# SPH Simulation of Hydraulic Jump on Corrugated Riverbeds
## 3 Benchmark Study
### Wave Profiles and Velocity Distribution  
The hydraulic jumps in the other five cases develop in a similar manner as that in Case A. The wave profiles at the steady state in these cases are presented in Figure 5, in which the SPH results are in general good agreement with the laboratory measurement in Ead and Rajaratnam [1]. When the water in the reservoir goes through the gate, rapid flows are generated just downstream of the gate. The wave profiles predicted by SPH in all the cases are disordered. However, the experimental wave profiles published as presented in Ead and Rajaratnam [1] are smooth. In practice, the hydraulic jump contains bubble flows. Thus, the experimental results are interfered owing to the splashing of fragmentary flow caused by the roller. Moreover, the SPH numerical model proposed in this paper does not consider the influence of air on water motion, resulting in the error of results. Consequently, deviations between the experimental and SPH wave profiles exist. Figure 4: Snapshots of hydraulic jump evolvement. Figure 5: Wave profiles and velocity vector of different cases. (**a**) Wave profile of Case A; (**b**) Wave profile of Case B; (**c**) Wave profile of Case C; (**d**) Wave profile of Case D; (**e**) Wave profile of Case E; (**f**) Wave profile of Case F; (**g**) Velocity vector of Case A; (**h**) Velocity vector Case B; (**i**) Velocity vector of Case C; (**j**) Velocity vector of Case D; (**k**) Velocity vector of Case E; (**l**) Velocity vector of Case F.  
It can be seen from the velocity vectors of Cases A, B, C, and D (Figure 5g-j) that when the corrugation shapes and gate heights \(e\) are the same, the flow becomes more fragmentary as the Fr number increases and there is a distinct eddy structure. The formation of roller is mainly caused by the uneven distribution of flow velocity within the hydraulic jump section. The water particles between the flow layers are in relative motion, leading to the generation of internal friction shear stress. In addition, this stress between flow layers is manifested as the downstream flow on the surface and the reverse flow at the bottom, namely, the clockwise roller. Under the same conditions, both the slope and size (see the ellipses in the right column of Figure 5) of the roller increase with the Fr number. It means that the influencing region of roller expands gradually. The contour of the horizontal velocity of Case A is presented in Figure 6. The flows within the hydraulic jump section contain the downstream flow of positive velocity and the reverse flow of negative velocity. The reverse flow mainly exists in the surface and bottom areas. The velocity of flow within the hydraulic jump section differs significantly. Specifically, the flows near the wall and water surface are at low velocity, while the flow in the middle layer is at high velocity. Thus, a torque generates in the hydraulic jump section, manifested as the clockwise roller. In the open channel behind the hydraulic jump section, the flow on the surface has the maximum velocity, while that near the wall has the minimum velocity. The countercurrent near the water surface is caused by the backflow of water, while that at the bottom is mainly resulted from the clockwise roller. For clarity, the steady-state wave profiles for the six cases are compared in Figure 7, with the result of an error analysis in Table 3. After comparing the wave profiles of Cases A, B, C and D (of the same corrugated riverbed), it is found that when the corrugation shapes and the gate heights are the same, the hydraulic jumps become more violent for higher Fr numbers. The corrugation changes the local fluid motion, which then affects free-surface profile, as can be seen from the comparison between Cases E and F. The relative errors of each case are varied between \(-9.1\%\) and \(9.6\%\). A more detailed comparison study on the effect of corrugation shape on the hydraulic jump process will be presented in Section 5.