Application of a Discontinuous Galerkin Finite Element Method to Liquid Sloshing

Abstract

A Runge-Kutta discontinuous Galerkin (RKDG) finite element method is applied to the liquid sloshing problem using the depth-averaged shallow water equations in a rotating frame of reference. A weak statement formulation is developed by multiplying the equations by a test function and integrating over a typical element. The basis functions are Legendre polynomials of degree one, resulting in formally second-order spatial accuracy. Second-order time integration is achieved using a second-order Runge-Kutta method. A minmod slope limiter is incorporated into the solution near discontinuities to control nonphysical oscillations and to ensure nonlinear total variation bounded stability. The method is first applied to the dam-breaking problem with zero rotation to validate the basic numerical implementation. Grid independence of the solutions is established and solution error is quantified by computing the L1 norm and comparing the estimated convergence rates to theoretical convergence rates. Stability is demonstrated subject to a Courant-Fredricks-Lewey restriction. Sloshing in a nonrotating tank with a prescribed initial water surface elevation is first investigated to demonstrate the ability of the method to capture the wave speed of traveling waves, followed by a tank undergoing sinusoidal rotation. Time histories of water surface elevation at selected locations, as well as pressure distribution on the tank walls and the corresponding moment about the tank centerline are computed and compared to experimental data and to previous computations. Finally, a limited parameter study is performed to determine the effect of varying roll angle, depth to width ratio, and forcing frequency on the resulting maximum moment about the tank centerline.