A depth-averaged -rheology for shallow granular free-surface flows

Abstract

The µ(I)-rheology is a nonlinear viscous law, with a strain-rate invariant and pressure-dependent viscosity, that has proved to be effective at modelling dry granular flows in the intermediate range of the inertial number, I. This paper shows how to incorporate the rheology into depth-averaged granular avalanche models. To leading order, the rheology generates an effective basal friction, which is equivalent to a rough bed friction law. A depth-averaged viscous-like term can be derived by integrating the in-plane deviatoric stress through the avalanche depth, using pressure and velocity profiles from a steady-uniform solution to the full µ(I)-rheology. The resulting viscosity is proportional to the thickness to the three halves power, with a coefficient of proportionality that is angle dependent. When substituted into the depth-averaged momentum balance this term generates second-order derivatives of the depth-averaged velocity, which are multiplied by a small parameter. Its inclusion therefore represents a singular perturbation to the equations. It is shown that a granular front propagating down a rough inclined plane is completely unaffected by the rheology, but, discontinuities, which naturally develop in inviscid roll-wave solutions, are smoothed out. By comparison with existing experimental data, it is shown that the depth-averaged µ(I)-rheology accurately predicts the growth rate of spatial instabilities to steady-uniform flow, as well as the dependence of the cutoff frequency on the Froude number and inclination angle. This provides strong evidence that, in the steady-uniform flow regime, the predicted decrease in the viscosity with increasing slope is correct. Outside the range of angles where steady-uniform flows develop, the viscosity becomes negative, which implies that the equations are ill-posed. This is a signature of the ill-posedness of the full µ(I)-rheology at both high and low inertial numbers. The depth-averaged µ(I)-rheology therefore cannot be used outside the valid range of angles without additional regularization.

Debris flows: Experiments and modelling

“…(21) has been carefully constructed to ensure that when it is summed over all constituents it yields the bulk incompressibility condition ∇ · u = 0. This condition and the lithostatic pressure distribution (8), are the key assumptions underlying most models for granular avalanches and geophysical mass flows (e.g., [1,20,21,30,[78][79][80][81][82][83][84]). The bulk threedimensional velocity field u may be reconstructed from depth-averaged avalanche models (e.g., [58]), or it may also be computed directly using the μ(I) rheology [85].…”

“…Woodhouse et al [38] showed that this was due to ill-posedness [107] when the characteristics of the two systems coincide and strict hyperbolicity is lost. Efforts are now underway to use the μ(I)-rheology [65,85] to introduce a depth-averaged viscous term [84] into the equations to set the wavelength.…”

“…J.M.N.T. Gray et al / C. R. Physique 16 (2015) [73][74][75][76][77][78][79][80][81][82][83][84][85] jacente de grains statiques ou en déplacement lent. La combinaison de la ségrégation et des changements de phase solide-fluide granulaire crée des motifs complexes dans les dépôts résultants, mais la compréhension complète de ces effets est pour le moment hors de portée.…”

A laboratory-numerical approach for modelling scale effects in dry granular slides

“…In this study, the Froude number is defined as Fr ¼ u= ffiffiffiffiffi gh p , where u is the slide velocity and h is the slide thickness. h was chosen as a characteristic length for the slide in analogy to MiDi (2004) and Pouliquen and Forterre (2009), as it represents the criticality of the flow, providing insight into the influence of wave speed relative to flow-speed and general flow dynamics (Gray and Edwards 2014;Heller 2011).…”

Stopping dynamics of a steady uniform granular flow over a rough incline

“…(3) are the gravity along the slope, the friction and the pressure gradient. The friction term can be interpreted either as a basal friction [6] or as the averaged internal friction over the thickness [17]. The friction μ(I) [4] is expressed as:…”