Figure 1. Three-dimensional finite element model of local scouring of semi-exposed submarine cable.

반노출 해저케이블의 국부 정련과정 및 영향인자에 대한 수치적 연구

Numerical Study of the Local Scouring Process and Influencing Factors of Semi-Exposed Submarine Cables

by Qishun Li,Yanpeng Hao^*,Peng Zhang,Haotian Tan,Wanxing Tian,Linhao Chen andLin Yang

School of Electric Power Engineering, South China University of Technology, Guangzhou 510640, China

^*Author to whom correspondence should be addressed.J. Mar. Sci. Eng.2023, 11(7), 1349; https://doi.org/10.3390/jmse11071349

Received: 10 June 2023 / Revised: 19 June 2023 / Accepted: 27 June 2023 / Published: 1 July 2023(This article belongs to the Section Ocean Engineering)

일부 수식이 손상되어 표시될 수 있습니다. 이 경우 원문을 참조하시기 바랍니다.

Abstract

Local scouring might result in the spanning of submarine cables, endangering their mechanical and electrical properties. In this contribution, a three-dimensional computational fluid dynamics simulation model is developed using FLOW-3D, and the scouring process of semi-exposed submarine cables is investigated. The effects of the sediment critical Shields number, sediment density, and ocean current velocity on local scouring are discussed, and variation rules for the submarine cables’ spanning time are provided. The results indicate that three scouring holes are formed around the submarine cables. The location of the bottom of the holes corresponds to that of the maximum shear velocity. The continuous development of scouring holes at the wake position leads to the spanning of the submarine cables. The increase in the sediment’s critical Shields number and sediment density, as well as the decrease in the ocean current velocity, will extend the time for maintaining the stability of the upstream scouring hole and retard the development velocity of the wake position and downstream scouring holes. The spanning time has a cubic relationship with the sediment’s critical Shields number, a linear relationship with the sediment density, and an exponential relationship with the ocean current velocity. In this paper, the local scouring process of semi-exposed submarine cables is studied, which provides a theoretical basis for the operation and maintenance of submarine cables.

Keywords:

submarine cable; local scouring; numerical simulation; computational fluid dynamics

1. Introduction

As a key piece of equipment in cross-sea power grids, submarine cables are widely used to connect autonomous power grids, supply power to islands or offshore platforms, and transmit electric power generated by marine renewable energy installations to onshore substations [1]. Once submarine cables break down due to natural disasters or human-made damage, the normal operation of other marine electric power equipment connected to them may be affected. These chain reactions will cause great economic losses and serious social impacts [2].

To protect submarine cables, they are usually buried 1 to 3 m below the seabed [3]. However, submarine cables are still confronted with potential threats from the complex subsea environment. Under the influence of fishing, anchor damage, ocean current scouring, and other factors, the sediment above submarine cables will always inevitably migrate. When a submarine cable is partially exposed, the scouring at this position will be exacerbated; eventually, it will cause the submarine cable to span. According to a field investigation of the 500 kV oil-filled submarine cable that is part of the Hainan networking system, the total length of the span is 49 m [4]. Under strong ocean currents, spanning submarine cables may experience vortex-induced vibrations. Fatigue stress caused by vortex-induced vibrations may lead to metal sheath rupture [5], which endangers the mechanical and electrical properties of submarine cables. Therefore, understanding the local scouring processes of partially exposed submarine cables is crucial for predicting scouring patterns. This is the basis for developing effective operation and maintenance strategies for submarine cables.

The mechanism and influencing factors of sediment erosion have been examined by researchers around the world. In 1988, Sumer [6] conducted experiments to show that the shedding vortex in the wake of a pipeline would increase the Shields parameter by 3–4 times, which would result in severe scouring. In 1991, Chiew [7] performed experiments to prove that the maximum scouring depth could be obtained when the pipeline was located on a flat bed and was scoured by a unidirectional water flow. Based on the test results, they provided a prediction formula for the maximum scouring depth. In 2003, Mastbergen [8] proposed a one-dimensional, steady-state numerical model of turbidity currents, which considered the negative pore pressures in the seabed. The calculated results of this model were basically consistent with the actual scouring of a submarine canyon. In 2007, Dey [9] presented a semitheoretical model for the computation of the maximum clear-water scour depth below underwater pipelines in uniform sediments under a steady flow, and the predicted scour depth in clear water satisfactorily agreed with the observed values. In 2008, Dey [10] conducted experiments on clear-water scour below underwater pipelines under a steady flow and obtained a variation pattern of the depth of the scouring hole. In 2008, Liang [11] used a two-dimensional numerical simulation to study the scouring process of a tube bundle under the action of currents and waves. They discovered that, compared with the scouring of a single tube, the scouring depth of the tube bundle was deeper, and the scouring time was longer. In 2012, Yang [12] found that placing rubber sheets under pipes can greatly accelerate their self-burial. The rubber sheets had the best performance when their length was about 1.5 times the size of the pipe. In 2020, Li [13] investigated the two-dimensional local scour beneath two submarine pipelines in tandem under wave-plus-current conditions via numerical simulation. They found that for conditions involving waves plus a low-strength current, the scour pattern beneath the two pipelines behaved like that in the pure-wave condition. Conversely, when the current had equal strength to the wave-induced flow, the scour pattern beneath the two pipelines resembled that in the pure-current condition. In 2020, Guan [14] studied and discussed the interactive coupling effects among a vibrating pipeline, flow field, and scour process through experiments, and the experimental data showed that the evolution of the scour hole had significant influences on the pipeline vibrations. In 2021, Liu [15] developed a two-dimensional finite element numerical model and researched the local scour around a vibrating pipeline. The numerical results showed that the maximum vibration amplitude of the pipeline could reach about 1.2 times diameter, and the maximum scour depth occurred on the wake side of the vibrating pipeline. In 2021, Huang [16] carried out two-dimensional numerical simulations to investigate the scour beneath a single pipeline and piggyback pipelines subjected to an oscillatory flow condition at a KC number of 11 and captured typical steady-streaming structures around the pipelines due to the oscillatory flow condition. In 2021, Cui [17] investigated the characteristics of the riverbed scour profile for a pipeline buried at different depths under the condition of riverbed sediments with different particle sizes. The results indicated that, in general, the equilibrium scour depth changed in a spoon shape with the gradual increase in the embedment ratio. In 2022, Li [18] used numerical simulation to study the influence of the burial depth of partially buried pipelines on the surrounding flow field, but they did not investigate the scour depth. In 2022, Zhu [19] performed experiments to prove that the scour hole propagation rate under a pipeline decreases with an increasing pipeline embedment ratio and rises with the KC number. In 2022, Najafzadeh [20] proposed equations for the prediction of the scouring propagation rate around pipelines due to currents based on a machine learning model, and the prediction results were consistent with the experimental data. In 2023, Ma [21] used the computational fluid dynamics coarse-grained discrete element method to simulate the scour process around a pipeline. The results showed that this method can effectively reduce the considerable need for computing resources and excessive computation time. In 2023, through numerical simulations, Hu [22] discovered that the water velocity and the pipeline diameter had a significant effect on the depth of scouring.

In the preceding works, the researchers investigated the mechanism of sediment scouring and the effect of various factors on the local scouring of submarine pipelines. However, submarine cables are buried beneath the seabed, while submarine pipelines are erected above the seabed. The difference in laying methods leads to a large discrepancy between their local scouring processes. Therefore, the conclusions of the above investigations are not applicable to the local scouring of submarine cables. Currently, there is no report on the research of the local scouring of partially exposed submarine cables.

In this paper, a three-dimensional computational fluid dynamics (CFD) finite element model, based on two-phase flow, is established using FLOW-3D. The local scouring process of semi-exposed submarine cables under steady-state ocean currents is studied, and the variation rules of the depth and the shape of the scouring holes, as well as the shear velocity with time, are obtained. By setting different critical Shields numbers of the sediment, different sediment densities, and different ocean current velocities, the change rule of the scouring holes’ development rate and the time required for the spanning of submarine cables are explored.

2. Sediment Scouring Model

In the sediment scouring model, the sediment is set as the dispersed particle, which is regarded as a kind of quasifluid. In this context, sediment scouring is considered as a two-phase flow process between the liquid phase and solid particle phase. The sediment in this process is further divided into two categories: one is suspended in the fluid, and the other is deposited on the bottom.When the local Shields number of sediment is greater than the critical Shields number, the deposited sediment will be transformed into the suspended sediment under the action of ocean currents. The calculation formulae of the local Shields numbers θ and the critical Shields numbers

θ_cr of sediment is given as [23,24

]

𝜃=𝑈2𝑓(𝜌𝑠/𝜌𝑓−1)𝑔𝑑50,�=��2(��/��−1)��50,(1)

𝜃𝑐𝑟=0.31+1.2𝐷∗+0.055(1−𝑒−0.02𝐷∗),��=0.31+1.2�*+0.055(1−�−0.02�*),(2)

𝐷∗=𝑑50𝜌𝑓(𝜌𝑠−𝜌𝑓)𝑔/𝜇2−−−−−−−−−−−−−−√3,�*=�50��(��−��)�/�23,(3)where

U_f is the shearing velocity of bed surface,

ρ_s is the density of the sediment particle,

ρ_f is the fluid density, g is the acceleration of gravity, d

₅₀ is the median size of sediment, and μ is the dynamic viscosity of sediment.And each sediment particle suspended in the fluid obeys the equations for mass conservation and energy conservation

∂𝑐𝑠∂𝑡+∇⋅(𝑢𝑐𝑠)=0,∂��∂�+∇⋅(�¯��)=0,(4)

∂𝑢𝑠∂𝑡+𝑢⋅∇𝑢𝑠=−1𝜌𝑠∇𝑃+𝐹−𝐾𝑓𝑠𝜌𝑠𝑢𝑟,∂��∂�+�¯⋅∇��=−1��∇�+�−��,(5)where

c_s is the concentration of the sediment particle,

𝑢�¯ is the mean velocity vector of the fluid and the sediment particle,

u_s is the velocity of the sediment particle,

f_s is the volume fraction of the sediment particle, P is the pressure, F is the volumetric and viscous force, K is the drag force, and

u_r is the relative velocity.

3. Numerical Setup and Modeling

In this paper, a three-dimensional submarine cable local scouring simulation model is established by FLOW-3D. Based on the numerical simulation, the process of the submarine cable, which gradually changes from semi-exposed to the spanning state under the steady-state ocean current, is studied. The geometric modeling, the mesh division, the physical field setup, and the grid independent test of CFD numerical model are as follows.

3.1. Geometric Modeling and Mesh Division

A three-dimensional (3D) numerical model of the local scouring of a semi-exposed submarine cable is established, which is shown in Figure 1. The dimensions of the model are marked in Figure 1. The inlet direction of the ocean current is defined as the upstream of the submarine cable (referred to as upstream), and the outlet direction of the ocean current is defined as the downstream of the submarine cable (referred to as downstream).

Figure 1. Three-dimensional finite element model of local scouring of semi-exposed submarine cable.

The submarine cable with a diameter of 0.2 m is positioned on sediment that is initially in a semi-exposed state. When the length of the span is short, the submarine cable will not show obvious deformation due to gravity or scouring from the ocean current. Therefore, the submarine cable surface is set as the fixed boundary. The model’s left boundary is set as the inlet, the right boundary is set as the outlet, the front and rear boundaries are set as symmetry, and the bottom boundary is set as the non-slip wall. Since the water depth above the submarine cable is more than 0.6 m in practice, the top boundary of the model is also set as symmetry. The sediment near the inlet and the outlet will be carried by ocean currents, which leads to the abnormal scouring terrain. At each end of the sediment, a baffle (thickness of 3 cm) is installed to ensure that the simulation results can reflect the real situation.

Due to the fact that the flow field around the semi-exposed submarine cable is not a simple two-dimensional symmetrical distribution, it should be solved by three-dimensional numerical simulation. Considering the accuracy and efficiency of the calculation, the size of mesh is set to 0.02 m. The total number of meshes after the dissection is 133,254.

3.2. Physical Field Setup

The CFD finite element model contains four physical field modules: sediment scouring module, gravity and non-inertial reference frame module, density evaluation module, and viscosity and turbulence module. In this paper, the renormalization group (RNG) k–ε turbulence model is used, which has high computational accuracy for turbulent vortices. Therefore, this turbulence model is suitable for calculating the sediment scouring process around the semi-exposed submarine cable [25]. The key parameters of the numerical simulation are referring to the survey results of submarine sediments in the Korean Peninsula [26], as listed in Table 1.Table 1. Key parameters of numerical simulation.

3.3. Mesh Independent Test

In order to eliminate errors caused by the quantity of grids in the calculation process, two sizes of mesh are set on the validation model, and the scour profiles under different mesh sizes are compared. The validation model is shown in Figure 2, and the scouring terrain under different mesh size is given in Figure 3.

Figure 2. Validation model.

Figure 3. Scouring terrain under different mesh sizes.

It can be seen from Figure 3 that with the increase in the number of meshes, the scouring terrain of the verification model changes slightly, and the scouring depth is basically unchanged. Considering the accuracy of the numerical simulation and the calculation’s time cost, it is reasonable to consider setting the mesh size to 0.02 m.

4. Results and Analysis

4.1. Analysis of Local Scouring Process

Based on the CFD finite element numerical simulation, the local scouring process of the submarine cable under the steady-state ocean current is analyzed. The end time of the simulation is 9 h, the initial time step is 0.01 s, and the fluid velocity is 0.40 m/s. Simulation results are saved every minute. Figure 4 illustrates the scouring terrain around the semi-exposed submarine cable, which has been scoured by the steady-state current for 5 h.

Figure 4. Scouring terrain around semi-exposed submarine cable (scour for 5 h).

As can be seen from Figure 4, three scouring holes were separately formed in the upstream wake position and downstream of the semi-exposed submarine cable. The scouring holes are labeled according to their locations. The variation of the scouring terrain around the semi-exposed submarine cable over time is given in Figure 5. The red circle in the picture corresponds to the position of the submarine cable, and the red box in the legend marks the time when the submarine cable is spanning.

Figure 5. Variation of scouring terrain around semi-exposed submarine cable adapted to time.

From Figure 5, in the first hour of scouring, the upstream (−0.5 m to −0.1 m) and downstream (0.43 m to 1.5 m) scouring holes appeared. The upstream scouring hole was relatively flat with depth of 0.04 m. The depth of the downstream scouring hole increased with the increase in distance, and the maximum depth was 0.13 m. The scouring hole that developed at the wake position was very shallow, and its depth was only 0.007 m.

In the second hour of scouring, the upstream scouring hole’s depth remained nearly constant. The depth of the downstream scouring hole only increased by 0.002 m. The scouring hole at the wake position developed steadily, and its depth increased from 0.007 m to 0.014 m.

The upstream and downstream scouring holes did not continue to develop during the third to the sixth hour. Compared to the first two hours, the development of scouring holes at the wake position accelerated significantly, with an average growth rate of 0.028 m/h. The growth rate in the fifth hour of the scouring hole at the wake position was slightly faster than the other times. After 6 h of scouring, the sediment on the right side of the submarine cable had been hollowed out.

In the seventh and the eighth hour of scouring, the upstream scouring hole’s depth increased slightly, the downstream scouring hole still remained stable, and the depth of the scouring hole at wake position increased by 0.019 m. The sediment under the submarine cable was gradually eroded as well. By the end of the eighth hour, the lower right part of the submarine cable had been exposed to water as well.

At 8 h 21 min of the scouring, the submarine cable was completely spanned, and the scouring holes were connected to each other. Within the next 10 min, the development of the scouring holes sped up significantly, and the maximum depth of scouring holes increased greatly to 0.27 m.

In reference [17], researchers have studied the local scouring process of semi-buried pipelines in sandy riverbeds through experiments. The test results show that the scouring process can be divided into a start-up stage, micropore formation stage, extension stage, and equilibrium stage. In this paper, the first three stages are simulated, and the results are in good agreement with the experiment, which proves the accuracy of the present numerical model.

In this research, the velocity of ocean currents at the sediment surface is defined as the shear velocity, which plays an important role in the process of local scouring. Figure 6 provides visual data on how the shear velocity varies over time.

Figure 6. Shear velocity changes in the scouring process.

The semi-exposed submarine cable protrudes from the seabed, which makes the shear velocity of its surface much higher than other locations. After the submarine cable is spanned, the shear velocity of the scouring hole surface below it is taken. This is the reason for the sudden change of shear velocity at the submarine cable’s location in Figure 6.The shear velocity in the initial state of the upstream scouring hole is obviously greater than in subsequent times. After 1 h of scouring, the shear velocity in the upstream scouring hole rapidly decreased from 1.1 × 10

⁻² m/s to 3.98 × 10

⁻³ m/s and remained stable until the end of the sixth hour. This phenomenon explains why the upstream scouring hole developed rapidly in the first hour but remained stable for the following 5 h.The shear velocity in the downstream scouring hole reduced at first and then increased; its initial value was 1.41 × 10

⁻² m/s. It took approximately 5 h for the shear velocity to stabilize, and the stable shear velocity was 2.26 × 10

⁻³ m/s. Therefore, compared with the upstream scouring hole, the downstream scouring hole was deeper and required more time to reach stability.The initial shear velocity in the scouring hole at the wake position was only 7.1 × 10

⁻³ m/s, which almost does not change in the first hour. This leads to a very slow development of the scouring hole at the wake position in the early stages. The maximum shear velocity in this scouring hole gradually increased to 1.05 × 10

⁻² m/s from the second to the fifth hour, and then decreased to 6.61 × 10

⁻³ m/s by the end of the eighth hour. This is why the scouring hole at the wake position grows fastest around the fifth hour. Consistent with the pattern of change in the scouring hole’s terrain, the location of the maximal shear velocity also shifted to the right with time.

The shear velocity of all three scouring holes rose dramatically in the last hour. Combined with the terrain in Figure 5, this can be attributed to the complete spanning of the submarine cable.

From Equations (3)–(5), one can see the movement of the sediment is related directly with the sediment’s critical Shields number, sediment density, and ocean current velocity. Based on the parameters in Table 1, the influence of the above parameters on the local scouring process of semi-exposed submarine cables will be discussed.

4.2. Influence Factors

4.2.1. Sediment’s Critical Shields Number

The sediment’s critical Shields number

θ_cr is set as 0.02, 0.03, 0.04, 0.05, 0.06, and 0.07, and the variations of scouring terrain over time under each

θ_cr are displayed in Figure 7.

Figure 7. Influence of sediment’s critical Shields number

θ_cr on local scouring around semi-exposed submarine cable: (a

) θ_cr = 0.02; (b

) θ_cr = 0.03; (c

) θ_cr = 0.04; (d

) θ_cr = 0.05; (e

) θ_cr = 0.06; and (f

) θ_cr = 0.07.From Figure 7, one can see that a change in

θ_cr will affect the depth of the upstream scouring hole and the development speed of the scouring hole at the wake position, but it will have no significant impact on the expansion of the downstream scouring hole.Under conditions of different

θ_cr, the upstream scouring hole will reach a temporary plateau within 1 h, at which time the stable depth will be about 0.04 m. When

θ_cr ≤ 0.05, the upstream scouring hole will continue to expand after a few hours. The stable time is obviously affected by

θ_cr, which will gradually increase from 1 h to 11 h with the increase in

θ_cr. The terrain of the upstream scouring hole will gradually convert to deep on the left and to shallow on the right. Since the scouring hole at the wake position has not been stable, its state at the time of submarine cable spanning is studied emphatically. In the whole process of scouring, the scouring hole at the wake position continues to develop and does not reach a stable state. With the increase in

θ_cr, the development velocity of the scouring hole at the wake position will decrease considerably. Its average evolution velocity decreases from 3.88 cm/h to 1.62 cm/h, and its depth decreases from 21.9 cm to 18.8 cm. Under the condition of each

θ_cr, the downstream scouring hole will stabilize within 1 h, and the stable depth will be basically unchanged (all about 13.5 cm).As

θ_cr increases, so does the sediment’s ability to withstand shearing forces, which will cause it to become increasingly difficult to be eroded or carried away by ocean currents. This effect has been directly reflected in the depth of scouring holes (upstream and wake position). Due to the blocking effect of semi-exposed submarine cables, the wake is elongated, which is why the downstream scouring hole develops before the scouring hole at the wake position and quickly reaches a stable state. However, due to the high wake intensity, this process is not significantly affected by the change of

θ_cr.

4.2.2. Sediment Density

The density of sediment

ρ_s is set as 1550 kg/m

³, 1600 kg/m

³, 1650 kg/m

³, 1700 kg/m

³, 1750 kg/m

³, and 1800 kg/m

³, and the variation of scouring terrain over time under each

ρ_s are displayed in Figure 8.

