A general solution of Benjamin-type gravity current in a channel of non-rectangular cross-section

摘要

We consider the steady-state propagation of a high-Reynolds-number gravity current in a horizontal channel 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 form −f 1(z) ≤ y ≤ f 2(z) for 0 ≤ z ≤ H, where f 1,2 are piecewise continuous functions and f 1 + f 2 0 for ${z in(0,H)}$ . The interface of the current is horizontal, the (maximum) thickness is h, its density is ρ c . The reduced gravity g′ = |ρ c /ρ a − 1|g (where ${- ghat{z}}$ is the gravity acceleration and ρ a the density of the ambient) drives the current with speed U into the stationary ambient fluid. We show that the dimensionless Fr = U/(g′ h)1/2, the rate of energy dissipation (scaled with the rate of pressure work), and the dimensionless head-loss Δ/h, can be expressed by compact formulas which involve three integrals over the cross-section areas of the current and ambient. By some standard manipulations these integrals are simplified into quite simple line-integrals of the shape-function of the channel, f(z) = f 1(z) + f 2(z), and of z f(z). This theory applies to Boussinesq and non-Boussinesq currents of “heavy” (bottom) and “light” (top) type. The classical results of Benjamin (J Fluid Mech 31:209–248, 1968) for a rectangular channel are fully recovered. We also recover the Fr results of Marino and Thomas (J Fluid Eng 131(5):051201, 2009) for channels of shape y = ±b z α (where b, α are positive constants); in addition, we consider the energy dissipation of these flows. The results provide insights into the effect of the cross-section shape on the behavior of the steady-state current, in quite general cases, for both heavy-into-light and light-into-heavy fluid systems, Boussinesq and non-Boussinesq. In particular, we show that a very deep current displays ${Fr = sqrt{2}}$ , and is dissipative; the value of Fr and rate of dissipation (absolute value) decrease when the thickness of the current increases. However, in general, energy considerations restrict the thickness of the current by a clear-cut condition of the form h/H ≤ a max 1.