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Structure and stability of non-symmetric Burgers vortices

Prochazka, Aurelius and Pullin, D. I. (1998) Structure and stability of non-symmetric Burgers vortices. Journal of Fluid Mechanics, 363 . pp. 199-228. ISSN 0022-1120.

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We investigate, numerically and analytically, the structure and stability of steady and quasi-steady solutions of the Navier–Stokes equations corresponding to stretched vortices embedded in a uniform non-symmetric straining field, ([alpha]x, [beta]y, [gamma]z), [alpha]+[beta]+[gamma]=0, one principal axis of extensional strain of which is aligned with the vorticity. These are known as non-symmetric Burgers vortices (Robinson & Saffman 1984). We consider vortex Reynolds numbers R=[Gamma]/(2[pi]v) where [Gamma] is the vortex circulation and v the kinematic viscosity, in the range R=1[minus sign]104, and a broad range of strain ratios [lambda]=([beta][minus sign][alpha])/([beta]+[alpha]) including [lambda]>1, and in some cases [lambda][dbl greater-than sign]1. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady and quasi-steady vortex states over our whole (R, [lambda]) parameter space including [lambda] where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When [lambda][dbl greater-than sign]1, R[dbl greater-than sign]1 and [epsilon][identical with][lambda]/R[double less-than sign]1, we find an accurate asymptotic form for the vorticity in a region 1<r/(2v/[gamma])1/2[less-than-or-eq, slant][epsilon]1/2, giving very good agreement with our numerical solutions. This suggests the existence of an extended region where the exponentially small vorticity is confined to a nearly cat's-eye-shaped region of the almost two-dimensional flow, and takes a constant value nearly equal to [Gamma][gamma]/(4[pi]v)exp[[minus sign]1/(2e[epsilon])] on bounding streamlines. This allows an estimate of the leakage rate of circulation to infinity as [partial partial differential][Gamma]/[partial partial differential]t =(0.48475/4[pi])[gamma][epsilon][minus sign]1[Gamma] exp ([minus sign]1/2e[epsilon]) with corresponding exponentially slow decay of the vortex when [lambda]>1. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case [lambda]>0. Stability is found for 0[less-than-or-eq, slant][lambda][less-than-or-eq, slant]1, and a neutrally convective mode of instability is found and analysed for [lambda]>1. Our general conclusion is that the generalized non-symmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all R, 0[less-than-or-eq, slant][lambda][less-than-or-eq, slant]1, and that when [lambda]>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case.

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Additional Information:"Reprinted with the permission of Cambridge University Press." (Received February 2 1997; Revised December 2 1997) We gratefully acknowledge helpful discussions with David Hill, James Buntine, and Ron Henderson. This research was partial supported by NSF Grant CTS-9634222.
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Deposited On:18 May 2006
Last Modified:21 Sep 2016 22:36

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