A rational approximation to the evolution of a free surface during fluid withdrawal through a point sink

Abstract

The time varying flow in which fluid is withdrawn from a reservoir through a point sink of variable strength beneath a free surface is considered. Asymptotic techniques are used to derive an approximate solution to the flow that is valid at intermediate times, giving a simple rational approximation to track changes in the free surface for any temporal variations in the sink strength. Comparisons with numerical simulations are given, showing that the approximation has wide applicability.

A line sink in a flowing stream with surface tension effects

“…The cases of infinite or finite depth fluid, two-dimensional or three-dimensional and steady or unsteady flows have been considered in single-and multi-layered fluids. The problems of the three-dimensional flow due to a point sink, and the unsteady flow due to a line or point sink have been considered by Hocking et al [10,11], Stokes et al [19][20][21], Xue and Yue [30] and references therein. Of particular interest in this paper is the steady two-dimensional flow due to a line source or sink in a single layer of fluid, for which two main solution types exist.…”

Unsteady flows induced by a point source or sink in a fluid of finite depth

“…In general, the behaviour was found to be qualitatively very similar to the infinite depth line sink case [26]. In [15], Hocking et al showed that a simple rational approximation could be obtained based on the solution to the linearised equations and that this performed well in comparison with the full non-linear solution at small values of the flow rate and if the variation in the flow rate was small. The question arises, given the differences between finite and infinite depth in the two-dimensional flow, whether a similar difference exists in the case of a point sink in axisymmetric flow.…”

“…For example, the change in local depth in the two-dimensional finite depth case [28] is well described by a first-order linearised solution. Likewise, in the infinite depth point sink problem [15], a Froude number expansion provided an excellent representation of the development of the free surface as the flow rate was increased and a rational approximation was obtained that performed very well in simulating a variety of subcritical flows. It is therefore worthwhile to consider such solutions here, both for the insight provided and also to verify the numerical scheme.…”

“…φˆt + 1 2 (û 2 +v 2 ) + gη = 0 onẑ =η(r,t), (2.1) and the kinematic condition ηˆt +φrηr −φẑ = 0 onẑ =η(r,t), (2.2) stating that the fluid may not cross its own boundary. To allow the sink strength m to be viewed as a function of time, we do not include it in our non-dimensionalisation; this is the approach taken in [15] and [28]. As in [15], which concerned linearised solutions to the axisymmetric infinite depth withdrawal problem in which the sink strength was a function of time, we non-dimensionalise using the length scale h, the time scale h/g, and hence a velocity scale of √ gh, and a velocity potential scale of h √ gh.…”

“…These ideas were extended in [15,27] to consider the case of a point sink in a fluid of infinite depth. Again, the results showed that the critical drawdown parameters depended on the history of the flow, and that past steady and experimental work needs to be examined in this light.…”