Investigating Supercritical Bended Flow Using Physical Model and CFD

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Introduction
It is well known that topographical and geological conditions make it sometimes inevitable to construct spillways with bends.These bends in spillway chute play a major role in creating perturbations, shock waves [1], and vibrations.This can lead to serious hydraulic problems like cavitation in the chute of spillways and after the gates of the bottom outlet [13][14].
In the aim to understand the behaviour of these transverse waves called also rooster tail waves [2][3][4] for supercritical flows in the bend chute, a plethora of studies were done by several authors.
Ippen and Knapp [5] presented the first study.They conducted several tests with different relative radii of curvature, and they gave the first formula in order to estimate the maximum and minimum of the wave height that occurs in the bend channel.
Reinauer and Hager [6][7][8][9] studied the supercritical bend flow using experimentations.They described the flow in these conditions and the wave extrema as a function of the bend number and wave profiles with the help of three channels.
Some other researchers who focused on the same topic were introduced by [10][11][12] using 2D shallow water equations in order to compute the hydraulic parameters of the bend flow.
In the literature, we found two types of bend flow: (i) Weak bend flow: The profile of water between the outer and inner wall is almost trapezoidal and continuous.
(ii) Strong bend flow: The flow is separated, and we observe high water levels in the outer wall in comparison with the inner wall.Also, the transverse surface profile almost has the shape of a triangle [13,14].
Furthermore, the bend supercritical flow creates shock waves; thus, the flow breaks into high unstable hydraulic jumps with huge amount of air entrainment, and this results in significant hydraulic problems, namely cavitation, that can damage the chutes [15][16][17][18].
Moreover, the height of these rooster tail waves increases with the increase of the Froude number and the amount of aeration on the flow [19], [1].
The goal of this research is to simulate case studies: (i) Three curved channels with different ρ.
(ii) Bottom outlet with two curvatures downstream the gates.
The aim is to comprehend the flow behaviour, the effect of slope, and relative curvature ρ = b/R for different head discharges (H = 1 -4 m).And compare it with numerical modelling using Flow 3D Hydro in terms of the hydraulic parameters such as the water surface, velocity field, and Froude number.The second objective is to validate the solution of the separating wall.The dividing wall plays a crucial role for balancing the water profile and avoiding the creation of a vortex or a zone of stagnation of flow by the formation of a hydraulic jump.The zone of stagnation, which is created for the low head discharges, generates a problem of detachment of the jet in the ski jump, because the flow does not have enough kinetic energy to take off further away from the dam.
Five physical models were used in order to investigate the flow behaviour, patterns, velocity field, and maximum height in the curvature and to validate these results.(ii) The downstream part is also a rectangular basin in order to study the problem in relation with scour issues; this part is beyond the scope of this article.

Instrumentation
(i) Rolling point instrumentation with high accuracy is used to measure the flow height above the weir (±1 mm).
Physical model of the spillway is based on the Froude similitude defined by: To ensure that we have similar flows between model and prototype with less scale effect, we must verify certain  2 Modelling and Simulation in Engineering criteria that depend on two numbers Re (Reynolds number) and We (Weber number), defined by the following equations: We = ρhV 2 σ : ð3Þ We know that the air transport in the model is influenced essentially by scale effect; therefore, the following conditions must be verified: (i) Re model >1 to 2:5 × 10 5 (ii) We 0:5 > 110 to 170 ½19 The geometry of side channel spillway with a curved chute shown in Figure 1 for our case study is composed of: (i) Side channel trough with L-shaped weir.
(ii) Curved channel with radius of curvature of 400 m.
(iii) Ski jump at the end as an energy dissipator.
For the second case study, the curved shown in Figure 2 is composed of: (i) A channel with a width of 8.8 m and two radii of curvature of 100 m and 70 m.
(ii) The length is 133 m and has a slope of 1%.
(iii) Two walls on each side of the channel with a height of 5 m.
In order to examine and explore the flow patterns and to test the supercritical flow for bend channel, five models with a scale of 1/40 were used.
To understand the effect of curvature and bottom slope in the channels of hydraulic structures, we varied the relative curvature from 0.075 to 0.125, and the bottom slope from 1% to 18%.
We opted for the choice of testing several variants numerically and then validating the results using the experimental data.
In the first part, we will test several geometries (see Table 1) for different slopes and different ρ, to try to understand the behaviour of the collisions of the water surface with the walls of the curved channels, by theoretical calculations, numerical, and physical modelling.
We will give the maximum values of the water height for each case, the water surface for each wave angle θ varied from 0 °to 30 °.
For the curved spillway chute, the head discharges varied between 0.5 m and 4.8 m [8].
For the case study of the bottom outlet, we tested the head discharge H = 5 m.
Samples of the laboratory data will be given in comparison with the numerical approaches for each part of the present research.
We study, for the case of the bottom outlet, the part after the gates, the free surface part that is composed of two curvatures with two radii of curvatures 100 m and 70 m, respectively (Figure 2).
A calculation of the increase of the water level in this part is made in order to verify the passage of the design head discharge.The flow is supercritical, and so gravity waves manifest in the curvature.
A comparison between the water surfaces from the physical model with that of the computational fluid dynamics 3 Modelling and Simulation in Engineering (CFD) is done, and the verification of the maximum and minimum depth of water surface in order to avoid overflow over the walls.

