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On the formation of axial corner vortices during spin-up in a cylinder of square cross-section

Munro, Richard J.; Hewitt, R.E; Foster, M.R.

Authors

R.E Hewitt

M.R. Foster



Abstract

We present experimental and theoretical results for the adjustment of a fluid (homogeneous or linearly stratified), which is initially rotating as a solid body with angular frequency Ω−ΔΩ, to a nonlinear increase ΔΩ in the angular frequency of all bounding surfaces. The fluid is contained in a cylinder of square cross-section which is aligned centrally along the rotation axis, and we focus on the O(Ro−1Ω−1) time scale, where Ro=ΔΩ/Ω is the Rossby number. The flow development is shown to be dominated by unsteady separation of a viscous sidewall layer, leading to an eruption of vorticity that becomes trapped in the four vertical corners of the container. The longer-time evolution on the standard ‘spin-up’ time scale, E−1/2Ω−1 (where E is the associated Ekman number), has been described in detail for this geometry by Foster & Munro (J. Fluid Mech., vol. 712, 2012, pp. 7–40), but only for small changes in the container’s rotation rate (i.e. Ro≪1). In the linear case, for Ro≪E1/2≪1, there is no sidewall separation. In the present investigation we focus on the fully nonlinear problem, Ro=O(1), for which the sidewall viscous layers are Prandtl boundary layers and (somewhat unusually) periodic around the container’s circumference. Some care is required in the corners of the container, but we show that the sidewall boundary layer breaks down (separates) shortly after an impulsive change in rotation rate. These theoretical boundary-layer results are compared with two-dimensional Navier–Stokes results which capture the eruption of vorticity, and these are in turn compared to laboratory observations and data. The experiments show that when the Burger number, S=(N/Ω)2 (where N is the buoyancy frequency), is relatively large – corresponding to a strongly stratified fluid – the flow remains (horizontally) two-dimensional on the O(Ro−1Ω−1) time scale, and good quantitative predictions can be made by a two-dimensional theory. As S was reduced in the experiments, three-dimensional effects were observed to become important in the core of each corner vortex, on this time scale, but only after the breakdown of the sidewall layers.

Journal Article Type Article
Publication Date Jun 1, 2015
Journal Journal of Fluid Mechanics
Print ISSN 0022-1120
Electronic ISSN 1469-7645
Publisher Cambridge University Press (CUP)
Peer Reviewed Peer Reviewed
Volume 772
APA6 Citation Munro, R. J., Hewitt, R., & Foster, M. (2015). On the formation of axial corner vortices during spin-up in a cylinder of square cross-section. Journal of Fluid Mechanics, 772, https://doi.org/10.1017/jfm.2015.219
DOI https://doi.org/10.1017/jfm.2015.219
Keywords boundary-layer separation; rotating flows; stratified flows
Publisher URL http://dx.doi.org/10.1017/jfm.2015.219
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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