Shallow-Water Equations

Abstract

This chapter focuses on the derivation of shallow-water equations commonly used to describe water movement through flood plains and river channel networks. The Cauchy momentum equation is derived by combining a three-dimensional stress balance with Newton’s second law. The Navier Stokes equations are developed by incorporating a viscosity relationship between deviatoric stress and fluid velocity. Shallow water equations are derived by vertically averaging the Navier Stokes equations using Leibniz’s theorem for differentiation of an integral. Eddy viscosities are used to obtain simplified approximations for the associated internal deviatoric stress terms and differential advection terms. Wall shear stress is represented using a Darcy–Weisbach-type skin friction model. The Saint Venant equations are derived by assuming one-dimensional open channel flow. Three additional approximations are considered including the gradually varied flow equation (referred to as a quasi-steady approximation), the diffusion wave equation and the kinematic wave equation. The Ogata–Banks solution is presented as an analytical solution for the diffusion wave equation and used to explore some practical applications, presented through a problem sheet with worked solutions.