Theoretical Calculations.
The change in the direction of flow generated in the curved channels, in supercritical regime, gives rise to two shock waves: a positive producing an increase in water depth and a negative involving a depth less than the uniform depth [6] (see Figure 3(a)).
Due to the transverse waves, Figure 3 shows the formation of shock waves in the walls in the form of zigzags (wave interference principle).It has been shown that these interferences are created in positions where they are the multiples of the wave angle θ, for example, 2θ, 3θ… (see Figure 3).
The angle β formed by the two wave fronts with the initial direction of flow depends on the Froude number upstream of the curvature F 0 and is given approximately by the following equations: The angle θ is determined, as a function of the width of the spillway chute b and the radius of the curve R, by: The maximum and minimum water heights are estimated using the following formula: The angle θ indicates the place where, along the curved channel, a maximum and a minimum of the height of water on the sidewalls are established for the first time.[6] carried out experimental work on the hydraulic characteristics of supercritical flow in a curved rectangular channel, whose objective is to define the velocity distribution and the flow area across the curve.

Hydraulic Tests. Reinauer and Hager
Concerning the velocity distribution, these tests show that the velocity remains constant across the channel and is equal to the approach velocity V 0 with the exception of the boundary layer and the wall separation zones.
As for the water surface, the torrential flow leads to significant variations across the channel.For small values of mean curvatureure ρ a = b/R a , the maximum heights (index e) along the inner and outer walls are given by: This shows that the product in equation ( 7) is involved in this phenomenon, and that the Froude number dominates the structure of the shock waves.
From an experimental point of view, only the number B 0 , called the Bend number, must vary instead of an independent variation of the Froude number and the relative curvature.
In practice, we are mainly interested in the maximum heights of shock waves to size the walls.Let h M be the maximum height along the curve and h m be the corresponding The location of extreme waves defined by the angles, θ M and θ m , respectively depends on the product ρ a F 0 and are as follow: For ρ a F 0 > 0:35 The minimum wave height is given by the exact relationship (giving Y e ):

Numerical Model
There are many steps in order to build the numerical model [20][21][22]: (i) Definition of the objective of the modelling: this step is crucial for understanding our system and the initial parameters.(iv) Choosing the parameters that influence our system, sensitivity analysis, the modelling approach, and methods of resolution.
(vi) Fixing the criteria in terms of convergence, stability, and accuracy in the aim to guarantee the performance of the model.
(vii) Verifying the results and the outcomes of the system by testing and checking.
(viii) Estimating the uncertainties that depend on the purposes of the model.
For the geometry, we use the Free CAD software [23] in order to draw the geometry of the case studies.
We used the software Flow 3D Hydro, which is the best CFD software, for free surface flows in the objective to model the flow for supercritical bend flow.
For the totality of the numerical simulations, the following models were used: (i) The order of accuracy is second order.
(v) Fluid flow solver: solve momentum and continuity equations.
3.1.Grid Sensitivity Analysis.A grid convergence analysis involving different size of mesh was done for each case study in order to select the adequate proper cell size.
The correct value of the cell size is found by comparing the computed values of the hydraulic parameters with those found in the physical modelling.
For the whole side spillway with curved chute, we used two cell sizes of 0.5 m and 0.7 m, and we found that the proper cell size is 0.5 m.
The total number of meshes varies between 80,000 cells and 16 million.
Table 2 summarizes the tested cases with the size and number of cells for each case tested.

