Journal of Fluid Mechanics

Structure and stability of non-symmetric Burgers vortices

a1 Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA


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, (αx, βy, γz), α+β+γ=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=Γ/(2πv) where Γ is the vortex circulation and v the kinematic viscosity, in the range R=1−104, and a broad range of strain ratios λ=(β−α)/(β+α) including λ>1, and in some cases λ[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, λ) parameter space including λ where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When λ[dbl greater-than sign]1, R[dbl greater-than sign]1 and ε[identical with]λ/R[double less-than sign]1, we find an accurate asymptotic form for the vorticity in a region 1<r/(2v/γ)1/2[less-than-or-eq, slant]ε1/2, 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 Γγ/(4πv)exp[−1/(2eε)] on bounding streamlines. This allows an estimate of the leakage rate of circulation to infinity as [partial partial differential]Γ/[partial partial differential]t =(0.48475/4π)γε−1Γ exp (−1/2eε) with corresponding exponentially slow decay of the vortex when λ>1. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case λ>0. Stability is found for 0[less-than-or-eq, slant]λ[less-than-or-eq, slant]1, and a neutrally convective mode of instability is found and analysed for λ>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]λ[less-than-or-eq, slant]1, and that when λ>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case.

(Received February 2 1997)
(Revised December 2 1997)