)����=0.03(1−0.35�cr/�)(��−6)(7)

where, u is near-bed orbital velocity amplitude, uc is critical velocity corresponding the onset of sediment motion.

S/D=1.3{1−exp[−0.03(KC2lnn+36)1/2−6]}�/�=1.31−exp−0.03(��2ln�+36)1/2−6(8)

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 [16] 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 [35] 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 [16] if the 1/10′th highest wave was used. To author’s knowledge, the stochastic approach proposed by Myrhaug and Rue [37] 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 [34] and the stochastic model developed by Myrhaug and Rue [37] 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:

∂u∂t+1VF(uAx∂u∂x+vAy∂u∂y+wAz∂u∂z)=−1ρf∂p∂x+Gx+fx∂�∂�+1��(���∂�∂�+���∂�∂�+���∂�∂�)=−1�f∂�∂�+��+��(9)

∂v∂t+1VF(uAx∂v∂x+vAy∂v∂y+wAz∂v∂z)=−1ρf∂p∂y+Gy+fy∂�∂�+1��(���∂�∂�+���∂�∂�+���∂�∂�)=−1�f∂�∂�+��+��(10)

∂w∂t+1VF(uAx∂w∂x+vAy∂w∂y+wAz∂w∂z)=−1ρf∂p∂z+Gz+fz∂�∂�+1��(���∂�∂�+���∂�∂�+���∂�∂�)=−1�f∂�∂�+��+��(11)

where, VF is the volume fraction; uv, and w are the velocity components in xyz 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 = xyz).

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].

∂kT∂T+1VF(uAx∂kT∂x+vAy∂kT∂y+wAz∂kT∂z)=PT+GT+DiffkT−εkT∂��∂�+1��(���∂��∂�+���∂��∂�+���∂��∂�)=��+��+������−���(12)

∂εT∂T+1VF(uAx∂εT∂x+vAy∂εT∂y+wAz∂εT∂z)=CDIS1εTkT(PT+CDIS3GT)+Diffε−CDIS2ε2TkT∂��∂�+1��(���∂��∂�+���∂��∂�+���∂��∂�)=����1����(��+����3��)+�����−����2��2��(13)

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 VFAiCDIS1CDIS2 and CDIS3 are dimensionless parameters, and CDIS1CDIS3 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 [43]:

ulift,i=αinsd0.3∗(θ−θcr)1.5∥g∥di(ρi−ρf)ρf−−−−−−−−−−−−√�lift,i=�����*0.3(�−�cr)1.5���(��−�f)�f(14)

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.

d∗=di(∥g∥ρf(ρi−ρf)μ2f)1/3�*=��(��f(��−�f)�f2)1/3(15)

where μf is the fluid dynamic viscosity.

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 [43]. 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:

θ=U2f,m(ρi/ρf−1)gd50�=�f,m2(��/�f−1)��50(16)

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 [44].

The critical shields parameter θcr was obtained from the Equation (17) [44]

θcr=0.31+1.2d∗+0.055[1−exp(−0.02d∗)]�cr=0.31+1.2�*+0.0551−exp(−0.02�*)(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 [44]:

usettling,i=νfdi[(10.362+1.049d3∗)0.5−10.36]�settling,�=�f��(10.362+1.049�*3)0.5−10.36(18)

where νf is the fluid kinematic viscosity.

2.3.2. Bed Load Transport

This is called bed load transport when the sand particles roll or bounce over the seabed and always have contact with seabed. The bed load transport velocity was computed by [45]:

ubedload,i=qb,iδicb,ifb�bedload,�=�b,����b,��b(19)

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.

qb,i=8[∥g∥(ρi−ρfρf)d3i]1/2�b,�=8�(��−�f�f)��31/2(20)

δi=0.3d0.7∗(θθcr−1)0.5di��=0.3�*0.7(��cr−1)0.5��(21)

2.3.3. Suspended Load Transport

Through the following transport equation, the suspended sediment concentration could be acquired.

∂Cs,i∂t+∇(us,iCs,i)=∇∇(DfCs,i)∂�s,�∂�+∇(�s,��s,�)=∇∇(�f�s,�)(22)

where, Cs,i is the suspended sand particles mass concentration of sand i in the multiple species, us,i is the sand particles velocity of sand iDf is the diffusivity.

The velocity of sand i in the multiple species could be obtained from the following equation:

us,i=u¯¯+usettling,ics,i�s,�=�¯+�settling,��s,�(23)

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).

cs,i=Cs,iρi�s,�=�s,���(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.

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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.

Table

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.

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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 [46]. 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 [47]:

S(ω)=αg2ω5exp[−54(ωpω)4]γexp[−(ω−ωp)22σ2ω2p]�(�)=��2�5exp−54(�p�)4�exp−(�−�p)22�2�p2(25)

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.

α=0.0076(gXU2)−0.22�=0.0076(���2)−0.22(26)

ωp=22(gU)(gXU2)−0.33�p=22(��)(���2)−0.33(27)

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

ε=2πgHsT2a�=2���s�a2(28)

Ur=Hsk2h3w�r=�s�2ℎw3(29)

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.

Table

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. [48], 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. [49] and Corvaro et al. [50]. 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.

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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 [51].

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. [52]. 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. [52]. 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. [52], although scour depth in the present model is slightly larger than the experimental results at initial stage.

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Figure 5. Comparison of time evolution of scour between the present study and Khosronejad et al. [52], Petersen et al. [17].

Secondly, another comparison was further conducted between the results of present study and the experimental data of Petersen et al. [17]. 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. [17]. 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 [53] 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.

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Figure 6. Comparison of surface elevation between the present study and Stahlmann [53].

Finally, another comparison was conducted for equilibrium scour depth or maximum scour depth under random waves with the experimental data of Sumer and Fredsøe [16] and Schendel et al. [22]. The Figure 7 shows the comparison between the numerical results and experimental data of Run01, Run05, Run21 and Run22 in Sumer and Fredsøe [16] and test A05 and A09 in Schendel et al. [22]. 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.

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Figure 7. Comparison of equilibrium (or maximum) scour depth between the present study and Sumer and Fredsøe [16], Schendel et al. [22].

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 [16] proposed the following formula for the scour development under waves

St=Seq(1−exp(−t/Tc))�t=�eq(1−exp(−�/�c))(30)

where Tc is time scale of scour process.

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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 [16], 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.

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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.

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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.

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Figure 11. Sketch of scour mechanism around USAF under random waves.

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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. [48] 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 [21]. Briefly speaking, the vortex system at wave crest phase was mainly related to the scour process around USAF under random waves.

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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 [16] 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

Table 4. Uwm,s and KC for case 1~9.

Table

Raaijmakers and Rudolph [34] 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 [54], 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.

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Figure 14. Sketch of the position where the Seq was evaluated.

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Figure 15. Comparison of the equilibrium scour depth between the present model and the model of Raaijmakers and Rudolph [34]: (aKCrms,sKCrms,a; (bKCrms,pKCrms,m; (cKCrms,n = 1/10KCrms,n = 1/5; (dKCs,sKCs,a; (eKCs,pKCs,m; (fKCs,n = 1/10KCs,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

S′eq/D=1.95[tanh(hD)](1−exp(−0.012KCs,p))+ΔS/D�eq′/�=1.95tanh(ℎ�)(1−exp(−0.012��s,p))+∆�/�(31)

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 [21]. 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. [21] 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.

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Figure 16. Comparison of Seq between the simulating results and the predicting values by Equation (31).

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Figure 17. Comparison of Seq/D between the Experimental results of Corvaro et al. [21] 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 [37] 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.

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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).

ΔSeq/D=0.052*exp(−n/6.566)+0.068∆�eq/�=0.052*exp(−�/6.566)+0.068(32)

S′eq¯/D=S′eq/D+0.052*exp(−n/6.566)+0.068�eq′¯/�=�eq′/�+0.052*exp(−�/6.566)+0.068(33)

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Figure 19. The fitting line between ∆S′and n.

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 [37] for equilibrium scour depth predicting around slender pile in case of random waves.

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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 [16] and Corvaro et al. [21]. The details of experiments conducted by Corvaro et al. [21] were described in above section. Sumer and Fredsøe [16] 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 [16] and Corvaro et al. [21] 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.

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Figure 21. Comparison of Seq between the experimental results of Sumer and Fredsøe [16] and Corvaro et al. [21] 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. [21] and Schendel et al. [55], 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 [42]

Fr=UwgD−−−√�r=�w��(34)

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.

Uw=1T/4∫0T/4Uwmsin(t/T)dt=2πUwm�w=1�/4∫0�/4�wmsin(�/�)��=2��wm(35)

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Figure 22. Sketch of flow field at upstream USAF edges.

Tavouktsoglou et al. [25] proposed the following formula between Fr and the vertical location of the stagnation y

yh∝Fer�ℎ∝�r�(36)

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. [21] 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. [21]. 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. [25] 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 [19] 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

lg(Seq/D)=Aexp(B/Fr)+Clg(�eq/�)=�exp(�/�r)+�(37)

where AB and C are constant.

Jmse 09 00886 g023 550

Figure 23. The fitting curve between Seq/D and Fr.

Jmse 09 00886 g024 550

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 [25]. The root-mean-square (RMS) value of near-bed velocity amplitude Um,rms is used for calculating Um.

Eu=U2mgD�u=�m2��(38)

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 [16] and Corvaro et al. [21] 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 [16] and Corvaro et al. [21]. 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.

Jmse 09 00886 g025 550

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.

5. Conclusions

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. [52], Petersen et al. [17], Sumer and Fredsøe [16] and Schendel et al. [22]. 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 [34] and the stochastic model developed by Myrhaug and Rue [37] 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. [52], Petersen et al. [17], Sumer and Fredsøe [16] and Schendel et al. [22], 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 [37] 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 [16] and Corvaro et al. [21].(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.

Author Contributions

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.

Funding

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

Not applicable.

Informed Consent Statement

Not applicable.

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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Hu, R.; Liu, H.; Leng, H.; Yu, P.; Wang, X. Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves. J. Mar. Sci. Eng. 20219, 886. https://doi.org/10.3390/jmse9080886

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Hu R, Liu H, Leng H, Yu P, Wang X. Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves. Journal of Marine Science and Engineering. 2021; 9(8):886. https://doi.org/10.3390/jmse9080886Chicago/Turabian Style

Hu, Ruigeng, Hongjun Liu, Hao Leng, Peng Yu, and Xiuhai Wang. 2021. “Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves” Journal of Marine Science and Engineering 9, no. 8: 886. https://doi.org/10.3390/jmse9080886

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Strain rate magnitude at the free surface, illustrating Kelvin-Helmoltz (KH) shear instabilities.

On the reef scale hydrodynamics at Sodwana Bay, South Africa

Environmental Fluid Mechanics (2022)Cite this article

Abstract

The hydrodynamics of coral reefs strongly influences their biological functioning, impacting processes such as nutrient availability and uptake, recruitment success and bleaching. For example, coral reefs located in oligotrophic regions depend on upwelling for nutrient supply. Coral reefs at Sodwana Bay, located on the east coast of South Africa, are an example of high latitude marginal reefs. These reefs are subjected to complex hydrodynamic forcings due to the interaction between the strong Agulhas current and the highly variable topography of the region. In this study, we explore the reef scale hydrodynamics resulting from the bathymetry for two steady current scenarios at Two-Mile Reef (TMR) using a combination of field data and numerical simulations. The influence of tides or waves was not considered for this study as well as reef-scale roughness. Tilt current meters with onboard temperature sensors were deployed at selected locations within TMR. We used field observations to identify the dominant flow conditions on the reef for numerical simulations that focused on the hydrodynamics driven by mean currents. During the field campaign, southerly currents were the predominant flow feature with occasional flow reversals to the north. Northerly currents were associated with greater variability towards the southern end of TMR. Numerical simulations showed that Jesser Point was central to the development of flow features for both the northerly and southerly current scenarios. High current variability in the south of TMR during reverse currents is related to the formation of Kelvin-Helmholtz type shear instabilities along the outer edge of an eddy formed north of Jesser Point. Furthermore, downward vertical velocities were computed along the offshore shelf at TMR during southerly currents. Current reversals caused a change in vertical velocities to an upward direction due to the orientation of the bathymetry relative to flow directions.

Highlights

  • A predominant southerly current was measured at Two-Mile Reef with occasional reversals towards the north.
  • Field observations indicated that northerly currents are spatially varied along Two-Mile Reef.
  • Simulation of reverse currents show the formation of a separated flow due to interaction with Jesser Point with Kelvin–Helmholtz type shear instabilities along the seaward edge.

지금까지 Sodwana Bay에서 자세한 암초 규모 유체 역학을 모델링하려는 시도는 없었습니다. 이러한 모델의 결과는 규모가 있는 산호초 사이의 흐름이 산호초 건강에 어떤 영향을 미치는지 탐색하는 데 사용할 수 있습니다. 이 연구에서는 Sodwana Bay의 유체역학을 탐색하는 데 사용할 수 있는 LES 모델을 개발하기 위한 단계별 접근 방식을 구현합니다. 여기서 우리는 이 초기 단계에서 파도와 조수의 영향을 배제하면서 Agulhas 해류의 유체역학에 초점을 맞춥니다. 이 접근법은 흐름의 첫 번째 LES를 제시하고 Sodwana Bay의 산호초에서 혼합함으로써 향후 연구의 기초를 제공합니다.

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Text and image taken from Deoraj, et al. (2022), On the reef scale hydrodynamics at Sodwana Bay, South Africa. Preprint courtesy the authors.

Figure 2: 3D (left) and 2D (right) views of wave elevation using case C

CFD 접근법을 사용하여 파도에서 하이드로포일의 SEAKEEPING 성능

SYAFIQ ZIKRYAND FITRIADHY*
Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala
Terengganu, Terengganu, Malaysia
*
Corresponding author: naoe.afit@gmail.com http://doi.org/10.46754/umtjur.2021.07.017

Abstract

수중익선은 일반적으로 열악한 환경 조건으로 인해 승객의 편안함에 영향을 미칠 수 있는 높은 저항과 과도한 수직 운동(히브 및 피치)을 경험합니다. 따라서 복잡한 유체역학적 현상이 존재하기 때문에 파랑에서 수중익선의 내항성능을 규명할 필요가 있다.

이를 위해 수중익선 운동에 대한 CFD(Computational Fluid Dynamic) 해석을 제안한다. Froude Number 및 포일 받음각과 같은 여러 매개변수가 고려되었습니다.

그 결과 Froude Number의 후속 증가는 히브 및 피치 운동에 반비례한다는 것이 밝혀졌습니다. 본질적으로 이것은 높은 응답 진폭 연산자(RAO)의 형태로 제공되는 수중익선 항해 성능의 업그레이드로 이어졌습니다.

또한 포일 선수의 증가하는 각도는 히브 운동에 비례하는 반면, 포일 선미는 7.5o에서 낮은 히브 운동을 보였고, 그 다음으로 5o, 10o 순으로 나타났다. 피치모션의 경우 포일 보우의 증가는 5o에서 더 낮았고, 그 다음이 10o, 7.5o 순이었다. 포일 선미의 증가는 수중익선에 의한 피치 모션 경험에 비례했습니다.

일반적으로 이 CFD 시뮬레이션은 앞서 언급한 설계 매개변수와 관련하여 공해 상태에서 수중익선 설계의 운영 효율성을 보장하는 데 매우 유용합니다.

Keywords

CFD, hydrofoil, foil angle of attack, heave, pitch.

Figure 1: Overall mesh block being used in simulation
Figure 1: Overall mesh block being used in simulation
Figure 2: 3D (left) and 2D (right) views of wave elevation using case C
Figure 2: 3D (left) and 2D (right) views of wave elevation using case C

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Fig. 2. Design of the grate inlet types studied: (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, (f) R6, (g) R7 (source: based on geometries of Chaparro Andrade and Abaunza Tabares, 2021)

Three-dimensional Numerical Evaluation of Hydraulic Efficiency and Discharge Coefficient in Grate Inlets

쇠창살 격자 유입구의 수리효율 및 배출계수에 대한 3차원 수치적 평가

Melquisedec Cortés Zambrano*, Helmer Edgardo Monroy González,
Wilson Enrique Amaya Tequia
Faculty of Civil Engineering, Santo Tomas Tunja University. Address Av. Universitaria No. 45-202.
Tunja – Boyacá – Colombia

Abstract

홍수는 지반이동 및 이동의 원인 중 하나이며, 급속한 도시화 및 도시화로 인해 이전보다 빈번하게 발생할 수 있다. 도시 배수 시스템의 특성은 집수 요소가 결정적인 역할을 하는 범람의 발생 및 범위를 정의할 수 있습니다. 이 문서는 7가지 유형의 화격자 유입구의 수력 유입 효율 및 배출 계수에 대한 수치 조사를 제시합니다. FLOW-3D® 시뮬레이터는 Q = 24, 34.1, 44, 100, 200 및 300 L/s의 유속에서 풀 스케일로 격자를 테스트하는 데 사용되며 종방향 기울기가 1.0인 실험 프로토타입의 구성을 유지합니다. %, 1.5% 및 2.0% 및 고정 횡단 경사, 총 126개 모델. 그 결과를 바탕으로 종류별 및 종단경사 조건에 따른 수력유입구 효율곡선과 토출계수를 구성하였다. 결과는 다른 조사에서 제안된 경험적 공식으로 조정되어 프로토타입의 물리적 테스트 결과를 검증하는 역할을 합니다.

Floods are one of the causes of ground movement and displacement, and due to rapid urbanization and urban growth may occur more frequently than before. The characteristics of an urban drainage system can define the occurrence and extent of flooding, where catchment elements have a determining role. This document presents the numerical investigation of the hydraulic inlet efficiency and the discharge coefficient of seven types of grate inlets. The FLOW-3D® simulator is used to test the gratings at a full scale, under flow rates of Q = 24, 34.1, 44, 100, 200 and 300 L/s, preserving the configuration of the experimental prototype with longitudinal slopes of 1.0%, 1.5% and 2.0% and a fixed cross slope, for a total of 126 models. Based on the results, hydraulic inlet efficiency curves and discharge coefficients are constructed for each type and a longitudinal slope condition. The results are adjusted with empirical formulations proposed in other investigations, serving to verify the results of physical testing of prototypes.

Keywords

grate inlet, inlet efficiency, discharge coefficient, computational fluid dynamic, 3D modelling.

Fig. 1. Physical model of the experimental campaign (source: Chaparro Andrade and Abaunza Tabares, 2021)
Fig. 1. Physical model of the experimental campaign (source: Chaparro Andrade and Abaunza Tabares, 2021)
Fig. 2. Design of the grate inlet types studied: (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, (f) R6, (g) R7 (source: based on geometries of Chaparro Andrade
and Abaunza Tabares, 2021)
Fig. 2. Design of the grate inlet types studied: (a) R1, (b) R2, (c) R3, (d) R4, (e) R5, (f) R6, (g) R7 (source: based on geometries of Chaparro Andrade and Abaunza Tabares, 2021)
Fig. 4. Comparison between the results obtained during physical experimentation in prototype 7 and simulation results with FLOW-3D® (source:
made with FlowSight® and photographic record by Chaparro Andrade and Abaunza Tabares, 2021)
Fig. 4. Comparison between the results obtained during physical experimentation in prototype 7 and simulation results with FLOW-3D® (source: made with FlowSight® and photographic record by Chaparro Andrade and Abaunza Tabares, 2021)
Fig. 6. Example of the results of flow depth and velocity vectors in the xy plane, for a stable flow condition in a grate inlet type and free surface
configuration and flow regime, of some grating types (source: produced with FlowSight®)
Fig. 6. Example of the results of flow depth and velocity vectors in the xy plane, for a stable flow condition in a grate inlet type and free surface configuration and flow regime, of some grating types (source: produced with FlowSight®)

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측수로 물넘이 수위별 해석 결과

저수지 측수로형 여수로 불완전월류 정밀안전진단 수리 해석 ( 3차원 전산 수치해석 )

불완전 월류 조건의 저수지 측수로형 여수로에 대한 3차원 전산 추치해석

현재 농어촌공사와 농어촌연구원, 수자원공사, 학계 등에서는 전 세계에서 오랜 기간 학계의 연구활동을 통한 수많은 논문 검증과 현장 사용을 통해 검증된 FLOW-3D 수치해석 프로그램을 이용하고 있습니다.

한국농어촌공사 재난안전진단본부 FLOW-3D 수치해석 교육 장면
2024년 한국농어촌공사 안전진단본부 여수로 불완전월류 정밀안전진단 FLOW-3D 수치 해석교육 장면

농어촌공사 정밀안전진단 업무 수행시 수치해석이 필요하십니까? 수치해석에 대해 궁금하신 사항이나 용역 의뢰가 필요하시면 언제든지 아래 연락처로 연락 주시기 바랍니다.


