Compressible Shallow Granular Flow over a Rough Plane

Most existing depth-averaged granular flow theories assume that dry, cohesionless granular materials are incompressible, with the void ratio among grains remaining spatially and temporally invariant. However, recent large-scale experiments showed that the pore space among grains varies both spatially and temporally. This study, therefore, incorporates the effects of granular dilatancy to perform analytical and numerical investigations of granular flows down inclined planes. A high-resolution shock-capturing scheme is employed to numerically solve the compressible depth-averaged equations for temporal and spatial evolution of the flow thickness and depth-averaged velocity, as well as depth-averaged volume fraction. Additionally, a traveling wave solution is constructed. The comparison between analytical and numerical solutions confirms the accuracy of the numerical solution and also reveals that the gradient of the solids volume fraction, induced by granular dilatancy, results in a gentler slope of the granular front, in agreement with experimental observations. Furthermore, this numerical framework is applied to investigate granular flows transitioning from an inclined plane onto a horizontal run-out pad. The numerical solution shows that the incorporation of granular dilatancy causes the shock wave to propagate upstream more rapidly. As a result, the position and morphology of the mass deposit exhibit closer alignment with experimental data.