A Finite Volume Implementation of the Shallow Water Equations for Boussinesq Gravity Currents

Abstract

The shallow water equations (SWE) are a powerful tool for modeling the propagation of gravity currents (GC) because of their relative simplicity, computational efficiency and accuracy. Finite difference solutions, either based on the method of characteristics (MOC) or the implementation of numerical schemes such as Lax-Wendroff ($LxW$) have been traditionally used in such flow computations. On the other hand, the finite volume method (FVM) has been gaining popularity in several other hydraulic applications, being favored in cases when flow discontinuities are anticipated. This work is focused on an implementation of the finite volume method (FVM) to the solution of Boussinesq GC using the one and two-layer SWE models. The proposed two-layer mathematical model is a modification of the the work by \cite{Rottman1983}, adapted to express such equations in a vectorial conservative format, amenable for FVM implementation. The traditional solution for the GC front boundary condition (BC), using a characteristic equation and a front condition, is compared to a new formulation that explicitly enforces local mass and momentum conservation. Linear numerical schemes ($LxW$ and FORCE) and non-linear schemes based on the approximate solution of the Riemann problem (Roe and HLL) are implemented in this framework along with various front conditions. The proposed modeling framework is tested against experimental data collected by this investigation, and is also compared to previous investigations. Results indicate that this proposed model has a comparably simple and robust implementation, being flexible enough to be applied in a wide range of GC flow conditions and presents good accuracy.