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Withdrawal of a fluid of finite depth through a line sink with a cusp in the free surface
PII:
Computers & Fluids Vol. 27, No. 7, pp. 797±806, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0045-7930(98)00006-1 0045-7930/98 $19.00 + 0.00
WITHDRAWAL OF A FLUID OF FINITE DEPTH THROUGH A LINE SINK WITH A CUSP IN THE FREE SURFACE G. C. HOCKING1 and J-M. VANDEN-BROECK2 Mathematics and Statistics, Division of Science, Murdoch University, Murdoch 6150, Australia 2 Department of Mathematics, University of WisconsinÐMadison, WI 53706, U.S.A.
1
(Received 5 December 1996; in revised form 15 December 1997) AbstractÐThe steady withdrawal of a ¯uid of ®nite depth into a line sink is considered. The problem is solved numerically by a boundary integral equation method. It is shown that the ¯ow depends on the Froude number F = m(gH3)ÿ1/2 and the nondimensional sink depth b = HS/H, where m is the sink strength, g the acceleration of gravity, H is the total depth and HS is the depth of the sink. For given values of b and F there is a one-parameter family of solutions with a cusp on the free surface above the sink. It is found that in general there is a train of steady waves on the free surface. For particular values of the parameters the amplitude of the waves vanishes and the solutions reduce to those computed by Vanden-Broeck and Keller. These ®ndings con®rm and generalize the calculations of VandenBroeck where the free surface was covered by a lid everywhere but close to the sink. # 1998 Elsevier Science Ltd. All rights reserved
1. PRELIMINARY DISCUSSION
Problems involving withdrawal from water bodies have been of interest to engineers for many years. In particular, there is a range of problems involving withdrawal of water, either from single layers or from one of several layers of dierent density which impinge on such questions as water quality and reservoir management [1]. In particular, it has long been known that when water is withdrawn from a basin containing several layers of dierent density, the water ¯ows from the layer adjacent to the outlet until some critical ¯ow rate is exceeded, after which water ¯ows from both (or several) layers (see [2± 13]). Experiments and semi-analytical and numerical solutions of this problem have given dierent values for this critical withdrawal ¯ow rate, often with quite a large variation in values for the same geometric con®guration. This is true in the case of both two-dimensional withdrawal (slot or line sink), and in the case of three-dimensional withdrawal (pipe or line sink). The reason for these discrepancies is still unclear, despite a great deal of work. Numerical solutions of the steady ¯ow with a cusp [13±22] (see ``Cusp solution'' in Fig. 1) on the interface (or free surface) have long been thought to correspond to the critical value, and Hocking [23] recently provided strong evidence that this is indeed true for the case in which there is a line sink in an unbounded ¯uid. However, for two-dimensional withdrawal when the ¯uid is of ®nite depth, there are several complications to this. Firstly, there is a range of ¯ow rates over which the cusp solutions exist (for identical geometry). Secondly, even the smallest of these ¯ow rates is much larger than the critical drawdown values observed experimentally [2,7,9,12]. Finally, there are two branches of solution with a cusp, one of which only exists when the link sink is a long distance above the bottom, i.e. greater than 170% of the total ¯uid height [16]. It is necessary to examine these ¯ows in greater detail in order to obtain a clearer understanding of their behaviour, with the ultimate goal of understanding the process of critical drawdown. In this paper we examine one of the branches of solution with a cusp in the interface (or on the free surface), for withdrawal of a ¯uid of ®nite depth through a line sink. No attempt will be made to look at the full unsteady problem, which has been considered recently by other researchers [24,25]. 797
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Fig. 1. The two main free surface shapes for ¯ow into a line sink from a layer of ®nite depth.
This ¯ow can be characterized by the Froude numbers m F p gH 3 and m FS q : gHS3 Here m is the strength of the line sink (i.e. the ¯ux of the far ®eld), g is gravity, H is the total depth of the layer in the far ®eld (or the average depth if waves are present on the free surface) and HS is the (average) depth of the sink. If we wish to examine a two-layer ¯ow, the equations are identical if we assume the upper layer to be stagnant, and replace gravity, g, by the eective gravity g' = (Dr/r)g, where r is some reference density and Dr the density dierence between the two layers. In that case the Froude number is modi®ed to be the densimetric Froude number, i.e. FDS=m/(g'H3S)1/2. The Froude number F determines whether or not there can be waves on the free surface. The linear theory of water waves shows that if F < 1, then waves can occur on the free surface, while if Fe 1, waves are not possible. Flows with F < 1 and F>1 are called subcritical and supercritical, respectively, for this reason. This distinction is based on linear theory. The terms subcritical and supercritical are also commonly used in nonlinear theory, although nonlinear waves are possible not only for F < 1 but also for 1 < F < 1.3 (the value 1.3 corresponds to the highest solitary wave). Tuck and Vanden-Broeck [15] obtained numerically the ``cusp solution'' of Fig. 1 in water of in®nite depth (i.e. H = 1). They found a unique solution, at FS=12.622. It was presumed at that time that this corresponded to a critical situation; if the ¯ow rate was increased, the free surface would be pulled down into the sink, as when air is pulled into a water pump, thus invalidating this con®guration for higher ¯ow rates. Strong evidence that this was indeed the case was provided by Hocking [23], who used numerical techniques to solve the corresponding problem with two layers of ¯uid being drawn into a line sink. As the ¯ow rate was decreased, the solutions obtained approached the single layer solutions of Tuck and Vanden-Broeck [15]. In addition to these papers, there has been a large amount of work in which many free surface ¯ows due to submerged sinks were computed. Hocking [17] computed solutions, similar to those of Tuck and Vanden-Broeck [15], but in which there was a boundary beneath the sink sloping away without bound. He also obtained unique cusped solutions, in which the ¯ow rate depends on the angle of the slope. The result when the angle of the wall is 308 downward matches an exact solution ®rst obtained by Sautreaux [13], and subsequently by Craya [14]. ``Stagnation point'' solutions (see Fig. 1), in which the free surface rises up to a stagnation point above the sink, were obtained by Peregrine [26], Hocking and Forbes [27] and Dun and Hocking [28]. Further stagnation point solutions, but in ¯uid of ®nite depth, were obtained by Mekias and Vanden-Broeck [29,30], Hocking and Forbes [31], and recently by Vanden-Broeck [32]. These were found to exist for both subcritical and supercritical ¯ows and in some subcritical cases steady waves were found on the free surface.
