Omega-Luitex법을 이용한 수력점프 발생시 러프 베드의 와류 진화 예측 및 영향 분석

Cong Trieu Tran, Cong Ty Trinh

Abstract

The dissipation of energy downstream of hydropower projects is a significant issue. The hydraulic jump is exciting and widely applied in practice to dissipate energy. Many hydraulic jump characteristics have been studied, such as length of jump Lj and sequent flow depth y2. However, understanding the evolution of the vortex structure in the hydraulic jump shows a significant challenge. This study uses the RNG k-e turbulence model to simulate hydraulic jumps on the rough bed. The Omega-Liutex method is compared with Q-criterion for capturing vortex structure in the hydraulic jump. The formation, development, and shedding of the vortex structure at the rough bed in the hydraulic jumper are analyzed. The vortex forms and rapidly reduces strength on the rough bed, resulting in fast dissipation of energy. At the rough block rows 2nd and 3rd, the vortex forms a vortex rope that moves downstream and then breaks. The vortex-shedding region represents a significant energy attenuation of the flow. Therefore, the rough bed dissipates kinetic energy well. Adding reliability to the vortex determined by the Liutex method, the vorticity transport equation is used to compare the vorticity distribution with the Liutex distribution. The results show a further comprehension of the hydraulic jump phenomenon and its energy dissipation.

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[11] Tran, C. T. & Pham, D. C. (2022). Application of Liutex and Entropy Production to Analyze the Influence of Vortex Rope in the Francis-99 Turbine Draft Tube. Tehnički vjesnik, 29(4), 1177-1183. https://doi.org/10.17559/TV-20210821070801

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Waqed H. Hassan^{}| Zahraa Mohammad Fadhe^{*}| Rifqa F. Thiab^{}| Karrar Mahdi^{} Civil Engineering Department, Faculty of Engineering, University of Warith Al-Anbiyaa, Kerbala 56001, Iraq Civil Engineering Department, Faculty of Engineering, University of Kerbala, Kerbala 56001, Iraq Corresponding Author Email: Waqed.hammed@uowa.edu.iq

OPEN ACCESS

Abstract:

This work investigates numerically a local scour moves in irregular waves around tripods. It is constructed and proven to use the numerical model of the seabed-tripod-fluid with an RNG k turbulence model. The present numerical model then examines the flow velocity distribution and scour characteristics. After that, the suggested computational model Flow-3D is a useful tool for analyzing and forecasting the maximum scour development and the flow field in random waves around tripods. The scour values affecting the foundations of the tripod must be studied and calculated, as this phenomenon directly and negatively affects the structure of the structure and its design life. The lower diagonal braces and the main column act as blockages, increasing the flow accelerations underneath them. This increases the number of particles that are moved, which in turn creates strong scouring in the area. The numerical model has a good agreement with the experimental model, with a maximum percentage of error of 10% between the experimental and numerical models. In addition, Based on dimensional analysis parameters, an empirical equation has been devised to forecast scour depth with flow depth, median size ratio, Keulegan-Carpenter (Kc), Froud number flow, and wave velocity that the results obtained in this research at various flow velocities and flow depths demonstrated that the maximum scour depth rate depended on wave height with rising velocities and decreasing particle sizes (d_{50}) and the scour depth attains its steady-current value for V_{w} < 0.75. As the Froude number rises, the maximum scour depth will be large.

Keywords:

local scour, tripod foundation, Flow-3D, waves

1. Introduction

New energy sources have been used by mankind since they become industrialized. The main energy sources have traditionally been timber, coal, oil, and gas, but advances in the science of new energies, such as nuclear energy, have emerged [1, 2]. Clean and renewable energy such as offshore wind has grown significantly during the past few decades. There are numerous different types of foundations regarding offshore wind turbines (OWTs), comprising the tripod, jacket, gravity foundation, suction anchor (or bucket), and monopile [3, 4]. When the water depth is less than 30 meters, Offshore wind farms usually employ the monopile type [4]. Engineers must deal with the wind’s scouring phenomenon turbine foundations when planning and designing wind turbines for an offshore environment [5]. Waves and currents generate scour, this is the erosion of soil near a submerged foundation and at its location [6]. To predict the regional scour depth at a bridge pier, Jalal et al. [7-10] developed an original gene expression algorithm using artificial neural networks. Three monopiles, one main column, and several diagonal braces connecting the monopiles to the main column make up the tripod foundation, which has more complicated shapes than a single pile. The design of the foundation may have an impact on scour depth and scour development since the foundation’s form affects the flow field [11, 12]. Stahlmann [4] conducted several field investigations. He discovered that the main column is where the greatest scour depth occurred. Under the main column is where the maximum scour depth occurs in all experiments. The estimated findings show that higher wave heights correspond to higher flow velocities, indicating that a deeper scour depth is correlated with finer silt granularity [13] recommends as the design value for a single pile. These findings support the assertion that a tripod may cause the seabed to scour more severely than a single pile. The geography of the scour is significantly more influenced by the KC value (Keulegan–Carpenter number)

The capability of computer hardware and software has made computational fluid dynamics (CFD) quite popular to predict the behavior of fluid flow in industrial and environmental applications has increased significantly in recent years [14].

Finding an acceptable piece of land for the turbine’s construction and designing the turbine pile precisely for the local conditions are the biggest challenges. Another concern related to working in a marine environment is the effect of sea waves and currents on turbine piles and foundations. The earth surrounding the turbine’s pile is scoured by the waves, which also render the pile unstable.

In this research, the main objective is to investigate numerically a local scour around tripods in random waves. It is constructed and proven to use the tripod numerical model. The present numerical model is then used to examine the flow velocity distribution and scour characteristics.

2. Numerical Model

To simulate the scouring process around the tripod foundation, the CFD code Flow-3D was employed. By using the fractional area/volume method, it may highlight the intricate boundaries of the solution domain (FAVOR).

This model was tested and validated utilizing data derived experimentally from Schendel et al. [15] and Sumer and Fredsøe [6]. 200 runs were performed at different values of parameters.

2.1 Momentum equations

The incompressible viscous fluid motion is described by the three RANS equations listed below [16]:

where, respectively, u, v, and w represent the x, y, and z flow velocity components; volume fraction (VF), area fraction (A_{i}; I=x, y, z), water density (f), viscous force (f_{i}), and body force (G_{i}) are all used in the formula.

2.2 Model of turbulence

Several turbulence models would be combined to solve the momentum equations. A two-equation model of turbulence is the RNG k-model, which has a high efficiency and accuracy in computing the near-wall flow field. Therefore, the flow field surrounding tripods was captured using the RNG k-model.

2.3 Model of sediment scour

2.3.1 Induction and deposition

Eq. (4) can be used to determine the particle entrainment lift velocity [17].

α𝛼_{i }is the Induction parameter, n_{s} the normal vector is parallel to the seafloor, and for the present numerical model, n_{s}=(0,0,1), θ𝜃_{cr} is the essential Shields variable, g is the accelerated by gravity, d_{i} is the size of the particles, ρ_{i} is species density in beds, and d_{∗} The diameter of particles without dimensions; these values can be obtained in Eq. (5).

f_{b}is the essential particle packing percentage, q_{b}, i is the bed load transportation rate, and c_{b}, I the percentage of sand by volume i. These variables can be found in Eq. (9), Eq. (10), f_{b}, δ𝛿_{i} the bed load thickness.

In this paper, after the calibration of numerous trials, the selection of parameters for sediment scour is crucial. Maximum packing fraction is 0.64 with a shields number of 0.05, entrainment coefficient of 0.018, the mass density of 2650, bed load coefficient of 12, and entrainment coefficient of 0.01.

3. Model Setup

To investigate the scour characteristics near tripods in random waves, the seabed-tripod-fluid numerical model was created as shown in Figure 1. The tripod basis, a seabed, and fluid and porous medium were all components of the model. The seabed was 240 meters long, 40 meters wide, and three meters high. It had a median diameter of d_{50} and was composed of uniformly fine sand. The 2.5-meter main column diameter D. The base of the main column was three dimensions above the original seabed. The center of the seafloor was where the tripod was, 130 meters from the offshore and 110 meters from the onshore. To prevent wave reflection, the porous media were positioned above the seabed on the onshore side.

Figure 1. An illustration of the numerical model for the seabed-tripod-fluid

3.1 Generation of meshes

Figure 2 displays the model’s mesh for the Flow-3D software grid. The current model made use of two different mesh types: global mesh grid and nested mesh grid. A mesh grid with the following measurements was created by the global hexahedra mesh grid: 240m length, 40m width, and 32m height. Around the tripod, a finer nested mesh grid was made, with dimensions of 0 to 32m on the z-axis, 10 to 30 m on the x-axis, and 25 to 15 m on the y-axis. This improved the calculation’s precision and mesh quality.

To increase calculation efficiency, the top side, The model’s two x-z plane sides, as well as the symmetry boundaries, were all specified. For u, v, w=0, the bottom boundary wall was picked. The offshore end of the wave boundary was put upstream. For the wave border, random waves were generated using the wave spectrum from the Joint North Sea Wave Project (JONSWAP). Boundary conditions are shown in Figure 3.

Figure 3. Boundary conditions of the typical problem

The wave spectrum peak enhancement factor (=3.3 for this work) and can be used to express the unidirectional JONSWAP frequency spectrum.

3.3 Mesh sensitivity

Before doing additional research into scour traits and scour depth forecasting, mesh sensitivity analysis is essential. Three different mesh grid sizes were selected for this section: Mesh 1 has a 0.45 by 0.45 nested fine mesh and a 0.6 by 0.6 global mesh size. Mesh 2 has a 0.4 global mesh size and a 0.35 nested fine mesh size, while Mesh 3 has a 0.25 global mesh size and a nested fine mesh size of 0.15. Comparing the relative fine mesh size (such as Mesh 2 or Mesh 3) to the relatively coarse mesh size (such as Mesh 1), a larger scour depth was seen; this shows that a finer mesh size can more precisely represent the scouring and flow field action around a tripod. Significantly, a lower mesh size necessitates a time commitment and a more difficult computer configuration. Depending on the sensitivity of the mesh guideline utilized by Pang et al., when Mesh 2 is applied, the findings converge and the mesh size is independent [20]. In the next sections, scouring the area surrounding the tripod was calculated using Mesh 2 to ensure accuracy and reduce computation time. The working segment generates a total of 14, 800,324 cells.

3.4 Model validation

Comparisons between the predicted outcomes from the current model and to confirm that the current numerical model is accurate and suitably modified, experimental data from Sumer and Fredsøe [6] and Schendel et al. [15] were used. For the experimental results of Run 05, Run 15, and Run 22 from Sumer and Fredsøe [6], the experimental A9, A13, A17, A25, A26, and A27 results from Schendel et al. [15], and the numerical results from the current model are shown in Figure 4. The present model had d_{50}=0.051cm, the height of the water wave(h)=10m, and wave velocity=0.854 m.s^{-1}.

Figure 5. Comparison of the present study’s maximum scour depth with that authored by Sumer and Fredsøe [6] and Schendel et al. [15]

According to Figure 5, the highest discrepancy between the numerical results and experimental data is about 10%, showing that overall, there is good agreement between them. The ability of the current numerical model to accurately depict the scour process and forecast the maximum scour depth (S) near foundations is demonstrated by this. Errors in the simulation were reduced by using the calibrated values of the parameter. Considering these results, a suggested simulated scouring utilizing a Flow-3D numerical model is confirmed as a superior way for precisely forecasting the maximum scour depth near a tripod foundation in random waves.

3.5 Dimensional analysis

The variables found in this study as having the greatest impacts, variables related to flow, fluid, bed sediment, flume shape, and duration all had an impact on local scouring depth (t). Hence, scour depth (S) can be seen as a function of these factors, shown as:

With the aid of dimensional analysis, the 14-dimensional parameters in Eq. (11) were reduced to 6 dimensionless variables using Buckingham’s -theorem. D, V, and were therefore set as repetition parameters and others as constants, allowing for the ignoring of their influence. Eq. (12) thus illustrates the relationship between the effect of the non-dimensional components on the depth of scour surrounding a tripod base.

(12)

\frac{S}{D}=f\left(\frac{h}{D}, \frac{d 50}{D}, \frac{V}{V W}, F r, K c\right)

where, SD𝑆𝐷 are scoured depth ratio, VVw𝑉𝑉𝑤 is flow wave velocity, d50D𝑑50𝐷 median size ratio, $Fr representstheFroudnumber,and𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠𝑡ℎ𝑒𝐹𝑟𝑜𝑢𝑑𝑛𝑢𝑚𝑏𝑒𝑟,𝑎𝑛𝑑Kc$is the Keulegan-Carpenter.

4. Result and Discussion

4.1 Development of scour

Similar to how the physical model was used, this numerical model was also used. The numerical model’s boundary conditions and other crucial variables that directly influence the outcomes were applied (flow depth, median particle size (d_{50}), and wave velocity). After the initial 0-300 s, the scour rate reduced as the scour holes grew quickly. The scour depths steadied for about 1800 seconds before reaching an asymptotic value. The findings of scour depth with time are displayed in Figure 6.

4.2 Features of scour

Early on (t=400s), the scour hole began to appear beneath the main column and then began to extend along the diagonal bracing connecting to the wall-facing pile. Gradually, the geography of the scour; of these results is similar to the experimental observations of Stahlmann [4] and Aminoroayaie Yamini et al. [1]. As the waves reached the tripod, there was an enhanced flow acceleration underneath the main column and the lower diagonal braces as a result of the obstructing effects of the structural elements. More particles are mobilized and transported due to the enhanced near-bed flow velocity, it also increases bed shear stress, turbulence, and scour at the site. In comparison to a single pile, the main column and structural components of the tripod have a significant impact on the flow velocity distribution and, consequently, the scour process and morphology. The main column and seabed are separated by a gap, therefore the flow across the gap may aid in scouring. The scour hole first emerged beneath the main column and subsequently expanded along the lower structural components, both Aminoroayaie Yamini et al. [1] and Stahlmann [4] made this claim. Around the tripod, there are several different scour morphologies and the flow velocity distribution as shown in Figures 7 and 8.

Figure 8. Random waves of flow velocity distribution around a tripod

4.3 Wave velocity’s (V_{w}) impact on scour depth

In this study’s section, we looked at how variations in wave current velocity affected the scouring depth. Bed scour pattern modification could result from an increase or decrease in waves. As a result, the backflow area produced within the pile would become stronger, which would increase the depth of the sediment scour. The quantity of current turbulence is the primary cause of the relationship between wave height and bed scour value. The current velocity has increased the extent to which the turbulence energy has changed and increased in strength now present. It should be mentioned that in this instance, the Jon swap spectrum random waves are chosen. The scour depth attains its steady-current value for Vw<0.75, Figure 9 (a) shows that effect. When (V) represents the mean velocity=0.5 m.s^{-1}.

Figure 9. Main effects on maximum scour depth (S_{max}) as a function of column diameter (D)

4.4 Impact of a median particle (d_{50}) on scour depth

In this section of the study, we looked into how variations in particle size affected how the bed profile changed. The values of various particle diameters are defined in the numerical model for each run numerical modeling, and the conditions under which changes in particle diameter have an impact on the bed scour profile are derived. Based on Figure 9 (b), the findings of the numerical modeling show that as particle diameter increases the maximum scour depth caused by wave contact decreases. When (d_{50}) is the diameter of Sediment (d_{50}). The Shatt Al-Arab soil near Basra, Iraq, was used to produce a variety of varied diameters.

4.5 Impact of wave height and flow depth (h) on scour depth

One of the main elements affecting the scour profile brought on by the interaction of the wave and current with the piles of the wind turbines is the height of the wave surrounding the turbine pile causing more turbulence to develop there. The velocity towards the bottom and the bed both vary as the turbulence around the pile is increased, modifying the scour profile close to the pile. According to the results of the numerical modeling, the depth of scour will increase as water depth and wave height in random waves increase as shown in Figure 9 (c).

4.6 Froude number’s (Fr) impact on scour depth

No matter what the spacing ratio, the Figure 9 shows that the Froude number rises, and the maximum scour depth often rises as well increases in Figure 9 (d). Additionally, it is crucial to keep in mind that only a small portion of the findings regarding the spacing ratios with the smallest values. Due to the velocity acceleration in the presence of a larger Froude number, the range of edge scour downstream is greater than that of upstream. Moreover, the scouring phenomena occur in the region farthest from the tripod, perhaps as a result of the turbulence brought on by the collision of the tripod’s pile. Generally, as the Froude number rises, so does the deposition height and scour depth.

4.7 Keulegan-Carpenter (KC) number

The geography of the scour is significantly more influenced by the KC value. Greater KC causes a deeper equilibrium scour because an increase in KC lengthens the horseshoe vortex’s duration and intensifies it as shown in Figure 10.

The result can be attributed to the fact that wave superposition reduced the crucial KC for the initiation of the scour, particularly under small KC conditions. The primary variable in the equation used to calculate This is the depth of the scouring hole at the bed. The following expression is used to calculate the Keulegan-Carpenter number:

Kc=Vw∗TpD𝐾𝑐=𝑉𝑤∗𝑇𝑝𝐷 (13)

where, the wave period is T_{p} and the wave velocity is shown by V_{w}_{.}

Figure 10. Relationship between the relative maximum scour depth and KC

5. Conclusion

(1) The existing seabed-tripod-fluid numerical model is capable of faithfully reproducing the scour process and the flow field around tripods, suggesting that it may be used to predict the scour around tripods in random waves.

(2) Their results obtained in this research at various flow velocities and flow depths demonstrated that the maximum scour depth rate depended on wave height with rising velocities and decreasing particle sizes (d_{50}).

(3) A diagonal brace and the main column act as blockages, increasing the flow accelerations underneath them. This raises the magnitude of the disturbance and the shear stress on the seafloor, which in turn causes a greater number of particles to be mobilized and conveyed, as a result, causes more severe scour at the location.

(4) The Froude number and the scouring process are closely related. In general, as the Froude number rises, so does the maximum scour depth and scour range. The highest maximum scour depth always coincides with the bigger Froude number with the shortest spacing ratio.

Since the issue is that there aren’t many experiments or studies that are relevant to this subject, therefore we had to rely on the monopile criteria. Therefore, to gain a deeper knowledge of the scouring effect surrounding the tripod in random waves, further numerical research exploring numerous soil, foundation, and construction elements as well as upcoming physical model tests will be beneficial.