Figure 8. Influence of sediment density

ρ_s on local scouring around semi-exposed submarine cable: (a

) ρ_s = 1550 kg/m

³; (b) ρ_s = 1600 kg/m

³; (c) ρ_s = 1650 kg/m

³; (d) ρ_s = 1700 kg/m

³; (e) ρ_s = 1750 kg/m

³; and (f

) ρ_s = 1800 kg/m

³.From Figure 8, one can see that a change in

ρ_s will also affect the depth of the upstream scouring hole and the development speed of the scouring hole at the wake position. In addition, it can even have an impact on the downstream scouring hole depth.Under different

ρ_s conditions, the upstream scouring hole will always reach a temporary stable state in 1 h, at which time the stable depth will be 0.04 m. When

ρ_s ≤ 1750 kg/m

³, the upstream scouring hole will continue to expand after a few hours. The stabilization time of upstream scouring hole is more clearly affected by

ρ_s, which will gradually increase from 3 h to 13 h with the increase in

ρ_s. The terrain of the upstream scouring hole will gradually change to deep on the left and to shallow on the right. Since the scouring hole at the wake position has not been stable, its state at the time of the submarine cable spanning is studied emphatically, too. In the whole process of scouring, the scouring hole at the wake position continues to develop and does not reach a stable state. When

ρ_s is large, the development rate of scouring hole obviously decreased with time. With the increase in

ρ_s, the development velocity of the scouring hole at the wake position reduces from 3.38 cm/h to 1.14 cm/h, and the depth of this scouring hole declines from 20 cm to 15 cm. As

ρ_s increases, the stabilization time of the downstream scouring hole increases from less than 1 h to about 2 h, but the stabilization depth of the downstream scouring hole remains essentially the same (all around 13.5 cm).As can be seen from Equation (1), the increase in

ρ_s will reduce the Shields number, thus weakening the shear action of the sediment by the ocean current, which explains the extension of the stability time of the upstream scouring hole. At the same time, with the increase in the depth of scouring hole at the wake position, its shear velocity will decreases. Therefore, under a larger

ρ_s value, the development speed of scouring hole at the wake position will decrease significantly with time. Possibly for the same reason,

ρ_s can affect the development rate of downstream scouring hole.

4.2.3. Ocean Current Velocity

The ocean current velocity v is set as 0.35 m/s, 0.40 m/s, 0.45 m/s, 0.50 m/s, 0.55 m/s, and 0.60 m/s. Figure 9 presents the variation in scouring terrain with time for each v.

Figure 9. Influence of ocean current velocity v on local scouring around semi-exposed submarine cable: (a) v = 0.35 m/s; (b) v = 0.40 m/s; (c) v = 0.45 m/s; (d) v = 0.50 m/s; (e) v = 0.55 m/s; and (f) v = 0.60 m/s.

Changes in v affect the depth of the upstream and downstream scouring holes, as well as the development velocity of the wake position and downstream scouring holes.

When v ≤ 0.45 m/s, the upstream scouring hole will reach a temporary stable state within 1 h, at which point the stable depth will be 0.04 m. The stabilization time of the upstream scouring hole is affected by v, which will gradually decrease from 15 h to 3 h with the increase in v. When v > 0.45 m/s, the upstream scouring hole is going to expand continuously. With the increase in v, its average development velocity increases from 6.68 cm/h to 8.66 cm/h, and its terrain changes to deep on the left and to shallow on the right. When the submarine cable is spanning, special attention should be paid to the depth of the scouring hole at the wake position. Throughout whole scouring process, the scouring hole at the wake position continues to develop and does not reach a stable state. With the increase in v, the depth of scouring hole at the wake position will increase from 14 cm to 20 cm, and the average development velocity will increase from 0.91 cm/h to 10.43 cm/h. As v increases, the time required to stabilize the downstream scouring hole is shortened from 1to 2 h to less than 1 h, but the stable depth is remains nearly constant at 13.5 cm.

An increase in v will increase the shear velocity. Therefore, when the depth of the scouring hole increases, the shear velocity in the hole will also increase, which can deepen both the upstream and downstream scouring hole. According to Equation (1), the Shields number is proportional to the square of the shear velocity. The increase in shear velocity significantly intensifies local scouring, which increases the development rate of scouring holes at the wake position and downstream.

4.3. Variation Rule of Spanning Time

In this paper, the spanning time is defined as the time taken for a semi-exposed submarine cable (initial state) to become a spanning submarine cable. Figure 10 illustrates the effect of the above parameters on the spanning time of the semi-exposed submarine cable.

Figure 10. Influence of different parameters on spanning time of the semi-exposed submarine cable: (a) Sediment critical Shields number; (b) Sediment density; and (c) Ocean current velocity.From Figure 10a, the spanning time monotonically increases with the increase in the critical Shields number of sediment. However, the slope of the curve decreases first and then increases, and the inflection point is at

θ_cr = 4.59 × 10

⁻². The relationship between spanning time t and sediment’s critical Shields number

θ_cr can be formulated by a cubic function as shown in Equation (6):

𝑡=−2.98+6.76𝜃𝑐𝑟−1.45𝜃2𝑐𝑟+0.11𝜃3𝑐𝑟.�=−2.98+6.76��−1.45��2+0.11��3.(6)It can be seen from Figure 10b that with the increase in the sediment density, the spanning time increases monotonically and linearly. The relationship between the spanning time t and the sediment’s density

ρ_s can be formulated by the first order function as shown in Equation (7):

𝑡=−41.59+30.54𝜌𝑠.�=−41.59+30.54��.(7)Figure 10c shows that with the increase in the ocean current velocity, the spanning time decreases monotonically. The slope of the curve increases with the increase in the ocean current velocity, so it can be considered that there is saturation of the ocean current velocity effect. The relationship between the spanning time t and the ocean current velocity v can be formulated by the exponential function

𝑡=0.15𝑣−4.38.�=0.15�−4.38.(8)

5. Conclusions

In this paper, a three-dimensional CFD finite element numerical simulation model is established, which is used to research the local scouring process of the semi-exposed submarine cable under the steady-state ocean current. The relationship between shear velocity and scouring terrain is discussed, the influence of sediment critical Shields number, sediment density and ocean current velocity on the local scouring process is analyzed, and the variation rules of the spanning time of the semi-exposed submarine cable is given. The conclusions are as follows:

Under the steady-state ocean currents, scouring holes will be formed at the upstream, wake position and downstream of the semi-exposed submarine cable. The upstream and downstream scouring holes develop faster, which will reach a temporary stable state at about 1 h after the start of the scouring. The scouring hole at the wake position will continue to expand at a slower rate and eventually lead to the spanning of the submarine cable.
There is a close relationship between the distribution of shear velocity and the scouring terrain. As the local scouring process occurs, the location of the maximum shear velocity within the scouring hole shifts and causes the bottom of the hole to move as well.
When the sediment’s critical Shields number and density are significantly large and ocean current velocity is sufficiently low, the duration of the stable state of the upstream scouring hole will be prolonged, and the average development velocity of the scouring holes at the wake position and downstream will be reduced.
The relationship between the spanning time and the critical Shields number θ_cr can be formulated as a cubic function, in which the curve’s inflection point is θ_cr = 4.59 × 10⁻². The relationship between spanning time and sediment density can be formulated as a linear function. The relationship between spanning time and ocean current velocity can be formulated by exponential function.

Based on the conclusions of this paper, even when it is too late to take measures or when the exposed position of the submarine cable cannot be located, the degree of burial depth development still can be predicted. This prediction is important for the operation and maintenance of the submarine cable. However, the study still leaves something to be desired. Only the local scouring process under the steady-state ocean current was studied, which is an extreme condition. In practice, exposed submarine cables are more likely to be scoured by reciprocating ocean currents. In the future, we will investigate the local scouring of submarine cables under the reciprocating ocean current.

Author Contributions

Conceptualization, Y.H. and Q.L.; methodology, Q.L., P.Z. and H.T.; software, Q.L.; validation, Q.L., L.C. and W.T.; writing—original draft preparation, Q.L.; writing—review and editing, Y.H. and Q.L.; supervision, Y.H. and L.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the [Smart Grid Joint Fund Key Project between National Natural Science Foundation of China and State Grid Corporation] grant number [U1766220].

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data supporting the reported results cannot be shared at this time, as they have been used in producing more publications on this research.

Acknowledgments

This work is supported by the Smart Grid Joint Fund Key Project of the National Natural Science Foundation of China and State Grid Corporation (Grant No. U1766220).

Conflicts of Interest

The authors declare no conflict of interest.

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Validity evaluation of popular liquid-vapor phase change models for cryogenic self-pressurization process

극저온 자체 가압 공정을 위한 인기 있는 액체-증기 상 변화 모델의 타당성 평가

액체-증기 상 변화 모델은 밀폐된 용기의 자체 가압 프로세스 시뮬레이션에 매우 큰 영향을 미칩니다. Hertz-Knudsen 관계, 에너지 점프 모델 및 그 파생물과 같은 널리 사용되는 액체-증기 상 변화 모델은 실온 유체를 기반으로 개발되었습니다. 액체-증기 전이를 통한 극저온 시뮬레이션에 널리 적용되었지만 각 모델의 성능은 극저온 조건에서 명시적으로 조사 및 비교되지 않았습니다. 본 연구에서는 171가지 일반적인 액체-증기 상 변화 모델을 통합한 통합 다상 솔버가 제안되었으며, 이를 통해 이러한 모델을 실험 데이터와 직접 비교할 수 있습니다. 증발 및 응축 모델의 예측 정확도와 계산 속도를 평가하기 위해 총 <>개의 자체 가압 시뮬레이션이 수행되었습니다. 압력 예측은 최적화 전략이 서로 다른 모델 계수에 크게 의존하는 것으로 나타났습니다. 에너지 점프 모델은 극저온 자체 가압 시뮬레이션에 적합하지 않은 것으로 나타났습니다. 평균 편차와 CPU 소비량에 따르면 Lee 모델과 Tanasawa 모델은 다른 모델보다 안정적이고 효율적인 것으로 입증되었습니다.

원문 바로가기

International Journal of Heat and Mass Transfer

Volume 181, December 2021, 121879

Author links open overlay panelZhongqi Zuo, Jingyi Wu, Yonghua HuangShow moreAdd to MendeleyShareCite

https://doi.org/10.1016/j.ijheatmasstransfer.2021.121879 Get rights and content

Abstract

Liquid-vapor phase change models vitally influence the simulation of self-pressurization processes in closed containers. Popular liquid-vapor phase change models, such as the Hertz-Knudsen relation, energy jump model, and their derivations were developed based on room-temperature fluids. Although they had widely been applied in cryogenic simulations with liquid-vapor transitions, the performance of each model was not explicitly investigated and compared yet under cryogenic conditions. A unified multi-phase solver incorporating four typical liquid-vapor phase change models has been proposed in the present study, which enables direct comparison among those models against experimental data. A total number of 171 self-pressurization simulations were conducted to evaluate the evaporation and condensation models’ prediction accuracy and calculation speed. It was found that the pressure prediction highly depended on the model coefficients, whose optimization strategies differed from each other. The energy jump model was found inadequate for cryogenic self-pressurization simulations. According to the average deviation and CPU consumption, the Lee model and the Tanasawa model were proven to be more stable and more efficient than the others.

Introduction

The liquid-vapor phase change of cryogenic fluids is widely involved in industrial applications, such as the hydrogen transport vehicles [1], shipborne liquid natural gas (LNG) containers [2] and on-orbit cryogenic propellant tanks [3]. These applications require cryogenic fluids to be stored for weeks to months. Although high-performance insulation measures are adopted, heat inevitably enters the tank via radiation and conduction. The self-pressurization in the tank induced by the heat leakage eventually causes the venting loss of the cryogenic fluids and threatens the safety of the craft in long-term missions. To reduce the boil-off loss and extend the cryogenic storage duration, a more comprehensive understanding of the self-pressurization mechanism is needed.

Due to the difficulties and limitations in implementing cryogenic experiments, numerical modeling is a convenient and powerful way to study the self-pressurization process of cryogenic fluids. However, how the phase change models influence the mass and heat transfer under cryogenic conditions is still unsettled [4]. As concluded by Persad and Ward [5], a seemingly slight variation in the liquid-vapor phase change models can lead to erroneous predictions.

Among the liquid-vapor phase change models, the kinetic theory gas (KTG) based models and the energy jump model are the most popular ones used in recent self-pressurization simulations [6]. The KTG based models, also known as the Hertz-Knudsen relation models, were developed on the concept of the Maxwell-Boltzmann distribution of the gas molecular [7]. The Hertz-Knudsen relation has evolved to several models, including the Schrage model [8], the Tanasawa model [9], the Lee model [10] and the statistical rate theory (SRT) [11], which will be described in Section 2.2. Since the Schrage model and the Lee model are embedded and configured as the default ones in the commercial CFD solvers Flow-3D® and Ansys Fluent® respectively, they have been widely used in self-pressurization simulations for liquid nitrogen [12], [13] and liquid hydrogen [14], [15]. The major drawback of the KTG models lies in the difficulty of selecting model coefficients, which were reported in a considerably wide range spanning three magnitudes even for the same working fluid [16], [17], [18], [19], [20], [21]. Studies showed that the liquid level, pressure and mass transfer rate are directly influenced by the model coefficients [16], [22], [23], [24], [25]. Wrong coefficients will lead to deviation or even divergence of the results. The energy jump model is also known as the thermal limitation model. It assumes that the evaporation and condensation at the liquid-vapor interface are induced only by heat conduction. The model is widely adopted in lumped node simulations due to its simplicity [6], [26], [27]. To improve the accuracy of mass flux prediction, the energy jump model was modified by including the convection heat transfer [28], [29]. However, the convection correlations are empirical and developed mainly for room-temperature fluids. Whether the correlation itself can be precisely applied in cryogenic simulations still needs further investigation.

Fig. 1 summarizes the cryogenic simulations involving the modeling of evaporation and condensation processes in recent years. The publication has been increasing rapidly. However, the characteristics of each evaporation and condensation model are not explicitly revealed when simulating self-pressurization. A comparative study of the phase change models is highly needed for cryogenic fluids for a better simulation of the self-pressurization processes.

In the present paper, a unified multi-phase solver incorporating four typical liquid-vapor phase change models, namely the Tanasawa model, the Lee model, the energy jump model, and the modified energy jump model has been proposed, which enables direct comparison among different models. The models are used to simulate the pressure and temperature evolutions in an experimental liquid nitrogen tank in normal gravity, which helps to evaluate themselves in the aspects of accuracy, calculation speed and robustness.

Section snippets

Governing equations for the self-pressurization tank

In the present study, both the fluid domain and the solid wall of the tank are modeled and discretized. The heat transportation at the solid boundaries is considered to be irrelevant with the nearby fluid velocity. Consequently, two sets of the solid and the fluid governing equations can be decoupled and solved separately. The pressures in the cryogenic container are usually from 100 kPa to 300 kPa. Under these conditions, the Knudsen number is far smaller than 0.01, and the fluids are

Self-pressurization results and phase change model comparison

This section compares the simulation results by different phase change models. Section 3.1 compares the pressure and temperature outputs from two KTG based models, namely the Lee model and the Tanasawa model. Section 3.2 presents the pressure predictions from the energy transport models, namely the energy jump model and the modified energy jump model, and compares pressure prediction performances between the KTG based models and the energy transport models. Section 3.3 evaluates the four models

Conclusion

A unified vapor-liquid-solid multi-phase numerical solver has been accomplished for the self pressurization simulation in cryogenic containers. Compared to the early fluid-only solver, the temperature prediction in the vicinity of the tank wall improves significantly. Four liquid-vapor phase change models were integrated into the solver, which enables fair and effective comparison for performances between each other. The pressure and temperature prediction accuracies, and the calculation speed

CRediT authorship contribution statement

Zhongqi Zuo: Data curation, Formal analysis, Writing – original draft, Validation. Jingyi Wu: Conceptualization, Writing – review & editing, Validation. Yonghua Huang: Conceptualization, Formal analysis, Writing – review & editing, Validation.

Declaration of Competing Interest

Authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Validity evaluation of popular liquid-vapor phase change models for cryogenic self-pressurization process”.

Acknowledgement

This project is supported by the National Natural Science Foundation of China (No. 51936006).

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Cited by (7)

Theoretical investigation on heat leakage distribution between vapor and liquid in liquid hydrogen tanks2023, International Journal of Hydrogen EnergyShow abstract
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The distribution of the computed maximum current speed during the entire duration of the NAMI DANCE and FLOW-3D simulations. The resolution of computational domain is 10 m

Performance Comparison of NAMI DANCE and FLOW-3D® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems

NTHMP 벤치마크 문제를 사용하여 쓰나미 전파, 침수 및 해류에서 NAMI DANCE 및 FLOW-3D® 모델의 성능 비교

Pure and Applied Geophysics volume 176, pages3115–3153 (2019)Cite this article

Abstract

Field observations provide valuable data regarding nearshore tsunami impact, yet only in inundation areas where tsunami waves have already flooded. Therefore, tsunami modeling is essential to understand tsunami behavior and prepare for tsunami inundation. It is necessary that all numerical models used in tsunami emergency planning be subject to benchmark tests for validation and verification. This study focuses on two numerical codes, NAMI DANCE and FLOW-3D^®, for validation and performance comparison. NAMI DANCE is an in-house tsunami numerical model developed by the Ocean Engineering Research Center of Middle East Technical University, Turkey and Laboratory of Special Research Bureau for Automation of Marine Research, Russia. FLOW-3D^® is a general purpose computational fluid dynamics software, which was developed by scientists who pioneered in the design of the Volume-of-Fluid technique. The codes are validated and their performances are compared via analytical, experimental and field benchmark problems, which are documented in the ‘‘Proceedings and Results of the 2011 National Tsunami Hazard Mitigation Program (NTHMP) Model Benchmarking Workshop’’ and the ‘‘Proceedings and Results of the NTHMP 2015 Tsunami Current Modeling Workshop”. The variations between the numerical solutions of these two models are evaluated through statistical error analysis.

현장 관찰은 연안 쓰나미 영향에 관한 귀중한 데이터를 제공하지만 쓰나미 파도가 이미 범람한 침수 지역에서만 가능합니다. 따라서 쓰나미 모델링은 쓰나미 행동을 이해하고 쓰나미 범람에 대비하는 데 필수적입니다.

쓰나미 비상 계획에 사용되는 모든 수치 모델은 검증 및 검증을 위한 벤치마크 테스트를 받아야 합니다. 이 연구는 검증 및 성능 비교를 위해 NAMI DANCE 및 FLOW-3D®의 두 가지 숫자 코드에 중점을 둡니다.

NAMI DANCE는 터키 중동 기술 대학의 해양 공학 연구 센터와 러시아 해양 연구 자동화를 위한 특별 조사국 연구소에서 개발한 사내 쓰나미 수치 모델입니다. FLOW-3D®는 Volume-of-Fluid 기술의 설계를 개척한 과학자들이 개발한 범용 전산 유체 역학 소프트웨어입니다.

코드의 유효성이 검증되고 분석, 실험 및 현장 벤치마크 문제를 통해 코드의 성능이 비교되며, 이는 ‘2011년 NTHMP(National Tsunami Hazard Mitigation Program) 모델 벤치마킹 워크숍의 절차 및 결과’와 ”절차 및 NTHMP 2015 쓰나미 현재 모델링 워크숍 결과”. 이 두 모델의 수치 해 사이의 변동은 통계적 오류 분석을 통해 평가됩니다.

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Acknowledgements

The authors wish to thank Dr. Andrey Zaytsev due to his undeniable contributions to the development of in-house numerical model, NAMI DANCE. The Turkish branch of Flow Science, Inc. is also acknowledged. Finally, the National Tsunami Hazard Mitigation Program (NTHMP), who provided most of the benchmark data, is appreciated. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Deniz Velioglu SogutPresent address: 1212 Computer Science, Department of Civil Engineering, Stony Brook University, Stony Brook, NY, 11794, USA

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Middle East Technical University, 06800, Ankara, TurkeyDeniz Velioglu Sogut & Ahmet Cevdet Yalciner

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Velioglu Sogut, D., Yalciner, A.C. Performance Comparison of NAMI DANCE and FLOW-3D^® Models in Tsunami Propagation, Inundation and Currents using NTHMP Benchmark Problems. Pure Appl. Geophys. 176, 3115–3153 (2019). https://doi.org/10.1007/s00024-018-1907-9

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Received22 December 2017
Revised16 May 2018
Accepted24 May 2018
Published07 June 2018
Issue Date01 July 2019
DOIhttps://doi.org/10.1007/s00024-018-1907-9

Keywords

Tsunami
depth-averaged shallow water
Reynolds-averaged Navier–Stokes
benchmarking
NAMI DANCE
FLOW-3D^®

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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 Hu¹,Hongjun Liu²,Hao Leng¹,Peng Yu³ 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. 2021, 9(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 S_eq around USAF. At last, a parametric study was carried out to study the effects of the Froude number F_r and Euler number E_u for the S_eq. 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 KC_s,p < 8. The predicting results of the revised stochastic model are the most favorable for n = 10 when KC_rms,a < 4. The higher F_r and E_u both lead to the more intensive horseshoe vortex and larger S_eq.

Keywords:

scour; numerical investigation; random waves; equilibrium scour depth; KC 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].

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 < θ_cr) or live bed scour (θ > θ_cr). 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(R_d) (R_d is pile Reynolds), which means the turbulence intensity induced by the flow-structure interaction would decrease with R_d increases, but the effects of R_d 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 θ, KC has a greater influence on the scour morphology [21,27,28]. The influence mechanism of KC on the scour around the pile is reflected in two aspects: the horseshoe vortex at upstream and wake vortex shedding at downstream.

KC=UwmTD��=�wm��(1)

where, U_wm 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, K_wave is correction factor considering wave action, K_hw is correction factor considering water depth.

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

where, h_w 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, θ_cr 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, u_c 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 U_m and peak wave period T_P to calculate KC. Khalfin [35] recommended the RMS wave height H_rms and peak wave period T_P 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 U_m and mean zero-crossing wave period T_z. 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 (F_r) and Euler number (E_u) 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, V_F is the volume fraction; u, v, and w are the velocity components in x, y, z direction respectively with Cartesian coordinates; A_i is the area fraction; ρ_f is the fluid density, f_i is the viscous fluid acceleration, G_i is the fluid body acceleration (i = x, y, z).