저수지 정밀안전진단 수치해석 과업 예시

과업의 범위

  • 3차원 수치해석을 통한 OO저수지의 측수로부 수면 검토
  • 측수로 불완전 월류 발생 여부 및 제방 여유고 검토

수치해석 과업 세부내용

가능최대홍수량과 200년, 100년 빈도의 홍수량에 대해 각각의 측수로부 3차원 수치해석

경계조건

가. 수위

  • 만수위
  • 홍수위
    – 100년 빈도
    – 200년 빈도
    – 가능최대홍수량(PMF)

나. 홍수량

  • 100년 빈도의 홍수량
  • 200년 빈도의 홍수량
  • 가능최대홍수량(PMF)

저수지 수위별 방류량 검토 및 제방 여유고 검토

  • 경계조건에 대해 측수로부 물넘이 수면 형상 검토
  • 수위별 방류량을 제공된 수리계산값과 수치해석 결과값을 비교하여 방류 능력 검토
  • 수위에 따른 물넘이 수위를 검토하여 제방 여유고 검토

※ 수위별 수리계산값은 발주처에서 제공

성과물

  • 100년빈도, 200년빈도 및 가능최대홍수량(PMF) 유입에 따른 측수로부 불완전 월류 여부로 인한 제방 여유고 안정성 검토
  • 가능최대홍수량(PMF)을 고려 할 경우 검증된 3차원 수치해석 모델 Data 구축
  • 과업보고서, 보고서 원본 파일 및 PDF 파일, 수치해석 원본 입력 파일 및 결과 파일
  • 기타
    ※ 모든 성과물은 CD 및 이동저장장치에 별도 저장하여 납품

경주 저수지 붕괴 "많은 저수량에 따른 수압 탓"(속보) | 연합뉴스
경주 저수지 붕괴 “많은 저수량에 따른 수압 탓” | 연합뉴스

보고서 예

OO저수지 측수로 및 주변 지형 형상을 반영하고자 지형 및 구조물 등에 대한 차원 3
CAD , FLOW-3D 3 형상을 작성하였으며 수치모형을 이용하여 구룡저수지 차원 형상에 대
한 측수로 수위별 유동상황을 재현하고자 수치해석을 수행하였다.
금회 수치해석 결과 수위조 , EL.224.82m 건 이상에서는 측수로부 방류능력이 부족하여
불완전 월류가 발생하는 것으로 나타났으며 수리계산 대비 수치해석의 , 오차율은 대략
-16.2% +7 ∼ 2.8% 정도로 나타났다.
수치해석 결과로 산정한 수위별 방류량 관계식은 아래와 같다.
y = 3.2583E+00x4 – 2.9427E+03x3 – 9.9663E+05x2 – 1.5001E+08x + 8.4674E+09
여기서 는 저수지수위 는 방류량을 나타낸다 x , y

측수로 수위 방류량 및 불완전월류율
측수로 수위 방류량 및 불완전월류율

수위 방류량 데이터

수위 방류량 데이터
수위 방류량 데이터

측수로 물넘이 수위별 해석 결과

측수로 물넘이 수위별 해석 결과
측수로 물넘이 수위별 해석 결과

측수로 측벽 수위 분포

측수로 측벽 수위 분포
측수로 측벽 수위 분포

측수로 측벽 여유고

측수로 측벽은 저수지 수위 에서부터 여유고가 부족하여 월류 발생 EL. 225.32m 예상

측수로 측벽 여유고
측수로 측벽 여유고

<담당자 연락처>

  • 전화 :   02-2026-0455
  • Email : flow3d@stikorea.co.kr
Figure 3: Wave pattern at sea surface at 20 knots (10.29 m/s) for mesh 1

Flow-3D에서 CFD 시뮬레이션을 사용한 선박 저항 분석

Ship resistance analysis using CFD simulations in Flow-3D

Author

Deshpande, SujaySundsbø, Per-ArneDas, Subhashis

Abstract

선박의 동력 요구 사항을 설계할 때 고려해야 할 가장 중요한 요소는 선박 저항 또는 선박에 작용하는 항력입니다. 항력을 극복하는 데 필요한 동력이 추진 시스템의 ‘손실’에 기여하기 때문에 추진 시스템을 설계하는 동안 선박 저항을 추정하는 것이 중요합니다. 선박 저항을 계산하는 세 가지 주요 방법이 있습니다:

Holtrop-Mennen(HM) 방법과 같은 통계적 방법, 수치 분석 또는 CFD(전산 유체 역학) 시뮬레이션 및 모델 테스트, 즉 예인 탱크에서 축소된 모델 테스트. 설계 단계 초기에는 기본 선박 매개변수만 사용할 수 있을 때 HM 방법과 같은 통계 모델만 사용할 수 있습니다.

수치 해석/CFD 시뮬레이션 및 모델 테스트는 선박의 완전한 3D 설계가 완료된 경우에만 수행할 수 있습니다. 본 논문은 Flow-3D 소프트웨어 패키지를 사용하여 CFD 시뮬레이션을 사용하여 잔잔한 수상 선박 저항을 예측하는 것을 목표로 합니다.

롤온/롤오프 승객(RoPax) 페리에 대한 사례 연구를 조사했습니다. 선박 저항은 다양한 선박 속도에서 계산되었습니다. 메쉬는 모든 CFD 시뮬레이션의 결과에 영향을 미치기 때문에 메쉬 민감도를 확인하기 위해 여러 개의 메쉬가 사용되었습니다. 시뮬레이션의 결과를 HM 방법의 추정치와 비교했습니다.

시뮬레이션 결과는 낮은 선박 속도에 대한 HM 방법과 잘 일치했습니다. 더 높은 선속을 위한 HM 방법에 비해 결과의 차이가 상당히 컸다. 선박 저항 분석을 수행하는 Flow-3D의 기능이 시연되었습니다.

While designing the power requirements of a ship, the most important factor to be considered is the ship resistance, or the sea drag forces acting on the ship. It is important to have an estimate of the ship resistance while designing the propulsion system since the power required to overcome the sea drag forces contribute to ‘losses’ in the propulsion system. There are three main methods to calculate ship resistance: Statistical methods like the Holtrop-Mennen (HM) method, numerical analysis or CFD (Computational Fluid Dynamics) simulations, and model testing, i.e. scaled model tests in towing tanks. At the start of the design stage, when only basic ship parameters are available, only statistical models like the HM method can be used. Numerical analysis/ CFD simulations and model tests can be performed only when the complete 3D design of the ship is completed. The present paper aims at predicting the calm water ship resistance using CFD simulations, using the Flow-3D software package. A case study of a roll-on/roll-off passenger (RoPax) ferry was investigated. Ship resistance was calculated at various ship speeds. Since the mesh affects the results in any CFD simulation, multiple meshes were used to check the mesh sensitivity. The results from the simulations were compared with the estimate from the HM method. The results from simulations agreed well with the HM method for low ship speeds. The difference in the results was considerably high compared to the HM method for higher ship speeds. The capability of Flow-3D to perform ship resistance analysis was demonstrated.

Figure 1: Simplified ship geometry
Figure 1: Simplified ship geometry
Figure 3: Wave pattern at sea surface at 20 knots (10.29 m/s) for mesh 1
Figure 3: Wave pattern at sea surface at 20 knots (10.29 m/s) for mesh 1
Figure 4: Ship Resistance (kN) vs Ship Speed (knots)
Figure 4: Ship Resistance (kN) vs Ship Speed (knots)

Publisher

International Society of Multiphysics

Citation

Deshpande SR, Sundsbø P, Das S. Ship resistance analysis using CFD simulations in Flow-3D. The International Journal of Multiphysics. 2020;14(3):227-236

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Figure 5. Schematic view of flap and support structure [32]

Design Optimization of Ocean Renewable Energy Converter Using a Combined Bi-level Metaheuristic Approach

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

Abstract

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.

Keywords

Wave Energy Converter

OSWEC

Hydrodynamic Effects

Geometric Design

Metaheuristic Optimization

Multi-Verse Optimizer

1Introduction

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 [1][2][3]. 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) [4][5]and technology readiness level (TRL) of wave energy converters (WECs). This has been achieved using novel modeling techniques [6][7][8][9][10][11][12][13][14] to gain the following advantages [15]: (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. [16] (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. [17] (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) [18].

In general, the energy conversion process can be divided into three stages in a WEC device, including primary, secondary, and tertiary stages [19][20]. 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 [10][13][12]. 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 [21][22]. 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 [15][23], we focus on oscillating surge wave energy converters (OSWECs) in this paper due to its high capacity for industrialization [24].

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 [25], 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 [26][27]. 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 [28][29]. 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 [30]. 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) [31]. 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 [32]. 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 [33]. 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 [19]. In addition, Guo and Ringwood surveyed geometric optimization methods to improve the hydrodynamic performance of OSWECs at the primary stage [14]. 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 [34]. 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 [35].

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.

2.1Model 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 [36]. 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 [37]. 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 [21]. 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.

2.2Verification

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 [38]. 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 [32]. 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 [32][39]:(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 [40].Table 2.

Table 1. Constant coefficients in RNGK- model

Factors�0�1�2������
Quantity0.0124.381.421.681.391.390.084

Table 2. Flap properties

Joint height (m)0.476
Height of the center of mass (m)0.53
Weight (Kg)10.77

It is worth mentioning that the volume of fluid method is used to separate water and air phases in this software [41]. Below is the equation of this method [40].(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 [42][34][43]:(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 [44]:(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 [15], �� 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.

3Sensitivity 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.

4Design 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 [45]. 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: [46].(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 [47]. 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 [48]. 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 [49]. 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 [50]. 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 [51]. Black holes, which behave completely in contrast to white holes, attract everything including light beams with their extremely high gravitational force [52]. 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) [53]. 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. [54]

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:

Assume that

(16)���=����1<��(��)����1≥��(��)

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 [54]. 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 [1][54].

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 [55]. 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 [56][55]. 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��)+�∗→(�)

Where:(19)�′→=|�∗→(�)-�→(�)|

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 [55].

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 [57]. 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(�).

5. Conclusion

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

Empty CellAlgorithm 1: Hill Climb Multiverse Optimization
01:procedure HCMVO
02:�=30,�=5▹���������������������������������
03:�=〈F1,B1,N,R,H1〉,…〈FN,B2,N,R,HN〉⇒lb1N⩽�⩽ubN
04:Initialize parameters�ER,�DR,�EP,Best�,���ite��▹Wormhole existence probability (WEP)
05:��=����(��)
06:��=Normalize the inflation rate��
07:for iter in[1,⋯,���iter]do
08:for�in[1,⋯,�]do
09:Update�EP,�DR,Black����Index=�
10:for���[1,⋯,�]��
11:�1=����()
12:if�1≤��(��)then
13:White HoleIndex=Roulette�heelSelection(-��)
14:�(Black HoleIndex,�)=��(White HoleIndex,�)
15:end if
16:�2=����([0,�])
17:if�2≤�EPthen
18:�3=����(),�4=����()
19:if�3<0.5then
20:�1=((��(�)-��(�))�4+��(�))
21:�(�,�)=Best�(�)+�DR�
22:else
23:�(�,�)=Best�(�)-�DR�
24:end if
25:end if
26:end for
27:end for
28:�HD=����([�1,�2,⋯,�Np])
29:Bes�TH�itr=����HD
30:ΔBestTHD=∑�=1�BestTII��-BestTII��-1�
31:ifΔBestTHD<��then▹Perform hill climbing local search
32:BestTHD=����-�lim��������THD
33:end if
34:end for
35:return�,BestTHD▹Final configuration
36:end procedure

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.

Empty CellAlgorithm 1: Hill Climb Multiverse Optimization
01:procedure HCMVO
02:Initialization
03:Initialize the constraints��1�,��1�
04:�1�=Mi�1�+���1�/�▹Compute the step size,�is search resolution
05:So�1=〈�,�,�,�,�〉▹���������������
06:�������1=����So�1▹���������ℎ���������
07:Main loop
08:for iter≤���ita=do
09:���=���±��
10:while�≤���(Sol1)do
11:���=���+�,▹����ℎ���ℎ��������ℎ
12:fitness��iter=�������
13:t = t+1
14:end while
15:〈�����,������max〉=����������
16:���itev=���Inde�max▹�������ℎ�������������������������������ℎ�������
17:��=��-����Max��+1▹�����������������
18:end for
19:return���iter,����
20:end procedure

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.

Acknowledgement

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.

Data availability

Data will be made available on request.

References

Figure 15. Velocity distribution of impinging jet on a wall under different Reynolds numbers.

Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment

by Hongbo Mi 1,2, Chuan Wang 1,3, Xuanwen Jia 3,*, Bo Hu 2, Hongliang Wang 4, Hui Wang 3 and Yong Zhu 5

1College of Mechatronics Engineering, Hainan Vocational University of Science and Technology, Haikou 571126, China

2Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China

3College of Hydraulic Science and Engineering, Yangzhou University, Yangzhou 225009, China

4School of Aerospace and Mechanical Engineering/Flight College, Changzhou Institute of Technology, Changzhou 213032, China

5National Research Center of Pumps, Jiangsu University, Zhenjiang 212013, China

*Author to whom correspondence should be addressed.Sustainability202315(6), 5159; https://doi.org/10.3390/su15065159

Received: 30 January 2023 / Revised: 4 March 2023 / Accepted: 10 March 2023 / Published: 14 March 2023(This article belongs to the Special Issue Advanced Technologies of Renewable Energy and Water Management for Sustainable Environment

Abstract

Due to their high efficiency, low heat loss and associated sustainability advantages, impinging jets have been used extensively in marine engineering, geotechnical engineering and other engineering practices. In this paper, the flow structure and impact characteristics of impinging jets with different Reynolds numbers and impact distances are systematically studied by Flow-3D based on PIV experiments. In the study, the relevant state parameters of the jets are dimensionlessly treated, obtaining not only the linear relationship between the length of the potential nucleation zone and the impinging distance, but also the linear relationship between the axial velocity and the axial distance in the impinging zone. In addition, after the jet impinges on the flat plate, the vortex action range caused by the wall-attached flow of the jet gradually decreases inward with the increase of the impinging distance. By examining the effect of Reynolds number Re on the hydraulic characteristics of the submerged impact jet, it can be found that the structure of the continuous submerged impact jet is relatively independent of the Reynolds number. At the same time, the final simulation results demonstrate the applicability of the linear relationship between the length of the potential core region and the impact distance. This study provides methodological guidance and theoretical support for relevant engineering practice and subsequent research on impinging jets, which has strong theoretical and practical significance.

Keywords: 

PIVFlow-3Dimpinging jethydraulic characteristicsimpinging distance

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Figure 1. Geometric model.

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Figure 2. Model grid schematic.

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Figure 3. (a) Schematic diagram of the experimental setup; (b) PIV images of vertical impinging jets with velocity fields.

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Figure 4. (a) Velocity distribution verification at the outlet of the jet pipe; (b) Distribution of flow angle in the mid-axis of the jet [39].

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Figure 5. Along-range distribution of the dimensionless axial velocity of the jet at different impact distances.Figure 6 shows the variation of H

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Figure 6. Relationship between the distribution of potential core region and the impact height H/D.

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Figure 7. The relationship between the potential core length 

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Figure 8. Along-range distribution of the flow angle φ of the jet at different impact distances.

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Figure 9. Velocity distribution along the axis of the jet at different impinging regions.

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Figure 10. The absolute value distribution of slope under different impact distances.

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Figure 11. Velocity distribution of impinging jet on wall under different impinging distances.

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Figure 12. Along-range distribution of the dimensionless axial velocity of the jet at different Reynolds numbers.

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Figure 13. Along-range distribution of the flow angle φ of the jet at different Reynolds numbers.

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Figure 14. Velocity distribution along the jet axis at different Reynolds numbers.

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Figure 15. Velocity distribution of impinging jet on a wall under different Reynolds numbers.

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MDPI and ACS Style

Mi, H.; Wang, C.; Jia, X.; Hu, B.; Wang, H.; Wang, H.; Zhu, Y. Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment. Sustainability 202315, 5159. https://doi.org/10.3390/su15065159

AMA Style

Mi H, Wang C, Jia X, Hu B, Wang H, Wang H, Zhu Y. Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment. Sustainability. 2023; 15(6):5159. https://doi.org/10.3390/su15065159Chicago/Turabian Style

Mi, Hongbo, Chuan Wang, Xuanwen Jia, Bo Hu, Hongliang Wang, Hui Wang, and Yong Zhu. 2023. “Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment” Sustainability 15, no. 6: 5159. https://doi.org/10.3390/su15065159

Figure 5 A schematic of the water model of reactor URO 200.

Physical and Numerical Modeling of the Impeller Construction Impact on the Aluminum Degassing Process

알루미늄 탈기 공정에 미치는 임펠러 구성의 물리적 및 수치적 모델링

Kamil Kuglin,1 Michał Szucki,2 Jacek Pieprzyca,3 Simon Genthe,2 Tomasz Merder,3 and Dorota Kalisz1,*

Mikael Ersson, Academic Editor

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Data Availability Statement

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Abstract

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.

Keywords: aluminum, impeller construction, degassing process, numerical modeling, physical modeling

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1. Introduction

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.

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2. Materials and Methods

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 1Figure 2Figure 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 1Figure 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.

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Figure 1

A 3D model—impeller with four holes—variant B4.

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Figure 2

A 3D model—impeller with eight holes—variant B8.

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Figure 3

A 3D model—impeller with twelve holes—variant B12.

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Figure 4

A 3D model—‘red triangle’ impeller with three holes—variant RT3.

2.2. Physical Models

Investigations were carried out on a water model of the URO 200 reactor of the barbotage refining process (see Figure 5).

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Figure 5

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.

Table 1

Values of parameters used in the calculations.

ParameterValueUnit
Maximum number of gas particles1,000,000
Rate of particle generation20001·s−1
Specific gas constant287.058J·kg−1·K−1
Atmospheric pressure1.013 × 105Pa
Water density1000kg·m−3
Water viscosity0.001kg·m−1·s−1
Boundary condition on the wallsNo-slip
Size of computational cell0.0034m

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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.

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Figure 6

The velocity of the water depending on the size of the computational grid.

The quality of the generated computational meshes was checked using the criterion skewness angle QEAS [18]. This criterion is described by the following relationship:

QEAS=max{βmax−βeq180−βeq,βeq−βminβeq},

(1)

where βmaxβmin 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).

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Figure 7

Structured equidistant mesh used in numerical calculations: (a) mesh with smoothed, surface cells (the so-called FAVOR method) used in Flow-3D; (b) visualization of the applied mesh resolution.

The calculations were conducted with an explicit scheme. The timestep was selected by the program automatically and controlled by stability and convergence. From the moment of the initial velocity field generation (start of particle generation), it was 0.0001 s.

When modeling the degassing process, three fluids are present in the system: water, gas supplied through the rotor head (impeller), and the surrounding air. Modeling such a multiphase flow is a numerically very complex issue. The necessity to overcome the liquid backpressure by the gas flowing out from the impeller leads to the formation of numerical instabilities in the volume of fluid (VOF)-based approach used by Flow-3D software. Therefore, a mixed description of the analyzed flow was used here. In this case, water was treated as a continuous medium, while, in the case of gas bubbles, the discrete phase model (DPM) model was applied. The way in which the air surrounding the system was taken into account is later described in detail.

The following additional assumptions were made in the modeling:

  • —The liquid phase was considered as an incompressible Newtonian fluid.
  • —The effect of chemical reactions during the refining process was neglected.
  • —The composition of each phase (gas and liquid) was considered homogeneous; therefore, the viscosity and surface tension were set as constants.
  • —Only full turbulence existed in the liquid, and the effect of molecular viscosity was neglected.
  • —The gas bubbles were shaped as perfect spheres.
  • —The mutual interaction between gas bubbles (particles) was neglected.

2.3.1. Modeling of Liquid Flow 

The motion of the real fluid (continuous medium) is described by the Navier–Stokes Equation [20].

dudt=−1ρ∇p+ν∇2u+13ν∇(∇⋅ u)+F,

(2)

where du/dt is the time derivative, u is the velocity vector, t is the time, and F is the term accounting for external forces including gravity (unit components denoted by XYZ).

In the simulations, the fluid flow was assumed to be incompressible, in which case the following equation is applicable:

∂u∂t+(u⋅∇)u=−1ρ∇p+ν∇2u+F.

(3)

Due to the large range of liquid velocities during flows, the turbulence formation process was included in the modeling. For this purpose, the k–ε model turbulence kinetic energy k and turbulence dissipation ε were the target parameters, as expressed by the following equations [21]:

∂(ρk)∂t+∂(ρkvi)∂xi=∂∂xj[(μ+μtσk)⋅∂k∂xi]+Gk+Gb−ρε−Ym+Sk,

(4)

∂(ρε)∂t+∂(ρεui)∂xi=∂∂xj[(μ+μtσε)⋅∂k∂xi]+C1εεk(Gk+G3εGb)+C2ερε2k+Sε,

(5)

where ρ is the gas density, σκ and σε are the Prandtl turbulence numbers, k and ε are constants of 1.0 and 1.3, and Gk and Gb are the kinetic energy of turbulence generated by the average velocity and buoyancy, respectively.

As mentioned earlier, there are two gas phases in the considered problem. In addition to the gas bubbles, which are treated here as particles, there is also air, which surrounds the system. The boundary of phase separation is in this case the free surface of the water. The shape of the free surface can change as a result of the forming velocity field in the liquid. Therefore, it is necessary to use an appropriate approach to free surface tracking. The most commonly used concept in liquid–gas flow modeling is the volume of fluid (VOF) method [22,23], and Flow-3D uses a modified version of this method called TrueVOF. It introduces the concept of the volume fraction of the liquid phase fl. This parameter can be used for classifying the cells of a discrete grid into areas filled with liquid phase (fl = 1), gaseous phase, or empty cells (fl = 0) and those through which the phase separation boundary (fl ∈ (0, 1)) passes (free surface). To determine the local variations of the liquid phase fraction, it is necessary to solve the following continuity equation:

dfldt=0.

(6)

Then, the fluid parameters in the region of coexistence of the two phases (the so-called interface) depend on the volume fraction of each phase.

ρ=flρl+(1−fl)ρg,

(7)

ν=flνl+(1−fl)νg,

(8)

where indices l and g refer to the liquid and gaseous phases, respectively.

The parameter of fluid velocity in cells containing both phases is also determined in the same way.

u=flul+(1−fl)ug.

(9)

Since the processes taking place in the surrounding air can be omitted, to speed up the calculations, a single-phase, free-surface model was used. This means that no calculations were performed in the gas cells (they were treated as empty cells). The liquid could fill them freely, and the air surrounding the system was considered by the atmospheric pressure exerted on the free surface. This approach is often used in modeling foundry and metallurgical processes [24].

2.3.2. Modeling of Gas Bubble Flow 

As stated, a particle model was used to model bubble flow. Spherical particles (gas bubbles) of a given size were randomly generated in the area marked with green in Figure 7b. In the simulations, the gas bubbles were assumed to have diameters of 0.016 and 0.02 m corresponding to the gas flow rates of 10 and 30 dm3·min−1, respectively.

Experimental studies have shown that, as a result of turbulent fluid motion, some of the bubbles may burst, leading to the formation of smaller bubbles, although merging of bubbles into larger groupings may also occur. Therefore, to be able to observe the behavior of bubbles of different sizes (diameter), the calculations generated two additional particle types with diameters twice smaller and twice larger, respectively. The proportion of each species in the system was set to 33.33% (Table 2).

Table 2

Data assumed for calculations.

