Direct numerical simulation of roll waves on landslide mudflow

Roll waves on landslide muds are nonlinear instabilities on non-Newtonian laminar free-surface flow. Existing theories of the roll waves on the mudflow have been developed using long-wavelength approximations and different rheological models. The differences in the approximations and models have led to vastly different parameterizations and expressions for describing the waves. In this paper, we analyze rheological data of landslide mud and conduct direct numerical simulations (DNS) of the roll waves to reconcile the limitations and differences between existing theories. The DNS identifies for the first time the existence of a leading wave emerging as a “front runner” in mudflow instability development. Depending on the Froude and Reynolds numbers, the front runner can be a smooth or breaking wave. For comparison, we also conduct shallow-layer simulation for the instability using a two-equation depth-averaged model. We analyze temporal and spatial instabilities using two rheological models: the Herschel–Bulkley (HB) and power-law models. While HB and PL models have different rheological parameters, they produce the same roll waves on the landslide mudflow. The DNS shows the limitation of the widely used long-wavelength approximations for the dependence of the wave and instability on the Froude and Reynolds numbers. We find universality in the celerity–amplitude relationship in the nonlinear development of roll waves. The amplitude dispersion of the roll waves on the non-Newtonian mudflow follows a linear relationship remarkably similar to roll waves on laminar and turbulent flows of Newtonian fluid.