2.2. Turbulent Model

The turbulence closure is available by the turbulent model, such as one-equation, the one-equation k-ε model, the standard k-ε model, RNG k-ε turbulent model and large eddy simulation (LES) model. The LES model requires very fine mesh grid, so the computational time is large, which hinders the LES model application in engineering. The RNG k-ε model can reduce computational time greatly with high accuracy in the near-wall region. Furthermore, the RNG k-ε model computes the maximum turbulent mixing length dynamically in simulating sediment scour model. Therefore, the RNG k-ε model was adopted to study the scour around anchor under random waves [41,42].

∂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, k_T is specific kinetic energy involved with turbulent velocity, G_T is the turbulent energy generated by buoyancy; ε_T is the turbulent energy dissipating rate, P_T is the turbulent energy, Diff_ε and Diff_kT are diffusion terms associated with V_F, A_i; CDIS1, CDIS2 and CDIS3 are dimensionless parameters, and CDIS1, CDIS3 have default values of 1.42, 0.2 respectively. CDIS2 can be obtained from P_T and k_T.

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, α_i is the entrainment parameter, n_s is the outward point perpendicular to the seabed, d_* is the dimensionless diameter of sand particles, which was calculated by Equation (15), θ_cr is the critical Shields parameter, g is the gravity acceleration, d_i is the diameter of sand particles, ρ_i is the density of seabed species.

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

where μ_f is the fluid dynamic viscosity.

In Equation (14), the entrainment parameter α_i 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]. n_s is the outward pointing normal to the seabed interface, and n_s = (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, U_f,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 θ_cr 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 ν_f 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, q_b,i is the bed load transport rate, which was obtained from Equation (20), δ_i is the bed load thickness, which was calculated by Equation (21), c_b,i is the volume fraction of sand i in the multiple species, f_b 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, C_s,i is the suspended sand particles mass concentration of sand i in the multiple species, u_s,i is the sand particles velocity of sand i, D_f 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, c_s,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.

Figure 2. (a) The sketch of seabed-USAF-wave three-dimensional model; (b) boundary condation:Wv-wave boundary, S-symmetric boundary, O-outflow boundary; (c) USAF model.

Table 1. Numerical simulating cases.

3.1. Mesh Geometric Dimensions

In the simulation of the scour under the random waves, the model includes the umbrella suction anchor foundation, seabed and fluid. As shown in Figure 3, the model mesh includes global mesh grid and nested mesh grid, and the total number of grids is 1,812,000. The basic procedure for building mesh grid consists of two steps. Step 1: Divide the global mesh using regular hexahedron with size of 0.6 × 0.6. The global mesh area is cubic box, embracing the seabed and whole fluid volume, and the dimensions are 210 m in length, 30 m in width and 32 m in height. The details of determining the grid size can see the following mesh sensitivity section. Step 2: Set nested fine mesh grid in vicinity of the USAF with size of 0.3 × 0.3 so as to shorten the computation cost and improve the calculation accuracy. The encryption range is −15 m to 15 m in x direction, −15 m to 15 m in y direction and 0 m to 32 m in z direction, respectively. In order to accurately capture the free-surface dynamics, such as the fluid-air interface, the volume of fluid (VOF) method was adopted for tracking the free water surface. One specific algorithm called FAVORTM (Fractional Area/Volume Obstacle Representation) was used to define the fractional face areas and fractional volumes of the cells which are open to fluid flow.

Figure 3. The sketch of mesh grid.

3.2. Boundary Conditions

As shown in Figure 2, the initial fluid length is 210 m as long as seabed. A wave boundary was specified at the upstream offshore end. The details of determining the random wave spectrum can see the following wave parameters section. The outflow boundary was set at the downstream onshore end. The symmetry boundary was used at the top and two sides of the model. The symmetric boundaries were the better strategy to improve the computation efficiency and save the calculation cost [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, ω_p 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 ≤ ω_p and 0.09 for ω > ω_p 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 U_r were acquired form Equations (28) and (29) respectively

ε=2πgHsT2a�=2��s�a2(28)

Ur=Hsk2h3w�r=�s�2ℎw3(29)

where, H_s is significant wave height, T_a is average wave period, k is wave number, h_w is water depth. The Shield parameter θ satisfies θ _> θ_cr for all simulations in current study, indicating the live bed scour prevails.

Table 2. Numerical simulating cases.

3.4. Mesh Sensitivity

In this section, a mesh sensitivity analysis was conducted to investigate the influence of mesh grid size to results and make sure the calculation is mesh size independent and converged. Three mesh grid size were chosen: Mesh 1—global mesh grid size of 0.75 × 0.75, nested fine mesh grid size of 0.4 × 0.4, and total number of grids 1,724,000, Mesh 2—global mesh grid size of 0.6 × 0.6, nested fine mesh grid size of 0.3 × 0.3, and total number of grids 1,812,000, Mesh 3—global mesh grid size of 0.4 × 0.4, nested fine mesh grid size of 0.2 × 0.2, and total number of grids 1,932,000. The near-bed shear velocity U_* is an important factor for influencing scour process [1,15], so U_* at the position of (4,0,11.12) was evaluated under three mesh sizes. As the Figure 4 shown, the maximum error of shear velocity ∆U_*1,2 is about 39.8% between the mesh 1 and mesh 2, and 4.8% between the mesh 2 and mesh 3. According to the mesh sensitivity criterion adopted by Pang et al. [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.

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.

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.

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.

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 T_c is time scale of scour process.

Figure 8. Time evolution of scour for case 1–9: (a) Case 1–5; (b) Case 6–9.

The computing time is 3600 s and the scour development curves in Figure 8 kept fluctuating, meaning it’s still not in equilibrium scour stage in these cases. According to Sumer and Fredsøe [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.

Figure 9. Scour morphology under different times for case 7.

From Figure 9, at the initial stage (t < 1200 s), the scour occurred at upstream foundation edges between neighboring anchor branches. The maximum scour depth appeared at the lee-side of the USAF. Correspondingly, the sediments deposited at the periphery of the USAF, and the location of the maximum accretion depth was positioned at an angle of about 45° symmetrically with respect to the wave propagating direction in the lee-side of the USAF. After that, when t > 2400 s, the location of the maximum scour depth shifted to the upside of the USAF at an angle of about 45° with respect to the wave propagating direction.

According to previous studies [1,15,16,19,30,31], the horseshoe vortex, streamline compression and wake vortex shedding were responsible for scour around foundation. The Figure 10 displays the distribution of flow velocity in vicinity of foundation, which reflects the evolving processes of horseshoe vertex.

Figure 10. Velocity profile around USAF: (a) Flow runup and down stream at upstream anchor edges; (b) Horseshoe vortex at upstream anchor edges; (c) Flow reversal during wave through stage at lee side.

As shown in Figure 10, the inflow tripped to the upstream edges of the USAF and it was blocked by the upper tube of USAF. Then, the downflow formed the horizontal axis clockwise vortex and rolled on the seabed bypassing the tube, that is, the horseshoe vortex (Figure 11). The Figure 12 displays the turbulence intensity around the tube on the seabed. From Figure 12, it can be seen that the turbulence intensity was high-intensity with respect to the region of horseshoe vortex. This phenomenon occurred because of drastic water flow momentum exchanging in the horseshoe vortex. As a result, it created the prominent shear stress on the seabed, causing the local scour at the upstream edges of USAF. Besides, the horseshoe vortex moved downstream gradually along the periphery of the tube and the wake vortex shed off continually at the lee-side of the USAF, i.e., wake vortex.

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

Figure 12. Turbulence intensity: (a) Turbulence intensity of horseshoe vortex; (b) Turbulence intensity of wake vortex; (c) Turbulence intensity of accretion area.

The core of wake vortex is a negative pressure center, liking a vacuum cleaner [11,42]. Hence, the soil particles were swirled into the negative pressure core and carried away by wake vortex. At the same time, the onset of scour at rear side occurred. Finally, the wake vortex became downflow at the downside of USAF. As is shown in Figure 12, the turbulence intensity was low where the downflow occurred at lee-side, which means the turbulence energy may not be able to support the survival of wake vortex, leading to accretion happening. As mentioned in previous section, the formation of horseshoe vortex was dependent with adverse pressure gradient at upside of foundation. As shown in Figure 13, the evaluated range of pressure distribution is −15 m to 15 m in x direction. The t = 450 s and t = 1800 s indicate that the wave crest and trough arrived at the upside and lee-side of the foundation respectively, and the t = 350 s was neither the wave crest nor trough. The adverse gradient pressure reached the maximum value at t = 450 s corresponding to the wave crest phase. In this case, it’s helpful for the wave boundary separating fully from seabed, which leads to the formation of horseshoe vortex with high turbulence intensity. Therefore, the horseshoe vortex is responsible for the local scour between neighboring anchor branches at upside of USAF. What’s more, due to the combination of the horseshoe vortex and streamline compression, the maximum scour depth occurred at the upside of the USAF with an angle of about 45° corresponding to the wave propagating direction. This is consistent with the findings of Pang et al. [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.

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 U_wm and wave period T. For random waves, the U_wm can be denoted by the root-mean-square (RMS) value of near-bed velocity amplitude U_wm,rms or the significant value of near-bed velocity amplitude U_wm,s. The U_wm,rms and U_wm,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 T_a, peak wave period T_p, significant wave period T_s, the maximum wave period T_m, 1/10′th highest wave period T_{n = 1/10} and 1/5′th highest wave period T_{n = 1/5} for random waves, so the different combinations of U_wm and T will acquire different KC. The Table 3 and Table 4 list 12 types of KC, for example, the KC_rms,s was calculated by U_wm,rms and T_s. 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 KC_rms,p. It should be noted that the Equation (2) is only suitable for KC > 6 under regular waves or KC_rms,p > 6 under random waves.

Table 3. U_wm,rms and KC for case 1~9.

Table 4. U_wm,s and KC for case 1~9.

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 S_eq between the present study and Raaijmakers’s equation was conducted. The position where the scour depth S_eq was evaluated is the location of the maximum scour depth, and it was depicted in Figure 14. The Figure 15 displays the comparison of S_eq with different KC between the present study and Raaijmakers’s model.

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

Figure 15. Comparison of the equilibrium scour depth between the present model and the model of Raaijmakers and Rudolph [34]: (a) KC_rms,s, KC_rms,a; (b) KC_rms,p, KC_rms,m; (c) KC_{rms,n = 1/10}, KC_{rms,n = 1/5}; (d) KC_s,s, KC_s,a; (e) KC_s,p, KC_s,m; (f) KC_{s,n = 1/10}, KC_{s,n = 1/5}.

As shown in Figure 15, there is an error in predicting S_eq between the present study and Raaijmakers’s model, and Raaijmakers’s model underestimates the results generally. Although the error exists, the varying trend of S_eq 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 KC_s,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 (KC_s,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 KC_s,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 KC_s,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 KC_s,p > 8.

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

Figure 17. Comparison of S_eq/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 KC_rms,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 S_eq with KC_rms,a obtained from the stochastic model is consistent with the present study basically. What’s more, the gap between the predicting values by stochastic model and the simulating results decreases with the increase of n, but for large n, for example n = 500, the varying trend diverges between the predicting values and simulating results, meaning it’s not feasible only by increasing n in stochastic model to predict the equilibrium scour depth around USAF.

Figure 18. Comparison of S_eq between the simulating results and the predicting values by Equation (8).

The Figure 19 lists the deviation value ∆S_eq/D′ between the predicting values and simulating results with different KC_rms,a and n. Then, fitted the relationship between the ∆S′and n under different KC_rms,a, and the fitting curve can be written by Equation (32). The revised stochastic model (Equation (33)) can be acquired by adding ∆S_eq/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)

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 S_eq with KC_rms,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.

Figure 20. Comparison of S_eq 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 (KC_rms,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 KC_rms,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 KC_rms,a (KC_rms,a < 4), the error between the predicting values and experimental results increases with decreasing of n for KC_rms,a > 4. Therefore, the predicting results are the most favorable for n = 2 when KC_rms,a > 4.

Figure 21. Comparison of S_eq 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 F_r is the key parameter to influence the scale and intensity of horseshoe vortex. The F_r under waves can be calculated by the following formula [42]

Fr=UwgD−−−√�r=�w��(34)

where U_w 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 U_wm,rms is used for calculating U_wm.

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

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

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

yh∝Fer�ℎ∝�r�(36)

where e is constant.

The Figure 23 displays the relationship between S_eq/D and F_r 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 F_r 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 F_r, 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 F_r 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 F_r 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 S_eq/D and F_r as following

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

where A, B and C are constant.

Figure 23. The fitting curve between S_eq/D and F_r.

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 S_eq/D and F_r in present study is consistent with Equation (37) basically, meaning the Equation (37) is applicable to express the relationship of S_eq/D with F_r around USAF under random waves.

4.4.2. Influence of Euler Number

The Euler number E_u is the influencing factor for the hydrodynamic field around foundation. The E_u under waves can be calculated by the following formula. The E_u can be represented by the Equation (38) for uniform cylinders [25]. The root-mean-square (RMS) value of near-bed velocity amplitude U_m,rms is used for calculating U_m.

Eu=U2mgD�u=�m2��(38)

where U_m is depth-averaged flow velocity.

The Figure 25 displays the relationship between S_eq/D and E_u 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 S_eq/D and F_r, the equilibrium scour depth appears a logarithmic increase as E_u 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 E_u, which is benefit for the wave boundary layer separating from seabed, inducing the high-intensity horseshoe vortex, hence, causing intensive scour around USAF.

Figure 25. The fitting curve between S_eq/D and E_u.

Therefore, the variation of F_r and E_u 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 S_eq/D and E_u in present study is consistent with Equation (37) basically, meaning the Equation (37) is also applicable to express the relationship of S_eq/D with E_u around USAF under random waves. Additionally, according to the above description of F_r, it can be inferred that the higher F_r and E_u 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 F_r and Euler number E_u 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 KC_s,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 KC_s,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 KC_rms,a < 4. However, contrary to the case of low KC_rms,a, the predicting results are the most favorable for n = 2 when KC_rms,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 S_eq/D with E_u or F_r, and it can be inferred that the higher F_r and E_u both lead to the more intensive horseshoe vortex and larger S_eq.

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. 2021, 9, 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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Study on Hydrodynamic Performance of Unsymmetrical Double Vertical Slotted Barriers

침수된 강성 식생을 갖는 개방 수로 흐름의 특성에 대한 3차원 수치 시뮬레이션

A 3-D numerical simulation of the characteristics of open channel flows with submerged rigid vegetation

Journal of Hydrodynamics volume 33, pages833–843 (2021)Cite this article

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Abstract

이 백서는 Flow-3D를 적용하여 다양한 흐름 배출 및 식생 시나리오가 흐름 속도(세로, 가로 및 수직 속도 포함)에 미치는 영향을 조사합니다.

실험적 측정을 통한 검증 후 식생직경, 식생높이, 유량방류량에 대한 민감도 분석을 수행하였다. 종방향 속도의 경우 흐름 구조에 가장 큰 영향을 미치는 것은 배출보다는 식생 직경에서 비롯됩니다.

그러나 식생 높이는 수직 분포의 변곡점을 결정합니다. 식생지 내 두 지점, 즉 상류와 하류의 횡속도를 비교하면 수심에 따른 대칭적인 패턴을 확인할 수 있다. 식생 지역의 가로 및 세로 유체 순환 패턴을 포함하여 흐름 또는 식생 시나리오와 관계없이 수직 속도에 대해서도 동일한 패턴이 관찰됩니다.

또한 식생의 직경이 클수록 이러한 패턴이 더 분명해집니다. 상부 순환은 초목 캐노피 근처에서 발생합니다. 식생지역의 가로방향과 세로방향의 순환에 관한 이러한 발견은 침수식생을 통한 3차원 유동구조를 밝혀준다.

This paper applies the Flow-3D to investigate the impacts of different flow discharge and vegetation scenarios on the flow velocity (including the longitudinal, transverse and vertical velocities). After the verification by using experimental measurements, a sensitivity analysis is conducted for the vegetation diameter, the vegetation height and the flow discharge. For the longitudinal velocity, the greatest impact on the flow structure originates from the vegetation diameter, rather than the discharge. The vegetation height, however, determines the inflection point of the vertical distribution. Comparing the transverse velocities at two positions in the vegetated area, i.e., the upstream and the downstream, a symmetric pattern is identified along the water depth. The same pattern is also observed for the vertical velocity regardless of the flow or vegetation scenario, including both transverse and vertical fluid circulation patterns in the vegetated area. Moreover, the larger the vegetation diameter is, the more evident these patterns become. The upper circulation occurs near the vegetation canopy. These findings regarding the circulations along the transverse and vertical directions in the vegetated region shed light on the 3-D flow structure through the submerged vegetation.

Key words

Submerged rigid vegetation
longitudinal velocity
transverse velocity
vertical velocity
open channel

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원문보기

Figure 2 Modeling the plant with cylindrical tubes at the bottom of the canal.

Optimized Vegetation Density to Dissipate Energy of Flood Flow in Open Canals

열린 운하에서 홍수 흐름의 에너지를 분산시키기 위해 최적화된 식생 밀도

Mahdi Feizbahr,¹Navid Tonekaboni,²Guang-Jun Jiang,^3,4and Hong-Xia Chen^3,4
Academic Editor: Mohammad Yazdi

Abstract

강을 따라 식생은 조도를 증가시키고 평균 유속을 감소시키며, 유동 에너지를 감소시키고 강 횡단면의 유속 프로파일을 변경합니다. 자연의 많은 운하와 강은 홍수 동안 초목으로 덮여 있습니다. 운하의 조도는 식물의 영향을 많이 받기 때문에 홍수시 유동저항에 큰 영향을 미친다. 식물로 인한 흐름에 대한 거칠기 저항은 흐름 조건과 식물에 따라 달라지므로 모델은 유속, 유속 깊이 및 수로를 따라 식생 유형의 영향을 고려하여 유속을 시뮬레이션해야 합니다. 총 48개의 모델을 시뮬레이션하여 근관의 거칠기 효과를 조사했습니다. 결과는 속도를 높임으로써 베드 속도를 감소시키는 식생의 영향이 무시할만하다는 것을 나타냅니다.

Abstract

Vegetation along the river increases the roughness and reduces the average flow velocity, reduces flow energy, and changes the flow velocity profile in the cross section of the river. Many canals and rivers in nature are covered with vegetation during the floods. Canal’s roughness is strongly affected by plants and therefore it has a great effect on flow resistance during flood. Roughness resistance against the flow due to the plants depends on the flow conditions and plant, so the model should simulate the current velocity by considering the effects of velocity, depth of flow, and type of vegetation along the canal. Total of 48 models have been simulated to investigate the effect of roughness in the canal. The results indicated that, by enhancing the velocity, the effect of vegetation in decreasing the bed velocity is negligible, while when the current has lower speed, the effect of vegetation on decreasing the bed velocity is obviously considerable.

1. Introduction

Considering the impact of each variable is a very popular field within the analytical and statistical methods and intelligent systems [1–14]. This can help research for better modeling considering the relation of variables or interaction of them toward reaching a better condition for the objective function in control and engineering [15–27]. Consequently, it is necessary to study the effects of the passive factors on the active domain [28–36]. Because of the effect of vegetation on reducing the discharge capacity of rivers [37], pruning plants was necessary to improve the condition of rivers. One of the important effects of vegetation in river protection is the action of roots, which cause soil consolidation and soil structure improvement and, by enhancing the shear strength of soil, increase the resistance of canal walls against the erosive force of water. The outer limbs of the plant increase the roughness of the canal walls and reduce the flow velocity and deplete the flow energy in vicinity of the walls. Vegetation by reducing the shear stress of the canal bed reduces flood discharge and sedimentation in the intervals between vegetation and increases the stability of the walls [38–41].

One of the main factors influencing the speed, depth, and extent of flood in this method is Manning’s roughness coefficient. On the other hand, soil cover [42], especially vegetation, is one of the most determining factors in Manning’s roughness coefficient. Therefore, it is expected that those seasonal changes in the vegetation of the region will play an important role in the calculated value of Manning’s roughness coefficient and ultimately in predicting the flood wave behavior [43–45]. The roughness caused by plants’ resistance to flood current depends on the flow and plant conditions. Flow conditions include depth and velocity of the plant, and plant conditions include plant type, hardness or flexibility, dimensions, density, and shape of the plant [46]. In general, the issue discussed in this research is the optimization of flood-induced flow in canals by considering the effect of vegetation-induced roughness. Therefore, the effect of plants on the roughness coefficient and canal transmission coefficient and in consequence the flow depth should be evaluated [47, 48].

Current resistance is generally known by its roughness coefficient. The equation that is mainly used in this field is Manning equation. The ratio of shear velocity to average current velocity is another form of current resistance. The reason for using the ratio is that it is dimensionless and has a strong theoretical basis. The reason for using Manning roughness coefficient is its pervasiveness. According to Freeman et al. [49], the Manning roughness coefficient for plants was calculated according to the Kouwen and Unny [50] method for incremental resistance. This method involves increasing the roughness for various surface and plant irregularities. Manning’s roughness coefficient has all the factors affecting the resistance of the canal. Therefore, the appropriate way to more accurately estimate this coefficient is to know the factors affecting this coefficient [51].

To calculate the flow rate, velocity, and depth of flow in canals as well as flood and sediment estimation, it is important to evaluate the flow resistance. To determine the flow resistance in open ducts, Manning, Chézy, and Darcy–Weisbach relations are used [52]. In these relations, there are parameters such as Manning’s roughness coefficient (n), Chézy roughness coefficient (C), and Darcy–Weisbach coefficient (f). All three of these coefficients are a kind of flow resistance coefficient that is widely used in the equations governing flow in rivers [53].

