The steady flow between reservoirs with different density and level over a sill

Abstract

This paper presents a complete exact analytical solution of steady gravity flow between two reservoirs containing fluids of different densities and levels and connected by a channel of slowly varying depth. Six different regimes are possible depending on the value of the parameter δ = γε which is the ratio of relative lighter and denser reservoir level difference, γ, to positive relative density difference, e. If the level of denser fluid is equal to or higher than the level of the lighter fluid, σ≤0, then the lighter fluid is at rest. Expressions for the thickness of the lighter and denser layers at the smallest cross-section and for the discharge coefficient of the denser fluid, q 2(ε,σ), are the same as for a channel with a flat bottom and variable width. If the level of lighter fluid is higher than the level of denser fluid, δ > 0, then the lighter fluid is in motion. If δ≥1/(2+ε), then the level of the denser fluid is equal to or lower than the sill crest, the denser fluid is motionless and the interface is a horizontal plane. If 0<δ< 1/(2+ε), exchange flow takes place. The discharge coefficient of the lighter fluid has a simple form q 1 =2δ 3/2. The discharge coefficient of the denser fluid q 2(ε,δ) depends on δ and ε; the approximate formula q 2apr(0, δ) = 2(1 − 6δ + 2(2δ) 3/2)/√27 is shown be valid. The comparison of the solutions of exchange flows over a sill and through a contraction is made. Some examples of application of the theory for prediction of mass and volume transport through a strait or an estuary entrance are given.