Nomenclature

CFD

Computational fluid dynamics

FAVOR

Fractional Area/Volume Obstacle Representation

VOF

Volume of Fluid

RNG

Renormalized Group

OWTs

Offshore wind turbines

Greek Symbols

ε, ω

Dissipation rate of the turbulent kinetic energy, m^{2}s^{-3}

Subscripts

d_{50}

Median particle size

V_{f}

Volume fraction

G_{T}

Turbulent energy of buoyancy

K_{T}

Turbulent velocity

P_{T}

Kinetic energy of the turbulence

Α_{i}

Induction parameter

n_{s}

Induction parameter

ΘΘ_{cr}

The essential Shields variable

D_{i}

Diameter of sediment

d_{∗}

The diameter of particles without dimensions

µ_{f}

Dynamic viscosity of the fluid

q_{b},_{i}

The bed load transportation rate

C_{s,i}

Sand particle’s concentration of mass

D

Diameter of pile

D_{f}

Diffusivity

D

Diameter of main column

Fr

Froud number

Kc

Keulegan–Carpenter number

G

Acceleration of gravity g

H

Flow depth

Vw

Wave Velocity

V

Mean Velocity

Tp

Wave Period

S

Scour depth

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[18] Soulsby, R. (1997). Dynamics of marine sands. https://doi.org/10.1680/doms.25844

[19] Van Rijn, L.C. (1984). Sediment transport, part I: Bed load transport. Journal of Hydraulic Engineering, 110(10): 1431-1456. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:10(1431)

[20] Pang, A.L.J., Skote, M., Lim, S.Y., Gullman-Strand, J., Morgan, N. (2016). A numerical approach for determining equilibrium scour depth around a mono-pile due to steady currents. Applied Ocean Research, 57: 114-124. https://doi.org/10.1016/j.apor.2016.02.010

Mohammad Raze Raeisi Dehkordi^{1}*, Amir Hossein Yeganeh Mazhar^{1} , Farzaneh Kheradzare^{2} ^{1}– PhD. Student in the Department of Construction and Water Management, Science and Research Unit, Islamic Azad University, Tehran, Iran ^{2}– M.Sc. Graduate Water resource management, Department of Civil Engineering and Mechanics, Ghiaseddin Jamshid Kashani University, Qazvin, Iran

One of the key issues in river engineering is analyzing the flow properties at the intersection of natural rivers and canals. The flow of the side channel moves away from the intersection of the two channels as a result of the exchange of input force from the side channel with the main flow after coming into contact with it. One of the most evident properties of the flow in these sections is the development of a revolving region with low pressure and even negative pressure close to the inner wall of the side channel. One advantage of the whirling flow in this low-pressure region is that it gives the flow enough space to sediment, but it also increases flow speed near the channel’s bottom and outside wall by lowering the intersectional area of the flow. One of the most crucial considerations in the design of these intersections is minimizing sedimentation in the rotating region and scouring in the area above the shear plane.

Materials and methods:

The channel (flume) created in the laboratory based on Weber et al., (2001) model, was employed in the current investigation to confirm the validity and examine other study objectives. The main channel is 21. 95 meters long, while the side channel, which is at a 90-degree angle to the main channel, is 3. 66 meters long. The total downstream discharge is approximately 0. 17 m3/s, with the upstream velocities of the main channel being 0. 166 m/s and the side channel being 0. 5 m/s. In both channels, the flow depth and width are 0. 91 meters and 0. 296 meters, respectively. In this study, 6 various models’ angles of intersection between the main and side channels, inlet flow velocity, intersectional area, and side channel length have been examined. Models 2 and 3 have intersection angles of 60 and 30 degrees, respectively, and share the rest of their attributes with the fundamental model, or model number 1. Model 1 is the same as Weber’s experimental model. The length of the side channel in model 4 is different from model 1. The only difference between model 6 and the basic model is the side channel intake speed.

Results and Discussion

Analyzing the intersection angle The angle between the main channel and the side channel is investigated in this section of the findings. Models 1, 2, and 3 are assessed using the intersection angles of 90, 60, and 30 degrees, respectively. In some studies, the impact of the intersection angle has been examined, but in this study, three-dimensional investigation in transverse and longitudinal sections as well as the plan of the intersection is discussed, as can be observed from the literature review. Considering three models with intersection angles of 90, 60, and 30 degrees, the kinetic energy contours at the channel’s middle height can be obtained for each model. The channel with a 30-degree intersection angle (model 3) has the maximum kinetic energy in the flow. The channel with a 60-degree intersection has the minimum kinetic energy. As a result of the maximum deviation of the flow in the main channel caused by the flow of the side channel, the channel with a 90-degree intersection also has the maximum kinetic energy near the wall in front of the side channel.

Examining the side channel length In model 1, the side channel is 3. 66 meters long, whereas in model 4, it is 5. 52 meters long. This study aims to determine how changing the side channel’s length affects the flow pattern where two channels intersect. The kinetic energy contours were obtained for two states of the channel length, which are known to extend the lateral channel, increase the energy of the flow after the intersection, and shorten the length of the high-kinetic energy zone. When compared to model 1 with a shorter length of the side channel, the width of the flow separation zone is reduced by approximately 20%, which results in less flow sedimentation. Figure 12 illustrates the rotating zones in the flow separation area. The flow separation region’s length is essentially unchanged. Studying the intersection of the lateral channel After determining the lateral channel’s length, its width and, consequently, its intersectional area should be evaluated.

This section compares model 1 width of 0. 91 meters to model 5 width of 1. 40 meters. One of the most recent topics related to the intersection of the main and side channels is examining the intersection of the side channel. In model 5, the side channel’s flow rate has also increased due to an increase in the width or intersection of the channel. The flow rate through the intersection and the momentum of the flow from the side channel and the main channel increase when the side channel flow rate rises. The findings indicate that when flow width and side channel flow rise, energy increases after the inlet.

Investigating the value of inlet speed in the side channel Unlike the preceding sections, which were all concerned with the channel geometry, the inlet velocity in the side channel is one of the hydraulic parameters of the flow. In this section, models 1 and 6 with inlet velocities of the side channel of 0. 5 and 0. 75 m/s are evaluated. According to the modeling, the flow is somewhat horst before and immediately on the intersection of the flow level, but it undergoes a substantial prolapse just after the intersection. Model 6 has a larger volume and height of flow, but a smaller and softer prolapse after the intersection.

Conclusion

Some hydraulic and geometric properties of the intersection of channels have been examined using Flow-3D software. The RNG turbulence model was used for three-dimensional modeling. Some of the results are listed below. The flow is uniform upstream of the main and minor channels and only slightly becomes horst at the intersection. The analysis of the lengthening of the side channel revealed a 20% reduction in the separation zone’s width and a considerable reduction in the kinetic energy at the intersection. The input flow rate of this channel to the intersection increases with the speed and width of the side channel, which accounts for the local drop in the width of the main channel flow.

References

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Behdarvandi, M., Hajipour, M., Parsi, E. & Ansari ghojghar, M. (2022). Investigation of Velocity Changes in a Straight Asymmetric pattern at river bend. Water and Soil Conservation, 22(6), 81-89. (In Persian).

Ghobadian, R. & Seyedi tabar, Z. (2016). Numerical investigating of the effect of lateral channel junction position on flow Rectangular Composite Channel Using Flow3D Software. Irrigation and Water Engineering, 13(1), 1-16. Doi: 10.22125/iwe.2022.158503 (In Persian).

Burqaʻi, S. M. & Nazari, A. (2003). Laboratory investigation of sediment pattern at the intersection of channels. 6th International Civil Engineering Conference, Amirkabir University of Technology, Tehran, Iran (In Persian).

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Khosravinia, P., Hosseini, S.H. & Hosseinzadeh Dalir, A. (2018). Numerical analyzing of flow in open channel junction with effect of side slope of channel. Irrigation and Water Engineering, 10(1), 1-16. Doi: 10.22125/iwe.2019.95871 (In Persian).

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Raeisi Dehkordi, M. (2022). Description of types of pollution in water resources and protection of water resources, New Approaches in Civil Engineering, 6(1), 42- 52. Doi: 10.30469/jnace.2022.154373 (In Persian).

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Thiong’o, J.W. (2011). Investigations of fluid flows in open rectangular and triangular channels. Master’s thesis, Jomo Kenyatta University of Agriculture and Technology, Juja, Kenya.

Weber, L.J., Schumate, E.D. & Mawer, N. (2001). Experiments on Flow at a 90° Open-Channel Junction. Journal of hydraulic engineering, 127(5), 340-350.

•Landslide travel distance is considered for the first time in a predictive equation.

•Predictive equation derived from databases using 3D physical and numerical modeling.

•The equation was successfully tested on the 2018 Anak Krakatau tsunami event.

•The developed equation using three-dimensional data exhibits a 91 % fitting quality.

Abstract

Landslide tsunamis, responsible for thousands of deaths and significant damage in recent years, necessitate the allocation of sufficient time and resources for studying these extreme natural hazards. This study offers a step change in the field by conducting a large number of three-dimensional numerical experiments, validated by physical tests, to develop a predictive equation for the maximum initial amplitude of tsunamis generated by subaerial landslides. We first conducted a few 3D physical experiments in a wave basin which were then applied for the validation of a 3D numerical model based on the Flow3D-HYDRO package. Consequently, we delivered 100 simulations using the validated model by varying parameters such as landslide volume, water depth, slope angle and travel distance. This large database was subsequently employed to develop a predictive equation for the maximum initial tsunami amplitude. For the first time, we considered travel distance as an independent parameter for developing the predictive equation, which can significantly improve the predication accuracy. The predictive equation was tested for the case of the 2018 Anak Krakatau subaerial landslide tsunami and produced satisfactory results.

The Anak Krakatau landslide tsunami on 22nd December 2018 was a stark reminder of the dangers posed by subaerial landslide tsunamis (Ren et al., 2020; Mulia et al. 2020a; Borrero et al., 2020; Heidarzadeh et al., 2020; Grilli et al., 2021). The collapse of the volcano’s southwest side into the ocean triggered a tsunami that struck the Sunda Strait, leading to approximately 450 fatalities (Syamsidik et al., 2020; Mulia et al., 2020b) (Fig. 1). As shown in Fig. 1, landslide tsunamis (both submarine and subaerial) have been responsible for thousands of deaths and significant damage to coastal communities worldwide. These incidents underscored the critical need for advanced research into landslide-generated waves to aid in hazard prediction and mitigation. This is further emphasized by recent events such as the 28th of November 2020 landslide tsunami in the southern coast mountains of British Columbia (Canada), where an 18 million m^{3} rockslide generated a massive tsunami, with over 100 m wave run-up, causing significant environmental and infrastructural damage (Geertsema et al., 2022).

Physical modelling and numerical simulation are crucial tools in the study of landslide-induced waves due to their ability to replicate and analyse the complex dynamics of landslide events (Kim et al., 2020). In two-dimensional (2D) modelling, the discrepancy between dimensions can lead to an artificial overestimation of wave amplification (e.g., Heller and Spinneken, 2015). This limitation is overcome with 3D modelling, which enables the scaled-down representation of landslide-generated waves while avoiding the simplifications inherent in 2D approaches (Erosi et al., 2019). Another advantage of 3D modelling in studying landslide-generated waves is its ability to accurately depict the complex dynamics of wave propagation, including lateral and radial spreading from the slide impact zone, a feature unattainable with 2D models (Heller and Spinneken, 2015).

Physical experiments in tsunami research, as presented by authors such as Romano et al. (2020), McFall and Fritz (2016), and Heller and Spinneken (2015), have supported 3D modelling works through validation and calibration of the numerical models to capture the complexities of wave generation and propagation. Numerical modelling has increasingly complemented experimental approach in tsunami research due to the latter’s time and resource-intensive nature, particularly for 3D models (Li et al., 2019; Kim et al., 2021). Various numerical approaches have been employed, from Eulerian and Lagrangian frameworks to depth-averaged and Navier–Stokes models, enhancing our understanding of tsunami dynamics (Si et al., 2018; Grilli et al., 2019; Heidarzadeh et al., 2017, 2020; Iorio et al., 2021; Zhang et al., 2021; Kirby et al., 2022; Wang et al., 2021, 2022; Hu et al., 2022). The sophisticated numerical techniques, including the Particle Finite Element Method and the Immersed Boundary Method, have also shown promising results in modelling highly dynamic landslide scenarios (Mulligan et al., 2020; Chen et al., 2020). Among these methods and techniques, FLOW-3D HYDRO stands out in simulating landslide-generated tsunami waves due to its sophisticated technical features such as offering Tru Volume of Fluid (VOF) method for precise free surface tracking (e.g., Sabeti and Heidarzadeh 2022a). TruVOF distinguishes itself through a split Lagrangian approach, adeptly reducing cumulative volume errors in wave simulations by dynamically updating cell volume fractions and areas with each time step. Its intelligent adaptation of time step size ensures precise capture of evolving free surfaces, offering unparalleled accuracy in modelling complex fluid interfaces and behaviour (Flow Science, 2023).

Predictive equations play a crucial role in assessing the potential hazards associated with landslide-generated tsunami waves due to their ability to provide risk assessment and warnings. These equations can offer swift and reasonable evaluations of potential tsunami impacts in the absence of detailed numerical simulations, which can be time-consuming and expensive to produce. Among multiple factors and parameters within a landslide tsunami generation, the initial maximum wave amplitude (Fig. 1) stands out due to its critical role. While it is most likely that the initial wave generated by a landslide will have the highest amplitude, it is crucial to clarify that the term “initial maximum wave amplitude” refers to the highest amplitude within the first set of impulse waves. This parameter is essential in determining the tsunami’s impact severity, with higher amplitudes signalling a greater destructive potential (Sabeti and Heidarzadeh 2022a). Additionally, it plays a significant role in tsunami modelling, aiding in the prediction of wave propagation and the assessment of potential impacts.

In this study, we initially validate the FLOW-3D HYDRO model through a series of physical experiments conducted in a 3D wave tank at University of Bath (UK). Upon confirmation of the model’s accuracy, we use it to systematically vary parameters namely landslide volume, water depth, slope angle, and travel distance, creating an extensive database. Alongside this, we perform a sensitivity analysis on these variables to discern their impacts on the initial maximum wave amplitude. The generated database was consequently applied to derive a non-dimensional predictive equation aimed at estimating the initial maximum wave amplitude in real-world landslide tsunami events.

Two innovations of this study are: (i) The predictive equation of this study is based on a large number of 3D experiments whereas most of the previous equations were based on 2D results, and (ii) For the first time, the travel distance is included in the predictive equation as an independent parameter. To evaluate the performance of our predictive equation, we applied it to a previous real-world subaerial landslide tsunami, i.e., the Anak Krakatau 2018 event. Furthermore, we compare the performance of our predictive equation with other existing equations.

2. Data and methods

The methodology applied in this research is a combination of physical and numerical modelling. Limited physical modelling was performed in a 3D wave basin at the University of Bath (UK) to provide data for calibration and validation of the numerical model. After calibration and validation, the numerical model was employed to model a large number of landslide tsunami scenarios which allowed us to develop a database for deriving a predictive equation.

2.1. Physical experiments

To validate our numerical model, we conducted a series of physical experiments including two sets in a 3D wave basin at University of Bath, measuring 2.50 m in length (W_{L}), 2.60 m in width (W_{W}), and 0.60 m in height (W_{H}) (Fig. 2a). Conducting two distinct sets of experiments (Table 1), each with different setups (travel distance, location, and water depth), provided a robust framework for validation of the numerical model. For wave measurement, we employed a twin wire wave gauge from HR Wallingford (https://equipit.hrwallingford.com). In these experiments, we used a concrete prism solid block, the dimensions of which are outlined in Table 2. In our experiments, we employed a concrete prism solid block with a density of 2600 kg/m^{3}, chosen for its similarity to the natural density of landslides, akin to those observed with the 2018 Anak Krakatau tsunami, where the landslide composition is predominantly solid rather than granular. The block’s form has also been endorsed in prior studies (Watts, 1998; Najafi-Jilani and Ataie-Ashtiani, 2008) as a suitable surrogate for modelling landslide-induced waves. A key aspect of our methodology was addressing scale effects, following the guidelines proposed by Heller et al. (2008) as it is described in Table 1. To enhance the reliability and accuracy of our experimental data, we conducted each physical experiment three times which revealed all three experimental waveforms were identical. This repetition was aimed at minimizing potential errors and inconsistencies in laboratory measurements.

Table 1. The locations and other information of the laboratory setups for making landslide-generated waves in the physical wave basin. This table details the specific parameters for each setup, including slope range (α), slide volume (V), kinematic viscosity (ν), water depth (h), travel distance (D), surface tension coefficient of water (σ), Reynolds number (R), Weber number (W), and the precise coordinates of the wave gauges (WG).

The acceptable ranges for avoiding scale effects are based on the study by Heller et al. (2008).⁎⁎

The Reynolds number (R) is given by g^{0.5}h^{1.5}/ν, with ν denoting the kinematic viscosity. The Weber number (W) is W = ρgh^{2}/σ, where σ represents surface tension coefficient and ρ = 1000kg/m^{3} is the density of water. In our experiments, conducted at a water temperature of approximately 20 °C, the kinematic viscosity (ν) and the surface tension coefficient of water (σ) are 1.01 × 10^{−6} m²/s and 0.073 N/m, respectively (Kestin et al., 1978).

Table 2. Specifications of the solid block used in physical experiments for generating subaerial landslides in the laboratory.

Solid-block attributes

Property metrics

Geometric shape

Slide width (b_{s})

0.26 m

Slide length (l_{s})

0.20 m

Slide thickness (s)

0.10 m

Slide volume (V)

2.60 × 10^{−3} m^{3}

Specific gravity, (γ_{s})

2.60

Slide weight (m_{s})

6.86 kg

2.2. Numerical simulations applying FLOW-3D hydro

The detailed theoretical framework encompassing the governing equations, the computational methodologies employed, and the specific techniques used for tracking the water surface in these simulations are thoroughly detailed in the study by Sabeti et al. (2024). Here, we briefly explain some of the numerical details. We defined a uniform mesh for our flow domain, carefully crafted with a fine spatial resolution of 0.005 m (i.e., grid size). The dimensions of the numerical model directly matched those of our wave basin used in the physical experiment, being 2.60 m wide, 0.60 m deep, and 2.50 m long (Fig. 2). This design ensures comprehensive coverage of the study area. The output intervals of the numerical model are set at 0.02 s. This timing is consistent with the sampling rates of wave gauges used in laboratory settings. The friction coefficient in the FLOW-3D HYDRO is designated as 0.45. This value corresponds to the Coulombic friction measurements obtained in the laboratory, ensuring that the simulation accurately reflects real-world physical interactions.