The three relations that express the relationship between the average flow velocity (V) and the resistance and geometric and hydraulic coefficients of the canal are as follows:where n, f, and c are Manning, Darcy–Weisbach, and Chézy coefficients, respectively. V = average flow velocity, R = hydraulic radius, S_f = slope of energy line, which in uniform flow is equal to the slope of the canal bed, = gravitational acceleration, and K_n is a coefficient whose value is equal to 1 in the SI system and 1.486 in the English system. The coefficients of resistance in equations (1) to (3) are related as follows:

Based on the boundary layer theory, the flow resistance for rough substrates is determined from the following general relation:where f = Darcy–Weisbach coefficient of friction, y = flow depth, K_s = bed roughness size, and A = constant coefficient.

On the other hand, the relationship between the Darcy–Weisbach coefficient of friction and the shear velocity of the flow is as follows:

By using equation (6), equation (5) is converted as follows:

Investigation on the effect of vegetation arrangement on shear velocity of flow in laboratory conditions showed that, with increasing the shear Reynolds number (), the numerical value of the ratio also increases; in other words the amount of roughness coefficient increases with a slight difference in the cases without vegetation, checkered arrangement, and cross arrangement, respectively [54].

Roughness in river vegetation is simulated in mathematical models with a variable floor slope flume by different densities and discharges. The vegetation considered submerged in the bed of the flume. Results showed that, with increasing vegetation density, canal roughness and flow shear speed increase and with increasing flow rate and depth, Manning’s roughness coefficient decreases. Factors affecting the roughness caused by vegetation include the effect of plant density and arrangement on flow resistance, the effect of flow velocity on flow resistance, and the effect of depth [45, 55].

One of the works that has been done on the effect of vegetation on the roughness coefficient is Darby [56] study, which investigates a flood wave model that considers all the effects of vegetation on the roughness coefficient. There are currently two methods for estimating vegetation roughness. One method is to add the thrust force effect to Manning’s equation [47, 57, 58] and the other method is to increase the canal bed roughness (Manning-Strickler coefficient) [45, 59–61]. These two methods provide acceptable results in models designed to simulate floodplain flow. Wang et al. [62] simulate the floodplain with submerged vegetation using these two methods and to increase the accuracy of the results, they suggested using the effective height of the plant under running water instead of using the actual height of the plant. Freeman et al. [49] provided equations for determining the coefficient of vegetation roughness under different conditions. Lee et al. [63] proposed a method for calculating the Manning coefficient using the flow velocity ratio at different depths. Much research has been done on the Manning roughness coefficient in rivers, and researchers [49, 63–66] sought to obtain a specific number for n to use in river engineering. However, since the depth and geometric conditions of rivers are completely variable in different places, the values of Manning roughness coefficient have changed subsequently, and it has not been possible to choose a fixed number. In river engineering software, the Manning roughness coefficient is determined only for specific and constant conditions or normal flow. Lee et al. [63] stated that seasonal conditions, density, and type of vegetation should also be considered. Hydraulic roughness and Manning roughness coefficient n of the plant were obtained by estimating the total Manning roughness coefficient from the matching of the measured water surface curve and water surface height. The following equation is used for the flow surface curve:where is the depth of water change, S₀ is the slope of the canal floor, Sf is the slope of the energy line, and Fr is the Froude number which is obtained from the following equation:where D is the characteristic length of the canal. Flood flow velocity is one of the important parameters of flood waves, which is very important in calculating the water level profile and energy consumption. In the cases where there are many limitations for researchers due to the wide range of experimental dimensions and the variety of design parameters, the use of numerical methods that are able to estimate the rest of the unknown results with acceptable accuracy is economically justified.

FLOW-3D software uses Finite Difference Method (FDM) for numerical solution of two-dimensional and three-dimensional flow. This software is dedicated to computational fluid dynamics (CFD) and is provided by Flow Science [67]. The flow is divided into networks with tubular cells. For each cell there are values of dependent variables and all variables are calculated in the center of the cell, except for the velocity, which is calculated at the center of the cell. In this software, two numerical techniques have been used for geometric simulation, FAVOR™ (Fractional-Area-Volume-Obstacle-Representation) and the VOF (Volume-of-Fluid) method. The equations used at this model for this research include the principle of mass survival and the magnitude of motion as follows. The fluid motion equations in three dimensions, including the Navier–Stokes equations with some additional terms, are as follows:where are mass accelerations in the directions x, y, z and are viscosity accelerations in the directions x, y, z and are obtained from the following equations:

Shear stresses in equation (11) are obtained from the following equations:

The standard model is used for high Reynolds currents, but in this model, RNG theory allows the analytical differential formula to be used for the effective viscosity that occurs at low Reynolds numbers. Therefore, the RNG model can be used for low and high Reynolds currents.

Weather changes are high and this affects many factors continuously. The presence of vegetation in any area reduces the velocity of surface flows and prevents soil erosion, so vegetation will have a significant impact on reducing destructive floods. One of the methods of erosion protection in floodplain watersheds is the use of biological methods. The presence of vegetation in watersheds reduces the flow rate during floods and prevents soil erosion. The external organs of plants increase the roughness and decrease the velocity of water flow and thus reduce its shear stress energy. One of the important factors with which the hydraulic resistance of plants is expressed is the roughness coefficient. Measuring the roughness coefficient of plants and investigating their effect on reducing velocity and shear stress of flow is of special importance.

Roughness coefficients in canals are affected by two main factors, namely, flow conditions and vegetation characteristics [68]. So far, much research has been done on the effect of the roughness factor created by vegetation, but the issue of plant density has received less attention. For this purpose, this study was conducted to investigate the effect of vegetation density on flow velocity changes.

In a study conducted using a software model on three density modes in the submerged state effect on flow velocity changes in 48 different modes was investigated (Table 1).

Table 1

The studied models.

The number of cells used in this simulation is equal to 1955888 cells. The boundary conditions were introduced to the model as a constant speed and depth (Figure 1). At the output boundary, due to the presence of supercritical current, no parameter for the current is considered. Absolute roughness for floors and walls was introduced to the model (Figure 1). In this case, the flow was assumed to be nonviscous and air entry into the flow was not considered. After seconds, this model reached a convergence accuracy of .

Figure 1

The simulated model and its boundary conditions.

Due to the fact that it is not possible to model the vegetation in FLOW-3D software, in this research, the vegetation of small soft plants was studied so that Manning’s coefficients can be entered into the canal bed in the form of roughness coefficients obtained from the studies of Chow [69] in similar conditions. In practice, in such modeling, the effect of plant height is eliminated due to the small height of herbaceous plants, and modeling can provide relatively acceptable results in these conditions.

48 models with input velocities proportional to the height of the regular semihexagonal canal were considered to create supercritical conditions. Manning coefficients were applied based on Chow [69] studies in order to control the canal bed. Speed profiles were drawn and discussed.

Any control and simulation system has some inputs that we should determine to test any technology [70–77]. Determination and true implementation of such parameters is one of the key steps of any simulation [23, 78–81] and computing procedure [82–86]. The input current is created by applying the flow rate through the VFR (Volume Flow Rate) option and the output flow is considered Output and for other borders the Symmetry option is considered.

Simulation of the models and checking their action and responses and observing how a process behaves is one of the accepted methods in engineering and science [87, 88]. For verification of FLOW-3D software, the results of computer simulations are compared with laboratory measurements and according to the values of computational error, convergence error, and the time required for convergence, the most appropriate option for real-time simulation is selected (Figures 2 and 3 ).

Figure 2

Modeling the plant with cylindrical tubes at the bottom of the canal.

Figure 3

Velocity profiles in positions 2 and 5.

The canal is 7 meters long, 0.5 meters wide, and 0.8 meters deep. This test was used to validate the application of the software to predict the flow rate parameters. In this experiment, instead of using the plant, cylindrical pipes were used in the bottom of the canal.

The conditions of this modeling are similar to the laboratory conditions and the boundary conditions used in the laboratory were used for numerical modeling. The critical flow enters the simulation model from the upstream boundary, so in the upstream boundary conditions, critical velocity and depth are considered. The flow at the downstream boundary is supercritical, so no parameters are applied to the downstream boundary.

The software well predicts the process of changing the speed profile in the open canal along with the considered obstacles. The error in the calculated speed values can be due to the complexity of the flow and the interaction of the turbulence caused by the roughness of the floor with the turbulence caused by the three-dimensional cycles in the hydraulic jump. As a result, the software is able to predict the speed distribution in open canals.

2. Modeling Results

After analyzing the models, the results were shown in graphs (Figures 4–14 ). The total number of experiments in this study was 48 due to the limitations of modeling.

(a)
(b)
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(d)

Figure 4

Flow velocity profiles for canals with a depth of 1 m and flow velocities of 3–3.3 m/s. Canal with a depth of 1 meter and a flow velocity of (a) 3 meters per second, (b) 3.1 meters per second, (c) 3.2 meters per second, and (d) 3.3 meters per second.

Figure 5

Canal diagram with a depth of 1 meter and a flow rate of 3 meters per second.

Figure 6

Canal diagram with a depth of 1 meter and a flow rate of 3.1 meters per second.

Figure 7

Canal diagram with a depth of 1 meter and a flow rate of 3.2 meters per second.

Figure 8

Canal diagram with a depth of 1 meter and a flow rate of 3.3 meters per second.

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(b)
(c)
(d)

Figure 9

Flow velocity profiles for canals with a depth of 2 m and flow velocities of 4–4.3 m/s. Canal with a depth of 2 meters and a flow rate of (a) 4 meters per second, (b) 4.1 meters per second, (c) 4.2 meters per second, and (d) 4.3 meters per second.

Figure 10

Canal diagram with a depth of 2 meters and a flow rate of 4 meters per second.

Figure 11

Canal diagram with a depth of 2 meters and a flow rate of 4.1 meters per second.

Figure 12

Canal diagram with a depth of 2 meters and a flow rate of 4.2 meters per second.

Figure 13

Canal diagram with a depth of 2 meters and a flow rate of 4.3 meters per second.

(a)
(b)
(c)
(d)

Figure 14

Flow velocity profiles for canals with a depth of 3 m and flow velocities of 5–5.3 m/s. Canal with a depth of 2 meters and a flow rate of (a) 4 meters per second, (b) 4.1 meters per second, (c) 4.2 meters per second, and (d) 4.3 meters per second.

To investigate the effects of roughness with flow velocity, the trend of flow velocity changes at different depths and with supercritical flow to a Froude number proportional to the depth of the section has been obtained.

According to the velocity profiles of Figure 5, it can be seen that, with the increasing of Manning’s coefficient, the canal bed speed decreases.

According to Figures 5 to 8, it can be found that, with increasing the Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the models 1 to 12, which can be justified by increasing the speed and of course increasing the Froude number.

According to Figure 10, we see that, with increasing Manning’s coefficient, the canal bed speed decreases.

According to Figure 11, we see that, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of Figures 5–10, which can be justified by increasing the speed and, of course, increasing the Froude number.

With increasing Manning’s coefficient, the canal bed speed decreases (Figure 12). But this deceleration is more noticeable than the deceleration of the higher models (Figures 5–8 and 10, 11), which can be justified by increasing the speed and, of course, increasing the Froude number.

According to Figure 13, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of Figures 5 to 12, which can be justified by increasing the speed and, of course, increasing the Froude number.

According to Figure 15, with increasing Manning’s coefficient, the canal bed speed decreases.

Figure 15

Canal diagram with a depth of 3 meters and a flow rate of 5 meters per second.

According to Figure 16, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the higher model, which can be justified by increasing the speed and, of course, increasing the Froude number.

Figure 16

Canal diagram with a depth of 3 meters and a flow rate of 5.1 meters per second.

According to Figure 17, it is clear that, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the higher models, which can be justified by increasing the speed and, of course, increasing the Froude number.

Figure 17

Canal diagram with a depth of 3 meters and a flow rate of 5.2 meters per second.

According to Figure 18, with increasing Manning’s coefficient, the canal bed speed decreases. But this deceleration is more noticeable than the deceleration of the higher models, which can be justified by increasing the speed and, of course, increasing the Froude number.

Figure 18

Canal diagram with a depth of 3 meters and a flow rate of 5.3 meters per second.

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(b)
(c)

Figure 19

Comparison of velocity profiles with the same plant densities (depth 1 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 1 m; (b) plant densities of 50%, depth 1 m; and (c) plant densities of 75%, depth 1 m.

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(b)
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Figure 20

Comparison of velocity profiles with the same plant densities (depth 2 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 2 m; (b) plant densities of 50%, depth 2 m; and (c) plant densities of 75%, depth 2 m.

According to Figure 21, it can be seen that, with increasing speed, the effect of vegetation on reducing the bed flow rate becomes more noticeable and the role of the input current does not have much effect. In general, it can be seen that, by increasing the speed of the input current, the slope of the profiles increases from the bed to the water surface and due to the fact that, in software, the roughness coefficient applies to the channel floor only in the boundary conditions, this can be perfectly justified. Of course, it can be noted that, due to the flexible conditions of the vegetation of the bed, this modeling can show acceptable results for such grasses in the canal floor. In the next directions, we may try application of swarm-based optimization methods for modeling and finding the most effective factors in this research [2, 7, 8, 15, 18, 89–94]. In future, we can also apply the simulation logic and software of this research for other domains such as power engineering [95–99].

(a)
(b)
(c)

Figure 21

Comparison of velocity profiles with the same plant densities (depth 3 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 3 m; (b) plant densities of 50%, depth 3 m; and (c) plant densities of 75%, depth 3 m.

3. Conclusion

The effects of vegetation on the flood canal were investigated by numerical modeling with FLOW-3D software. After analyzing the results, the following conclusions were reached:(i)Increasing the density of vegetation reduces the velocity of the canal floor but has no effect on the velocity of the canal surface.(ii)Increasing the Froude number is directly related to increasing the speed of the canal floor.(iii)In the canal with a depth of one meter, a sudden increase in speed can be observed from the lowest speed and higher speed, which is justified by the sudden increase in Froude number.(iv)As the inlet flow rate increases, the slope of the profiles from the bed to the water surface increases.(v)By reducing the Froude number, the effect of vegetation on reducing the flow bed rate becomes more noticeable. And the input velocity in reducing the velocity of the canal floor does not have much effect.(vi)At a flow rate between 3 and 3.3 meters per second due to the shallow depth of the canal and the higher landing number a more critical area is observed in which the flow bed velocity in this area is between 2.86 and 3.1 m/s.(vii)Due to the critical flow velocity and the slight effect of the roughness of the horseshoe vortex floor, it is not visible and is only partially observed in models 1-2-3 and 21.(viii)As the flow rate increases, the effect of vegetation on the rate of bed reduction decreases.(ix)In conditions where less current intensity is passing, vegetation has a greater effect on reducing current intensity and energy consumption increases.(x)In the case of using the flow rate of 0.8 cubic meters per second, the velocity distribution and flow regime show about 20% more energy consumption than in the case of using the flow rate of 1.3 cubic meters per second.

Nomenclature

n:	Manning’s roughness coefficient
C:	Chézy roughness coefficient
f:	Darcy–Weisbach coefficient
V:	Flow velocity
R:	Hydraulic radius
g:	Gravitational acceleration
y:	Flow depth
Ks:	Bed roughness
A:	Constant coefficient
:	Reynolds number
∂y/∂x:	Depth of water change
S₀:	Slope of the canal floor
Sf:	Slope of energy line
Fr:	Froude number
D:	Characteristic length of the canal
G:	Mass acceleration
:	Shear stresses.

Data Availability

All data are included within the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under Contract no. 71761030 and Natural Science Foundation of Inner Mongolia under Contract no. 2019LH07003.

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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

¹. ^Introduction

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.

^2.1. ^{Model Setup}

FLOW-3D is a powerful and comprehensive CFD simulation platform for studying fluid dynamics. This software has several modules to solve many complex engineering problems. In addition, modeling complex flows is simple and effective using FLOW-3D’s robust meshing capabilities [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.

^2.2. ^Verification

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	��	��	��
Quantity	0.012	4.38	1.42	1.68	1.39	1.39	0.084

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.

³. ^{Sensitivity Analysis}

Based on previous studies, in addition to the flap design, the location of the flap relative to the water surface (freeboard) and its elevation relative to the flume bed (flap bottom elevation) play a significant role in extracting energy from the wave energy converter. This study measures the sensitivity of the model to various parameters related to the flap design including upper part width of the flap, lower part width of the flap, the freeboard, and the flap bottom elevation. Moreover, as a novel idea, we propose that the flap widths differ in the lower and upper parts. In Figure10, as an example, a flap with an upper thickness of 100 (mm) and a lower thickness of 50 (mm) and a flap with an upper thickness of 50 (mm) and a lower thickness of 100 (mm) are shown. The influence of such discrepancy between the widths of the upper and lower parts on the interaction between the wave and the flap, or in other words on the capture factor, is evaluated. To do so, other parameters are remained constant, such as the freeboard, the distance between the flap and the flume bed, and the wave properties.

In Figure11, models are simulated with distinct upper and lower widths. As it is clear in this figure, the first axis depicts the lower part width of the flap, the second axis indicates the upper part width of the flap, and the colors represent the capture factor values. Additionally, in order to consider a sufficient range of change, the flap thickness varies from half to double the value of the primary model for each part.

According to this study, the greater the discrepancy in these two parts, the lower the capture factor. It is on account of the fact that when the lower part of the flap is thicker than the upper part, and this thickness difference in these two parts is extremely conspicuous, the inertia against the motion is significant at zero degrees of rotation. Consequently, it is difficult to move the flap, which results in a low capture factor. Similarly, when the upper part of the flap is thicker than the lower part, and this thickness difference in these two parts is exceedingly noticeable, the inertia is so great that the flap can not reverse at the maximum degree of rotation. As the results indicate, the discrepancy can enhance the performance of the converter if the difference between these two parts is around 20%. As it is depicted in the Figure11, the capture factor reaches its own maximum amount, when the lower part thickness is from 5 to 6 (cm), and the upper part thickness is between 6 and 7 (cm). Consequently, as a result of this discrepancy, less material will be used, and therefore there will be less cost.

As illustrated in Figure12, this study examines the effects of freeboard (level difference between the flap top and water surface) and the flap bottom elevation (the distance between the flume bed and flap bottom) on the converter performance. In this diagram, the first axis demonstrates the freeboard and the second axis on the left side displays the flap bottom elevation, while the colors indicate the capture factor. In addition, the feasible range of freeboard is between -15 to 15 (cm) due to the limitation of the numerical model, so that we can take the wave slamming and the overtopping into consideration. Additionally, based on the Schmitt model and its scaled model of 1:40 of the base height, the flap bottom should be at least 9 (cm) high. Since the effect of surface waves is distributed over the depth of the flume, it is imperative to maintain a reasonable flap height exposed to incoming waves. Thus, the maximum flap bottom elevation is limited to 19 (cm). As the Figure12 pictures, at constant negative values of the freeboard, the capture factor is in inverse proportion with the flap bottom elevation, although slightly.

Furthermore, at constant positive values of the freeboard, the capture factor fluctuates as the flap bottom elevation decreases while it maintains an overall increasing trend. This is on account of the fact that increasing the flap bottom elevation creates turbulence flow behind the flap, which encumbers its rotation, as well as the fact that the flap surface has less interaction with the incoming waves. Furthermore, while keeping the flap bottom elevation constant, the capture factor increases by raising the freeboard. This is due to the fact that there is overtopping with adverse impacts on the converter performance when the freeboard is negative and the flap is under the water surface. Besides, increasing the freeboard makes the wave slam more vigorously, which improves the converter performance.

Adding ribs to the flap surface, as shown in Figure13, is a novel idea that is investigated in the next section. To achieve an optimized design for the proposed geometry of the flap, we determine the optimal number and dimensions of ribs based on the flap properties as our decision variables in the optimization process. As an example, Figure13 illustrates a flap with 3 ribs on each side with specific dimensions.

Figure14 shows the flow velocity field around the flap jointed to the flume bed. During the oscillation of the flap, the pressure on the upper and lower surfaces of the flap changes dynamically due to the changing angle of attack and the resulting change in the direction of fluid flow. As the flap moves upwards, the pressure on the upper surface decreases, and the pressure on the lower surface increases. Conversely, as the flap moves downwards, the pressure on the upper surface increases, and the pressure on the lower surface decreases. This results in a cyclic pressure variation around the flap. Under certain conditions, the pressure field around the flap can exhibit significant variations in magnitude and direction, forming vortices and other flow structures. These flow structures can affect the performance of the OSWEC by altering the lift and drag forces acting on the flap.

⁴. ^{Design Optimization}

We consider optimizing the design parameters of the flap of converter using a nature-based swarm optimization method, that fall in the category of metaheuristic algorithms [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 x_k 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 Cell	Algorithm 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 Cell	Algorithm 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.

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원문보기

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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 Hu², Hongliang Wang⁴, Hui Wang³ and Yong Zhu⁵

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.Sustainability2023, 15(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:

PIV; Flow-3D; impinging jet; hydraulic characteristics; impinging distance

원문 보기

Figure 1. Geometric model.

Figure 2. Model grid schematic.

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

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].

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

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

Figure 7. The relationship between the potential core length

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

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

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

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

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

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

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

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 2023, 15, 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

Flow Hydrodynamics Influences Due to Flood Plain Sand Mining in a Meandering Channel

Abstract

Flow hydrodynamics in the main channel due to floodplain sand mining is important for a better understanding of maintaining the natural habitat or the reliance between the flood plain and the main channel for the river’s long-term survival and also facilitates more effective river restoration engineering. Day by day anthropogenic stresses are increasing in the river corridor system, indiscriminate sand mining is one of them. In this study, a computational fluid dynamics (CFD)-based software Flow-3D hydro (renormalized group K-ε turbulence model used) is used to study the flow hydrodynamics of sinuous (sinuosity index = 1.25) channel 18 m long, 1 m width, and 0.3 m height with floodplain sand mining pit. Sand mining additionally increases the secondary current near the outer bank of the channel, therefore leading to scouring or erosion at the outer bank, as a result, rivers migrate laterally. The turbulence kinetic energy (TKE) is concentrated in the mining pit and near the inner bank. This study result can be used to understand the flow hydrodynamic of the river system due to the series of sand mining.