Withdrawal of a ¯uid of ®nite depth
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``Cusp solutions'' in water of ®nite depth were obtained by, amongst others, Collings [18], Vanden-Broeck and Keller [16], King and Bloor [20], and Hocking [19,21]. It was generally found that cusped solutions were supercritical and obtained over a range of values of the Froude number, F, above some minimum, usually greater than one, and in many cases could be computed from this minimum value up to in®nite Froude number. However, there was one branch of solutions obtained by Vanden-Broeck and Keller [16] which was dierent from the other solutions in two ways. Firstly, the solutions were subcritical and secondly, it was a one parameter family of solutions. In particular there was only one value of the Froude number for a given depth of the sink. Furthermore, as the ¯uid became deeper, it approached the solution of Tuck and Vanden-Broeck [15]. In a recent paper, Vanden-Broeck [22] computed solutions using a series truncation method in which a submerged sink was placed under a free surface in water of ®nite depth, but in which a rigid lid was placed on the free surface everywhere except very close to the region just above the sink. He found that he could reproduce qualitatively the results obtained in Vanden-Broeck and Keller [16], i.e. unique, subcritical cusped solutions with a waveless free surface for each sink to bottom depth ratio. In addition, however, he obtained solutions near to these waveless solutions which contained steady waves on the free surface. As the Froude number was increasingly distanced from the value which gave waveless solutions, the amplitude of the steady waves was found to grow until the waves approached a limiting con®guration with a stagnation point and a 1208 angle at their crest. Beyond this range of values, no further solutions were found. The work of Vanden-Broeck [22] was restricted by the nature of the series truncation method to problems in which the rigid lid began close to the region of the free surface above the line sink. In this paper we extend this investigation using an integral equation technique which allows us to move the rigid lid as far out as we wish, thus simulating an unbounded free surface. The results in Ref. [22] are con®rmed and shown to indicate the correct pattern of behaviour for the less restricted case. In particular it is found that there is a three parameter family of solutions which includes the waveless one parameter family of Ref. [16]. In Section 2 of this paper we will formulate the problem as a set of non-linear, singular integral equations, and in Section 3 we will describe the method used for the numerical solution of these equations. The results of the calculations are described and discussed in Sections 4 and 5. 2. PROBLEM FORMULATION
The steady, irrotational motion of an inviscid, incompressible ¯uid due to a submerged sink is to be examined. The ¯ow is assumed to be two-dimensional and gravity is acting vertically downwards (see Fig. 1). We concentrate in this paper on subcritical ¯ows for which waves can be present on the free surface. Following Vanden-Broeck [22], we truncate the free surface at some distance from the sink and extend it to the right and to the left by rigid horizontal lids. We de®ne the average depth H as the distance between the lids and the horizontal bottom. The average velocity U in the far ®eld is then de®ned by U = m/H. We choose Cartesian coordinates so that the sink is at x = 0 and the bottom at y = ÿ H. Then the level of the lids is y = 0. We denote by y = ÿ HS the ordinate of the sink. In the formulation given here we will use the symmetry of the ¯ow about x = 0, and solve for only the left hand half of the ¯ow Let z = x + iy be the physical plane (see Fig. 2(a)). The mathematical problem is to ®nd a complex potential w = f(x, y) + ic(x, y), which is analytic in the ¯ow domain and satis®es the conditions of no ¯ow across the solid boundaries. Without loss of generality, we choose c = 0 on the free surface and f = 0 at the cusp. The surface of the water must also be at constant pressure, a condition provided by Bernoulli's equation, which if we nondimensionalise with respect to the velocity U and the depth H, takes the form 2F ÿ2 Z q2 q20
1
where y = Z(x) is the equation for the elevation of the free surface, q is the ¯uid velocity, and F is the Froude number de®ned above. The value of the quantity q0 is to be determined, but is the velocity of the ¯uid at the free surface as it attaches to the rigid lid. Let hS be the depth of the sink, so that b = HS/H is the nondimensional sink depth. In nondimensional variables, U = 1
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Fig. 2. Mapped planes used in the problem formulating: (a) the physical z-plane; (b) the complex velocity potential w-plane; and (c) the upper-half z-plane.
and H = 1, so the incoming ¯ux far upstream, under the rigid lid, is also of unit magnitude. Since the only term involving the velocity is squared, the equations are independent of the direction of the ¯ow and therefore solutions are equally valid for a ¯ow into (sink) or out (source) of the slot. However, we will discuss the ¯ow as if it were into a sink (even though the actual formulation is for a source ¯ow). To derive an integral equation for this problem we follow a procedure similar to that used in Hocking [21]. The transformation eÿpw z
2
maps the in®nite strip between c = ÿ 1 and c = 0 in the w-plane to the upper-half of the zplane (see Fig. 2(c)). The point z = 0 corresponds to the sink at y = ÿ b, so that the free surface CD and the lid DI lie along the real z-axis where ze 1. The line between z = 0 and z = 1 corresponds to the vertical wall above the sink, or to the line of symmetry of the ¯ow if both sides are being considered. The points between z = 0 and z = zB corresponds to the vertical wall beneath the sink, i.e. the line of symmetry of the ¯ow, and z < zB to the bottom of the channel, which goes away horizontally to x = ÿ 1. The ¯ow domain is the upper half z-plane (see Fig. 2(c)). The case in which zB=0 corresponds to the case of a line source or sink on the bottom of the channel. We de®ne a new function O(z) = d(z) + it(z), related to the complex conjugate of the velocity ®eld by w0 z z eÿiO z ;
3
where the prime denotes dierentiation with respect to z. The magnitude of the velocity at any point is then given by vw'(z)v = et(z), and the angle any streamline makes with the horizontal is d(z). Since the total ¯ux is one, and the upstream ¯uid depth approaches one, O 4 0 as vzv 4 1. Since there is a cusp at C, we have d = ÿ p/2 at z = 1.
