Discrete Boltzmann Equation model of polydisperse shallow granular flows

Abstract

A polydisperse shallow mixture consisting of Np solid phases (Np ≥ 1) and one fluid phase (the ambient fluid) is a reliable model for several industrial and environmental flows, as e.g. landslides, avalanches, debris flows and fluidized beds. The description and prediction of these flows is of primary importance, mainly with respect to the mitigation and protection from natural hazards. This paper is aimed at deriving the polydisperse shallow granular flow equations by depth-averaging mass and momentum equations of the mixture and at formulating an equivalent Discrete Boltzmann Equation (hereinafter DBE) model as solution method. The reason is the simplicity and the versatility of the DBE, which consists of a set of purely advective, linear, first order partial differential equations, whose numerical integration does not need sophisticated methods. Both 1D and 2D benchmarks, concerning with the propagation of discontinuities in three-phase shallow granular flows, are obtained by applying the finite differences Lax-Friedrichs (hereinafter LF) method to the polydisperse shallow granular flow equations. The overall agreement is good, showing that the DBE and the LF numerical results are equivalent.