Extensions to the theory of selective withdrawal

Abstract

Most reservoirs contain stratified fluid and selective withdrawal is used to obtain water with the desired properties. We initially deal with a layered density distribution. The theory for the critical discharge for a single layer and a point sink is reviewed and extended to cover the case where there is gate discharge (a line sink). The theory for the case when the upper layer depth is large and the flow is coming from both layers is reviewed and it is shown that the valve controls the discharge and a virtual control determines the ratio of the discharge in each layer. This virtual control moves further from the valve as the total discharge increases. We determine the position of the virtual control and the criteria for the maximum for two layers when the upper layer is finite and below a stationary layer. Before this maximum, we show that when the discharge is increased above the critical discharge for the single layer, the finite upper layer does not affect the ratio of the flows from each layer until the virtual control reaches that for the maximum discharge. At this stage, the upper layer becomes tangential to the dam face and this condition and the smoothness of the lower interface determine both the total discharge and the ratio of the flow from each layer. Indeed, at this stage, virtual control and the control of the discharge are at the same section.In a similar way, with a stationary layer above and below two flowing layers, we derive the maximum discharge from the two flowing layers. For this case, the solution is self-similar. This is then extended to a stable stratified continuous density distribution. The experiments of Gariel (1949) and Lawrence & Imberger (1979) suggest that the predictions of the theory are within the experimental errors.