Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-08T19:29:49.146Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  22 July 2016

T. E. STOKES ,
G. C. HOCKING  and
L. K. FORBES
T. E. STOKES
Affiliation:
Department of Mathematics, University of Waikato, Hamilton, New Zealand email: [email protected]
G. C. HOCKING
Affiliation:
Mathematics and Statistics, Murdoch University, Murdoch, Western Australia, 6150, Australia email: [email protected]
L. K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania, GPO Box 252-37, Hobart 7001, Australia email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The time-varying flow in which fluid is withdrawn from or added to a reservoir of infinite or arbitrary finite depth through a point sink or source of variable strength beneath a free surface is considered. Backed up by some analytic work, a numerical method is used, and the results are compared with previous work on steady and unsteady flows. In the case of withdrawal for an impulsively started flow, it is found that the critical flow rate increases with reservoir depth, although it changes little as the depth increases beyond double the sink submergence depth. The largest flow rate at which steady solutions can evolve in source flows follows a similar pattern although at a considerably higher value. Simulations indicate that some of the previously calculated steady state solutions at higher flow rates may be unstable, if they exist at all.

Keywords

Free-surface potential flowsideal fluids
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 3 , June 2017 , pp. 357 - 379
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Craya, A. (1949) Theoretical research on the flow of nonhomogeneous fluids. La Houille Blanche 4 (2), 44–55.Google Scholar
[2] Forbes, L. K. & Hocking, G. C. (1990) Flow caused by a point sink in a fluid having a free surface. J. Austral. Math. Soc. Ser. B 32 (2), 233–252.Google Scholar
[3] 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 (3), 377–391.Google Scholar
[4] Forbes, L. K. & Hocking, G. C. (1995) The bath-plug vortex. J. Fluid Mech. 284, 43–62.Google 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 (3), 385–401.Google 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 (3), 406–416.Google Scholar
[7] Gariel, P. (1949) Experimental research on the flow of nonhomogeneous fluids. La Houille Blanche 4, 56–65.Google 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. Aust. Math Soc. Ser. B 26 (APR), 470–486.Google Scholar
[9] Hocking, G. C. (1991) Withdrawal from two-layer fluid through line sink. J. Hydr. Engng ASCE 117 (6), 800–805.Google Scholar
[10] Hocking, G. C. (1995) Supercritical withdrawal from a two-layer fluid through a line sink. J. Fluid Mech. 297, 37–47.Google Scholar
[11] Hocking, G. C. & Forbes, L. K. (1991) A note on the flow induced by a line sink beneath a free surface. J. Aust. Math Soc. Ser. B 32 (3), 251–260.Google 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.Google Scholar
[13] Hocking, G. C., Vanden Broeck, J.-M. & Forbes, L. K. (2002) A note on withdrawal from a fluid of finite depth through a point sink. ANZIAM J. 44 (2), 181–191.Google Scholar
[14] Hocking, G. C., Forbes, L. K. & Stokes, T. E. (2014) A note on steady flow into a submerged point sink. ANZIAM J. 56 (2), 150–159.CrossRefGoogle Scholar
[15] Hocking, G. C., Stokes, T. E. & Forbes, L. K. (2010) A rational approximation to the evolution of a free surface during fluid withdrawal through a point sink. ANZIAM J. Ser. E 51, E31–E36.Google Scholar
[16] Huber, D. G. (1960) Irrotational motion of two fluid strata towards a line sink. J. Engng. Mech. Div. Proc. ASCE, 86, EM4, 71–85.Google Scholar
[17] Imberger, J. & Hamblin, P. F. (1982) Dynamics of lakes, reservoirs and cooling ponds. Ann. Rev. Fluid Mech. 14, 153–187.Google 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. Hyd. Res. 17 (1), 53–62.Google Scholar
[19] Landrini, M. & Tyvand, P. A. (2001) Generation of water waves and bores by impulsive bottom flux. J. Engng. Maths 39 (1–4), 131–171.CrossRefGoogle Scholar
[20] Lawrence, G. A. & Imberger, J. (1979) Selective Withdrawal Through a Point Sink in a Continuously Stratified Fluid with a Pycnocline. Tech. Report No. ED-79-002, Dept. of Civil Eng., University of Western Australia, Australia.Google Scholar
[21] 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.Google Scholar
[22] Lustri, C. J., McCue, S. W. & Chapman, S. J. (2013) Exponential asymptotics of free surface flow due to a line source. IMA J. Appl. Math. 78 (4), 697–713.Google Scholar
[23] Miloh, T. & Tyvand, P. A. (1993) Nonlinear transient free-surface flow and dip formation due to a point sink. Phys. Fluids A 5 (6), 1368–1375.Google Scholar
[24] Sautreaux, C. (1901) Mouvement d'un liquide parfait soumis à lapesanteur. Dé termination des lignes de courant. J. Math. Pures Appl. 7, 125–159.Google Scholar
[25] Scullen, D. & Tuck, E. O. (1995) Non-linear free-surface flow computations for submerged cylinders. J. Ship Res. 39 (3), 185–193.Google Scholar
[26] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2002) Unsteady free surface flow induced by a line sink. J. Eng. Math. 47 (2), 137–160.Google Scholar
[27] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2005) Unsteady flow induced by a withdrawal point beneath a free surface. ANZIAM J. 47 (2), 185–202.Google Scholar
[28] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2008) Unsteady flow induced by withdrawal in a fluid of finite depth. Comput. Fluids 37 (3), 236–249.Google Scholar
[29] Tuck, E. O. (1997) Solution of nonlinear free-surface problems by boundary and desingularised integral equation techniques. In: Noye, J. et al. (editors), Proc. 8th Biennial Computational Techniques and Applications Conference, World Scientific, Singapore, pp. 11–26.Google Scholar
[30] Tuck, E. O. & Vanden Broeck, J.-M. (1984) A cusp-like free-surface flow due to a submerged source or sink. J. Aust. Math Soc. Ser. B 25 (APR), 443–450.Google Scholar
[31] Vanden Broeck, J.-M. & Keller, J. B. (1987) Free surface flow due to a sink. J. Fluid Mech. 175, 109–117.Google Scholar
[32] Wehausen, J. V. & Laitone, E. V. (1960) Surface waves. In: Encyclopaedia of Physics, Vol. IX, Springer-Verlag, Berlin, pp. 446–778.Google Scholar
[33] Xue, M. & Yue, D. K. P. (1998) Nonlinear free-surface flow due to an impulsively started submerged point sink. J. Fluid Mech. 364, 325–347.Google Scholar
[34] Yih, C. S. (1980) Stratified Flows, Academic Press, New York, pp. 110–121.Google Scholar