NoRotor Speed (Rotational Speed)
rpm
Bubbles Diameter
m
Corresponding Gas Flow Rate
dm3·min−1
NoRotor Speed (Rotational Speed)
rpm
Bubbles Diameter
m
Corresponding Gas Flow Rate
dm3·min−1
A2000.01610D2000.0230
0.0080.01
0.0320.04
B3000.01610E3000.0230
0.0080.01
0.0320.04
C5000.01610F5000.0230
0.0080.01
0.0320.04

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The velocity of the particle results from the generated velocity field (calculated from Equation (3) in the liquid ul around it and its velocity resulting from the buoyancy force ub. The effect of particle radius r on the terminal velocity associated with buoyancy force can be determined according to Stokes’ law.

ub=29 (ρg−ρl)μlgr2,

(10)

where g is the acceleration (9.81).

The DPM model was used for modeling the two-phase (water–air) flow. In this model, the fluid (water) is treated as a continuous phase and described by the Navier–Stokes equation, while gas bubbles are particles flowing in the model fluid (discrete phase). The trajectories of each bubble in the DPM system are calculated at each timestep taking into account the mass forces acting on it. Table 3 characterizes the DPM model used in our own research [18].

Table 3

Characteristic of the DPM model.

MethodEquations
Euler–LagrangeBalance 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.

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3. Results and Discussion

3.1. Calculations of Power and Mixing Time by the Flowing Gas Bubbles

One of the most important parameters of refining with a rotor is the mixing power induced by the spinning rotor and the outflowing gas bubbles (via impeller). The mixing power of liquid metal in a ladle of height (h) by gas injection can be determined from the following relation [15]:

pgVm=ρ⋅g⋅uB,

(11)

where pg is the mixing power, Vm is the volume of liquid metal in the reactor, ρ is the density of liquid aluminum, and uB is the average speed of bubbles, given below.

uB=n⋅R⋅TAc⋅Pm⋅t,

(12)

where n is the number of gas moles, R is the gas constant (8.314), Ac is the cross-sectional area of the reactor vessel, T is the temperature of liquid aluminum in the reactor, and Pm is the pressure at the middle tank level. The pressure at the middle level of the tank is calculated by a function of the mean logarithmic difference.

Pm=(Pa+ρ⋅g⋅h)−Paln(Pa+ρ⋅g⋅h)Pa,

(13)

where Pa is the atmospheric pressure, and h is the the height of metal in the reactor.

Themelis and Goyal [25] developed a model for calculating mixing power delivered by gas injection.

pg=2Q⋅R⋅T⋅ln(1+m⋅ρ⋅g⋅hP),

(14)

where Q is the gas flow, and m is the mass of liquid metal.

Zhang [26] proposed a model taking into account the temperature difference between gas and alloy (metal).

pg=QRTgVm[ln(1+ρ⋅g⋅hPa)+(1−TTg)],

(15)

where Tg is the gas temperature at the entry point.

Data for calculating the mixing power resulting from inert gas injection into liquid aluminum are given below in Table 4. The design parameters were adopted for the model, the parameters of which are shown in Figure 5.

Table 4

Data for calculating mixing power introduced by an inert gas.

ParameterValueUnit
Height of metal column0.7m
Density of aluminum2375kg·m−3
Process duration20s
Gas temperature at the injection site940K
Cross-sectional area of ladle0.448m2
Mass of liquid aluminum546.25kg
Volume of ladle0.23M3
Temperature of liquid aluminum941.15K

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Table 5 presents the results of mixing power calculations according to the models of Themelis and Goyal and of Zhang for inert gas flows of 10, 20, and 30 dm3·min−1. The obtained calculation results significantly differed from each other. The difference was an order of magnitude, which indicates that the model is highly inaccurate without considering the temperature of the injected gas. Moreover, the calculations apply to the case when the mixing was performed only by the flowing gas bubbles, without using a rotor, which is a great simplification of the phenomenon.

Table 5

Mixing power calculated from mathematical models.

Mathematical ModelMixing Power (W·t−1)
for a Given Inert Gas Flow (dm3·min−1)
102030
Themelis and Goyal11.4923.3335.03
Zhang0.821.662.49

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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.

Table 6

Models for calculating mixing time.

AuthorsModelRemarks
Szekely [31]τ=800ε−0.4ε—W·t−1
Chiti and Paglianti [27]τ=CVQlV—volume of reactor, m3
Ql—flow intensity, m3·s−1
Iguchi and Nakamura [32]τ=1200⋅Q−0.4D1.97h−1.0υ0.47υ—kinematic viscosity, m2·s−1
D—diameter of ladle, m
h—height of metal column, m
Q—liquid flow intensity, m3·s−1

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Figure 8 and Figure 9 show the mixing time as a function of gas flow rate for various heights of the liquid column in the ladle and mixing power values.

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Figure 8

Mixing time as a function of gas flow rate for various heights of the metal column (Iguchi and Nakamura model).

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Figure 9

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 kA, and uB can be calculated using the below equations [33].

k=2D⋅uBdB⋅π−−−−−−√,

(16)

A=6Q⋅hdB⋅uB,

(17)

uB=1.02g⋅dB,−−−−−√

(18)

where D is the diffusion coefficient, and dB is the bubble diameter.

After substituting appropriate values, we get

dB=3.03×104(πD)−2/5g−1/5h4/5Q0.344N−1.48.

(19)

According to the last equation, the size of the gas bubble decreases with the increasing rotational speed (see Figure 10).

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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]:

  • —Sevik and Park:

dBmax=We0.6kr⋅(σ⋅103ρ⋅10−3)0.6⋅(10⋅ε)−0.4⋅10−2.

(20)

  • —Evans:

dBmax=⎡⎣Wekr⋅σ⋅1032⋅(ρ⋅10−3)13⎤⎦35 ⋅(10⋅ε)−25⋅10−2.

(21)

The results of calculating the maximum diameter of the bubble dBmax determined from Equation (21) are given in Table 7.

Table 7

The results of calculating the maximum diameter of the bubble using Equation (21).

ModelMixing Energy
ĺ (m2·s−3)
Weber Number (Wekr)
0.591.01.2
Zhang and Taniguchi
dmax
0.10.01670.02300.026
0.50.00880.01210.013
1.00.00670.00910.010
1.50.00570.00780.009
Sevik and Park
dBmax
0.10.2650.360.41
0.50.1390.190.21
1.00.1060.140.16
1.50.0900.120.14
Evans
dBmax
0.10.2470.3400.38
0.50.1300.1780.20
1.00.0980.1350.15
1.50.0840.1150.13

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3.3. Physical Modeling

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 11Figure 12Figure 13 and Figure 14.

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Figure 11

Impeller variant B4—gas bubbles dispersion registered for a gas flow rate of 10 dm3·min−1 and rotor speed of (a) 200, (b) 300, (c) 400, and (d) 500 rpm.

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Figure 12

Impeller variant B8—gas bubbles dispersion registered for a gas flow rate of 10 dm3·min−1 and rotor speed of (a) 200, (b) 300, (c) 400, and (d) 500 rpm.

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Figure 13

Gas bubble dispersion registered for different processing parameters (impeller variant B12).

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Figure 14

Gas bubble dispersion registered for different processing parameters (impeller variant RT3).

The analysis of the refining variants presented in Figure 11Figure 12Figure 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.

An external file that holds a picture, illustration, etc.
Object name is materials-15-05273-g015.jpg

Figure 15

Simulation results of the impeller RT3, for given flows and rotational speeds after a time of 1 s: simulation variants (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F.

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Object name is materials-15-05273-g016.jpg

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.

An external file that holds a picture, illustration, etc.
Object name is materials-15-05273-g017.jpg

Figure 17

Compilations of model research results (CFD and physical): A—single gas bubbles formed on the surface of the modeling liquid, B—excessive formation of gas chains and swirls, C—uniform distribution of gas bubbles in the entire volume of the tank, and D—dead zones without gas bubbles, no dispersion. (a) Variant B; (b) variant F.

It should be mentioned here that, in numerical simulations, it is necessary to make certain assumptions and simplifications. The calculations assumed three particle size classes (Table 2), which represent the different gas bubbles that form due to different gas flow rates. The maximum number of particles/bubbles (Table 1) generated was assumed in advance and related to the computational capabilities of the computer. Too many particles can also make it difficult to visualize and analyze the results. The size of the particles, of course, affects their behavior during simulation, while, in the figures provided in the article, the bubbles are represented by spheres (visualization of the results) of the same size. Please note that, due to the adopted Lagrangian–Eulerian approach, the simulation did not take into account phenomena such as bubble collapse or fusion. However, the obtained results allow a comprehensive analysis of the behavior of gas bubbles in the system under consideration.

The comparative analysis of the visualization (quantitative) results obtained with the water model and CFD simulations (see Figure 17) generated a sufficient agreement from the point of view of the trends. A precise quantitative evaluation is difficult to perform because of the lack of a refraction compensating system in the water model. Furthermore, in numerical simulations, it is not possible to determine the geometry of the forming gas bubbles and their interaction with each other as opposed to the visualization in the water model. The use of both research methods is complementary. Thus, a direct comparison of images obtained by the two methods requires appropriate interpretation. However, such an assessment gives the possibility to qualitatively determine the types of the present gas bubble dispersion, thus ultimately validating the CFD results with the water model.

A summary of the visualization results for impellers RT3, i.e., analysis of the occurring gas bubble dispersion types, is presented in Table 8.

Table 8

Summary of visualization results (impeller RT3)—different types of gas bubble dispersion.

No Exp.ABCDEF
Gas flow rate, dm3·min−11030
Impeller speed, rpm200300500200300500
Type of dispersionAccurateUniformUniform/excessiveMinimalExcessiveExcessive

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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.

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4. Conclusions

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.

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Funding Statement

This paper was created with the financial support grants from the AGH-UST, Faculty of Foundry Engineering, Poland (16.16.170.654 and 11/990/BK_22/0083) for the Faculty of Materials Engineering, Silesian University of Technology, Poland.

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Author Contributions

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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Institutional Review Board Statement

Not applicable.

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Informed Consent Statement

Not applicable.

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Data Availability Statement

Data are contained within the article.

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Conflicts of Interest

The authors declare no conflict of interest.

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Footnotes

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Figure 4.24 - Model with virtual valves in the extremities of the geometries to simulate the permeability of the mold promoting a more uniformed filling

Optimization of filling systems for low pressure by Flow-3D

Dissertação de Mestrado
Ciclo de Estudos Integrados Conducentes ao
Grau de Mestre em Engenharia Mecânica
Trabalho efectuado sob a orientação do
Doutor Hélder de Jesus Fernades Puga
Professor Doutor José Joaquim Carneiro Barbosa

ABSTRACT

논문의 일부로 튜터 선택 가능성과 해결해야 할 주제가 설정되는 매개변수를 염두에 두고 개발 주제 ‘Flow- 3D ®에 의한 저압 충전 시스템 최적화’가 선택되었습니다. 이를 위해서는 달성해야 할 목표와 이를 달성하기 위한 방법을 정의하는 것이 필요했습니다.

충전 시스템을 시뮬레이션하고 검증할 수 있는 광범위한 소프트웨어에도 불구하고 Flow-3D®는 시장에서 최고의 도구 중 하나로 표시되어 전체 충전 프로세스 및 행동 표현과 관련하여 탁월한 정확도로 시뮬레이션하는 능력을 입증했습니다.

이를 위해 관련 프로세스를 더 잘 이해하고 충진 시스템 시뮬레이션을 위한 탐색적 기반 역할을 하기 위해 이 도구를 탐색하는 것이 중요합니다. 지연 및 재료 낭비에 반영되는 실제적인 측면에서 충전 장치의 치수를 완벽하게 만드는 비용 및 시간 낭비. 이러한 방식으로 저압 주조 공정에서 충진 시스템을 설계하고 물리적 모델을 탐색하여 특성화하는 방법론을 검증하기 위한 것입니다.

이를 위해 다음 주요 단계를 고려하십시오.

시뮬레이션 소프트웨어 Flow 3D® 탐색;
충전 시스템 모델링;
모델의 매개변수를 탐색하여 모델링된 시스템의 시뮬레이션, 검증 및 최적화.

따라서 연구 중인 압력 곡선과 주조 분석에서 가장 관련성이 높은 정보의 최종 마이닝을 검증하기 위한 것입니다.

사용된 압력 곡선은 수집된 문헌과 이전에 수행된 실제 작업을 통해 얻었습니다. 결과를 통해 3단계 압력 곡선이 층류 충진 체계의 의도된 목적과 관련 속도가 0.5 𝑚/𝑠를 초과하지 않는다는 결론을 내릴 수 있었습니다.

충전 수준이 2인 압력 곡선은 0.5 𝑚/𝑠 이상의 속도로 영역을 채우는 더 난류 시스템을 갖습니다. 열전달 매개변수는 이전에 얻은 값이 주물에 대한 소산 거동을 확증하지 않았기 때문에 연구되었습니다.

이러한 방식으로 주조 공정에 더 부합하는 새로운 가치를 얻었습니다. 달성된 결과는 유사한 것으로 나타난 NovaFlow & Solid®에 의해 생성된 결과와 비교되어 시뮬레이션에서 설정된 매개변수를 검증했습니다. Flow 3D®는 주조 부품 시뮬레이션을 위한 강력한 도구로 입증되었습니다.

As part of the dissertation and bearing in mind the parameters in which the possibility of a choice of tutor and the subject to be addressed is established, the subject for development ’Optimization of filling systems for low pressure by Flow 3D ®’ was chosen. For this it was necessary to define the objectives to achieve and the methods to attain them. Despite the wide range of software able to simulate and validate filling systems, Flow 3D® has been shown as one of the best tools in the market, demonstrating its ability to simulate with distinctive accuracy with respect to the entire process of filling and the behavioral representation of the fluid obtained. To this end, it is important to explore this tool for a better understanding of the processes involved and to serve as an exploratory basis for the simulation of filling systems, simulation being one of the great strengths of the current industry due to the need to reduce costs and time waste, in practical terms, that lead to the perfecting of the dimensioning of filling devices, which are reflected in delays and wasted material. In this way it is intended to validate the methodology to design a filling system in lowpressure casting process, exploring their physical models and thus allowing for its characterization. For this, consider the following main phases: The exploration of the simulation software Flow 3D®; modeling of filling systems; simulation, validation and optimization of systems modeled by exploring the parameters of the models. Therefore, it is intended to validate the pressure curves under study and the eventual mining of the most relevant information in a casting analysis. The pressure curves that were used were obtained through the gathered literature and the practical work previously performed. Through the results it was possible to conclude that the pressure curve with 3 levels meets the intended purpose of a laminar filling regime and associated speeds never exceeding 0.5 𝑚/𝑠. The pressure curve with 2 filling levels has a more turbulent system, having filling areas with velocities above 0.5 𝑚/𝑠. The heat transfer parameter was studied due to the values previously obtained didn’t corroborate the behavior of dissipation regarding to the casting. In this way, new values, more in tune with the casting process, were obtained. The achieved results were compared with those generated by NovaFlow & Solid®, which were shown to be similar, validating the parameters established in the simulations. Flow 3D® was proven a powerful tool for the simulation of casting parts.

키워드

저압, Flow 3D®, 시뮬레이션, 파운드리, 압력-시간 관계,Low Pressure, Flow 3D®, Simulation, Foundry, Pressure-time relation

Figure 4.24 - Model with virtual valves in the extremities of the geometries to simulate the permeability of the mold promoting a more uniformed filling
Figure 4.24 – Model with virtual valves in the extremities of the geometries to simulate the permeability of the mold promoting a more uniformed filling
Figure 4.39 - Values of temperature contours using full energy heat transfer parameter for simula
Figure 4.39 – Values of temperature contours using full energy heat transfer parameter for simula
Figure 4.40 – Comparison between software simulations (a) Flow 3D® simulation,
(b) NovaFlow & Solid® simulation
Figure 4.40 – Comparison between software simulations (a) Flow 3D® simulation, (b) NovaFlow & Solid® simulation

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Figure 1: Mold drawings

3D Flow and Temperature Analysis of Filling a Plutonium Mold

플루토늄 주형 충전의 3D 유동 및 온도 분석

Authors: Orenstein, Nicholas P. [1]

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)

Introduction

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.

Setup

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.

Results

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.

Figure 1: Mold drawings
Figure 1: Mold drawings
Figure 2: Mold Assembly
Figure 2: Mold Assembly
Figure 4: Actual mold and cast Pu
Figure 4: Actual mold and cast Pu
Figure 5: Bottom cavity filling
from runner
Figure 5: Bottom cavity filling from runner
Figure 6: Pouring and filling
Figure 6: Pouring and filling
Figure 8: Edge detection of cavity fill geometry. Two similar depressed areas form
about the centerline. Top cavity shown; same pressure scale as other figures
Figure 8: Edge detection of cavity fill geometry. Two similar depressed areas form about the centerline. Top cavity shown; same pressure scale as other figures
Figure 10: Cavity fill times,from first fluid contact with pouring basin, Figure 11:Fluid temperature remains essentially constant
Figure 10: Cavity fill times,from first fluid contact with pouring basin, Figure 11:Fluid temperature remains essentially constant

Conclusions

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.

Extratropical cyclone damage to the seawall in Dawlish, UK: eyewitness accounts, sea level analysis and numerical modelling

영국 Dawlish의 방파제에 대한 온대 저기압 피해: 목격자 설명, 해수면 분석 및 수치 모델링

Extratropical cyclone damage to the seawall in Dawlish, UK: eyewitness accounts, sea level analysis and numerical modelling

Natural Hazards (2022)Cite this article

Abstract

2014년 2월 영국 해협(영국)과 특히 Dawlish에 영향을 미친 온대 저기압 폭풍 사슬은 남서부 지역과 영국의 나머지 지역을 연결하는 주요 철도에 심각한 피해를 입혔습니다.

이 사건으로 라인이 두 달 동안 폐쇄되어 5천만 파운드의 피해와 12억 파운드의 경제적 손실이 발생했습니다. 이 연구에서는 폭풍의 파괴력을 해독하기 위해 목격자 계정을 수집하고 해수면 데이터를 분석하며 수치 모델링을 수행합니다.

우리의 분석에 따르면 이벤트의 재난 관리는 성공적이고 효율적이었으며 폭풍 전과 도중에 인명과 재산을 구하기 위해 즉각적인 조치를 취했습니다. 파도 부이 분석에 따르면 주기가 4–8, 8–12 및 20–25초인 복잡한 삼중 봉우리 바다 상태가 존재하는 반면, 조위계 기록에 따르면 최대 0.8m의 상당한 파도와 최대 1.5m의 파도 성분이 나타났습니다.

이벤트에서 가능한 기여 요인으로 결합된 진폭. 최대 286 KN의 상당한 임펄스 파동이 손상의 시작 원인일 가능성이 가장 높았습니다. 수직 벽의 반사는 파동 진폭의 보강 간섭을 일으켜 파고가 증가하고 최대 16.1m3/s/m(벽의 미터 너비당)의 상당한 오버탑핑을 초래했습니다.

이 정보와 우리의 공학적 판단을 통해 우리는 이 사고 동안 다중 위험 계단식 실패의 가장 가능성 있는 순서는 다음과 같다고 결론을 내립니다. 조적 파괴로 이어지는 파도 충격력, 충전물 손실 및 연속적인 조수에 따른 구조물 파괴.

The February 2014 extratropical cyclonic storm chain, which impacted the English Channel (UK) and Dawlish in particular, caused significant damage to the main railway connecting the south-west region to the rest of the UK. The incident caused the line to be closed for two months, £50 million of damage and an estimated £1.2bn of economic loss. In this study, we collate eyewitness accounts, analyse sea level data and conduct numerical modelling in order to decipher the destructive forces of the storm. Our analysis reveals that the disaster management of the event was successful and efficient with immediate actions taken to save lives and property before and during the storm. Wave buoy analysis showed that a complex triple peak sea state with periods at 4–8, 8–12 and 20–25 s was present, while tide gauge records indicated that significant surge of up to 0.8 m and wave components of up to 1.5 m amplitude combined as likely contributing factors in the event. Significant impulsive wave force of up to 286 KN was the most likely initiating cause of the damage. Reflections off the vertical wall caused constructive interference of the wave amplitudes that led to increased wave height and significant overtopping of up to 16.1 m3/s/m (per metre width of wall). With this information and our engineering judgement, we conclude that the most probable sequence of multi-hazard cascading failure during this incident was: wave impact force leading to masonry failure, loss of infill and failure of the structure following successive tides.

Introduction

The progress of climate change and increasing sea levels has started to have wide ranging effects on critical engineering infrastructure (Shakou et al. 2019). The meteorological effects of increased atmospheric instability linked to warming seas mean we may be experiencing more frequent extreme storm events and more frequent series or chains of events, as well as an increase in the force of these events, a phenomenon called storminess (Mölter et al. 2016; Feser et al. 2014). Features of more extreme weather events in extratropical latitudes (30°–60°, north and south of the equator) include increased gusting winds, more frequent storm squalls, increased prolonged precipitation and rapid changes in atmospheric pressure and more frequent and significant storm surges (Dacre and Pinto 2020). A recent example of these events impacting the UK with simultaneous significant damage to coastal infrastructure was the extratropical cyclonic storm chain of winter 2013/2014 (Masselink et al. 2016; Adams and Heidarzadeh 2021). The cluster of storms had a profound effect on both coastal and inland infrastructure, bringing widespread flooding events and large insurance claims (RMS 2014).