Turbulence Modelling: RANS Governing Equations.
As in most engineering applications, we use RANS model (Reynolds-averaged Navier-Stokes equations) in order to model turbulence.
For our simulation, we utilize the RANS equations (RNG k − ε), in order to close the system and obtain the algebraic equations using finite volume approach.Flow 3D computes the mean values for the hydraulic parameters for each cell using a staggered grid system [21].
The resolution of these governing (), with the help of the constants from Table 3, gives the velocity, pressure field, and the tracking of the water surface (equation ( 23)).
The governing equations and the constants used are as follow.
7 Modelling and Simulation in Engineering 3.3.Free Surface Modelling.The volume of fluid (VOF) approach can estimate the water surface.
The VOF method utilizes the fraction function α as an indicator in order to determine the percentage of the cell full of water, air, or both [12].The software utilizes the Tru-VOF method for tracking the interface between water and air.
The VOF equation for interface tracking is: 3.4.Boundary Conditions.The four frequent boundary conditions are: (i) S (symmetry): This is the default condition indicating that you have no conditions to check on this facet or that you are working with a control volume.
(ii) W (wall): Indicates that the facet is a wall containing the flow.
(iii) P (specified pressure): Represents the pressure in meters that may correspond to the elevation of the fluid on a facet or the fraction of the fluid equal to 0 (total absence of water) to designate the atmosphere on the facet at the top of the mesh.
(iv) O (outflow): Illustrates the facets that can discharge the outflow.To avoid water, build up in the block.
Figure 4 shows the boundary conditions used in the tested cases.
We have opted for the choice of two blocks in order to optimize the solution and the numerical calculation time.
For block 1: the inlet boundary was specified with a fixed pressure (head discharge), the bottom and sidewalls were set as no-slip condition with equivalent roughness k s = 0:015 mm according to the Plexiglas surface properties.The top boundary is fixed as atmospheric pressure (p = 0).The outlet boundary was configured as outflow is set to zero-gradient condition in order to let the flow enter to the block 2.
For block 2: the inlet boundary was specified with outlet condition, the sidewalls were set as symmetry condition.The top boundary is fixed as atmospheric pressure (p = 0).The outlet boundary was configured as outflow and is set to zero-gradient condition (see Figure 4).

Results and Discussion
4.1.The Water Level in the Cross-Sections.The simulated water surface profiles at three cross-sections for each case tested.
The water surface for each section is found for θ values 10 °, 20 °, and 30 °, respectively.
We have chosen these positions because in these regions, the maximum height of the water surface is located.
For the relative curvature = 0:125 with a slope of 1% and for a curved channel with a width of 8.8 m, for the three sections, the water level in the cross-section increases with the increase of the head discharges, and a maximum of the water height is observed at the right wall (see Figures 5(a), 5(b), and 5(c)).
In the first section, we observe that the water level increase sharply unlike the other Sections 2 and 3, the variation of the water surface is moderate.
For the relative curvature ρ = 0:125 with a slope of 2% and for a curved channel with a width of 8.8 m, for the three sections, the water level in the cross-section increases with the increase of the head discharges (see Figures 5(d), 5(e), and 5(f)).
Only small deviations can be found in comparison with the water surface with the same geometrical parameters.
The slope accelerates the flow but reduces the water level slightly.
The same flow patterns can be seen in section number 3. The minimum height is recorded in this section, because the flow has dissipated a portion of its hydraulic energy due to friction with the curved channel sidewalls.
For the relative curvature ρ = 0:075 with a slope of 10% and for a curved channel with a width of 3 m, for the three sections, the water level in the cross-section increases with the increase of the head discharges (see Figures 5(g), 5(h), and 5(i)).
The water level for the first section jumped rapidly with the increase of the head discharges.
For the second section, for head discharge H = 1 m, it can be noticed that the water profile rose up quickly in comparison with discharges higher than 1 m, the other water surfaces remain nearly steady.For Section 3, for H < m, the water profile went up sharply, and reaches the maximum height.For head discharges H > 2 m, the water level shows some fluctuations; furthermore, the water is unstable for these charges due to the presence of shock waves and water wings.
For the relative curvature = 0:04 with a slope of 10% and for a curved channel with a width of 2 m, for the three sections, the water level in the cross-section increases with the increase of the head discharges (see Figures 5(j), 5(k), and  5(l)).
The water level for the first section dropped rapidly with the increase of the head discharges.
For the second section, for head discharge H = 1 m, it can be seen that the water profile maintains a constant level comparison with discharges higher than 1 m, the other water surfaces climbed steeply.
For section 3, for H < 2 m, the water profile remains slightly constant.For head discharges H > 2 m, the water level shows some fluctuations and water wings (rooster tail waves).
The novelty of these abacuses is that they allow an estimation of the wave height in these cross-sections for the aim of sizing the sidewalls for the initial design.

