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

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DOI: 10.1007/s10652-011-9232-1

- Cite this article as:
- Ungarish, M. Environ Fluid Mech (2012) 12: 251. doi:10.1007/s10652-011-9232-1

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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*_{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 \({- g\hat{z}}\) 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 *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.