An extension to the theory of steady selective withdrawal for a two layer fluid

Abstract

Most reservoirs contain stratified fluid and selective withdrawal is used to obtain water of the desired properties. Initially we review the case with an infinite upper layer with a sharp interface. When the total discharge is specified, then the ratio of the discharge from each layer is determined by the criteria of smoothness at the virtual control (i.e. the critical point). At this point, the long wave velocity on the interface is zero. For the case when the upper layer depth is large, we show that the control is in the valve and the virtual control (which determines the ratio of the discharge in each layer) moves further from the source as the total discharge increases. When there is a finite upper layer, a portion of the flow is in the duct and a portion of the flow is in the free surface. We derive the criteria for the virtual control in the free surface flow and show that the duct control occurs first. If we then assume that the flow is not over-specified, we determine the necessary conditions for a smooth transition between the duct and the free surface flow. This enables us to determine the minimum ratio of the upper layer depth to the lower layer depth for the steady duct solution to be valid. This contrasts with the conclusions of Bryant and Wood (1976).