Keywords

Flow hydrodynamics
Turbulence modeling
Flow-3D
Sinuosity
Sand mining

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Flow Hydrodynamics Influences Due to Flood Plain Sand Mining in a Meandering Channel

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First Online: 15 March 2023

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Abstract

Keywords

Flow hydrodynamics
Turbulence modeling
Flow-3D
Sinuosity
Sand mining

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References

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Khan, S., Sugie, A.: Sand mining and its social impacts on local society in rural Bangladesh: a case study of a village in Tangail district. J. Urban Reg. Stud. Contemp. India 2(1), 1–11 (2015)Google Scholar
Daneshfaraz, R. et al.: The experimental study of the effects of river mining holes on the bridge piers. Iranian J. Soil Water Res. 50(7), 1619–1633 (2019)Google Scholar
Hackney, C. R., Darby, S. E., Parsons, D. R., Leyland, J., Best, J. L., Aalto, R., … & Houseago, R. C.: River bank instability from unsustainable sand mining in the lower Mekong River. Nat. Sustain. 3(3), 217–225 (2020)Google Scholar
Callander, R.A.: River meandering. Annu. Rev. Fluid Mech. 10(1), 129–158 (1978)CrossRef Google Scholar
Koehnken, L., Rintoul, M.: Impacts of sand mining on ecosystem structure, process and biodiversity in rivers. World Wildlife Fund International (2018)Google Scholar
Gavriletea, M.D.: Environmental impacts of sand exploitation. Analysis of sand market. Sustainability 9(7), 1118 (2017)Google Scholar
Koehnken, L., et al.: Impacts of riverine sand mining on freshwater ecosystems: a review of the scientific evidence and guidance for future research. River Res. Appl. 36(3), 362–370 (2020)Google Scholar
Myers, W.R.C.: Momentum transfer in a compound channel. J. Hydraul. Res. 16(2), 139–150 (1978)CrossRef Google Scholar
Rajaratnam, N., Ahmadi, R.M.: Interaction between main channel and flood-plain flows. J. Hydraul. Div. 105(5), 573–588 (1979)CrossRef Google Scholar
Sellin, R.H.J.: A laboratory investigation into the interaction between the flow in the channel of a river and that over its flood plain. La Houille Blanche 7, 793–802 (1964)CrossRef Google Scholar
Karami, H., et al.: Verification of numerical study of scour around spur dikes using experimental data. Water Environ. J. 28(1), 124–134 (2014)Google Scholar
Bathurst, J.C., et al.: Overbank sediment deposition patterns for straight and meandering flume channels. Earth Surf. Proc. Land. 27(6), 659–665 (2002)CrossRef Google Scholar
Xu, D., Bai, Y.: Experimental study on the bed topography evolution in alluvial meandering rivers with various sinuousnesses. J. Hydro-Environ. Res. 7(2), 92–102 (2013)CrossRef Google Scholar
Priego-Hernández, G.A., Rivera-Trejo, F.: Secondary currents: measurement and analysis. Atmósfera 29(1), 23–34 (2016)Google Scholar
Alshamani, K.M.M.: Correlations among turbulent shear stress, turbulent kinetic energy, and axial turbulence intensity. AIAA J. 16(8), 859–861 (1978)CrossRef Google Scholar
Biron, P.M., et al.: Comparing different methods of bed shear stress estimates in simple and complex flow fields. Earth Surface Process. Landforms: J. British Geomorphol. Res. Group 29(11), 1403–1415 (2004)Google Scholar
Clark, L.A., Theresa, M.W.: Boundary Shear Stress Along Vegetated Streambanks (2007)Google Scholar
Kim, S.-C., et al.: Estimating bottom stress in tidal boundary layer from acoustic Doppler velocimeter data. J. Hydraul. Eng. 126(6), 399–406 (2000)CrossRef Google Scholar

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Authors and Affiliations

Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, IndiaO. P. Maurya, K. K. Nandi, S. Modalavalasa & S. Dutta

Corresponding author

Correspondence to O. P. Maurya .

Editor information

Editors and Affiliations

Centre for the Environment, Indian Institute of Technology Guwahati, Guwahati, IndiaDeepmoni Deka
Department of Chemical engineering, Indian Institute of Technology Guwahati, Guwahati, IndiaSubrata Kumar Majumder
Department of Chemical engineering, Indian Institute of Technology Guwahati, Guwahati, IndiaMihir Kumar Purkait

Figure 4. Field gate discharge experiment.

FLOW-3D Model Development for the Analysis of the Flow Characteristics of Downstream Hydraulic Structures

하류 유압 구조물의 유동 특성 분석을 위한 FLOW-3D 모델 개발

Beom-Jin Kim ¹, Jae-Hong Hwang ² and Byunghyun Kim ³,*
¹ Advanced Structures and Seismic Safety Research Division, Korea Atomic Energy Research Institute,
Daejeon 34057, Korea
² Korea Water Resources Corporation (K-Water), Daejeon 34350, Korea
³Department of Civil Engineering, Kyungpook National University, Daegu 41566, Korea

Correspondence: bhkimc@knu.ac.kr; Tel.: +82-53-950-7819

Abstract

Hydraulic structures installed in rivers inevitably create a water level difference between upstream and downstream regions. The potential energy due to this difference in water level is converted into kinetic energy, causing high-velocity flow and hydraulic jumps in the river. As a result, problems such as scouring and sloping downstream may occur around the hydraulic structures. In this study, a FLOW-3D model was constructed to perform a numerical analysis of the ChangnyeongHaman weir in the Republic of Korea. The constructed model was verified based on surface velocity measurements from a field gate operation experiment. In the simulation results, the flow discharge differed from the measured value by 9–15 m3/s, from which the accuracy was evaluated to be 82–87%. The flow velocity was evaluated with an accuracy of 92% from a difference of 0.01 to 0.16 m/s. Following this verification, a flow analysis of the hydraulic structures was performed according to boundary conditions and operation conditions for numerous scenarios. Since 2018, the ChangnyeongHaman weir gate has been fully opened due to the implementation of Korea’s eco-environmental policy; therefore, in this study, the actual gate operation history data prior to 2018 was applied and evaluated. The evaluation conditions were a 50% open gate condition and the flow discharge of two cases with a large difference in water level. As a result of the analysis, the actual operating conditions showed that the velocity and the Froude number were lower than the optimal conditions, confirming that the selected design was appropriate. It was also found that in the bed protection section, the average flow velocity was high when the water level difference was large, whereas the bottom velocity was high when the gate opening was large. Ultimately, through the reviewed status survey data in this study, the downstream flow characteristics of hydraulic structures along with adequacy verification techniques, optimal design techniques such as procedures for design, and important considerations were derived. Based on the current results, the constructed FLOW-3D-based model can be applied to creating or updating flow analysis guidelines for future repair and reinforcement measures as well as hydraulic structure design.

하천에 설치되는 수력구조물은 필연적으로 상류와 하류의 수위차를 발생시킨다. 이러한 수위차로 인한 위치에너지는 운동에너지로 변환되어 하천의 고속유동과 수압점프를 일으킨다. 그 결과 수력구조물 주변에서 하류의 세굴, 경사 등의 문제가 발생할 수 있다.

본 연구에서는 대한민국 창녕함안보의 수치해석을 위해 FLOW-3D 모델을 구축하였다. 구축된 모델은 현장 게이트 작동 실험에서 표면 속도 측정을 기반으로 검증되었습니다.

시뮬레이션 결과에서 유량은 측정값과 9~15 m3/s 차이가 나고 정확도는 82~87%로 평가되었다. 유속은 0.01~0.16m/s의 차이에서 92%의 정확도로 평가되었습니다.

검증 후 다양한 시나리오에 대한 경계조건 및 운전조건에 따른 수리구조물의 유동해석을 수행하였다. 2018년부터 창녕함안보 문은 한국의 친환경 정책 시행으로 전면 개방되었습니다.

따라서 본 연구에서는 2018년 이전의 실제 게이트 운영 이력 데이터를 적용하여 평가하였다. 평가조건은 50% open gate 조건과 수위차가 큰 2가지 경우의 유수방류로 하였다. 해석 결과 실제 운전조건은 속도와 Froude수가 최적조건보다 낮아 선정된 설계가 적합함을 확인하였다.

또한 베드보호구간에서는 수위차가 크면 평균유속이 높고, 수문개구가 크면 저저유속이 높은 것으로 나타났다. 최종적으로 본 연구에서 검토한 실태조사 자료를 통해 적정성 검증기법과 함께 수력구조물의 하류 유동특성, 설계절차 등 최적 설계기법 및 중요 고려사항을 도출하였다.

현재의 결과를 바탕으로 구축된 FLOW-3D 기반 모델은 수력구조 설계뿐만 아니라 향후 보수 및 보강 조치를 위한 유동해석 가이드라인 생성 또는 업데이트에 적용할 수 있습니다.

Figure 1. Effect of downstream riverbed erosion according to the type of weir foundation.

Figure 2. Changnyeong-Haman weir depth survey results (June 2015)

Figure 16. Analysis results for Case 7 and Case 8

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전체 원문보기

Fig. 9. Simulated separation regions for surface mounted cylinder

Investigation on the Local Scour Beneath Piggyback Pipelines Under Clear-Water Conditions

China Ocean Engineering volume 35, pages422–431 (2021)Cite this article

Abstract

피기백 파이프라인은 2개의 파이프로 구성되어 2차 라인이 2개의 파이프 사이의 길이가 고정된 거리로 메인 파이프에 탑승합니다. 새로운 전략은 단일 흐름 라인 대신 연안 지역에서 활용됩니다.

이와 관련하여 정상 전류에서 피기백 파이프라인 아래의 세굴 효과를 조사하는 실험 및 수치 연구는 소수에 불과합니다. 따라서 본 연구에서는 수치모사 및 실험적 실험을 통해 관직경, 관간격 등 정류에 의한 세굴에 영향을 미치는 요인을 살펴보고자 합니다.

따라서 연구의 첫 번째 단계에서 단일 파이프를 설치하고 실험식의 결과와 결과를 비교하기 위해 실험실에서 테스트했습니다. 실험적 검증을 마친 후, 피기백 파이프라인도 조립하여 안정된 전류 조건에서 정련을 연구했습니다. 파이프 사이의 간격을 늘리면 최대 세굴 깊이가 감소한다는 결론이 내려졌습니다.

그러나 작은 파이프의 직경이 증가하면 최대 세굴 깊이가 커집니다. 둘째, 본 연구의 수치적 조사에 적합한 도구인 FLOW-3D 소프트웨어를 사용하여 수치해석을 수행하였습니다.

마지막으로, 수치 결과를 해당 실험 데이터와 비교했으며, 이들 사이에 비교적 좋은 일치가 달성되었습니다.

A piggyback pipeline consists of two pipes such that the secondary line rides on the main pipe with a fixed distance between two pipes in length. The novel strategy is utilized in offshore areas instead of a single flow line. In this regard, there are only a handful of experimental and numerical studies investigating the effect of scour below a piggyback pipeline under steady current. Hence, this study focuses on examining the influential factors on scouring due to steady current including the pipe diameter and the gap between pipes through numerical simulations and experimental tests. Accordingly, at the first phase of the research, a single pipe was established and tested in laboratory to compare the results with those of an empirical equation. After finishing experimental verifications, piggyback pipelines were also assembled to study the scouring under steady current conditions. It was concluded that by increasing the gap distance between the pipes, the maximum scour depth decreases; however, an increase in the small pipe’s diameter results in a larger maximum scour depth. Secondly, numerical simulations were carried out using the FLOW-3D software which was found to be a suitable tool for the numerical investigation of this study. Finally, the numerical results have been compared with the corresponding experimental data and a relatively good agreement was achieved between them.

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Fig. 1. (a) Arrangement of piggyback pipeline, (b) Plan view of experimental flume.

Fig. 3. Initial photos of two mounted piggyback pipelines in experimental setup for d/D=0.25.

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Gating System Design Based on Numerical Simulation and Production Experiment Verification of Aluminum Alloy Bracket Fabricated by Semi-solid Rheo-Die Casting Process

반고체 레오 다이 캐스팅 공정으로 제작된 알루미늄 합금 브래킷의 수치 시뮬레이션 및 생산 실험 검증을 기반으로 한 게이팅 시스템 설계

International Journal of Metalcasting volume 16, pages878–893 (2022)Cite this article

Abstract

In this study a gating system including sprue, runner and overflows for semi-solid rheocasting of aluminum alloy was designed by means of numerical simulations with a commercial software. The effects of pouring temperature, mold temperature and injection speed on the filling process performance of semi-solid die casting were studied. Based on orthogonal test analysis, the optimal die casting process parameters were selected, which were metal pouring temperature 590 °C, mold temperature 260 °C and injection velocity 0.5 m/s. Semi-solid slurry preparation process of Swirled Enthalpy Equilibration Device (SEED) was used for die casting production experiment. Aluminum alloy semi-solid bracket components were successfully produced with the key die casting process parameters selected, which was consistent with the simulation result. The design of semi-solid gating system was further verified by observing and analyzing the microstructure of different zones of the casting. The characteristic parameters, particle size and shape factor of microstructure of the produced semi-solid casting showed that the semi-solid aluminum alloy components are of good quality.

이 연구에서 알루미늄 합금의 반고체 레오캐스팅을 위한 스프루, 러너 및 오버플로를 포함하는 게이팅 시스템은 상용 소프트웨어를 사용한 수치 시뮬레이션을 통해 설계되었습니다. 주입 온도, 금형 온도 및 사출 속도가 반고체 다이캐스팅의 충전 공정 성능에 미치는 영향을 연구했습니다. 직교 테스트 분석을 기반으로 금속 주입 온도 590°C, 금형 온도 260°C 및 사출 속도 0.5m/s인 최적의 다이 캐스팅 공정 매개변수가 선택되었습니다. Swirled Enthalpy Equilibration Device(SEED)의 반고체 슬러리 제조 공정을 다이캐스팅 생산 실험에 사용하였다. 알루미늄 합금 반고체 브래킷 구성 요소는 시뮬레이션 결과와 일치하는 주요 다이 캐스팅 공정 매개변수를 선택하여 성공적으로 생산되었습니다. 반고체 게이팅 시스템의 설계는 주조의 다른 영역의 미세 구조를 관찰하고 분석하여 추가로 검증되었습니다. 생산된 반고체 주조물의 특성 매개변수, 입자 크기 및 미세 구조의 형상 계수는 반고체 알루미늄 합금 부품의 품질이 양호함을 보여주었습니다.

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Keywords

semi-solid rheo-die casting
gating system
process parameters
numerical simulation
microstructure

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Figure 6. Circular section of the viscosity and shear-rate clouds.

Simulation and Visual Tester Verification of Solid Propellant Slurry Vacuum Plate Casting

Wu Yue,Li Zhuo,Lu RongFirst published: 26 February 2020 https://doi.org/10.1002/prep.201900411Citations: 3

Abstract

Using an improved Carreau constitutive model, a numerical simulation of the casting process of a type of solid propellant slurry vacuum plate casting was carried out using the Flow3D software. Through the flow process in the orifice flow channel and the combustion chamber, the flow velocity of the slurry passing through the plate flow channel was quantitatively analyzed, and the viscosity, shear rate, and leveling characteristics of the slurry in the combustion chamber were qualitatively analyzed and predicted. The pouring time, pouring quality, and flow state predicted by the numerical simulation were verified using a visual tester consisting of a vacuum plate casting system in which a pouring experiment was carried out. Studies have shown that HTPB three-component propellant slurry is a typical yielding pseudoplastic fluid. When the slurry flows through the flower plate and the airfoil, the fluid shear rate reaches its maximum value and the viscosity of the slurry decreases. The visual pouring platform was built and the experiment was controlled according to the numerically-calculated parameters, ensuring the same casting speed. The comparison between the predicted casting quality and the one obtained in the verification test resulted in an error less than 10 %. Moreover, the error between the simulated casting completion time and the process verification test result was also no more than 10 %. Last, the flow state of the slurry during the simulation was consistent with the one during the experimental test. The overall leveling of the slurry in the combustion chamber was adequate and no relatively large holes and flaws developed during the pouring process.

개선된 Carreau 구성 모델을 사용하여 FLOW-3D 소프트웨어를 사용하여 고체 추진제 슬러리 진공판 유형의 Casting Process에 대한 수치 시뮬레이션을 수행했습니다. 오리피스 유로와 연소실에서의 유동과정을 통해 판 유로를 통과하는 슬러리의 유속을 정량적으로 분석하고, 연소실에서 슬러리의 점도, 전단율, 레벨링 특성을 정성적으로 분석하하고, 예측하였습니다.

타설시간, 타설품질, 수치해석으로 예측된 유동상태는 타설실험을 수행한 진공판주조시스템으로 구성된 비주얼 테스터를 이용하여 검증하였습니다.

연구에 따르면 HTPB 3성분 추진제 슬러리는 전형적인 생성 가소성 유체입니다. 슬러리가 플라워 플레이트와 에어포일을 통과할 때 유체 전단율이 최대값에 도달하고 슬러리의 점도가 감소합니다.

시각적 주입 플랫폼이 구축되었고 동일한 주조 속도를 보장하기 위해 수치적으로 계산된 매개변수에 따라 실험이 제어되었습니다. 예측된 casting 품질과 검증 테스트에서 얻은 품질을 비교한 결과 10 % 미만의 오류가 발생했습니다.

또한 모의 casting 완료시간과 공정검증시험 결과의 오차도 10 % 이하로 나타났습니다.

마지막으로 시뮬레이션 중 슬러리의 흐름 상태는 실험 테스트 시와 일치하였다. 연소실에서 슬러리의 전체 레벨링은 적절했으며 주입 과정에서 상대적으로 큰 구멍과 결함이 발생하지 않았습니다.

Figure 1. The equipment used in the vacuum flower-plate pouring process.

Figure 4. Circular section of the speed cloud.

Figure 5. Viscosity and shear rate distribution cloud pattern flowing through the plate holes.

Figure 7. Volume fraction cloud chart at different time.

Figure 9. Emulation experimental device.

Figure 10. Visualization of the flow state of the pulp inside the tester.

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Full Test View

Figure 8: Instantaneous flow structures extracted using the Q-criterion (Qcriterion=1200) and colored by the magnitude of flow velocity.

Hybrid modeling on 3D hydraulic features of a step-pool unit

Chendi Zhang¹
, Yuncheng Xu^1,2, Marwan A Hassan³
, Mengzhen Xu1
, Pukang He1
¹State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, 100084, China. 2
College of Water Resources and Civil Engineering, China Agricultural University, Beijing, 100081, China.
^{5 3}Department of Geography, University of British Columbia, 1984 West Mall, Vancouver BC, V6T1Z2, Canada.
Correspondence to: Chendi Zhang (chendinorthwest@163.com) and Mengzhen Xu (mzxu@mail.tsinghua.edu.cn)

Abstract

스텝 풀 시스템은 계류의 일반적인 기반이며 전 세계의 하천 복원 프로젝트에 활용되었습니다. 스텝 풀 장치는 스텝 풀 기능의 형태학적 진화 및 안정성과 밀접하게 상호 작용하는 것으로 보고된 매우 균일하지 않은 수력 특성을 나타냅니다.

그러나 스텝 풀 형태에 대한 3차원 수리학의 자세한 정보는 측정의 어려움으로 인해 부족했습니다. 이러한 지식 격차를 메우기 위해 SfM(Structure from Motion) 및 CFD(Computational Fluid Dynamics) 기술을 기반으로 하이브리드 모델을 구축했습니다. 이 모델은 CFD 시뮬레이션을 위한 입력으로 6가지 유속의 자연석으로 만든 인공 스텝 풀 장치가 있는 침대 표면의 3D 재구성을 사용했습니다.

하이브리드 모델은 스텝 풀 장치에 대한 3D 흐름 구조의 고해상도 시각화를 제공하는 데 성공했습니다. 결과는 계단 아래의 흐름 영역의 분할, 즉 수면에서의 통합 점프, 침대 근처의 줄무늬 후류 및 그 사이의 고속 제트를 보여줍니다.

수영장에서 난류 에너지의 매우 불균일한 분포가 밝혀졌으며 비슷한 용량을 가진 두 개의 에너지 소산기가 수영장에 공존하는 것으로 나타났습니다. 흐름 증가에 따른 풀 세굴 개발은 점프 및 후류 와류의 확장으로 이어지지만 이러한 증가는 스텝 풀 실패에 대한 임계 조건에 가까운 높은 흐름에서 점프에 대해 멈춥니다.

음의 경사면에서 발달된 곡물 20 클러스터와 같은 미세 지반은 국부 수력학에 상당한 영향을 주지만 이러한 영향은 수영장 바닥에서 억제됩니다. 스텝 스톤의 항력은 가장 높은 흐름이 사용되기 전에 배출과 함께 증가하는 반면 양력은 더 큰 크기와 더 넓은 범위를 갖습니다. 우리의 결과는 계단 풀 형태의 복잡한 흐름 특성을 조사할 때 물리적 및 수치적 모델링을 결합한 하이브리드 모델 접근 방식의 가능성과 큰 잠재력을 강조합니다.

