Polydisperse particle-driven gravity currents in non-rectangular cross section channels

We consider a high-Reynolds-number gravity current generated by polydisperse suspension of n types of particles distributed in a fluid of density ρi. Each class of particles in suspension has a different settling velocity. The current propagates along a channel of non-rectangular cross section into an ambient fluid of constant density ρa. The bottom and top of the channel are at z = 0, H, and the cross section is given by the quite general form −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. The flow is modeled by the one-layer shallow-water equations obtained for the time-dependent motion. We solve the problem by a finite-difference numerical code to present typical height h, velocity u, and mass fractions of particle (concentrations) (ϕ( j), j = 1, …, n) profiles. The runout length of suspensions in channels of power-law cross sections is analytically predicted using a simplified depth-averaged “box” model. We demonstrate that any degree of polydispersivity adds to the runout length of the currents, relative to that of equivalent monodisperse currents with an average settling velocity. The theoretical predictions are supported by the available experimental data. The present approach is a significant generalization of the particle-driven gravity current problem: on the one hand, now the monodisperse current in non-rectangular channels is a particular case of n = 1. On the other hand, the classical formulation of polydisperse currents for a rectangular channel is now just a particular case, f(z) = const., in the wide domain of cross sections covered by this new model.