A moment model of shallow granular flows with variable friction laws

TL;DR

The paper develops a general friction-aware shallow water moment framework for granular flows on inclined planes by deriving a transformed, vertically resolved velocity expansion and embedding diverse friction laws through a three-step procedure. It introduces a hyperbolic regularized moment system (HSWME) and a path-conservative finite-volume scheme to handle nonconservative terms and stiff friction sources, enabling robust numerical simulations including wet-dry fronts. The authors explicitly treat Newtonian, Manning, Savage-Hutter, Coulomb-type, and $\mu(I)$-rheology friction, with analytical results for special cases (notably $N=1$) and equilibrium analyses, complemented by numerical experiments that illustrate the influence of incline, bed friction, and vertical moments on runout and velocity profiles. The framework provides a flexible, analytically tractable tool for predicting granular-flow dynamics in geophysical and industrial contexts, with potential extensions to well-balanced schemes and higher-order discretizations.

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 $μ(I)$-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.

Figures (8)

• Figure 1: Shallow flow geometry on an inclined plane with angle $\theta$ and bottom topography $b(x)$ and two velocity profiles: constant (left) and more accurate polynomial (right).
• Figure 2: Stresses in the physical (left) and transformed (right) coordinate systems.
• Figure 3: Example 1: Inclination effect on the height $h$ (top row), average velocity $u_m$ (middle row) and bottom velocity $\widetilde{u}(0)$ (bottom row) for the Newtonian-slip friction at $t=0.4$ (left column), $t=0.6$ (middle column) and $t=1$ (right column). In all the examples $\Lambda = 10^{-4}$ and $\eta=0.01$.
• Figure 4: Example 2: Bottom friction effect on the bottom velocity for the Newtonian-slip (top row) and Newtonian-Manning (middle row) for $\theta = \pi/4$ and different values of $\Lambda$ and $n$ at $t=0.4$ (left column), $t=0.6$ (middle column) and $t=1$ (right column). Bottom row: comparison of the height $h$ between Newtonian-slip with $\Lambda=0.0015$ and Newtonian-Manning with $n = 0.0165$.
• Figure 5: Example 3: Comparison of the height $h$ (top row) and bottom velocity $\widetilde{u}(0)$ (bottom row) computed with the Savage-Hutter friction model for two different bed friction angles $\delta=15^\circ$ (red) and $\delta = 18^\circ$ (blue) at $t=0.4$ (left column), $t=0.6$ (middle column) and $t=1$ (right column). The inclination angle used is $\theta = \pi/4$ and inner friction angle $\varphi = 20^\circ$.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6