Fitting the Maximum Height in Curved Channels Using
Power and Second Order Polynomial Laws.We have shown that the maximum height in the studied sections of the curved channels follows a power law for more than 83% of the sections (10 sections out of 12), and the others follow a polynomial law as a function of the bend number B 0 .
It has been shown that the maximum wave height h M follows two fitting laws for the case tested with correlation coefficient R 2 between 0.72 and 0.997 (see Tables 4 and 5).
The wave height is given by the empirical formula (equation ( 24)): The novelty of these fitting laws is that we can estimate the maximum height h M using Figures 6(a), 6(b), and 6(c), in order to estimate the height of the sidewalls in the curved channels, the local velocity at each section, the Froude number in the phase of initial design, and thereafter validate the optimal variant by the physical model with an optimization of the cost of construction of the physical models and also the computational cost.
The figures give a comparison between the values calculated by the numerical modelling noted CFD and values derived from the physical modelling noted exp.

Inner Wall
Outer Wall   Modelling and Simulation in Engineering graphs allow giving an estimation of the maximum height at the level of the curved channels.
Table 4 gives the values of the fitting constants α, β, A, B, C for the two laws the range of applicability.

Bottom Outlet
4.3.1.Theoretical and Numerical Calculations.The theoretical calculations for the imaximum and minimum water level for the bottom outlet were done using equations ( 9)- (11).The results are shown in Table 6.
In the case of the bottom outlet, we observe from the results of the numerical model (Figure 7) that we have two maximum heights H max-1 and H max-2 .The water level near the outer wall is high in comparison to the inner wall, due to the effect of centrifugal force [18].We can clearly see in the transition zone between the two curved channels the presence of the rooster tail waves, and this generates huge amount of air and turbulence.
Relative mean error (RME) is given by this formula: The measured value is obtained from the physical model.The computed value is derived from the CFD calculations.Table 6 shows that the theoretical maximum height at the level of the first curvature is 3.9 m, whereas a maximum height of 4.39 m was measured for the physical model, which represents a RME of 11.16%.Moreover, for the second curvature a value of 4.3 m was calculated and a corresponding value of 4.71 m was measured, with a RME of 8.7% 4.3.2.Flow Patterns.From Figures 7 and 8, which illustrate the water level, it is clear that we have fluctuations in the bend channel.
In the tests of the physical model, we have noticed an increase in water level in the outer wall in comparison with the inner wall (see Figure 8).
In addition, we observed a zone of disturbance in the intersection of the two curvatures with some water wings.The perturbation due to the first curvature propagates in the direction of the flow and finds a second curvature, which will amplify the effect of this disturbance.Hence, we have high level of water in the physical model as against theoretical calculations and CFD approaches (see Figure 7).
Tables 7 and 8 show the water level in the bottom outlet for the left and right walls for the tested head discharge of the project.
The values in Tables 5 and 6 validate the numerical modelling.The experimental measurements show a good agreement with the values found numerically in terms of the water level, but there are some small deviations that can be explained by the swelling of the water surface due to the interference of the shock waves in the perturbation zone.
For the right part of the wall, the maximum height is 4.35 m and for the left part, the value is 4.48 m.
These results show that we can use the results of the numerical modelling to estimate the height of the walls and optimize them using the physical modelling.11 Modelling and Simulation in Engineering Figure 9 gives RME for the and numerically computed water heights for both walls.
It is noticeable from Figure 9 that the RME does not exceed 15% between the values of the numerical modelling and the experimentation.
This difference can be explained by the huge amount of air generated this zone, due to the water-air mixing.
The study of the air quantity generated in this critical zone (the intersection of the two curvatures is beyond the scope of this paper).

4.4.
Modelling the Spillway without Guide Wall.The curvature in the spillway chute creates a flow dissymmetry at the curvature (see Figure 10).This phenomenon induces eddies, vortices, and hydraulic jump that will impede the flow and cause erosion of the left wall of spillway chute.This empty area visible in the ski jump (Figure 10) has been confirmed by the results of the physical model (Figures 12 and 14).This zone of stagnation was created essentially for low head discharges, because the flow doesn't have enough kinetic energy to combat the centrifugal force [18] due to the curvature upstream the ski jump.Therefore, the jet doesn't take off downstream the spillway and generate a vortex (Figure 12).4.5.Water Surface with Guide Wall.To remedy this phenomenon, and given the great width of the chute, the proposed solution consists of the installation of a separation wall, 1 m thick, with the same height as the inner and outer walls.The dividing wall is located in the middle of the width of the curved chute spillway in order to have an equilibrium distribution of the water surface in this zone (see Figure 13).
After the adding the separating wall, we notice that the results of the numerical and physical modelling present a good agreement with some minimal deviations.
The numerical results can be used to approximately design the sidewalls, while their validation against the experimental data can be used to optimize the height of the walls.Modelling the Spillway with Guide Wall.14 shows the geometry of the separating wall and its size.
The simulation of the spillway with the dividing wall gave a satisfactory result for the resolution of the problem.The flow, shown in Figure 15, became symmetrical to the left and right of the guide wall.
The addition of the guide wall reduced the separation in the inner wall and improved the flow behaviour in the chute and the ski jump.It was also noted that the water jet becomes well distributed over the entire width of the ski jump, which will allow it to take off as far as possible downstream of the dam.

