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

作者：Ungarish Marius^*

来源：Environmental Fluid Mechanics, 2012, 12(3): 251-263.

DOI：10.1007/s10652-011-9232-1

• Gravity current
• Froude number
• Front condition

We consider the steady-state propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate . The bottom and top of the channel are at = 0, , and the cross-section is given by the quite general form - (1)() a parts per thousand currency sign a parts per thousand currency sign (2)() for 0 a parts per thousand currency sign a parts per thousand currency sign , where (1,2) are piecewise continuous functions and (1) + (2) %26gt; 0 for . The interface of the current is horizontal, the (maximum) thickness is , its density is . The reduced gravity %26apos; = | / - 1| (where is the gravity acceleration and the density of the ambient) drives the current with speed into the stationary ambient fluid. We show that the dimensionless = /(%26apos; )(1/2), the rate of energy dissipation (scaled with the rate of pressure work), and the dimensionless head-loss Delta/, 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, () = (1)() + (2)(), and of (). This theory applies to Boussinesq and non-Boussinesq currents of %26quot;heavy%26quot; (bottom) and %26quot;light%26quot; (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 results of Marino and Thomas (J Fluid Eng 131(5):051201, 2009) for channels of shape = +/- (where , 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 , and is dissipative; the value of 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 / a parts per thousand currency signA %26lt; 1.

• 出版日期2012-6