Front dynamics of supercritical non‐Boussinesq gravity currents

Abstract

[1] In this paper, we seek similarity solutions to the shallow water (Saint-Venant) equations for describing the motion of a non-Boussinesq, gravity-driven current in an inertial regime. The current is supplied in fluid by a source placed at the inlet of a horizontal plane. Gratton and Vigo (1994) found similarity solutions to the Saint-Venant equations when a Benjamin-like boundary condition was imposed at the front (i.e., nonzero flow depth); the Benjamin condition represents the resisting effect of the ambient fluid for a Boussinesq current (i.e., a small-density mismatch between the current and the surrounding fluid). In contrast, for non-Boussinesq currents the flow depth is expected to be zero at the front in absence of friction. In this paper, we show that the Saint-Venant equations also admit similarity solutions in the case of non-Boussinesq regimes provided that there is no shear in the vertical profile of the streamwise velocity field. In that case, the front takes the form of an acute wedge with a straight free boundary and is separated from the body by a bore.

Mathematical Modeling of Powder‐Snow Avalanche Flows

“…The avalanches arise from an instability in a pile of granular material like sand or snow [2]. The destabilization phase of an avalanche life is still a challenging problem.…”

An exact solution for ideal dam‐break floods on steep slopes

“…The more realistic case involving a rough bed (represented by a Chézy-like friction force) has been addressed by a number of authors, including Whitham [1954], Dressler [1952], and Hogg and Pritchard [2004], but only asymptotic solutions have been developed to date. Taking into account a nonuniform velocity distribution in the vertical direction leads to mathematical difficulties, but exact self-similar solutions can still be obtained for floods with variable inflow (i.e., the released volume is a function of time) [Ancey et al, 2006[Ancey et al, , 2007.…”

Existence and features of similarity solutions for non-Boussinesq gravity currents

“…The head is in a subcritical regime (Fr < 1), while the body is in a supercritical regime (Fr > 1). Since a jump is associated with a significant change in the velocity profile, it is also possible to construct solutions where the Boussinesq solution is unity within the leading edge, but is in excess of unity within the body [4].…”

“…It is worth noting that for δ > 1 and α > 1, we may also hypothesize that within the body, there is shear in the vertical direction (i.e., γ > 1), whereas within the tip region, the Boussinesq coefficient drops to unity since the hydraulic jump leads to profoundly altering the velocity profile [4]. With these assumptions, we can construct solutions where the flow conditions are supercritical at the source and the velocity profile is sheared.…”

“…(2). In a recent paper [4], however, we have shown that this asymptotic limit poses severe mathematical difficulties since the only regular self-similar solution to the shallow-water equation is the trivial solution h = 0 for the entire domain of flow. An exceptional solution may, however, be worked out on some occasions.…”