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

We consider the propagation of a gravity current of density ρ_c from a lock length x₀ and height h₀ into an ambient fluid of density ρ_a 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 = ρ_c/ρ_a, 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 ρ_c/ρ_a ≪ 1, Boussinesq ρ_c/ρ_a ≈ 1 and heavy ρ_c/ρ_a ≫ 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) = bz^α 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.