Withdrawal of a ¯uid of ®nite depth
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On the portions of the real z-axis which correspond to the solid boundaries of the ¯ow domain, the streamlines must be parallel to the walls, so that the condition that there be no ¯ow normal to the solid boundaries is satis®ed if we choose d(z) to be the angle of the wall to the horizontal, i.e. 8 0 if ÿ 1 < z < zB > > > > p=2 if zB < z < 0 < 4 ÿp=2 if 0 < z 1 : d z > > unknown if 1 < z < z > D > : 0 if z > zD Note that d is unknown on 1 < z < zD because this is the free surface. d is zero on z>zD since this corresponds to the rigid horizontal lid. The only singularities of the function O(z) in the z-plane are those at the origin and at z = zB, which correspond respectively to the ¯ow into the sink, and to the corner point beneath the sink. Both of these singularities can be shown to be weaker than a simple pole, so that Cauchy's integral formula can be applied to O(z) on a path consisting of the real z-axis, a semicircle at vzv 4 1 in the upper-half plane, and a circle of vanishing radius about the point z. If we then let Im{z} 4 0+ and take real and imaginary parts, we obtain I 1 1 d z0 dz ; 5a t z ÿ p ÿ1 z0 ÿ z 0 and d z
1 p
I
1 ÿ1
t z0 dz ; z0 ÿ z 0
where the integrals are of Cauchy-principal-value form. Substituting the known values of d(z) given by Equation (4) into Equation (5a) gives I 1 z ÿ 1 z ÿ zB 1 zD d z0 dz : ÿ t z ln 2 p 1 z0 ÿ z 0 z2
5b
6
To this we must add the equation for constant pressure on the free surface, which can be obtained by combining Equations (1)±(3) on 1 E z < 1 to give Z ÿ2F ÿ2 z eÿt z0 sin d z0 dz0 e2t z q20 : 7 z0 p zD Using Equations (6) and (7) on the free surface gives a nonlinear system of integral equations for d(z) on 1 E z < zD. The value of d is known elsewhere on the boundary from the boundary conditions, and hence we can obtain t from Equation (6). Using d and t, it is possible to integrate Equation (3) to obtain the location of points on the free surface. These may be written as Z 1 z eÿt z0 cos d z0 dz0 ; 8a x z ÿ p 1 z0 and y z ÿhC ÿ
1 p
Z
z ÿt z0
e
1
sin d z0 dz0 : z0
Since y 4 0 as z 4 1, the depth at which the two in¯owing free surfaces meet in a cusp is Z 1 zD eÿt z0 sin d z0 dz0 : hC ÿ p 1 z0
8b
9
Choosing larger values of zD moves the beginning of the rigid lid further and further outwards.
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3. NUMERICAL SOLUTION
No closed form solution is known for the full nonlinear system of equations given by Equations (6) and (7) (except for the case F = 1, see refs [16,18±20]), but it can be solved quite well using collocation. A set of N equations can be obtained by discretizing the equations at a set of values of z. The corresponding unknown values of d at each of the points on the interval can then be obtained using a Newton±Raphson iteration scheme. The behaviour of z is like ef, so we choose to make the transformation z = ea. Points were chosen at N equally spaced values of a, so that 0 = a0
We found that there is a three-parameter family of solutions. The three parameters can be chosen as the Froude number, F, and the values of zB and zD. To reproduce the results of Vanden-Broeck and Keller [16], we used a variant of the scheme in which we ®xed q0=1 and took F as one of the unknowns. We found that the pro®les are then waveless and essentially independent of zD for zD suciently large. Thus the solutions depend essentially on one parameter (chosen here as zB) in accordance with the ®ndings of Vanden-Broeck and Keller. Returning to the original scheme, we used one of the waveless solutions as the initial guess, and once a solution was obtained, gradually varied the sink depth b. As b was moved away Table 1. Convergence of the numerical method for increasing values of N for the case F = 0.45, b = 0.324 and truncation at f = at=17.3. The two numbers in the ``Crest Height'' column are for the last two wave crests. The total depth of the ¯ow has an exact value of 1 N 50 100 200 400 800
Crest height 0.00797,0.00698 0.04595,0.04055 0.05127,0.05021 0.05204,0.05152 0.05218,0.05180
Cusp location ÿ0.27363 ÿ0.24365 ÿ0.23502 ÿ0.23205 ÿ0.23087
Total depth 1.03090 1.00681 1.00140 1.00024 1.00002
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Fig. 3. Free surface pro®les for F = 0.44 with nondimensional sink depth b = 0.308, less than the value of waveless solutions, and b = 0.363, greater than the waveless value.
from the value which gave waveless solutions, steady waves appeared on the free surface. As b was moved further away from this waveless value, the amplitude grew until it approached the limiting Stokes con®guration with a 1208 angle at the crest. If b was chosen larger than the waveless value, the bulk of the oscillation amplitude occurred at a height greater than the level of the rigid lid, while if b was less than the waveless value, the free surface level remained mainly below the level of the plate. This is illustrated in Fig. 3 where surface pro®les for F = 0.44 are shown for dierent values of sink height, b. The waveless solution corresponds to b = 0.33 for this case. These ®ndings are consistent with the work of Vanden-Broeck [22]. It was found that for ®xed values of Froude number, F, and sink depth, b, the waves had almost identical amplitude and wavelength. However, it was also found that the phase of the steady waves depended upon the location of the rigid lid, and it appears that the phase is such that the attachment to the plate occurs at either a crest or a trough. Figure 4(a) shows solutions calculated with the plate distance set so that the attachment is almost exactly in phase for the case F = 0.44, b = 0.32. The solutions are almost identical except very near to the upstream attachment point. Figure 4(b) on the other hand shows two plots for identical parameter values in which the upstream attachment point is almost exactly out of phase. These ®ndings suggest there is a three parameter family of solutions for the problem without a lid, i.e. the ¯ow with a free surface of unlimited horizontal extent. Figure 5 shows the region in parameter space in which these solutions exist, and how these relate to other solutions to this problem which have been calculated. The dashed lines correspond to con®gurations close to the limiting con®guration with a corner on the crest. The exact solutions with breaking waves could not be computed with the present scheme due to a singularity associated with the 1208angle. For each point between the two lines, there is a one-parameter family of solutions. The solid line represents the waveless solutions of Ref. [16]. As the depth of the ¯uid increases, the width of the solution region decreases until the solution of Tuck and Vanden-Broeck [15] (in in®nite depth) is reached. At that point there cannot be waves on the free surface because the depth of the ¯uid is in®nite, meaning that the velocity far upstream must approach zero, and is consequently unable to support steady waves.
5. COMMENTS
In this paper, we have used a set of integral equations to compute solutions to the problem of steady withdrawal from a layer of ®nite depth through a line sink in which there is a cusp in the free surface. It is shown that there are solutions with waves which exist near to the previously calculated sub-critical branch of solutions of Vanden-Broeck and Keller [16] which have a cusp but no waves. As the Froude number is changed away from the value at which there are waveless solutions, the amplitude of the steady waves grows until a limiting con®guration with a sharp crest with a 1208 angle is approached.
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Fig. 4. Solutions with F = 0.44, b = 0.324 with truncation at (a) f = at=25,21.5 and 17.3, i.e. steady waves in phase, (b) f = at=21.5 and 19, i.e. steady waves out of phase, at is the truncation point in the real potential variable f.
