Numerical investigation of flow characteristics over stepped spillways

Spillways are constructed to evacuate flood discharge safely so that a flood wave does not overtop the dam body. There are different types of spillways, with the ogee type being the conventional one. A stepped spillway is an example of a nonconventional spillway. The turbulent flow over a stepped spillway was studied numerically by using the Flow-3D package. Different fluid flow characteristics such as longitudinal flow velocity, temperature distribution, density and chemical concentration can be well simulated by Flow-3D. In this study, the influence of slope changes on flow characteristics such as air entrainment, velocity distribution and dynamic pressures distribution over a stepped spillway was modelled by Flow-3D. The results from the numerical model were compared with an experimental study done by others in the literature. Two models of a stepped spillway with different discharge for each model were simulated. The turbulent flow in the experimental model was simulated by the Renormalized Group (RNG) turbulence scheme in the numerical model. A good agreement was achieved between the numerical results and the observed ones, which are exhibited in terms of graphics and statistical tables.


INTRODUCTION
Dam structures are the most important projects around the world to store water or to transport water because protection of water is the key to living. And the spillway is classified as one of the most important parts of a dam. A spillway is constructed to protect a dam from destruction or damage by flood. Dam building and flood control can be considered a very important issue across the world given the importance of hydroelectric power generation, navigation, recreation and fishing. There are many types of spillway, but the most common types are: ogee spillways, free over-fall spillways, siphon spillways, chute spillways, side channel spillways, tunnel spillways, shaft spillways and stepped spillways. And every spillway consists of four necessary components: an inlet channel, control structure, discharge carrier and outlet channel. A large number of stepped spillways were constructed through the recent decades, as especially associated with the technique of roller compacted concrete (RCC) dam construction and construction of stepped spillways classified as an easier, quicker and cheaper technique of construction (Chanson ; Felder & Chanson ). A stepped spillway structure increases the rate of energy dissipation which decreases cavitation risk (Boes & Hager b). And stepped spillways have advantages which make them more attractive under various conditions.
The flow behaviour in stepped spillways is generally classi- Computational fluid dynamics (CFD), namely numerical models of hydraulic engineering, generally reduce the amount of total cost and time that will be spent on physical models. So numerical models are classified as faster and cheaper than experimental models and can also be used for more than one purpose at the same time. There are many CFD software packages available but the most widely used one is Flow-3D. In this study, Flow 3D software is used to simulate the air concentration, velocity distribution and dynamic pressure distribution on a stepped spillway for two different models with different flow rates.
Roshan et al. () studied the investigation of flow regimes and energy dissipation over two physical models of stepped spillways with different numbers of steps and discharges. The slope of the experimental models was 19.2% and the number of steps 12 and 23, respectively. The results illustrated that the observed flow regimes in the 23-step physical model was considered more acceptable than the 12-step model. However, the energy dissipation on the 12-step model was more than on the 23-step model. And the experiments observed that in the skimming flow regime the energy dissipation in the 23-step model was less than in the 12-step model by about 12%. Ghaderi et al. (a) carried out experimental studies of stepped spillways to investigate the influence of scouring parameters with different step sizes and flow rates. The results showed that the flow regime affected the scour-hole dimensions, such as the minimum scouring depth which happened under the nappe flow regime. Moreover, the tailwater depth and step size are actual parameters for maximum scouring depth. By increasing the depth of tailwater from 6.31 cm to 8.54 and 11.82 cm this increased the scouring depth by 18.56% and 11.42% respectively. Also, this increasing tailwater depth decreases the scouring length by 31.43% and 16.55% respectively. In addition, the Froude number increases by increasing the flow rate, and the increased momentum of the flow promotes scouring. Also, the results indicated that the scouring in the middle is less than at the sidewalls of the cross-section. An empirical formula was suggested to predict the maximum depth of scouring downstream of stepped spillways and then compared with the experimental results. And the comparison illustrated that the suggested formula can predict the depth of scouring within 3.86% and 9.31% relative and maximum errors, respectively. observed that these types of spillways have better performance because they increase the magnification ratio L T /W t (L T is the total edge length, W t is the width of spillway).
Also, the trapezoidal labyrinth shaped stepped spillway has a larger friction factor and a lower residual head. The friction factor differs from 0.79 to 1.33 for various magnification ratios, while for the flat stepped spillway it is approximately equal to 0.66. Also, the ratio of the residual head (H res /dc) is approximately 2.89 in a TLS stepped spillway, while it is approximately equal to 4.32 for a flat stepped spillway.   (1) and (2) are RANS and continuity equations with FAVOR variables that are applied for incompressible flows.
where u i is the velocity in x i direction, t is the time, In this research the RNG model was selected because this model is more commonly used than other models in dealing with particles; moreover, it is more accurate to work with air entrainment and other particles. In general, the RNG model is classified as a more widely-used application than the standard k-ε model. And in particular, the RNG model is more accurate in flows that have strong shear regions than the standard k-ε model and it is defined to describe low intensity turbulent flows. For the turbulent dissipation ε T it solves an additional transport equation: where CDIS1, CDIS2, and CDIS3 are dimensionless parameters and the user can modify them. The diffusion of dissipation, Diff ε, is where u, v and w are the x, y and z coordinates of the fluid velocity; A x , A y , A z and V F , are FLOW-3D's FAVOR TM defined terms; P T and G T are turbulence due to shearing and buoyancy effects, respectively. R and ξ are related to the cylindrical coordinate system. The default values of RMTKE, CDIS1 and CNU differ, being 1.39, 1.42 and 0.085 respectively. And CDIS2 is calculated from turbulent production (P T ) and turbulent kinetic energy (k T ).
The kinematic turbulent viscosity is the same in all turbulence transport models and is calculated from where v T : is the turbulent kinematic viscosity. ε T is defined as the numerical challenge between the RNG and the twoequation k-ε models, found in the equation below. To avoid an unphysically large result for v T in Equation (3), since this equation could produce a value for ε T very close to zero and also because the physical value of k T may approach to zero in such cases, the value of ε T is calculated from the following equation: where TLEN: the turbulent length scale.
VOF and FAVOR are classifications of volume-fraction methods. In these two methods, firstly the area should be subdivided into a control volume grid or a small element.
where ρ is the density of the fluid, R DIF is a turbulent diffusion term, R SOR is a mass source, V F is the fractional volume open to flow. The components of velocity (u, v, w) are in the direction of coordinates (x, y, z) or (r, R SOR , z). Another fractional volume can be used to define the solid surface. Then, this information is used to determine the boundary conditions of the wall that the flow should be adapted for.

