On the formation of axial corner vortices during spinup in a cylinder of square crosssectionTools Munro, Richard J. and Hewitt, R.E and Foster, M.R. (2015) On the formation of axial corner vortices during spinup in a cylinder of square crosssection. Journal of Fluid Mechanics, 772 . pp. 246271. ISSN 14697645
Official URL: http://dx.doi.org/10.1017/jfm.2015.219
AbstractWe 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 crosssection 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 longertime evolution on the standard ‘spinup’ 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 boundarylayer results are compared with twodimensional 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) twodimensional on the O(Ro−1Ω−1) time scale, and good quantitative predictions can be made by a twodimensional theory. As S was reduced in the experiments, threedimensional 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.
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