One could argue that the waves are due to the presence of the lid, and that if it were not present there would be no waves. However, the results of Fig. 4(a) show that it is possible to move the lid further and further away in such a way that the solution remains essentially unchanged near the cusp. As the lid moves further away, more and more oscillations appear on the free surface. In the limit as the lid moves to vxv 4 1, we expect the solutions near the cusp to be independent of the position of the lid. This suggests the existence of a three-parameter family of
Fig. 5. Sketch of parameter space in which cusped solutions with steady waves exist. The solutions of Vanden-Broeck and Keller [16] are shown as a solid line. The dashed lines show the limiting values at which solutions with steady waves were computed. These are approximate due to the diculty of calculating the limiting con®guration of standing wave solutions.
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805
solutions for a sink submerged below a free surface of in®nite horizontal extent. This family includes the waveless one-parameter family of Vanden-Broeck and Keller as particular cases. There is a further question which must remain unanswered for the moment about the stability of these solutions. Since all of the calculations are of steady ¯ow, it is not possible to determine whether the solutions are stable to small perturbations. A full time-dependent scheme, such as in refs [24,25], would need to be derived to test this. In concluding this work, we should return to the comments in Section 1. It is interesting to note that it is possible to compute solutions near to the cusp solutions which are not supercritical from the selective withdrawal point of view, i.e. which maintain ¯ow from a single layer only. This means that although the cusp ¯ow in in®nite depth seems to be the limit of singlelayer ¯ow, this may not be the case if the ¯uid is of ®nite depth. If it is, then there must be some non-uniqueness in the problem. Alternatively, the solutions with breaking waves may be the precursor to the collapse of the ¯ow into the two-layer ¯ow and drawdown of the interface. Work is under way to compute the full two-layer, steady withdrawal ¯ows and answer this question. REFERENCES 1. Imberger, J. and Hamblin, P. F., Dynamics of lakes, reservoirs and cooling ponds, Rev. Fluid Mech., 1982, 14, 153± 187. 2. Gariel, P., Experimental research on the ¯ow of nonhomogeneous ¯uids, La Houille Blanche, 1949, 4, 56±65. 3. Huber, D. G., Irrotational motion of two ¯uid strata towards a line sink, J. Engng Mech. Div. Proc. ASCE, 1960, 71(85), . 4. Harleman, D. R. F. and Elder, R. E., Withdrawal from two-layer strati®ed ¯ow, Proc. ASCE, 1965, 91, HY4. 5. Lubin, B. T. and Springer, G. S., The formation of a dip on the surface of a liquid draining from a tank, J. Fluid Mech, 1967, 29, 385±390. 6. Wood, I. R. and Lai, K. K., Selective withdrawal from a two-layer ¯uid, J. Hyd. Res., 1972, 10, 475±496. 7. Jirka, G. H., Supercritical withdrawal from two-layered ¯uid systems, Part 1Ðtwo-dimensional skimmer wall, J. Hyd. Res., 1979, 17(1), 43±51. 8. Jirka, G. H. and Katavola, D. S., Supercritical withdrawal from two-layered ¯uid systems, Part 2Ðthree-dimensional ¯ow into a round intake, J. Hyd. Res., 1979, 17(1), 53±62. 9. Lawrence, G. A. and Imberger, J., Selective withdrawal through a point sink in a continuously strati®ed ¯uid with a pycnocline. University of Western Australia, Center for Water Research, Environmental Dynamics Report, ED-79-002. 1979. 10. Imberger, J., Selective withdrawal: a review. In 2nd International Symposium on Strati®ed Flows. Trondheim, Norway, 1980. 11. Zhou, Q-N. and Graebel, W. P., Axisymmetric draining of a cylindrical tank with a free surface, J. Fluid Mech., 1990, 221, 511±532. 12. Hocking, G. C., Withdrawal from two-layer ¯uid through line sink, J. Hyd. Engng ASCE, 1991, 117(6), 800±805. 13. Sautreaux, C., Movement d'un liquide parfait soumis aÁ la pesanteur. DeÂtermination des lignes de courant, J. Math. Pures Appl., 1901, 7, 125±159. 14. Craya, A., Theoretical research on the ¯ow of nonhomogeneous ¯uids, La Houille Blanche, 1949, 4, 44±55. 15. Tuck, E. O. and Vanden-Broeck, J. M., A cusp-like free-surface ¯ow due to a submerged source or sink, J. Aust. Math Soc. Ser. B, 1984, 25, 443±450. 16. Vanden-Broeck, J. M. and Keller, J. B., Free surface ¯ow due to a sink, J. Fluid Mech., 1987, 175, 109±117. 17. Hocking, G. C., Cusp-like free-surface ¯ows due to a submerged source or sink in the presence of a ¯at or sloping bottom, J. Aust. Math Soc. Ser. B, 1985, 26, 470±486. 18. Collings, I. L., Two in®nite Froude number cusped free surface ¯ows due to a submerged line source or sink, J. Aust. Math Soc. Ser. B, 1986, 28, 260±270. 19. Hocking, G. C., In®nite Froude number solutions to the problem of a submerged source or sink, J. Aust. Math Soc. Ser. B, 1988, 29, 401±409. 20. King, A. C. and Bloor, M. I. G., A note on the free surface induced by a submerged source at in®nite Froude number, J. Aust. Math Soc. Ser. B, 1988, 30, 147±156. 21. Hocking, G. C., Critical withdrawal from a two-layer ¯uid through a line sink, J. Engng Math., 1991, 25, 1±11. 22. Vanden-Broeck, J. M., Cusp ¯ow due to a submerged source with a free surface partially covered by a lid., Eur. J. Mech. B (Fluids), 1997, 16(2), 249±255. 23. Hocking, G. C., Supercritical withdrawal from a two-layer ¯uid through a line sink, J. Fluid Mech., 1995, 297, 37± 45. 24. Singler, T. J. and Geer, J. F., A hybrid perturbation-Galerkin solution to a problem in selective withdrawal, Phys. Fluids Ser. A, 1993, 5, 1156±1166. 25. Miloh, T. and Tyvand, P. A., Nonlinear transient free-surface ¯ow and dip formation due to a point sink, Phys. Fluids Ser. A, 1993, 5, 1368±1375. 26. Peregrine, D. H., A line source beneath a free surface. University of Wisconsin, Mathematical Research Center Technical Summary Report, 1248, 1972. 27. Hocking, G. C. and Forbes, L. K., A note on the ¯ow induced by a line sink beneath a free surface, J. Aust. Math Soc. Ser. B, 1991, 32, 251±260. 28. Dun, C. and Hocking, G. C., Withdrawal of ¯uid through a line sink beneath a free surface above a sloping boundary, J. Engng Math., 1995, 29, 1±10.
