Graeme C. Hocking

FLOW CAUSED BY A POINT SINK IN A FLUID HAVING A FREE SURFACE

The flow caused by a point sink immersed in an otherwise stationary fluid is investigated. Low Froude number solutions are sought, in which the flow is radially symmetric and possesses a stagnation point at the surface, directly above the sink. A small-Froude-number expansion is derived and compared with the results of a numerical solution to the fully nonlinear problem. It is found that solutions of this type exist for all Froude numbers less than some maximum value, at which a secondary circular stagnation line is formed at the surface. The nonlinear solutions are reasonably well predicted by the small-Froude-number expansion, except for Froude numbers close to this maximum.

Supercritical withdrawal from a two-layer fluid through a line sink

Accurate numerical solutions to the problem of finding the location of the interface between two unconfined regions of fluid of different density during the withdrawal process are presented. Supercritical flows are considered, in which the interface is drawn directly into the sink. As the flow rate is reduced, the interface enters the sink more steeply, until the solution method breaks down just before the interface enters the sink vertically from above, and becomes flow from the lower layer only. This lower bound on supercritical flow is compared with the upper bound on single-layer (free surface) flow with good agreement.

A note on the flow induced by a line sink beneath a free surface

The problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.

Subcritical free-surface flow caused by a line source in a fluid of finite depth

An integral equation is derived and solved numerically to compute the flow and the free surface shape generated when water flows from a line source into a fluid of finite depth. At very low values of the Froude number, stagnation point solutions are found to exist over a continuous range in the parameter space. For each value of the source submergence depth to free stream depth ratio, an upper bound on the existence of stagnation point solutions is found. These results are compared with existing known solutions. A second integral equation formulation is discussed which investigates the hypothesis that these upper bounds correspond to the formation of waves on the free surface. No waves are found, however, and the results of the first method are confirmed.

A numerical model for withdrawal from a two-layer fluid

This paper reports the results of several direct numerical simulations of the withdrawal of a two-layer fluid with a finite-thickness interface through a slot in the base of a finite rectangular cavity. Particular attention is paid to the role of long (basin scale) interfacial waves on the processes leading to drawdown of the interface into the slot. It is shown that these waves play an important role and can either delay or accelerate drawdown. This means that drawdown can occur over a range of Froude numbers. The results are compared with previous work for ideal flow and experimental results.

Super-critical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth

The steady response of a fluid consisting of two regions of different density, the lower of which is of finite depth, is considered during withdrawal. Super-critical flows are considered in which water from both layers is being withdrawn, meaning that the interface is drawn down directly into the sink. The results indicate that if the flow rate is above some minimum, the angle of entry of the interface depends more strongly on the relative depth of the sink than on the flow rate. This has quite dramatic consequences for the understanding of selective withdrawal from layered fluids.

Unsteady free-surface flow induced by a line sink

The unsteady flow of fluid from a deep reservoir through a line sink beneath a free surface with surface tension is considered. Two different initial conditions are discussed; the first effectively represents impulsive withdrawal from rest, and the second can be regarded as a disturbance to an existing steady flow. Small-time expansions and numerical methods are used to investigate both the movement to steady states and the critical drawdown of the free surface in the two situations. It is shown that there are several different critical values of flow parameters at which the flow changes its nature. In the zero-surface-tension case, the situation is not fully resolved, but the addition of surface tension clarifies the flow behaviour greatly, and drawdown or movement to a steady state becomes evident. For the second class of initial conditions, it appears that either movement to a steady state or drawdown are the only subcritical possibilities.

Control of algal blooms in reservoirs with a curtain: a numerical analysis

Two vertical curtains, having depths to cover the epilimnion thickness, were installed across the Terauchi Dam Reservoir in the western island of Japan to curtail the nutrient supply from nutrient-rich inflows to the downstream epilimnion of the reservoir. The withdrawal level was also regulated to keep the downstream epilimnion away from the nutrient supply. This method markedly reduced algal blooming in the reservoir downstream of the curtains during spring and summer. The physical and biological processes in the reservoir ecosystem were analysed using the 2-D reservoir model DYRESM and chemical and biological submodels, to predict the water quality and algal species composition in the reservoir. The horizontal variability was maintained in the model by dividing the horizontal layer into parcels. Temperature, chlorophyll-a, soluble phosphorus, nitrate, ammonium, dissolved oxygen, biochemical oxygen demand, internal nitrogen, internal phosphorus were considered as state variables in the model. The simulated results revealed the mechanism of how algal blooming is reduced, during early spring high algal concentrations consume large amounts of nutrients, which reduces the nutrient supply to the downstream zone of the reservoir, whereas during late spring and summer, nutrient dispersion from the upstream epilimnion to the downstream epilimnion is curtailed by the curtains, markedly reducing algal blooming in the downstream zone.

Flow induced by a line sink in a quiescent fluid with surface-tension effects

When a line sink is placed beneath the free surface of an otherwise quiescent fluid of infinite depth, two different flow types are now known to be possible. One type of flow involves the fluid being drawn down toward the sink, and in the other type, a stagnation point forms at the surface immediately above the position of the sink. This paper investigates the second of these two flow types, which involves a free-surface stagnation point. The effects of surface tension are included, and even when small, these are shown to have a very significant effect on the overall solution behaviour. We demonstrate by direct numerical calculation that there are regions of genuine non-uniqueness in the nonlinear solution, when the surface-tension parameter does not vanish. In addition, an asymptotic solution valid for small Froude number is derived.

An analytical solution for critical withdrawal of layered fluid through a line sink in a porous medium

Fluid withdrawn through a line sink from a layered fluid in a vertically confined porous medium is considered. A hodograph method is used to obtain the shape of the interface for a given sink position at the critical flow rate. The analytical solution is compared with a more general numerical solution developed in earlier work. It was found that the surface profiles obtained by the two methods are in close agreement. However, the present work has the advantage that it gives a fully explicit solution.