Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

Abstract

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.

A generalized vortex ring model

“…In the case of classical vortex rings generated in liquids (e.g. water) their basic properties have been explained in terms of the conventional models of laminar rings (Saffman 1970;Saffman 1992;Rott & Cantwell 1993a,b;Wang et al 1997;Fukumoto & Moffatt 2000). In what follows, some of the theoretical results described so far are compared with published experimental data.…”

“…The properties of the vortex rings have been studied for over a century both theoretically and experimentally (Helmholtz 1858;Lamb 1932;Phillips 1956;Norbury 1973;Kambe & Oshima 1975;Saffman 1992;Shariff & Leonard 1992;Lim & Nickels 1995). Recent developments on the modelling side include Stanaway, Cantwell & Spalart (1988), Rott & Cantwell (1993a, b), Mohseni & Gharib (1998), Kaplanski & Rudi (1999, 2005, Fukumoto & Moffatt (2000), Shusser & Gharib (2000), Mohseni (2001Mohseni ( , 2006, Linden & Turner (2001) and Fukumoto & Kaplanski (2008).…”

A model for confined vortex rings with elliptical-core vorticity distribution

“…This model was analytically derived as a first-order solution of the linearised axisymmetric Navier-Stokes equation. The measurements of unconfined or confined vortex ring vorticity, however, show that the vorticity field of the vortex ring deforms during its development (Fukumoto & Moffatt 2000); at later stages it becomes elongated due to the Reynolds number effects (Weigand & Gharib (1997); Danaila & Helie (2008)). This deformation is accompanied by the modification of the vorticity distribution function and leads to changes in vortex ring integral characteristics.…”

Remark on well-posedness for the fourth order nonlinear Schrödinger type equation

“…The equation in (1.1) describes the three-dimensional motion of an isolated vortex filament embedded in an inviscid incompressible fluid filling an infinite region. This equation is proposed by Fukumoto and Moffatt [8] as some detailed model taking account of the effect from the higher order corrections of the Da Rios model (cubic nonlinear Schrödinger equation):…”