The extreme storms of February 2014, which had a catastrophic effect on the seawall of the south Devon stretch of the UK’s south-west mainline, caused a two-month closure of the line and significant disruption to the local and regional economy (Fig. 1b) (Network Rail 2014; Dawson et al. 2016; Adams and Heidarzadeh 2021). Restoration costs were £35 m, and economic effects to the south-west region of England were estimated up to £1.2bn (Peninsula Rail Taskforce 2016). Adams and Heidarzadeh (2021) investigated the disparate cascading failure mechanisms which played a part in the failure of the railway through Dawlish and attempted to put these in the context of the historical records of infrastructure damage on the line. Subsequent severe storms in 2016 in the region have continued to cause damage and disruption to the line in the years since 2014 (Met Office 2016). Following the events of 2014, Network Rail Footnote1 who owns the network has undertaken a resilience study. As a result, it has proposed a £400 m refurbishment of the civil engineering assets that support the railway (Fig. 1) (Network Rail 2014). The new seawall structure (Fig. 1a,c), which is constructed of pre-cast concrete sections, encases the existing Brunel seawall (named after the project lead engineer, Isambard Kingdom Brunel) and has been improved with piled reinforced concrete foundations. It is now over 2 m taller to increase the available crest freeboard and incorporates wave return features to minimise wave overtopping. The project aims to increase both the resilience of the assets to extreme weather events as well as maintain or improve amenity value of the coastline for residents and visitors.

figure 1
Fig. 1

In this work, we return to the Brunel seawall and the damage it sustained during the 2014 storms which affected the assets on the evening of the 4th and daytime of the 5th of February and eventually resulted in a prolonged closure of the line. The motivation for this research is to analyse and model the damage made to the seawall and explain the damage mechanisms in order to improve the resilience of many similar coastal structures in the UK and worldwide. The innovation of this work is the multidisciplinary approach that we take comprising a combination of analysis of eyewitness accounts (social science), sea level and wave data analysis (physical science) as well as numerical modelling and engineering judgement (engineering sciences). We investigate the contemporary wave climate and sea levels by interrogating the real-time tide gauge and wave buoys installed along the south-west coast of the English Channel. We then model a typical masonry seawall (Fig. 2), applying the computational fluid dynamics package FLOW3D-Hydro,Footnote2 to quantify the magnitude of impact forces that the seawall would have experienced leading to its failure. We triangulate this information to determine the probable sequence of failures that led to the disaster in 2014.

figure 2
Fig. 2

Data and methods

Our data comprise eyewitness accounts, sea level records from coastal tide gauges and offshore wave buoys as well as structural details of the seawall. As for methodology, we analyse eyewitness data, process and investigate sea level records through Fourier transform and conduct numerical simulations using the Flow3D-Hydro package (Flow Science 2022). Details of the data and methodology are provided in the following.

Eyewitness data

The scale of damage to the seawall and its effects led the local community to document the first-hand accounts of those most closely affected by the storms including residents, local businesses, emergency responders, politicians and engineering contractors involved in the post-storm restoration work. These records now form a permanent exhibition in the local museum in DawlishFootnote3, and some of these accounts have been transcribed into a DVD account of the disaster (Dawlish Museum 2015). We have gathered data from the Dawlish Museum, national and international news reports, social media tweets and videos. Table 1 provides a summary of the eyewitness accounts. Overall, 26 entries have been collected around the time of the incident. Our analysis of the eyewitness data is provided in the third column of Table 1 and is expanded in Sect. 3.Table 1 Eyewitness accounts of damage to the Dawlish railway due to the February 2014 storm and our interpretations

Full size table

Sea level data and wave environment

Our sea level data are a collection of three tide gauge stations (Newlyn, Devonport and Swanage Pier—Fig. 5a) owned and operated by the UK National Tide and Sea Level FacilityFootnote4 for the Environment Agency and four offshore wave buoys (Dawlish, West Bay, Torbay and Chesil Beach—Fig. 6a). The tide gauge sites are all fitted with POL-EKO (www.pol-eko.com.pl) data loggers. Newlyn has a Munro float gauge with one full tide and one mid-tide pneumatic bubbler system. Devonport has a three-channel data pneumatic bubbler system, and Swanage Pier consists of a pneumatic gauge. Each has a sampling interval of 15 min, except for Swanage Pier which has a sampling interval of 10 min. The tide gauges are located within the port areas, whereas the offshore wave buoys are situated approximately 2—3.3 km from the coast at water depths of 10–15 m. The wave buoys are all Datawell Wavemaker Mk III unitsFootnote5 and come with sampling interval of 0.78 s. The buoys have a maximum saturation amplitude of 20.5 m for recording the incident waves which implies that every wave larger than this threshold will be recorded at 20.5 m. The data are provided by the British Oceanographic Data CentreFootnote6 for tide gauges and the Channel Coastal ObservatoryFootnote7 for wave buoys.

Sea level analysis

The sea level data underwent quality control to remove outliers and spikes as well as gaps in data (e.g. Heidarzadeh et al. 2022; Heidarzadeh and Satake 2015). We processed the time series of the sea level data using the Matlab signal processing tool (MathWorks 2018). For calculations of the tidal signals, we applied the tidal package TIDALFIT (Grinsted 2008), which is based on fitting tidal harmonics to the observed sea level data. To calculate the surge signals, we applied a 30-min moving average filter to the de-tided data in order to remove all wind, swell and infra-gravity waves from the time series. Based on the surge analysis and the variations of the surge component before the time period of the incident, an error margin of approximately ± 10 cm is identified for our surge analysis. Spectral analysis of the wave buoy data is performed using the fast Fourier transform (FFT) of Matlab package (Mathworks 2018).

Numerical modelling

Numerical modelling of wave-structure interaction is conducted using the computational fluid dynamics package Flow3D-Hydro version 1.1 (Flow Science 2022). Flow3D-Hydro solves the transient Navier–Stokes equations of conservation of mass and momentum using a finite difference method and on Eulerian and Lagrangian frameworks (Flow Science 2022). The aforementioned governing equations are:

∇.u=0∇.u=0

(1)

∂u∂t+u.∇u=−∇Pρ+υ∇2u+g∂u∂t+u.∇u=−∇Pρ+υ∇2u+g

(2)

where uu is the velocity vector, PP is the pressure, ρρ is the water density, υυ is the kinematic viscosity and gg is the gravitational acceleration. A Fractional Area/Volume Obstacle Representation (FAVOR) is adapted in Flow3D-Hydro, which applies solid boundaries within the Eulerian grid and calculates the fraction of areas and volume in partially blocked volume in order to compute flows on corresponding boundaries (Hirt and Nichols 1981). We validated the numerical modelling through comparing the results with Sainflou’s analytical equation for the design of vertical seawalls (Sainflou 1928; Ackhurst 2020), which is as follows:

pd=ρgHcoshk(d+z)coshkdcosσtpd=ρgHcoshk(d+z)coshkdcosσt

(3)

where pdpd is the hydrodynamic pressure, ρρ is the water density, gg is the gravitational acceleration, HH is the wave height, dd is the water depth, kk is the wavenumber, zz is the difference in still water level and mean water level, σσ is the angular frequency and tt is the time. The Sainflou’s equation (Eq. 3) is used to calculate the dynamic pressure from wave action, which is combined with static pressure on the seawall.

Using Flow3D-Hydro, a model of the Dawlish seawall was made with a computational domain which is 250.0 m in length, 15.0 m in height and 0.375 m in width (Fig. 3a). The computational domain was discretised using a single uniform grid with a mesh size of 0.125 m. The model has a wave boundary at the left side of the domain (x-min), an outflow boundary on the right side (x-max), a symmetry boundary at the bottom (z-min) and a wall boundary at the top (z-max). A wall boundary implies that water or waves are unable to pass through the boundary, whereas a symmetry boundary means that the two edges of the boundary are identical and therefore there is no flow through it. The water is considered incompressible in our model. For volume of fluid advection for the wave boundary (i.e. the left-side boundary) in our simulations, we utilised the “Split Lagrangian Method”, which guarantees the best accuracy (Flow Science, 2022).

figure 3
Fig. 3

The stability of the numerical scheme is controlled and maintained through checking the Courant number (CC) as given in the following:

C=VΔtΔxC=VΔtΔx

(4)

where VV is the velocity of the flow, ΔtΔt is the time step and ΔxΔx is the spatial step (i.e. grid size). For stability and convergence of the numerical simulations, the Courant number must be sufficiently below one (Courant et al. 1928). This is maintained by a careful adjustment of the ΔxΔx and ΔtΔt selections. Flow3D-Hydro applies a dynamic Courant number, meaning the program adjusts the value of time step (ΔtΔt) during the simulations to achieve a balance between accuracy of results and speed of simulation. In our simulation, the time step was in the range ΔtΔt = 0.0051—0.051 s.

In order to achieve the most efficient mesh resolution, we varied cell size for five values of ΔxΔx = 0.1 m, 0.125 m, 0.15 m, 0.175 m and 0.20 m. Simulations were performed for all mesh sizes, and the results were compared in terms of convergence, stability and speed of simulation (Fig. 3). A linear wave with an amplitude of 1.5 m and a period of 6 s was used for these optimisation simulations. We considered wave time histories at two gauges A and B and recorded the waves from simulations using different mesh sizes (Fig. 3). Although the results are close (Fig. 3), some limited deviations are observed for larger mesh sizes of 0.20 m and 0.175 m. We therefore selected mesh size of 0.125 m as the optimum, giving an extra safety margin as a conservative solution.

The pressure from the incident waves on the vertical wall is validated in our model by comparing them with the analytical equation of Sainflou (1928), Eq. (3), which is one of the most common set of equations for design of coastal structures (Fig. 4). The model was tested by running a linear wave of period 6 s and wave amplitude of 1.5 m against the wall, with a still water level of 4.5 m. It can be seen that the model results are very close to those from analytical equations of Sainflou (1928), indicating that our numerical model is accurately modelling the wave-structure interaction (Fig. 4).

figure 4
Fig. 4

Eyewitness account analysis

Contemporary reporting of the 4th and 5th February 2014 storms by the main national news outlets in the UK highlights the extreme nature of the events and the significant damage and disruption they were likely to have on the communities of the south-west of England. In interviews, this was reinforced by Network Rail engineers who, even at this early stage, were forecasting remedial engineering works to last for at least 6 weeks. One week later, following subsequent storms the cascading nature of the events was obvious. Multiple breaches of the seawall had taken place with up to 35 separate landslide events and significant damage to parapet walls along the coastal route also were reported. Residents of the area reported extreme effects of the storm, one likening it to an earthquake and reporting water ingress through doors windows and even through vertical chimneys (Table 1). This suggests extreme wave overtopping volumes and large wave impact forces. One resident described the structural effects as: “the house was jumping up and down on its footings”.

Disaster management plans were quickly and effectively put into action by the local council, police service and National Rail. A major incident was declared, and decisions regarding evacuation of the residents under threat were taken around 2100 h on the night of 4th February when reports of initial damage to the seawall were received (Table 1). Local hotels were asked to provide short-term refuge to residents while local leisure facilities were prepared to accept residents later that evening. Initial repair work to the railway line was hampered by successive high spring tides and storms in the following days although significant progress was still made when weather conditions permitted (Table 1).

Sea level observations and spectral analysis

The results of surge and wave analyses are presented in Figs. 5 and 6. A surge height of up to 0.8 m was recorded in the examined tide gauge stations (Fig. 5b-d). Two main episodes of high surge heights are identified: the first surge started on 3rd February 2014 at 03:00 (UTC) and lasted until 4th of February 2014 at 00:00; the second event occurred in the period 4th February 2014 15:00 to 5th February 2014 at 17:00 (Fig. 5b-d). These data imply surge durations of 21 h and 26 h for the first and the second events, respectively. Based on the surge data in Fig. 5, we note that the storm event of early February 2014 and the associated surges was a relatively powerful one, which impacted at least 230 km of the south coast of England, from Land’s End to Weymouth, with large surge heights.

figure 5
Fig. 5
figure 6
Fig. 6

Based on wave buoy records, the maximum recorded amplitudes are at least 20.5 m in Dawlish and West Bay, 1.9 m in Tor Bay and 4.9 m in Chesil (Fig. 6a-b). The buoys at Tor Bay and Chesil recorded dual peak period bands of 4–8 and 8–12 s, whereas at Dawlish and West Bay registered triple peak period bands at 4–8, 8–12 and 20–25 s (Fig. 6c, d). It is important to note that the long-period waves at 20–25 s occur with short durations (approximately 2 min) while the waves at the other two bands of 4–8 and 8–12 s appear to be present at all times during the storm event.

The wave component at the period band of 4–8 s can be most likely attributed to normal coastal waves while the one at 8–12 s, which is longer, is most likely the swell component of the storm. Regarding the third component of the waves with long period of 20 -25 s, which occurs with short durations of 2 min, there are two hypotheses; it is either the result of a local (port and harbour) and regional (the Lyme Bay) oscillations (eg. Rabinovich 1997; Heidarzadeh and Satake 2014; Wang et al. 1992), or due to an abnormally long swell. To test the first hypothesis, we consider various water bodies such as Lyme Bay (approximate dimensions of 70 km × 20 km with an average water depth of 30 m; Fig. 6), several local bays (approximate dimensions of 3.6 km × 0.6 km with an average water depth of 6 m) and harbours (approximate dimensions of 0.5 km × 0.5 km with an average water depth of 4 m). Their water depths are based on the online Marine navigation website.Footnote8 According to Rabinovich (2010), the oscillation modes of a semi-enclosed rectangle basin are given by the following equation:

Tmn=2gd−−√[(m2L)2+(nW)2]−1/2Tmn=2gd[(m2L)2+(nW)2]−1/2

(5)

where TmnTmn is the oscillation period, gg is the gravitational acceleration, dd is the water depth, LL is the length of the basin, WW is the width of the basin, m=1,2,3,…m=1,2,3,… and n=0,1,2,3,…n=0,1,2,3,…; mm and nn are the counters of the different modes. Applying Eq. (5) to the aforementioned water bodies results in oscillation modes of at least 5 min, which is far longer than the observed period of 20–25 s. Therefore, we rule out the first hypothesis and infer that the long period of 20–25 s is most likely a long swell wave coming from distant sources. As discussed by Rabinovich (1997) and Wang et al. (2022), comparison between sea level spectra before and after the incident is a useful method to distinguish the spectrum of the weather event. A visual inspection of Fig. 6 reveals that the forcing at the period band of 20–25 s is non-existent before the incident.

Numerical simulations of wave loading and overtopping

Based on the results of sea level data analyses in the previous section (Fig. 6), we use a dual peak wave spectrum with peak periods of 10.0 s and 25.0 s for numerical simulations because such a wave would be comprised of the most energetic signals of the storm. For variations of water depth (2.0–4.0 m), coastal wave amplitude (0.5–1.5 m) (Fig. 7) and storm surge height (0.5–0.8 m) (Fig. 5), we developed 20 scenarios (Scn) which we used in numerical simulations (Table 2). Data during the incident indicated that water depth was up to the crest level of the seawall (approximately 4 m water depth); therefore, we varied water depth from 2 to 4 m in our simulation scenarios. Regarding wave amplitudes, we referred to the variations at a nearby tide gauge station (West Bay) which showed wave amplitude up to 1.2 m (Fig. 7). Therefore, wave amplitude was varied from 0.5 m to 1.5 m by considering a factor a safety of 25% for the maximum wave amplitude. As for the storm surge component, time series of storm surges calculated at three coastal stations adjacent to Dawlish showed that it was in the range of 0.5 m to 0.8 m (Fig. 5). These 20 scenarios would help to study uncertainties associated with wave amplitudes and pressures. Figure 8 shows snapshots of wave propagation and impacts on the seawall at different times.

figure 7
Fig. 7

Table 2 The 20 scenarios considered for numerical simulations in this study

Full size table

figure 8
Fig. 8

Results of wave amplitude simulations

Large wave amplitudes can induce significant wave forcing on the structure and cause overtopping of the seawall, which could eventually cascade to other hazards such as erosion of the backfill and scour (Adams and Heidarzadeh, 2021). The first 10 scenarios of our modelling efforts are for the same incident wave amplitudes of 0.5 m, which occur at different water depths (2.0–4.0 m) and storm surge heights (0.5–0.8 m) (Table 2 and Fig. 9). This is because we aim at studying the impacts of effective water depth (deff—the sum of mean sea level and surge height) on the time histories of wave amplitudes as the storm evolves. As seen in Fig. 9a, by decreasing effective water depth, wave amplitude increases. For example, for Scn-1 with effective depth of 4.5 m, the maximum amplitude of the first wave is 1.6 m, whereas it is 2.9 m for Scn-2 with effective depth of 3.5 m. However, due to intensive reflections and interferences of the waves in front of the vertical seawall, such a relationship is barely seen for the second and the third wave peaks. It is important to note that the later peaks (second or third) produce the largest waves rather than the first wave. Extraordinary wave amplifications are seen for the Scn-2 (deff = 3.5 m) and Scn-7 (deff = 3.3 m), where the corresponding wave amplitudes are 4.5 m and 3.7 m, respectively. This may indicate that the effective water depth of deff = 3.3–3.5 m is possibly a critical water depth for this structure resulting in maximum wave amplitudes under similar storms. In the second wave impact, the combined wave height (i.e. the wave amplitude plus the effective water depth), which is ultimately an indicator of wave overtopping, shows that the largest wave heights are generated by Scn-2, 7 and 8 (Fig. 9a) with effective water depths of 3.5 m, 3.3 m and 3.8 m and combined heights of 8.0 m, 7.0 m and 6.9 m (Fig. 9b). Since the height of seawall is 5.4 m, the combined wave heights for Scn-2, 7 and 8 are greater than the crest height of the seawall by 2.6 m, 1.6 m and 1.5 m, respectively, which indicates wave overtopping.

figure 9
Fig. 9

For scenarios 11–20 (Fig. 10), with incident wave amplitudes of 1.5 m (Table 2), the largest wave amplitudes are produced by Scn-17 (deff = 3.3 m), Scn-13 (deff = 2.5 m) and Scn-12 (deff = 3.5 m), which are 5.6 m, 5.1 m and 4.5 m. The maximum combined wave heights belong to Scn-11 (deff = 4.5 m) and Scn-17 (deff = 3.3 m), with combined wave heights of 9.0 m and 8.9 m (Fig. 10b), which are greater than the crest height of the seawall by 4.6 m and 3.5 m, respectively.

figure 10
Fig. 10

Our simulations for all 20 scenarios reveal that the first wave is not always the largest and wave interactions, reflections and interferences play major roles in amplifying the waves in front of the seawall. This is primarily because the wall is fully vertical and therefore has a reflection coefficient of close to one (i.e. full reflection). Simulations show that the combined wave height is up to 4.6 m higher than the crest height of the wall, implying that severe overtopping would be expected.

Results of wave loading calculations

The pressure calculations for scenarios 1–10 are given in Fig. 11 and those of scenarios 11–20 in Fig. 12. The total pressure distribution in Figs. 1112 mostly follows a triangular shape with maximum pressure at the seafloor as expected from the Sainflou (1928) design equations. These pressure plots comprise both static (due to mean sea level in front of the wall) and dynamic (combined effects of surge and wave) pressures. For incident wave amplitudes of 0.5 m (Fig. 11), the maximum wave pressure varies in the range of 35–63 kPa. At the sea surface, it is in the range of 4–20 kPa (Fig. 11). For some scenarios (Scn-2 and 7), the pressure distribution deviates from a triangular shape and shows larger pressures at the top, which is attributed to the wave impacts and partial breaking at the sea surface. This adds an additional triangle-shaped pressure distribution at the sea surface elevation consistent with the design procedure developed by Goda (2000) for braking waves. The maximum force on the seawall due to scenarios 1–10, which is calculated by integrating the maximum pressure distribution over the wave-facing surface of the seawall, is in the range of 92–190 KN (Table 2).

figure 11
Fig. 11
figure 12
Fig. 12

For scenarios 11–20, with incident wave amplitude of 1.5 m, wave pressures of 45–78 kPa and 7–120 kPa, for  the bottom and top of the wall, respectively, were observed (Fig. 12). Most of the plots show a triangular pressure distribution, except for Scn-11 and 15. A significant increase in wave impact pressure is seen for Scn-15 at the top of the structure, where a maximum pressure of approximately 120 kPa is produced while other scenarios give a pressure of 7–32 kPa for the sea surface. In other words, the pressure from Scn-15 is approximately four times larger than the other scenarios. Such a significant increase of the pressure at the top is most likely attributed to the breaking wave impact loads as detailed by Goda (2000) and Cuomo et al. (2010). The wave simulation snapshots in Fig. 8 show that the wave breaks before reaching the wall. The maximum force due to scenarios 11–20 is 120–286 KN.

The breaking wave impacts peaking at 286 KN in our simulations suggest destabilisation of the upper masonry blocks, probably by grout malfunction. This significant impact force initiated the failure of the seawall which in turn caused extensive ballast erosion. Wave impact damage was proposed by Adams and Heidarzadeh (2021) as one of the primary mechanisms in the 2014 Dawlish disaster. In the multi-hazard risk model proposed by these authors, damage mechanism III (failure pathway 5 in Adams and Heidarzadeh, 2021) was characterised by wave impact force causing damage to the masonry elements, leading to failure of the upper sections of the seawall and loss of infill material. As blocks were removed, access to the track bed was increased for inbound waves allowing infill material from behind the seawall to be fluidised and subsequently removed by backwash. The loss of infill material critically compromised the stability of the seawall and directly led to structural failure. In parallel, significant wave overtopping (discussed in the next section) led to ballast washout and cascaded, in combination with masonry damage, to catastrophic failure of the wall and suspension of the rails in mid-air (Fig. 1b), leaving the railway inoperable for two months.

Wave Overtopping

The two most important factors contributing to the 2014 Dawlish railway catastrophe were wave impact forces and overtopping. Figure 13 gives the instantaneous overtopping rates for different scenarios, which experienced overtopping. It can be seen that the overtopping rates range from 0.5 m3/s/m to 16.1 m3/s/m (Fig. 13). Time histories of the wave overtopping rates show that the phenomenon occurs intermittently, and each time lasts 1.0–7.0 s. It is clear that the longer the overtopping time, the larger the volume of the water poured on the structure. The largest wave overtopping rates of 16.1 m3/s/m and 14.4 m3/s/m belong to Scn-20 and 11, respectively. These are the two scenarios that also give the largest combined wave heights (Fig. 10b).

figure 13
Fig. 13

The cumulative overtopping curves (Figs. 1415) show the total water volume overtopped the structure during the entire simulation time. This is an important hazard factor as it determines the level of soil saturation, water pore pressure in the soil and soil erosion (Van der Meer et al. 2018). The maximum volume belongs to Scn-20, which is 65.0 m3/m (m-cubed of water per metre length of the wall). The overtopping volumes are 42.7 m3/m for Scn-11 and 28.8 m3/m for Scn-19. The overtopping volume is in the range of 0.7–65.0 m3/m for all scenarios.

figure 14
Fig. 14
figure 15
Fig. 15

For comparison, we compare our modelling results with those estimated using empirical equations. For the case of the Dawlish seawall, we apply the equation proposed by Van Der Meer et al. (2018) to estimate wave overtopping rates, based on a set of decision criteria which are the influence of foreshore, vertical wall, possible breaking waves and low freeboard:

qgH3m−−−−√=0.0155(Hmhs)12e(−2.2RcHm)qgHm3=0.0155(Hmhs)12e(−2.2RcHm)

(6)

where qq is the mean overtopping rate per metre length of the seawall (m3/s/m), gg is the acceleration due to gravity, HmHm is the incident wave height at the toe of the structure, RcRc is the wall crest height above mean sea level, hshs is the deep-water significant wave height and e(x)e(x) is the exponential function. It is noted that Eq. (6) is valid for 0.1<RcHm<1.350.1<RcHm<1.35. For the case of the Dawlish seawall and considering the scenarios with larger incident wave amplitude of 1.5 m (hshs= 1.5 m), the incident wave height at the toe of the structure is HmHm = 2.2—5.6 m, and the wall crest height above mean sea level is RcRc = 0.6–2.9 m. As a result, Eq. (6) gives mean overtopping rates up to approximately 2.9 m3/s/m. A visual inspection of simulated overtopping rates in Fig. 13 for Scn 11–20 shows that the mean value of the simulated overtopping rates (Fig. 13) is close to estimates using Eq. (6).