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29. Mekias, H. and Vanden-Broeck, J. M., Supercritical free-surface ¯ow with a stagnation point due to a submerged source, Phys. Fluids A, 1989, 1(10), 1694±1697. 30. Mekias, H. and Vanden-Broeck, J. M., Subcritical ¯ow with a stagnation point due to a source beneath a free surface, Phys. Fluids A, 1991, 3, 2652±2658. 31. Hocking, G. C. and Forbes, L. K., Subcritical free-surface ¯ow caused by a line source in a ¯uid of ®nite depth, J. Engng Math., 1992, 26, 455±466. 32. Vanden-Broeck, J. M., Waves generated by a source below a free surface in water of ®nite depth, J. Engng Math., 1996, 30, 603±609.
Computers & Fluids Vol. 27, No. 7, pp. 797±806, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0045-7930(98)00006-1 0045-7930/98 $19.00 + 0.00
WITHDRAWAL OF A FLUID OF FINITE DEPTH THROUGH A LINE SINK WITH A CUSP IN THE FREE SURFACE G. C. HOCKING1 and J-M. VANDEN-BROECK2 Mathematics and Statistics, Division of Science, Murdoch University, Murdoch 6150, Australia 2 Department of Mathematics, University of WisconsinÐMadison, WI 53706, U.S.A.
1
(Received 5 December 1996; in revised form 15 December 1997) AbstractÐThe steady withdrawal of a ¯uid of ®nite depth into a line sink is considered. The problem is solved numerically by a boundary integral equation method. It is shown that the ¯ow depends on the Froude number F = m(gH3)ÿ1/2 and the nondimensional sink depth b = HS/H, where m is the sink strength, g the acceleration of gravity, H is the total depth and HS is the depth of the sink. For given values of b and F there is a one-parameter family of solutions with a cusp on the free surface above the sink. It is found that in general there is a train of steady waves on the free surface. For particular values of the parameters the amplitude of the waves vanishes and the solutions reduce to those computed by Vanden-Broeck and Keller. These ®ndings con®rm and generalize the calculations of VandenBroeck where the free surface was covered by a lid everywhere but close to the sink. # 1998 Elsevier Science Ltd. All rights reserved
1. PRELIMINARY DISCUSSION
Problems involving withdrawal from water bodies have been of interest to engineers for many years. In particular, there is a range of problems involving withdrawal of water, either from single layers or from one of several layers of dierent density which impinge on such questions as water quality and reservoir management [1]. In particular, it has long been known that when water is withdrawn from a basin containing several layers of dierent density, the water ¯ows from the layer adjacent to the outlet until some critical ¯ow rate is exceeded, after which water ¯ows from both (or several) layers (see [2± 13]). Experiments and semi-analytical and numerical solutions of this problem have given dierent values for this critical withdrawal ¯ow rate, often with quite a large variation in values for the same geometric con®guration. This is true in the case of both two-dimensional withdrawal (slot or line sink), and in the case of three-dimensional withdrawal (pipe or line sink). The reason for these discrepancies is still unclear, despite a great deal of work. Numerical solutions of the steady ¯ow with a cusp [13±22] (see ``Cusp solution'' in Fig. 1) on the interface (or free surface) have long been thought to correspond to the critical value, and Hocking [23] recently provided strong evidence that this is indeed true for the case in which there is a line sink in an unbounded ¯uid. However, for two-dimensional withdrawal when the ¯uid is of ®nite depth, there are several complications to this. Firstly, there is a range of ¯ow rates over which the cusp solutions exist (for identical geometry). Secondly, even the smallest of these ¯ow rates is much larger than the critical drawdown values observed experimentally [2,7,9,12]. Finally, there are two branches of solution with a cusp, one of which only exists when the link sink is a long distance above the bottom, i.e. greater than 170% of the total ¯uid height [16]. It is necessary to examine these ¯ows in greater detail in order to obtain a clearer understanding of their behaviour, with the ultimate goal of understanding the process of critical drawdown. In this paper we examine one of the branches of solution with a cusp in the interface (or on the free surface), for withdrawal of a ¯uid of ®nite depth through a line sink. No attempt will be made to look at the full unsteady problem, which has been considered recently by other researchers [24,25]. 797
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Fig. 1. The two main free surface shapes for ¯ow into a line sink from a layer of ®nite depth.
This ¯ow can be characterized by the Froude numbers m F p gH 3 and m FS q : gHS3 Here m is the strength of the line sink (i.e. the ¯ux of the far ®eld), g is gravity, H is the total depth of the layer in the far ®eld (or the average depth if waves are present on the free surface) and HS is the (average) depth of the sink. If we wish to examine a two-layer ¯ow, the equations are identical if we assume the upper layer to be stagnant, and replace gravity, g, by the eective gravity g' = (Dr/r)g, where r is some reference density and Dr the density dierence between the two layers. In that case the Froude number is modi®ed to be the densimetric Froude number, i.e. FDS=m/(g'H3S)1/2. The Froude number F determines whether or not there can be waves on the free surface. The linear theory of water waves shows that if F < 1, then waves can occur on the free surface, while if Fe 1, waves are not possible. Flows with F < 1 and F>1 are called subcritical and supercritical, respectively, for this reason. This distinction is based on linear theory. The terms subcritical and supercritical are also commonly used in nonlinear theory, although nonlinear waves are possible not only for F < 1 but also for 1 < F < 1.3 (the value 1.3 corresponds to the highest solitary wave). Tuck and Vanden-Broeck [15] obtained numerically the ``cusp solution'' of Fig. 1 in water of in®nite depth (i.e. H = 1). They found a unique solution, at FS=12.622. It was presumed at that time that this corresponded to a critical situation; if the ¯ow rate was increased, the free surface would be pulled down into the sink, as when air is pulled into a water pump, thus invalidating this con®guration for higher ¯ow rates. Strong evidence that this was indeed the case was provided by Hocking [23], who used numerical techniques to solve the corresponding problem with two layers of ¯uid being drawn into a line sink. As the ¯ow rate was decreased, the solutions obtained approached the single layer solutions of Tuck and Vanden-Broeck [15]. In addition to these papers, there has been a large amount of work in which many free surface ¯ows due to submerged sinks were computed. Hocking [17] computed solutions, similar to those of Tuck and Vanden-Broeck [15], but in which there was a boundary beneath the sink sloping away without bound. He also obtained unique cusped solutions, in which the ¯ow rate depends on the angle of the slope. The result when the angle of the wall is 308 downward matches an exact solution ®rst obtained by Sautreaux [13], and subsequently by Craya [14]. ``Stagnation point'' solutions (see Fig. 1), in which the free surface rises up to a stagnation point above the sink, were obtained by Peregrine [26], Hocking and Forbes [27] and Dun and Hocking [28]. Further stagnation point solutions, but in ¯uid of ®nite depth, were obtained by Mekias and Vanden-Broeck [29,30], Hocking and Forbes [31], and recently by Vanden-Broeck [32]. These were found to exist for both subcritical and supercritical ¯ows and in some subcritical cases steady waves were found on the free surface.
