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https://resolver.caltech.edu/CaltechETD:etd-01142008-091416
Year: 1997
DOI: 10.7907/sc0x-1g40
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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 steady stretched vortices embedded in a uniform nonsymmetric straining field, [...], [...], one principal axis of extensional strain of which is aligned with the vorticity. These are known as nonsymmetric Burgers vortices studied first by Robinson and Saffman (1984). We consider vortex Reynolds numbers [...] where [...] is the vortex circulation and [...] the kinematic vorticity, in the range [...], and a broad range of strain ratios [...] including [...], and in some cases [...]. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady vortex states over our whole ([...] parameter space including [...], where arguments proposed by Moffatt, Kida, and Ohkitani (1994) suggest the nonexistence of steady solutions. When [...] and [...], we find an accurate asymptotic form for the vorticity in a region [...], giving, in some cases, near machine-precision agreement with our numerical solutions. This suggests the existence of an extended region where the exponentially small vorticity is confined to a near cat's-eye shaped region of the almost two-dimensional flow, and takes a constant value nearly equal to [...] on bounding streamlines. This allows an estimate of the leakage rate of circulation to infinity as [...] = [...] with corresponding exponentially slow decay of the vortex when [...]. This leakage rate differs substantially from that estimated by Moffatt, Kida, and Ohkitani. The normal-mode linear stability of the axisymmetric Burgers vortex [...] to two-dimensional disturbances is calculated in detail and the vortex is found to be stable at all Reynolds numbers. An iterative technique based on the Power method is used to estimate the largest eigenvalues for the nonsymmetric case [...]. Stability is found for [...], and a neutrally convective mode of instability is found and analyzed analytically for [...]. Our general conclusion is that the generalized nonsymmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all [...], and that the vortex will tend to move with the background strain when [...], but maintain its structure which will change only through exponentially slow leakage of vorticity, indicating extreme robustness in this case.
https://resolver.caltech.edu/CaltechETD:etd-01142008-091416