In order to simulate the landslide motion, we applied coupled motion objects in FLOW-3D-HYDRO where the dynamics are predominantly driven by gravity and surface friction. This methodology stands in contrast to other models that necessitate explicit inputs of force and torque. This approach ensures that the simulation more accurately reflects the natural movement of landslides, which is heavily reliant on gravitational force and the interaction between sliding surfaces. The stability of the numerical simulations is governed by the Courant Number criterion (Courant et al., 1928), which dictates the maximum time step (Δt) for a given mesh size (Δx) and flow speed (U). According to Courant et al. (1928), this number is required to stay below one to ensure stability of numerical simulations. In our simulations, the Courant number is always maintained below one.

In alignment with the parameters of physical experiments, we set the fluid within the mesh to water, characterized by a density of 1000 kg/m³ at a temperature of 20 °C. Furthermore, we defined the top, front, and back surfaces of the mesh as symmetry planes. The remaining surfaces are designated as wall types, incorporating no-slip conditions to accurately simulate the interaction between the fluid and the boundaries. In terms of selection of an appropriate turbulence model, we selected the k–ω model that showed a better performance than other turbulence methods (e.g., Renormalization-Group) in a previous study (Sabeti et al., 2024). The simulations are conducted using a PC Intel® Core™ i7-10510U CPU with a frequency of 1.80 GHz, and a 16 GB RAM. On this PC, completion of a 3-s simulation required approximately 12.5 h.

2.3. Validation

The FLOW-3D HYDRO numerical model was validated using the two physical experiments (Fig. 3) outlined in Table 1. The level of agreement between observations (O_{i}) and simulations (S_{i}) is examined using the following equation:(1)�=|��−����|×100where ε represents the mismatch error, O_{i} denotes the observed laboratory values, and S_{i} represents the simulated values from the FLOW-3D HYDRO model. The results of this validation process revealed that our model could replicate the waves generated in the physical experiments with a reasonable degree of mismatch (ε): 14 % for Lab 1 and 8 % for Lab 2 experiments, respectively (Fig. 3). These values indicate that while the model is not perfect, it provides a sufficiently close approximation of the real-world phenomena.

In terms of mesh efficiency, we varied the mesh size to study sensitivity of the numerical results to mesh size. First, by halving the mesh size and then by doubling it, we repeated the modelling by keeping other parameters unchanged. This analysis guided that a mesh size of ∆x = 0.005 m is the most effective for the setup of this study. The total number of computational cells applying mesh size of 0.005 m is 9.269 × 10^{6}.

2.4. The dataset

The validated numerical model was employed to conduct 100 simulations, incorporating variations in four key landslide parameters namely water depth, slope angle, slide volume, and travel distance. This methodical approach was essential for a thorough sensitivity analysis of these variables, and for the creation of a detailed database to develop a predictive equation for maximum initial tsunami amplitude. Within the model, 15 distinct slide volumes were established, ranging from 0.10 × 10^{−3} m^{3} to 6.25 × 10^{−3} m^{3} (Table 3). The slope angle varied between 35° and 55°, and water depth ranged from 0.24 m to 0.27 m. The travel distance of the landslides was varied, spanning from 0.04 m to 0.07 m. Detailed configurations of each simulation, along with the maximum initial wave amplitudes and dominant wave periods are provided in Table 4.

Table 3. Geometrical information of the 15 solid blocks used in numerical modelling for generating landslide tsunamis. Parameters are: l_{s}, slide length; b_{s}, slide width; s, slide thickness; γ_{s}, specific gravity; and V, slide volume.

Solid block

l_{s} (m)

b_{s} (m)

s (m)

V (m^{3})

γ_{s}

Block-1

0.310

0.260

0.155

6.25 × 10^{−3}

2.60

Block-2

0.300

0.260

0.150

5.85 × 10^{−3}

2.60

Block-3

0.280

0.260

0.140

5.10 × 10^{−3}

2.60

Block-4

0.260

0.260

0.130

4.39 × 10^{−3}

2.60

Block-5

0.240

0.260

0.120

3.74 × 10^{−3}

2.60

Block-6

0.220

0.260

0.110

3.15 × 10^{−3}

2.60

Block-7

0.200

0.260

0.100

2.60 × 10^{−3}

2.60

Block-8

0.180

0.260

0.090

2.11 × 10^{−3}

2.60

Block-9

0.160

0.260

0.080

1.66 × 10^{−3}

2.60

Block-10

0.140

0.260

0.070

1.27 × 10^{−3}

2.60

Block-11

0.120

0.260

0.060

0.93 × 10^{−3}

2.60

Block-12

0.100

0.260

0.050

0.65 × 10^{−3}

2.60

Block-13

0.080

0.260

0.040

0.41 × 10^{−3}

2.60

Block-14

0.060

0.260

0.030

0.23 × 10^{−3}

2.60

Block-15

0.040

0.260

0.020

0.10 × 10^{−3}

2.60

Table 4. The numerical simulation for the 100 tests performed in this study for subaerial solid-block landslide-generated waves. Parameters are a_{M}, maximum wave amplitude; α, slope angle; h, water depth; D, travel distance; and T, dominant wave period. The location of the wave gauge is X=1.030 m, Y=1.210 m, and Z=0.050 m. The properties of various solid blocks are presented in Table 3.

Test-

Block No

α (°)

h (m)

D (m)

T(s)

a_{M} (m)

1

Block-7

45

0.246

0.029

0.510

0.0153

2

Block-7

45

0.246

0.030

0.505

0.0154

3

Block-7

45

0.246

0.031

0.505

0.0156

4

Block-7

45

0.246

0.032

0.505

0.0158

5

Block-7

45

0.246

0.033

0.505

0.0159

6

Block-7

45

0.246

0.034

0.505

0.0160

7

Block-7

45

0.246

0.035

0.505

0.0162

8

Block-7

45

0.246

0.036

0.505

0.0166

9

Block-7

45

0.246

0.037

0.505

0.0167

10

Block-7

45

0.246

0.038

0.505

0.0172

11

Block-7

45

0.246

0.039

0.505

0.0178

12

Block-7

45

0.246

0.040

0.505

0.0179

13

Block-7

45

0.246

0.041

0.505

0.0181

14

Block-7

45

0.246

0.042

0.505

0.0183

15

Block-7

45

0.246

0.043

0.505

0.0190

16

Block-7

45

0.246

0.044

0.505

0.0197

17

Block-7

45

0.246

0.045

0.505

0.0199

18

Block-7

45

0.246

0.046

0.505

0.0201

19

Block-7

45

0.246

0.047

0.505

0.0191

20

Block-7

45

0.246

0.048

0.505

0.0217

21

Block-7

45

0.246

0.049

0.505

0.0220

22

Block-7

45

0.246

0.050

0.505

0.0226

23

Block-7

45

0.246

0.051

0.505

0.0236

24

Block-7

45

0.246

0.052

0.505

0.0239

25

Block-7

45

0.246

0.053

0.510

0.0240

26

Block-7

45

0.246

0.054

0.505

0.0241

27

Block-7

45

0.246

0.055

0.505

0.0246

28

Block-7

45

0.246

0.056

0.505

0.0247

29

Block-7

45

0.246

0.057

0.505

0.0248

30

Block-7

45

0.246

0.058

0.505

0.0249

31

Block-7

45

0.246

0.059

0.505

0.0251

32

Block-7

45

0.246

0.060

0.505

0.0257

33

Block-1

45

0.246

0.045

0.505

0.0319

34

Block-2

45

0.246

0.045

0.505

0.0294

35

Block-3

45

0.246

0.045

0.505

0.0282

36

Block-4

45

0.246

0.045

0.505

0.0262

37

Block-5

45

0.246

0.045

0.505

0.0243

38

Block-6

45

0.246

0.045

0.505

0.0223

39

Block-7

45

0.246

0.045

0.505

0.0196

40

Block-8

45

0.246

0.045

0.505

0.0197

41

Block-9

45

0.246

0.045

0.505

0.0198

42

Block-10

45

0.246

0.045

0.505

0.0184

43

Block-11

45

0.246

0.045

0.505

0.0173

44

Block-12

45

0.246

0.045

0.505

0.0165

45

Block-13

45

0.246

0.045

0.404

0.0153

46

Block-14

45

0.246

0.045

0.404

0.0124

47

Block-15

45

0.246

0.045

0.505

0.0066

48

Block-7

45

0.202

0.045

0.404

0.0220

49

Block-7

45

0.204

0.045

0.404

0.0219

50

Block-7

45

0.206

0.045

0.404

0.0218

51

Block-7

45

0.208

0.045

0.404

0.0217

52

Block-7

45

0.210

0.045

0.404

0.0216

53

Block-7

45

0.212

0.045

0.404

0.0215

54

Block-7

45

0.214

0.045

0.505

0.0214

55

Block-7

45

0.216

0.045

0.505

0.0214

56

Block-7

45

0.218

0.045

0.505

0.0213

57

Block-7

45

0.220

0.045

0.505

0.0212

58

Block-7

45

0.222

0.045

0.505

0.0211

59

Block-7

45

0.224

0.045

0.505

0.0208

60

Block-7

45

0.226

0.045

0.505

0.0203

61

Block-7

45

0.228

0.045

0.505

0.0202

62

Block-7

45

0.230

0.045

0.505

0.0201

63

Block-7

45

0.232

0.045

0.505

0.0201

64

Block-7

45

0.234

0.045

0.505

0.0200

65

Block-7

45

0.236

0.045

0.505

0.0199

66

Block-7

45

0.238

0.045

0.404

0.0196

67

Block-7

45

0.240

0.045

0.404

0.0194

68

Block-7

45

0.242

0.045

0.404

0.0193

69

Block-7

45

0.244

0.045

0.404

0.0192

70

Block-7

45

0.246

0.045

0.505

0.0190

71

Block-7

45

0.248

0.045

0.505

0.0189

72

Block-7

45

0.250

0.045

0.505

0.0187

73

Block-7

45

0.252

0.045

0.505

0.0187

74

Block-7

45

0.254

0.045

0.505

0.0186

75

Block-7

45

0.256

0.045

0.505

0.0184

76

Block-7

45

0.258

0.045

0.505

0.0182

77

Block-7

45

0.259

0.045

0.505

0.0183

78

Block-7

45

0.260

0.045

0.505

0.0191

79

Block-7

45

0.261

0.045

0.505

0.0192

80

Block-7

45

0.262

0.045

0.505

0.0194

81

Block-7

45

0.263

0.045

0.505

0.0195

82

Block-7

45

0.264

0.045

0.505

0.0195

83

Block-7

45

0.265

0.045

0.505

0.0197

84

Block-7

45

0.266

0.045

0.505

0.0197

85

Block-7

45

0.267

0.045

0.505

0.0198

86

Block-7

45

0.270

0.045

0.505

0.0199

87

Block-7

30

0.246

0.045

0.505

0.0101

88

Block-7

35

0.246

0.045

0.505

0.0107

89

Block-7

36

0.246

0.045

0.505

0.0111

90

Block-7

37

0.246

0.045

0.505

0.0116

91

Block-7

38

0.246

0.045

0.505

0.0117

92

Block-7

39

0.246

0.045

0.505

0.0119

93

Block-7

40

0.246

0.045

0.505

0.0121

94

Block-7

41

0.246

0.045

0.505

0.0127

95

Block-7

42

0.246

0.045

0.404

0.0154

96

Block-7

43

0.246

0.045

0.404

0.0157

97

Block-7

44

0.246

0.045

0.404

0.0162

98

Block-7

45

0.246

0.045

0.505

0.0197

99

Block-7

50

0.246

0.045

0.505

0.0221

100

Block-7

55

0.246

0.045

0.505

0.0233

In all these 100 simulations, the wave gauge was consistently positioned at coordinates X=1.09 m, Y=1.21 m, and Z=0.05 m. The dominant wave period for each simulation was determined using the Fast Fourier Transform (FFT) function in MATLAB (MathWorks, 2023). Furthermore, the classification of wave types was carried out using a wave categorization graph according to Sorensen (2010), as shown in Fig. 4a. The results indicate that the majority of the simulated waves are on the border between intermediate and deep-water waves, and they are categorized as Stokes waves (Fig. 4a). Four sample waveforms from our 100 numerical experiments are provided in Fig. 4b.

The dataset in Table 4 was used to derive a new predictive equation that incorporates travel distance for the first time to estimate the initial maximum tsunami amplitude. In developing this equation, a genetic algorithm optimization technique was implemented using MATLAB (MathWorks 2023). This advanced approach entailed the use of genetic algorithms (GAs), an evolutionary algorithm type inspired by natural selection processes (MathWorks, 2023). This technique is iterative, involving selection, crossover, and mutation processes to evolve solutions over several generations. The goal was to identify the optimal coefficients and powers for each landslide parameter in the predictive equation, ensuring a robust and reliable model for estimating maximum wave amplitudes. Genetic Algorithms excel at optimizing complex models by navigating through extensive combinations of coefficients and exponents. GAs effectively identify highly suitable solutions for the non-linear and complex relationships between inputs (e.g., slide volume, slope angle, travel distance, water depth) and the output (i.e., maximum initial wave amplitude, a_{M}). MATLAB’s computational environment enhances this process, providing robust tools for GA to adapt and evolve solutions iteratively, ensuring the precision of the predictive model (Onnen et al., 1997). This approach leverages MATLAB’s capabilities to fine-tune parameters dynamically, achieving an optimal equation that accurately estimates a_{M}. It is important to highlight that the nondimensionalized version of this dataset is employed to develop a predictive equation which enables the equation to reproduce the maximum initial wave amplitude (a_{M}) for various subaerial landslide cases, independent of their dimensional differences (e.g., Heler and Hager 2014; Heller and Spinneken 2015; Sabeti and Heidarzadeh 2022b). For this nondimensionalization, we employed the water depth (h) to nondimensionalize the slide volume (V/h^{3}) and travel distance (D/h). The slide thickness (s) was applied to nondimensionalize the water depth (h/s).

2.5. Landslide velocity

In discussing the critical role of landslide velocity for simulating landslide-generated waves, we focus on the mechanisms of landslide motion and the techniques used to record landslide velocity in our simulations (Fig. 5). Also, we examine how these methods were applied in two distinct scenarios: Lab 1 and Lab 2 (see Table 1 for their details). Regarding the process of landslide movement, a slide starts from a stationary state, gaining momentum under the influence of gravity and this acceleration continues until the landslide collides with water, leading to a significant reduction in its speed before eventually coming to a stop (Fig. 5) (e.g., Panizzo et al. 2005).

To measure the landslide’s velocity in our simulations, we attached a probe at the centre of the slide, which supplied a time series of the velocity data. The slide’s velocity (v_{s}) peaks at the moment it enters the water (Fig. 5), a point referred to as the impact time (t_{Imp}). Following this initial impact, the slides continue their underwater movement, eventually coming to a complete halt (t_{Stop}). Given the results in Fig. 5, it can be seen that Lab 1, with its longer travel distance (0.070 m), exhibits a higher peak velocity of 1.89 m/s. This increase in velocity is attributed to the extended travel distance allowing more time for the slide to accelerate under gravity. Whereas Lab 2, featuring a shorter travel distance (0.045 m), records a lower peak velocity of 1.78 m/s. This difference underscores how travel distance significantly influences the dynamics of landslide motion. After reaching the peak, both profiles show a sharp decrease in velocity, marking the transition to submarine motion until the slides come to a complete stop (t_{Stop}). There are noticeable differences observable in Fig. 5 between the Lab-1 and Lab-2 simulations, including the peaks at 0.3 s . These variations might stem from the placement of the wave gauge, which differs slightly in each scenario, as well as the water depth’s minor discrepancies and, the travel distance.

2.6. Effect of air entrainment

In this section we examine whether it is required to consider air entrainment for our modelling or not as the FLOW-3D HYDRO package is capable of modelling air entrainment. The process of air entrainment in water during a landslide tsunami and its subsequent transport involve two key components: the quantification of air entrainment at the water surface, and the simulation of the air’s transport within the fluid (Hirt, 2003). FLOW-3D HYDRO employs the air entrainment model to compute the volume of air entrained at the water’s surface utilizing three approaches: a constant density model, a variable density model accounting for bulking, and a buoyancy model that adds the Drift-FLUX mechanism to variable density conditions (Flow Science, 2023). The calculation of the entrainment rate is based on the following equation:(2)�������=������[2(��−�����−2�/���)]1/2where parameters are: V_{air}, volume of air; C_{air}, entrainment rate coefficient; A_{s}, surface area of fluid; ρ, fluid density; k, turbulent kinetic energy; g_{n}, gravity normal to surface; L_{t}, turbulent length scale; and σ, surface tension coefficient. The value of k is directly computed from the Reynolds-averaged Navier-Stokes (RANS) (k–w) calculations in our model.

In this study, we selected the variable density + Drift-FLUX model, which effectively captures the dynamics of phase separation and automatically activates the constant density and variable density models. This method simplifies the air-water mixture, treating it as a single, homogeneous fluid within each computational cell. For the phase volume fractions f_{1}and f_{2}, the velocities are expressed in terms of the mixture and relative velocities, denoted as u and u_{r}, respectively, as follows:(3)��1��+�.(�1�)=��1��+�.(�1�)−�.(�1�2��)=0(4)��2��+�.(�2�)=��2��+�.(�2�)−�.(�1�2��)=0

The outcomes from this simulation are displayed in Fig. 6, which indicates that the influence of air entrainment on the generated wave amplitude is approximately 2 %. A value of 0.02 for the entrained air volume fraction means that, in the simulated fluid, approximately 2 % of the volume is composed of entrained air. In other words, for every unit volume of the fluid-air mixture at that location, 2 % is air and the remaining 98 % is water. The configuration of Test-17 (Table 4) was employed for this simulation. While the effect of air entrainment is anticipated to be more significant in models of granular landslide-generated waves (Fritz, 2002), in our simulations we opted not to incorporate this module due to its negligible impact on the results.

3. Results

In this section, we begin by presenting a sequence of our 3D simulations capturing different time steps to illustrate the generation process of landslide-generated waves. Subsequently, we derive a new predictive equation to estimate the maximum initial wave amplitude of landslide-generated waves and assess its performance.

