Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

A simple radial basis function (RBF) meshless method is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular, frictional topography involving wetting and drying. At first, we construct the RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge–Kutta scheme to approximate the temporal derivative of the differential equation. For the problems involving shock or discontinuity solutions, we use an artificial viscosity for shock capturing. Then, we apply our scheme for several theoretical two-dimensional numerical experiments involving dam break flows over nonuniform beds and moving wet-dry fronts over irregular bed topography. Promising results are obtained.

Simulation of Shallow Flows in Nonuniform Open Channels

This paper presents a new formulation of the 2D shallow water equations, based on which a numerical model (referred to as NewChan) is developed for simulating complex flows in nonuniform open channels. The new shallow water equations mathematically balance the flux and source terms and can be directly applied to predict flows over irregular bed topography without any necessity for a special numerical treatment of source terms. The balanced governing equations are solved on uniform Cartesian grids using a finite-volume Godunov-type scheme, enabling automatic capture of transcritical flows. A high-order numerical scheme is achieved using a second-order Runge–Kutta integration method. A very simple immersed boundary approach is used to deal with an irregular domain geometry. This method can be easily implemented in a Cartesian model and does not have any influence on computational efficiency. The numerical model is validated against several benchmark tests. The computed results are compared with analytical solutions, previously published predictions, and experimental measurements and excellent agreements are achieved.