Step-pool systems are common bedforms in mountain streams and have been utilized in river restoration projects around the world. Step-pool units exhibit highly non-uniform hydraulic characteristics which have been reported to closely 10 interact with the morphological evolution and stability of step-pool features. However, detailed information of the threedimensional hydraulics for step-pool morphology has been scarce due to the difficulty of measurement. To fill in this knowledge gap, we established a hybrid model based on the technologies of Structure from Motion (SfM) and computational fluid dynamics (CFD). The model used 3D reconstructions of bed surfaces with an artificial step-pool unit built by natural stones at six flow rates as inputs for CFD simulations. The hybrid model succeeded in providing high-resolution visualization 15 of 3D flow structures for the step-pool unit. The results illustrate the segmentation of flow regimes below the step, i.e., the integral jump at the water surface, streaky wake vortexes near the bed, and high-speed jets in between. The highly non-uniform distribution of turbulence energy in the pool has been revealed and two energy dissipaters with comparable capacity are found to co-exist in the pool. Pool scour development under flow increase leads to the expansion of the jump and wake vortexes but this increase stops for the jump at high flows close to the critical condition for step-pool failure. The micro-bedforms as grain 20 clusters developed on the negative slope affect the local hydraulics significantly but this influence is suppressed at pool bottom. The drag forces on the step stones increase with discharge before the highest flow is used while the lift force has a larger magnitude and wider varying range. Our results highlight the feasibility and great potential of the hybrid model approach combining physical and numerical modeling in investigating the complex flow characteristics of step-pool morphology.

Figure 1: Workflow of the hybrid modeling. SfM-MVS refers to the technology of Structure from Motion with Multi View Stereo. DSM is short for digital surface model. RNG-VOF is short for Renormalized Group (RNG) k-ε turbulence model coupled with Volume of Fluid method.

Figure 2: Flume experiment settings in Zhang et al., (2020): (a) the artificially built-up step-pool model using natural stones, with stone number labelled; (b) the unsteady hydrograph of the run of CIFR (continually-increasing-flow-rate) T2 used in this study.

Figure 3: Setup of the CFD model: (a) three-dimensional digital surface model (DSM) of the step-pool unit by structure from motion with multi view stereo (SfM-MVS) method as the input to the 3D computational fluid dynamics (CFD) modeling; (b) extruded bed 160 surface model connected to the extra downstream component (in purple blue) and rectangular columns to fill leaks (in green), with the boundary conditions shown on mesh planes; (c) recognized geometry with mesh grids of two mesh blocks shown where MS is short for mesh size; (d) sampling volumes to capture the flow forces acting on each step stone at X, Y, and Z directions; and (e) an example for the simulated 3D flow over the step-pool unit colored by velocity magnitude at the discharge of 49.9 L/s. The abbreviations for boundary conditions in (b) are: V for specified velocity; C for continuative; P for specific pressure; and W for wall 165 condition. The contraction section in Figure (e) refers to the edge between the jet and jump at water surface.

Figure 4: Distribution of time-averaged velocity magnitude (VM_mean) and vectors in three longitudinal sections. The section at Y = 0 cm goes across the keystone while the other two (Y = -18 and 13.5 cm) are located at the step stones beside the keystone with 265 lower top elevations. Q refers to the discharge at the inlet of the computational domain. The spacing for X, Y, and Z axes are all 10 cm in the plots. — Figure 4: Distribution of time-averaged velocity magnitude (VM_mean) and vectors in three longitudinal sections. The section at Y = 0 cm goes across the keystone while the other two (Y = -18 and 13.5 cm) are located at the step stones beside the keystone with lower top elevations. Q refers to the discharge at the inlet of the computational domain. The spacing for X, Y, and Z axes are all 10 cm in the plots.

Figure 5: Distribution of time-averaged flow velocity at five cross sections which are set according to the reference section (x0). The reference cross section x0 is located at the downstream end of the keystone (KS). The five sections are located at 18 cm and 6 cm upstream of the reference section (x0-18 and x0-6), and 2 cm, 15 cm and 40 cm downstream of the reference section (x0+2, x0+15, x0+40). The spacing for X, Y, and Z axes are all 10 cm in the plots.

Figure 6: Distribution of the time-averaged turbulence kinetic energy (TKE) at the five cross sections same with Figure 3.

Figure 7: Boxplots for the distributions of the mass-averaged flow kinetic energy (KE, panels a-f), turbulence kinetic energy (TKE, panels g-l), and turbulent dissipation (εT, panels m-r) in the pool for all the six tested discharges (the plots at the same discharge are in the same row). The mass-averaged values were calculated every 2 cm in the streamwise direction. The flow direction is from left to right in all the plots. The general locations of the contraction section for all the flow rates are marked by the dashed lines, except for Q = 5 L/s when the jump is located too close to the step. The longitudinal distance taken up by negative slope in the pool for the inspected range is shown by shaded area in each plot.

Figure 9: Time-averaged dynamic pressure (DP_mean) on the bed surface in the step-pool model under the two highest discharges, with the step numbers marked. The negative values in the plots result from the setting of standard atmospheric pressure = 0 Pa, whose absolute value is 1.013×105 Pa.

Figure 10: Time-averaged shear stress (SS_mean) on bed surface in the step-pool model, with the step numbers marked. The standard atmospheric pressure is set as 0 Pa.

Figure 11: Variation of fluid force components and magnitude of resultant flow force acting on step stones with flow rate. The stone 4 is the keystone. Stone numbers are consistent with those in Fig. 9-10. The upper limit of the sampling volumes for flow force calculation is higher than water surface while the lower limit is set at 3 cm lower than the keystone crest.

Figure 12: Variation of drag (CD) and lift (CL) coefficient of the step stones along with flow rate. Stone numbers are consistent with those in Fig. 8-9. KS is short for keystone. The negative values of CD correspond to the drag forces towards the upstream while the negative values of CL correspond to lift forces pointing downwards.

Figure 13: Longitudinal distributions of section-averaged and -integral turbulent kinetic energy (TKE) for the jump and wake vortexes at the largest three discharges. The flow direction is from left to right in all the plots. The general locations of the contraction sections under the three flow rates are marked by dashed lines in figures (d) to (f).

Figure A1: Water surface profiles of the simulations with different mesh sizes at the discharge of 43.6 L/s at the longitudinal sections at: (a) Y = 24.5 cm (left boundary); (b) Y = 0.3 cm (middle section); (c) Y = -24.5 cm (right boundary). MS is short for mesh size. The flow direction is from left to right in each plot.

Figure A2: Contours of velocity magnitude in the longitudinal section at Y = 0 cm at different mesh sizes (MSs) under the flow condition with the discharge of 43.6 L/s: (a) 0.50 cm; (b) 0.375 cm; (c) 0.30 cm; (d) 0.27 cm; (e) 0.25 cm; (f) 0.24 cm. The flow direction is from left to right.

Figure A3: Measurements of water surfaces (orange lines) used in model verification: (a) water surface profiles from both sides of the flume; (b) upstream edge of the jump regime from top view. KS refers to keystone in figure (b).

Figure A15. Figure (a) shows the locations of the cross sections and target coarse grains at Q = 49.9 L/s. Figures (b) to (e) show the distribution of velocity magnitude (VM_mean) in the four chosen cross sections: (a) x0+8.0; (b) x0+14.0; (c) x0+21.5; (d) x0+42.5. G1 to G6 refer to 6 protruding grains in the micro-bedforms in the pool.

Figure A16. The distribution of turbulent kinetic energy (TKE) in the same cross sections as in figure S15: (a) x0+8.0; (b) x0+14.0; (c) x0+21.5; (d) x0+42.5.

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Figure 3 Simulation PTC pipes enhanced with copper foam and nanoparticles in FLOW-3D software.

다공성 미디어 및 나노유체에 의해 강화된 수집기로 태양광 CCHP 시스템의 최적화

Optimization of Solar CCHP Systems with Collector Enhanced by Porous Media and Nanofluid

Navid Tonekaboni,¹Mahdi Feizbahr,² Nima Tonekaboni,¹Guang-Jun Jiang,^3,4 and Hong-Xia Chen^3,4

Abstract

태양열 집열기의 낮은 효율은 CCHP(Solar Combined Cooling, Heating, and Power) 사이클의 문제점 중 하나로 언급될 수 있습니다. 태양계를 개선하기 위해 나노유체와 다공성 매체가 태양열 집열기에 사용됩니다.

다공성 매질과 나노입자를 사용하는 장점 중 하나는 동일한 조건에서 더 많은 에너지를 흡수할 수 있다는 것입니다. 이 연구에서는 평균 일사량이 1b인 따뜻하고 건조한 지역의 600 m2 건물의 전기, 냉방 및 난방을 생성하기 위해 다공성 매질과 나노유체를 사용하여 태양열 냉난방 복합 발전(SCCHP) 시스템을 최적화했습니다.

본 논문에서는 침전물이 형성되지 않는 lb = 820 w/m²(이란) 정도까지 다공성 물질에서 나노유체의 최적량을 계산하였다. 이 연구에서 태양열 집열기는 구리 다공성 매체(95% 다공성)와 CuO 및 Al2O3 나노 유체로 향상되었습니다.

나노유체의 0.1%-0.6%가 작동 유체로 물에 추가되었습니다. 나노유체의 0.5%가 태양열 집열기 및 SCCHP 시스템에서 가장 높은 에너지 및 엑서지 효율 향상으로 이어지는 것으로 밝혀졌습니다.

본 연구에서 포물선형 집열기(PTC)의 최대 에너지 및 엑서지 효율은 각각 74.19% 및 32.6%입니다. 그림 1은 태양 CCHP의 주기를 정확하게 설명하기 위한 그래픽 초록으로 언급될 수 있습니다.

The low efficiency of solar collectors can be mentioned as one of the problems in solar combined cooling, heating, and power (CCHP) cycles. For improving solar systems, nanofluid and porous media are used in solar collectors. One of the advantages of using porous media and nanoparticles is to absorb more energy under the same conditions. In this research, a solar combined cooling, heating, and power (SCCHP) system has been optimized by porous media and nanofluid for generating electricity, cooling, and heating of a 600 m² building in a warm and dry region with average solar radiation of Ib = 820 w/m² in Iran. In this paper, the optimal amount of nanofluid in porous materials has been calculated to the extent that no sediment is formed. In this study, solar collectors were enhanced with copper porous media (95% porosity) and CuO and Al₂O₃ nanofluids. 0.1%–0.6% of the nanofluids were added to water as working fluids; it is found that 0.5% of the nanofluids lead to the highest energy and exergy efficiency enhancement in solar collectors and SCCHP systems. Maximum energy and exergy efficiency of parabolic thermal collector (PTC) riches in this study are 74.19% and 32.6%, respectively. Figure 1 can be mentioned as a graphical abstract for accurately describing the cycle of solar CCHP.

1. Introduction

Due to the increase in energy consumption, the use of clean energy is one of the important goals of human societies. In the last four decades, the use of cogeneration cycles has increased significantly due to high efficiency. Among clean energy, the use of solar energy has become more popular due to its greater availability [1]. Low efficiency of energy production, transmission, and distribution system makes a new system to generate simultaneously electricity, heating, and cooling as an essential solution to be widely used. The low efficiency of the electricity generation, transmission, and distribution system makes the CCHP system a basic solution to eliminate waste of energy. CCHP system consists of a prime mover (PM), a power generator, a heat recovery system (produce extra heating/cooling/power), and thermal energy storage (TES) [2]. Solar combined cooling, heating, and power (SCCHP) has been started three decades ago. SCCHP is a system that receives its propulsive force from solar energy; in this cycle, solar collectors play the role of propulsive for generating power in this system [3].

Increasing the rate of energy consumption in the whole world because of the low efficiency of energy production, transmission, and distribution system causes a new cogeneration system to generate electricity, heating, and cooling energy as an essential solution to be widely used. Building energy utilization fundamentally includes power required for lighting, home electrical appliances, warming and cooling of building inside, and boiling water. Domestic usage contributes to an average of 35% of the world’s total energy consumption [4].

Due to the availability of solar energy in all areas, solar collectors can be used to obtain the propulsive power required for the CCHP cycle. Solar energy is the main source of energy in renewable applications. For selecting a suitable area to use solar collectors, annual sunshine hours, the number of sunny days, minus temperature and frosty days, and the windy status of the region are essentially considered [5]. Iran, with an average of more than 300 sunny days, is one of the suitable countries to use solar energy. Due to the fact that most of the solar radiation is in the southern regions of Iran, also the concentration of cities is low in these areas, and transmission lines are far apart, one of the best options is to use CCHP cycles based on solar collectors [6]. One of the major problems of solar collectors is their low efficiency [7]. Low efficiency increases the area of collectors, which increases the initial cost of solar systems and of course increases the initial payback period. To increase the efficiency of solar collectors and improve their performance, porous materials and nanofluids are used to increase their workability.

There are two ways to increase the efficiency of solar collectors and mechanical and fluid improvement. In the first method, using porous materials or helical filaments inside the collector pipes causes turbulence of the flow and increases heat transfer. In the second method, using nanofluids or salt and other materials increases the heat transfer of water. The use of porous materials has grown up immensely over the past twenty years. Porous materials, especially copper porous foam, are widely used in solar collectors. Due to the high contact surface area, porous media are appropriate candidates for solar collectors [8]. A number of researchers investigated Solar System performance in accordance with energy and exergy analyses. Zhai et al. [9] reviewed the performance of a small solar-powered system in which the energy efficiency was 44.7% and the electrical efficiency was 16.9%.

Abbasi et al. [10] proposed an innovative multiobjective optimization to optimize the design of a cogeneration system. Results showed the CCHP system based on an internal diesel combustion engine was the applicable alternative at all regions with different climates. The diesel engine can supply the electrical requirement of 31.0% and heating demand of 3.8% for building.

Jiang et al. [11] combined the experiment and simulation together to analyze the performance of a cogeneration system. Moreover, some research focused on CCHP systems using solar energy. It integrated sustainable and renewable technologies in the CCHP, like PV, Stirling engine, and parabolic trough collector (PTC) [2, 12–15].

Wang et al. [16] optimized a cogeneration solar cooling system with a Rankine cycle and ejector to reach the maximum total system efficiency of 55.9%. Jing et al. analyzed a big-scale building with the SCCHP system and auxiliary heaters to produced electrical, cooling, and heating power. The maximum energy efficiency reported in their work is 46.6% [17]. Various optimization methods have been used to improve the cogeneration system, minimum system size, and performance, such as genetic algorithm [18, 19].

Hirasawa et al. [20] investigated the effect of using porous media to reduce thermal waste in solar systems. They used the high-porosity metal foam on top of the flat plate solar collector and observed that thermal waste decreased by 7% due to natural heat transfer. Many researchers study the efficiency improvement of the solar collector by changing the collector’s shapes or working fluids. However, the most effective method is the use of nanofluids in the solar collector as working fluid [21]. In the experimental study done by Jouybari et al. [22], the efficiency enhancement up to 8.1% was achieved by adding nanofluid in a flat plate collector. In this research, by adding porous materials to the solar collector, collector efficiency increased up to 92% in a low flow regime. Subramani et al. [23] analyzed the thermal performance of the parabolic solar collector with Al₂O₃ nanofluid. They conducted their experiments with Reynolds number range 2401 to 7202 and mass flow rate 0.0083 to 0.05 kg/s. The maximum efficiency improvement in this experiment was 56% at 0.05 kg/s mass flow rate.

Shojaeizadeh et al. [24] investigated the analysis of the second law of thermodynamic on the flat plate solar collector using Al₂O₃/water nanofluid. Their research showed that energy efficiency rose up to 1.9% and the exergy efficiency increased by a maximum of 0.72% compared to pure water. Tiwari et al. [25] researched on the thermal performance of solar ﬂat plate collectors for working fluid water with different nanoﬂuids. The result showed that using 1.5% (optimum) particle volume fraction of Al₂O₃ nanoﬂuid as an absorbing medium causes the thermal efﬁciency to enhance up to 31.64%.

The effect of porous media and nanofluids on solar collectors has already been investigated in the literature but the SCCHP system with a collector embedded by both porous media and nanofluid for enhancing the ratio of nanoparticle in nanofluid for preventing sedimentation was not discussed. In this research, the amount of energy and exergy of the solar CCHP cycles with parabolic solar collectors in both base and improved modes with a porous material (copper foam with 95% porosity) and nanofluid with different ratios of nanoparticles was calculated. In the first step, it is planned to design a CCHP system based on the required load, and, in the next step, it will analyze the energy and exergy of the system in a basic and optimize mode. In the optimize mode, enhanced solar collectors with porous material and nanofluid in different ratios (0.1%–0.7%) were used to optimize the ratio of nanofluids to prevent sedimentation.

2. Cycle Description

CCHP is one of the methods to enhance energy efficiency and reduce energy loss and costs. The SCCHP system used a solar collector as a prime mover of the cogeneration system and assisted the boiler to generate vapor for the turbine. Hot water flows from the expander to the absorption chiller in summer or to the radiator or fan coil in winter. Finally, before the hot water wants to flow back to the storage tank, it flows inside a heat exchanger for generating domestic hot water [26].

For designing of solar cogeneration system and its analysis, it is necessary to calculate the electrical, heating (heating load is the load required for the production of warm water and space heating), and cooling load required for the case study considered in a residential building with an area of 600 m² in the warm region of Iran (Zahedan). In Table 1, the average of the required loads is shown for the different months of a year (average of electrical, heating, and cooling load calculated with CARRIER software).Table 1 The average amount of electric charges, heating load, and cooling load used in the different months of the year in the city of Zahedan for a residential building with 600 m².

According to Table 1, the maximum magnitude of heating, cooling, and electrical loads is used to calculate the cogeneration system. The maximum electric load is 96 kW, the maximum amount of heating load is 62 kW, and the maximum cooling load is 118 kW. Since the calculated loads are average, all loads increased up to 10% for the confidence coefficient. With the obtained values, the solar collector area and other cogeneration system components are calculated. The cogeneration cycle is capable of producing 105 kW electric power, 140 kW cooling capacity, and 100 kW heating power.

2.1. System Analysis Equations

An analysis is done by considering the following assumptions:(1)The system operates under steady-state conditions(2)The system is designed for the warm region of Iran (Zahedan) with average solar radiation Ib = 820 w/m²(3)The pressure drops in heat exchangers, separators, storage tanks, and pipes are ignored(4)The pressure drop is negligible in all processes and no expectable chemical reactions occurred in the processes(5)Potential, kinetic, and chemical exergy are not considered due to their insignificance(6)Pumps have been discontinued due to insignificance throughout the process(7)All components are assumed adiabatic

Schematic shape of the cogeneration cycle is shown in Figure 1 and all data are given in Table 2.

Figure 1 Schematic shape of the cogeneration cycle.Table 2 Temperature and humidity of different points of system.

Based on the first law of thermodynamic, energy analysis is based on the following steps.

First of all, the estimated solar radiation energy on collector has been calculated:where α is the heat transfer enhancement coefficient based on porous materials added to the collector’s pipes. The coefficient α is increased by the porosity percentage, the type of porous material (in this case, copper with a porosity percentage of 95), and the flow of fluid to the collector equation.

Collector efficiency is going to be calculated by the following equation [9]:

Total energy received by the collector is given by [9]

Also, the auxiliary boiler heat load is [2]

Energy consumed from vapor to expander is calculated by [2]

The power output form by the screw expander [9]:

The efficiency of the expander is 80% in this case [11].

In this step, cooling and heating loads were calculated and then, the required heating load to reach sanitary hot water will be calculated as follows:

First step: calculating the cooling load with the following equation [9]:

Second step: calculating heating loads [9]:

Then, calculating the required loud for sanitary hot water will be [9]

According to the above-mentioned equations, efficiency is [9]

In the third step, calculated exergy analysis as follows.

First, the received exergy collector from the sun is calculated [9]:

In the previous equation, f is the constant of air dilution.

The received exergy from the collector is [9]

In the case of using natural gas in an auxiliary heater, the gas exergy is calculated from the following equation [12]:

Delivering exergy from vapor to expander is calculated with the following equation [9]:

In the fourth step, the exergy in cooling and heating is calculated by the following equation:

Cooling exergy in summer is calculated [9]:

Heating exergy in winter is calculated [9]:

In the last step based on thermodynamic second law, exergy efficiency has been calculated from the following equation and the above-mentioned calculated loads [9]:

3. Porous Media

The porous medium that filled the test section is copper foam with a porosity of 95%. The foams are determined in Figure 2 and also detailed thermophysical parameters and dimensions are shown in Table 3.

Figure 2 Copper foam with a porosity of 95%.Table 3 Thermophysical parameters and dimensions of copper foam.

In solar collectors, copper porous materials are suitable for use at low temperatures and have an easier and faster manufacturing process than ceramic porous materials. Due to the high coefficient conductivity of copper, the use of copper metallic foam to increase heat transfer is certainly more efficient in solar collectors.

Porous media and nanofluid in solar collector’s pipes were simulated in FLOW-3D software using the finite-difference method [27]. Nanoparticles Al₂O₃ and CUO are mostly used in solar collector enhancement. In this research, different concentrations of nanofluid are added to the parabolic solar collectors with porous materials (copper foam with porosity of 95%) to achieve maximum heat transfer in the porous materials before sedimentation. After analyzing PTC pipes with the nanofluid flow in FLOW-3D software, for energy and exergy efficiency analysis, Carrier software results were used as EES software input. Simulation PTC with porous media inside collector pipe and nanofluids sedimentation is shown in Figure 3.

Figure 3 Simulation PTC pipes enhanced with copper foam and nanoparticles in FLOW-3D software.

