Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-19T18:38:26.729Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex evolution in non-axisymmetric impulsive spin-up from rest

Published online by Cambridge University Press:  26 April 2006

D. M. Henderson ,
J. M. Lopez  and
D. L. Stewart
D. M. Henderson
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
J. M. Lopez
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA Earth System Science Center, The Pennsylvania State University, University Park, PA 16802, USA
D. L. Stewart
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA Present address: ICAM, Virginia Tech University, Blacksburg, VA 24061, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow evolution of water in a completely filled rectangular container, impulsively rotated from rest to a steady angular speed, is investigated experimentally and numerically. The pathlines of the fluid from rest to solid-body rotation primarily follow one of two possible configurations that have been described previously in the literature. The first, consisting of two cyclones that form following the separation and roll-up of the sidewall boundary layers and an anticyclone that forms subsequently, results in a pattern on the path to spin-up of cyclonic–anticyclonic–cyclonic vorticity. In the second configuration the cyclones migrate into the interior of the container and merge, resulting in a pattern on the path to spin-up of anticyclonic–cyclonic–anticyclonic vorticity. The experiments provide a parameterization of the possible evolutionary configurations as a function of horizontal and vertical aspect ratios and Reynolds numbers. Critical Reynolds numbers for vortex merger are determined experimentally. Evolutionary configurations in addition to the primary two are observed; in particular symmetry breaking occurs at high Reynolds numbers causing complicated patterns of flow evolution. For some flow conditions at high Reynolds numbers, more than one evolutionary pattern is observed for the same external parameters. The experiments are conducted with a rigid lid showing that a free surface is not required for vortex merger. Numerical integrations of the two-dimensional Navier–Stokes equations (a situation corresponding to the limiting case of a container of infinite depth, where there are no effects from the top and bottom and all flow is horizontal) reproduce qualitatively many of the features of the experimental observations, in particular the merger events. The numerical results show that neither vertical flow due to Ekman boundary layers at the top and bottom nor a free surface are necessary for the observed vortex merger.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 324 , 10 October 1996 , pp. 109 - 134
Copyright
© 1996 Cambridge University Press

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

Benton, E. R. & Clark, A. 1974 Spin-up. Ann. Rev. Fluid Mech. 6, 257–280.Google Scholar
Buntine, J. D. & Pullin, D. I. 1989 Merger and cancellation of strained vortices. J. Fluid Mech. 205, 263–295.Google Scholar
Dritschel, D. G. 1986 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–182.Google Scholar
Griffiths, R. W. & Hopfinger, E. J. 1991 Coalescing of geostrophic vortices. J. Fluid Mech. 178, 73–97.Google Scholar
Heijst, G.J.F. van 1989 Spin-up phenomena in non-axisymmetric containers. J. Fluid Mech. 206, 171–191.Google Scholar
Heijst, G.J.F. van, Davies, P.A. & Davis, R.G. 1990 Spin-up in a rectangular container. Phys. Fluids A 2, 150–159.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Kitchens, C. W. 1980 Navier-Stokes solutions for spin-up in a filled cylinder. AIAA J. 18, 929–934.Google Scholar
Konijnenberg, J. A. van de 1995 Spin-up in non-axisymmetric containers. PhD thesis, Fluid Dynamics Laboratory, Department of Physics, Eindhoven University of Technology, The Netherlands.
Konijnenberg, J. A. van de, Andersson, H. I., Billdal, J. T. & Heijst, G.J.F. van 1994 Spin-up in a rectangular tank with low angular velocity. Phys. Fluids 6, 1168–1176.Google Scholar
Melander, M. V., Zabusky, N.J. & McWilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Pedlosky, J. 1984 Geophysical Fluid Dynamics. Springer.
Stewart, D. L. 1993 Spin-up in a rectangular cylinder. Master of Arts Thesis, Department of Mathematics, The Pennsylvania State University.
Suh, Y.K. 1994 Numerical study on two-dimensional spin-up in a rectangle. Phys. Fluids 6, 2333–2344.Google Scholar
Sweet, R. A. 1974 A generalized cyclic reduction algorithm. SIAM J. Numer. Anal. 10, 506–592.Google Scholar
Weast, R. C. (ed.) 1976 Handbook of Chemistry and Physics, 57th Edn. CRC Press.
Weidman, P.D. 1976a On the spin-up and spin-down of a rotating fluid. Part 1. Extending the Wedemeyer model. J. Fluid Mech. 77, 658–708.Google Scholar
Weidman, P.D. 1976b On the spin-up and spin-down of a rotating fluid. Part 2. Measurements and stability. J. Fluid Mech. 77, 709–735.Google Scholar
Williams, G. P. 1967 Thermal convection in a rotating fluid annulus: Part 1. The basic axisymmetric flow. J. Atmos. Sci. 24, 144–161.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96–106.Google Scholar