# SPH Simulation of Hydraulic Jump on Corrugated Riverbeds
## 3 Benchmark Study
### Jump Length, Water Depth and Energy Dissipation Rate  
Jump length, water depth after hydraulic jump, and energy dissipation rate are important indicators for project design. The jump length determines the length of stilling basin, and the water depth after jump determines the height of apron. In practical applications, the stilling basin should be designed to be as short as possible while satisfying the energy dissipation rate to save construction costs. When hydraulic jump occurs, the flow swirls wildly, leading to a dramatic change in the surface water. The water at the end of the roller constantly flows into the mainstream and thus is gradually submerged and finally develops into open-channel flow when the water surface cannot be greatly affected by the swirling vortex. As is shown in Figure 8, the roller length \(L_{rj}\) is from the beginning of hydraulic jump to Location 2 (i.e., the end of the roller); the jump length \(L_{j}\), namely the range of water surface affected by hydraulic jump, is from the beginning of hydraulic jump to Location 3 where the water flows back to the free surface after tumbling over the roller. The experimental and simulated roller lengths (\(L_{rj}\)) and jump lengths (\(L_{j}\)) of the six studied cases as well as the length of hydraulic jump on smooth riverbed (\(L_{rj}\)) are presented in Table 4. The predicted \(L_{rj}\) and \(L_{j}\) by SPH are in good agreement with the experimental results. The maximum relative errors for \(L_{rj}\) and \(L_{j}\) are 6.5% and 5.7%, respectively. This shows the good accuracy of the present SPH model. For all the cases, \(L_{j}\) is 30% smaller than \(L_{j*}\), and this number for Case C reaches 40%. It indicates that corrugated sheet can improve energy dissipation rate and reduce jump length as well. In practice, therefore, the length of a stilling basin can be reduced by using corrugated bed. Figure 8: Definitions of roller length and jump length. Figure 7: Wave profiles of Cases A to F.  
The water depth at Location 3 in Figure 8 (the starting location of open-channel flow) is considered to be the water depth after hydraulic jump (\(y_{2}\)). The experimental and simulated water depths (\(y_{2}\)) of the six studied cases as well as the water depth of smooth riverbed case (\(y_{2}\)*) are listed in Table 5. The SPH model over predicts the water depth, with the maximum error around 11% in the six cases. The discrepancy is partly attributed to the experimental errors in measuring the energetic free surface of the hydraulic jump. The water depths after hydraulic jump of the six cases with corrugated riverbeds are about 30% smaller than that of the smooth riverbed case. The hydraulic jump dissipation is one of the major energy dissipation modes of outlet structures. When flow at high velocity discharges into stilling basin, a hydraulic jump occurs, after which, the rapid flow will slow down rapidly, dissipating most of the energy of upstream flow. Figure 9 shows the mean flow velocities (\(\overline{U}\)) of the studied cases along the channel. In all the cases, the flow has the maximum velocity while passing through the gate (at the beginning of hydraulic jump), and starts to decline progressively thereafter (the decreased amount reaches 85%), and eventually becomes stable in the open-channel region. With the water depths and wave velocities at the reservoir gate and Location 3 in Figure 8, the wave energies at these two locations can be computed by using the formulae as presented in Wen [36]. The difference of these two energies is the energy dissipated by the hydraulic process, which divided by the wave energy at the reservoir gate is the energy dissipation rate \(\eta\). Energy dissipation rate \(\eta\) on horizontal bed of rectangular channel was computed by the following equations [36].