Author(s): M. M. Ali; P. M. Steffler
Linked Author(s): Peter Steffler
Keywords: Open channel flow; Numerical modeling; Finite element method; Fourieranalysis; Small depth; Source term dominated
Abstract: The Saint-Venant equations are the fundamental mathematical description governing the depth and average velocity in one-dimensional (1D) and twodimensional (2D) open channel flow. There are three main situations where stability problems are observed while solving the Saint-Venant equations numerically: convection dominated, wetting/drying, and small depth or source term dominated. In this research the numerical issues associated with source term are studied using Fourier analysis of the linearized and non-dimensionalized Saint-Venant equations. A sinusoidal bed perturbation is introduced and its effects on the solution variables for the final steady state are observed. The discrete non-dimensional equations are found using characteristic dissipative Galerkin (CDG) finite element scheme and the nondimensional parameter groups identified from these equations are: the uniform flow Froude number, the uniform flow Courant number, the ratio of the perturbation wavelength to the discretization length, and the ratio of the discretization length to a characteristic length scale which is defined as the ratio of the uniform depth and the average bed slope. The Courant number has no effect for the final steady state solution and the Froude number effect is not significant compared to the other two parameters. When the last non-dimensional parameter becomes greater than unity, the errors in the solution variables can be of the order of 100 percent or greater for the shortest wavelengths and can lead to stability or convergence problems. This analysis can be used to find minimum depths to be modeled or to do mesh refinement or to switch to alternate sets of equations. The analysis also can be used as a framework to study the source term dominated cases for other numerical schemes or for other forms of Saint-Venant equations.
Year: 2009