Discussion and conclusions

We applied a combination of eyewitness account analysis, sea level data analysis and numerical modelling in combination with our engineering judgement to explain the damage to the Dawlish railway seawall in February 2014. Main findings are:

  • Eyewitness data analysis showed that the extreme nature of the event was well forecasted in the hours prior to the storm impact; however, the magnitude of the risks to the structures was not well understood. Multiple hazards were activated simultaneously, and the effects cascaded to amplify the damage. Disaster management was effective, exemplified by the establishment of an emergency rendezvous point and temporary evacuation centre during the storm, indicating a high level of hazard awareness and preparedness.
  • Based on sea level data analysis, we identified triple peak period bands at 4–8, 8–12 and 20–25 s in the sea level data. Storm surge heights and wave oscillations were up to 0.8 m and 1.5 m, respectively.
  • Based on the numerical simulations of 20 scenarios with different water depths, incident wave amplitudes, surge heights and peak periods, we found that the wave oscillations at the foot of the seawall result in multiple wave interactions and interferences. Consequently, large wave amplitudes, up to 4.6 m higher than the height of the seawall, were generated and overtopped the wall. Extreme impulsive wave impact forces of up to 286 KN were generated by the waves interacting with the seawall.
  • We measured maximum wave overtopping rates of 0.5–16.1 m3/s/m for our scenarios. The cumulative overtopping water volumes per metre length of the wall were 0.7–65.0 m3/m.
  • Analysis of all the evidence combined with our engineering judgement suggests that the most likely initiating cause of the failure was impulsive wave impact forces destabilising one or more grouted joints between adjacent masonry blocks in the wall. Maximum observed pressures of 286 KN in our simulations are four times greater in magnitude than background pressures leading to block removal and initiating failure. Therefore, the sequence of cascading events was :1) impulsive wave impact force causing damage to masonry, 2) failure of the upper sections of the seawall, 3) loss of infill resulting in a reduction of structural strength in the landward direction, 4) ballast washout as wave overtopping and inbound wave activity increased and 5) progressive structural failure following successive tides.

From a risk mitigation point of view, the stability of the seawall in the face of future energetic cyclonic storm events and sea level rise will become a critical factor in protecting the rail network. Mitigation efforts will involve significant infrastructure investment to strengthen the civil engineering assets combined with improved hazard warning systems consisting of meteorological forecasting and real-time wave observations and instrumentation. These efforts must take into account the amenity value of coastal railway infrastructure to local communities and the significant number of tourists who visit every year. In this regard, public awareness and active engagement in the planning and execution of the project will be crucial in order to secure local stakeholder support for the significant infrastructure project that will be required for future resilience.

Notes

  1. https://www.networkrail.co.uk/..
  2. https://www.flow3d.com/products/flow-3d-hydro/.
  3. https://www.devonmuseums.net/Dawlish-Museum/Devon-Museums/.
  4. https://ntslf.org/.
  5. https://www.datawell.nl/Products/Buoys/DirectionalWaveriderMkIII.aspx.
  6. https://www.bodc.ac.uk/.
  7. https://coastalmonitoring.org/cco/.
  8. https://webapp.navionics.com/#boating@8&key=iactHlwfP.

References

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Acknowledgements

We are grateful to Brunel University London for administering the scholarship awarded to KA. The Flow3D-Hydro used in this research for numerical modelling is licenced to Brunel University London through an academic programme contract. We sincerely thank Prof Harsh Gupta (Editor-in-Chief) and two anonymous reviewers for their constructive review comments.

Funding

This project was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) through a PhD scholarship to Keith Adams.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Brunel University London, Uxbridge, UB8 3PH, UKKeith Adams
  2. Department of Architecture and Civil Engineering, University of Bath, Bath, BA2 7AY, UKMohammad Heidarzadeh

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Correspondence to Keith Adams.

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Adams, K., Heidarzadeh, M. Extratropical cyclone damage to the seawall in Dawlish, UK: eyewitness accounts, sea level analysis and numerical modelling. Nat Hazards (2022). https://doi.org/10.1007/s11069-022-05692-2

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  • Received17 May 2022
  • Accepted17 October 2022
  • Published14 November 2022
  • DOIhttps://doi.org/10.1007/s11069-022-05692-2

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Keywords

  • Storm surge
  • Cyclone
  • Railway
  • Climate change
  • Infrastructure
  • Resilience
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FLOW-3D (x) 는 FLOW-3D 툴킷에 추가된 강력한 기능으로 사용자가 CAD 매개변수 정의에서 자동화된 시뮬레이션 및 후처리 전체 주기 워크플로우를 통해 많은 시뮬레이션 요소를 쉽게 연결, 자동화 및 최적화할 수 있습니다. 이 교육에서 사용자는 견고한 시뮬레이션 환경을 만들기 위해 다른 소프트웨어 노드와 함께 FLOW-3D (x) 를 사용하는 방법을 배우게 됩니다.

참석자는 컨퍼런스 후 FLOW-3D (x) 의 3개월 무료 라이선스를 받게 됩니다 .

Rescale: FLOW-3D 사용자가 클라우드 기반 고성능 컴퓨팅(HPC) 리소스를 활용할 수 있는 새로운 플랫폼

Flow Science는 고객 이 다양한 원격 하드웨어에서 FLOW-3D 모델 을 실행할 수 있도록 새로운 클라우드 기반 리소스인 Rescale 을 제공하고 있습니다. 이 교육은 다음 세 가지 주제로 구성됩니다. 

  1. Rescale 계정 개설, 모델 실행 및 데이터 후처리 
  2. 명령줄 모드에서 Rescale에서 실행하는 것과 사용자 인터페이스 기반 환경에서 Rescale을 사용하는 것 비교. 그리고 
  3. Rescale에서 사용할 수 있는 다양한 유형의 하드웨어 아키텍처에 대한 자세한 벤치마킹을 통해 하드웨어 선택 및 HPC 배포 전략과 관련된 비용 성능 고려 사항을 명확히 합니다. 교육 세션이 끝나면 사용자는 Rescale 플랫폼에서 모델을 실행하는 비용과 실용성을 모두 명확하게 이해할 수 있습니다.

발표자 정보

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소피텔 스트라스부르
소피텔 스트라스부르 로비
소피텔 스트라스부르 테라스

Fig. 1. Schematic figure showing the PREP with additional gas flowing on the end face of electrode.

플라즈마 회전 전극 공정 중 분말 형성에 대한 공정 매개변수 및 냉각 가스의 영향

Effects of process parameters and cooling gas on powder formation during the plasma rotating electrode process

Yujie Cuia Yufan Zhaoa1 Haruko Numatab Kenta Yamanakaa Huakang Biana Kenta Aoyagia AkihikoChibaa
aInstitute for Materials Research, Tohoku University, Sendai 980-8577, JapanbDepartment of Materials Processing, Graduate School of Engineering, Tohoku University, Sendai 980-8577, Japan

Highlights

•The limitation of increasing the rotational speed in decreasing powder size was clarified.

•Cooling and disturbance effects varied with the gas flowing rate.

•Inclined angle of the residual electrode end face affected powder formation.

•Additional cooling gas flowing could be applied to control powder size.

Abstract

The plasma rotating electrode process (PREP) is rapidly becoming an important powder fabrication method in additive manufacturing. However, the low production rate of fine PREP powder limits the development of PREP. Herein, we investigated different factors affecting powder formation during PREP by combining experimental methods and numerical simulations. The limitation of increasing the rotation electrode speed in decreasing powder size is attributed to the increased probability of adjacent droplets recombining and the decreased tendency of granulation. The effects of additional Ar/He gas flowing on the rotational electrode on powder formation is determined through the cooling effect, the disturbance effect, and the inclined effect of the residual electrode end face simultaneously. A smaller-sized powder was obtained in the He atmosphere owing to the larger inclined angle of the residual electrode end face compared to the Ar atmosphere. Our research highlights the route for the fabrication of smaller-sized powders using PREP.

플라즈마 회전 전극 공정(PREP)은 적층 제조 에서 중요한 분말 제조 방법으로 빠르게 자리잡고 있습니다. 그러나 미세한 PREP 분말의 낮은 생산율은 PREP의 개발을 제한합니다. 여기에서 우리는 실험 방법과 수치 시뮬레이션을 결합하여 PREP 동안 분말 형성에 영향을 미치는 다양한 요인을 조사했습니다. 분말 크기 감소에서 회전 전극 속도 증가의 한계는 인접한 액적 재결합 확률 증가 및 과립화 경향 감소에 기인합니다.. 회전 전극에 흐르는 추가 Ar/He 가스가 분말 형성에 미치는 영향은 냉각 효과, 외란 효과 및 잔류 전극 단면의 경사 효과를 통해 동시에 결정됩니다. He 분위기에서는 Ar 분위기에 비해 잔류 전극 단면의 경사각이 크기 때문에 더 작은 크기의 분말이 얻어졌다. 우리의 연구는 PREP를 사용하여 더 작은 크기의 분말을 제조하는 경로를 강조합니다.

Keywords

Plasma rotating electrode process

Ti-6Al-4 V alloy, Rotating speed, Numerical simulation, Gas flowing, Powder size

Introduction

With the development of additive manufacturing, there has been a significant increase in high-quality powder production demand [1,2]. The initial powder characteristics are closely related to the uniform powder spreading [3,4], packing density [5], and layer thickness observed during additive manufacturing [6], thus determining the mechanical properties of the additive manufactured parts [7,8]. Gas atomization (GA) [9–11], centrifugal atomization (CA) [12–15], and the plasma rotating electrode process (PREP) are three important powder fabrication methods.

Currently, GA is the dominant powder fabrication method used in additive manufacturing [16] for the fabrication of a wide range of alloys [11]. GA produces powders by impinging a liquid metal stream to droplets through a high-speed gas flow of nitrogen, argon, or helium. With relatively low energy consumption and a high fraction of fine powders, GA has become the most popular powder manufacturing technology for AM.

The entrapped gas pores are generally formed in the powder after solidification during GA, in which the molten metal is impacted by a high-speed atomization gas jet. In addition, satellites are formed in GA powder when fine particles adhere to partially molten particles.

The gas pores of GA powder result in porosity generation in the additive manufactured parts, which in turn deteriorates its mechanical properties because pores can become crack initiation sites [17]. In CA, a molten metal stream is poured directly onto an atomizer disc spinning at a high rotational speed. A thin film is formed on the surface of the disc, which breaks into small droplets due to the centrifugal force. Metal powder is obtained when these droplets solidify.

Compared with GA powder, CA powder exhibits higher sphericity, lower impurity content, fewer satellites, and narrower particle size distribution [12]. However, very high speed is required to obtain fine powder by CA. In PREP, the molten metal, melted using the plasma arc, is ejected from the rotating rod through centrifugal force. Compared with GA powder, PREP-produced powders also have higher sphericity and fewer pores and satellites [18].

For instance, PREP-fabricated Ti6Al-4 V alloy powder with a powder size below 150 μm exhibits lower porosity than gas-atomized powder [19], which decreases the porosity of additive manufactured parts. Furthermore, the process window during electron beam melting was broadened using PREP powder compared to GA powder in Inconel 718 alloy [20] owing to the higher sphericity of the PREP powder.

In summary, PREP powder exhibits many advantages and is highly recommended for powder-based additive manufacturing and direct energy deposition-type additive manufacturing. However, the low production rate of fine PREP powder limits the widespread application of PREP powder in additive manufacturing.

Although increasing the rotating speed is an effective method to decrease the powder size [21,22], the reduction in powder size becomes smaller with the increased rotating speed [23]. The occurrence of limiting effects has not been fully clarified yet.

Moreover, the powder size can be decreased by increasing the rotating electrode diameter [24]. However, these methods are quite demanding for the PREP equipment. For instance, it is costly to revise the PREP equipment to meet the demand of further increasing the rotating speed or electrode diameter.

Accordingly, more feasible methods should be developed to further decrease the PREP powder size. Another factor that influences powder formation is the melting rate [25]. It has been reported that increasing the melting rate decreases the powder size of Inconel 718 alloy [26].

In contrast, the powder size of SUS316 alloy was decreased by decreasing the plasma current within certain ranges. This was ascribed to the formation of larger-sized droplets from fluid strips with increased thickness and spatial density at higher plasma currents [27]. The powder size of NiTi alloy also decreases at lower melting rates [28]. Consequently, altering the melting rate, varied with the plasma current, is expected to regulate the PREP powder size.

Furthermore, gas flowing has a significant influence on powder formation [27,29–31]. On one hand, the disturbance effect of gas flowing promotes fluid granulation, which in turn contributes to the formation of smaller-sized powder [27]. On the other hand, the cooling effect of gas flowing facilitates the formation of large-sized powder due to increased viscosity and surface tension. However, there is a lack of systematic research on the effect of different gas flowing on powder formation during PREP.

Herein, the authors systematically studied the effects of rotating speed, electrode diameter, plasma current, and gas flowing on the formation of Ti-6Al-4 V alloy powder during PREP as additive manufactured Ti-6Al-4 V alloy exhibits great application potential [32]. Numerical simulations were conducted to explain why increasing the rotating speed is not effective in decreasing powder size when the rotation speed reaches a certain level. In addition, the different factors incited by the Ar/He gas flowing on powder formation were clarified.

Fig. 1. Schematic figure showing the PREP with additional gas flowing on the end face of electrode.
Fig. 1. Schematic figure showing the PREP with additional gas flowing on the end face of electrode.

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1115/1.3422970

Fig. 1. Model geometry with the computational domain, extrusion nozzle, toolpath, and boundary conditions. The model is presented while printing the fifth layer.

재료 압출 적층 제조에서 증착된 층의 안정성 및 변형

Md Tusher Mollah Raphaël 사령관 Marcin P. Serdeczny David B. Pedersen Jon Spangenberg덴마크 공과 대학 기계 공학과, Kgs. 덴마크 링비

2020년 12월 22일 접수, 2021년 5월 1일 수정, 2021년 7월 15일 수락, 2021년 7월 21일 온라인 사용 가능, 기록 버전 2021년 8월 17일 .

Abstract

이 문서는 재료 압출 적층 제조 에서 여러 레이어를 인쇄하는 동안 증착 흐름의 전산 유체 역학 시뮬레이션 을 제공합니다 개발된 모델은 증착된 레이어의 형태를 예측하고 점소성 재료 를 인쇄하는 동안 레이어 변형을 캡처합니다 . 물리학은 일반화된 뉴턴 유체 로 공식화된 Bingham 구성 모델의 연속성 및 운동량 방정식에 의해 제어됩니다. . 증착된 층의 단면 모양이 예측되고 재료의 다양한 구성 매개변수에 대해 층의 변형이 연구됩니다. 층의 변형은 인쇄물의 정수압과 압출시 압출압력으로 인한 것임을 알 수 있다. 시뮬레이션에 따르면 항복 응력이 높을수록 변형이 적은 인쇄물이 생성되는 반면 플라스틱 점도 가 높을수록 증착된 레이어에서변형이 커 집니다 . 또한, 인쇄 속도, 압출 속도 의 영향, 층 높이 및 인쇄된 층의 변형에 대한 노즐 직경을 조사합니다. 마지막으로, 이 모델은 후속 인쇄된 레이어의 정수압 및 압출 압력을 지원하기 위해 증착 후 점소성 재료가 요구하는 항복 응력의 필요한 증가에 대한 보수적인 추정치를 제공합니다.

This paper presents computational fluid dynamics simulations of the deposition flow during printing of multiple layers in material extrusion additive manufacturing. The developed model predicts the morphology of the deposited layers and captures the layer deformations during the printing of viscoplastic materials. The physics is governed by the continuity and momentum equations with the Bingham constitutive model, formulated as a generalized Newtonian fluid. The cross-sectional shapes of the deposited layers are predicted, and the deformation of layers is studied for different constitutive parameters of the material. It is shown that the deformation of layers is due to the hydrostatic pressure of the printed material, as well as the extrusion pressure during the extrusion. The simulations show that a higher yield stress results in prints with less deformations, while a higher plastic viscosity leads to larger deformations in the deposited layers. Moreover, the influence of the printing speed, extrusion speed, layer height, and nozzle diameter on the deformation of the printed layers is investigated. Finally, the model provides a conservative estimate of the required increase in yield stress that a viscoplastic material demands after deposition in order to support the hydrostatic and extrusion pressure of the subsequently printed layers.

Fig. 1. Model geometry with the computational domain, extrusion nozzle, toolpath, and boundary conditions. The model is presented while printing the fifth layer.
Fig. 1. Model geometry with the computational domain, extrusion nozzle, toolpath, and boundary conditions. The model is presented while printing the fifth layer.

키워드

점성 플라스틱 재료, 재료 압출 적층 제조(MEX-AM), 다층 증착, 전산유체역학(CFD), 변형 제어
Viscoplastic Materials, Material Extrusion Additive Manufacturing (MEX-AM), Multiple-Layers Deposition, Computational Fluid Dynamics (CFD), Deformation Control

Introduction

Three-dimensional printing of viscoplastic materials has grown in popularity over the recent years, due to the success of Material Extrusion Additive Manufacturing (MEX-AM) [1]. Viscoplastic materials, such as ceramic pastes [2,3], hydrogels [4], thermosets [5], and concrete [6], behave like solids when the applied load is below their yield stress, and like a fluid when the applied load exceeds their yield stress [7]. Viscoplastic materials are typically used in MEX-AM techniques such as Robocasting [8], and 3D concrete printing [9,10]. The differences between these technologies lie in the processing of the material before the extrusion and in the printing scale (from microscale to big area additive manufacturing). In these extrusion-based technologies, the structure is fabricated in a layer-by-layer approach onto a solid surface/support [11, 12]. During the process, the material is typically deposited on top of the previously printed layers that may be already solidified (wet-on-dry printing) or still deformable (wet-on-wet printing) [1]. In wet-on-wet printing, control over the deformation of layers is important for the stability and geometrical accuracy of the prints. If the material is too liquid after the deposition, it cannot support the pressure of the subsequently deposited layers. On the other hand, the material flowability is a necessity during extrusion through the nozzle. Several experimental studies have been performed to analyze the physics of the extrusion and deposition of viscoplastic materials, as reviewed in Refs. [13–16]. The experimental measurements can be supplemented with Computational Fluid Dynamics (CFD) simulations to gain a more complete picture of MEX-AM. A review of the CFD studies within the material processing and deposition in 3D concrete printing was presented by Roussel et al. [17]. Wolfs et al. [18] predicted numerically the failure-deformation of a cylindrical structure due to the self-weight by calculating the stiffness and strength of the individual layers. It was found that the deformations can take place in all layers, however the most critical deformation occurs in the bottom layer. Comminal et al. [19,20] presented three-dimensional simulations of the material deposition in MEX-AM, where the fluid was approximated as Newtonian. Subsequently, the model was experimentally validated in Ref. [21] for polymer-based MEX-AM, and extended to simulate the deposition of multiple layers in Ref. [22], where the previously printed material was assumed solid. Xia et al. [23] simulated the influence of the viscoelastic effects on the shape of deposited layers in MEX-AM. A numerical model for simulating the deposition of a viscoplastic material was recently presented and experimentally validated in Refs. [24] and [25]. These studies focused on predicting the cross-sectional shape of a single printed layer for different processing conditions (relative printing speed, and layer height). Despite these research efforts, a limited number of studies have focused on investigating the material deformations in wet-on-wet printing when multiple layers are deposited on top of each other. This paper presents CFD simulations of the extrusion-deposition flow of a viscoplastic material for several subsequent layers (viz. three- and five-layers). The material is continuously printed one layer over another on a fixed solid surface. The rheology of the viscoplastic material is approximated by the Bingham constitutive equation that is formulated using the Generalized Newtonian Fluid (GNF) model. The CFD model is used to predict the cross-sectional shapes of the layers and their deformations while printing the next layers on top. Moreover, the simulations are used to quantify the extrusion pressure applied by the deposited material on the substrate, and the previously printed layers. Numerically, it is investigated how the process parameters (i.e., the extrusion speed, printing speed, nozzle diameter, and layer height) and the material rheology affect the deformations of the deposited layers. Section 2 describes the methodology of the study. Section 3 presents and discusses the results. The study is summarized and concluded in Section 4.

