Numerical Simulation For Discontinuous Shallow Water Flow And Its Application To The Dam-Break Flow

One of the primary trends of the modern shallow water flows simulation is to make use of the mathematic similarity between the homogeneous shallow water equation and Euler equation,as well the high performance algorithm of computational aerodynamics such as Osher,Roe,FVS, HLL,HLLC,ENO and WENO to simulate flow that contains discontinuity or weak discontinuity such as dam-break and bore.The quotations will be adjusted in accordance with the particularity of shallow water flows.Based on the research of other scholars,this paper uses the finite volume method with high performance schemes to two-dimensional shallow water equation and builds a mathematical model that can stimulate two-dimensional shallow water flows on unstructured grids(triangular).In the space discreteness of the model,this paper applies the finite volume framework,the two-dimensional unstructured grid high-precision,high-resolution model is presented with nonlinear shallow water equations with source terms,and successfully applied to discontinuous flow and the dam-break flow of the numerical simulation with complicated geometry and topography.Based on the unstructured grid,the Roe's approximate Riemann solver is used for the computation of inviscid numerical flux functions,well-balanced between the Godunov scheme and the two-dimensional model is presented for the shallow water equation with source terms.The source terms are decomposed in the characteristic directions,which keep the flux balance at the interface and protected the scheme harmonious.Balancing flux gradients and source terms in equations makes the model can solve the shallow water flows and discontinuous flows with variable depth.So the harmonious Roe-Upwind finite volume model is constructed finally.Moving boundary shallow water simulation is a key problem.A wetting-drying condition for unsteady shallow water flow in two-dimensional leading to zero numerical error in mass conservation is presented in this work.It is shown that this numerical technique reproduces exactly steady state of still water and enables to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope.Numerical results are shown which demonstrate the effectiveness of the wetting-drying condition in flood propagation and dam break flows over real complex geometries and bottom slope variation. The 2D,Roe-Upwind finite volume model is applied to numerical simulation of some classic discontinuous flow examples to prove the validity and applicability.The examples include Stoker problem,2D dam-break problem,oblique hydraulic jump problem and mixed flow problem.All the computed results are good agreement the analytic solutions,and present steep jump without non-physical oscillations near the shock.Model also successfully applied to the actual water flow or dam-break flow,the better to verify the results.All results show that the model is efficient,accurate,stable,and can be used widely.