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[Gt ]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[Gt ]1, R[Gt ]1 and
ε≡λ/R[Lt ]1, we find an accurate
asymptotic form for the vorticity in a region
1<r/(2v/γ)1/2[les ]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
∂Γ/∂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[les ]λ[les ]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[les ]λ[les ]1, and that when λ>1, the vortex
will decay only through exponentially slow leakage of vorticity, indicating
extreme
robustness in this case.
A calculation procedure for the partially-parabolized Navier-Stokes equations in transformed coordinates for two-dimensional flow in channels of variable cross-section
