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Steady free surface flows induced by a submerged ring source or sink

Published online by Cambridge University Press:  24 January 2012

T. E. Stokes ,
G. C. Hocking  and
L. K. Forbes
T. E. Stokes
Affiliation:
Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand
G. C. Hocking*
Affiliation:
Mathematics and Statistics, Murdoch University, Perth, WA 6150, Australia
L. K. Forbes
Affiliation:
School of Mathematics and Physics, University of Tasmania, Hobart, TAS 7001, Australia
*
†Email address for correspondence: G.Hocking@murdoch.edu.au
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Abstract

The steady axisymmetric flow induced by a ring sink (or source) submerged in an unbounded inviscid fluid is computed and the resulting deformation of the free surface is obtained. Solutions are obtained analytically in the limit of small Froude number (and hence small surface deformation) and numerically for the full nonlinear problem. The small Froude number solutions are found to have the property that if the non-dimensional radius of the ring sink is less than , there is a central stagnation point on the surface surrounded by a dip which rises to the stagnation level in the far distance. However, as the radius of the ring sink increases beyond , a surface stagnation ring forms and moves outward as the ring sink radius increases. It is also shown that as the radius of the sink increases, the solutions in the vicinity of the ring sink/source change continuously from those due to a point sink/source () to those due to a line sink/source (). These properties are confirmed by the numerical solutions to the full nonlinear equations for finite Froude numbers. At small values of the Froude number and sink or source radius, the nonlinear solutions look like the approximate solutions, but as the flow rate increases a limiting maximum Froude number solution with a secondary stagnation ring is obtained. At large values of sink or source radius, however, this ring does not form and there is no obvious physical reason for the limit on solutions. The maximum Froude numbers at which steady solutions exist for each radius are computed.

JFM classification

Waves/Free-surface Flows: Critical layers Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 694 , 10 March 2012 , pp. 352 - 370
Copyright
Copyright © Cambridge University Press 2012

