On Non-Boussinesq Gravity Currents in Non-Rectangular Cross-Section Channels

We consider the propagation of a gravity current of density ρc from a lock length x0 and height h0 into an ambient fluid of density ρa in a horizontal channel of height H along the horizontal coordinate x. The bottom and top of the channel are at z = 0, H, and the cross-section is given by the quite general −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. When the Reynolds number is large, the resulting flow is governed by the parameters R = ρc/ρa, H* = H/h0 and f(z) = f1(z) + f2(z). We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness h and speed u of the current. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1 and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio in various cross-section geometries. We obtain analytical solutions for the initial dam-break stage of propagation with constant speed, which appears for any cross-section geometry, and derive explicitly the trend for small and large values of the governing parameters. For large time, t, a self-similar propagation is feasible for f(z) = bzα cross-sections only, with t(2+2α)/(3+2α). The present approach is a significant generalization of the classical non-Boussinesq gravity current problem. The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, f(z) = const., in the wide domain of cross-sections covered by this new model.Copyright © 2014 by ASME