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Structure of a steady drain-hole vortex in a viscous fluid

Published online by Cambridge University Press:  10 June 2010

L. BØHLING ,
A. ANDERSEN  and
D. FABRE
L. BØHLING
Affiliation:
Department of Physics and Center for Fluid Dynamics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
A. ANDERSEN*
Affiliation:
Department of Physics and Center for Fluid Dynamics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
D. FABRE
Affiliation:
Université de Toulouse, INPT, UPS; Institut de Mécanique des Fluides de Toulouse (IMFT), Allée du Professeur Camille Soula, F-31400, Toulouse, France
*
†Email address for correspondence: aanders@fysik.dtu.dk
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Abstract

We use direct numerical simulations to study a steady bathtub vortex in a cylindrical tank with a central drain-hole, a flat stress-free surface and velocity prescribed at the inlet. We find that the qualitative structure of the meridional flow does not depend on the radial Reynolds number, whereas we observe a weak overall rotation at a low radial Reynolds number and a concentrated vortex above the drain-hole at a high radial Reynolds number. We introduce a simple analytically integrable model that shows the same qualitative dependence on the radial Reynolds number as the simulations and compares favourably with the results for the radial velocity and the azimuthal velocity at the surface. Finally, we describe the height dependence of the radius of the vortex core and the maximum of the azimuthal velocity at a high radial Reynolds number, and we show that the data on the radius of the vortex core and the maximum of the azimuthal velocity as functions of height collapse on single curves by appropriate scaling.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 177 - 188
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Alekseenko, S. V., Kuibin, P. A. & Okulov, V. L. 2007 Theory of Concentrated Vortices. Springer.Google Scholar
Andersen, A., Bohr, T., Stenum, B., Juul Rasmussen, J. & Lautrup, B. 2003 Anatomy of a bathtub vortex. Phys. Rev. Lett. 91, 104502.CrossRefGoogle ScholarPubMed
Auguste, F., Fabre, D. & Magnaudet, J. 2010 Bifurcations in the wake of a thick circular disk. Theor. Comput. Fluid Dyn. 24, 305–313.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Burgers, J. M. 1940 Application of a model system to illustrate some points of the statistical theory of free turbulence. Proc. Acad. Sci. Amsterdam 43, 2–12.Google Scholar
De Felice, V. F. 2007 Il vortice a superficie libera in quanto instabilità. PhD thesis, Università degli Studi di Salerno.Google Scholar
Donaldson, C. duP. & Sullivan, R. D. 1960 Behaviour of solutions of the Navier–Stokes equations for a complete class of three-dimensional viscous vortices. In Proc. Heat Transfer and Fluid Mech. Institute, Stanford University, pp. 16–30.Google Scholar
Einstein, H. A. & Li, H. 1951 Steady vortex flow in a real fluid. In Proc. Heat Transfer and Fluid Mech. Institute, Stanford University, pp. 33–43.Google Scholar
Guyon, E., Hulin, J. P., Petit, L. & Mitescu, C. D. 2001 Physical Hydrodynamics. Oxford University Press.CrossRefGoogle Scholar
Hamel, G. 1917 Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresbericht d. Deutschen Mathem.-Vereinigung 25, 34–60.Google Scholar
Ito, K., Sakai, T. & Yamaguchi, A. 2003 Numerical simulation of free surface vortex in cylindrical tank. In The 10th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-10) Seoul, Korea, October 5–9, 2003, pp. 1–13.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn.Butterworth-Heinemann.Google Scholar
Lewellen, W. S. 1962 A solution for three-dimensional vortex flows with strong circulation. J. Fluid Mech. 14, 420–432.CrossRefGoogle Scholar
Lugt, H. J. 1995 Vortex Flow in Nature and Technology. Krieger.Google Scholar
Magnaudet, J., Rivero, M. & Fabre, J. 1995 Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow. J. Fluid Mech. 284, 97–135.CrossRefGoogle Scholar
Miles, J. 1998 A note on the Burgers–Rott vortex with a free surface. Z. Angew. Math. Phys. 49, 162–165.CrossRefGoogle Scholar
Preston, J. H. 1950 The steady circulatory flow about a circular cylinder with uniformly distributed suction at the surface. Aeronaut. Q. 1, 319–338.CrossRefGoogle Scholar
Rossi, M., Bottausci, F., Maurel, A. & Petitjeans, P. 2004 A Nonuniformly stretched vortex. Phys. Rev. Lett. 92, 054504.CrossRefGoogle ScholarPubMed
Rott, N. 1958 On the viscous core of a line vortex. Z. Angew. Math. Phys. 9, 543–553.CrossRefGoogle Scholar
Shapiro, A. H. 1962 Bath-tub vortex. Nature 196, 1080–1081.CrossRefGoogle Scholar
Shingubara, S., Hagiwara, K., Fukushima, R., Kawakubo, T. 1988 Vortices around a sinkhole: phase diagram for one-celled and two-celled vortices. J. Phys. Soc. Japan 57, 88–94.CrossRefGoogle Scholar
Stepanyants, Y. A. & Yeoh, G. H. 2008 a Burgers–Rott vortices with surface tension. Z. Angew. Math. Phys. 59, 1–12.CrossRefGoogle Scholar
Stepanyants, Y. A. & Yeoh, G. H. 2008 b Stationary bathtub vortices and a critical regime of liquid discharge. J. Fluid Mech. 604, 77–98.CrossRefGoogle Scholar
Sullivan, R. D. 1959 A two-cell vortex solution of the Navier–Stokes equations. J. Aerosp. Sci. 26, 767–768.CrossRefGoogle Scholar
Thwaites, B. 1950 Note on the circulatory flow about a circular cylinder through which the normal velocity is large. Q. J. Mech. Appl. Math. 3, 74–79.CrossRefGoogle Scholar
Trefethen, L. M., Bilger, R. W., Fink, P. T., Luxton, R. E., Tanner, R. I. 1965 The bath-tub vortex in the southern hemisphere. Nature 207, 1084–1085.CrossRefGoogle Scholar
Tyvand, P. A. & Haugen, K. B. 2005 An impulsive bathtub vortex. Phys. Fluids 17, 062105.CrossRefGoogle Scholar