Segregation-induced finger formation in granular free-surface flows

Abstract

Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregationmobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the µ(I)-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.

Self-channelisation and levee formation in monodisperse granular flows

“…Numerical solutions are calculated using a high-resolution semi-discrete nonoscillatory central (NOC) scheme for convection-diffusion equations (Kurganov & Tadmor 2000). This method has proved its ability to solve similar systems of conservation laws for erosion-deposition waves (Edwards & Gray 2015;Edwards et al 2017), segregation-induced finger formation (Baker et al 2016b) and bi-disperse roll waves (Viroulet et al 2018). The equations are solved in conservative form, where U = (h, hū, hv) T is the vector of conserved fields, with U x and U y being the derivatives of U with respect to x and y, respectively.…”

“…Both of these sets of experiments are dry and have a very narrow range of particle sizes, but, as Félix & Thomas (2004) showed, static levees still form. This suggests that neither interstitial fluid nor particle-size segregation are essential to the self-channelisation process, although they may strongly enhance its effects (Pouliquen, Delour & Savage 1997;Pouliquen & Vallance 1999;Félix & Thomas 2004;Goujon, Dalloz-Dubrujeaud & Thomas 2007;Iverson et al 2010;Woodhouse et al 2012;Kokelaar et al 2014;Baker, Johnson & Gray 2016b).…”

Formation of levees, troughs and elevated channels by avalanches on erodible slopes

“…Strong evidence for these viscous terms is provided by the fact that, unlike the inviscid avalanche model, they predict a downslope velocity profile across a channel with rough side walls (Baker et al 2016a). The viscous terms are also vital in regularizing depth-averaged models of segregation-induced fingering (Pouliquen, Delour & Savage 1997;Pouliquen & Vallance 1999;Woodhouse et al 2012;Baker, Johnson & Gray 2016b). In addition, the functional dependence on the slope inclination ζ and thickness h to the three-halves power is crucial in matching the cutoff frequency of roll waves without any additional fitting parameters .…”

Effect of geometric base roughness on size segregation

“…For instance, the minimum thickness necessary to sustain a steady flow down incline is a function of base roughness [1]; the fingering instability developed at the front of granular avalanches is due to the change of basal resistance upon segregation [2][3][4]; an inadequate base roughness may be responsible for the unsteady flow regime and crystallisation in chute flows [5][6][7]; in smallscale experimental debris flows, base roughness is the key to produce a realistic deposition with levee formation [8].…”