Ben R. Hodges
B.R. Hodges, “Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage,” Hydrology and Earth System Sciences Discussion, in review, 2018. https://doi.org/10.5194/hess-2018-242
Publication year: 2018

ABSTRACT: New finite-volume forms of the Saint-Venant equations for one-dimensional (1D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and serve to transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. The derivation introduces an analytical approximation of the free surface across a finite volume element (e.g. linear, parabolic) as well as an analytical approximation of the bottom topography. Integration of the product of these provides an approximation of a piezometric pressure gradient term that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, water surface elevations, and the channel bottom elevation (but without using any volume-averaged bottom slope). The new conservative form should be more tractable for large-scale simulations of river networks and urban drainage systems than the traditional conservative form of the Saint-Venant equations where it is difficult to maintain a well-balanced discretization for highly-variable topography.

RESULTS:

We start with the traditional differential Cunge-Liggett form of the Saint Venant Equations (SVE), where momentum is

A conservative finite-volume C-L form (or variants) is typically preferred for solutions of the SVE in natural channels. We show that we can write a new finite-volume conservative form that has some significant advantages in the simplicity of the source term containing the free surface. Three different forms are presented as

The above correspond to different approximations of the relationship between the spatially-varying bottom bathymetry and the free surface in the development of the discrete pressure term.  These terms occur because of the divergence/convergence of the free surface and bottom, e.g.

We can discretize the above by considering a stair-step bathymetry:

As the number of stair steps goes to infinity, we can represent the bottom pressure by an analytical function. Similarly, we can use an analytical function (e.g. linear slope) for the free surface. The quadrature of these functions provides the piezometric pressure contributions from the bottom that influence the finite volume.