3.1. Nano Fluid

In this research, copper and silver nanofluids (Al₂O₃, CuO) have been added with percentages of 0.1%–0.7% as the working fluids. The nanoparticle properties are given in Table 4. Also, system constant parameters are presented in Table 4, which are available as default input in the EES software.Table 4 Properties of the nanoparticles [9].

System constant parameters for input in the software are shown in Table 5.Table 5 System constant parameters.

The thermal properties of the nanofluid can be obtained from equations (18)–(21). The basic fluid properties are indicated by the index (bf) and the properties of the nanoparticle silver with the index (np).

The density of the mixture is shown in the following equation [28]:where ρ is density and ϕ is the nanoparticles volume fraction.

The specific heat capacity is calculated from the following equation [29]:

The thermal conductivity of the nanofluid is calculated from the following equation [29]:

The parameter β is the ratio of the nanolayer thickness to the original particle radius and, usually, this parameter is taken equal to 0.1 for the calculated thermal conductivity of the nanofluids.

The mixture viscosity is calculated as follows [30]:

In all equations, instead of water properties, working fluids with nanofluid are used. All of the above equations and parameters are entered in the EES software for calculating the energy and exergy of solar collectors and the SCCHP cycle. All calculation repeats for both nanofluids with different concentrations of nanofluid in the solar collector’s pipe.

4. Results and Discussion

In the present study, relations were written according to Wang et al. [16] and the system analysis was performed to ensure the correctness of the code. The energy and exergy charts are plotted based on the main values of the paper and are shown in Figures 4 and 5. The error rate in this simulation is 1.07%.

Figure 4 Verification charts of energy analysis results.

Figure 5 Verification charts of exergy analysis results.

We may also investigate the application of machine learning paradigms [31–41] and various hybrid, advanced optimization approaches that are enhanced in terms of exploration and intensification [42–55], and intelligent model studies [56–61] as well, for example, methods such as particle swarm optimizer (PSO) [60, 62], differential search (DS) [63], ant colony optimizer (ACO) [61, 64, 65], Harris hawks optimizer (HHO) [66], grey wolf optimizer (GWO) [53, 67], differential evolution (DE) [68, 69], and other fusion and boosted systems [41, 46, 48, 50, 54, 55, 70, 71].

At the first step, the collector is modified with porous copper foam material. 14 cases have been considered for the analysis of the SCCHP system (Table 6). It should be noted that the adding of porous media causes an additional pressure drop inside the collector [9, 22–26, 30, 72]. All fourteen cases use copper foam with a porosity of 95 percent. To simulate the effect of porous materials and nanofluids, the first solar PTC pipes have been simulated in the FLOW-3D software and then porous media (copper foam with porosity of 95%) and fluid flow with nanoparticles (AL₂O₃ and CUO) are generated in the software. After analyzing PTC pipes in FLOW-3D software, for analyzing energy and exergy efficiency, software outputs were used as EES software input for optimization ratio of sedimentation and calculating energy and exergy analyses.Table 6 Collectors with different percentages of nanofluids and porous media.

In this research, an enhanced solar collector with both porous media and Nanofluid is investigated. In the present study, 0.1–0.5% CuO and Al₂O₃ concentration were added to the collector fully filled by porous media to achieve maximum energy and exergy efficiencies of solar CCHP systems. All steps of the investigation are shown in Table 6.

Energy and exergy analyses of parabolic solar collectors and SCCHP systems are shown in Figures 6 and 7.

Figure 6 Energy and exergy efficiencies of the PTC with porous media and nanofluid.

Figure 7 Energy and exergy efficiency of the SCCHP.

Results show that the highest energy and exergy efficiencies are 74.19% and 32.6%, respectively, that is achieved in Step 12 (parabolic collectors with filled porous media and 0.5% Al₂O₃). In the second step, the maximum energy efficiency of SCCHP systems with fourteen steps of simulation are shown in Figure 7.

In the second step, where 0.1, −0.6% of the nanofluids were added, it is found that 0.5% leads to the highest energy and exergy efficiency enhancement in solar collectors and SCCHP systems. Using concentrations more than 0.5% leads to sediment in the solar collector’s pipe and a decrease of porosity in the pipe [73]. According to Figure 7, maximum energy and exergy efficiencies of SCCHP are achieved in Step 12. In this step energy efficiency is 54.49% and exergy efficiency is 18.29%. In steps 13 and 14, with increasing concentration of CUO and Al₂O₃ nanofluid solution in porous materials, decreasing of energy and exergy efficiency of PTC and SCCHP system at the same time happened. This decrease in efficiency is due to the formation of sediment in the porous material. Calculations and simulations have shown that porous materials more than 0.5% nanofluids inside the collector pipe cause sediment and disturb the porosity of porous materials and pressure drop and reduce the coefficient of performance of the cogeneration system. Most experience showed that CUO and AL₂O₃ nanofluids with less than 0.6% percent solution are used in the investigation on the solar collectors at low temperatures and discharges [74]. One of the important points of this research is that the best ratio of nanofluids in the solar collector with a low temperature is 0.5% (AL₂O₃ and CUO); with this replacement, the cost of solar collectors and SCCHP cycle is reduced.

5. Conclusion and Future Directions

In the present study, ways for increasing the efficiency of solar collectors in order to enhance the efficiency of the SCCHP cycle are examined. The research is aimed at adding both porous materials and nanofluids for estimating the best ratio of nanofluid for enhanced solar collector and protecting sedimentation in porous media. By adding porous materials (copper foam with porosity of 95%) and 0.5% nanofluids together, high efficiency in solar parabolic collectors can be achieved. The novelty in this research is the addition of both nanofluids and porous materials and calculating the best ratio for preventing sedimentation and pressure drop in solar collector’s pipe. In this study, it was observed that, by adding 0.5% of AL₂O₃ nanofluid in working fluids, the energy efficiency of PTC rises to 74.19% and exergy efficiency is grown up to 32.6%. In SCCHP cycle, energy efficiency is 54.49% and exergy efficiency is 18.29%.

In this research, parabolic solar collectors fully filled by porous media (copper foam with a porosity of 95) are investigated. In the next step, parabolic solar collectors in the SCCHP cycle were simultaneously filled by porous media and different percentages of Al₂O₃ and CuO nanofluid. At this step, values of 0.1% to 0.6% of each nanofluid were added to the working fluid, and the efficiency of the energy and exergy of the collectors and the SCCHP cycle were determined. In this case, nanofluid and the porous media were used together in the solar collector and maximum efficiency achieved. 0.5% of both nanofluids were used to achieve the biggest efficiency enhancement.

In the present study, as expected, the highest efficiency is for the parabolic solar collector fully filled by porous material (copper foam with a porosity of 95%) and 0.5% Al₂O₃. Results of the present study are as follows:(1)The average enhancement of collectors’ efficiency using porous media and nanofluids is 28%.(2)Solutions with 0.1 to 0.5% of nanofluids (CuO and Al₂O₃) are used to prevent collectors from sediment occurrence in porous media.(3)Collector of solar cogeneration cycles that is enhanced by both porous media and nanofluid has higher efficiency, and the stability of output temperature is more as well.(4)By using 0.6% of the nanofluids in the enhanced parabolic solar collectors with copper porous materials, sedimentation occurs and makes a high-pressure drop in the solar collector’s pipe which causes decrease in energy efficiency.(5)Average enhancement of SCCHP cycle efficiency is enhanced by both porous media and nanofluid 13%.

Nomenclature

:	Solar radiation
a:	Heat transfer augmentation coefficient
A:	Solar collector area
Bf:	Basic fluid
:	Specific heat capacity of the nanofluid
F:	Constant of air dilution
:	Thermal conductivity of the nanofluid
:	Thermal conductivity of the basic fluid
:	Viscosity of the nanofluid
:	Viscosity of the basic fluid
:	Collector efficiency
:	Collector energy receives
:	Auxiliary boiler heat
:	Expander energy
:	Gas energy
:	Screw expander work
:	Cooling load, in kilowatts
:	Heating load, in kilowatts
:	Solar radiation energy on collector, in Joule
:	Sanitary hot water load
Np:	Nanoparticle
:	Energy efficiency
:	Heat exchanger efficiency
:	Sun exergy
:	Collector exergy
:	Natural gas exergy
:	Expander exergy
:	Cooling exergy
:	Heating exergy
:	Exergy efficiency
:	Steam mass flow rate
:	Hot water mass flow rate
:	Specific heat capacity of water
:	Power output form by the screw expander
T_am:	Average ambient temperature
:	Density of the mixture.

Greek symbols

ρ:	Density
ϕ:	Nanoparticles volume fraction
β:	Ratio of the nanolayer thickness.

Abbreviations

CCHP:	Combined cooling, heating, and power
EES:	Engineering equation solver.

Data Availability

For this study, data were generated by CARRIER software for the average electrical, heating, and cooling load of a residential building with 600 m² in the city of Zahedan, Iran.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under Contract no. 71761030 and Natural Science Foundation of Inner Mongolia under Contract no. 2019LH07003.

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Fig. 6. Configuration of Johnson (1958) hydraulic experiment.

전체 수심 범위에서 선박 파고에 대한 방정식

Equation for ship wave crests in the entire range of water depths

Byeong Wook Lee ^a
, Changhoon Lee ^b,
*^a Coastal Development and Ocean Energy Research Center, Korea Institute of Ocean Science & Technology, 385 Haeyang-ro, Busan, 49111, Republic of Korea
^b Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul, 05006, Republic of Korea

ABSTRACT

An equation for ship wave crests y/x in the entire range of water depths is developed using the linear dispersion relation. In deep water, the developed equation is reduced to the equation of Kelvin (1906). The locations of ship wave crests in the x – and y -directions are obtained using a dimensionless constant C. The wave ray angle θc at the cusp locus is determined using the condition that θc is maximal at the cusp locus and the cusp locus angle is determined as αc=−tan−1(y/x)max. Numerical experiments are conducted using the FLOW-3D to simulate ship wave propagation. The cusp locus angles of the FLOW-3D are similar to both those of the present theory and Havelock (1908) theory in the entire range of the Froude number. Both the present theory and the FLOW-3D yield that, with the increase of ship speed, the Froude number increases and does the wavelength. For the Froude number equal to or greater than unity, the wavelength becomes infinitely large and the transverse waves disappear. The wavelengths of the FLOW-3D are slightly smaller than those of the present theory because the FLOW-3D considers the decrease of wavelength due to energy dissipation which happens because of viscosity of water and turbulence of high-speed particle velocities.

Fig. 8. Comparison of ship wave crest patterns: (a) Fr ¼ 0:66 (Us ¼ 6:5m=s, kh � 0:724π), (b) Fr ¼ 0:86 (Us ¼ 8:5m=s, kh � 0:342π), (c) Fr ¼ 1:21 (Us ¼ 12:0m=s, kh � 0:003π). Line definition: red solid line ¼ present theory; yellow dashed line ¼ Kelvin theory; white dot ¼ FLOW-3D solution. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) — Fig. 8. Comparison of ship wave crest patterns: (a) Fr ¼ 0:66 (Us ¼ 6:5m=s, kh >= 0:724π), (b) Fr ¼ 0:86 (Us ¼ 8:5m=s, kh >= 0:342π), (c) Fr ¼ 1:21 (Us ¼ 12:0m=s, kh >= 0:003π). Line definition: red solid line ¼ present theory; yellow dashed line ¼ Kelvin theory; white dot ¼ FLOW-3D solution. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Keywords

Ship wave crests
Cusp locus angle
Entire range of water depths
Theoretical solution
Numerical experiment

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원문보기

Fig. 11. Velocity vectors along x-direction through the center of the box culvert for B0, B30, B50, and B70 respectively.

Numerical investigation of scour characteristics downstream of blocked culverts

막힌 암거 하류의 세굴 특성 수치 조사

NesreenTaha^bMaged M.El-Feky^aAtef A.El-Saiad^aIsmailFathy^a
^aDepartment of Water and Water Structures Engineering, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt
^bLab Manager, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt

Abstract

횡단 구조물을 통한 막힘은 안정성을 위협하는 위험한 문제 중 하나입니다. 암거의 막힘 형상 및 하류 세굴 특성에 미치는 영향에 관한 연구는 거의 없습니다.

이 연구의 목적은 수면과 세굴 모두에서 상자 암거를 통한 막힘의 작용을 수치적으로 논의하는 것입니다. 이를 위해 FLOW 3D v11.1.0을 사용하여 퇴적물 수송 모델을 조사했습니다.

상자 암거를 통한 다양한 차단 비율이 연구되었습니다. FLOW 3D 모델은 실험 데이터로 보정되었습니다. 결과는 FLOW 3D 프로그램이 세굴 다운스트림 상자 암거를 정확하게 시뮬레이션할 수 있음을 나타냅니다.

막힌 경우에 대한 속도 분포, 최대 세굴 깊이 및 수심을 플롯하고 비차단된 사례(기본 사례)와 비교했습니다.

그 결과 암거 높이의 70% 차단율은 상류의 수심을 암거 높이의 2.3배 증가시키고 평균 유속은 기본 경우보다 3배 더 증가시키는 것으로 입증되었다. 막힘 비율의 함수로 상대 최대 세굴 깊이를 추정하는 방정식이 만들어졌습니다.

Blockage through crossing structures is one of the dangerous problems that threaten its stability. There are few researches concerned with blockage shape in culverts and its effect on characteristics of scour downstream it.

The study’s purpose is to discuss the action of blockage through box culvert on both water surface and scour numerically. A sediment transport model has been investigated for this purpose using FLOW 3D v11.1.0. Different ratios of blockage through box culvert have been studied. The FLOW 3D model was calibrated with experimental data.

The results present that the FLOW 3D program was capable to simulate accurately the scour downstream box culvert. The velocity distribution, maximum scour depth and water depths for blocked cases have been plotted and compared with the non-blocked case (base case).

The results proved that the blockage ratio 70% of culvert height makes the water depth upstream increases by 2.3 times of culvert height and mean velocity increases by 3 times more than in the base case. An equation has been created to estimate the relative maximum scour depth as a function of blockage ratio.

1. Introduction

Local scour is the removal of granular bed material by the action of hydrodynamic forces. As the depth of scour hole increases, the stability of the foundation of the structure may be endangered, with a consequent risk of damage and failure [1]. So the prediction and control of scour is considered to be very important for protecting the water structures from failure. Most previous studies were designed to study the different factors that impact on scour and their relationship with scour hole dimensions like fluid characteristics, flow conditions, bed properties, and culvert geometry. Many previous researches studied the effect of flow rate on scour hole by information Froude number or modified Froude number [2], [3], [4], [5], [6]. Cesar Mendoza [6] found a good correlation between the scour depth and the discharge Intensity (Qg^−.5D^−2.5). Breusers and Raudkiv [7] used shear velocity in the outlet-scour prediction procedure. Ali and Lim [8] used the densimetric Froude number in estimation of the scour depth [1], [8], [9], [10], [11], [12], [13], [14]. “The densimetric Froude number presents the ratio of the tractive force on sediment particle to the submerged specific weight of the sediment” [15](1)Fd=uρsρ-1gD50

Ali and Lim [8] pointed to the consequence of tailwater depth on scour behavior [1], [2], [8], [13]. Abida and Townsend [2] indicated that the maximum depth of local scour downstream culvert was varying with the tailwater depth in three ways: first, for very shallow tailwater depths, local scouring decreases with a decrease in tailwater depth; second, when the ratio of tailwater depth to culvert height ranged between 0.2 and 0.7, the scour depth increases with decreasing tailwater depth; and third for a submerged outlet condition. The tailwater depth has only a marginal effect on the maximum depth of scour [2]. Ruff et al. [16] observed that for materials having similar mean grain sizes (d₅₀) but different standard deviations (σ). As (σ) increased, the maximum scour hole depth decreased. Abt et al. [4] mentioned to role of soil type of maximum scour depth. It was noticed that local scour was more dangerous for uniform sands than for well-graded mixtures [1], [2], [4], [9], [17], [18]. Abt et al [3], [19] studied the culvert shape effect on scour hole. The results evidenced that the culvert shape has a limited effect on outlet scour. Under equivalent discharge conditions, it was noted that a square culvert with height equal to the diameter of a circular culvert would reduce scour [16], [20]. The scour hole dimension was also effected by the culvert slope. Abt et al. [3], [21] showed that the culvert slope is a key element in estimating the culvert flow velocity, the discharge capacity, and sediment transport capability. Abt et al. [21], [22] tested experimentally culvert drop height effect on maximum scour depth. It was observed that as the drop height was increasing, the depth of scour was also increasing. From the previous studies, it could have noticed that the most scour prediction formula downstream unblocked culvert was the function of densimetric Froude number, soil properties (d₅₀, σ), tailwater depth and culvert opening size. Blockage is the phenomenon of plugging water structures due to the movement of water flow loaded with sediment and debris. Water structures blockage has a bad effect on water flow where it causes increasing of upstream water level that may cause flooding around the structure and increase of scour rate downstream structures [23], [24]. The blockage phenomenon through was studied experimentally and numerical [15], [25], [26], [27], [28], [29], [30], [31], [32], [33]. Jaeger and Lucke [33] studied the debris transport behavior in a natural channel in Australia. Froude number scale model of an existing culvert was used. It was noticed that through rainfall event, the mobility of debris was impressed by stream shape (depth and width). The condition of the vegetation (size and quantities) through the catchment area was the main factor in debris transport. Rigby et al. [26] reported that steep slope was increasing the ability to mobilize debris that form field data of blocked culverts and bridges during a storm in Wollongong city.

Streftaris et al. [32] studied the probability of screen blockage by debris at trash screens through a numerical model to relate between the blockage probability and nature of the area around. Recently, many commercial computational fluid programs (CFD) such as SSIIM, Fluent, and FLOW 3D are used in the analysis of the scour process. Scour and sediment transport numerical model need to validate by using experimental data or field data [34], [35], [36], [37], [38]. Epely-Chauvin et al. [36] investigated numerically the effect of a series of parallel spur diked. The experimental data were compared by SSIIM and FLOW 3D program. It was found that the accuracy of calibrated FLOW 3D model was better than SSIIM model. Nielsen et al. [35] used the physical model and FLOW 3D model to analyze the scour process around the pile. The soil around the pile was uniform coarse stones in the physical models that were simulated by regular spheres, porous media, and a mixture of them. The calibrated porous media model can be used to determine the bed shear stress. In partially blocked culverts, there aren’t many studies that explain the blockage impact on scour dimensions. Sorourian et al. [14], [15] studied the effect of inlet partial blockage on scour characteristics downstream box culvert. It resulted that the partial blockage at the culvert inlet could be the main factor in estimating the depth of scour. So, this study is aiming to investigate the effects of blockage through a box culvert on flow and scour characteristics by different blockage ratios and compares the results with a non-blocked case. Create a dimensionless equation relates the blockage ratio of the culvert with scour characteristics downstream culvert.

2. Experimental data

The experimental work of the study was conducted in the Hydraulics and Water Engineering Laboratory, Faculty of Engineering, Zagazig University, Egypt. The flume had a rectangular cross-section of 66 cm width, 65.5 cm depth, and 16.2 m long. A rectangular culvert was built with 0.2 m width, 0.2 m height and 3.00 m long with θ = 25° gradually outlet and 0.8 m fixed apron. The model was located on the mid-point of the channel. The sediment part was extended for a distance 2.20 m with 0.66 m width and 0.20 m depth of coarse sand with specific weight 1.60 kg/cm³, d₅₀ = 2.75 mm and σ (d₉₀/d₅₀) = 1.50. The particle size distribution was as shown in Fig. 1. The experimental model was tested for different inlet flow (Q) of 25, 30, 34, 40 l/s for different submerged ratio (S) of 1.25, 1.50, 1.75.

3. Dimensional analysis

A dimensional analysis has been used to reduce the number of variables which affecting on the scour pattern downstream partial blocked culvert. The main factors affecting the maximum scour depth are:(2)ds=f(b.h.L.hb.lb.Q.ud.hu.hd.D50.ρ.ρs.g.ls.dd.ld)

Fig. 2 shows a definition sketch of the experimental model. The maximum scour depth can be written in a dimensionless form as:(3)dsh=f(B.Fd.S)where the d_s/h is the relative maximum scour depth.

4. Numerical work

The FLOW 3D is (CFD) program used by many researchers and appeared high accuracy in solving hydrodynamic and sediment transport models in the three dimensions. Numerical simulation with FLOW 3D was performed to study the impacts of blockage ratio through box culvert on shear stress, velocity distribution and the sediment transport in terms of the hydrodynamic features (water surface, velocity and shear stress) and morphological parameters (scour depth and sizes) conditions in accurately and efficiently. The renormalization group (RNG) turbulence model was selected due to its high ability to predict the velocity profiles and turbulent kinetic energy for the flow through culvert [39]. The one-fluid incompressible mode was used to simulate the water surface. Volume of fluid (VOF) method was employed in FLOW 3D to tracks a liquid interface through arbitrary deformations and apply the correct boundary conditions at the interface [40].1.

Governing equations

Three-dimensional Reynolds-averaged Navier Stokes (RANS) equation was applied for incompressible viscous fluid motion. The continuity equation is as following:(4)VF∂ρ∂t+∂∂xρuAx+∂∂yρvAy+∂∂zρwAz=RDIF(5)∂u∂t+1VFuAx∂u∂x+vAy∂u∂y+ωAz∂u∂z=-1ρ∂P∂x+Gx+fx(6)∂v∂t+1VFuAx∂v∂x+vAy∂v∂y+ωAz∂v∂z=-1ρ∂P∂y+Gy+fy(7)∂ω∂t+1VFuAx∂ω∂x+vAy∂ω∂y+ωAz∂ω∂z=-1ρ∂P∂z+Gz+fz

ρ is the fluid density,

V_F is the volume fraction,

(x,y,z) is the Cartesian coordinates,

(u,v,w) are the velocity components,

(A_x,A_y,A_z) are the area fractions and

R_DIF is the turbulent diffusion.

