# SPH Simulation of Hydraulic Jump on Corrugated Riverbeds
## 6 Conclusions  
In the present paper, a 2D SPHysics model is used to simulate the hydraulic jump on corrugated riverbed. Validated by an experimental example, this model is shown to be of general good accuracy in simulating the hydraulic jump process. By using this model, the dynamic evolvement of a hydraulic jump over corrugated riverbed is analyzed by looking into the velocity distribution, jump length, water depth after hydraulic jump, and energy dissipation rate. It is found that a clockwise vortex forms in the hydraulic jump area. For the same corrugation shape, the slope of the vortex roller increases with the Fr number, and so does the influencing region of the vortex. The roller length, jump length, and water depth after hydraulic jump increase with the Fr number. In all the studied cases, the hydraulic jump over corrugated riverbed dissipates over 45% of the wave energy, which is 10% more than that over the smooth riverbed.  
A parametric study is conducted to investigate the effect of the corrugation shape on the wave energy dissipation efficiency. It is found that the corrugated riverbed is more effective when the inflow has larger velocity (and hence more wave energy). Within the studied range, the following observations are made: 1) when the corrugation length and the reservoir gate height are fixed, the corrugation height displays a positive correlation with the water depth after hydraulic jump and the jump length, while the corrugation height has an insignificant effect on the energy dissipation rate; and 2) when corrugation height and the reservoir gate height are fixed, the water depth after hydraulic jump reduces  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Particle Spacing (m)**} & **Experimental Discharge** & **SPH-Estimated** & **Deviation** \\  & **(m\({}^{2}\)/s)** & **Discharge (m\({}^{2}\)/s)** & **(\%)** \\ \hline
0.005 & 0.051 & 0.05 & 2 \\
0.006 & 0.051 & 0.052 & \(-2\) \\
0.008 & 0.051 & 0.049 & 3.9 \\
0.01 & 0.051 & 0.044 & 13.7 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Discharge deviation with different particle spacing.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Particle Spacing (m)**} & **Mean Absolute** & **Relative Error (\%)** & **Root Mean Squared** \\  & **Deviation (10\({}^{-2}\) m)** & **Error (10\({}^{-2}\) m)** \\ \hline
0.005 & 0.79 & 3.54 & 0.91 \\
0.006 & 3.14 & \(-11.12\) & 3.65 \\
0.008 & 2.12 & \(-14.32\) & 2.29 \\
0.01 & 2.57 & \(-24.6\) & 2.82 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Errors of the SPH-estimated water-surface profiles with respect to the experimental data.  
as the corrugation length increases, while the jump length and energy dissipation rate increase as the corrugation length increases.  
As a further study, by investigating the application of variable smoothing length, the authors aim to enhance the accuracy of the model in the areas with high Fr number, such as the jet at the downstream of the gate where the flow depth is shallow, and the computational resolution is not sufficient to capture the detailed velocity and pressure fields.  
Conceptualization, S.G., F.B. and J.W.; methodology, S.G., F.B. and M.L.; simulation and data analysis, S.G., F.B., M.L., Y.Z. and E.K.; writing--original draft preparation, S.G., F.B. and Y.Z.; writing--review and editing, S.G., F.B., M.L. and E.K.  
This research work is supported by the National Key R&D Program of China (No. 2017 YFC0404303), National Natural Science Foundation of China (No.51869025 and No.51479087), Qinghai Science & Technology Projects (No.2018-ZJ-710), High-end Innovative Talent Program of Qinghai Province and the Open Funding SKHLI710 and SKHLI712 from the State Key Laboratory of Hydraulics and Mountain River Engineering in Sichuan University, China.  
The authors thank [PERSON] from the University of Sheffield for his constructive comments in the preparation of this manuscript.  
The authors declare no conflict of interest.