3.1. Wave generation and propagation

To demonstrate the wave generation process in our simulation, we reference Test-17 from Table 4, where we employed Block-7 (Tables 3, 4). In this configuration, the slope angle was set to 45°, with a water depth of 0.246 m and a travel distance at 0.045 m (Fig. 7). At 0.220 s, the initial impact of the moving slide on the water is depicted, marking the onset of the wave generation process (Fig. 7a). Disturbances are localized to the immediate area of impact, with the rest of the water surface remaining undisturbed. At this time, a maximum water particle velocity of 1.0 m/s – 1.2 m/s is seen around the impact zone (Fig. 7d). Moving to 0.320 s, the development of the wave becomes apparent as energy transfer from the landslide to the water creates outwardly radiating waves with maximum water particle velocity of up to around 1.6 m/s – 1.8 m/s (Fig. 7b, e). By the time 0.670 s, the wave has fully developed and is propagating away from the impact point exhibiting maximum water particle velocity of up to 2.0 m/s – 2.1 m/s. Concentric wave fronts are visible, moving outwards in all directions, with a colour gradient signifying the highest wave amplitude near the point of landslide entry, diminishing with distance (Fig. 7c, f).

3.2. Influence of landslide parameters on tsunami amplitude

In this section, we investigate the effects of various landslide parameters namely slide volume (V), water depth (h), slipe angle (α) and travel distance (D) on the maximum initial wave amplitude (a_{M}). Fig. 8 presents the outcome of these analyses. According to Fig. 8, the slide volume, slope angle, and travel distance exhibit a direct relationship with the wave amplitude, meaning that as these parameters increase, so does the amplitude. Conversely, water depth is inversely related to the maximum initial wave amplitude, suggesting that the deeper the water depth, the smaller the maximum wave amplitude will be (Fig. 8b).

Fig. 8a highlights the pronounced impact of slide volume on the a_{M}, demonstrating a direct correlation between the two variables. For instance, in the range of slide volumes we modelled (Fig. 8a), The smallest slide volume tested, measuring 0.10 × 10^{−3} m^{3}, generated a low initial wave amplitude (a_{M}= 0.0066 m) (Table 4). In contrast, the largest volume tested, 6.25 × 10^{−3} m^{3}, resulted in a significantly higher initial wave amplitude (a_{M}= 0.0319 m) (Table 4). The extremities of these results emphasize the slide volume’s paramount impact on wave amplitude, further elucidated by their positions as the smallest and largest a_{M} values across all conducted tests (Table 4). This is corroborated by findings from the literature (e.g., Murty, 2003), which align with the observed trend in our simulations.

The slope angle’s influence on a_{M} was smooth. A steady increase of wave amplitude was observed as the slope angle increased (Fig. 8c). In examining travel distance, an anomaly was identified. At a travel distance of 0.047 m, there was an unexpected dip in a_{M}, which deviates from the general increasing trend associated with longer travel distances. This singular instance could potentially be attributed to a numerical error. Beyond this point, the expected pattern of increasing a_{M} with longer travel distances resumes, suggesting that the anomaly at 0.047 m is an outlier in an otherwise consistent trend, and thus this single data point was overlooked while deriving the predictive equation. Regarding the inverse relationship between water depth and wave amplitude, our result (Fig. 8b) is consistent with previous reports by Fritz et al. (2003), (2004), and Watts et al. (2005).

The insights from Fig. 8 informed the architecture of the predictive equation in the next Section, with slide volume, travel distance, and slope angle being multiplicatively linked to wave amplitude underscoring their direct correlations with wave amplitude. Conversely, water depth is incorporated as a divisor, representing its inverse relationship with wave amplitude. This structure encapsulates the dynamics between the landslide parameters and their influence on the maximum initial wave amplitude as discussed in more detail in the next Section.

3.3. Predictive equation

Building on our sensitivity analysis of landslide parameters, as detailed in Section 3.2, and utilizing our nondimensional dataset, we have derived a new predictive equation as follows:(5)��/ℎ=0.015(tan�)0.10(�ℎ3)0.90(�ℎ)0.10(ℎ�)−0.11where, V is sliding volume, h is water depth, α is slope angle, and s is landslide thickness. It is important to note that this equation is valid only for subaerial solid-block landslide tsunamis as all our experiments were for this type of waves. The performance of this equation in predicting simulation data is demonstrated by the satisfactory alignment of data points around a 45° line, indicating its accuracy and reliability with regard to the experimental dataset (Fig. 9). The quality of fit between the dataset and Eq. (5) is 91 % indicating that Eq. (5) represents the dataset very well. Table 5 presents Eq. (5) alongside four other similar equations previously published. Two significant distinctions between our Eq. (5) and these others are: (i) Eq. (5) is derived from 3D experiments, whereas the other four equations are based on 2D experiments. (ii) Unlike the other equations, our Eq. (5) incorporates travel distance as an independent parameter.

Table 5. Performance comparison among our newly-developed equation and existing equations for estimating the maximum initial amplitude (a_{M}) of the 2018 Anak Krakatau subaerial landslide tsunami. Parameters: a_{M}, initial maximum wave amplitude; h, water depth; v_{s}, landslide velocity; V, slide volume; b_{s}, slide width; l_{s}, slide length; s, slide thickness; α, slope angle; and ����, volume of the final immersed landslide. We considered ����= V as the slide volume.

Geometrical and kinematic parameters of the 2018 Anak Krakatau subaerial landslide based on Heidarzadeh et al. (2020), Grilli et al. (2019) and Grilli et al. (2021): V=2.11 × 10^{7} m^{3}, h= 50 m; s= 114 m; α= 45°; l_{s}=1250 m; b_{s}= 2700 m; v_{s}=44.9 m/s; D= 2500 m; a_{M}= 100 m −150 m.⁎⁎

a_{M}= An average value of a_{M} = 134 m is considered in this study.⁎⁎⁎

The equation of Bolin et al. (2014) is based on the reformatted one reported by Lindstrøm (2016).⁎⁎⁎⁎

Error is calculated using Eq. (1), where the calculated a_{M} is assumed as the simulated value.

Additionally, we evaluated the performance of this equation using the real-world data from the 2018 Anak Krakatau subaerial landslide tsunami. Based on previous studies (Heidarzadeh et al., 2020; Grilli et al., 2019, 2021), we were able to provide a list of parameters for the subaerial landslide and associated tsunami for the 2018 Anak Krakatau event (see footnote of Table 5). We note that the data of the 2018 Anak Krakatau event was not used while deriving Eq. (5). The results indicate that Eq. (5) predicts the initial amplitude of the 2018 Anak Krakatau tsunami as being 130 m indicating an error of 2.9 % compared to the reported average amplitude of 134 m for this event. This performance indicates an improvement compared to the previous equation reported by Sabeti and Heidarzadeh (2022a) (Table 5). In contrast, the equations from Robbe-Saule et al. (2021) and Bolin et al. (2014) demonstrate higher discrepancies of 4200 % and 77 %, respectively (Table 5). Although Noda’s (1970) equation reproduces the tsunami amplitude of 134 m accurately (Table 5), it is crucial to consider its limitations, notably not accounting for parameters such as slope angle and travel distance.

It is essential to recognize that both travel distance and slope angle significantly affect wave amplitude. In our model, captured in Eq. (5), we integrate the slope angle (α) through the tangent function, i.e., tan α. This choice diverges from traditional physical interpretations that often employ the cosine or sine function (e.g., Heller and Hager, 2014; Watts et al., 2003). We opted for the tangent function because it more effectively reflects the direct impact of slope steepness on wave generation, yielding superior estimations compared to conventional methods.

The significance of this study lies in its application of both physical and numerical 3D experiments and the derivation of a predictive equation based on 3D results. Prior research, e.g. Heller et al. (2016), has reported notable discrepancies between 2D and 3D wave amplitudes, highlighting the important role of 3D experiments. It is worth noting that the suitability of applying an equation derived from either 2D or 3D data depends on the specific geometry and characteristics inherent in the problem being addressed. For instance, in the case of a long, narrow dam reservoir, an equation derived from 2D data would likely be more suitable. In such contexts, the primary dynamics of interest such as flow patterns and potential wave propagation are predominantly two-dimensional, occurring along the length and depth of the reservoir. This simplification to 2D for narrow dam reservoirs allows for more accurate modelling of these dynamics.

This study specifically investigates waves initiated by landslides, focusing on those characterized as solid blocks instead of granular flows, with slope angles confined to a range of 25° to 60°. We acknowledge the additional complexities encountered in real-world scenarios, such as dynamic density and velocity of landslides, which could affect the estimations. The developed equation in this study is specifically designed to predict the maximum initial amplitude of tsunamis for the aforementioned specified ranges and types of landslides.

4. Conclusions

Both physical and numerical experiments were undertaken in a 3D wave basin to study solid-block landslide-generated waves and to formulate a predictive equation for their maximum initial wave amplitude. At the beginning, two physical experiments were performed to validate and calibrate a 3D numerical model, which was subsequently utilized to generate 100 experiments by varying different landslide parameters. The generated database was then used to derive a predictive equation for the maximum initial wave amplitude of landslide tsunamis. The main features and outcomes are:

•The predictive equation of this study is exclusively derived from 3D data and exhibits a fitting quality of 91 % when applied to the database.

•For the first time, landslide travel distance was considered in the predictive equation. This inclusion provides more accuracy and flexibility for applying the equation.

•To further evaluate the performance of the predictive equation, it was applied to a real-world subaerial landslide tsunami (i.e., the 2018 Anak Krakatau event) and delivered satisfactory performance.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

RS is supported by the Leverhulme Trust Grant No. RPG-2022-306. MH is funded by open funding of State Key Lab of Hydraulics and Mountain River Engineering, Sichuan University, grant number SKHL2101. We acknowledge University of Bath Institutional Open Access Fund. MH is also funded by the Great Britain Sasakawa Foundation grant no. 6217 (awarded in 2023).

Acknowledgements

Authors are sincerely grateful to the laboratory technician team, particularly Mr William Bazeley, at the Faculty of Engineering, University of Bath for their support during the laboratory physical modelling of this research. We appreciate the valuable insights provided by Mr. Brian Fox (Senior CFD Engineer at Flow Science, Inc.) regarding air entrainment modelling in FLOW-3D HYDRO. We acknowledge University of Bath Institutional Open Access Fund.

Data availability

All data used in this study are given in the body of the article.

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Gonzalo Duró, Mariano De Dios, Alfredo López, Sergio O. Liscia

ABSTRACT

This study presents comparisons between the results of a commercial CFD code and physical model measurements. The case study is a hydro-combined power station operating in spillway mode for a given scenario. Two turbulence models and two scales are implemented to identify the capabilities and limitations of each approach and to determine the selection criteria for CFD modeling for this kind of structure. The main flow characteristics are considered for analysis, but the focus is on a fluctuating frequency phenomenon for accurate quantitative comparisons. Acceptable representations of the general hydraulic functioning are found in all approaches, according to physical modeling. The k-ε RNG, and LES models give good representation of the discharge flow, mean water depths, and mean pressures for engineering purposes. The k-ε RNG is not able to characterize fluctuating phenomena at a model scale but does at a prototype scale. The LES is capable of identifying the dominant frequency at both prototype and model scales. A prototype-scale approach is recommended for the numerical modeling to obtain a better representation of fluctuating pressures for both turbulence models, with the complement of physical modeling for the ultimate design of the hydraulic structures.

본 연구에서는 상용 CFD 코드 결과와 물리적 모델 측정 결과를 비교합니다. 사례 연구는 주어진 시나리오에 대해 배수로 모드에서 작동하는 수력 복합 발전소입니다.

각 접근 방식의 기능과 한계를 식별하고 이러한 종류의 구조에 대한 CFD 모델링의 선택 기준을 결정하기 위해 두 개의 난류 모델과 두 개의 스케일이 구현되었습니다. 주요 흐름 특성을 고려하여 분석하지만 정확한 정량적 비교를 위해 변동하는 주파수 현상에 중점을 둡니다.

일반적인 수리학적 기능에 대한 허용 가능한 표현은 물리적 모델링에 따라 모든 접근 방식에서 발견됩니다. k-ε RNG 및 LES 모델은 엔지니어링 목적을 위한 배출 유량, 평균 수심 및 평균 압력을 잘 표현합니다.

k-ε RNG는 모델 규모에서는 변동 현상을 특성화할 수 없지만 프로토타입 규모에서는 특성을 파악합니다. LES는 프로토타입과 모델 규모 모두에서 주요 주파수를 식별할 수 있습니다.

수력학적 구조의 궁극적인 설계를 위한 물리적 모델링을 보완하여 두 난류 모델에 대한 변동하는 압력을 더 잘 표현하기 위해 수치 모델링에 프로토타입 규모 접근 방식이 권장됩니다.

ADRIAN R. J. (2007). “Hairpin vortex organization in wall turbulence.” Phys. Fluids 19(4), 041301. DEWALS B., ARCHAMBEAU P., RULOT F., PIROTTON M. and ERPICUM S. (2013). “Physical and Numerical Modelling in Low-Head Structures Design.” Proc. International Workshop on Hydraulic Design of Low-Head Structures, Aachen, Germany, Bundesanstalt für Wasserbau Publ., D.B. BUNG and S. PAGLIARA Editors, pp.11-30. GRENANDER, U. (1959). Probability and Statistics: The Harald Cramér Volume. Wiley. HIRT, C. W. and NICHOLS B. D. (1981). “Volume of fluid (VOF) method for the dynamics of free boundaries.” Journal of Computational Physics 39(1): 201-225. JOHNSON M. C. and SAVAGE B. M. (2006). “Physical and numerical comparison of flow over ogee spillway in the presence of tailwater.” J. Hydraulic Eng. 132(12): 1353–1357. KHAN L.A., WICKLEIN E.A., RASHID M., EBNER L.L. and RICHARDS N.A. (2004). “Computational fluid dynamics modeling of turbine intake hydraulics at a hydropower plant.” Journal of Hydraulic Research, 42:1, 61-69 LAROCQUE L.A., IMRAN J. and CHAUDHRY M. (2013). “3D numerical simulation of partial breach dam-break flow using the LES and k–ϵ turbulence models.” Jl of Hydraulic Research, 51:2, 145-157 LI S., LAI Y., WEBER L., MATOS SILVA J. and PATEL V.C. (2004). “Validation of a threedimensional numerical model for water-pump intakes.” Journal of Hydraulic Research, 42:3, 282-292 NOVAK P., GUINOT V., JEFFREY A. and REEVE D.E. (2010). “Hydraulic modelling – An introduction.” Spon Press, London and New York, ISBN 978-0-419-25010-4, 616 pp.

This work investigates numerically a local scour moves in irregular waves around tripods. It is constructed and proven to use the numerical model of the seabed-tripodfluid with an RNG k turbulence model. The present numerical model then examines the flow velocity distribution and scour characteristics. After that, the suggested computational model Flow-3D is a useful tool for analyzing and forecasting the maximum scour development and the flow field in random waves around tripods. The scour values affecting the foundations of the tripod must be studied and calculated, as this phenomenon directly and negatively affects the structure of the structure and its design life. The lower diagonal braces and the main column act as blockages, increasing the flow accelerations underneath them. This increases the number of particles that are moved, which in turn creates strong scouring in the area. The numerical model has a good agreement with the experimental model, with a maximum percentage of error of 10% between the experimental and numerical models. In addition, Based on dimensional analysis parameters, an empirical equation has been devised to forecast scour depth with flow depth, median size ratio, Keulegan-Carpenter (Kc), Froud number flow, and wave velocity that the results obtained in this research at various flow velocities and flow depths demonstrated that the maximum scour depth rate depended on wave height with rising velocities and decreasing particle sizes (d50) and the scour depth attains its steady-current value for Vw < 0.75. As the Froude number rises, the maximum scour depth will be large.

Weirs are essential structures used to manage excess water flow from behind dams to downstream areas. Enhancing discharge efficiency often involves extending the effective length of Piano Key Weirs (PKW) in dams or regulating flow within irrigation and drainage networks. This study employed both numerical and laboratory investigations to assess the impact of different base nose shapes installed beneath the outlet keys and varying Input to output key width ratios (Wi/Wo) on discharges ranging from 5 to 80 liters per second. Furthermore, the study aimed to achieve research objectives and compare the performance of Piano Key Weirs with Ogee Weir. For numerical simulation, the optimal number of cells for meshing was determined, and an appropriate turbulence model was selected. The results indicated that the numerical model accurately simulated the laboratory sample with a high degree of precision. Moreover, the numerical model closely approximated PKW for all parameters Q, H, and Cd compared to the laboratory sample. The findings revealed that in laboratory models with a maximum discharge area of 80 liters per second, the weir with Wi/Wo=1.2 and a flow head value of 285 mm exhibited the lowest value, whereas the weir with Wi/Wo=0.71 and a flow head value of 305 mm showed the highest, attributed to the higher discharge in the input-output ratio. Additionally, as the ratio of flow head to weir height H/P increased, the discharge coefficient Cd decreased. Comparing the flow conditions in weirs with different base nose shapes, it was observed that the weir with a spindle nose shape (PKW1.2S) outperformed the PKW with a flat (PKW1.2), semi-cylindrical (PKW1.2CL) and triangular base nose (PKW1.2TR). The results emphasized that models featuring semi-cylindrical and flat noses exhibited notable flow deviation and abrupt disruption upon impact with the nose. However, this effect was significantly reduced in models equipped with triangular and spindle-shaped noses. Also, the coefficient of discharge in PKW1.2S and PKW1.2TR weirs, compared to the PKW1.20 weir, increased by 27% and 20%, respectively.

웨어는 댐 뒤에서 하류 지역으로의 과도한 물 흐름을 관리하는 데 사용되는 필수 구조물입니다. 배출 효율을 높이는 데에는 댐의 피아노 키 위어(PKW) 유효 길이를 연장하거나 관개 및 배수 네트워크 내 흐름을 조절하는 것이 포함됩니다.

이 연구에서는 콘센트 키 아래에 설치된 다양한 베이스 노즈 모양과 초당 5~80리터 범위의 배출에 대한 다양한 입력 대 출력 키 너비 비율(Wi/Wo)의 영향을 평가하기 위해 수치 및 실험실 조사를 모두 사용했습니다. 또한 본 연구에서는 연구 목적을 달성하고 Piano Key Weir와 Ogee Weir의 성능을 비교하는 것을 목표로 했습니다.

수치 시뮬레이션을 위해 메시 생성을 위한 최적의 셀 수를 결정하고 적절한 난류 모델을 선택했습니다. 결과는 수치 모델이 높은 정밀도로 실험실 샘플을 정확하게 시뮬레이션했음을 나타냅니다. 더욱이, 수치 모델은 실험실 샘플과 비교하여 모든 매개변수 Q, H 및 Cd에 대해 PKW에 매우 근접했습니다.

