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Scaling of decaying shallow axisymmetric swirl flows

Published online by Cambridge University Press:  07 April 2010

M. DURAN-MATUTE ,
L. P. J. KAMP ,
R. R. TRIELING  and
G. J. F. van HEIJST
M. DURAN-MATUTE*
Affiliation:
Department of Applied Physics and J. M. Burgers Centre, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
L. P. J. KAMP
Affiliation:
Department of Applied Physics and J. M. Burgers Centre, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
R. R. TRIELING
Affiliation:
Department of Applied Physics and J. M. Burgers Centre, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
G. J. F. van HEIJST
Affiliation:
Department of Applied Physics and J. M. Burgers Centre, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
*
†Email address for correspondence: m.duran.matute@tue.nl
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Abstract

There is a lack of rigour in the usual explanation for the scaling of the vertical velocity of shallow flows based on geometrical arguments and the continuity equation. In this paper we show, by studying shallow axisymmetric swirl flows, that the dynamics of the flow are crucial to determine the proper scaling. In addition, we present two characteristic scaling parameters for such flows: Reδ2 for the radial velocity and Reδ3 for the vertical velocity, where Re is the Reynolds number of the swirl flow and δ=H/L is the flow aspect ratio with H the fluid depth and L a typical horizontal length scale. This scaling contradicts the common assumption that the vertical velocity should scale with the primary motion proportional to the aspect ratio δ. Moreover, if this scaling applies, then the primary flow can be considered as quasi-two-dimensional. Numerical simulations of a decaying Lamb–Oseen vortex served to test the analytical results and to determine their range of validity. It was found that the primary flow can be considered as quasi-two-dimensional only if δRe1/2≲3 and δRe1/3≲1.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 471 - 484
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Akkermans, R. A. D., Kamp, L. P. J., Clercx, H. J. H. & van Heijst, G. J. F. 2008 Intrinsic three-dimensionality in electromagnetically driven shallow flows. Europhys. Lett. 83, 24001.CrossRefGoogle Scholar
Berger, S. A., Talbot, L. & Yao, L.-S. 1983 Flow in curved pipes. Annu. Rev. Fluid Mech. 15, 461–512.Google Scholar
Clercx, H. J. H., van Heijst, G. J. F. & Zoeteweij, M. L. 2003 Quasi-two-dimensional turbulence in shallow fluid layers: the role of bottom friction and fluid layer depth. Phys. Rev. E 67, 066303.CrossRefGoogle ScholarPubMed
Comsol AB 2009 Comsol 3.5, User's Guide. Tegnérgatan 23, SE-111 40 Stockholm, Sweden. http://www.comsol.com.Google Scholar
Dean, W. R. 1927 XVI. Note on the motion of fluid in a curved pipe. Phil. Mag. Ser. 7 4, 208–223.CrossRefGoogle Scholar
Dolzhanskii, F. V., Krymov, V. A. & Manin, D. Yu. 1992 An advanced experimental investigation of quasi-two-dimensional shear flows. J. Fluid Mech. 241, 705–722.CrossRefGoogle Scholar
Hopfinger, E. J. & van Heijst, G. J. F. 1993 Vortices in rotating fluids. Annu. Rev. Fluid Mech. 25, 241–289.CrossRefGoogle Scholar
Jirka, G. H. & Uijttewaal, W. S. J. 2004 Shallow flows: a definition. In Shallow Flows (ed. Jirka, G. H. & Uijttewaal, W. S. J.), pp. 3–11. Taylor & Francis.CrossRefGoogle Scholar
Paret, J. & Tabeling, P. 1997 Experimental observation of the two-dimensional inverse energy cascade. Phys. Rev. Lett. 79, 4162–4165.CrossRefGoogle Scholar
Satijn, M. P., Cense, A. W., Verzicco, R., Clercx, H. J. H. & van Heijst, G. J. F. 2001 Three-dimensional structure and decay properties of vortices in shallow fluid layers. Phys. Fluids 13, 1932–1945.CrossRefGoogle Scholar
Sous, D., Bonneton, N. & Sommeria, J. 2005 Transition from deep to shallow water layer: formation of vortex dipoles. Eur. J. Mech B 24, 19–32.CrossRefGoogle Scholar
Tabeling, P., Burkhart, S., Cardoso, O. & Willaime, H. 1991 Experimental study of freely decaying two-dimensional turbulence. Phys. Rev. Lett. 67, 3772–3775.CrossRefGoogle ScholarPubMed