Original Article

Environmental Fluid Mechanics

, Volume 12, Issue 3, pp 251-263

First online:

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

  • Marius UngarishAffiliated withDepartment of Computer Science, Technion Email author 

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Abstract

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 1f 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 \({- g\hat{z}}\) 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 FrU/(gh)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.

Keywords

Gravity current Froude number Front condition