Maximal Exchange in Channels with Nonrectangular Cross Sections

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Abstract
The functional formalism of Gill and Dalziel is extended for two-layer flows in nonrectangular cross sections. Exchange flow through channels with a range of different cross sections is investigated in detail. Hydraulic transitions are found to play the same role in controlling the flow as in channels with rectangular cross sections. The response of the flow to different alongchannel geometries and net flow rates is qualitatively similar to that for rectangular cross sections but exhibits a number of novel features. In a channel of constant center line depth, the two hydraulic transitions do not coincide at the narrowest point unless the channel is symmetric about its middepth plane. For flow over a sill, the location of the virtual control is found to be a discontinuous function of the net flow rate, undergoing one or two jumps in position (depending on the precise geometry) as the net flow is altered. In the absence of friction and instabilities, it is not possible to eliminate the lower layer from the contraction by increasing the strength of the net flow out of the less dense reservoir if the channel narrows to a point at its bottom.

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