WAF And MUSCL-HLLC Finite Volume Scheme And The Applications In Dam-break Problem

The Godunov schemes of WAF and MUSCL-HLLC are proposed for the shallow water equations based on unstructured mesh. At the same time, our attention is also paid to the generation and development of dam-breaking and flooding waves. For the purpose of dealing properly with shock wave, irregular topography and wet/dry fronts, the main work is as following:(1) Considering the problem that the regular mesh can’t manage the com-plex topography and irregular domain in dam-break, the well-balanced WAF and MUSCL-HLLC schemes are firstly proposed on the unstructured mesh for the shallow water equations which are written in the form of water surface el-evation and discharge per unit width. Firstly, the form of the finite volume scheme on the unstructured mesh has been developed. Secondly, the WAF method, MUSCL data reconstruction and HLLC Riemann solver are used to approximate the numerical flux at the interface. The second order Runge-Kutta integration method is used to improve the accuracy in time. For the treatment of source terms, the rectangular rule is applied to the line integral of source terms due to topography.(2) In order to develop a numerical scheme which can reproduce the exact solution for the steady flow, the shallow water considered are presented in the form of water surface elevation and discharge per unit width. And for source terms arising from topography, the Green rule is used to transform the surface integral of the bed slope terms over the element to the line integral along the sides of the corresponding element and the rectangle formula is employed to approximate the line integral of the bed slope terms. The method of treatment of source terms is first used for the WAF and MUSCL-HLLC schemes on the unstructured mesh, following which the WAF and MUSCL-HLLC schemes are proved to satisfy C-property. For the treatment of wet/dry fronts, the method of the redefinition of the discretized bottom based on the Roe scheme is first extended to the schemes of the WAF and MUSCL-HLLC. The modified WAF and MUSCL-HLLC schemes are proved to satisfy the extended C-property in the theory and the computational tests of one and two dimensional stationary flows.(3) The problems of dam-break and flooding waves are discussed by the schemes of WAF and MUSCL-HLLC. In one dimension, the steady flow over a bump, the dam-break on the smooth and discontinuous channel, the dry chan-nel generated by the two rarefaction waves and the pulse wave propagation are considered and the oblique shock wave, partial dam-break, dam-break over the channel with three humps and dam break in channel with 45°and 90°bend are discussed for the 2D cases. The computational results by the WAF and MUSCL-HLLC schemes agree with the analytical results and the existing or experimental results. In one aspect, the WAF and MUSCL-HLLC proposed are high accurate and high resolution, can deal with wet/dry fronts, have the advantages of ro-bustness, reliability and practicability. In the other aspect, the mechanism of the dam-break, the procedure of the dam-break and flooding waves and the distri-bution of complex flow field are considered. The effects of the factors, such as the initial dam-break conditions, irregular topography and wet/dry fronts, on the problem of dam-break and flooding waves are analyzed.