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``Cusp solutions'' in water of ®nite depth were obtained by, amongst others, Collings [18], Vanden-Broeck and Keller [16], King and Bloor [20], and Hocking [19,21]. It was generally found that cusped solutions were supercritical and obtained over a range of values of the Froude number, F, above some minimum, usually greater than one, and in many cases could be computed from this minimum value up to in®nite Froude number. However, there was one branch of solutions obtained by Vanden-Broeck and Keller [16] which was dierent from the other solutions in two ways. Firstly, the solutions were subcritical and secondly, it was a one parameter family of solutions. In particular there was only one value of the Froude number for a given depth of the sink. Furthermore, as the ¯uid became deeper, it approached the solution of Tuck and Vanden-Broeck [15]. In a recent paper, Vanden-Broeck [22] computed solutions using a series truncation method in which a submerged sink was placed under a free surface in water of ®nite depth, but in which a rigid lid was placed on the free surface everywhere except very close to the region just above the sink. He found that he could reproduce qualitatively the results obtained in Vanden-Broeck and Keller [16], i.e. unique, subcritical cusped solutions with a waveless free surface for each sink to bottom depth ratio. In addition, however, he obtained solutions near to these waveless solutions which contained steady waves on the free surface. As the Froude number was increasingly distanced from the value which gave waveless solutions, the amplitude of the steady waves was found to grow until the waves approached a limiting con®guration with a stagnation point and a 1208 angle at their crest. Beyond this range of values, no further solutions were found. The work of Vanden-Broeck [22] was restricted by the nature of the series truncation method to problems in which the rigid lid began close to the region of the free surface above the line sink. In this paper we extend this investigation using an integral equation technique which allows us to move the rigid lid as far out as we wish, thus simulating an unbounded free surface. The results in Ref. [22] are con®rmed and shown to indicate the correct pattern of behaviour for the less restricted case. In particular it is found that there is a three parameter family of solutions which includes the waveless one parameter family of Ref. [16]. In Section 2 of this paper we will formulate the problem as a set of non-linear, singular integral equations, and in Section 3 we will describe the method used for the numerical solution of these equations. The results of the calculations are described and discussed in Sections 4 and 5. 2. PROBLEM FORMULATION
The steady, irrotational motion of an inviscid, incompressible ¯uid due to a submerged sink is to be examined. The ¯ow is assumed to be two-dimensional and gravity is acting vertically downwards (see Fig. 1). We concentrate in this paper on subcritical ¯ows for which waves can be present on the free surface. Following Vanden-Broeck [22], we truncate the free surface at some distance from the sink and extend it to the right and to the left by rigid horizontal lids. We de®ne the average depth H as the distance between the lids and the horizontal bottom. The average velocity U in the far ®eld is then de®ned by U = m/H. We choose Cartesian coordinates so that the sink is at x = 0 and the bottom at y = ÿ H. Then the level of the lids is y = 0. We denote by y = ÿ HS the ordinate of the sink. In the formulation given here we will use the symmetry of the ¯ow about x = 0, and solve for only the left hand half of the ¯ow Let z = x + iy be the physical plane (see Fig. 2(a)). The mathematical problem is to ®nd a complex potential w = f(x, y) + ic(x, y), which is analytic in the ¯ow domain and satis®es the conditions of no ¯ow across the solid boundaries. Without loss of generality, we choose c = 0 on the free surface and f = 0 at the cusp. The surface of the water must also be at constant pressure, a condition provided by Bernoulli's equation, which if we nondimensionalise with respect to the velocity U and the depth H, takes the form 2F ÿ2 Z q2 q20
1
where y = Z(x) is the equation for the elevation of the free surface, q is the ¯uid velocity, and F is the Froude number de®ned above. The value of the quantity q0 is to be determined, but is the velocity of the ¯uid at the free surface as it attaches to the rigid lid. Let hS be the depth of the sink, so that b = HS/H is the nondimensional sink depth. In nondimensional variables, U = 1
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Fig. 2. Mapped planes used in the problem formulating: (a) the physical z-plane; (b) the complex velocity potential w-plane; and (c) the upper-half z-plane.
and H = 1, so the incoming ¯ux far upstream, under the rigid lid, is also of unit magnitude. Since the only term involving the velocity is squared, the equations are independent of the direction of the ¯ow and therefore solutions are equally valid for a ¯ow into (sink) or out (source) of the slot. However, we will discuss the ¯ow as if it were into a sink (even though the actual formulation is for a source ¯ow). To derive an integral equation for this problem we follow a procedure similar to that used in Hocking [21]. The transformation eÿpw z
2
maps the in®nite strip between c = ÿ 1 and c = 0 in the w-plane to the upper-half of the zplane (see Fig. 2(c)). The point z = 0 corresponds to the sink at y = ÿ b, so that the free surface CD and the lid DI lie along the real z-axis where ze 1. The line between z = 0 and z = 1 corresponds to the vertical wall above the sink, or to the line of symmetry of the ¯ow if both sides are being considered. The points between z = 0 and z = zB corresponds to the vertical wall beneath the sink, i.e. the line of symmetry of the ¯ow, and z < zB to the bottom of the channel, which goes away horizontally to x = ÿ 1. The ¯ow domain is the upper half z-plane (see Fig. 2(c)). The case in which zB=0 corresponds to the case of a line source or sink on the bottom of the channel. We de®ne a new function O(z) = d(z) + it(z), related to the complex conjugate of the velocity ®eld by w0 z z eÿiO z ;
3
where the prime denotes dierentiation with respect to z. The magnitude of the velocity at any point is then given by vw'(z)v = et(z), and the angle any streamline makes with the horizontal is d(z). Since the total ¯ux is one, and the upstream ¯uid depth approaches one, O 4 0 as vzv 4 1. Since there is a cusp at C, we have d = ÿ p/2 at z = 1.
