Exact Riemann Solution and Weighted Average Flux Method for Power-law Channel Section

Abstract

The aim of this work is to extend the Weighted Averaged Flux (WAF) method by E.F. Toro from rectangular to generic regular cross sections, in order to represent the geometric effects of the cross-section on wave propagation along channels. Assuming a power-law variation of the channel width, the 1D conservative shallow water equations, their characteristic form and the shock propagation equations are presented. The exact Riemann solver is derived and is applied to the dam-break problem in valleys with different shape in order to test its efficiency and to check the accuracy order of solutions obtained by approximating the real cross-section with an equivalent rectangle. The WAF method is extended to a power-law channel section and is used to solve the 1D proposed equations taking into account all the source terms that are incorporated into the local Riemann problem. A code based on this method has been developed and results of numerical applications to a Venturi channel and to the attenuation of waves are presented, in order to verify, for well known situations, how accurately the source terms are represented. The code is applied to reproduce one of Brock's experiments (1969; 1970) on roll waves generation in a rectangular channel and results are compared with those obtained with a Godunov-type code developed at CEMAGREF, which is based on a Roe scheme with a later average evaluation of source terms.