Selective withdrawal and the effect of surface tension on free surface flow

Ha Nguyen

Selective withdrawal from stratified or layered fluid is important in the management of reservoirs and cooling ponds, and it has been studied over a long period of time. In many practical situations, the body of fluid in question has a free surface or an interface between layers of different density, the location of which is unknown. Extensive work has been done to understand the effect of a submerged outlet (called a ‘sink’) that withdraws fluid at different flow rates. Earlier works started with a line sink (eg. a slot or skimmer wall in a reservoir) and much is understood about the types of flow generated, both in the steady and unsteady cases for fluids of finite depth or unbounded. Similar works were done for the case of a point sink (eg. a withdrawal pipe) with less comprehensive results. The study of withdrawal flows is of interest because it helps determine the most effective rate at which fluid can be removed, for example from a reservoir, and the maximum rate at which a particular, desired layer, e.g. clean or polluted water, can be withdrawn. In this thesis, I will start with the discussion of two-dimensional flows caused by a line sink in a fluid of finite depth. A vertical wall near the sink will influence the flow. This work is an extension from the work of Vanden-Broeck and Keller* on two-dimensional flows caused by a sink located on a vertical wall. I extended this to look at the case where the Froude number, F, which is representative of the flow rate, becomes large, and developed a method for finding solutions when F is finite, with the sink located on a vertical wall or on a horizontal bottom. We were able to recalculate and confirm the results from Vanden-Broeck and Keller for the case where the sink is on the vertical wall and observed new results for when the sink is on the horizontal bottom of the fluid. In the latter case, we found that there is a minimum F value below which there is no steady solution. As the sink moves away from the wall, this Fmin value increases. Furthermore, there are no subcritical solutions (F < 1) when the sink is on the bottom. In the next two chapters, we move on to three-dimensional models of flow caused by a point sink where the effect of surface tension is taken into consideration. We start with an investigation of the steady, axisymmetric flow induced by a point sink (or source) submerged in an inviscid fluid of infinite depth. We computed maximum Froude numbers at which steady solutions exist and found that the determining factor in reaching the critical flow changes as more surface tension is included. If there is zero or a very small amount of surface tension, the limiting factor appears to be the formation of small wavelets on the free surface; but, as the surface tension increases, this is replaced by a tendency for the lowest point on the free surface to descend sharply as the Froude number is increased. We continued our work with the case where the fluid is of finite depth. Using both a spectral method and an integral equation approach, results are confirmed for the maximum-flow-rate steady solution for each configuration. It is found that surface tension has the effect of increasing the maximum flow rate at which steady-state solutions can exist. The two chapters on three-dimensional flow mentioned above (chapters 3 and 4) have already been published ([18], [19]) and are reproduced with only minor modification in this thesis. This work has increased understanding of the behaviour of withdrawal flows in reservoirs and is useful in improving techniques to maintain water quality. *Vanden Broeck, J.-M. & Keller, J.B. 1987 Free surface flow due to a sink, J. Fluid Mech. 175, 109–117

Selective withdrawal and the effect of surface tension on free surface flow

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