Criteria for the choice of flood-routing methods in natural channels

Numerical Modeling in Open Channel Hydraulics

The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow. A variance of the Galerkin method is proposed that exhibits highly selective damping characteristics. The dissipation affects only the numerically-generated high-frequency parasitic waves, while maintaining remarkable accuracy in the approximation to the true solution of the problem. In fact, it is shown that the phase error of the finite element simulation is improved by the introduction of dissipation. The resulting model is second-order accurate with respect to the time step and produces a clean, sharp jump structure that agrees favorably with the exact solution of some test problems. The method is based on discontinuous weighting functions that produce “upwind” effects but at the same time maintain the accuracy of a central difference scheme. The dissipation level is selected by analytical investigations, so that the numerical error is minimized. No second-order pseudo viscosity terms are required, which relaxes the inter-element continuity conditions and results in a very simple and inexpensive scheme.

A Dissipative Galerkin Scheme for Open‐Channel Flow

The diffusive wave equation is generally used in flood routing in rivers. The two parameters of the equation, celerity and diffusivity, are usually taken as functions of the discharge. If these two parameters can be assumed to be constant without lateral inflow, the diffusive wave equation may have an analytical solution: the Hayami model. A general analytical method, based on ‘Hayami’s hypothesis, is developed here which resolves the diffusive wave flood routing equation with lateral inflow or outflow uniformly distributed over a channel reach. Flood routing parameters are then identified using observed inflow and outflow and the Hayami model used to simulate outflow. Two examples are discussed. Firstly, the prediction of the hydrograph at a downstream section on the basis of a knowledge of the hydrograph at an upstream section and the lateral inflow. The second example concerns lateral inflow identification between an upstream and a downstream section on the basis of a knowledge of hydrographs at the upstream and downstream sections. The new general Hayami model was applied to flood routing simulation and for lateral inflow identification of the River Allier in France. The major advantages of the method relate to computer simulation, real-time forecasting and control applications in examples where numerical instabilities, in the solution of the partial differential equations must be avoided.

Analytical hayami solution for the diffusive wave flood routing problem with lateral inflow

Groundwater management, the use of numerical models

A finite element solution based on the Galerkin method was developed for the Saint-Venant equations that approximately govern unsteady flow in rigid open channels. A predictor-corrector solution scheme produced theoretically stable and convergent results, and applications to test problems confirmed stability provided that time steps were not so large as to render invalid the Newton linearization scheme used for the friction factor. Linear elements were adequate for all problems solved. Time step limitations to produce accurate results were closely related to: (1) The degree to which piecewise linear functions describing the variation of depth and discharge in time could be fit to the actual curvilinear functions; and (2) the value given to the time step weighting factor. Values of this factor near 0.5 gave the best results.

Richard L. Cooley

Papers

Finite Element Solution of Saint-Venant Equations