연구 결과, 최대 배출 면적이 초당 80리터인 실험실 모델에서는 Wi/Wo=1.2, 플로우 헤드 값이 285mm인 웨어가 가장 낮은 값을 나타냈고, Wi/Wo=0.71 및 a인 웨어는 가장 낮은 값을 나타냈습니다. 플로우 헤드 값은 305mm로 가장 높은 것으로 나타났는데, 이는 입출력 비율의 높은 토출량에 기인합니다. 또한, 웨어 높이에 대한 유수두 비율 H/P가 증가함에 따라 유출계수 Cd는 감소하였다.

베이스 노즈 모양이 다른 웨어의 흐름 조건을 비교해 보면, 스핀들 노즈 모양(PKW1.2S)의 웨어가 평면(PKW1.2), 반원통형(PKW1.2CL) 및 삼각형 모양의 PKW보다 성능이 우수한 것으로 관찰되었습니다. 베이스 노즈(PKW1.2TR) 결과는 반원통형 및 편평한 노즈를 특징으로 하는 모델이 노즈에 충격을 가할 때 눈에 띄는 흐름 편차와 급격한 중단을 나타냄을 강조했습니다.

그러나 삼각형 및 방추형 노즈를 장착한 모델에서는 이러한 효과가 크게 감소했습니다. 또한 PKW1.20보에 비해 PKW1.2S보와 PKW1.2TR보의 유출계수는 각각 27%, 20% 증가하였다.

Keywords

Piano Key Weir, Base Nose Shape, Flow Hydraulics, Numerical Model, Triangular Nose Shape, Flat Nose Shape, Semi-Cylindrical Nose Shape, Spindle Nose Shape

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Alireza Khoshkonesh^{1}, Blaise Nsom^{2}, Saeid Okhravi^{3}*, Fariba Ahmadi Dehrashid^{4}, Payam Heidarian^{5}, Silvia DiFrancesco^{6} ^{1 }Department of Geography, School of Social Sciences, History, and Philosophy, Birkbeck University of London, London, UK. ^{2} Université de Bretagne Occidentale. IRDL/UBO UMR CNRS 6027. Rue de Kergoat, 29285 Brest, France. ^{3} Institute of Hydrology, Slovak Academy of Sciences, Dúbravská cesta 9, 84104, Bratislava, Slovak Republic. ^{4}Department of Water Science and Engineering, Faculty of Agriculture, Bu-Ali Sina University, 65178-38695, Hamedan, Iran. ^{5 }Department of Civil, Environmental, Architectural Engineering and Mathematics, University of Brescia, 25123 Brescia, Italy.^{ } ^{6}Niccol`o Cusano University, via Don C. Gnocchi 3, 00166 Rome, Italy. * Corresponding author. Tel.: +421-944624921. E-mail: saeid.okhravi@savba.sk

Abstract

This study aimed to comprehensively investigate the influence of substrate level difference and material composition on dam break wave evolution over two different erodible beds. Utilizing the Volume of Fluid (VOF) method, we tracked free surface advection and reproduced wave evolution using experimental data from the literature. For model validation, a comprehensive sensitivity analysis encompassed mesh resolution, turbulence simulation methods, and bed load transport equations. The implementation of Large Eddy Simulation (LES), non-equilibrium sediment flux, and van Rijn’s (1984) bed load formula yielded higher accuracy compared to alternative approaches. The findings emphasize the significant effect of substrate level difference and material composition on dam break morphodynamic characteristics. Decreasing substrate level disparity led to reduced flow velocity, wavefront progression, free surface height, substrate erosion, and other pertinent parameters. Initial air entrapment proved substantial at the wavefront, illustrating pronounced air-water interaction along the bottom interface. The Shields parameter experienced a one-third reduction as substrate level difference quadrupled, with the highest near-bed concentration observed at the wavefront. This research provides fresh insights into the complex interplay of factors governing dam break wave propagation and morphological changes, advancing our comprehension of this intricate phenomenon.

이 연구는 두 개의 서로 다른 침식층에 대한 댐 파괴파 진화에 대한 기질 수준 차이와 재료 구성의 영향을 종합적으로 조사하는 것을 목표로 했습니다. VOF(유체량) 방법을 활용하여 자유 표면 이류를 추적하고 문헌의 실험 데이터를 사용하여 파동 진화를 재현했습니다.

모델 검증을 위해 메쉬 해상도, 난류 시뮬레이션 방법 및 침대 하중 전달 방정식을 포함하는 포괄적인 민감도 분석을 수행했습니다. LES(Large Eddy Simulation), 비평형 퇴적물 플럭스 및 van Rijn(1984)의 하상 부하 공식의 구현은 대체 접근 방식에 비해 더 높은 정확도를 산출했습니다.

연구 결과는 댐 붕괴 형태역학적 특성에 대한 기질 수준 차이와 재료 구성의 중요한 영향을 강조합니다. 기판 수준 차이가 감소하면 유속, 파면 진행, 자유 표면 높이, 기판 침식 및 기타 관련 매개변수가 감소했습니다.

초기 공기 포집은 파면에서 상당한 것으로 입증되었으며, 이는 바닥 경계면을 따라 뚜렷한 공기-물 상호 작용을 보여줍니다. 기판 레벨 차이가 4배로 증가함에 따라 Shields 매개변수는 1/3로 감소했으며, 파면에서 가장 높은 베드 근처 농도가 관찰되었습니다.

이 연구는 댐 파괴파 전파와 형태학적 변화를 지배하는 요인들의 복잡한 상호 작용에 대한 새로운 통찰력을 제공하여 이 복잡한 현상에 대한 이해를 향상시킵니다.

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Farhoud Kalateh ^{a},*, Ehsan Aminvash ^{a} and Rasoul Daneshfaraz ^{b} ^{a} Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran ^{b} Faculty of Engineering, University of Maragheh, Maragheh, Iran *Corresponding author. E-mail: f.kalateh@gmail.com

ABSTRACT

The main goal of the present study is to investigate the effects of macro-roughnesses downstream of the inclined drop through numerical models. Due to the vital importance of geometrical properties of the macro-roughnesses in the hydraulic performance and efficient energy dissipation downstream of inclined drops, two different geometries of macro-roughnesses, i.e., semi-circular and triangular geometries, have been investigated using the Flow-3D model. Numerical simulation showed that with the flow rate increase and relative critical depth, the flow energy consumption has decreased. Also, relative energy dissipation increases with the increase in height and slope angle, so that this amount of increase in energy loss compared to the smooth bed in semi-circular and triangular elements is 86.39 and 76.80%, respectively, in the inclined drop with a height of 15 cm and 86.99 and 65.78% in the drop with a height of 20 cm. The Froude number downstream on the uneven bed has been dramatically reduced, so this amount of reduction has been approximately 47 and 54% compared to the control condition. The relative depth of the downstream has also increased due to the turbulence of the flow on the uneven bed with the increase in the flow rate.

본 연구의 주요 목표는 수치 모델을 통해 경사 낙하 하류의 거시 거칠기 효과를 조사하는 것입니다. 수력학적 성능과 경사 낙하 하류의 효율적인 에너지 소산에서 거시 거칠기의 기하학적 특성이 매우 중요하기 때문에 두 가지 서로 다른 거시 거칠기 형상, 즉 반원형 및 삼각형 형상이 Flow를 사용하여 조사되었습니다.

3D 모델 수치 시뮬레이션을 통해 유량이 증가하고 상대 임계 깊이가 증가함에 따라 유동 에너지 소비가 감소하는 것으로 나타났습니다. 또한, 높이와 경사각이 증가함에 따라 상대적인 에너지 소산도 증가하는데, 반원형 요소와 삼각형 요소에서 평활층에 비해 에너지 손실의 증가량은 경사낙하에서 각각 86.39%와 76.80%입니다.

높이 15cm, 높이 20cm의 드롭에서 86.99%, 65.78%입니다. 고르지 못한 베드 하류의 프루드 수가 극적으로 감소하여 이 감소량은 대조 조건에 비해 약 47%와 54%였습니다. 유속이 증가함에 따라 고르지 못한 층에서의 흐름의 난류로 인해 하류의 상대적 깊이도 증가했습니다.

Key words

flow energy dissipation, Froude number, inclined drop, numerical simulation

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웨어의 두 가지 서로 다른 배열(즉, 직선형 웨어와 직사각형 미로 웨어)을 사용하여 웨어 모양, 웨어 간격, 웨어의 오리피스 존재, 흐름 영역에 대한 바닥 경사와 같은 기하학적 매개변수의 영향을 평가했습니다.

유량과 수심의 관계, 수심 평균 속도의 변화와 분포, 난류 특성, 어도에서의 에너지 소산. 흐름 조건에 미치는 영향을 조사하기 위해 FLOW-3D® 소프트웨어를 사용하여 전산 유체 역학 시뮬레이션을 수행했습니다.

수치 모델은 계산된 표면 프로파일과 속도를 문헌의 실험적으로 측정된 값과 비교하여 검증되었습니다. 수치 모델과 실험 데이터의 결과, 급락유동의 표면 프로파일과 표준화된 속도 프로파일에 대한 평균 제곱근 오차와 평균 절대 백분율 오차가 각각 0.014m와 3.11%로 나타나 수치 모델의 능력을 확인했습니다.

수영장과 둑의 흐름 특성을 예측합니다. 각 모델에 대해 L/B = 1.83(L: 웨어 거리, B: 수로 폭) 값에서 급락 흐름이 발생할 수 있고 L/B = 0.61에서 스트리밍 흐름이 발생할 수 있습니다. 직사각형 미로보 모델은 기존 모델보다 무차원 방류량(Q+)이 더 큽니다.

수중 흐름의 기존 보와 직사각형 미로 보의 경우 Q는 각각 1.56과 1.47h에 비례합니다(h: 보 위 수심). 기존 웨어의 풀 내 평균 깊이 속도는 직사각형 미로 웨어의 평균 깊이 속도보다 높습니다.

그러나 주어진 방류량, 바닥 경사 및 웨어 간격에 대해 난류 운동 에너지(TKE) 및 난류 강도(TI) 값은 기존 웨어에 비해 직사각형 미로 웨어에서 더 높습니다. 기존의 웨어는 직사각형 미로 웨어보다 에너지 소산이 더 낮습니다.

더 낮은 TKE 및 TI 값은 미로 웨어 상단, 웨어 하류 벽 모서리, 웨어 측벽과 채널 벽 사이에서 관찰되었습니다. 보와 바닥 경사면 사이의 거리가 증가함에 따라 평균 깊이 속도, 난류 운동 에너지의 평균값 및 난류 강도가 증가하고 수영장의 체적 에너지 소산이 감소했습니다.

둑에 개구부가 있으면 평균 깊이 속도와 TI 값이 증가하고 풀 내에서 가장 높은 TKE 범위가 감소하여 두 모델 모두에서 물고기를 위한 휴식 공간이 더 넓어지고(TKE가 낮아짐) 에너지 소산율이 감소했습니다.

Two different arrangements of the weir (i.e., straight weir and rectangular labyrinth weir) were used to evaluate the effects of geometric parameters such as weir shape, weir spacing, presence of an orifice at the weir, and bed slope on the flow regime and the relationship between discharge and depth, variation and distribution of depth-averaged velocity, turbulence characteristics, and energy dissipation at the fishway. Computational fluid dynamics simulations were performed using FLOW-3D® software to examine the effects on flow conditions. The numerical model was validated by comparing the calculated surface profiles and velocities with experimentally measured values from the literature. The results of the numerical model and experimental data showed that the root-mean-square error and mean absolute percentage error for the surface profiles and normalized velocity profiles of plunging flows were 0.014 m and 3.11%, respectively, confirming the ability of the numerical model to predict the flow characteristics of the pool and weir. A plunging flow can occur at values of L/B = 1.83 (L: distance of the weir, B: width of the channel) and streaming flow at L/B = 0.61 for each model. The rectangular labyrinth weir model has larger dimensionless discharge values (Q+) than the conventional model. For the conventional weir and the rectangular labyrinth weir at submerged flow, Q is proportional to 1.56 and 1.47h, respectively (h: the water depth above the weir). The average depth velocity in the pool of a conventional weir is higher than that of a rectangular labyrinth weir. However, for a given discharge, bed slope, and weir spacing, the turbulent kinetic energy (TKE) and turbulence intensity (TI) values are higher for a rectangular labyrinth weir compared to conventional weir. The conventional weir has lower energy dissipation than the rectangular labyrinth weir. Lower TKE and TI values were observed at the top of the labyrinth weir, at the corner of the wall downstream of the weir, and between the side walls of the weir and the channel wall. As the distance between the weirs and the bottom slope increased, the average depth velocity, the average value of turbulent kinetic energy and the turbulence intensity increased, and the volumetric energy dissipation in the pool decreased. The presence of an opening in the weir increased the average depth velocity and TI values and decreased the range of highest TKE within the pool, resulted in larger resting areas for fish (lower TKE), and decreased the energy dissipation rates in both models.

1 Introduction

Artificial barriers such as detour dams, weirs, and culverts in lakes and rivers prevent fish from migrating and completing the upstream and downstream movement cycle. This chain is related to the life stage of the fish, its location, and the type of migration. Several riverine fish species instinctively migrate upstream for spawning and other needs. Conversely, downstream migration is a characteristic of early life stages [1]. A fish ladder is a waterway that allows one or more fish species to cross a specific obstacle. These structures are constructed near detour dams and other transverse structures that have prevented such migration by allowing fish to overcome obstacles [2]. The flow pattern in fish ladders influences safe and comfortable passage for ascending fish. The flow’s strong turbulence can reduce the fish’s speed, injure them, and delay or prevent them from exiting the fish ladder. In adult fish, spawning migrations are usually complex, and delays are critical to reproductive success [3].

Various fish ladders/fishways include vertical slots, denil, rock ramps, and pool weirs [1]. The choice of fish ladder usually depends on many factors, including water elevation, space available for construction, and fish species. Pool and weir structures are among the most important fish ladders that help fish overcome obstacles in streams or rivers and swim upstream [1]. Because they are easy to construct and maintain, this type of fish ladder has received considerable attention from researchers and practitioners. Such a fish ladder consists of a sloping-floor channel with series of pools directly separated by a series of weirs [4]. These fish ladders, with or without underwater openings, are generally well-suited for slopes of 10% or less [1, 2]. Within these pools, flow velocities are low and provide resting areas for fish after they enter the fish ladder. After resting in the pools, fish overcome these weirs by blasting or jumping over them [2]. There may also be an opening in the flooded portion of the weir through which the fish can swim instead of jumping over the weir. Design parameters such as the length of the pool, the height of the weir, the slope of the bottom, and the water discharge are the most important factors in determining the hydraulic structure of this type of fish ladder [3]. The flow over the weir depends on the flow depth at a given slope S0 and the pool length, either “plunging” or “streaming.” In plunging flow, the water column h over each weir creates a water jet that releases energy through turbulent mixing and diffusion mechanisms [5]. The dimensionless discharges for plunging (Q+) and streaming (Q*) flows are shown in Fig. 1, where Q is the total discharge, B is the width of the channel, w is the weir height, S_{0} is the slope of the bottom, h is the water depth above the weir, d is the flow depth, and g is the acceleration due to gravity. The maximum velocity occurs near the top of the weir for plunging flow. At the water’s surface, it drops to about half [6].

Extensive experimental studies have been conducted to investigate flow patterns for various physical geometries (i.e., bed slope, pool length, and weir height) [2]. Guiny et al. [7] modified the standard design by adding vertical slots, orifices, and weirs in fishways. The efficiency of the orifices and vertical slots was related to the velocities at their entrances. In the laboratory experiments of Yagci [8], the three-dimensional (3D) mean flow and turbulence structure of a pool weir fishway combined with an orifice and a slot is investigated. It is shown that the energy dissipation per unit volume and the discharge have a linear relationship.

Considering the beneficial characteristics reported in the limited studies of researchers on the labyrinth weir in the pool-weir-type fishway, and knowing that the characteristics of flow in pool-weir-type fishways are highly dependent on the geometry of the weir, an alternative design of the rectangular labyrinth weir instead of the straight weirs in the pool-weir-type fishway is investigated in this study [7, 9]. Kim [10] conducted experiments to compare the hydraulic characteristics of three different weir types in a pool-weir-type fishway. The results show that a straight, rectangular weir with a notch is preferable to a zigzag or trapezoidal weir. Studies on natural fish passes show that pass ability can be improved by lengthening the weir’s crest [7]. Zhong et al. [11] investigated the semi-rigid weir’s hydraulic performance in the fishway’s flow field with a pool weir. The results showed that this type of fishway performed better with a lower invert slope and a smaller radius ratio but with a larger pool spacing.

Considering that an alternative method to study the flow characteristics in a fishway with a pool weir is based on numerical methods and modeling from computational fluid dynamics (CFD), which can easily change the geometry of the fishway for different flow fields, this study uses the powerful package CFD and the software FLOW-3D to evaluate the proposed weir design and compare it with the conventional one to extend the application of the fishway. The main objective of this study was to evaluate the hydraulic performance of the rectangular labyrinth pool and the weir with submerged openings in different hydraulic configurations. The primary objective of creating a new weir configuration for suitable flow patterns is evaluated based on the swimming capabilities of different fish species. Specifically, the following questions will be answered: (a) How do the various hydraulic and geometric parameters relate to the effects of water velocity and turbulence, expressed as turbulent kinetic energy (TKE) and turbulence intensity (TI) within the fishway, i.e., are conventional weirs more affected by hydraulics than rectangular labyrinth weirs? (b) Which weir configurations have the greatest effect on fish performance in the fishway? (c) In the presence of an orifice plate, does the performance of each weir configuration differ with different weir spacing, bed gradients, and flow regimes from that without an orifice plate?

2 Materials and Methods

2.1 Physical Model Configuration

This paper focuses on Ead et al. [6]’s laboratory experiments as a reference, testing ten pool weirs (Fig. 2). The experimental flume was 6 m long, 0.56 m wide, and 0.6 m high, with a bottom slope of 10%. Field measurements were made at steady flow with a maximum flow rate of 0.165 m^{3}/s. Discharge was measured with magnetic flow meters in the inlets and water level with point meters (see Ead et al. [6]. for more details). Table 1 summarizes the experimental conditions considered for model calibration in this study.