P is the average hydrodynamic pressure,

(G_x, G_y, G_z) are the body accelerations and

(f_x, f_y, f_z) are the viscous accelerations.

The motion of sediment transport (suspended, settling, entrainment, bed load) is estimated by predicting the erosion, advection and deposition process as presented in [41].

The critical shields parameter is (θ_cr) is defined as the critical shear stress τ_cr at which sediments begin to move on a flat and horizontal bed [41]:(8)θcr=τcrgd50(ρs-ρ)

The Soulsby–Whitehouse [42] is used to predict the critical shields parameter as:(9)θcr=0.31+1.2d∗+0.0551-e(-0.02d∗)(10)d∗=d50g(Gs-1ν3where:

d_* is the dimensionless grain size

G_s is specific weight (G_s = ρ_s/ρ)

The entrainment coefficient (0.005) was used to scale the scour rates and fit the experimental data. The settling velocity controls the Soulsby deposition equation. The volumetric sediment transport rate per width of the bed is calculated using Van Rijn [43].2.

Meshing and geometry of model

After many trials, it was found that the uniform cell size with 0.03 m cell size is the closest to the experimental results and takes less time. As shown in Fig. 3. In x-direction, the total model length in this direction is 700 cm with mesh planes at −100, 0, 300, 380 and 600 cm respectively from the origin point, in y-direction, the total model length in this direction is 66 cm at distances 0, 23, 43 and 66 cm respectively from the origin point. In z-direction, the total model length in this direction is 120 cm. with mesh planes at −20, 0, 20 and 100 cm respectively.3.

Boundary condition

As shown in Fig. 4, the boundary conditions of the model have been defined to simulate the experimental flow conditions accurately. The upstream boundary was defined as the volume flow rate with a different flow rate. The downstream boundary was defined as specific pressure with different fluid elevation. Both of the right side, the left side, and the bottom boundary were defined as a wall. The top boundary defined as specified pressure with pressure value equals zero.

5. Validation of experimental results and numerical results

The experimental results investigated the flow and scour characteristics downstream culvert due to different flow conditions. The measured value of maximum scour depth is compared with the simulated depth from FLOW 3D model as shown in Fig. 5. The scour results show that the simulated results from the numerical model is quite close to the experimental results with an average error of 3.6%. The water depths in numerical model results is so close to the experimental results as shown in Fig. 6 where the experiment and numerical results are compared at different submerged ratios and flow rates. The results appear maximum error percentage in water depths upstream and downstream the culvert is about 2.37%. This indicated that the FLOW 3D is efficient for the prediction of maximum scour depth and the flow depths downstream box culvert.

6. Computation time

The run time was chosen according to reaching to the stability limit. Hydraulic stability was achieved after 50 s, where the scour development may still go on. For run 1, the numerical simulation was run for 1000 s as shown in Fig. 7 where it mostly reached to scour stability at 800 s. The simulation time was taken 500 s at about 95% of scour stability.

7. Analysis and discussions

Fig. 8 shows the study sections where sec 1 represents to upstream section, sec2 represents to inside section and sec3 represents to downstream stream section. Table 1 indicates the scour hole dimensions at different blockage case. The symbol (B) represents to blockage and the number points to blockage ratio. B0 case signifies to the non-blocked case, B30 is that blockage height is 30% to the culvert height and so on.

Table 1. The scour results of different blockage ratio.

Case	h_{b cm}	B = h_b/h	Q lit/s	S	F_d	d_{50 mm}	d_s/h measured	l_s/h	d_d/h	l_d/h	d_s/h estimated
B0	0	0	35	1.26	1.69	2.5	0.58	1.50	0.27	5.00	0.46
B30	6	0.30	35	1.26	1.68	2.5	0.48	1.25	0.27	4.25	0.40
B50	10	0.50	35	1.22	1.74	2.5	0.45	1.10	0.24	4.00	0.37
B70	14	0.70	35	1.23	1.73	2.5	0.43	1.50	0.16	5.50	0.33

7.1. Scour hole geometry

The scour hole geometry mainly depends on the properties of soil of the bed downstream the fixed apron. From Table 1, the results show that the maximum scour depth in B0 case is about 0.58 of culvert height while the maximum deposition in B0 is 0.27 culvert height. There is a symmetric scour hole as shown in Fig. 9 in B0 case. An asymmetric scour hole is created in B50 and B70 due to turbulences that causes the deviation of the jet direction from the center of the flume where appear in Fig. 11 and Fig. 19.

7.2. Flow water surface

Fig. 10 presents the relative free surface water (h_w/h) along the x-direction at center of the box culvert. From the mention Figure, it is easy to release the effect of different blockage ratios. The upstream water level rises by increasing the blockage ratio. Increasing upstream water level may cause flooding over the banks of the waterway. In the 70% blockage case, the upstream water level rises to 2.3 times of culvert height more than the non-blocked case at the same discharge and submerged ratio. The water surface profile shows an increase in water level upstream the culvert due to a decrease in transverse velocity. Because of decreasing velocity downstream culvert, there is an increase in water level before it reaches its uniform depth.

7.3. Velocity vectors

Scour downstream hydraulic structures mainly affects by velocities distribution and bed shear stress. Fig. 11 shows the velocity vectors and their magnitude in xz plane at the same flow conditions. The difference in the upstream water level due to the different blockage ratios is so clear. The maximum water level is in B70 and the minimum level is in B0. The inlet mean velocity value is about 0.88 m/s in B0 increases to 2.86 m/s in B70. As the blockage ratio increases, the inlet velocity increases. The outlet velocity in B0 case makes downward jet causes scour hole just after the fixed apron in the middle of the bed while the blockage causes upward water flow that appears clearly in B70. The upward jet decreases the scour depth to 0.13 culvert height less than B0 case. After the scour hole, the velocity decreases and the flow becomes uniform.

7.4. Velocity distribution

Fig. 12 represents flow velocity (Vx) distribution along the vertical depth (z/h_u) upstream the inlet for the different blockage ratios at the same flow conditions. From the Figure, the maximum velocity creates closed to bed in B0 while in blocked case, the maximum horizontal velocity creates at 0.30 of relative vertical depth (z/h_u). Fig. 13 shows the (Vz) distribution along the vertical depth (z/h_u) upstream culvert at sec 1. From the mentioned Figure, it is easy to note that the maximum vertical is in B70 which appears that as the blockage ratio increases the vertical ratio also increases. In the non-blocked case. The vertical velocity (Vz) is maximum at (z/h_u) equals 0.64. At the end of the fixed apron (sec 3), the horizontal velocity (Vx) is slowly increasing to reach the maximum value closed to bed in B0 and B30 while the maximum horizontal velocity occurs near to the top surface in B50 and B70 as shown in Fig. 14. The vertical velocity component along the vertical depth (z/h_d) is presented in Fig. 15. The vertical velocity (Vz) is maximum in B0 at vertical depth (z/h_d) 0.3 with value 0.45 m/s downward. Figs. 16 and 17 observe velocity components (Vx, Vz) along the vertical depth just after the end of blockage length at the centerline of the culvert barrel. It could be noticed the uniform velocity distribution in B0 case with horizontal velocity (Vx) closed to 1.0 m/s and vertical velocity closed to zero. In the blocked case, the maximum horizontal velocity occurs in depth more than the blockage height.

7.5. Bed velocity distribution

Fig. 18 presents the x-velocity vectors at 1.5 cm above the bed for different blockage ratios from the velocity vectors distribution and magnitude, it is easy to realize the position of the scour hole and deposition region. In B0 and B30, the flow is symmetric so that the scour hole is created around the centerline of flow while in B50 and B70 cases, the flow is asymmetric and the scour hole creates in the right of flow direction in B50. The maximum scour depth is found in the left of flow direction in B70 case where the high velocity region is found.

8. Maximum scour depth prediction

Regression analysis is used to estimate maximum scour depth downstream box culvert for different ratios of blockage by correlating the maximum relative scour by other variables that affect on it in one formula. An equation is developed to predict maximum scour depth for blocked and non-blocked. As shown in the equation below, the relative maximum scour depth(d_s/h_d) is a function of densimetric Froude number (F_d), blockage ratio (B) and submerged ratio (S)(11)dsh=0.56Fd-0.20B+0.45S-1.05

In this equation the coefficient of correlation (R²) is 0.82 with standard error equals 0·08. The developed equation is valid for F_d = [0.9 to 2.10] and submerged ratio (S) ≥ 1.00. Fig. 19 shows the comparison between relative maximum scour depths (d_s/h) measured and estimated for different blockage ratios. Fig. 20 clears the comparison between residuals and d_s/h estimated for the present study. From these figures, it could be noticed that there is a good agreement between the measured and estimated relative scour depth.

9. Comparison with previous scour equations

Many previous scour formulae have been produced for calculation the maximum scour depth downstream non-blockage culvert. These equations have been included the effect of flow regime, culvert shape, soil properties and the flow rate on maximum scour depth. Two of previous experimental studies data have been chosen to be compared with the present study results in non-blocked study data. Table 2 shows comparison of culvert shape, densmetric Froude number, median particle size and scour equations for these previous studies. By applying the present study data in these studies scour formula as shown in Fig. 21, it could be noticed that there are a good agreement between present formula results and others empirical equations results. Where that Lim [44] and Abt [4] are so closed to the present study data.

Table 2. Comparison of some previous scour formula.

Researchers	F_d	Culvert shape	d₅₀(mm)	Proposed equation	Submerged ratio
Present study	0.9–2.11	square	2.75	dsh=0.56Fd-0.20B+0.45S-1.05	1.25–1.75
Lim [44]	1–10	Circular	1.65	dsh=0.45Fd	0.47
Abt [4]	Fd ≥ 1	Circular	0.22–7.34-	dsh=3.67Fd0.57∗D500.4∗σ-0.4	–

10. Conclusions

The present study has shown that the FLOW 3D model can accurately simulate water surface and the scour hole characteristics downstream the box culvert with error percentage in water depths does not exceed 2.37%. Velocities distribution through and outlets culvert barrel helped on understanding the scour hole shape.

The blockage through culvert had caused of increasing of water surface upstream structure where the upstream water level in B70 was 2.3 of culvert height more than non-blocked case at the same discharge that could be dangerous on the stability of roads above. The depth averaged velocity through culvert barrel increased by 3 times its value in non-blocked case.

On the other hand, blockage through culvert had a limited effect on the maximum scour depth. The little effect of blockage on maximum scour depth could be noticed in Fig. 11. From this Figure, it could be noted that the residual part of culvert barrel after the blockage part had made turbulences. These turbulences caused the deviation of the flow resulting in the formation of asymmetric scour hole on the side of channel. This not only but in B70 the blockage height caused upward jet which made a wide far scour hole as cleared from the results in Table 1.

An empirical equation was developed from the results to estimate the maximum scour depth relative to culvert height function of blockage ratio (B), submerged ratio (S), and densimetric Froude number (F_d). The equation results was compared with some scour formulas at the same densimetric Froude number rang where the present study results was in between the other equations results as shown in Fig. 21.

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.

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Peer review under responsibility of Faculty of Engineering, Alexandria University.

e) 표시 탭에서 결과를 볼 수 있으며 필요한 경우 슬라이스 옵션을 사용하여 특정 영역을 분석할 수 있습니다.

유체 역학 및 응용 유압 분야에서 사용하기 위한 수치 모델링(CFD)을 적용한 가상 실험실 실습 매뉴얼

This manual was developed with the purpose of presenting and executing basic numerical models in the software known as Flow 3D within the virtual laboratories of Fluid Mechanics and Applied Hydraulics, to complement and reinforce what was learned in class, the development of the manual covers a theoretical content and an exemplified práctical part for the handling of the software, besides including some feedback for the students, in order to mark the characteristics that the software has. With the handling of the Flow 3D program, the student will be introduced to the concept of Computational Fluid Dynamics or CFD, and a simple procedure to represent numerically and graphically the behavior of hydraulic structures. The hydraulic structures presented in the laboratory manual are: thin and thick wall orifices, gates with free and submerged discharge, thin and thick wall spillways with free and submerged discharge, WES type spillway, submerged intake with pressure conduction and as a complement, hydrostatic pressures on vertical, curved and inclined walls were added. Each of the mentioned hydraulic structures obtained a práctical verification as a verification within the Flow 3D software, presenting a consistency in the results obtained in both ways.

이 매뉴얼은 Fluid Mechanics 및 Applied Hydraulics의 가상 연구실 내에서 Flow 3D로 알려진 소프트웨어에서 기본 수치 모델을 제시하고 실행하기 위해 개발되었으며, 수업에서 배운 내용을 보완하고 강화하기 위해 개발되었으며, 매뉴얼 개발은 이론적인 내용을 다룹니다. 소프트웨어의 특성을 표시하기 위해 학생들을 위한 일부 피드백을 포함하는 것 외에도 소프트웨어 처리에 대한 내용 및 예시된 실제적인 부분. Flow 3D 프로그램을 다루면서 학생은 전산유체역학(Computational Fluid Dynamics) 또는 CFD의 개념과 수력학적 구조의 거동을 수치 및 그래픽으로 표현하는 간단한 절차를 소개합니다. 실험실 매뉴얼에 제시된 유압 구조는 얇고 두꺼운 벽 오리피스, 자유 및 수중 배출이 있는 수문, 자유 및 수중 배출이 있는 얇고 두꺼운 벽 여수로, WES 유형 방수로, 압력 전도 및 보완으로 수중 유입이 있는 수중 흡입구입니다. 수직, 곡선 및 경사 벽에 추가되었습니다. 언급된 각 수력학적 구조는 Flow 3D 소프트웨어 내에서 검증으로 실제 검증을 획득하여 두 가지 방식에서 얻은 결과의 일관성을 나타냅니다.

Keywords: Flow 3D, numerical modeling, manual, practice, Fluid Mechanics.

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Flow velocity profiles for canals with a depth of 3 m and flow velocities of 5–5.3 m/s.

Optimization Algorithms and Engineering: Recent Advances and Applications

Mahdi Feizbahr,¹ Navid Tonekaboni,²Guang-Jun Jiang,^3,4 and Hong-Xia Chen^3,4Show moreAcademic Editor: Mohammad YazdiReceived08 Apr 2021Revised18 Jun 2021Accepted17 Jul 2021Published11 Aug 2021

Abstract

강의 식생은 거칠기를 증가시키고 평균 유속을 감소시키며, 유속 에너지를 감소시키고 강의 단면에서 유속 프로파일을 변경합니다. 자연의 많은 운하와 강은 홍수 동안 초목으로 덮여 있습니다. 운하의 조도는 식물의 영향을 많이 받으므로 홍수시 유동저항에 큰 영향을 미칩니다. 식물로 인한 흐름에 대한 거칠기 저항은 흐름 조건 및 식물에 따라 다르므로 모델은 유속, 흐름 깊이 및 운하를 따라 식생 유형의 영향을 고려하여 현재 속도를 시뮬레이션해야 합니다. 근관의 거칠기의 영향을 조사하기 위해 총 48개의 모델이 시뮬레이션되었습니다. 결과는 유속을 높임으로써 유속을 감소시키는 식생의 영향은 무시할 수 있는 반면, 해류가 더 낮은 유속일 때 유속을 감소시키는 식생의 영향은 분명히 상당함을 나타냈다.

1. Introduction

On the other hand, the relationship between the Darcy–Weisbach coefficient of friction and the shear velocity of the flow is as follows:

By using equation (6), equation (5) is converted as follows:

Shear stresses in equation (11) are obtained from the following equations:

In a study conducted using a software model on three density modes in the submerged state effect on flow velocity changes in 48 different modes was investigated (Table 1).Table 1 The studied models.

Figure 1 The simulated model and its boundary conditions.

Figure 2 Modeling the plant with cylindrical tubes at the bottom of the canal.

Figure 3 Velocity profiles in positions 2 and 5.

2. Modeling Results

After analyzing the models, the results were shown in graphs (Figures 4–14 ). The total number of experiments in this study was 48 due to the limitations of modeling.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)Figure 4 Flow velocity profiles for canals with a depth of 1 m and flow velocities of 3–3.3 m/s. Canal with a depth of 1 meter and a flow velocity of (a) 3 meters per second, (b) 3.1 meters per second, (c) 3.2 meters per second, and (d) 3.3 meters per second.

Figure 5 Canal diagram with a depth of 1 meter and a flow rate of 3 meters per second.

Figure 6 Canal diagram with a depth of 1 meter and a flow rate of 3.1 meters per second.

Figure 7 Canal diagram with a depth of 1 meter and a flow rate of 3.2 meters per second.

Figure 8 Canal diagram with a depth of 1 meter and a flow rate of 3.3 meters per second.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)Figure 9 Flow velocity profiles for canals with a depth of 2 m and flow velocities of 4–4.3 m/s. Canal with a depth of 2 meters and a flow rate of (a) 4 meters per second, (b) 4.1 meters per second, (c) 4.2 meters per second, and (d) 4.3 meters per second.

Figure 10 Canal diagram with a depth of 2 meters and a flow rate of 4 meters per second.

Figure 11 Canal diagram with a depth of 2 meters and a flow rate of 4.1 meters per second.

Figure 12 Canal diagram with a depth of 2 meters and a flow rate of 4.2 meters per second.

Figure 13 Canal diagram with a depth of 2 meters and a flow rate of 4.3 meters per second.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)Figure 14 Flow velocity profiles for canals with a depth of 3 m and flow velocities of 5–5.3 m/s. Canal with a depth of 2 meters and a flow rate of (a) 4 meters per second, (b) 4.1 meters per second, (c) 4.2 meters per second, and (d) 4.3 meters per second.

According to the velocity profiles of Figure 5, it can be seen that, with the increasing of Manning’s coefficient, the canal bed speed decreases.

According to Figure 10, we see that, with increasing Manning’s coefficient, the canal bed speed decreases.

According to Figure 15, with increasing Manning’s coefficient, the canal bed speed decreases.

Figure 15 Canal diagram with a depth of 3 meters and a flow rate of 5 meters per second.

Figure 16 Canal diagram with a depth of 3 meters and a flow rate of 5.1 meters per second.

Figure 17 Canal diagram with a depth of 3 meters and a flow rate of 5.2 meters per second.

Figure 18 Canal diagram with a depth of 3 meters and a flow rate of 5.3 meters per second.

According to Figure 19, it can be seen that the vegetation placed in front of the flow input velocity has negligible effect on the reduction of velocity, which of course can be justified due to the flexibility of the vegetation. The only unusual thing is the unexpected decrease in floor speed of 3 m/s compared to higher speeds.
(a)
(b)
(c)
(a)
(b)
(c)Figure 19 Comparison of velocity profiles with the same plant densities (depth 1 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 1 m; (b) plant densities of 50%, depth 1 m; and (c) plant densities of 75%, depth 1 m.

According to Figure 20, by increasing the speed of vegetation, the effect of vegetation on reducing the flow rate becomes more noticeable. And the role of input current does not have much effect in reducing speed.
(a)
(b)
(c)
(a)
(b)
(c)Figure 20 Comparison of velocity profiles with the same plant densities (depth 2 m). Comparison of velocity profiles with (a) plant densities of 25%, depth 2 m; (b) plant densities of 50%, depth 2 m; and (c) plant densities of 75%, depth 2 m.

3. Conclusion

Nomenclature

n:	Manning’s roughness coefficient
C:	Chézy roughness coefficient
f:	Darcy–Weisbach coefficient
V:	Flow velocity
R:	Hydraulic radius
g:	Gravitational acceleration
y:	Flow depth
Ks:	Bed roughness
A:	Constant coefficient
:	Reynolds number
∂y/∂x:	Depth of water change
S₀:	Slope of the canal floor
Sf:	Slope of energy line
Fr:	Froude number
D:	Character

[태그:] Verification

Numerical Study of the Local Scouring Process and Influencing Factors of Semi-Exposed Submarine Cables

Abstract

1. Introduction

2. Sediment Scouring Model

3. Numerical Setup and Modeling

3.1. Geometric Modeling and Mesh Division

3.2. Physical Field Setup

3.3. Mesh Independent Test

4. Results and Analysis

4.1. Analysis of Local Scouring Process

4.2. Influence Factors

4.2.1. Sediment’s Critical Shields Number

4.2.2. Sediment Density

4.2.3. Ocean Current Velocity

4.3. Variation Rule of Spanning Time

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Validity evaluation of popular liquid-vapor phase change models for cryogenic self-pressurization process

Abstract

Introduction

Section snippets

Governing equations for the self-pressurization tank

Self-pressurization results and phase change model comparison

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgement

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Keywords

Abstract

1. Introduction

2. Numerical Method

2.1. Governing Equations of Flow

2.2. Turbulent Model

2.3. Sediment Scour Model

2.3.1. Entrainment and Deposition

2.3.2. Bed Load Transport

2.3.3. Suspended Load Transport

3. Model Setup

3.1. Mesh Geometric Dimensions

3.2. Boundary Conditions

3.3. Wave Parameters

3.4. Mesh Sensitivity

3.5. Model Validation

4. Numerical Results and Discussions

4.1. Scour Evolution

4.2. Scour Mechanism under Random Waves

4.3. Equilibrium Scour Depth

4.4. Parametric Study

4.4.1. Influence of Froude Number

4.4.2. Influence of Euler Number

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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¹. ^Introduction

^2.1. ^{Model Setup}

^2.2. ^Verification

³. ^{Sensitivity Analysis}

⁴. ^{Design Optimization}