Continuum simulation for regularized non-local μ(I) model of dense granular flows

Abstract

In this work, we implement the local μ(I) rheology law and its non-local modification in a PISO-VOF numerical scheme with a sharp-interface-capturing THINC method for two-dimensional dense granular flows in both steady and transient states. The non-local effect on the constitutive relation is addressed by the gradient expansion model to account for additional momentum transport in view of the spatial variation of flow inertial number and the stress model singularity near the static state, I=0, is handled by the Bercovier-Engelman regularization scheme. Our simulations provide the first numerical evidence for the crucial role of non-local momentum transport on degrading the velocity profiles predicted with a local rheology model as those reported from two-dimensional discrete element simulations. For example, we successfully reproduce the sub-Bagnold velocity profile for steady inclined flows of height comparable to the Hstop and capture how the linear velocity profile in a simple shear creeping flow is degraded to a S-shape. More importantly, we exploit the current solver to study the layer-accumulated deviation, ε, between the local and the non-local velocity profile (Bagnold to its degradation) for inclined flows. For steady flows, ε decays monotonically with flow Froude number Fr and a critical Frc≈0.25 is detected below which ε rises rapidly necessitating a non-local constitutive relation. For a layer developing from rest to its steady state, the time evolution of ε(t) shows further dependence on flow height and can be collapsed into a monotonically growing trend with local instantaneous inertial number I(t). A critical Icr is also determined and the local model prediction can be erroneous whenever I(t)<Icr. Interestingly, the result for simple shear flows without gravity also confirms that non-local effect becomes non-negligible when the macroscopic inertial number drops to the order of Icr. Finally, we simulate the two-dimensional column collapse flows with both local and non-local rheology and find a shorter run-out distance with the presence of non-local momentum transport.