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On the portions of the real z-axis which correspond to the solid boundaries of the ¯ow domain, the streamlines must be parallel to the walls, so that the condition that there be no ¯ow normal to the solid boundaries is satis®ed if we choose d(z) to be the angle of the wall to the horizontal, i.e. 8 0 if ÿ 1 < z < zB > > > > p=2 if zB < z < 0 < 4 ÿp=2 if 0 < z 1 : d z > > unknown if 1 < z < z > D > : 0 if z > zD Note that d is unknown on 1 < z < zD because this is the free surface. d is zero on z>zD since this corresponds to the rigid horizontal lid. The only singularities of the function O(z) in the z-plane are those at the origin and at z = zB, which correspond respectively to the ¯ow into the sink, and to the corner point beneath the sink. Both of these singularities can be shown to be weaker than a simple pole, so that Cauchy's integral formula can be applied to O(z) on a path consisting of the real z-axis, a semicircle at vzv 4 1 in the upper-half plane, and a circle of vanishing radius about the point z. If we then let Im{z} 4 0+ and take real and imaginary parts, we obtain I 1 1 d z0 dz ; 5a t z ÿ p ÿ1 z0 ÿ z 0 and d z
1 p
I
1 ÿ1
t z0 dz ; z0 ÿ z 0
where the integrals are of Cauchy-principal-value form. Substituting the known values of d(z) given by Equation (4) into Equation (5a) gives I 1 z ÿ 1 z ÿ zB 1 zD d z0 dz : ÿ t z ln 2 p 1 z0 ÿ z 0 z2
5b
6
To this we must add the equation for constant pressure on the free surface, which can be obtained by combining Equations (1)±(3) on 1 E z < 1 to give Z ÿ2F ÿ2 z eÿt z0 sin d z0 dz0 e2t z q20 : 7 z0 p zD Using Equations (6) and (7) on the free surface gives a nonlinear system of integral equations for d(z) on 1 E z < zD. The value of d is known elsewhere on the boundary from the boundary conditions, and hence we can obtain t from Equation (6). Using d and t, it is possible to integrate Equation (3) to obtain the location of points on the free surface. These may be written as Z 1 z eÿt z0 cos d z0 dz0 ; 8a x z ÿ p 1 z0 and y z ÿhC ÿ
1 p
Z
z ÿt z0
e
1
sin d z0 dz0 : z0
Since y 4 0 as z 4 1, the depth at which the two in¯owing free surfaces meet in a cusp is Z 1 zD eÿt z0 sin d z0 dz0 : hC ÿ p 1 z0
8b
9
Choosing larger values of zD moves the beginning of the rigid lid further and further outwards.
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3. NUMERICAL SOLUTION
No closed form solution is known for the full nonlinear system of equations given by Equations (6) and (7) (except for the case F = 1, see refs [16,18±20]), but it can be solved quite well using collocation. A set of N equations can be obtained by discretizing the equations at a set of values of z. The corresponding unknown values of d at each of the points on the interval can then be obtained using a Newton±Raphson iteration scheme. The behaviour of z is like ef, so we choose to make the transformation z = ea. Points were chosen at N equally spaced values of a, so that 0 = a0
We found that there is a three-parameter family of solutions. The three parameters can be chosen as the Froude number, F, and the values of zB and zD. To reproduce the results of Vanden-Broeck and Keller [16], we used a variant of the scheme in which we ®xed q0=1 and took F as one of the unknowns. We found that the pro®les are then waveless and essentially independent of zD for zD suciently large. Thus the solutions depend essentially on one parameter (chosen here as zB) in accordance with the ®ndings of Vanden-Broeck and Keller. Returning to the original scheme, we used one of the waveless solutions as the initial guess, and once a solution was obtained, gradually varied the sink depth b. As b was moved away Table 1. Convergence of the numerical method for increasing values of N for the case F = 0.45, b = 0.324 and truncation at f = at=17.3. The two numbers in the ``Crest Height'' column are for the last two wave crests. The total depth of the ¯ow has an exact value of 1 N 50 100 200 400 800
Crest height 0.00797,0.00698 0.04595,0.04055 0.05127,0.05021 0.05204,0.05152 0.05218,0.05180
Cusp location ÿ0.27363 ÿ0.24365 ÿ0.23502 ÿ0.23205 ÿ0.23087
Total depth 1.03090 1.00681 1.00140 1.00024 1.00002
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Fig. 3. Free surface pro®les for F = 0.44 with nondimensional sink depth b = 0.308, less than the value of waveless solutions, and b = 0.363, greater than the waveless value.
from the value which gave waveless solutions, steady waves appeared on the free surface. As b was moved further away from this waveless value, the amplitude grew until it approached the limiting Stokes con®guration with a 1208 angle at the crest. If b was chosen larger than the waveless value, the bulk of the oscillation amplitude occurred at a height greater than the level of the rigid lid, while if b was less than the waveless value, the free surface level remained mainly below the level of the plate. This is illustrated in Fig. 3 where surface pro®les for F = 0.44 are shown for dierent values of sink height, b. The waveless solution corresponds to b = 0.33 for this case. These ®ndings are consistent with the work of Vanden-Broeck [22]. It was found that for ®xed values of Froude number, F, and sink depth, b, the waves had almost identical amplitude and wavelength. However, it was also found that the phase of the steady waves depended upon the location of the rigid lid, and it appears that the phase is such that the attachment to the plate occurs at either a crest or a trough. Figure 4(a) shows solutions calculated with the plate distance set so that the attachment is almost exactly in phase for the case F = 0.44, b = 0.32. The solutions are almost identical except very near to the upstream attachment point. Figure 4(b) on the other hand shows two plots for identical parameter values in which the upstream attachment point is almost exactly out of phase. These ®ndings suggest there is a three parameter family of solutions for the problem without a lid, i.e. the ¯ow with a free surface of unlimited horizontal extent. Figure 5 shows the region in parameter space in which these solutions exist, and how these relate to other solutions to this problem which have been calculated. The dashed lines correspond to con®gurations close to the limiting con®guration with a corner on the crest. The exact solutions with breaking waves could not be computed with the present scheme due to a singularity associated with the 1208angle. For each point between the two lines, there is a one-parameter family of solutions. The solid line represents the waveless solutions of Ref. [16]. As the depth of the ¯uid increases, the width of the solution region decreases until the solution of Tuck and Vanden-Broeck [15] (in in®nite depth) is reached. At that point there cannot be waves on the free surface because the depth of the ¯uid is in®nite, meaning that the velocity far upstream must approach zero, and is consequently unable to support steady waves.
