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Vortices in oscillating spin-up

Published online by Cambridge University Press:  05 February 2007

M. G. WELLS ,
H. J. H. CLERCX  and
G. J. F. VAN HEIJST
M. G. WELLS
Affiliation:
Department of Applied Physics, Eindhoven University of Technology, PO Box 513, NL-5600 MB Eindhoven, The Netherlands
H. J. H. CLERCX
Affiliation:
Department of Applied Physics, Eindhoven University of Technology, PO Box 513, NL-5600 MB Eindhoven, The Netherlands
G. J. F. VAN HEIJST
Affiliation:
Department of Applied Physics, Eindhoven University of Technology, PO Box 513, NL-5600 MB Eindhoven, The Netherlands
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Abstract

Laboratory experiments and numerical simulations of oscillating spin-up in a square tank have been conducted to investigate the production of small-scale vorticity near the no-slip sidewalls of the container and the formation and subsequent decay of wall-generated quasi-two-dimensional vortices. The flow is made quasi-two-dimensional by a steady background rotation, and a small sinusoidal perturbation to the background rotation leads to the periodic formation of eddies in the corners of the tank by the roll-up of vorticity generated along the sidewalls. When the oscillation period is greater than the time scale required to advect a full-grown corner vortex to approximately halfway along the sidewall, dipole structures are observed to form. These dipoles migrate away from the walls, and the interior of the tank is continually filled with new vortices. The average size of these vortices appears to be largely controlled by the initial formation mechanism. Their vorticity decays from interactions with other stronger vortices that strip off filaments of vorticity, and by Ekman pumping at the bottom of the tank. Subsequent interactions between the weaker ‘old’ vortices and the ‘young’ vortices result in the straining, and finally the destruction, of older vortices. This inhibits the formation of large-scale vortices with diameters comparable to the size of the container.

The laboratory experiments revealed a k−5/3 power law of the energy spectrum for small-to-intermediate wavenumbers. Measurements of the intensity spectrum of a passive scalar were consistent with the Batchelor prediction of a k−1 power law at large wavenumbers. Two-dimensional numerical simulations, under similar conditions to those in the experiments (with weak Ekman decay), were also performed and the simultaneous presence of a k−5/3 and k−3−ζ (with 0 < ζ « 1) power spectrum is observed, with the transition occurring at the wavenumber at which vorticity is injected from the viscous boundary layer into the interior. For higher Ekman decay rates, steeper spectra are obtained for the large wavenumber range, with ζ = O(1) and proportional to the Ekman decay rate. Movies are available with the online version of the paper.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 573 , February 2007 , pp. 339 - 369
Copyright
Copyright © Cambridge University Press 2007

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References

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Wells et al. supplementary movie

Movie 1. The movie above is taken from the same sequence of images shown in Figure 3. Dye was initially injected into the top left corner and this visualizes the generation  of vorticity at the boundaries. Dipoles are formed when opposite signed vorticies merge, and these dipoles subsequently propagate away from the boundaries, every forcing cycle, to fill the interior with a sea of interacting vorticies. The size and strength of the vortices can be seen to decrease with time due to a combination of Ekman damping and stripping of vorticity.

Download Wells et al. supplementary movie(Video)
Video 5 MB

Wells et al. supplementary movie

Movie 1. The movie above is taken from the same sequence of images shown in Figure 3. Dye was initially injected into the top left corner and this visualizes the generation  of vorticity at the boundaries. Dipoles are formed when opposite signed vorticies merge, and these dipoles subsequently propagate away from the boundaries, every forcing cycle, to fill the interior with a sea of interacting vorticies. The size and strength of the vortices can be seen to decrease with time due to a combination of Ekman damping and stripping of vorticity.

Download Wells et al. supplementary movie(Video)
Video 5.3 MB

Wells et al. supplementary movie

Movie 2. The movie above shows the evolution of vorticity over a 1.5 forcing periods in the numerical simulation described in Figure 12  (run 40 in Table 3). Vorticity is produced at the no-slip boundaries and rolls up to form vorticies. This vorticity detaches from the boundaries when dipoles form. These dipoles migrates into the interior  of the  domain.  Subsequent interactions strip filaments of vorticity  from the vortices and the interior of the tank become filled with intense filaments of vorticity. Ultimately the older vortices weaken with time as few of the interactions lead to vortex mergers, and they are replaced by new vortices in the thank interior.

Download Wells et al. supplementary movie(Video)
Video 9 MB

Wells et al. supplementary movie

Movie 2. The movie above shows the evolution of vorticity over a 1.5 forcing periods in the numerical simulation described in Figure 12  (run 40 in Table 3). Vorticity is produced at the no-slip boundaries and rolls up to form vorticies. This vorticity detaches from the boundaries when dipoles form. These dipoles migrates into the interior  of the  domain.  Subsequent interactions strip filaments of vorticity  from the vortices and the interior of the tank become filled with intense filaments of vorticity. Ultimately the older vortices weaken with time as few of the interactions lead to vortex mergers, and they are replaced by new vortices in the thank interior.

Download Wells et al. supplementary movie(Video)
Video 8 MB