Formation of standing waves in granular chute flows induced by mild basal topography

Abstract

This study demonstrates that standing waves can arise in dry granular flows within a chute with mild sloped basal topography, even when the applied Froude number remains subcritical ($Fr<1$) and below the critical threshold for surface wave instability ($Fr<F{r}_{cr}$). In the absence of basal topography, a stable uniform flow would be possible in this regime. By employing the Saint-Venant equations augmented with the $\mu \left(I\right)$ rheology, we numerically observe the formation of these standing waves and confirm the negligible influence of the viscous diffusive term. A key finding is that these standing waves can be described by a single non-linear inviscid ordinary differential equation. While this equation lacks an analytical solution, we introduce a modified Euler’s method, a semi-analytical approach based on the equation’s direction field, to accurately capture the wave profile. For the special case of very gentle slopes, an implicit analytical approximation can be derived directly from the curve that corresponds to the zero inclination direction field (nullcline). Finally, we conduct numerical simulations using the full Saint-Venant equations to demonstrate that in the opposite Froude regime, when $Fr>F{r}_{cr}$, even a very mild basal topography can induce the formation of roll waves and, furthermore, accelerate the coarsening process. It is shown that the generated roll waves can reach a steady state, even when the basal topography is present along the entire length of the chute. These results highlight the significant influence of topography on flow dynamics across different Froude number regimes.