It is difficult to capture detailed information about complex flow using the physical model experiment. To understand the hydraulic characteristics of the curved spillway more intuitively and comprehensively, the numerical model was used to conduct further study. The reliability of the numerical model was verified by comparison with the measured data.
3.1. Numerical Methodology
Water flow is an incompressible Newtonian fluid. The continuity equation and Unsteady Reynolds-Averaged Navier–Stokes (URANS) equation system were adopted as the hydrodynamic model.
Momentum equation:
where t is time,
is the density of water, the operation
is time averaging,
represents the different axes of the Cartesian coordinate system,
represents the mean velocity component in
coordinate,
and
represent the mean pressure and the kinematic viscosity coefficient, respectively, and
denotes the Reynolds stresses, which have to be resolved to close the momentum equations. The Reynolds stresses are solved by Boussinesq’s formula:
where
and
denote eddy viscosity and the Kronecker sign, respectively. The eddy viscosity is given by the k-ε model.
Turbulence kinetic energy, k, equation:
Energy dissipation, ε, equation:
In the above equations,
and
are different constants;
and the other details of the turbulence k-ε model are described by Rodi [
19].
The VOF method was used to track the free surface, as proposed by Hirt and Nichols [
20], which strictly complied with the mass conserving method. Cheng et al. [
21] and Boes et al. [
22] used the VOF method to track the free surface of air–water two-phase flow over a stepped spillway. The VOF method is performed by solving Equation (9):
where F is the average value in a cell volume fraction of water in the cell. A zero value of F corresponds to a cell with no fluid, and a unit value indicates that the cell is full of water. A cell with an F value of between zero and one must contain a portion of the free surface [
23,
24]. The isosurface of F = 0.55 is defined as the free surface location.
The governing equations were discretized based on the finite volume method (FVM) and solved by the semi-implicit method for pressure-inked equations consistent (SIMPLEC). The convection flux was computed by the second-order windward and the temporal discretization scheme was second-order implicit. The reconstruction of the free water surface adopted a geometry reconstruction method.
The inlet boundary that was located on the reservoir included the inlet of water and air. The uniform flow velocity boundary was used as the water inlet. The uniform velocity was calculated from the measured depth of the water on the weir crest and given the discharge. The pressure boundary adopted all air boundary conditions and the water outlet boundary condition. A no-slip boundary condition was applied at the sidewall and the base plate, respectively. The near-wall regions of the flow were analyzed using standard wall function
The computational domain was initially filled with water at rest, and the water was then gradually accelerated. The computations were continued until 500 s, with the time increment being 0.0016 s. The specific simulation conditions on the weir crest and the water flow into the curved section corresponding to the physical model of a flow rate of 272.0 m
3/s are shown in
Table 4 and
Table 5. The simulated results at a flow rate of 272.0 m
3/s that was converted to the prototype based on the similitude principle are presented. The simulation of air–water flow in the curved spillway with non-uniform-height steps was carried out using the ANSYS-FLUENT platform.
3.2. Mesh Tests and Model Validation
To ensure the accuracy of the numerical model validation, the geometric model was established with a length scale of 1:1 to the physical model. However, the results were nevertheless converted to the prototype based on the similitude (Froude) principle.
For all the cases, the cells were meshed by a hexahedral grid, as shown in
Figure 8. In the turbulent boundary layer zone near the wall, the distance between the first grid node and the wall was controlled by the dimensionless Y
+, the values of which were obtained by Equation (10). The value of this parameter is related to whether the grid density near the wall is appropriate. If the value is too large or too small, the computational accuracy of the flow field near the wall will be affected. Due to the complexity of the model and the flow, this value was maintained at between 30 and 300 in this study [
25]. The over-dense grid is not conducive to the calculation in this project.
where
is the distance between the first grid node and the wall, and
denotes friction velocity.
The vertical velocity at position [x = 1.4, y = 0, z = −0.3] of the smooth spillway for three distances
was used to test mesh independence, as shown in
Figure 9. H stands for the wall height. As can be seen from
Figure 9, the velocity distribution in the region of
obeys a logarithmic law for
and
. The velocity in the region of
, which is filled with air or a portion of water in a cell, is in accordance with the rule of the air–water interface given by Chanson et al. [
26]. For
, the distribution of velocity in
deviates from the logarithmic law. The relative error of velocity at the first node between
and
is 3.62%. Thus, for the step and smooth spillway, the distance
= 3.5 mm was adopted, and the numbers of total elements for the geometric model were 867,240 and 731,455, respectively.
Figure 10 shows the flow pattern on the weir crest of the smooth spillway and the curved section measured by the experiment, and the numerically simulated shape of the free water surface. The numerical simulation successfully captured the water wing formed at the pier, and the area not covered by water on the base plate in the curved section was basically consistent with the experimentally measured range. Therefore, the VOF method can successfully simulate the important hydraulic characteristics of free water surface.
The flow pattern from step No. 21 to step No. 24 is shown in
Figure 11, which further verifies the reliability of the flow characteristics of the curved spillway with non-uniform-height steps simulated by the numerical model in this study.
Figure 11a shows that the water flow on the steps presents the skimming flow regime. The results of the numerical simulation clearly show that the free surface of the flow is relatively smooth, and that air is entrained in the water on the steps. These flow characteristics captured by the numerical model in this study are similar to our experimental results and those described by Chanson et al., Tabbara et al. [
27] and Chatila et al. [
28] on the skimming flow.
Figure 12 shows the computed water level along the path compared to the corresponding measured data. When the volume fraction of water (F) is 0.55, the water level on the concave bank and the convex bank calculated by the numerical simulation is in good agreement with the measured data. Therefore, the VOF method can be used to accurately simulate the profile of the free surface, and the isosurface of a water volume fraction (F) of 0.55 is taken as the profile of free surface [
14].
Figure 13 gives a comparison of the flow velocity near the concave bank, the convex bank, and the central axis on the cross-section as determined by numerical simulation and experiment. As can be seen from
Figure 13, under the influence of centrifugal force, the velocity on the concave bank is greater than that on the convex bank, and the flow velocity calculated by numerical simulation is slightly larger than that of the experimental measurement at this point. The maximum difference between the velocity obtained in the numerical simulation and that obtained in the experiment is 8.5%, which meets the requirement of engineering accuracy.
In summary, the VOF model in conjunction with the Re-normalization group (RNG) k-ε turbulence model can successfully simulate the flow and the profile of the free surface. The numerical model is suitable for simulating the flow of the curved spillway with non-uniform-height steps.