Table 1 Experimental conditions considered for calibration

Computational fluid dynamics (CFD) simulations were performed using FLOW-3D® v11.2 to validate a series of experimental liner pool weirs by Ead et al. [6] and to investigate the effects of the rectangular labyrinth pool weir with an orifice. The dimensions of the channel and data collection areas in the numerical models are the same as those of the laboratory model. Two types of pool weirs were considered: conventional and labyrinth. The proposed rectangular labyrinth pool weirs have a symmetrical cross section and are sized to fit within the experimental channel. The conventional pool weir model had a pool length of l = 0.685 and 0.342 m, a weir height of w = 0.141 m, a weir width of B = 0.56 m, and a channel slope of S_{0} = 5 and 10%. The rectangular labyrinth weirs have the same front width as the offset, i.e., a = b = c = 0.186 m. A square underwater opening with a width of 0.05 m and a depth of 0.05 m was created in the middle of the weir. The weir configuration considered in the present study is shown in Fig. 3.

2.3 Governing Equations

FLOW-3D® software solves the Navier–Stokes–Reynolds equations for three-dimensional analysis of incompressible flows using the fluid-volume method on a gridded domain. FLOW -3D® uses an advanced free surface flow tracking algorithm (TruVOF) developed by Hirt and Nichols [12], where fluid configurations are defined in terms of a VOF function F (x, y, z, t). In this case, F (fluid fraction) represents the volume fraction occupied by the fluid: F = 1 in cells filled with fluid and F = 0 in cells without fluid (empty areas) [4, 13]. The free surface area is at an intermediate value of F. (Typically, F = 0.5, but the user can specify a different intermediate value.) The equations in Cartesian coordinates (x, y, z) applicable to the model are as follows:

�f∂�∂�+∂(���x)∂�+∂(���y)∂�+∂(���z)∂�=�SOR

(1)

∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�x+�x

(2)

∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�y+�y

(3)

∂�∂�+1�f(��x∂�∂�+��y∂�∂�+��z∂�∂�)=−1�∂�∂�+�z+�z

(4)

where (u, v, w) are the velocity components, (A_{x}, A_{y}, A_{z}) are the flow area components, (G_{x}, G_{y}, G_{z}) are the mass accelerations, and (f_{x}, f_{y}, f_{z}) are the viscous accelerations in the directions (x, y, z), ρ is the fluid density, R_{SOR} is the spring term, V_{f} is the volume fraction associated with the flow, and P is the pressure. The k–ε turbulence model (RNG) was used in this study to solve the turbulence of the flow field. This model is a modified version of the standard k–ε model that improves performance. The model is a two-equation model; the first equation (Eq. 5) expresses the turbulence’s energy, called turbulent kinetic energy (k) [14]. The second equation (Eq. 6) is the turbulent dissipation rate (ε), which determines the rate of dissipation of kinetic energy [15]. These equations are expressed as follows Dasineh et al. [4]:

In these equations, k is the turbulent kinetic energy, ε is the turbulent energy consumption rate, G_{k} is the generation of turbulent kinetic energy by the average velocity gradient, with empirical constants α_{ε} = α_{k} = 1.39, C_{1ε} = 1.42, and C_{2ε} = 1.68, eff is the effective viscosity, μ_{eff} = μ + μ_{t} [15]. Here, μ is the hydrodynamic density coefficient, and μ_{t} is the turbulent density of the fluid.

2.4 Meshing and the Boundary Conditions in the Model Setup

The numerical area is divided into three mesh blocks in the X-direction. The meshes are divided into different sizes, a containing mesh block for the entire spatial domain and a nested block with refined cells for the domain of interest. Three different sizes were selected for each of the grid blocks. By comparing the accuracy of their results based on the experimental data, the reasonable mesh for the solution domain was finally selected. The convergence index method (GCI) evaluated the mesh sensitivity analysis. Based on this method, many researchers, such as Ahmadi et al. [16] and Ahmadi et al. [15], have studied the independence of numerical results from mesh size. Three different mesh sizes with a refinement ratio (r) of 1.33 were used to perform the convergence index method. The refinement ratio is the ratio between the larger and smaller mesh sizes (r = G_{coarse}/G_{fine}). According to the recommendation of Celik et al. [17], the recommended number for the refinement ratio is 1.3, which gives acceptable results. Table 2 shows the characteristics of the three mesh sizes selected for mesh sensitivity analysis.Table 2 Characteristics of the meshes tested in the convergence analysis

The results of u_{1} = u_{max} (u_{1} = velocity component along the x_{1} axis and u_{max} = maximum velocity of u_{1} in a section perpendicular to the invert of the fishway) at Q = 0.035 m^{3}/s, × 1/l = 0.66, and Y_{1}/b = 0 in the pool of conventional weir No. 4, obtained from the output results of the software, were used to evaluate the accuracy of the calculation range. As shown in Fig. 4, x_{1} = the distance from a given weir in the x-direction, Y_{1} = the water depth measured in the y-direction, Y_{0} = the vertical distance in the Cartesian coordinate system, h = the water column at the crest, b = the distance between the two points of maximum velocity u_{max} and zero velocity, and l = the pool length.

The apparent index of convergence (p) in the GCI method is calculated as follows:

�=ln(�3−�2)(�2−�1)/ln(�)

(7)

f_{1}, f_{2}, and f_{3} are the hydraulic parameters obtained from the numerical simulation (f_{1} corresponds to the small mesh), and r is the refinement ratio. The following equation defines the convergence index of the fine mesh:

GCIfine=1.25|ε|��−1

(8)

Here, ε = (f_{2} − f_{1})/f_{1} is the relative error, and f_{2} and f_{3} are the values of hydraulic parameters considered for medium and small grids, respectively. GCI_{12} and GCI_{23} dimensionless indices can be calculated as:

GCI12=1.25|�2−�1�1|��−1

(9)

Then, the independence of the network is preserved. The convergence index of the network parameters obtained by Eqs. (7)–(9) for all three network variables is shown in Table 3. Since the GCI values for the smaller grid (GCI_{12}) are lower compared to coarse grid (GCI_{23}), it can be concluded that the independence of the grid is almost achieved. No further change in the grid size of the solution domain is required. The calculated values (GCI_{23}/r^{p}GCI_{12}) are close to 1, which shows that the numerical results obtained are within the convergence range. As a result, the meshing of the solution domain consisting of a block mesh with a mesh size of 0.012 m and a block mesh within a larger block mesh with a mesh size of 0.009 m was selected as the optimal mesh (Fig. 5).Table 3 GCI calculation

The boundary conditions applied to the area are shown in Fig. 6. The boundary condition of specific flow rate (volume flow rate-Q) was used for the inlet of the flow. For the downstream boundary, the flow output (outflow-O) condition did not affect the flow in the solution area. For the Z_{max} boundary, the specified pressure boundary condition was used along with the fluid fraction = 0 (P). This type of boundary condition considers free surface or atmospheric pressure conditions (Ghaderi et al. [19]). The wall boundary condition is defined for the bottom of the channel, which acts like a virtual wall without friction (W). The boundary between mesh blocks and walls were considered a symmetrical condition (S).

The convergence of the steady-state solutions was controlled during the simulations by monitoring the changes in discharge at the inlet boundary conditions. Figure 7 shows the time series plots of the discharge obtained from the Model A for the three main discharges from the numerical results. The 8 s to reach the flow equilibrium is suitable for the case of the fish ladder with pool and weir. Almost all discharge fluctuations in the models are insignificant in time, and the flow has reached relative stability. The computation time for the simulations was between 6 and 8 h using a personal computer with eight cores of a CPU (Intel Core i7-7700K @ 4.20 GHz and 16 GB RAM).

3 Results

3.1 Verification of Numerical Results

Quantitative outcomes, including free surface and normalized velocity profiles obtained using FLOW-3D software, were reviewed and compared with the results of Ead et al. [6]. The fourth pool was selected to present the results and compare the experiment and simulation. For each quantity, the percentage of mean absolute error (MAPE (%)) and root-mean-square error (RMSE) are calculated. Equations (10) and (11) show the method used to calculate the errors.

MAPE(%)100×1�∑1�|�exp−�num�exp|

(10)

RMSE(−)1�∑1�(�exp−�num)2

(11)

Here, X_{exp} is the value of the laboratory data, X_{num} is the numerical data value, and n is the amount of data. As shown in Fig. 8, let x_{1} = distance from a given weir in the x-direction and Y_{1} = water depth in the y-direction from the bottom. The trend of the surface profiles for each of the numerical results is the same as that of the laboratory results. The surface profiles of the plunging flows drop after the flow enters and then rises to approach the next weir. The RMSE and MAPE error values for Model A are 0.014 m and 3.11%, respectively, indicating acceptable agreement between numerical and laboratory results. Figure 9 shows the velocity vectors and plunging flow from the numerical results, where x and y are horizontal and vertical to the flow direction, respectively. It can be seen that the jet in the fish ladder pool has a relatively high velocity. The two vortices, i.e., the enclosed vortex rotating clockwise behind the weir and the surface vortex rotating counterclockwise above the jet, are observed for the regime of incident flow. The point where the jet meets the fish passage bed is shown in the figure. The normalized velocity profiles upstream and downstream of the impact points are shown in Fig. 10. The figure shows that the numerical results agree well with the experimental data of Ead et al. [6].

3.2 Flow Regime and Discharge-Depth Relationship

Depending on the geometric shape of the fishway, including the distance of the weir, the slope of the bottom, the height of the weir, and the flow conditions, the flow regime in the fishway is divided into three categories: dipping, transitional, and flow regimes [4]. In the plunging flow regime, the flow enters the pool through the weir, impacts the bottom of the fishway, and forms a hydraulic jump causing two eddies [2, 20]. In the streamwise flow regime, the surface of the flow passing over the weir is almost parallel to the bottom of the channel. The transitional regime has intermediate flow characteristics between the submerged and flow regimes. To predict the flow regime created in the fishway, Ead et al. [6] proposed two dimensionless parameters, Q_{t}* and L/w, where Q_{t}* is the dimensionless discharge, L is the distance between weirs, and w is the height of the weir:

��∗=���0���

(12)

Q is the total discharge, B is the width of the channel, S_{0} is the slope of the bed, and g is the gravity acceleration. Figure 11 shows different ranges for each flow regime based on the slope of the bed and the distance between the pools in this study. The results of Baki et al. [21], Ead et al. [6] and Dizabadi et al. [22] were used for this comparison. The distance between the pools affects the changes in the regime of the fish ladder. So, if you decrease the distance between weirs, the flow regime more likely becomes. This study determined all three flow regimes in a fish ladder. When the corresponding range of Q_{t}* is less than 0.6, the flow regime can dip at values of L/B = 1.83. If the corresponding range of Q_{t}* is greater than 0.5, transitional flow may occur at L/B = 1.22. On the other hand, when Q_{t}* is greater than 1, streamwise flow can occur at values of L/B = 0.61. These observations agree well with the results of Baki et al. [21], Ead et al. [6] and Dizabadi et al. [22].

For plunging flows, another dimensionless discharge (Q+) versus h/w given by Ead et al. [6] was used for further evaluation:

�+=��ℎ�ℎ=23�d�

(13)

where h is the water depth above the weir, and C_{d} is the discharge coefficient. Figure 12a compares the numerical and experimental results of Ead et al. [6]. In this figure, Rehbock’s empirical equation is used to estimate the discharge coefficient of Ead et al. [6].

�d=0.57+0.075ℎ�

(14)

The numerical results for the conventional weir (Model A) and the rectangular labyrinth weir (Model B) of this study agree well with the laboratory results of Ead et al. [6]. When comparing models A and B, it is also found that a rectangular labyrinth weir has larger Q + values than the conventional weir as the length of the weir crest increases for a given channel width and fixed headwater elevation. In Fig. 12b, Models A and B’s flow depth plot shows the plunging flow regime. The power trend lines drawn through the data are the best-fit lines. The data shown in Fig. 12b are for different bed slopes and weir geometries. For the conventional weir and the rectangular labyrinth weir at submerged flow, Q can be assumed to be proportional to 1.56 and 1.47h, respectively. In the results of Ead et al. [6], Q is proportional to 1.5h. If we assume that the flow through the orifice is Q_{o} and the total outflow is Q, the change in the ratio of Q_{o}/Q to total outflow for models A and B can be shown in Fig. 13. For both models, the flow through the orifice decreases as the total flow increases. A logarithmic trend line was also found between the total outflow and the dimensionless ratio Q_{o}/Q.

3.3 Depth-Averaged Velocity Distributions

To ensure that the target fish species can pass the fish ladder with maximum efficiency, the average velocity in the fish ladder should be low enough [4]. Therefore, the average velocity in depth should be as much as possible below the critical swimming velocities of the target fishes at a constant flow depth in the pool [20]. The contour plot of depth-averaged velocity was used instead of another direction, such as longitudinal velocity because fish are more sensitive to depth-averaged flow velocity than to its direction under different hydraulic conditions. Figure 14 shows the distribution of depth-averaged velocity in the pool for Models A and B in two cases with and without orifice plates. Model A’s velocity within the pool differs slightly in the spanwise direction. However, no significant variation in velocity was observed. The flow is gradually directed to the sides as it passes through the rectangular labyrinth weir. This increases the velocity at the sides of the channel. Therefore, the high-velocity zone is located at the sides. The low velocity is in the downstream apex of the weir. This area may be suitable for swimming target fish. The presence of an opening in the weir increases the flow velocity at the opening and in the pool’s center, especially in Model A. The flow velocity increase caused by the models’ opening varied from 7.7 to 12.48%. Figure 15 illustrates the effect of the inverted slope on the averaged depth velocity distribution in the pool at low and high discharge. At constant discharge, flow velocity increases with increasing bed slope. In general, high flow velocity was found in the weir toe sidewall and the weir and channel sidewalls.

On the other hand, for a constant bed slope, the high-velocity area of the pool increases due to the increase in runoff. For both bed slopes and different discharges, the most appropriate path for fish to travel from upstream to downstream is through the middle of the cross section and along the top of the rectangular labyrinth weirs. The maximum dominant velocities for Model B at S_{0} = 5% were 0.83 and 1.01 m/s; at S_{0} = 10%, they were 1.12 and 1.61 m/s at low and high flows, respectively. The low mean velocities for the same distance and S_{0} = 5 and 10% were 0.17 and 0.26 m/s, respectively.

Figure 16 shows the contour of the averaged depth velocity for various distances from the weir at low and high discharge. The contour plot shows a large variation in velocity within short distances from the weir. At L/B = 0.61, velocities are low upstream and downstream of the top of the weir. The high velocities occur in the side walls of the weir and the channel. At L/B = 1.22, the low-velocity zone displaces the higher velocity in most of the pool. Higher velocities were found only on the sides of the channel. As the discharge increases, the velocity zone in the pool becomes wider. At L/B = 1.83, there is an area of higher velocities only upstream of the crest and on the sides of the weir. At high discharge, the prevailing maximum velocities for L/B = 0.61, 1.22, and 1.83 were 1.46, 1.65, and 1.84 m/s, respectively. As the distance between weirs increases, the range of maximum velocity increases.

On the other hand, the low mean velocity for these distances was 0.27, 0.44, and 0.72 m/s, respectively. Thus, the low-velocity zone decreases with increasing distance between weirs. Figure 17 shows the pattern distribution of streamlines along with the velocity contour at various distances from the weir for Q = 0.05 m^{3}/s. A stream-like flow is generally formed in the pool at a small distance between weirs (L/B = 0.61). The rotation cell under the jet forms clockwise between the two weirs. At the distances between the spillways (L/B = 1.22), the transition regime of the flow is formed. The transition regime occurs when or shortly after the weir is flooded. The rotation cell under the jet is clockwise smaller than the flow regime and larger than the submergence regime. At a distance L/B = 1.83, a plunging flow is formed so that the plunging jet dips into the pool and extends downstream to the center of the pool. The clockwise rotation of the cell is bounded by the dipping jet of the weir and is located between the bottom and the side walls of the weir and the channel.

Figure 18 shows the average depth velocity bar graph for each weir at different bed slopes and with and without orifice plates. As the distance between weirs increases, all models’ average depth velocity increases. As the slope of the bottom increases and an orifice plate is present, the average depth velocity in the pool increases. In addition, the average pool depth velocity increases as the discharge increases. Among the models, Model A’s average depth velocity is higher than Model B’s. The variation in velocity ranged from 8.11 to 12.24% for the models without an orifice plate and from 10.26 to 16.87% for the models with an orifice plate.

3.4 Turbulence Characteristics

The turbulent kinetic energy is one of the important parameters reflecting the turbulent properties of the flow field [23]. When the k value is high, more energy and a longer transit time are required to migrate the target species. The turbulent kinetic energy is defined as follows:

�=12(�x′2+�y′2+�z′2)

(15)

where u_{x}, u_{y}, and u_{z} are fluctuating velocities in the x, y, and z directions, respectively. An illustration of the TKE and the effects of the geometric arrangement of the weir and the presence of an opening in the weir is shown in Fig. 19. For a given bed slope, in Model A, the highest TKE values are uniformly distributed in the weir’s upstream portion in the channel’s cross section. In contrast, for the rectangular labyrinth weir (Model B), the highest TKE values are concentrated on the sides of the pool between the crest of the weir and the channel wall. The highest TKE value in Models A and B is 0.224 and 0.278 J/kg, respectively, at the highest bottom slope (S_{0} = 10%). In the downstream portion of the conventional weir and within the crest of the weir and the walls of the rectangular labyrinth, there was a much lower TKE value that provided the best conditions for fish to recover in the pool between the weirs. The average of the lowest TKE for bottom slopes of 5 and 10% in Model A is 0.041 and 0.056 J/kg, and for Model B, is 0.047 and 0.064 J/kg. The presence of an opening in the weirs reduces the area of the highest TKE within the pool. It also increases the resting areas for fish (lower TKE). The highest TKE at the highest bottom slope in Models A and B with an orifice is 0.208 and 0.191 J/kg, respectively.

Figure 20 shows the effect of slope on the longitudinal distribution of TKE in the pools. TKE values significantly increase for a given discharge with an increasing bottom slope. Thus, for a low bed slope (S_{0} = 5%), a large pool area has expanded with average values of 0.131 and 0.168 J/kg for low and high discharge, respectively. For a bed slope of S_{0} = 10%, the average TKE values are 0.176 and 0.234 J/kg. Furthermore, as the discharge increases, the area with high TKE values within the pool increases. Lower TKE values are observed at the apex of the labyrinth weir, at the corner of the wall downstream of the weir, and between the side walls of the weir and the channel wall for both bottom slopes. The effect of distance between weirs on TKE is shown in Fig. 21. Low TKE values were observed at low discharge and short distances between weirs. Low TKE values are located at the top of the rectangular labyrinth weir and the downstream corner of the weir wall. There is a maximum value of TKE at the large distances between weirs, L/B = 1.83, along the center line of the pool, where the dip jet meets the bottom of the bed. At high discharge, the maximum TKE value for the distance L/B = 0.61, 1.22, and 1.83 was 0.246, 0.322, and 0.417 J/kg, respectively. In addition, the maximum TKE range increases with the distance between weirs.

