Dynamically consistent entrainment laws for depth-averaged avalanche models

Abstract

The bed entrainment rate in a gravity mass flow is uniquely determined by the properties of the bed and the flow. In depth-averaging, however, critical information on the flow variables near the bed is lost and empirical assumptions usually are made instead. We study the interplay between bed and flow assuming a perfectly brittle bed, characterized by its shear strength τ c , and erosion along the bottom surface of the flow; frontal entrainment is neglected here. The brittleness assumption implies that the shear stress at the bed surface cannot exceed τ c . For quasi-stationary flows in a simplified setting, analytic solutions are found for Bingham and frictional-collisional (FC) fluids. Extending this theory to non-stationary flows requires some assumptions for the velocity profile. For the Bingham fluid, the profile of a "proxy" quasi-stationary eroding flow is used; the rheological parameters are chosen to match the instantaneous velocity and shear-layer depth of the non-stationary flow. For the FC fluid, a two-parameter family of functions that closely match the profiles obtained in depth-resolved numerical simulations is assumed; the boundary conditions determine the instantaneous parameter values and allow computation of the erosion rate. Preliminary tests with the FC erosion formula incorporated in a simple slab model indicate that the non-stationary erosion formula matches the depthresolved simulations asymptotically, but differs in the start-up phase. The non-stationary erosion formulas are valid only up to a limit velocity (and to a limit flow depth if there is Coulomb friction). This appears to mark the transition to another erosion regime-to be described by a different model-where chunks of bed material are intermittently ripped out and gradually entrained into the flow.

A Review of Russian Snow Avalanche Models—From Analytical Solutions to Novel 3D Models

“…where q is the rate of volume entrainment per unit area at the flow bottom. Different formulas to determine q were suggested and used in [41,[43][44][45][46][47][48][49][51][52][53]. A review of various ways of incorporating the entrainment effect into models of flows of different physical nature can be found in [44].…”

“…In [103][104][105][106] the entrainment of the underlying material (the basal entrainment) and its influence on the avalanche dynamics are investigated. The approach that was suggested in [51,102] is adopted. Namely, it is assumed that the entrainment of the underlying material occurs if the shear stress at the bottom of the flow (τ b ) reaches the shear strength τ c of the underlying layer; τ b cannot exceed the value τ c .…”

Inferences on Mixed Snow Avalanches from Field Observations

“…Assumptions about the properties of the fluidized flows of the Vilan avalanche near the gully bend and the Albristhorn and Scex Rouge avalanches in the run-out zone (counter-slope, ↑) with Type 2 deposit, and the resulting mean erosion rates. A first-order estimate of the entrainment capacity of a gravity mass flow is [82,83] q e ≈ τ b − τ c u ≈…”

“…Similar problems plague present-day models as well. See [83] for a theoretical analysis of the constraints imposed on models of erosion by scour.…”

The dam-break problem for eroding viscoplastic fluids

“…As viscoplastic models deal with the solid-liquid transition, they also seem relevant for describing basal entrainment: part of the bed may yield under the effects of the normal and shear stresses exerted by the flow, and then be entrained in that flow. This is, for instance, what is thought to happen in snow avalanches [48][49][50].…”