Journal of Fluid Mechanics

Papers

The granular column collapse as a continuum: validity of a two-dimensional Navier–Stokes model with a μ(I)-rheology

P.-Y. Lagréea1 c1, L. Starona1 and S. Popineta1a2

a1 CNRS and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, Boîte 162, F-75005 Paris, France

a2 National Institute of Water and Atmospheric Research, PO Box 14-901 Kilbirnie, Wellington, New Zealand

Abstract

There is a large amount of experimental and numerical work dealing with dry granular flows (such as sand, glass beads, etc.) that supports the so-called $\ensuremath{\mu} (I)$-rheology. The reliability of the $\ensuremath{\mu} (I)$-rheology in the case of complex transient flows is not fully ascertained, however. From this perspective, the granular column collapse experiment provides an interesting benchmark. In this paper we implement the $\ensuremath{\mu} (I)$-rheology in a Navier–Stokes solver (Gerris) and compare the resulting solutions with both analytical solutions and two-dimensional contact dynamics discrete simulations. In a first series of simulations, we check the numerical model in the case of a steady infinite two-dimensional granular layer avalanching on an inclined plane. A second layer of Newtonian fluid is then added over the granular layer in order to recover a close approximation of a free-surface condition. Comparisons with analytical and semi-analytical solutions provide conclusive validation of the numerical implementation of the $\ensuremath{\mu} (I)$-rheology. In a second part, we simulate the unsteady two-dimensional collapse of granular columns over a wide range of aspect ratios. Systematic comparisons with discrete two-dimensional contact dynamics simulations show good agreement between the two methods for the inner deformations and the time evolution of the shape during most of the flow, while a systematic underestimation of the final run-out is observed. The experimental scalings of spreading of the column as a function of the aspect ratio available from the literature are also recovered. A discussion follows on the performances of other rheologies, and on the sensitivity of the simulations to the parameters of the $\ensuremath{\mu} (I)$-rheology.

(Received February 18 2011)

(Reviewed May 24 2011)

(Accepted August 03 2011)

(Online publication September 27 2011)

Key Words:

  • granular media;
  • Navier–Stokes equations;
  • non-Newtonian flows

Correspondence:

c1 Email address for correspondence: pierre-yves.lagree@upmc.fr

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