Dynamically consistent entrainment laws for depth-averaged avalanche models

Dieter Issler

doi:10.1017/jfm.2014.584

2014-10-28

PRE

Dieter Issler *

DOI：10.1017/jfm.2014.584

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摘要

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The bed entrainment rate in a gravity mass flow (GMF) 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 tau(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 tau(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 depth-resolved simulations asymptotically, but differs in the start-up phase. The non-stationary erosion formulae 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.

Keyword:

general fluid mechanics

geophysical and geological flows

mixing and dispersion