Dam break with Coulomb friction: a model for granular slumping

Axisymmetric and two-dimensional dam-break problems are solved under the shallow layer approximation and with basal Coulomb friction in a first attempt to model recent granular slumping experiments [G. Lube, H. E. Huppert, R. S. J. Sparks, and M. A. Hallworth, J. Fluid Mech. 508, 175 (2004); E. Lajeunesse, A. Mangeney-Castelnau, and J. P. Vilotte, Phys. Fluids 16, 2371 (2004); N. J. Balmforth and R. R. Kerswell, J. Fluid Mech. (in press)]. The axisymmetric model predicts a scaling law h(0)/h(infinity)proportional to a for the ratio of initial to final maximum heights which compares well with the slumping data over a wide range of a:=h(0)/r(0), the initial aspect ratio (h(0) and r(0) are the initial height and radius). The predicted runout scaling (r(infinity)-r(0))/r(0)proportional to a, however, significantly overestimates the data runout which behaves more like root a(r(infinity) being the maximum final radius). This situation may be improved by redefining the runout to take account of the finite granule size but a noticeable discrepancy still remains. The runout scaling prediction in the two-dimensional case mirrors the axisymmetric result, whereas in two dimensions h(0)/h(infinity)proportional to a(alpha) with alpha -> (2)/(3) as a ->infinity and a:=h(0)/l(0) where l(0) is the initial base length of the column. The limitations of the model and possible improvements are discussed.