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하류하천의 영향 최소화를 위한 보조 여수로 최적 활용방안 검토

The Optimal Operation on Auxiliary Spillway to Minimize the Flood Damage in Downstream River with Various Outflow Conditions

하류하천의 영향 최소화를 위한 보조 여수로 최적 활용방안 검토

Hyung Ju Yoo1, Sung Sik Joo2, Beom Jae Kwon3, Seung Oh Lee4*

유 형주1, 주 성식2, 권 범재3, 이 승오4*

1Ph.D Student, Dept. of Civil & Environmental Engineering, Hongik University
2Director, Water Resources & Environment Department, HECOREA
3Director, Water Resources Department, ISAN
4Professor, Dept. of Civil & Environmental Engineering, Hongik University

1홍익대학교 건설환경공학과 박사과정
2㈜헥코리아 수자원환경사업부 이사
3㈜이산 수자원부 이사
4홍익대학교 건설환경공학과 교수

ABSTRACT

최근 기후변화로 인해 강우강도 및 빈도의 증가에 따른 집중호우의 영향 및 기존 여수로의 노후화에 대비하여 홍수 시 하류 하천의 영향을 최소화할 수 있는 보조 여수로 활용방안 구축이 필요한 실정이다. 이를 위해, 수리모형 실험 및 수치모형 실험을 통하여 보조 여수로 운영에 따른 흐름특성 변화 검토에 관한 연구가 많이 진행되어 왔다. 그러나 대부분의 연구는 여수로에서의 흐름특성 및 기능성에 대한 검토를 수행하였을 뿐 보조 여수로의 활용방안에 따른 하류하천 영향 검토 및 호안 안정성 검토에 관한 연구는 미비한 실정이다. 이에 본 연구에서는 기존 여수로 및 보조 여수로 방류 조건에 따른 하류영향 분석 및 호안 안정성 측면에서 최적 방류 시나리오 검토를 3차원 수치모형인 FLOW-3D를 사용하여 검토하였다. 또한 FLOW-3D 수치모의 수행을 통한 유속, 수위 결과와 소류력 산정 결과를 호안 설계허용 기준과 비교하였다. 수문 완전 개도 조건으로 가정하고 계획홍수량 유입 시 다양한 보조 여수로 활용방안에 대하여 수치모의를 수행한 결과, 보조 여수로 단독 운영 시 기존 여수로 단독운영에 비하여 최대유속 및 최대 수위의 감소효과를 확인하였다. 다만 계획홍수량의 45% 이하 방류 조건에서 대안부의 호안 안정성을 확보하였고 해당 방류량 초과 경우에는 처오름 현상이 발생하여 월류에 대한 위험성 증가를 확인하였다. 따라서 기존 여수로와의 동시 운영 방안 도출이 중요하다고 판단하였다. 여수로의 배분 비율 및 총 허용 방류량에 대하여 검토한 결과 보조 여수로의 방류량이 기존 여수로의 방류량보다 큰 경우 하류하천의 흐름이 중심으로 집중되어 대안부의 유속 저감 및 수위 감소를 확인하였고, 계획 홍수량의 77% 이하의 조건에서 호안의 허용 유속 및 허용 소류력 조건을 만족하였다. 이를 통하여 본 연구에서 제안한 보조 여수로 활용방안으로는 기존 여수로와 동시 운영 시 총 방류량에 대하여 보조 여수로의 배분량이 기존 여수로의 배분량보다 크게 설정하는 것이 하류하천의 영향을 최소화 할 수 있는 것으로 나타났다. 그러나 본 연구는 여수로 방류에 따른 대안부에서의 영향에 대해서만 검토하였고 수문 전면 개도 조건에서 검토하였다는 한계점은 분명히 있다. 이에 향후에는 다양한 수문 개도 조건 및 방류 시나리오를 적용 및 검토한다면 보다 효율적이고, 효과적인 보조 여수로 활용방안을 도출이 가능할 것으로 기대 된다.

키워드 : 보조 여수로, FLOW-3D, 수치모의, 호안 안정성, 소류력

1. 서 론

최근 기후변화로 인한 집중호우의 영향으로 홍수 시 댐으로 유입되는 홍수량이 설계 홍수량보다 증가하여 댐 안정성 확보가 필요한 실정이다(Office for Government Policy Coordination, 2003). MOLIT & K-water(2004)에서는 기존댐의 수문학적 안정성 검토를 수행하였으며 이상홍수 발생 시 24개 댐에서 월류 등으로 인한 붕괴위험으로 댐 하류지역의 극심한 피해를 예상하여 보조여수로 신설 및 기존여수로 확장 등 치수능력 증대 기본계획을 수립하였고 이를 통하여 극한홍수 발생 시 홍수량 배제능력을 증대하여 기존댐의 안전성 확보 및 하류지역의 피해를 방지하고자 하였다. 여기서 보조 여수로는 기존 여수로와 동시 또는 별도 운영하는 여수로로써 비상상황 시 방류 기능을 포함하고 있고(K-water, 2021), 최근에는 기존 여수로의 노후화에 따라 보조여수로의 활용방안에 대한 관심이 증가하고 있다. 따라서 본 연구에서는 3차원 수치해석을 수행하여 기존 및 보조 여수로의 방류량 조합에 따른 하류 영향을 분석하고 하류 호안 안정성 측면에서 최적 방류 시나리오를 검토하고자 한다.

기존의 댐 여수로 검토에 관한 연구는 주로 수리실험을 통하여 방류조건 별 흐름특성을 검토하였으나 최근에는 수치모형 실험결과가 수리모형실험과 비교하여 근사한 것을 확인하는 등 점차 수치모형실험을 수리모형실험의 대안으로 활용하고 있다(Jeon et al., 2006Kim, 2007Kim et al., 2008). 국내의 경우, Jeon et al.(2006)은 수리모형 실험과 수치모의를 이용하여 임하댐 바상여수로의 기본설계안을 도출하였고, Kim et al.(2008)은 가능최대홍수량 유입 시 비상여수로 방류에 따른 수리학적 안정성과 기능성을 3차원 수치모형인 FLOW-3D를 활용하여 검토하였다. 또한 Kim and Kim(2013)은 충주댐의 홍수조절 효과 검토 및 방류량 변화에 따른 상·하류의 수위 변화를 수치모형을 통하여 검토하였다. 국외의 경우 Zeng et al.(2017)은 3차원 수치모형인 Fluent를 활용한 여수로 방류에 따른 흐름특성 결과와 측정결과를 비교하여 수치모형 결과의 신뢰성을 검토하였다. Li et al.(2011)은 가능 최대 홍수량(Probable Maximum Flood, PMF)조건에서 기존 여수로와 신규 보조 여수로 유입부 주변의 흐름특성에 대하여 3차원 수치모형 Fluent를 활용하여 검토하였고, Lee et al.(2019)는 서로 근접해있는 기존 여수로와 보조여수로 동시 운영 시 방류능 검토를 수리모형 실험 및 수치모형 실험(FLOW-3D)을 통하여 수행하였으며 기존 여수로와 보조 여수로를 동시운영하게 되면 배수로 간섭으로 인하여 총 방류량이 7.6%까지 감소되어 댐의 방류능력이 감소하였음을 확인하였다.

그러나 대부분의 여수로 검토에 대한 연구는 여수로 내에서의 흐름특성 및 기능성에 대한 검토를 수행하였고. 이에 기존 여수로와 보조 여수로 방류운영에 따른 하류하천의 흐름특성 변화 및 호안 안정성 평가에 관한 추가적인 검토가 필요한 실정이다. 따라서 본 연구에서는 기존 여수로 및 보조 여수로 방류 조건에 따른 하류하천의 흐름특성 및 호안 안정성분석을 3차원 수치모형인 FLOW-3D를 이용하여 검토하였다. 또한 다양한 방류 배분 비율 및 허용 방류량 조건 변화에 따른 하류하천의 흐름특성 및 소류력 분석결과를 호안 설계 허용유속 및 허용 소류력 기준과 비교하여 하류하천의 영향을 최소화 할 수 있는 최적의 보조 여수로 활용방안을 도출하고자 한다.

2. 본 론

2.1 이론적 배경

2.1.1 3차원 수치모형의 기본이론

FLOW-3D는 미국 Flow Science, Inc에서 개발한 범용 유체역학 프로그램(CFD, Computational Fluid Dynamics)으로 자유 수면을 갖는 흐름모의에 사용되는 3차원 수치해석 모형이다. 난류모형을 통해 난류 해석이 가능하고, 댐 방류에 따른 하류 하천의 흐름 해석에도 많이 사용되어 왔다(Flow Science, 2011). 본 연구에서는 FLOW-3D(version 12.0)을 이용하여 홍수 시 기존 여수로의 노후화에 대비하여 보조 여수로의 활용방안에 대한 검토를 하류하천의 호안 안정성 측면에서 검토하였다.

2.1.2 유동해석의 지배방정식

1) 연속 방정식(Continuity Equation)

FLOW-3D는 비압축성 유체에 대하여 연속방정식을 사용하며, 밀도는 상수항으로 적용된다. 연속 방정식은 Eqs. (1)(2)와 같다.

(1)

∇·v=0

(2)

∂∂x(uAx)+∂∂y(vAy)+∂∂z(wAz)=RSORρ

여기서, ρ는 유체 밀도(kg/m3), u, v, w는 x, y, z방향의 유속(m/s), Ax, Ay, Az는 각 방향의 요소면적(m2), RSOR는 질량 생성/소멸(mass source/sink)항을 의미한다.

2) 운동량 방정식(Momentum Equation)

각 방향 속도성분 u, v, w에 대한 운동방정식은 Navier-Stokes 방정식으로 다음 Eqs. (3)(4)(5)와 같다.

(3)

∂u∂t+1VF(uAx∂u∂x+vAy∂v∂y+wAz∂w∂z)=-1ρ∂p∂x+Gx+fx-bx-RSORρVFu

(4)

∂v∂t+1VF(uAx∂u∂x+vAy∂v∂y+wAz∂w∂z)=-1ρ∂p∂y+Gy+fy-by-RSORρVFv

(5)

∂w∂t+1VF(uAx∂u∂x+vAy∂v∂y+wAz∂w∂z)=-1ρ∂p∂z+Gz+fz-bz-RSORρVFw

여기서, Gx, Gy, Gz는 체적력에 의한 가속항, fx, fy, fz는 점성에 의한 가속항, bx, by, bz는 다공성 매체에서의 흐름손실을 의미한다.

2.1.3 소류력 산정

호안설계 시 제방사면 호안의 안정성 확보를 위해서는 하천의 흐름에 의하여 호안에 작용하는 소류력에 저항할 수 있는 재료 및 공법 선택이 필요하다. 국내의 경우 하천공사설계실무요령(MOLIT, 2016)에서 계획홍수량 유하 시 소류력 산정 방법을 제시하고 있다. 소류력은 하천의 평균유속을 이용하여 산정할 수 있으며, 소류력 산정식은 Eqs. (6)(7)과 같다.

1) Schoklitsch 공식

Schoklitsch(1934)는 Chezy 유속계수를 적용하여 소류력을 산정하였다.

(6)

τ=γRI=γC2V2

여기서, τ는 소류력(N/m2), R은 동수반경(m), γ는 물의 단위중량(10.0 kN/m3), I는 에너지경사, C는 Chezy 유속계수, V는 평균유속(m/s)을 의미한다.

2) Manning 조도계수를 고려한 공식

Chezy 유속계수를 대신하여 Manning의 조도계수를 고려하여 소류력을 산정할 수 있다.

(7)

τ=γn2V2R1/3

여기서, τ는 소류력(N/m2), R은 동수반경(m), γ는 물의 단위중량(10.0 kN/m3), n은 Manning의 조도계수, V는 평균유속(m/s)을 의미한다.

FLOW-3D 수치모의 수행을 통하여 하천의 바닥 유속을 도출할 수 있으며, 본 연구에서는 Maning 조도계수롤 고려하여 소류력을 산정하고자 한다. 소류력을 산정하기 위해서 여수로 방류에 따른 대안부의 바닥유속 변화를 검토하여 최대 유속 값을 이용하였다. 최종적으로 산정한 소류력과 호안의 재료 및 공법에 따른 허용 소류력과 비교하여 제방사면 호안의 안정성 검토를 수행하게 된다.

2.2 하천호안 설계기준

하천 호안은 계획홍수위 이하의 유수작용에 대하여 안정성이 확보되도록 계획하여야 하며, 호안의 설계 시에는 사용재료의 확보용이성, 시공상의 용이성, 세굴에 대한 굴요성(flexibility) 등을 고려하여 호안의 형태, 시공방법 등을 결정한다(MOLIT, 2019). 국내의 경우, 하천공사설계실무요령(MOLIT, 2016)에서는 다양한 호안공법에 대하여 비탈경사에 따라 설계 유속을 비교하거나, 허용 소류력을 비교함으로써 호안의 안정성을 평가한다. 호안에 대한 국외의 설계기준으로 미국의 경우, ASTM(미국재료시험학회)에서 호안블록 및 식생매트 시험방법을 제시하였고 제품별로 ASTM 시험에 의한 허용유속 및 허용 소류력을 제시하였다. 일본의 경우, 호안 블록에 대한 축소실험을 통하여 항력을 측정하고 이를 통해서 호안 블록에 대한 항력계수를 제시하고 있다. 설계 시에는 항력계수에 의한 블록의 안정성을 평가하고 있으나, 최근에는 세굴의 영향을 고려할 수 있는 호안 안정성 평가의 필요성을 제기하고 있다(MOLIT, 2019). 관련된 국내·외의 하천호안 설계기준은 Table 1에 정리하여 제시하였고, 본 연구에서 하천 호안 안정성 평가 시 하천공사설계실무요령(MOLIT, 2016)과 ASTM 시험에서 제시한 허용소류력 및 허용유속 기준을 비교하여 각각 0.28 kN/m2, 5.0 m/s 미만일 경우 호안 안정성을 확보하였다고 판단하였다.

Table 1.

Standard of Permissible Velocity and Shear on Revetment

Country (Reference)MaterialPermissible velocity (Vp, m/s)Permissible Shear (τp, kN/m2)
KoreaRiver Construction Design Practice Guidelines
(MOLIT, 2016)
Vegetated5.00.50
Stone5.00.80
USAASTM D’6460Vegetated6.10.81
Unvegetated5.00.28
JAPANDynamic Design Method of Revetment5.0

2.3. 보조여수로 운영에 따른 하류하천 영향 분석

2.3.1 모형의 구축 및 경계조건

본 연구에서는 기존 여수로의 노후화에 대비하여 홍수 시 보조여수로의 활용방안에 따른 하류하천의 흐름특성 및 호안안정성 평가를 수행하기 위해 FLOW-3D 모형을 이용하였다. 기존 여수로 및 보조 여수로는 치수능력 증대사업(MOLIT & K-water, 2004)을 통하여 완공된 ○○댐의 제원을 이용하여 구축하였다. ○○댐은 설계빈도(100년) 및 200년빈도 까지는 계획홍수위 이내로 기존 여수로를 통하여 운영이 가능하나 그 이상 홍수조절은 보조여수로를 통하여 조절해야 하며, 또한 2011년 기존 여수로 정밀안전진단 결과 사면의 표층 유실 및 옹벽 밀림현상 등이 확인되어 노후화에 따른 보수·보강이 필요한 상태이다. 이에 보조여수로의 활용방안 검토가 필요한 것으로 판단하여 본 연구의 대상댐으로 선정하였다. 하류 하천의 흐름특성을 예측하기 위하여 격자간격을 0.99 ~ 8.16 m의 크기로 하여 총 격자수는 49,102,500개로 구성하였으며, 여수로 방류에 따른 하류하천의 흐름해석을 위한 경계조건으로 상류는 유입유량(inflow), 바닥은 벽면(wall), 하류는 수위(water surface elevation)조건으로 적용하도록 하였다(Table 2Fig. 1 참조). FLOW-3D 난류모형에는 혼합길이 모형, 난류에너지 모형, k-ϵ모형, RNG(Renormalized Group Theory) k-ϵ모형, LES 모형 등이 있으며, 본 연구에서는 여수로 방류에 따른 복잡한 난류 흐름 및 높은 전단흐름을 정확하게 모의(Flow Science, 2011)할 수 있는 RNG k-ϵ모형을 사용하였고, 하류하천 호안의 안정성 측면에서 보조여수로의 활용방안을 검토하기 위하여 방류시나리오는 Table 3에 제시된 것 같이 설정하였다. Case 1 및 Case 2를 통하여 계획홍수량에 대하여 기존 여수로와 보조 여수로의 단독 운영이 하류하천에 미치는 영향을 확인하였고 보조 여수로의 방류량 조절을 통하여 호안 안정성 측면에서 보조 여수로 방류능 검토를 수행하였다(Case 3 ~ Case 6). 또한 기존 여수로와 보조 여수로의 방류량 배분에 따른 하류하천의 영향 검토(Case 7 ~ Case 10) 및 방류 배분에 따른 허용 방류량을 호안 안정성 측면에서 검토를 수행하였다(Case 11 ~ Case 14).

수문은 완전개도 조건으로 가정하였으며 하류하천의 계획홍수량에 대한 기존 여수로와 보조여수로의 배분량을 조절하여 모의를 수행하였다. 여수로는 콘크리트의 조도계수 값(Chow, 1959)을 채택하였고, 댐 하류하천의 조도계수는 하천기본계획(Busan Construction and Management Administration, 2009) 제시된 조도계수 값을 채택하였으며 FLOW-3D의 적용을 위하여 Manning-Strickler 공식(Vanoni, 2006)을 이용하여 조도계수를 조고값으로 변환하여 사용하였다. Manning-Strickler 공식은 Eq. (8)과 같으며, FLOW-3D에 적용한 조도계수 및 조고는 Table 4와 같다.

(8)

n=ks1/68.1g1/2

여기서, kS는 조고 (m), n은 Manning의 조도계수, g는 중력가속도(m/s2)를 의미한다.

시간에 따라 동일한 유량이 일정하게 유입되도록 모의를 수행하였으며, 시간간격(Time Step)은 0.0001초로 설정(CFL number < 1.0) 하였다. 또한 여수로 수문을 통한 유량의 변동 값이 1.0%이내일 경우는 연속방정식을 만족하고 있다고 가정하였다. 이는, 유량의 변동 값이 1.0%이내일 경우 유속의 변동 값 역시 1.0%이내이며, 수치모의 결과 1.0%의 유속변동은 호안의 유속설계기준에 크게 영향을 미치지 않는다고 판단하였다. 그 결과 모든 수치모의 Case에서 2400초 이내에 결과 값이 수렴하는 것을 확인하였다.

Table 2.

Mesh sizes and numerical conditions

MeshNumbers49,102,500 EA
Increment (m)DirectionExisting SpillwayAuxiliary Spillway
∆X0.99 ~ 4.301.00 ~ 4.30
∆Y0.99 ~ 8.161.00 ~ 5.90
∆Z0.50 ~ 1.220.50 ~ 2.00
Boundary ConditionsXmin / YmaxInflow / Water Surface Elevation
Xmax, Ymin, Zmin / ZmaxWall / Symmetry
Turbulence ModelRNG model
Table 3.

Case of numerical simulation (Qp : Design flood discharge)

CaseExisting Spillway (Qe, m3/s)Auxiliary Spillway (Qa, m3/s)Remarks
1Qp0Reference case
20Qp
300.58QpReview of discharge capacity on
auxiliary spillway
400.48Qp
500.45Qp
600.32Qp
70.50Qp0.50QpDetermination of optimal division
ratio on Spillways
80.61Qp0.39Qp
90.39Qp0.61Qp
100.42Qp0.58Qp
110.32Qp0.45QpDetermination of permissible
division on Spillways
120.35Qp0.48Qp
130.38Qp0.53Qp
140.41Qp0.56Qp
Table 4.

Roughness coefficient and roughness height

CriteriaRoughness coefficient (n)Roughness height (ks, m)
Structure (Concrete)0.0140.00061
River0.0330.10496
/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F1.jpg
Fig. 1

Layout of spillway and river in this study

2.3.2 보조 여수로의 방류능 검토

본 연구에서는 기존 여수로와 보조 여수로의 방류량 배분에 따른 하류하천 대안부의 유속분포 및 수위분포를 검토하기 위해 수치모의 Case 별 다음과 같이 관심구역을 설정하였다(Fig. 2 참조). 관심구역(대안부)의 길이(L)는 총 1.3 km로 10 m 등 간격으로 나누어 검토하였으며, Section 1(0 < X/L < 0.27)은 기존 여수로 방류에 따른 영향이 지배적인 구간, Section 2(0.27 < X/L < 1.00)는 보조 여수로 방류에 따른 영향이 지배적인 구간으로 각 구간에서의 수위, 유속, 수심결과를 확인하였다. 기존 여수로의 노후화에 따른 보조 여수로의 방류능 검토를 위하여 Case 1 – Case 6까지의 결과를 비교하였다.

보조 여수로의 단독 운영 시 기존 여수로 운영 시 보다 하류하천의 대안부의 최대 유속(Vmax)은 약 3% 감소하였으며, 이는 보조 여수로의 하천 유입각이 기존 여수로 보다 7°작으며 유입하천의 폭이 증가하여 유속이 감소한 것으로 판단된다. 대안부의 최대 유속 발생위치는 하류 쪽으로 이동하였으며 교량으로 인한 단면의 축소로 최대유속이 발생하는 것으로 판단된다. 또한 보조 여수로의 배분량(Qa)이 증가함에 따라 하류하천 대안부의 최대 유속이 증가하였다. 하천호안 설계기준에서 제시하고 있는 허용유속(Vp)과 비교한 결과, 계획홍수량(Qp)의 45% 이하(Case 5 & 6)를 보조 여수로에서 방류하게 되면 허용 유속(5.0 m/s)조건을 만족하여 호안안정성을 확보하였다(Fig. 3 참조). 허용유속 외에도 대안부에서의 소류력을 산정하여 하천호안 설계기준에서 제시한 허용 소류력(τp)과 비교한 결과, 유속과 동일하게 보조 여수로의 방류량이 계획홍수량의 45% 이하일 경우 허용소류력(0.28 kN/m2) 조건을 만족하였다(Fig. 4 참조). 각 Case 별 호안설계조건과 비교한 결과는 Table 5에 제시하였다.

하류하천의 수위도 기존 여수로 운영 시 보다 보조 여수로 단독 운영 시 최대 수위(ηmax)가 약 2% 감소하는 효과를 보였으며 최대 수위 발생위치는 수충부로 여수로 방류시 처오름에 의한 수위 상승으로 판단된다. 기존 여수로의 단독운영(Case 1)의 수위(ηref)를 기준으로 보조 여수로의 방류량이 증가함에 따라 수위는 증가하였으나 계획홍수량의 58%까지 방류할 경우 월류에 대한 안정성(ηmax/ηref<0.97(=기설제방고))은 확보되었다(Fig. 5 참조). 그러나 계획홍수량 조건에서는 월류에 대한 위험성이 존재하기 때문에 기존여수로와 보조여수로의 적절한 방류량 배분 조합을 도출하는 것이 중요하다고 판단되어 진다.

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F2.jpg
Fig. 2

Region of interest in this study

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F3.jpg
Fig. 3

Maximum velocity and location of Vmax according to Qa

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F4.jpg
Fig. 4

Maximum shear according to Qa

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F5.jpg
Fig. 5

Maximum water surface elevation and location of ηmax according to Qa

Table 5.

