Energy-Bounded Flow Approximation on a Cartesian-Product Grid over Rough Terrain

Abstract

We construct a method for modelling of three-dimensional, time dependent, compressible fluid flow in a gravitational field on a rotating cartesian-product grid with a spatially rough metric that bounds solutions by the total initial physical energy. Specifically: (1) the total physical energy is an /2 norm on the model state and (2) this total energy cannot increase provided the timestep does not exceed CFL limits. In particular, the first property means that our measure of the energy is always positive unless the mass, momentum, and internal energy are all everywhere zero. These conditions guarantee that no error can grow unchecked. This is thought to be a desirable property, although only in the case of linear systems is it sufficient for convergence of a consistent approximation to the true solution. The great merit of this choice of norm is that the method is applicable to a wide variety of real physical problems because, even in complex circumstances, the total physical energy is conserved and each component of this energy is in limited supply. We first note that conservation of energy is equivalent to antisymmetry of a particular tendency operator. Energy-bounded approximations of fluid flow are then constructed either from antisymmetric finite difference operators, or from antisymmetric Galerkin operators. The method may be particularly useful when reliability in difficult conditions is needed. For example, when the viscosity must be small in order to simulate flow separation or turbulence, a model of viscous dissipation may be chosen purely from physical considerations, uncompromised by any requirements of numerical stability. We demonstrate this for an "internal hydraulic jump" flow over a bell-shaped mountain, simulating an internal wave as it steepens and breaks to form a turbulent jump.