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References

1. Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.Google Scholar
2. Craya, A. 1949 Theoretical research on the flow of nonhomogeneous fluids. La Houille Blanche 4, 44–55.CrossRefGoogle Scholar
3. Forbes, L. K. & Hocking, G. C. 1990 Flow caused by a point sink in a fluid having a free surface. J. Austral. Math. Soc. B 32, 231–249.CrossRefGoogle Scholar
4. Forbes, L. K. & Hocking, G. C. 1993 Flow induced by a line sink in a quiescent fluid with surface-tension effects. J. Austral. Math. Soc. Ser. B 34, 377–391.CrossRefGoogle Scholar
5. Forbes, L. K. & Hocking, G. C. 2003 On the computation of steady axi-symmetric withdrawal from a two-layer fluid. Comput. Fluids 32, 385–401.CrossRefGoogle Scholar
6. Forbes, L. K., Hocking, G. C. & Chandler, G. A. 1996 A note on withdrawal through a point sink in fluid of finite depth. J. Austral. Math. Soc. Ser. B 37, 406–416.CrossRefGoogle Scholar
7. Harleman, D. R. F. & Elder, R. E. 1965 Withdrawal from two-layer stratified flow. J. Hydraul. Div. ASCE 91 (4), 43–58.CrossRefGoogle Scholar
8. Hocking, G. C. 1985 Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom. J. Austral. Math. Soc. Ser. B 26, 470–486.CrossRefGoogle Scholar
9. Hocking, G. C. 1991 Withdrawal from two-layer fluid through line sink. J. Hydraul. Engng ASCE 117 (6), 800–805.CrossRefGoogle Scholar
10. Hocking, G. C. 1995 Supercritical withdrawal from a two-layer fluid through a line sink. J. Fluid Mech. 297, 37–47.CrossRefGoogle Scholar
11. Hocking, G. C. & Forbes, L. K. 1991 A note on the flow induced by a line sink beneath a free surface. J. Austral. Math. Soc. Ser. B 32, 251–260.CrossRefGoogle Scholar
12. Hocking, G. C. & Forbes, L. K. 2001 Supercritical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth. J. Fluid Mech. 428, 333–348.CrossRefGoogle Scholar
13. Hocking, G. C., Vanden Broeck, J.-M. & Forbes, L. K. 2002 Withdrawal from a fluid of finite depth through a point sink. ANZIAM J. 44, 181–191.CrossRefGoogle Scholar
14. Huber, D. G. 1960 Irrotational motion of two fluid strata towards a line sink. J. Engng. Mech. Div. Proc. ASCE 86 (EM4), 71–85.CrossRefGoogle Scholar
15. Imberger, J. & Hamblin, P. F. 1982 Dynamics of lakes, reservoirs and cooling ponds. Annu. Rev. Fluid Mech. 14, 153–187.CrossRefGoogle Scholar
16. Imberger, J. & Patterson, J. C. 1990 Physical limnology. In Advances in Applied Mechanics (ed. Hutchinson, J. W. & Wu, T. ). vol. 27. pp. 303–475. Academic.Google Scholar
17. Jirka, G. H. 1979 Supercritical withdrawal from two-layered fluid systems. Part 1. Two-dimensional skimmer wall. J. Hydraul. Res. 17 (1), 43–51.CrossRefGoogle Scholar
18. Jirka, G. H. & Katavola, D. S 1979 Supercritical withdrawal from two-layered fluid systems. Part 2. Three dimensional flow into a round intake. J. Hydraul. Res. 17 (1), 53–62.CrossRefGoogle Scholar
19. Lubin, B. T. & Springer, G. S. 1967 The formation of a dip on the surface of a liquid draining from a tank. J. Fluid Mech. 29, 385–390.CrossRefGoogle Scholar
20. Mekias, H. & Vanden-Broeck, J.-M. 1991 Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids A 3, 2652–2658.CrossRefGoogle Scholar
21. Peregrine, H. 1972 A line source beneath a free surface. Rep. 1248. Mathematics Research Center, University of Wisconsin, Madison.Google Scholar
22. Sautreaux, C. 1901 Mouvement d’un liquide parfait soumis à lapesanteur. Dé termination des lignes de courant. J. Math. Pures Appl. 7 (5), 125–159.Google Scholar
23. Stokes, T. E., Hocking, G. C. & Forbes, L. K. 2002 Unsteady free surface flow induced by a line sink. J. Engng Maths 47, 137–160.CrossRefGoogle Scholar
24. Stokes, T. E., Hocking, G. C. & Forbes, L. K. 2005 Unsteady flow induced by a withdrawal point beneath a free surface. ANZIAM J. 47, 185–202.CrossRefGoogle Scholar
25. Tuck, E. O. 1975 On air flow over free surfaces of stationary water. J. Austral. Math. Soc. Ser. B 19, 66–80.CrossRefGoogle Scholar
26. Tuck, E. O. & Vanden Broeck, J.-M. 1984 A cusp-like free surface flow due to a submerged source or sink. J. Austral. Math. Soc. Ser. B 25, 443–450.CrossRefGoogle Scholar
27. Tyvand, P. A. 1992 Unsteady free-surface flow due to a line source. Phys. Fluids A 4, 671–676.CrossRefGoogle Scholar
28. Vanden Broeck, J.-M. & Keller, J. B. 1987 Free surface flow due to a sink. J. Fluid Mech. 175, 109–117.CrossRefGoogle Scholar
29. Vanden Broeck, J.-M., Schwartz, L. W. & Tuck, E. O. 1978 Divergent low-Froude-number series expansion of nonlinear free-surface flow problems. Proc. R. Soc. Lond. Ser. A 361, 207–224.Google Scholar
30. Wood, I. R. & Lai, K. K. 1972 Selective withdrawal from a two-layered fluid. J. Hydraul. Res. 10 (4), 475–496.CrossRefGoogle Scholar
31. Xue, X. & Yue, D. K. P 1998 Nonlinear free-surface flow due to an impulsively started submerged point sink. J. Fluid Mech. 364, 325–347.CrossRefGoogle Scholar