Numerical results for each cases (Case 1 ~ Case 6)

CaseMaximum Velocity
(Vmax, m/s)
Maximum Shear
(τmax, kN/m2)
Evaluation
in terms of Vp
Evaluation
in terms of τp
1
(Qa = 0)
9.150.54No GoodNo Good
2
(Qa = Qp)
8.870.56No GoodNo Good
3
(Qa = 0.58Qp)
6.530.40No GoodNo Good
4
(Qa = 0.48Qp)
6.220.36No GoodNo Good
5
(Qa = 0.45Qp)
4.220.12AccpetAccpet
6
(Qa = 0.32Qp)
4.040.14AccpetAccpet

2.3.3 기존 여수로와 보조 여수로 방류량 배분 검토

기존 여수로 및 보조 여수로 단독운영에 따른 하류하천 및 호안의 안정성 평가를 수행한 결과 계획홍수량 방류 시 하류하천 대안부에서 호안 설계 조건(허용유속 및 허용 소류력)을 초과하였으며, 처오름에 의한 수위 상승으로 월류에 대한 위험성 증가를 확인하였다. 따라서 계획 홍수량 조건에서 기존 여수로와 보조 여수로의 방류량 배분을 통하여 호안 안정성을 확보하고 하류하천에 방류로 인한 피해를 최소화할 수 있는 배분조합(Case 7 ~ Case 10)을 검토하였다. Case 7은 기존 여수로와 보조여수로의 배분 비율을 균등하게 적용한 경우이고, Case 8은 기존 여수로의 배분량이 보조 여수로에 비하여 많은 경우, Case 9는 보조 여수로의 배분량이 기존 여수로에 비하여 많은 경우를 의미한다. 최대유속을 비교한 결과 보조 여수로의 배분 비율이 큰 경우 기존 여수로의 배분량에 의하여 흐름이 하천 중심에 집중되어 대안부의 유속을 저감하는 효과를 확인하였다. 보조여수로의 방류량 배분 비율이 증가할수록 기존 여수로 대안부 측(0.00<X/L<0.27, Section 1) 유속 분포는 감소하였으나, 신규여수로 대안부 측(0.27<X/L<1.00, Section 2) 유속은 증가하는 것을 확인하였다(Fig. 6 참조). 그러나 유속 저감 효과에도 대안부 전구간에서 설계 허용유속 조건을 초과하여 제방의 안정성을 확보하지는 못하였다. 소류력 산정 결과 유속과 동일하게 보조 여수로의 방류량이 기존 여수로의 방류량 보다 크면 감소하는 것을 확인하였고 일부 구간에서는 허용 소류력 조건을 만족하는 것을 확인하였다(Fig. 7 참조).

따라서 유속 저감효과가 있는 배분 비율 조건(Qa>Qe)에서 Section 2에 유속 저감에 영향을 미치는 기존 여수로 방류량 배분 비율을 증가시켜 추가 검토(Case 10)를 수행하였다. 단독운영과 비교 시 하류하천에 유입되는 유량은 증가하였음에도 불구하고 기존 여수로 방류량에 의해 흐름이 하천 중심으로 집중되는 현상에 따라 대안부의 유속은 단독 운영에 비하여 감소하는 것을 확인하였고(Fig. 8 참조), 호안 설계 허용유속 및 허용 소류력 조건을 만족하는 구간이 발생하여 호안 안정성도 확보한 것으로 판단되었다. 최종적으로 각 Case 별 수위 결과의 경우 여수로 동시 운영을 수행하게 되면 대안부 전 구간에서 월류에 대한 안정성(ηmax/ηref<0.97(=기설제방고))은 확보하였다(Fig. 9 참조). 각 Case 별 대안부에서 최대 유속결과 및 산정한 소류력은 Table 6에 제시하였다.

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F6.jpg
Fig. 6

Maximum velocity on section 1 & 2 according to Qa

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F7.jpg
Fig. 7

Maximum shear on section 1 & 2 according to Qa

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F8.jpg
Fig. 8

Velocity results of FLOW-3D (a: auxiliary spillway operation only , b : simultaneous operation of spillways)

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F9.jpg
Fig. 9

Maximum water surface elevation on section 1 & 2 according to Qa

Table 6.

Numerical results for each cases (Case 7 ~ Case 10)

Case (Qe &amp; Qa)Maximum Velocity (Vmax, m/s)Maximum Shear
(τmax, kN/m2)
Evaluation in terms of VpEvaluation in terms of τp
Section 1Section 2Section 1Section 2Section 1Section 2Section 1Section 2
7
Qe : 0.50QpQa : 0.50Qp
8.106.230.640.30No GoodNo GoodNo GoodNo Good
8
Qe : 0.61QpQa : 0.39Qp
8.886.410.610.34No GoodNo GoodNo GoodNo Good
9
Qe : 0.39QpQa : 0.61Qp
6.227.330.240.35No GoodNo GoodAcceptNo Good
10
Qe : 0.42QpQa : 0.58Qp
6.394.790.300.19No GoodAcceptNo GoodAccept

2.3.4 방류량 배분 비율의 허용 방류량 검토

계획 홍수량 방류 시 기존 여수로와 보조 여수로의 배분 비율 검토 결과 Case 10(Qe = 0.42Qp, Qa = 0.58Qp)에서 방류에 따른 하류 하천의 피해를 최소화시킬 수 있는 것을 확인하였다. 그러나 대안부 전 구간에 대하여 호안 설계조건을 만족하지 못하였다. 따라서 기존 여수로와 보조 여수로의 방류 배분 비율을 고정시킨 후 총 방류량을 조절하여 허용 방류량을 검토하였다(Case 11 ~ Case 14).

호안 안정성 측면에서 검토한 결과 계획홍수량 대비 총 방류량이 감소하면 최대 유속 및 최대 소류력이 감소하고 최종적으로 계획 홍수량의 77%를 방류할 경우 하류하천의 대안부에서 호안 설계조건을 모두 만족하는 것을 확인하였다(Fig. 10Fig. 11 참조). 각 Case 별 대안부에서 최대 유속결과 및 산정한 소류력은 Table 7에 제시하였다. 또한 Case 별 수위 검토 결과 처오름으로 인한 대안부 전 구간에서 월류에 대한 안정성(ηmax/ηref<0.97(=기설제방고))은 확보하였다(Fig. 12 참조).

Table 7.

Numerical results for each cases (Case 11 ~ Case 14)

Case (Qe &amp; Qa)Maximum Velocity
(Vmax, m/s)
Maximum Shear
(τmax, kN/m2)
Evaluation in terms of VpEvaluation in terms of τp
Section 1Section 2Section 1Section 2Section 1Section 2Section 1Section 2
11
Qe : 0.32QpQa : 0.45Qp
3.634.530.090.26AcceptAcceptAcceptAccept
12
Qe : 0.35QpQa : 0.48Qp
5.745.180.230.22No GoodNo GoodAcceptAccept
13
Qe : 0.38QpQa : 0.53Qp
6.704.210.280.11No GoodAcceptAcceptAccept
14
Qe : 0.41QpQa : 0.56Qp
6.545.240.280.24No GoodNo GoodAcceptAccept
/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F10.jpg
Fig. 10

Maximum velocity on section 1 & 2 according to total outflow

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F11.jpg
Fig. 11

Maximum shear on section 1 & 2 according to total outflow

/media/sites/ksds/2021-014-02/N0240140207/images/ksds_14_02_07_F12.jpg
Fig. 12

Maximum water surface elevation on section 1 & 2 according to total outflow

3. 결 론

본 연구에서는 홍수 시 기존 여수로의 노후화로 인한 보조 여수로의 활용방안에 대하여 하류하천의 호안 안정성 측면에서 검토하였다. 여수로 방류로 인한 하류하천의 흐름특성을 검토하기 위하여 3차원 수치모형인 FLOW-3D를 활용하였고, 여수로 지형은 치수능력 증대사업을 통하여 완공된 ○○댐의 제원을 이용하였다. 하류하천 조도 계수 및 여수로 방류량은 하천기본계획을 참고하여 적용하였다. 최종적으로 여수로 방류로 인한 하류하천의 피해를 최소화 시킬 수 있는 적절한 보조 여수로의 활용방안을 도출하기 위하여 보조 여수로 단독 운영과 기존 여수로와의 동시 운영에 따른 하류 하천의 흐름특성 및 소류력의 변화를 검토하였다.

수문은 완전 개도 상태에서 방류한다는 가정으로 계획 홍수량 조건에서 보조 여수로 단독 운영 시 하류하천 대안부의 유속 및 수위를 검토한 결과 기존 여수로 단독운영에 비하여 최대 유속 및 최대 수위가 감소하는 것을 확인할 수 있었으며, 이는 보조 여수로 단독 운영 시 하류하천으로 유입각도가 작아지고, 유입되는 하천의 폭이 증가되기 때문이다. 그러나 계획 홍수량 조건에서 하천호안 설계기준에서 제시한 허용 유속(5.0 m/s)과 허용 소류력(0.28 kN/m2)과 비교하였을 때 호안 안정성을 확보하지 못하였으며, 계획홍수량의 45% 이하 방류 시에 대안부의 호안 안정성을 확보하였다. 수위의 경우 여수로 방류에 따른 대안부에서 처오름 현상이 발생하여 월류에 대한 위험성을 확인하였고 이를 통하여 기존 여수로와의 동시 운영 방안을 도출하는 것이 중요하다고 판단된다. 따라서 기존 여수로와의 동시 운영 측면에서 기존 여수로와 보조 여수로의 배분 비율 및 총 방류량을 변화시켜가며 하류 하천의 흐름특성 및 소류력의 변화를 검토하였다. 배분 비율의 경우 기존 여수로와 보조 여수로의 균등 배분(Case 7) 및 편중 배분(Case 8 & Case 9)을 검토하여 보조 여수로의 방류량이 기존 여수로의 방류량보다 큰 경우 하류하천의 중심부로 집중되어 대안부의 최대유속, 최대소류력 및 최대수위가 감소하는 것을 확인하였다. 이를 근거로 기존 여수로의 방류 비율을 증가(Qe=0.42Qp, Qa=0.58Qp)시켜 검토한 결과 대안부 일부 구간에서 허용 유속 및 허용소류력 조건을 만족하는 것을 확인하였다. 이를 통하여 기존 여수로와 보조 여수로의 동시 운영을 통하여 적절한 방류량 배분 비율을 도출하는 것이 방류로 인한 하류하천의 피해를 저감하는데 효과적인 것으로 판단된다. 그러나 설계홍수량 방류 시 전 구간에서 허용 유속 및 소류력 조건을 만족하지 못하였다. 최종적으로 전체 방류량에서 기존 여수로의 방류 비율을 42%, 보조 여수로의 방류 비율을 58%로 설정하여 허용방류량을 검토한 결과, 계획홍수량의 77%이하로 방류 시 대안부의 최대유속은 기존여수로 방류의 지배영향구간(section 1)에서 3.63 m/s, 기존 여수로와 보조 여수로 방류의 영향구간(section 2)에서 4.53 m/s로 허용유속 조건을 만족하였고, 산정한 소류력도 각각 0.09 kN/m2 및 0.26 kN/m2로 허용 소류력 조건을 만족하여 대안부 호안의 안정성을 확보하였다고 판단된다.

본 연구 결과는 기후변화 및 기존여수로의 노후화로 인하여 홍수 시 기존여수로의 단독운영으로 하류하천의 피해가 발생할 수 있는 현시점에서 치수증대 사업으로 완공된 보조 여수로의 활용방안에 대한 기초자료로 활용될 수 있고, 향후 계획 홍수량 유입 시 최적의 배분 비율 및 허용 방류량 도출에 이용할 수 있다. 다만 본 연구는 여수로 방류에 따른 제방에 작용하는 수충력은 검토하지 못하고, 허용 유속 및 허용소류력은 제방과 유수의 방향이 일정한 구간에 대하여 검토하였다. 또한 여수로 방류에 따른 대안부에서의 영향에 대해서만 검토하였고 수문 전면 개도 조건에서 검토하였다는 한계점은 분명히 있다. 이에 향후에는 다양한 수문 개도 조건 및 방류 시나리오를 적용 및 검토하여 보다 효율적이고, 효과적인 보조 여수로 활용방안을 도출하고자 한다.

Acknowledgements

본 결과물은 K-water에서 수행한 기존 및 신규 여수로 효율적 연계운영 방안 마련(2021-WR-GP-76-149)의 지원을 받아 연구되었습니다.

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Obrázek 44: Barevné rozlišení proudnic dle rychlosti

Numerické modelování proudění v bezpečnostním přelivu: Numerical modeling of flow in spilway

Abstract

졸업 논문의 목표는 보스코비체 댐의 계획된 방수로의 흐름을 수치적으로 모델링하는 것입니다. 이 졸업 논문은 유형과 프로필에 따라 기본 여수로를 설명하고 나눕니다. 비상용 배수로도 언급되어 있습니다. 그런 다음 논문에서는 범람량 계산에 대한 설명, 수학적 모델링 및 사용된 난류 모델에 대한 설명을 소개합니다. 다음 부분은 Boskovice 댐의 기술적 설명, AutoCAD 2020 소프트웨어에서 방수로 및 방수로 슈트의 가상 3D 모델 생성 및 Blender 소프트웨어에서 모델의 제어 및 수정과 관련되어 있습니다. 논문 말미에는 Flow-3D 소프트웨어를 통해 얻은 유동의 수치적 모델링 결과와 BUT 토목공학부 수구조연구소에서 시행한 수리학적 모델 연구와 비교한 결과를 언급하였다.

The goal of the diploma thesis is the numerical modelling of flow in planned spillway of the Boskovice dam. In the introduction of this diploma thesis are described and divided basic spillways according to their types and profiles. There are also mentioned emergency spillways. Then the thesis introduces the description of calculation of overflow quantity, the description of mathematic modelling and used turbulent models. The next part is concerned with the technical description of the Boskovice dam, the creation of virtual 3D model of spillway and spillway chute in the AutoCAD 2020 software and concerned with the control and revision of model in the Blender software. In the end of the thesis are mentioned results of numeric modelling of flow gained from the Flow-3D software and the comparison of results with the research of hydraulic model implemented at Water structures institute of Faculty of Civil Engineering of BUT.

AuthorSvoboda, Jiří
TitleNumerické modelování proudění v bezpečnostním přelivu: Numerical modeling of flow in spilway
URLhttp://hdl.handle.net/11012/195970
Publication Date2021
Date Accessioned2021-02-05 08:03:49
University/PublisherBrno University of Technology
AbstractThe goal of the diploma thesis is the numerical modelling of flow in planned spillway of the Boskovice dam. In the introduction of this diploma thesis are described and divided basic spillways according to their types and profiles. There are also mentioned emergency spillways. Then the thesis introduces the description of calculation of overflow quantity, the description of mathematic modelling and used turbulent models. The next part is concerned with the technical description of the Boskovice dam, the creation of virtual 3D model of spillway and spillway chute in the AutoCAD 2020 software and concerned with the control and revision of model in the Blender software. In the end of the thesis are mentioned results of numeric modelling of flow gained from the Flow-3D software and the comparison of results with the research of hydraulic model implemented at Water structures institute of Faculty of Civil Engineering of BUT.
Subjects/KeywordsBezpečnostní přeliv; numerický model; 3D model; FLOW-3D; VD Boskovice; sypaná kamenitá hráz.; Spillway; numerical model; 3D model; FLOW-3D; Boskovice dam; rockfill dam.
ContributorsJandora, Jan (advisor); Holomek, Petr (referee)
Languagecs
RightsStandardní licenční smlouva – přístup k plnému textu bez omezení
Country of Publicationcz
Record IDhandle:11012/195970
Repositorybrno-tech
Date Indexed2021-12-08
Note[mark] A;
Obrázek 18: Kašnový čelní bezpečnostní přeliv [24]
OFigure 18: Fountain front safety spillway [24]
Obrázek 20: Skluz a divergentní vývar bezpečnostního objektu VD Boskovice [24]
Figure 20: Slip and divergent broth of the security building VD Boskovice [24]
Obrázek 22: Půdorys bezpečnostního přelivu a části skluzu VD Boskovice [12]
Obrázek 22: Půdorys bezpečnostního přelivu a části skluzu VD Boskovice [12]
Obrázek 23: Podélný řez BP a spadiště v rovině symetrie [12]
Figure 23: Longitudinal section BP and drop in the plane of symmetry [12]
Obrázek 44: Barevné rozlišení proudnic dle rychlosti
Figure 44: Color resolution of jets according to speed
Obrázek 45: Průběh hladiny ve Flow-3D bez zobrazeného 3D modelu
Figure 45: Flow profile in Flow-3D without 3D model displayed
Figure 47: Level course on the physical model [22]
Figure 47: Level course on the physical model [22]

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[26] KOZUBKOVÁ, Milada, Modelování proudění tekutin, FLUENT, CFX. Vysoká škola
Báňská Technická univerzita Ostrava, 2008.

Fig. 5. The predicted shapes of initial breach (a) Rectangular (b) V-notch. Fig. 6. Dam breaching stages.

Investigating the peak outflow through a spatial embankment dam breach

공간적 제방댐 붕괴를 통한 최대 유출량 조사

Mahmoud T.GhonimMagdy H.MowafyMohamed N.SalemAshrafJatwaryFaculty of Engineering, Zagazig University, Zagazig 44519, Egypt

Abstract

Investigating the breach outflow hydrograph is an essential task to conduct mitigation plans and flood warnings. In the present study, the spatial dam breach is simulated by using a three-dimensional computational fluid dynamics model, FLOW-3D. The model parameters were adjusted by making a comparison with a previous experimental model. The different parameters (initial breach shape, dimensions, location, and dam slopes) are studied to investigate their effects on dam breaching. The results indicate that these parameters have a significant impact. The maximum erosion rate and peak outflow for the rectangular shape are higher than those for the V-notch by 8.85% and 5%, respectively. Increasing breach width or decreasing depth by 5% leads to increasing maximum erosion rate by 11% and 15%, respectively. Increasing the downstream slope angle by 4° leads to an increase in both peak outflow and maximum erosion rate by 2.0% and 6.0%, respectively.

유출 유출 수문곡선을 조사하는 것은 완화 계획 및 홍수 경보를 수행하는 데 필수적인 작업입니다. 본 연구에서는 3차원 전산유체역학 모델인 FLOW-3D를 사용하여 공간 댐 붕괴를 시뮬레이션합니다. 이전 실험 모델과 비교하여 모델 매개변수를 조정했습니다.

다양한 매개변수(초기 붕괴 형태, 치수, 위치 및 댐 경사)가 댐 붕괴에 미치는 영향을 조사하기 위해 연구됩니다. 결과는 이러한 매개변수가 상당한 영향을 미친다는 것을 나타냅니다. 직사각형 형태의 최대 침식율과 최대 유출량은 V-notch보다 각각 8.85%, 5% 높게 나타났습니다.

위반 폭을 늘리거나 깊이를 5% 줄이면 최대 침식률이 각각 11% 및 15% 증가합니다. 하류 경사각을 4° 증가시키면 최대 유출량과 최대 침식률이 각각 2.0% 및 6.0% 증가합니다.

Keywords

Spatial dam breach; FLOW-3D; Overtopping erosion; Computational fluid dynamics (CFD)

1. Introduction

There are many purposes for dam construction, such as protection from flood disasters, water storage, and power generationEmbankment failures may have a catastrophic impact on lives and infrastructure in the downstream regions. One of the most common causes of embankment dam failure is overtopping. Once the overtopping of the dam begins, the breach formation will start in the dam body then end with the dam failure. This failure occurs within a very short time, which threatens to be very dangerous. Therefore, understanding and modeling the embankment breaching processes is essential for conducting mitigation plans, flood warnings, and forecasting flood damage.

The analysis of the dam breaching process is implemented by different techniques: comparative methods, empirical models with dimensional and dimensionless solutions, physical-based models, and parametric models. These models were described in detail [1]Parametric modeling is commonly used to simulate breach growth as a time-dependent linear process and calculate outflow discharge from the breach using hydraulics principles [2]. Alhasan et al. [3] presented a simple one-dimensional mathematical model and a computer code to simulate the dam breaching process. These models were validated by small dams breaching during the floods in 2002 in the Czech Republic. Fread [4] developed an erosion model (BREACH) based on hydraulics principles, sediment transport, and soil mechanics to estimate breach size, time of formation, and outflow discharge. Říha et al. [5] investigated the dam break process for a cascade of small dams using a simple parametric model for piping and overtopping erosion, as well as a 2D shallow-water flow model for the flood in downstream areas. Goodarzi et al. [6] implemented mathematical and statistical methods to assess the effect of inflows and wind speeds on the dam’s overtopping failure.

Dam breaching studies can be divided into two main modes of erosion. The first mode is called “planar dam breach” where the flow overtops the whole dam width. While the second mode is called “spatial dam breach” where the flow overtops through the initial pilot channel (i.e., a channel created in the dam body). Therefore, the erosion will be in both vertical and horizontal directions [7].

The erosion process through the embankment dams occurs due to the shear stress applied by water flows. The dam breaching evolution can be divided into three stages [8][9], but Y. Yang et al. [10] divided the breach development into five stages: Stage I, the seepage erosion; Stage II, the initial breach formation; Stage III, the head erosion; Stage IV, the breach expansion; and Stage V, the re-equilibrium of the river channel through the breach. Many experimental tests have been carried out on non-cohesive embankment dams with an initial breach to examine the effect of upstream inflow discharges on the longitudinal profile evolution and the time to inflection point [11].

Zhang et al. [12] studied the effect of changing downstream slope angle, sediment grain size, and dam crest length on erosion rates. They noticed that increasing dam crest length and decreasing downstream slope angle lead to decreasing sediment transport rate. While the increase in sediment grain size leads to an increased sediment transport rate at the initial stages. Höeg et al. [13] presented a series of field tests to investigate the stability of embankment dams made of various materials. Overtopping and piping were among the failure tests carried out for the dams composed of homogeneous rock-fill, clay, or gravel with a height of up to 6.0 m. Hakimzadeh et al. [14] constructed 40 homogeneous cohesive and non-cohesive embankment dams to study the effect of changing sediment diameter and dam height on the breaching process. They also used genetic programming (GP) to estimate the breach outflow. Refaiy et al. [15] studied different scenarios for the downstream drain geometry, such as length, height, and angle, to minimize the effect of piping phenomena and therefore increase dam safety.

