, Volume 12, Issue 3, pp 251263
First online:
A general solution of Benjamintype gravity current in a channel of nonrectangular crosssection
 Marius UngarishAffiliated withDepartment of Computer Science, Technion Email author
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We consider the steadystate propagation of a highReynoldsnumber 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 crosssection 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 } − 1g (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 headloss Δ/h, can be expressed by compact formulas which involve three integrals over the crosssection areas of the current and ambient. By some standard manipulations these integrals are simplified into quite simple lineintegrals of the shapefunction of the channel, f(z) = f _{1}(z) + f _{2}(z), and of z f(z). This theory applies to Boussinesq and nonBoussinesq 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 crosssection shape on the behavior of the steadystate current, in quite general cases, for both heavyintolight and lightintoheavy fluid systems, Boussinesq and nonBoussinesq. 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 clearcut condition of the form h/H ≤ a _{ max } < 1.
Keywords
Gravity current Froude number Front condition Title
 A general solution of Benjamintype gravity current in a channel of nonrectangular crosssection
 Journal

Environmental Fluid Mechanics
Volume 12, Issue 3 , pp 251263
 Cover Date
 201206
 DOI
 10.1007/s1065201192321
 Print ISSN
 15677419
 Online ISSN
 15731510
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Gravity current
 Froude number
 Front condition
 Industry Sectors
 Authors

 Marius Ungarish ^{(1)}
 Author Affiliations

 1. Department of Computer Science, Technion, Haifa, 32000, Israel