Conservative numerical simulation of multi-component transport in two-dimensional unsteady shallow water flow

An explicit finite volume model to simulate two-dimensional shallow water flow with multi-component transport is presented. The governing system of coupled conservation laws demands numerical techniques to avoid unrealistic values of the transported scalars that cannot be avoided by decreasing the size of the time step. The presence of non conservative products such as bed slope and friction terms, and other source terms like diffusion and reaction, can make necessary the reduction of the time step given by the Courant number. A suitable flux difference redistribution that prevents instability and ensures conservation at all times is used to deal with the non-conservative terms and becomes necessary in cases of transient boundaries over dry bed. The resulting method belongs to the category of well-balanced Roe schemes and is able to handle steady cases with flow in motion. Test cases with exact solution, including transient boundaries, bed slope, friction, and reaction terms are used to validate the numerical scheme. Laboratory experiments are used to validate the techniques when dealing with complex systems as the κ–ϵκ–ϵ model. The results of the proposed numerical schemes are compared with the ones obtained when using uncoupled formulations.