For TKE size, the average value (TKE_{ave}) is plotted against q in Fig. 22. For all models, the TKE values increase with increasing q. For example, in models A and B with L/B = 0.61 and a slope of 10%, the TKE value increases by 41.66 and 86.95%, respectively, as q increases from 0.1 to 0.27 m^{2}/s. The TKE values in Model B are higher than Model A for a given discharge, bed slope, and weir distance. The TKE_{ave} in Model B is higher compared to Model A, ranging from 31.46 to 57.94%. The presence of an orifice in the weir reduces the TKE values in both weirs. The intensity of the reduction is greater in Model B. For example, in Models A and B with L/B = 0.61 and q = 0.1 m^{2}/s, an orifice reduces TKE_{ave} values by 60.35 and 19.04%, respectively. For each model, increasing the bed slope increases the TKE_{ave} values in the pool. For example, for Model B with q = 0.18 m^{2}/s, increasing the bed slope from 5 to 10% increases the TKE_{ave} value by 14.34%. Increasing the distance between weirs increases the TKE_{ave} values in the pool. For example, in Model B with S_{0} = 10% and q = 0.3 m^{2}/s, the TKE_{ave} in the pool increases by 34.22% if you increase the distance between weirs from L/B = 0.61 to L/B = 0.183.

Cotel et al. [24] suggested that turbulence intensity (TI) is a suitable parameter for studying fish swimming performance. Figure 23 shows the plot of TI and the effects of the geometric arrangement of the weir and the presence of an orifice. In Model A, the highest TI values are found upstream of the weirs and are evenly distributed across the cross section of the channel. The TI values increase as you move upstream to downstream in the pool. For the rectangular labyrinth weir, the highest TI values were concentrated on the sides of the pool, between the top of the weir and the side wall of the channel, and along the top of the weir. Downstream of the conventional weir, within the apex of the weir, and at the corners of the walls of the rectangular labyrinth weir, the percentage of TI was low. At the highest discharge, the average range of TI in Models A and B was 24–45% and 15–62%, respectively. The diversity of TI is greater in the rectangular labyrinth weir than the conventional weir. Fish swimming performance is reduced due to higher turbulence intensity. However, fish species may prefer different disturbance intensities depending on their swimming abilities; for example, Salmo trutta prefers a disturbance intensity of 18–53% [25]. Kupferschmidt and Zhu [26] found a higher range of TI for fishways, such as natural rock weirs, of 40–60%. The presence of an orifice in the weir increases TI values within the pool, especially along the middle portion of the cross section of the fishway. With an orifice in the weir, the average range of TI in Models A and B was 28–59% and 22–73%, respectively.

The effect of bed slope on TI variation is shown in Fig. 24. TI increases in different pool areas as the bed slope increases for a given discharge. For a low bed slope (S_{0} = 5%), a large pool area has increased from 38 to 63% and from 56 to 71% for low and high discharge, respectively. For a bed slope of S_{0} = 10%, the average values of TI are 45–67% and 61–73% for low and high discharge, respectively. Therefore, as runoff increases, the area with high TI values within the pool increases. A lower TI is observed for both bottom slopes in the corner of the wall, downstream of the crest walls, and between the side walls in the weir and channel. Figure 25 compares weir spacing with the distribution of TI values within the pool. The TI values are low at low flows and short distances between weirs. A maximum value of TI occurs at long spacing and where the plunging stream impinges on the bed and the area around the bed. TI ranges from 36 to 57%, 58–72%, and 47–76% for the highest flow in a wide pool area for L/B = 0.61, 1.22, and 1.83, respectively.

The average value of turbulence intensity (TI_{ave}) is plotted against q in Fig. 26. The increase in TI values with the increase in q values is seen in all models. For example, the average values of TI for Models A and B at L/B = 0.61 and slope of 10% increased from 23.9 to 33.5% and from 42 to 51.8%, respectively, with the increase in q from 0.1 to 0.27 m^{2}/s. For a given discharge, a given gradient, and a given spacing of weirs, the TI_{ave} is higher in Model B than Model A. The presence of an orifice in the weirs increases the TI values in both types. For example, in Models A and B with L/B = 0.61 and q = 0.1 m^{2}/s, the presence of an orifice increases TI_{ave} from 23.9 to 37.1% and from 42 to 48.8%, respectively. For each model, TI_{ave} in the pool increases with increasing bed slope. For Model B with q = 0.18 m^{2}/s, TI_{ave} increases from 37.5 to 45.8% when you increase the invert slope from 5 to 10%. Increasing the distance between weirs increases the TI_{ave} in the pool. In Model B with S_{0} = 10% and q = 0.3 m^{2}/s, the TI_{ave} in the pool increases from 51.8 to 63.7% as the distance between weirs increases from L/B = 0.61 to L/B = 0.183.

3.5 Energy Dissipation

To facilitate the passage of various target species through the pool of fishways, it is necessary to pay attention to the energy dissipation of the flow and to keep the flow velocity in the pool slow. The average volumetric energy dissipation (k) in the pool is calculated using the following basic formula:

�=����0��

(16)

where ρ is the water density, and H is the average water depth of the pool. The change in k versus Q for all models at two bottom slopes, S_{0} = 5%, and S_{0} = 10%, is shown in Fig. 27. Like the results of Yagci [8] and Kupferschmidt and Zhu [26], at a constant bottom slope, the energy dissipation in the pool increases with increasing discharge. The trend of change in k as a function of Q from the present study at a bottom gradient of S_{0} = 5% is also consistent with the results of Kupferschmidt and Zhu [26] for the fishway with rock weir. The only difference between the results is the geometry of the fishway and the combination of boulders instead of a solid wall. Comparison of the models shows that the conventional model has lower energy dissipation than the rectangular labyrinth for a given discharge. Also, increasing the distance between weirs decreases the volumetric energy dissipation for each model with the same bed slope. Increasing the slope of the bottom leads to an increase in volumetric energy dissipation, and an opening in the weir leads to a decrease in volumetric energy dissipation for both models. Therefore, as a guideline for volumetric energy dissipation, if the value within the pool is too high, the increased distance of the weir, the decreased slope of the bed, or the creation of an opening in the weir would decrease the volumetric dissipation rate.

To evaluate the energy dissipation inside the pool, the general method of energy difference in two sections can use:

ε=�1−�2�1

(17)

where ε is the energy dissipation rate, and E_{1} and E_{2} are the specific energies in Sects. 1 and 2, respectively. The distance between Sects. 1 and 2 is the same. (L is the distance between two upstream and downstream weirs.) Figure 28 shows the changes in ε relative to q (flow per unit width). The rectangular labyrinth weir (Model B) has a higher energy dissipation rate than the conventional weir (Model A) at a constant bottom gradient. For example, at S_{0} = 5%, L/B = 0.61, and q = 0.08 m^{3}/s.m, the energy dissipation rate in Model A (conventional weir) was 0.261. In Model B (rectangular labyrinth weir), however, it was 0.338 (22.75% increase). For each model, the energy dissipation rate within the pool increases as the slope of the bottom increases. For Model B with L/B = 1.83 and q = 0.178 m^{3}/s.m, the energy dissipation rate at S_{0} = 5% and 10% is 0.305 and 0.358, respectively (14.8% increase). Figure 29 shows an orifice’s effect on the pools’ energy dissipation rate. With an orifice in the weir, both models’ energy dissipation rates decreased. Thus, the reduction in energy dissipation rate varied from 7.32 to 9.48% for Model A and from 8.46 to 10.57 for Model B.

4 Discussion

This study consisted of entirely of numerical analysis. Although this study was limited to two weirs, the hydraulic performance and flow characteristics in a pooled fishway are highlighted by the rectangular labyrinth weir and its comparison with the conventional straight weir. The study compared the numerical simulations with laboratory experiments in terms of surface profiles, velocity vectors, and flow characteristics in a fish ladder pool. The results indicate agreement between the numerical and laboratory data, supporting the reliability of the numerical model in capturing the observed phenomena.

When the configuration of the weir changes to a rectangular labyrinth weir, the flow characteristics, the maximum and minimum area, and even the location of each hydraulic parameter change compared to a conventional weir. In the rectangular labyrinth weir, the flow is gradually directed to the sides as it passes the weir. This increases the velocity at the sides of the channel [21]. Therefore, the high-velocity area is located on the sides. In the downstream apex of the weir, the flow velocity is low, and this area may be suitable for swimming target fish. However, no significant change in velocity was observed at the conventional weir within the fish ladder. This resulted in an average increase in TKE of 32% and an average increase in TI of about 17% compared to conventional weirs.

In addition, there is a slight difference in the flow regime for both weir configurations. In addition, the rectangular labyrinth weir has a higher energy dissipation rate for a given discharge and constant bottom slope than the conventional weir. By reducing the distance between the weirs, this becomes even more intense. Finally, the presence of an orifice in both configurations of the weir increased the flow velocity at the orifice and in the middle of the pool, reducing the highest TKE value and increasing the values of TI within the pool of the fish ladder. This resulted in a reduction in volumetric energy dissipation for both weir configurations.

The results of this study will help the reader understand the direct effects of the governing geometric parameters on the hydraulic characteristics of a fishway with a pool and weir. However, due to the limited configurations of the study, further investigation is needed to evaluate the position of the weir’s crest on the flow direction and the difference in flow characteristics when combining boulders instead of a solid wall for this type of labyrinth weir [26]. In addition, hydraulic engineers and biologists must work together to design an effective fishway with rectangular labyrinth configurations. The migration habits of the target species should be considered when designing the most appropriate design [27]. Parametric studies and field observations are recommended to determine the perfect design criteria.

The current study focused on comparing a rectangular labyrinth weir with a conventional straight weir. Further research can explore other weir configurations, such as variations in crest position, different shapes of labyrinth weirs, or the use of boulders instead of solid walls. This would help understand the influence of different geometric parameters on hydraulic characteristics.

5 Conclusions

A new layout of the weir was evaluated, namely a rectangular labyrinth weir compared to a straight weir in a pool and weir system. The differences between the weirs were highlighted, particularly how variations in the geometry of the structures, such as the shape of the weir, the spacing of the weir, the presence of an opening at the weir, and the slope of the bottom, affect the hydraulics within the structures. The main findings of this study are as follows:

The calculated dimensionless discharge (Q_{t}*) confirmed three different flow regimes: when the corresponding range of Q_{t}* is smaller than 0.6, the regime of plunging flow occurs for values of L/B = 1.83. (L: distance of the weir; B: channel width). When the corresponding range of Q_{t}* is greater than 0.5, transitional flow occurs at L/B = 1.22. On the other hand, if Q_{t}* is greater than 1, the streaming flow is at values of L/B = 0.61.

For the conventional weir and the rectangular labyrinth weir with the plunging flow, it can be assumed that the discharge (Q) is proportional to 1.56 and 1.47h, respectively (h: water depth above the weir). This information is useful for estimating the discharge based on water depth in practical applications.

In the rectangular labyrinth weir, the high-velocity zone is located on the side walls between the top of the weir and the channel wall. A high-velocity variation within short distances of the weir. Low velocity occurs within the downstream apex of the weir. This area may be suitable for swimming target fish.

As the distance between weirs increased, the zone of maximum velocity increased. However, the zone of low speed decreased. The prevailing maximum velocity for a rectangular labyrinth weir at L/B = 0.61, 1.22, and 1.83 was 1.46, 1.65, and 1.84 m/s, respectively. The low mean velocities for these distances were 0.27, 0.44, and 0.72 m/s, respectively. This finding highlights the importance of weir spacing in determining the flow characteristics within the fishway.

The presence of an orifice in the weir increased the flow velocity at the orifice and in the middle of the pool, especially in a conventional weir. The increase ranged from 7.7 to 12.48%.

For a given bottom slope, in a conventional weir, the highest values of turbulent kinetic energy (TKE) are uniformly distributed in the upstream part of the weir in the cross section of the channel. In contrast, for the rectangular labyrinth weir, the highest TKE values were concentrated on the sides of the pool between the crest of the weir and the channel wall. The highest TKE value for the conventional and the rectangular labyrinth weir was 0.224 and 0.278 J/kg, respectively, at the highest bottom slope (S_{0} = 10%).

For a given discharge, bottom slope, and weir spacing, the average values of TI are higher for the rectangular labyrinth weir than for the conventional weir. At the highest discharge, the average range of turbulence intensity (TI) for the conventional and rectangular labyrinth weirs was between 24 and 45% and 15% and 62%, respectively. This reveals that the rectangular labyrinth weir may generate more turbulent flow conditions within the fishway.

For a given discharge and constant bottom slope, the rectangular labyrinth weir has a higher energy dissipation rate than the conventional weir (22.75 and 34.86%).

Increasing the distance between weirs decreased volumetric energy dissipation. However, increasing the gradient increased volumetric energy dissipation. The presence of an opening in the weir resulted in a decrease in volumetric energy dissipation for both model types.

Availability of data and materials

Data is contained within the article.

References

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

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

Keywords

Rubble mound breakwater

Computational fluid dynamics

Armour layer

Wave reflection coefficient

Wave transmission coefficient

Wave energy dissipation coefficient

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^{2} Researcher of Imam Hossein University, Faculty of Engineering and Passive Defense

^{3} Ivanki University, Iran.

10.22124/JCR.2023.24324.1618

Abstract

개방 수로의 심한 수력 구배는 침전으로 인한 심각한 침식과 문제를 일으킵니다. 패브릭 콘크리트는 기존의 콘크리트 표면을 대체할 수 있는 높은 실행 속도를 가진 독특한 제품입니다. 이 제품의 기계적 저항 매개변수에 따르면 부식 요인에 대한 우수한 내구성 외에도 직물 콘크리트의 응용 분야 중 하나는 운하 및 수로 암거 표면에 사용하는 것입니다. 이 연구에서는 먼저 사다리꼴 단면의 개방 채널 흐름을 직선 경로 상태의 3가지 공통 채널 형상, 굴곡 및 편차가 있는 경로, 마지막으로 채널 하단의 높이가 변경된 채널 경로를 포함하는 9가지 시나리오에서 시뮬레이션합니다. 각 주에서 flow-3d 소프트웨어를 사용한 흐름 난류 모델링과 함께 3개의 서로 다른 흐름 체제가 조사되었습니다.

FLOW-3D 소프트웨어를 사용한 유동 난류 모델링과 함께 다양한 유동 체제가 조사되었습니다. ABAQUS 소프트웨어를 사용하여 패브릭 콘크리트 구성요소와 연결 영역을 모델링하고, 콘크리트 표면과 취약한 연결 영역에 동일한 힘을 가하여 생성된 응력의 양을 확인했습니다. 결과는 생성된 응력이 직물 콘크리트의 인장 및 압축 응력 용량에 비해 매우 낮다는 것을 보여줍니다. 흐름과 콘크리트의 수력 연구를 검증하기 위해 관련 실험실 결과가 사용되었습니다.

Severe hydraulic gradients in open channels cause severe bed erosion and problems caused by sedimentation. Fabric concrete is a unique product with high execution speed that can replace traditional concrete surfaces. According to the mechanical resistance parameters of this product, in addition to its good durability against corrosive factors, one of the applications of fabric concrete is its use on the surface of canals and water course culverts. In this research, first, the flow of open channels in trapezoidal section is simulated under 9 scenarios, which include 3 common channel geometries in the state of a straight path, a path with bends and deviations, and finally, a channel path with a change in height at the bottom of the channel. In each of the states, 3 different flow regimes have been investigated along with flow turbulence modeling using flow-3d software.

Different flow regimes have been investigated along with flow turbulence modeling using flow-3d software. Using ABAQUS software, fabric concrete components and their connection areas have been modeled, and by applying forces equated to the concrete surface and vulnerable connection areas, the amount of created stresses has been checked. The results show that the created stresses are very low compared to the tensile and compressive stress capacity of fabric concrete. In order to validate the hydraulic studies of flow and concrete, the relevant laboratory results have been used.

해저 퇴적물 흐름은 퇴적물을 심해로 운반하는 주요 수단 중 하나이며, 종종 장거리를 이동하고 수십 또는 수백 킬로미터에 걸쳐 상당한 양의 퇴적물을 운반합니다. 그것의 강력한 파괴력은 종종 이동 과정에서 잠수함 유틸리티에 심각한 손상을 초래합니다.

퇴적물 흐름의 퇴적물 농도는 주변 해수와의 밀도차를 결정하며, 이 밀도 차이는 퇴적물 흐름의 흐름 능력을 결정하여 이송된 퇴적물의 최종 퇴적 위치에 영향을 미칩니다. 본 논문에서는 다양한 미사 및 점토 중량비(미사/점토 비율이라고 함)를 갖는 다양한 퇴적물 농도의 퇴적물 흐름을 수로 테스트를 통해 연구합니다.

우리의 테스트 결과는 특정 퇴적물 구성에 대해 퇴적물 흐름이 가장 빠르게 이동하는 임계 퇴적물 농도가 있음을 나타냅니다. 4가지 미사/점토 비율 각각에 대한 임계 퇴적물 농도와 이에 상응하는 최대 속도가 구해집니다. 결과는 점토 함량이 임계 퇴적물 농도와 선형적으로 음의 상관 관계가 있음을 나타냅니다.

퇴적물 농도가 증가함에 따라 퇴적물의 흐름 거동은 흐름 상태에서 붕괴된 상태로 변환되고 흐름 거동이 변화하는 두 탁한 현탁액의 유체 특성은 모두 Bingham 유체입니다.

또한 본 논문에서는 퇴적물 흐름 내 입자 배열을 분석하여 위에서 언급한 결과에 대한 미시적 설명도 제공합니다.

