Numerical modelling of rapidly varied river flow

Abstract

A new approach to solve shallow water flow problems over highly irregular geometry both correctly' and efficiently is presented in this thesis. Godunov-type schemes which are widely used with the finite volume technique cannot solve the shallow water equations correctly unless the source terms related to the bed slope and channel width variation are discretized properly, because Godunov-type schemes were developed on the basis of homogeneous governing equations which is not compatible with an inhomogeneous system. The main concept of the new approach is to avoid a fractional step method and transform the shallow water equations into homogeneous form equations. New definitions for the source terms which can be incorporated into the homogeneous form equations are also proposed in this thesis. The modification to the homogeneous form equations combines the source terms with the flux term and solves them by the same solution structure of the numerical scheme. As a result the source terms are automatically discretized to achieve perfect balance with the flux terms without any special treatment and the method does not introduce numerical errors. Another point considered to achieve well-balanced numerical schemes is that the channel geometry should be reconstructed in order to be compatible with the numerical flux term which is computed with piecewise constant initial data. In this thesis, the channel geometry has been changed to have constant state inside each cell and, consequently, each cell interface is considered as a discontinuity. The definition of the new flux related to the source terms has been obtained on the basis of the modified channel geometry. A simple and accurate algorithm to solve the moving boundary problem in two-dimensional modelling case has also been presented in this thesis. To solve the moving boundary condition, the locations of all the cell interfaces between the wet and dry cells have been detected first and the integrated numerical fluxes through the interfaces have been controlled according to the water surface level of the wet cells. The proposed techniques were applied to several well-known conservative schemes including Riemann solver based and verified against benchmark tests and natural river flow problems in the one and two dimensions. The numerical results shows good agreement with the analytical solutions, if available, and recorded data from other literature. The proposed approach features several advantages: 1) it can solve steady problems as well as highly unsteady ones over irregular channel geometry, 2) the numerical discretization of the source terms is always performed as the same way that the flux term is treated, 3) as a result, it shows strong applicability to various conservative numerical schemes, 4) it can solve the moving (wetting/drying) boundary problem correctly. The author believes that this new method can be a good option to simulate natural river flows over highly irregular geometries