# SPH Simulation of Hydraulic Jump on Corrugated Riverbeds
## 3 Benchmark Study
### Computational Parameters  
The test model for observing hydraulic jump was a 0.446 m \(\times\) 0.6 m \(\times\) 7.6 m rectangular flume made of polymethyl methacrylate whose bottom was installed with aluminum sinusoidal corrugated pipes. As shown in Figure 1, \(S\) and \(t\) are the corrugation wave length and height of corrugated riverbed, respectively, Fr number is defined by \(Fr=U_{1}/(gy_{1})^{1/2}\), \(U_{1}\) denotes the cross-section mean velocity before hydraulic jump, \(q\) denotes the rate of test flow, \(y_{1}(=e)\) and \(y_{2}\) respectively are the water depths before and after hydraulic jump, \(L_{j}\) denotes jump length, and \(L_{j}\) denotes roller length.  
The SPH computational model reproduces the whole experimental setup that includes reservoir, corrugated sheet, gate, and tail weir as shown in Figure 2. Red particles represent the fixed boundary, black particles the moving boundary and blue particles the water particles. The shape of the riverbed corrugation is described by \(y=0.5t\sin(2\pi x/S)\). When the gate height \(e\) is constant, the flow velocities of different cases are realized by regulating the water level in the reservoir. The water level is calculated by Equation (6)  
\[q=\mu e\sqrt{2g(H+\frac{\alpha_{0}U_{0}^{2}}{2g}-\epsilon e)} \tag{6}\]  
where \(H\) is the water level of reservoir, \(\alpha_{0}\) is kinetic energy correction factor (under normal flow conditions, \(\alpha_{0}=1.0\)) that considers the water head induced by the water velocity in the reservoir, \(U_{0}(=q/H)\) is the mean water velocity in the reservoir, \(\mu=\epsilon\varphi\) is the discharge coefficient of gate, \(\epsilon\) is the shrinkage coefficient of gate (related to the water depth in and the gate height of the reservoir, and determined according to Table 10-5 as presented in Wen [36]), and 0.95 is assigned to \(\varphi\).  
A tail weir is set at the end of the flume for controlling tail water, so that the hydraulic jump can form on the corrugated sheet near reservoir gate. The height of the tail weir \(p\) is calculated by \(y_{2}-h\). \(h\) is the weir head that can be obtained by solving the equation for the flow rate of sharp-crested weir, i.e., Equation (7) as,  
\[q=m_{0}\sqrt{2g}h^{3/2} \tag{7}\]  
where \(m_{0}\) is calculated by the following empirical formula as  
\[m_{0}=(0.405+\frac{0.0027}{h})[1+0.55(\frac{h}{h+p})] \tag{8}\]  
Figure 1: Schematic view and dimensions of the benchmark example (modified from Figure 1 in [PERSON] and Rajaratnam [1]).  
Figure 2: SPH computational domain of the benchmark example.  
In all the simulations of this study, the quantic spline kernel function is used as presented in [PERSON] [44]. The MLS density re-initialization scheme is adopted to filter out the numerical noises. The SPS turbulence model is used to consider the fluid turbulence. The solid wall boundaries are modeled by the dynamics boundary condition (DBC) as presented in [PERSON] and [PERSON] [40]. The symplectic (predictor-corrector) scheme is adopted for the time stepping. The time step is determined by the Courant condition as presented in [PERSON] and [PERSON] [38]. The initial particle spacing is selected to be \(5\times 10^{-3}\) m, the smoothing length \(h=3.68\times 10^{-3}\) m, and the simulation time of each case is 20 s. There is a total number of 96,476 particles in the model of Case A, and among them 4610 are boundary particles. The 6 test cases as presented in [PERSON] and [PERSON] [1] are studied with the parameters listed in Table 1. The simulations were performed on a cluster at the School of Water Resources and Electric Power, Qinghai University, China, where 32 cores (2 GHz CPU and 32 GB RAM) were used. Depending on the total number of particles for each test case, between 5 and 15 days of CPU time were spent for the computations.