Density variations in dry granular avalanches

Abstract

Dry granular avalanches exhibit bulk density variations. Understanding the physical mechanisms behind these density variations is especially important in the study of geophysical flows such as snow and rock avalanches. We performed small-scale chute experiments with glass beads to investigate how bulk density changes, measuring velocity profiles, flow height and basal normal stress in an Eulerian measurement frame. The chute inclination and the starting volume of glass beads were systematically varied. From the flow height and basal normal stress data, we could compute the depth-averaged density at the measurement location during the passing of the avalanches. We observed that the depth-averaged density is not constant, varying with chute inclination and starting volume. Furthermore, the depth-averaged density varies from the head to the tail within a single avalanche. We model changes in density by accounting for the energy associated with the velocity fluctuations of the grains, the density and the velocity fluctuations being related by the constitutive relation for the normal stress. We propose expressions for the conduction and decay coefficients of the fluctuation energy which allow us to model the observed density variations in the experiments.

Appendix

The interval $\left[ 0,h\right]$ is discretized in $N$ elements. The vector $y$ with size $N+$1 contains the $y$-coordinates of the elements edges. In a first step Eq. (18) is integrated for $n=N$ to 1 yielding the normal stress vector with size $N+$1:

$$\begin{aligned} p^n = p^{n+1} + \rho ^n g cos\theta \left( y^{n+1} - y^n\right) . \end{aligned}$$

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with the boundary conditions $p^{N+1} = 0$. In a second step Eq. (19) is integrated over the flow depth. First the vector $\frac{\partial T}{\partial y}$ with size $N+$1 is computed for $n=N$ to 1, second the vector $T$ with size $N+$1 is calculated for $n=$ 2 to $N+$1:

$$\begin{aligned} \frac{\partial T}{\partial y}^n&= \frac{\partial T}{\partial y}^{n+1} - \left( y^{n+1} - y^n\right) \Bigg (-\frac{\tau _0}{\kappa l^{n+1}} \left( d \frac{\partial u}{\partial y} \right) ^3\nonumber \\&- \frac{1}{\rho ^{n+1}} \frac{\partial \rho }{\partial y}^{n+1} \frac{\partial T}{\partial y}^{n+1} + \frac{\gamma _0}{\kappa l^{n+1}} T^{n+1 \frac{3}{2}} \Bigg )\nonumber \\ T^{n+1}&= \quad T^{n} + \left( y^{n+1} - y^n \right) \frac{\partial T}{\partial y}^n \end{aligned}$$

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with the boundary conditions $\frac{\partial T}{\partial y}^{N+1}=$ 0 and $T^1 = \xi \sqrt{u_s}$. In a third step, the normal stress and the granular temperature are evaluated at the centre of the elements (vectors with size $N$) and a new density vector with size $N$ is calculated. The size of the elements 1 to $N$ i.e. the vector $y$ is modified with respect to the new density vector so that the mass of each element is conserved.