Conclusion
In this paper, the flow patterns were investigated using CFD and experimental results, for two case studies: side channel spillway with curved chute and bottom outlet with two curvatures after the gates.
The analysis leads to the following observations and conclusions: (vi) The implementation of the dividing wall plays a crucial role in balancing the water surface between the walls; furthermore, the stagnation zone has disappeared, and this allowed flow to have kinetic energy required to take off as far as possible downstream the dam.
(vii) To the best of authors' knowledge for supercritical bended flow, there is still work to be done to study

V:
Flow velocity [m/s] g: Gravity constant [m/s 2 ] h: Flow depth [m] ρ: density [kg/m 3 ] σ: Water surface tension [N/m] ν: Water kinematic viscosity [m 2 /s] h: The water height upstream of the curvature [m] F 0 : The Froude number of the flow upstream of the curvature β: Angle formed by the two wave fronts with the initial direction of flow θ: Central angle determined by geometrical considerations k: Turbulent kinetic energy [m 2 /s 2 ] ε: The turbulent energy dissipation rate u i : Velocity component of x i [m/s] μ kef f , μ εef f : The effective diffusivity for k, ε, respectively ν t : The turbulent kinetic viscosity [m 2 /s] G: The production of turbulence due to shear RME: Relative mean error H max : Maximal height water level in the bottom outlet [m] H min : Minimal height water level in the bottom outlet [m] V 0 : The approach velocity ρ a : The mean curvature b: The width of the channel [m] R a : Radius of curvature [m] e : The theoretical maximum height [m] h 0 : The approach height of water [m] h m : The minimum wave height h M : The maximum wave height B 0 : The bend number θ m : The minimum wave angle θ M : The maximum wave angle.

2. 1 .
Physical Model.The experiments are completely based on a 3D physical model for this study in order to observe the effect of 3D conditions, and to have a better idea about the flow behaviour.The experimental facility is composed of: (i) The upstream part is a rectangular basin with dimensions (10 m × 20 m).

Figure 3 :
Figure 3: The model of shock waves in a curvature of a rectangular channel in supercritical regime [13].

Figure 5 :
Figure 5: Cross-section water level for the tested cases.

Figure 6 :
Figure 6: The maximum height as a function of the number for the three-tested section.

Figure 7 :
Figure 7: Numerical modeling of the bottom outlet-water surface.

Figure 8 :
Figure 8: Water surface and rooster tail waves for the bottom outlet in the physical model.

Figure 9 :
Figure 9: Comparison of RME between left and right walls.

Figure 10 :
Figure 10: The stagnation zone of flow in the ski jump-CFD.

Figure 12 :
Figure 12: Illustration of the dissymmetry of the flow in the chute due to the curvature R = 400 m [16].

Figures 16
Figures 16 and 17 show the validity and efficiency of the proposed solution in terms of the equilibrium of the distribution of the flow over the full width of the chute.It was also noted that the water jet becomes well distributed over the entire width of the ski jump, which will allow it to take off as far as possible downstream of the dam.
(i) 3D numerical model based on RNG k − ε turbulence model was used to study the flow behaviour.(ii) Supercritical bend flow contains a plethora of difficulties, shock waves, and vibrations.(iii) The numerical approaches and data measured show a fair agreement for the case of curved chute.(iv) In the case of bottom outlet, huge amount of air due to the interference of the flow in the perturbation zone.(v) The flow without guide wall presents an imbalanced water profile and creates vortex in the ski jump.It is recommended to use this solution for the final design of the spillway to avoid this major problem.

Figure 15 :
Figure 15: Comparison of the water surface for the sidewalls between the experimentation and the numerical methods.

Figure 16 :
Figure 16: Solving numerically the problem of stagnation zone by adding separator wall.

Figure 17 :
Figure 17: The flow in the chute after the addition of the dividing wall-physical model.

Table 1 :
Curved channels tested by CFD and theoretical calculations.

Table 3 :
The constants of turbulence model.

Table 4 :
Power law of the maximum wave height.

Table 5 :
Polynomial law of the maximum wave height.

Table 6 :
Water level for bottom outlet using theoretical calculations.

Table 7 :
Right water level-physical model.

Table 8 :
Left water level-physical model.