5. COMMENTS
In this paper, we have used a set of integral equations to compute solutions to the problem of steady withdrawal from a layer of ®nite depth through a line sink in which there is a cusp in the free surface. It is shown that there are solutions with waves which exist near to the previously calculated sub-critical branch of solutions of Vanden-Broeck and Keller [16] which have a cusp but no waves. As the Froude number is changed away from the value at which there are waveless solutions, the amplitude of the steady waves grows until a limiting con®guration with a sharp crest with a 1208 angle is approached.
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Fig. 4. Solutions with F = 0.44, b = 0.324 with truncation at (a) f = at=25,21.5 and 17.3, i.e. steady waves in phase, (b) f = at=21.5 and 19, i.e. steady waves out of phase, at is the truncation point in the real potential variable f.
One could argue that the waves are due to the presence of the lid, and that if it were not present there would be no waves. However, the results of Fig. 4(a) show that it is possible to move the lid further and further away in such a way that the solution remains essentially unchanged near the cusp. As the lid moves further away, more and more oscillations appear on the free surface. In the limit as the lid moves to vxv 4 1, we expect the solutions near the cusp to be independent of the position of the lid. This suggests the existence of a three-parameter family of
Fig. 5. Sketch of parameter space in which cusped solutions with steady waves exist. The solutions of Vanden-Broeck and Keller [16] are shown as a solid line. The dashed lines show the limiting values at which solutions with steady waves were computed. These are approximate due to the diculty of calculating the limiting con®guration of standing wave solutions.
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805
solutions for a sink submerged below a free surface of in®nite horizontal extent. This family includes the waveless one-parameter family of Vanden-Broeck and Keller as particular cases. There is a further question which must remain unanswered for the moment about the stability of these solutions. Since all of the calculations are of steady ¯ow, it is not possible to determine whether the solutions are stable to small perturbations. A full time-dependent scheme, such as in refs [24,25], would need to be derived to test this. In concluding this work, we should return to the comments in Section 1. It is interesting to note that it is possible to compute solutions near to the cusp solutions which are not supercritical from the selective withdrawal point of view, i.e. which maintain ¯ow from a single layer only. This means that although the cusp ¯ow in in®nite depth seems to be the limit of singlelayer ¯ow, this may not be the case if the ¯uid is of ®nite depth. If it is, then there must be some non-uniqueness in the problem. Alternatively, the solutions with breaking waves may be the precursor to the collapse of the ¯ow into the two-layer ¯ow and drawdown of the interface. Work is under way to compute the full two-layer, steady withdrawal ¯ows and answer this question. REFERENCES 1. Imberger, J. and Hamblin, P. F., Dynamics of lakes, reservoirs and cooling ponds, Rev. Fluid Mech., 1982, 14, 153± 187. 2. Gariel, P., Experimental research on the ¯ow of nonhomogeneous ¯uids, La Houille Blanche, 1949, 4, 56±65. 3. Huber, D. G., Irrotational motion of two ¯uid strata towards a line sink, J. Engng Mech. Div. Proc. ASCE, 1960, 71(85), . 4. Harleman, D. R. F. and Elder, R. E., Withdrawal from two-layer strati®ed ¯ow, Proc. ASCE, 1965, 91, HY4. 5. Lubin, B. T. and Springer, G. S., The formation of a dip on the surface of a liquid draining from a tank, J. Fluid Mech, 1967, 29, 385±390. 6. Wood, I. R. and Lai, K. 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Engng ASCE, 1991, 117(6), 800±805. 13. Sautreaux, C., Movement d'un liquide parfait soumis aÁ la pesanteur. DeÂtermination des lignes de courant, J. Math. Pures Appl., 1901, 7, 125±159. 14. Craya, A., Theoretical research on the ¯ow of nonhomogeneous ¯uids, La Houille Blanche, 1949, 4, 44±55. 15. Tuck, E. O. and Vanden-Broeck, J. M., A cusp-like free-surface ¯ow due to a submerged source or sink, J. Aust. Math Soc. Ser. B, 1984, 25, 443±450. 16. Vanden-Broeck, J. M. and Keller, J. B., Free surface ¯ow due to a sink, J. Fluid Mech., 1987, 175, 109±117. 17. Hocking, G. C., Cusp-like free-surface ¯ows due to a submerged source or sink in the presence of a ¯at or sloping bottom, J. Aust. Math Soc. Ser. B, 1985, 26, 470±486. 18. Collings, I. L., Two in®nite Froude number cusped free surface ¯ows due to a submerged line source or sink, J. Aust. Math Soc. Ser. B, 1986, 28, 260±270. 19. Hocking, G. C., In®nite Froude number solutions to the problem of a submerged source or sink, J. Aust. Math Soc. Ser. B, 1988, 29, 401±409. 20. King, A. C. and Bloor, M. I. G., A note on the free surface induced by a submerged source at in®nite Froude number, J. Aust. Math Soc. Ser. B, 1988, 30, 147±156. 21. Hocking, G. C., Critical withdrawal from a two-layer ¯uid through a line sink, J. Engng Math., 1991, 25, 1±11. 22. Vanden-Broeck, J. M., Cusp ¯ow due to a submerged source with a free surface partially covered by a lid., Eur. J. Mech. B (Fluids), 1997, 16(2), 249±255. 23. Hocking, G. C., Supercritical withdrawal from a two-layer ¯uid through a line sink, J. Fluid Mech., 1995, 297, 37± 45. 24. Singler, T. J. and Geer, J. F., A hybrid perturbation-Galerkin solution to a problem in selective withdrawal, Phys. Fluids Ser. A, 1993, 5, 1156±1166. 25. Miloh, T. and Tyvand, P. A., Nonlinear transient free-surface ¯ow and dip formation due to a point sink, Phys. Fluids Ser. A, 1993, 5, 1368±1375. 26. Peregrine, D. H., A line source beneath a free surface. 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29. Mekias, H. and Vanden-Broeck, J. M., Supercritical free-surface ¯ow with a stagnation point due to a submerged source, Phys. Fluids A, 1989, 1(10), 1694±1697. 30. Mekias, H. and Vanden-Broeck, J. M., Subcritical ¯ow with a stagnation point due to a source beneath a free surface, Phys. Fluids A, 1991, 3, 2652±2658. 31. Hocking, G. C. and Forbes, L. K., Subcritical free-surface ¯ow caused by a line source in a ¯uid of ®nite depth, J. Engng Math., 1992, 26, 455±466. 32. Vanden-Broeck, J. M., Waves generated by a source below a free surface in water of ®nite depth, J. Engng Math., 1996, 30, 603±609.