Case study
In this study, the experimental results of Ostad Mirza ( to þ15 on 50 -18.6 slope change, respectively).
Pressure sensors were arranged with the x/l values for different slope change as shown in Table 1, where x is the distance from the step edge, along the horizontal step face, and l is the length of the horizontal step face. The location of pressure sensors is shown in Table 1.       The bottom (Z-min) is prepared as a wall boundary condition and the top (Z-max) is computed as a pressure boundary condition, and for both (Y-min) and (Y-max) as symmetry.

RESULTS AND DISCUSSION
The air concentration distribution profiles in two models of stepped spillway were obtained at an acquisition time equal to 25 seconds in skimming flow for both upstream and downstream of a slope change 50 -18.6 and 50 -30 for different discharge as in Table 2, and as shown in Figure 4 for 50 -18.6 slope change and Figure          Flow-3D is a well modelled tool that deals with particles.
In this research, the model deals well with air entrainment particles by observing their results with experimental results.
And the reason for the small difference between the numerical and the experimental results is that the program deals with particles more accurately than the laboratory. In general, both numerical and experimental results showed that near to the slope change the flow bulking, air entrainment, velocity distribution and dynamic pressure are greatly affected by abrupt slope change on the steps. Although the extent of the slope change was relatively small, the influence of the slope change was major on flow characteristics.
The Renormalized Group (RNG) model was selected as a turbulence solver. For 3D modelling, orthogonal mesh was used as a computational domain and the mesh grid size used for X, Y, and Z direction was equal to 0.015 m. In CFD modelling, air concentration and velocity distribution were recorded for a period of 25 seconds, but dynamic pressure was recorded for a period of 70 seconds. The results showed that there is a good agreement between the numerical and the physical models. So, it can be concluded that the proposed CFD model is very suitable for use in simulating and analysing the design of hydraulic structures.

DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.