Zhu et al. [16] examined the effect of headcut erosion on dam breach growth, especially in the case of cohesive dams. They found that the breach growth in non-cohesive embankments is slower than cohesive embankments due to the little effect of headcut. Schmocker and Hager [7] proposed a relationship for estimating peak outflow from the dam breach process.(1)QpQin-1=1.7exp-20hc23d5013H0

where: Qp = peak outflow discharge.

Qin = inflow discharge.

hc = critical flow depth.

d50 = mean sediment diameter.

Ho = initial dam height.

Yu et al. [17] carried out an experimental study for homogeneous non-cohesive embankment dams in a 180° bending rectangular flume to determine the effect of overtopping flows on breaching formation. They found that the main factors influencing breach formation are water level, river discharge, and embankment material diameter.

Wu et al. [18] carried out a series of experiments to investigate the effect of breaching geometry on both non-cohesive and cohesive embankment dams in a U-bend flume due to overtopping flows. In the case of non-cohesive embankments, the non-symmetrical lateral expansion was noticed during the breach formation. This expansion was described by a coefficient ranging from 2.7 to 3.3.

The numerical models of the dam breach can be categorized according to different parameters, such as flow dimensions (1D, 2D, or 3D), flow governing equations, and solution methods. The 1D models are mainly used to predict the outflow hydrograph from the dam breach. Saberi et al. [19] applied the 1D Saint-Venant equation, which is solved by the finite difference method to investigate the outflow hydrograph during dam overtopping failure. Because of the ability to study dam profile evolution and breach formation, 2D models are more applicable than 1D models. Guan et al. [20] and Wu et al. [21] employed both 2D shallow water equations (SWEs) and sediment erosion equations, which are solved by the finite volume method to study the effect of the dam’s geometry parameters on outflow hydrograph and dam profile evolution. Wang et al. [22] also proposed a second-order hybrid-type of total variation diminishing (TVD) finite-difference to estimate the breach outflow by solving the 2D (SWEs). The accuracy of (SWEs) for both vertical flow contraction and surface roughness has been assessed [23]. They noted that the accuracy of (SWEs) is acceptable for milder slopes, but in the case of steeper slopes, modelers should be more careful. Generally, the accuracy of 2D models is still low, especially with velocity distribution over the flow depth, lateral momentum exchange, density-driven flows, and bottom friction [24]. Therefore, 3D models are preferred. Larocque et al. [25] and Yang et al. [26] started to use three-dimensional (3D) models that depend on the Reynolds-averaged Navier-Stokes (RANS) equations.

Previous experimental studies concluded that there is no clear relationship between the peak outflow from the dam breach and the initial breach characteristics. Some of these studies depend on the sharp-crested weir fixed at the end of the flume to determine the peak outflow from the breach, which leads to a decrease in the accuracy of outflow calculations at the microscale. The main goals of this study are to carry out a numerical simulation for a spatial dam breach due to overtopping flows by using (FLOW-3D) software to find an empirical equation for the peak outflow discharge from the breach and determine the worst-case that leads to accelerating the dam breaching process.

2. Numerical simulation

The current study for spatial dam breach is simulated by using (FLOW-3D) software [27], which is a powerful computational fluid dynamics (CFD) program.

2.1. Geometric presentations

A stereolithographic (STL) file is prepared for each change in the initial breach geometry and dimensions. The CAD program is useful for creating solid objects and converting them to STL format, as shown in Fig. 1.

2.2. Governing equations

The governing equations for water flow are three-dimensional Reynolds Averaged Navier-Stokes equations (RANS).

The continuity equation:(2)∂ui∂xi=0

The momentum equation:(3)∂ui∂t+1VFuj∂ui∂xj=1ρ∂∂xj-pδij+ν∂ui∂xj+∂uj∂xi-ρu`iu`j¯

where u is time-averaged velocity,ν is kinematic viscosity, VF is fractional volume open to flow, p is averaged pressure and -u`iu`j¯ are components of Reynold’s stress. The Volume of Fluid (VOF) technique is used to simulate the free surface profile. Hirt et al. [28] presented the VOF algorithm, which employs the function (F) to express the occupancy of each grid cell with fluid. The value of (F) varies from zero to unity. Zero value refers to no fluid in the grid cell, while the unity value refers to the grid cell being fully occupied with fluid. The free surface is formed in the grid cells having (F) values between zero and unity.(4)∂F∂t+1VF∂∂xFAxu+∂∂yFAyv+∂∂zFAzw=0

where (u, v, w) are the velocity components in (x, y, z) coordinates, respectively, and (AxAyAz) are the area fractions.

2.3. Boundary and initial conditions

To improve the accuracy of the results, the boundary conditions should be carefully determined. In this study, two mesh blocks are used to minimize the time consumed in the simulation. The boundary conditions for mesh block 1 are as follows: The inlet and sides boundaries are defined as a wall boundary condition (wall boundary condition is usually used for bound fluid by solid regions. In the case of viscous flows, no-slip means that the tangential velocity is equal to the wall velocity and the normal velocity is zero), the outlet is defined as a symmetry boundary condition (symmetry boundary condition is usually used to reduce computational effort during CFD simulation. This condition allows the flow to be transferred from one mesh block to another. No inputs are required for this boundary condition except that its location should be defined accurately), the bottom boundary is defined as a uniform flow rate boundary condition, and the top boundary is defined as a specific pressure boundary condition with assigned atmospheric pressure. The boundary conditions for mesh block 2 are as follows: The inlet is defined as a symmetry boundary condition, the outlet is defined as a free flow boundary condition, the bottom and sides boundaries are defined as a wall boundary condition, and the top boundary is defined as a specific pressure boundary condition with assigned atmospheric pressure as shown in Fig. 2. The initial conditions required to be set for the fluid (i.e., water) inside of the domain include configuration, temperature, velocities, and pressure distribution. The configuration of water depends on the dimensions and shape of the dam reservoir. While the other conditions have been assigned as follows: temperature is normal water temperature (25 °c) and pressure distribution is hydrostatic with no initial velocity.

2.4. Numerical method

FLOW-3D uses the finite volume method (FVM) to solve the governing equation (Reynolds-averaged Navier-Stokes) over the computational domain. A finite-volume method is an Eulerian approach for representing and evaluating partial differential equations in algebraic equations form [29]. At discrete points on the mesh geometry, values are determined. Finite volume expresses a small volume surrounding each node point on a mesh. In this method, the divergence theorem is used to convert volume integrals with a divergence term to surface integrals. After that, these terms are evaluated as fluxes at each finite volume’s surfaces.

2.5. Turbulent models

Turbulence is the chaotic, unstable motion of fluids that occurs when there are insufficient stabilizing viscous forces. In FLOW-3D, there are six turbulence models available: the Prandtl mixing length model, the one-equation turbulent energy model, the two-equation (k – ε) model, the Renormalization-Group (RNG) model, the two-equation (k – ω) models, and a large eddy simulation (LES) model. For simulating flow motion, the RNG model is adopted to simulate the motion behavior better than the k – ε and k – ω.

models [30]. The RNG model consists of two main equations for the turbulent kinetic energy KT and its dissipation.εT(5)∂kT∂t+1VFuAx∂kT∂x+vAy∂kT∂y+wAz∂kT∂z=PT+GT+DiffKT-εT(6)∂εT∂t+1VFuAx∂εT∂x+vAy∂εT∂y+wAz∂εT∂z=C1.εTKTPT+c3.GT+Diffε-c2εT2kT

where KT is the turbulent kinetic energy, PT is the turbulent kinetic energy production, GT is the buoyancy turbulence energy, εT is the turbulent energy dissipation rate, DiffKT and Diffε are terms of diffusion, c1, c2 and c3 are dimensionless parameters, in which c1 and c3 have a constant value of 1.42 and 0.2, respectively, c2 is computed from the turbulent kinetic energy (KT) and turbulent production (PT) terms.

2.6. Sediment scour model

The sediment scour model available in FLOW-3D can calculate all the sediment transport processes including Entrainment transport, Bedload transport, Suspended transport, and Deposition. The erosion process starts once the water flows remove the grains from the packed bed and carry them into suspension. It happens when the applied shear stress by water flows exceeds critical shear stress. This process is represented by entrainment transport in the numerical model. After entrained, the grains carried by water flow are represented by suspended load transport. After that, some suspended grains resort to settling because of the combined effect of gravity, buoyancy, and friction. This process is described through a deposition. Finally, the grains sliding motions are represented by bedload transport in the model. For the entrainment process, the shear stress applied by the fluid motion on the packed bed surface is calculated using the standard wall function as shown in Eq.7.(7)ks,i=Cs,i∗d50

where ks,i is the Nikuradse roughness and Cs,i is a user-defined coefficient. The critical bed shear stress is defined by a dimensionless parameter called the critical shields number as expressed in Eq.8.(8)θcr,i=τcr,i‖g‖diρi-ρf

where θcr,i is the critical shields number, τcr,i is the critical bed shear stress, g is the absolute value of gravity acceleration, di is the diameter of the sediment grain, ρi is the density of the sediment species (i) and ρf is the density of the fluid. The value of the critical shields number is determined according to the Soulsby-Whitehouse equation.(9)θcr,i=0.31+1.2d∗,i+0.0551-exp-0.02d∗,i

where d∗,i is the dimensionless diameter of the sediment, given by Eq.10.(10)d∗,i=diρfρi-ρf‖g‖μf213

where μf is the fluid dynamic viscosity. For the sloping bed interface, the value of the critical shields number is modified according to Eq.11.(11)θ`cr,i=θcr,icosψsinβ+cos2βtan2φi-sin2ψsin2βtanφi

where θ`cr,i is the modified critical shields number, φi is the angle of repose for the sediment, β is the angle of bed slope and ψ is the angle between the flow and the upslope direction. The effects of the rolling, hopping, and sliding motions of grains along the packed bed surface are taken by the bedload transport process. The volumetric bedload transport rate (qb,i) per width of the bed is expressed in Eq.12.(12)qb,i=Φi‖g‖ρi-ρfρfdi312

where Φi is the dimensionless bedload transport rate is calculated by using Meyer Peter and Müller equation.(13)Φi=βMPM,iθi-θ`cr,i1.5cb,i

where βMPM,i is the Meyer Peter and Müller user-defined coefficient and cb,i is the volume fraction of species i in the bed material. The suspended load transport is calculated as shown in Eq.14.(14)∂Cs,i∂t+∇∙Cs,ius,i=∇∙∇DCs,i

where Cs,i is the suspended sediment mass concentration, D is the diffusivity, and us,i is the grain velocity of species i. Entrainment and deposition are two opposing processes that take place at the same time. The lifting and settling velocities for both entrainment and deposition processes are calculated according to Eq.15 and Eq.16, respectively.(15)ulifting,i=αid∗,i0.3θi-θ`cr,igdiρiρf-1(16)usettling,i=υfdi10.362+1.049d∗,i3-10.36

where αi is the entrainment coefficient of species i and υf is the kinematic viscosity of the fluid.

2.7. Grid type

Using simple rectangular orthogonal elements in planes and hexahedral in volumes in the (FLOW-3D) program makes the mesh generation process easier, decreases the required memory, and improves numerical accuracy. Two mesh blocks were used in a joined form with a size ratio of 2:1. The first mesh block is coarser, which contains the reservoir water, and the second mesh block is finer, which contains the dam. For achieving accuracy and efficiency in results, the mesh size is determined by using a grid convergence test. The optimum uniform cell size for the first mesh block is 0.012 m and for the second mesh block is 0.006 m.

2.8. Time step

The maximum time step size is determined by using a Courant number, which controls the distance that the flow will travel during the simulation time step. In this study, the Courant number was taken equal to 0.25 to prevent the flow from traveling through more than one cell in the time step. Based on the Courant number, a maximum time step value of 0.00075 s was determined.

2.9. Numerical model validation

The numerical model accuracy was achieved by comparing the numerical model results with previous experimental results. The experimental study of Schmocker and Hager [7] was based on 31 tests with changes in six parameters (d50, Ho, Bo, Lk, XD, and Qin). All experimental tests were conducted in a straight open glass-sided flume. The horizontal flume has a rectangular cross-section with a width of 0.4 m and a height of 0.7 m. The flume was provided with a flow straightener and an intake with a length of 0.66 m. All tested dams were inserted at various distances (XD) from the intake. Test No.1 from this experimental program was chosen to validate the numerical model. The different parameters used in test No.1 are as follows:

(1) uniform sediment with a mean diameter (d50 = 0.31 mm), (2) Ho = 0.2 m, (3) Bo = 0.2 m, (4) Lk = 0.1 m,

(5) XD = 1.0 m, (6) Qin = 6.0 lit/s, (7) Su and Sd = 2:1, (8) mass density (ρs = 2650 kg/m3(9) Homogenous and non-cohesive embankment dam. As shown in Fig. 2, the simulation is contained within a rectangular grid with dimensions: 3.56 m in the x-direction (where 0.66 m is used as inlet, 0.9 m as dam base width, and 1.0 m as outlet), in y-direction 0.2 m (dam length), and in the z-direction 0.3 m, which represents the dam height (0.2 m) with a free distance (0.1 m) above the dam. There are two main reasons that this experimental program is preferred for the validation process. The first reason is that this program deals with homogenous, non-cohesive soil, which is available in FLOW-3D. The second reason is that this program deals with small-scale models which saves time for numerical simulation. Finally, some important assumptions were considered during the validation process. The flow is assumed to be incompressible, viscous, turbulent, and three-dimensional.

By comparing dam profiles at different time instants for the experimental test with the current numerical model, it appears that the numerical model gives good agreement as shown in Fig. 3 and Fig. 4, with an average error percentage of 9% between the experimental results and the numerical model.

3. Analysis and discussions

The current model is used to study the effects of different parameters such as (initial breach shapes, dimensions, locations, upstream and downstream dam slopes) on the peak outflow discharge, QP, time of peak outflow, tP, and rate of erosion, E.

This study consists of a group of scenarios. The first scenario is changing the shapes of the initial breach according to Singh [1], the most predicted shapes are rectangular and V-notch as shown in Fig. 5. The second scenario is changing the initial breach dimensions (i.e., width and depth). While the third scenario is changing the location of the initial breach. Eventually, the last scenario is changing the upstream and downstream dam slopes.

All scenarios of this study were carried out under the same conditions such as inflow discharge value (Qin=1.0lit/s), dimensions of the tested dam, where dam height (Ho=0.20m), crest width.

(Lk=0.1m), dam length (Bo=0.20m), and homogenous & non-cohesive soil with a mean diameter (d50=0.31mm).

3.1. Dam breaching process evolution

The dam breaching process is a very complex process due to the quick changes in hydrodynamic conditions during dam failure. The dam breaching process starts once water flows reach the downstream face of the dam. During the initial stage of dam breaching, the erosion process is relatively quiet due to low velocities of flow. As water flows continuously, erosion rates increase, especially in two main zones: the crest and the downstream face. As soon as the dam crest is totally eroded, the water levels in the dam reservoir decrease rapidly, accompanied by excessive erosion in the dam body. The erosion process continues until the water levels in the dam reservoir equal the remaining height of the dam.

According to Zhou et al. [11], the breaching process consists of three main stages. The first stage starts with beginning overtopping flow, then ends when the erosion point directed upstream and reached the inflection point at the inflection time (ti). The second stage starts from the end of the stage1 until the occurrence of peak outflow discharge at the peak outflow time (tP). The third stage starts from the end of the stage2 until the value of outflow discharge becomes the same as the value of inflow discharge at the final time (tf). The outflow discharge from the dam breach increases rapidly during stage1 and stage2 because of the large dam storage capacity (i.e., the dam reservoir is totally full of water) and excessive erosion. While at stage3, the outflow values start to decrease slowly because most of the dam’s storage capacity was run out. The end of stage3 indicates that the dam storage capacity was totally run out, so the outflow equalized with the inflow discharge as shown in Fig. 6 and Fig. 7.

3.2. The effect of initial breach shape

To identify the effect of the initial breach shape on the evolution of the dam breaching process. Three tests were carried out with different cross-section areas for each shape. The initial breach is created at the center of the dam crest. Each test had an ID to make the process of arranging data easier. The rectangular shape had an ID (Rec5h & 5b), which means that its depth and width are equal to 5% of the dam height, and the V-notch shape had an ID (V-noch5h & 1:1) which means that its depth is equal to 5% of the dam height and its side slope is equal to 1:1. The comparison between rectangular and V-notch shapes is done by calculating the ratio between maximum dam height at different times (ZMax) to the initial dam height (Ho), rate of erosion, and hydrograph of outflow discharge for each test. The rectangular shape achieves maximum erosion rate and minimum inflection time, in addition to a rapid decrease in the dam reservoir levels. Therefore, the dam breaching is faster in the case of a rectangular shape than in a V-notch shape, which has the same cross-section area as shown in Fig. 8.

Also, by comparing the hydrograph for each test, the peak outflow discharge value in the case of a rectangular shape is higher than the V-notch shape by 5% and the time of peak outflow for the rectangular shape is shorter than the V-notch shape by 9% as shown in Fig. 9.

3.3. The effect of initial breach dimensions

The results of the comparison between the different initial breach shapes indicate that the worst initial breach shape is rectangular, so the second scenario from this study concentrated on studying the effect of a change in the initial rectangular breach dimensions. Groups of tests were carried out with different depths and widths for the rectangular initial breach. The first group had a depth of 5% from the dam height and with three different widths of 5,10, and 15% from the dam height, the second group had a depth of 10% with three different widths of 5,10, and 15%, the third group had a depth of 15% with three different widths of 5,10, and 15% and the final group had a width of 15% with three different heights of 5, 10, and 15% for a rectangular breach shape. The comparison was made as in the previous section to determine the worst case that leads to the quick dam failure as shown in Fig. 10.

The results show that the (Rec 5 h&15b) test achieves a maximum erosion rate for a shorter period of time and a minimum ratio for (Zmax / Ho) as shown in Fig. 10, which leads to accelerating the dam failure process. The dam breaching process is faster with the minimum initial breach depth and maximum initial breach width. In the case of a minimum initial breach depth, the retained head of water in the dam reservoir is high and the crest width at the bottom of the initial breach (L`K) is small, so the erosion point reaches the inflection point rapidly. While in the case of the maximum initial breach width, the erosion perimeter is large.

3.4. The effect of initial breach location

The results of the comparison between the different initial rectangular breach dimensions indicate that the worst initial breach dimension is (Rec 5 h&15b), so the third scenario from this study concentrated on studying the effect of a change in the initial breach location. Three locations were checked to determine the worst case for the dam failure process. The first location is at the center of the dam crest, which was named “Center”, the second location is at mid-distance between the dam center and dam edge, which was named “Mid”, and the third location is at the dam edge, which was named “Edge” as shown in Fig. 11. According to this scenario, the results indicate that the time of peak outflow discharge (tP) is the same in the three cases, but the maximum value of the peak outflow discharge occurs at the center location. The difference in the peak outflow values between the three cases is relatively small as shown in Fig. 12.

The rates of erosion were also studied for the three cases. The results show that the maximum erosion rate occurs at the center location as shown in Fig. 13. By making a comparison between the three cases for the dam storage volume. The results show that the center location had the minimum values for the dam storage volume, which means that a large amount of water has passed to the downstream area as shown in Fig. 14. According to these results, the center location leads to increased erosion rate and accelerated dam failure process compared with the two other cases. Because the erosion occurs on both sides, but in the case of edge location, the erosion occurs on one side.

3.5. The effect of upstream and downstream dam slopes

The results of the comparison between the different initial rectangular breach locations indicate that the worst initial breach location is the center location, so the fourth scenario from this study concentrated on studying the effect of a change in the upstream (Su) and downstream (Sd) dam slopes. Three slopes were checked individually for both upstream and downstream slopes to determine the worst case for the dam failure process. The first slope value is (2H:1V), the second slope value is (2.5H:1V), and the third slope value is (3H:1V). According to this scenario, the results show that the decreasing downstream slope angle leads to increasing time of peak outflow discharge (tP) and decreasing value of peak outflow discharge. The difference in the peak outflow values between the three cases for the downstream slope is 2%, as shown in Fig. 15, but changing the upstream slope has a negligible impact on the peak outflow discharge and its time as shown in Fig. 16.

The rates of erosion were also studied in the three cases for both upstream and downstream slopes. The results show that the maximum erosion rate increases by 6.0% with an increasing downstream slope angle by 4°, as shown in Fig. 17. The results also indicate that the erosion rates aren’t affected by increasing or decreasing the upstream slope angle, as shown in Fig. 18. According to these results, increasing the downstream slope angle leads to increased erosion rate and accelerated dam failure process compared with the upstream slope angle. Because of increasing shear stress applied by water flows in case of increasing downstream slope.

According to all previous scenarios, the dimensionless peak outflow discharge QPQin is presented for a fixed dam height (Ho) and inflow discharge (Qin). Fig. 19 illustrates the relationship between QP∗=QPQin and.

Lr=ho2/3∗bo2/3Ho. The deduced relationship achieves R2=0.96.(17)QP∗=2.2807exp-2.804∗Lr

4. Conclusions

A spatial dam breaching process was simulated by using FLOW-3D Software. The validation process was performed by making a comparison between the simulated results of dam profiles and the dam profiles obtained by Schmocker and Hager [7] in their experimental study. And also, the peak outflow value recorded an error percentage of 12% between the numerical model and the experimental study. This model was used to study the effect of initial breach shape, dimensions, location, and dam slopes on peak outflow discharge, time of peak outflow, and the erosion process. By using the parameters obtained from the validation process, the results of this study can be summarized in eight points as follows.1.

The rectangular initial breach shape leads to an accelerating dam failure process compared with the V-notch.2.

The value of peak outflow discharge in the case of a rectangular initial breach is higher than the V-notch shape by 5%.3.

The time of peak outflow discharge for a rectangular initial breach is shorter than the V-notch shape by 9%.4.

The minimum depth and maximum width for the initial breach achieve maximum erosion rates (increasing breach width, b0, or decreasing breach depth, h0, by 5% from the dam height leads to an increase in the maximum rate of erosion by 11% and 15%, respectively), so the dam failure is rapid.5.

The center location of the initial breach leads to an accelerating dam failure compared with the edge location.6.

The initial breach location has a negligible effect on the peak outflow discharge value and its time.7.

Increasing the downstream slope angle by 4° leads to an increase in both peak outflow discharge and maximum rate of erosion by 2.0% and 6.0%, respectively.8.

The upstream slope has a negligible effect on the dam breaching process.

References

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