Shock-Capturing and Front-Tracking Methods for Granular Avalanches

Abstract

Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shock-capturing numerical scheme for the one-dimensional Savage-Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a front-tracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the front-tracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat run-out region, where a shock is formed as the avalanche comes to a halt. c 2002 Elsevier Science

Physically-based modelling of granular flows with Open Source GIS

“…Cell averages of h, u and v are computed using a staggered grid: the system is moved half of the cell size with every time step, the values at the corners of the cells and in the middle of the cells are computed alternatively. The TVD-NOC scheme with the minmod limiter has been successfully applied to a large number of avalanche and debris flow problems (Tai et al, 2002;Wang et al, 2004;Pudasaini et al, 2005a, b, 2008, Pudasaini and Kroener, 2008. Here, this scheme is implemented in the r.avalanche computational tool.…”

“…Zwinger et al (2003) adapted the Savage-Hutter model for arbitrarily shaped mountain terrain, introducing some geometric simplifications. Based on Nessyahu and Tadmor (1990) and Tai et al (2002), Wang et al (2004) devised a numerical scheme applicable to flows over complex shallow terrain along a horizontally straight flow path (Gray et al, 1999). Pudasaini and Hutter (2003) proposed an avalanche model for generally curved and twisted channels which was further developed by Pudasaini et al (2005aPudasaini et al ( , b, 2008.…”

Dense Granular Avalanches: Mathematical Description and Experimental Validation

“…4 of [24]. We sketch the method only and must direct the interested reader to the literature, see [24] and [25] for detail. Let us commence by recalling that the Lagrangian integration technique applied to the SH equations faced difficulties whenever a supercritical extending (diverging) flow became subcritical and contracting.…”

“…A third method is to treat the cells in the immediate neighbourhood of the margin separately by a special front-tracking method. For one-dimensional flows this has been done [25], and results for the spreading of a parabolic profile turned out to be very much improved; for two-dimensional situations the method must still be developed. We now present a few computational solutions.…”

On impinging near‐field granular jets

“…For example, for many materials, relatively large electrostatic charges can build up for grains below the one millimeter scale. For general overviews of different granular physical phenomena, we refer the reader to work by Behringer and collaborators [1][2][3], Hutter and collaborators [4][5][6][7][8][9][10][11][12][13][14][15][16][17], and Jaeger and collaborators [18][19][20][21][22][23][24][25][26].…”