Submarine sediment flows is one of the main means for transporting sediment to the deep sea, often traveling long-distance and transporting significant volumes of sediment for tens or even hundreds of kilometers. Its strong destructive force often causes serious damage to submarine utilities on its course of movement. The sediment concentration of the sediment flow determines its density difference with the ambient seawater, and this density difference determines the flow ability of the sediment flow, and thus affects the final deposition locations of the transported sediment. In this paper, sediment flows of different sediment concentration with various silt and clay weight ratios (referred to as silt/clay ratio) are studied using flume tests. Our test results indicate that there is a critical sediment concentration at which sediment flows travel the fastest for a specific sediment composition. The critical sediment concentrations and their corresponding maximum velocities for each of the four silt/clay ratios are obtained. The results further indicate that the clay content is linearly negatively correlated with the critical sediment concentration. As the sediment concentration increases, the flow behaviors of sediment flows transform from the flow state to the collapsed state, and the fluid properties of the two turbid suspensions with changing flow behaviors are both Bingham fluids. Additionally, this paper also provides a microscopic explanation of the above-mentioned results by analyzing the arrangement of particles within the sediment flow.

Introduction

Submarine sediment flows are important carriers for sea floor sediment movement and may carry and transport significant volumes of sediment for tens or even hundreds of kilometers (Prior et al., 1987; Pirmez and Imran, 2003; Zhang et al., 2018). Earthquakes, storms, and floods may all trigger submarine sediment flow events (Hsu et al., 2008; Piper and Normark, 2009; Pope et al., 2017b; Gavey et al., 2017). Sediment flows have strong forces during the movement, which will cause great harm to submarine structures such as cables and pipelines (Pope et al., 2017a). It was first confirmed that the cable breaking event caused by the sediment flow occurred in 1929. The sediment flow triggered by the Grand Banks earthquake damaged 12 cables. According to the time sequence of the cable breaking, the maximum velocity of the sediment flow is as high as 28 m/s (Heezen and Ewing, 1952; Kuenen, 1952; Heezen et al., 1954). Subsequent research shows that the lowest turbidity velocity that can break the cable also needs to reach 19 m/s (Piper et al., 1988). Since then, there have been many damage events of submarine cables and oil and gas pipelines caused by sediment flows in the world (Hsu et al., 2008; Carter et al., 2012; Cattaneo et al., 2012; Carter et al., 2014). During its movement, the sediment flow will gradually deposit a large amount of sediment carried by it along the way, that is, the deposition process of the sediment flow. On the one hand, this process brings a large amount of terrestrial nutrients and other materials to the ocean, while on the other hand, it causes damage and burial to benthic organisms, thus forming the largest sedimentary accumulation on Earth – submarine fans, which are highly likely to become good reservoirs for oil and gas resources (Daly, 1936; Yuan et al., 2010; Wu et al., 2022). The study on sediment flows (such as, the study of flow velocity and the forces acting on seabed structures) can provide important references for the safe design of seabed structures, the protection of submarine ecosystems, and exploration of turbidity sediments related oil and gas deposits. Therefore, it is of great significance to study the movement of sediment flows.

The sediment flow, as a highly sediment-concentrated fluid flowing on the sea floor, has a dense bottom layer and a dilute turbulent cloud. Observations at the Monterey Canyon indicated that the sediment flow can maintain its movement over long distances if its bottom has a relatively high sediment concentration. This dense bottom layer can be very destructive along its movement path to any facilities on the sea floor (Paull et al., 2018; Heerema et al., 2020; Wang et al., 2020). The sediment flow mentioned in this research paper is the general term of sediment density flow.

The sediment flow, which occurs on the seafloor, has the potential to cause erosion along its path. In this process, the suspended sediment is replenished, allowing the sediment flow to maintain its continuous flow capacity (Zhao et al., 2018). The dynamic force of sediment flow movement stem from its own gravity and density difference with surrounding water. In cases that the gravity drive of the slope is absent (on a flat sea floor), the flow velocity and distance of sediment flows are essentially determined by the sediment composition and concentration of the sediment flows as previous studies have demonstrated. Ilstad et al. (2004) conducted underwater flow tests in a sloped tank and employed high speed video camera to perform particle tracking. The results indicated that the premixed sand-rich and clay-rich slurries demonstrated different flow velocity and flow behavior. Using mixed kaolinite(d50 = 6 μm) and silica flour(d50 = 9 μm) in three compositions with total volumetric concentration ranged 22% or 28%, Felix and Peakall (2006) carried out underwater flow tests in a 5° slope Perspex channel and found that the flow ability of sediment flows is different depending on sediment compositions and concentrations. Sumner et al. (2009) used annular flume experiments to investigate the depositional dynamics and deposits of waning sediment-laden flows, finding that decelerating fast flows with fixed sand content and variable mud content resulted in four different deposit types. Chowdhury and Testik (2011) used lock-exchange tank, and experimented the kaolin clay sediment flows in the concentration range of 25–350 g/L, and predicted the fluid mud sediment flows propagation characteristics, but this study focused on giving sediment flows propagate phase transition time parameters, and is limited to clay. Lv et al. (2017) found through experiments that the rheological properties and flow behavior of kaolin clay (d50 = 3.7 μm) sediment flows were correlated to clay concentrations. In the field monitoring conducted by Liu et al. (2023) at the Manila Trench in the South China Sea in 2021, significant differences in the velocity, movement distance, and flow morphology of turbidity currents were observed. These differences may be attributed to variations in the particle composition of the turbidity currents.

On low and gentle slopes, although sediment flow with sand as the main sediment composition moves faster, it is difficult to propagate over long distances because sand has greater settling velocity and subaqueous angle of repose. Whereas the sediment flows with silt and clay as main composition may maintain relatively stable currents. Although its movement speed is slow, it has the ability to propagate over long distances because of the low settling rate of the fine particles (Ilstad et al., 2004; Liu et al., 2023). In a field observation at the Gaoping submarine canyon, the sediments collected from the sediment flows exhibited grain size gradation and the sediment was mostly composed of silt and clay (Liu et al., 2012). At the largest deltas in the world, for instance, the Mississippi River Delta, the sediments are mainly composed of silt and clay, which generally distributed along the coast in a wide range and provided the sediment sources for further distribution. The sediment flows originated and transported sediment from the coast to the deep sea are therefore share the same sediment compositions as delta sediments. To study the sediment flows composed of silt and clay is of great importance.

The sediment concentration of the sediment flows determines the density difference between the sediment flows and the ambient water and plays a key role in its flow ability. For the sediment flow with sediment composed of silt and clay, low sediment concentration means low density and therefore leads to low flow ability; however, although high sediment concentration results in high density, since there is cohesion between fine particles, it changes fluid properties and leads to low flow ability as well. Therefore, there should be a critical sediment concentration with mixed composition of silt and clay, at which the sediment flow maintains its strongest flow capacity and have the highest movement speed. In other words, the two characteristics of particle diameter and concentration of the sediment flow determine its own motion ability, which, if occurs, may become the most destructive force to submarine structures.

The objectives of this work was to study how the sediment composition (measured in relative weight of silt and clay, and referred as silt/clay ratio) and sediment concentration affect flow ability and behavior of the sediment flows, and to quantify the critical sediment concentration at which the sediment flows reached the greatest flow velocity under the experiment setting. We used straight flume without slope and conducted a series of flume tests with varying sediment compositions (silt-rich or clay-rich) and concentrations (96 to 1212 g/L). Each sediment flow sample was tested and analyzed for rheological properties using a rheometer, in order to characterize the relationship between flow behavior and rheological properties. Combined with the particle diameter, density and viscosity characteristics of the sediment flows measured in the experiment, a numerical modeling study is conducted, which are mutually validated with the experimental results.

The sediment concentration determines the arrangements of the sediment particles in the turbid suspension, and the arrangement impacts the fluid properties of the turbid suspension. The microscopic mode of particle arrangement in the turbid suspension can be constructed to further analyze the relationship between the fluid properties of turbid suspension and the flow behaviors of the sediment flow, and then characterize the critical sediment concentration at which the sediment flow runs the fastest. A simplified microscopic model of particle arrangement in turbid suspension was constructed to analyze the microscopic arrangement characteristics of sediment particles in turbid suspension with the fastest velocity.

Section snippets

Equipment and materials

The sediment flows flow experiments were performed in a Perspex channel with smooth transparent walls. The layout and dimensions of the experimental set-up were shown in Fig. 1. The bottom of the channel was flat and straight, and a gate was arranged to separate the two tanks. In order to study the flow capacity of turbidity currents from the perspective of their own composition (particle size distribution and concentration), we used a straight channel instead of an inclined one, to avoid any

Relationship between sediment flow flow velocity and sediment concentration

After the sediment flow is generated, its movement in the first half (50 cm) of the channel is relatively stable, and there is obvious shock diffusion in the second half. The reason is that the excitation wave (similar to the surge) will be formed during the sediment flow movement, and its speed is much faster than the speed of the sediment flow head. When the excitation wave reaches the tail of the channel, it will be reflected, thus affecting the subsequent flow of the sediment flow.

Sediment flows motion simulation based on FLOW-3D

As a relatively mature 3D fluid simulation software, FLOW-3D can accurately predict the free surface flow, and has been used to simulate the movement process of sediment flows for many times (Heimsund, 2007). The model adopted in this paper is RNG turbulence model, which can better deal with the flow with high strain rate and is suitable for the simulation of sediment flows with variable shape during movement. The governing equations of the numerical model involved include continuity equation,

Conclusions

In this study, we conducted a series of sediment flow flume tests with mixed silt and clay sediment samples in four silt/clay ratios on a flat slope. Rheological measurements were carried out on turbid suspension samples and microstructure analysis of the sediment particle arrangements was conducted, we concluded that:

(1)The flow velocity of the sediment flow is controlled by the sediment concentration and its own particle diameter composition, the flow velocity increased with the increase of the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China [Grant no. 42206055]; the National Natural Science Foundation of China [Grant no. 41976049]; and the National Natural Science Foundation of China [Grant no. 42272327].

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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.

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

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

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

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

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

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

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).

Progress in physical modelling and numerical simulation of phase transitions in cryogenic pool boiling and cavitation2023, Applied Mathematical ModellingCitation Excerpt :We will not delve into cryogenic evaporation phenomena, that predominantly drive the phase-change in well-insulated storage facilities, and thus are less relevant to spill scenarios. We instead refer the reader to the works of Zuo et al. [31–33]. If the static pressure at any location in a turbomachine drops below a fluid’s saturation pressure, localized evaporation events may occur, followed by rapid collapse of the vapour cavities in a process termed “cavitation” [34].Show abstract

Thermodynamic performance in a liquid oxygen tank during active-pressurization under different gas injection temperatures2023, International Communications in Heat and Mass TransferCitation Excerpt :The volume of fluid method is adopted to predict the tank pressurization performance. The associated governing equations could refer to previous published investigations [33–39,41,45,46]. Subjected to external heat input and gas injection, the phase change occurs at the interface and within the tank.Show abstract

Interfacial mass and energy transport during steady-state evaporation in liquid oxygen storage tanks2022, Applied EnergyCitation Excerpt :However, most of them simply used the Lee model for mass transport as did for regular fluids, and seldom focus on the evaporation itself related to the interfacial temperature distribution or were unable to validate their results against credible experimental data. A recent study proposed an optimized evaporation model for the cryogenic self-pressurization with a thorough comparison between popular phase change models [8], but still lacked of experimental data to validate the results. A series of experiments have been conducted on the heat and mass transport in a thin liquid layer in the vicinity of the liquid–vapor interface of room-temperature fluids [9–14].Show abstract

Thermal destratification of cryogenic liquid storage tanks by continuous bubbling of gases2022, International Journal of Hydrogen EnergyCitation Excerpt :It was concluded that a single injector with a larger diameter configuration showed a higher chance of developing a vertical temperature gradient. Zuo et al. [48] carried out a numerical analysis to investigate the temperature distribution within the LH2 storage tank with a self-pinning spraying bar. They used the SST turbulence model coupled with the 6-DOF model.Show abstract

Fine sediments enter into the river through various sources such as channel bed, bank, and catchment. It has been regarded as a type of pollution in river. Fine sediments present in a river have a significant effect on river health. Benthic micro-organism, plants, and large fishes, all are part of food chain of river biota. Any detrimental effect on any of these components of food chain misbalances the entire riverine ecosystem. Numerous studies have been carried out on the various environmental aspects of rivers considering the presence of fine sediment in river flow. The present paper critically reviews many of these aspects to understand the various environmental impacts of suspended sediment on river health, flora and fauna.

Introduction The existence of fine sediment in a river system is a natural phenomenon. But in many cases it is exacerbated by the manmade activities. The natural cause of fines being in flow generally keeps the whole system in equilibrium except during some calamites whereas anthropogenic activities leading to fines entering into the flow puts several adverse impacts on the entire river system and its ecology. Presence of fines in flow is considered as a type of pollution in water. In United States, the fine sediment in water along with other non point source pollution is considered as a major obstacle in providing quality water for fishes and recreation activities (Diplas and Parker 1985). Sediments in a river are broadly of two types, organic and inorganic, and they both move in two ways either along the bed of the channel called bed load or in suspension called suspended load and their movements depend upon fluid flow and sediment characteristics. Further many investigators have divided the materials in suspension into two different types. One which originates from channel bed and bank is called bed material suspended load and another that migrates from feeding catchment area is called wash load. A general perception is that wash loads are very fine materials like clay, silt but it may not always be true (Woo et al. 1986). In general, suspended materials are of size less than 2 mm. The impact of sand on the various aspects of river is comparatively less than that of silt and clay. The latter are chemically active and good carrier of many contaminants and nutrients such as dioxins, phosphorous, heavy and trace metals, polychlorinated biphenyl (PCBs), radionuclide, etc. (Foster and Charlesworth 1996; Horowitz et al. 1995; Owens et al. 2001; Salomons and Förstner 1984; Stone and Droppo 1994; Thoms 1987). Foy and Bailey-Watt (1998) reported that out of 129 lakes in England and Wales, 69% have phosphorous contamination. Ten percent lakes, rivers, and bays of United States have sediment contaminants with chemicals as reported by USEPA. Several field and experimental studies have been conducted considering, sand, silt, and clay as suspended material. Hence, the subject reported herein is based on considering the fine sediment size smaller than 2 mm. Fine sediments have the ability to alter the hydraulics of the flow. Presence of fines in flow can change the magnitude of turbulence, it can change the friction resistance to flow. Fines can change the mobility and permeability of the bed material. In some extreme cases, fines in flow may even change the morphology of the river (Doeg and Koehn 1994; Nuttall 1972; Wright and Berrie 1987). Fines in the flow adversely affect the producer by increasing the turbidity, hindering the photosynthesis process by limiting the light penetration. This is ultimately reflected in the entire food ecosystem of river (Davis-Colley et al. 1992; Van Niewenhuyre and Laparrieve 1986). In addition, abrasion due to flowing sediment kills the aquatic flora (Edwards 1969; Brookes 1986). Intrusion of fines into the pores of river bed reduces space for several invertebrates, affects the spawning process (Petts 1984; Richards and Bacon 1994; Schalchli 1992). There are several other direct or indirect, short-term or long-term impacts of fines in river. The present paper reports the physical/environmental significance of fines in river. The hydraulic significance of presence of fines in the river has been reviewed in another paper (Effect of fine sediments on river hydraulics – a research review – http://dx.doi.org/10.1080/09715010.2014.982001).

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이 연구에서는 세 가지 다른 말뚝 뚜껑 높이에서 직사각형 말뚝 캡이 있는 복잡한 부두 주변의 지역 세굴 및 관련 흐름 유체 역학을 조사합니다. 말뚝 캡 높이가 초기 모래층에 대해 선택되었으며, 말뚝 캡이 흐름에 노출되지 않고(사례 I), 부분적으로 노출되고(사례 II) 완전히 노출(사례 III)되도록 했습니다. 실험은 맑은 물 세굴 조건 하에서 재순환 수로에서 수행되었으며, 입자 이미지 유속계 (PIV) 기술을 사용하여 다른 수직면에서 순간 유속을 얻었습니다. 부분적으로 노출된 파일 캡 케이스는 최대 수세미 깊이(MSD)를 보여주었습니다. 사례 II에서 MSD가 발생한 이유는 난류 유동장 분석을 통해 밝혀졌는데, 이는 말뚝 캡이 흐름에 노출됨에 따라 더 높은 세굴 깊이를 담당하는 말뚝 가장자리에서 와류 생성에 지배적으로 영향을 미친다는 것을 보여주었습니다. 유동장에 대한 파일 캡의 영향은 평균 속도, 소용돌이, 레이놀즈 전단 응력 및 난류 운동 에너지 윤곽을 통해 사례 III에서 두드러지게 나타났지만 파일 캡이 베드에서 떨어져 있었기 때문에 파일 캡 모서리는 수세미에 직접적인 영향을 미치지 않았습니다.

In this study, the local scour and the associated flow hydrodynamics around a complex pier with rectangular pile-cap at three different pile-cap elevations are investigated. The pile-cap elevations were selected with respect to the initial sand bed, such that the pile-cap was unexposed (case I), partially exposed (case II), and fully exposed (case III) to the flow. The experiments were performed in a recirculating flume under clear-water scour conditions, and the instantaneous flow velocity was obtained at different vertical planes using the particle image velocimetry (PIV) technique. The partially exposed pile-cap case showed the maximum obtained scour-depth (MSD). The reason behind the MSD occurrence in case II was enunciated through the analysis of turbulent flow field which showed that as the pile-cap got exposed to the flow, it dominantly affected the generation of vortices from the pile-cap corners responsible for the higher scour depth. The effect of the pile-cap on the flow field was prominently seen in case III through the mean velocities, vorticity, Reynolds shear stresses and turbulent kinetic energy contours, but since the pile-cap was away from the bed, the pile-cap corners did not show any direct effect on the scour.

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A series of numerical simulation were conducted to study the local scour around umbrella suction anchor foundation (USAF) under random waves. In this study, the validation was carried out firstly to verify the accuracy of the present model. Furthermore, the scour evolution and scour mechanism were analyzed respectively. In addition, two revised models were proposed to predict the equilibrium scour depth 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}.

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.

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.

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:

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].

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]:

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.

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; d_{50} 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]

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]:

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]:

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.

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:

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 d_{50} = 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]:

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

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 d_{50} = 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).

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 d_{50} = 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}.

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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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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Mahdi Feizbahr,^{1}Navid Tonekaboni,^{2}Guang-Jun Jiang,^{3,4}and 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, S_{0} 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.

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.

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.

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.

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.

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.

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].

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

S_{0}:

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

Highlights

A predominant southerly current was measured at Two-Mile Reef with occasional reversals towards the north.

Field observations indicated that northerly currents are spatially varied along Two-Mile Reef.

Simulation of reverse currents show the formation of a separated flow due to interaction with Jesser Point with Kelvin–Helmholtz type shear instabilities along the seaward edge.

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

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