Flood routing methods

Abstract

The hierarchy of one-dimensional equations and numerical methods describing the motion of floods and disturbances in streams is studied, critically reviewed, and a number of results obtained. Initially the long wave equations are considered. When presented in terms of discharge and cross-sectional area they enable the development of simple fully-nonlinear advection-diffusion models whose only approximation is that disturbances be very long, easily satisfied in most flood routing problems. Then, making the approximation that changes in surface slope are relatively small such that diffusion terms in the equations are small, various advection-diffusion and Muskingum models are derived. Several well-known Muskingum formulations are tested; one is found to be in error. The three families of governing equations, the long wave equations, and the advection-diffusion and the Muskingum approximations, are linearised and analytical solutions obtained. A dimensionless diffusion-frequency number measures the accuracies of the approximate methods. Criteria for practical use are given, which reveal when they have difficulties, for streams of small slope, for fast-rising floods, and/or when shorter period waves are present in an inflow hydrograph. They can probably be used in most flood routing problems with an idealised smooth inflow. However the fact that they cannot be used for all problems requires a general alternative flood routing method, for which it is recommended to use the long wave equations themselves written in terms of discharge and cross-sectional area, when a surprisingly simple physical stream model can be used. An explicit finite difference numerical method is presented that can be used with different inflow specifications and downstream boundary conditions, and is recommended for general use.