An Analysis Framework to Study Steady State Friction Dominated Saint-Venant Equations

An analysis framework to study the friction term dominated steady state 1D Saint-Venant equations is developed using a non-uniform flow test case and Fourier analysis. In the non-uniform flow test, a sudden bed perturbation is introduced, and in the Fourier analysis, a periodic bed perturbation is used. In both analyses, the effects of the bed perturbations on the solution variables in a steady state case are observed. Both the shock capturing and non-shock capturing numerical schemes have been studied in this research. From the non-uniform flow test results, it is found that the oscillations in the discharge and/or depth solutions are apparent when discretization, roughness, and slope are large, and the errors in the solution variables increase with an increase in these parameters. From the Fourier analysis, the main non-dimensional parameter groups identified are: the number of discretization intervals per wavelength, the average flow Froude number, and the numerical Friction number. The Fourier analysis results show that the errors in both depth and discharge solutions or only in the depth solutions are observed whenever there is any perturbation in the bed topography. These errors increase with an increasing Froude number and increasing numerical Friction number. A combined friction parameter is proposed for practical modeling purposes which captures the variations of the separate parameters. The proposed combined friction parameter is capable of locating the friction dominated region in open channel flow modeling. The proposed combined friction parameter is easy to calculate and to implement in any open channel flow model. The proposed combined friction parameter is applied and used in 1D and 2D flow models as a mesh refinement indicator and as a minimum depth criterion. The results show that the proposed combined friction parameter is effective to identify and to eliminate or reduce spurious velocity vectors. The analysis presented in this study can be applicable to a wide range of numerical methods to study the friction dominated steady state Saint-Venant equations.