A moment model of shallow granular flows with variable friction laws

TL;DR

This paper introduces a new shallow water moment model for granular flows on inclined planes that incorporates variable friction laws and develops a numerical scheme to handle complex source term stiffness.

Contribution

It extends classical shallow water equations with polynomial velocity profiles and integrates various friction laws into a unified modeling framework.

Findings

Successfully incorporates multiple friction laws into the model

Develops a finite volume scheme for stiff source terms

Simulates wet-dry fronts in granular flows

Abstract

In this work, we develop a modelling framework for granular flows based on the shallow water moment equations on inclined planes. Under the assumption of a polynomial expansion of the velocity field, the model extends the classical shallow water equations to vertically variable velocity profiles. The friction effects, which are captured through the strain-rate tensor, are incorporated into the model in two terms, the bulk and bottom friction. We propose a modelling procedure to incorporate general friction laws into our framework and exemplify this combining the Manning, Coulomb, Savage-Hutter, and $\mu(I)$-rheology friction models in our modeling framework. Moreover, we develop a path-conservative finite volume numerical scheme based on the polynomial viscosity matrix method to properly handle the stiffness of the source terms. Numerical simulations are presented for different models of friction, including the case of wet-dry fronts.