# SPH Simulation of Hydraulic Jump on Corrugated Riverbeds
## 1 Introduction
[8] simulated the submerged hydraulic jump on smooth riverbed under the steady flow condition. However, this study could only simulate the hydraulic jump with slight change in free surface and failed to trace the development process of hydraulic jump. [PERSON] et al. [9] introduced the VOF model to trace the free water surface and simulate the hydraulic jumps on corrugated riverbed under five different conditions, and then verified the feasibility of the numerical simulation model. [PERSON] and [PERSON] [10] investigated the hydraulic jump on smooth riverbed with VOF method and turbulence model. Through numerical method, [PERSON] et al. [11] studied the hydraulic jump on discontinuous open channel, the hydraulic jump on corrugated riverbed, and the submerged hydraulic jump, respectively. Using a VOF model, [PERSON] et al. [12] studied the hydraulic jump characteristics in stilling basins with triangular and sinusoidal riverbeds. By comparing the numerical and experimental results, it was found that the energy dissipation rate was correlated with the Fr number. In the last decade, another category of numerical methods that get rid of meshes has undergone significant developments. That is the Lagrangian particle method. The most commonly used particle method is the SPH which was initially used in astrophysical simulations [13] and soon afterwards widely applied in continuous solid mechanics [14] and hydromechanics [15] where it matured rapidly. SPH algorithm solves problem domain based on arbitrarily distributed particle frameworks and has unique advantages in dealing with problems such as large deformation, kinematic interface, and free surface. It has been widely used in the hydrodynamic studies on dam break [16] landslide surge [17] multiphase flow [18] and spillway [19] and significant research results have been achieved. The pioneering study of mesh-free numerical modeling on the hydraulics could be attributed to [PERSON] et al. [20] who used the Moving Particle Semi-implicit (MPS) method to compute a turbulent jet. Although the model was not quantitatively validated by the experimental data, the proposed sub-particle scale (SPS) turbulence model and the soluble inflow wall boundary laid a milestone foundation for the numerous follow-on works in SPH turbulence [21; 22] and SPH open-channel flows [23; 24]. Another category of SPH models commonly used in the flooding hydraulics is based on the solution of Shallow Water Equations (SWEs), i.e., SWE-SPH, which solves the nonlinear SWEs by using the SPH interpolation principles and benchmark works in this field were documented by [25; 26]. In aspect of solving the oscillation in pressure field and diminishing difficulty of post-processing many works have been done. The Moving-Least-Squares (MLS) approach was developed by [PERSON] [27] and applied successfully to solve pressure oscillation by [PERSON] and [PERSON] [28]. [PERSON] et al. [29] used a filtering technique based on Wavelet Transform to remove the acoustic components for solving the problem of pressure oscillation, especially in the violent sloshing flow. [PERSON] et al. [30] also applied the Wavelet Transform technique to horizontal flow around circular cylinders to alleviate numerical noise in the pressure field. [PERSON] et al. [31] implemented \(\delta^{+}\)-SPH scheme to simulate the flows past bodies at large Reynolds numbers, whose results showed that it was effective in preventing the onset of tensile instability. The representative researches on hydraulic jump based on SPH algorithm are as follows. [PERSON] et al. [32] proposed three solutions to the discrepancies between the SPH simulated wave profile and the practical one (for example, the large fluctuation accompanied by pressure oscillation) and analyzed the feasibilities of the three solutions, through which the numerical model of modified turbulence model VOT(i) was determined and the wave profile and pressure distribution which agreed with the practical situation were obtained.