A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is
sought for the motion of an axisymmetric vortex ring of small cross-section embedded
in a viscous incompressible fluid. In order to take account of the influence of elliptical
deformation of the core due to the self-induced strain, the method of matched of matched
asymptotic expansions is extended to a higher order in a small parameter
ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and
Γ is the circulation. Alternatively, ε is regarded as a
measure of the ratio of the core radius to the ring radius, and our
scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third
order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and
reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to
Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an
infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand
the circles of stagnation points of radii Rs(t) and
R˜s(t) relative to the laboratory
frame and a comoving frame respectively, and that of peak vorticity of radius
R˜p(t) as
Rs ≈ R0
+ [2 log(4R0/√vt)
+ 1.4743424] vt/R0, R˜s ≈ R0
+ 2.5902739 vt/R0, and
Rp ≈ R0
+ 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in
time, is also deduced. The results are compatible with the experimental results of
Sallet & Widmayer (1974) and Weigand & Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture
of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a
line of dipoles along the core centreline, with their axes in the propagating direction,
subjected to the self-induced flow field. The strength of the dipole depends not only
on the curvature but also on the location of the core centre, and therefore should
be specified at the initial instant. This specification removes an indeterminacy of the
first-order theory. We derive a new asymptotic development of the Biot-Savart law
for an arbitrary distribution of vorticity, which makes the non-local induction velocity
from the dipoles calculable at third order.
Zero-Hopf Bifurcations of 3D Quadratic Jerk System