H. K. Moffatt

Viscous and resistive eddies near a sharp corner

Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.

The degree of knottedness of tangled vortex lines

Let u(x) be the velocity field in a fluid of infinite extent due to a vorticity distribution w(x) which is zero except in two closed vortex filaments of strengths K 1 , K 2 . It is first shown that the integral \[ I=\int{\bf u}.{\boldmath \omega}\,dV \] is equal to α K 1 K 2 where α is an integer representing the degree of linkage of the two filaments; α = 0 if they are unlinked, ± 1 if they are singly linked. The invariance of I for a continuous localized vorticity distribution is then established for barotropic inviscid flow under conservative body forces. The result is interpreted in terms of the conservation of linkages of vortex lines which move with the fluid. Some examples of steady flows for which I ± 0 are briefly described; in particular, attention is drawn to a family of spherical vortices with swirl (which is closely analogous to a known family of solutions of the equations of magnetostatics); the vortex lines of these flows are both knotted and linked. Two related magnetohydrodynamic invariants discovered by Woltjer (1958 a, b ) are discussed in ±5.

Helicity and the Calugareanu invariant

The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Călugăreanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Călugăreanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Călugăreanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

Stretched vortices - the sinews of turbulence; large-Reynolds-number asymptotics

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation r subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in E = 1/R, where R, = T/v. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(e2). The solution reveals that the stretched vortex can survive for a long time even when two of the principal rates of strain are positive, provided R, is large enough. The manner in which the theory may be extended to higher orders in E is indicated. The results are discussed in relation to the high-vorticity regions (here described as ‘sinews’) observed in many direct numerical simulations of turbulence.

On a class of steady confined Stokes flows with chaotic streamlines

The general incompressible flow uQ(x), quadratic in the space coordinates, and satisfying the condition uQ-n = 0 on a sphere r = 1, is considered. It is shown that this flow may be decomposed into the sum of three ingredients - a poloidal flow of Hill’s vortex structure, a quasi-rigid rotation, and a twist ingredient involving two parameters, the complete flow uQ(x) then involving essentially seven independent parameters. The flow, being quadratic, is a Stokes flow in the sphere. The streamline structure of the general flow is investigated, and the results illustrated with reference to a particular sub-family of ‘ stretch-twist-fold ’ (STF) flows that arise naturally in dynamo theory. When the flow is a small perturbation of a flow ul(x) with closed streamlines, the particle paths are constrained near surfaces defined by an ‘adiabatic invariant ’ associated with the perturbation field. When the flow u1 is dominated by its twist ingredient, the particles can migrate from one such surface to another, a phenomenon that is clearly evident in the computation of Poincar6 sections for the STF flow, and that we describe as ‘ trans-adiabatic drift ’. The migration occurs when the particles pass a neighbourhood of saddle points of the flow on r = 1, and leads to chaos in the streamline pattern in much the same way as the chaos that occurs near heteroclinic orbits of low-order dynamical systems. The flow is believed to be the first example of a steady Stokes flow in a bounded region exhibiting chaotic streamlines.

TURBULENT DYNAMO ACTION AT LOW MAGNETIC REYNOLDS NUMBER.

The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale 1 of the turbulence is considered. There are no external sources for the field, and in the abseilce of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number R, = u,l/h (where uo is the root-mean-square velocity and h the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities ZIL and R,, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order R;2Z and greater. I n the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t-iexp (a%/2h3) where a( = O(ugZ)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

Free-surface cusps associated with flow at low Reynolds number

When two cylinders are counter-rotated at low Reynolds number about parallel horizontal axes below the free surface of a viscous fluid, the rotation being such as to induce convergence of the flow on the free surface, then above a certain critical angular velocity Ω c , the free surface dips downwards and a cusp forms. We provide an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques

Fluid dynamical aspects of the levitation-melting process

When a piece of metal is placed above a coil carrying a high frequency current, the induced surface currents in the metal can provide a Lorentz force which can support it against gravity; at the same time the heat produced by Joule dissipation can melt the metal. This is the process of ‘levitation melting’, which is a well-established technique in fundamental work in physical and chemical metallurgy. Most theoretical studies of magnetic levitation have dealt only with solid conductors and so have a voided the interesting questions of interaction between the free surface, the magnetic field and the internal flow. These fluid dynamical aspects of the process are studied in this paper. A particular configuration that is studied in detail is a cylinder levitated by two equal parallel currents in phase; this is conceived as part of a toroidal configuration which avoids a difficulty of conventional configurations, viz the leakage of fluid through the ‘magnetic hole’ at a point on the metal surface where the surface tangential magnetic field vanishes. The equilibrium and stability of the solid circular cylinder is first considered; then the dynamics of the surface film when melting begins; then the equilibrium shape of the fully melted body (analysed by means of a general variational principle proved in § 5); and finally the dynamics of the interior flow, which, as argued in § 2, is likely to be turbulent when the levitated mass is of the order of a few grams or greater.

An approach to a dynamic theory of dynamo action in a rotating conducting fluid

Dynamo action associated with the motion generated by a random body force f(x, t) in a conducting fluid rotating with uniform angular velocity ω is considered. It is supposed that, in the Fourier decomposition off, only waves having a phase velocity V satisfying V . ω > 0 are present and that the Fourier amplitudes of f are isotropically distributed. The resulting velocity field then lacks reflexional symmetry, and energy is transferred to a magnetic field h 0 ( x , t ) provided the scale L of h 0 is sufficiently large. Attention is focused on a particular distribution of h 0 ( x , t ) (a circularly polarized wave) for which this dynamo action is most efficient. Under these conditions, the mean stresses acting on the fluid are irrotational and no mean flow develops. It is supposed that \[ \lambda \ll \Omega l^2,\quad \lambda\ll h_0l\quad{\rm and}\quad\nu/\lambda = O(1)\quad\hbox{or smaller}, \] where l ([Lt ] L ) is the scale of the f -field, and ν and λ are the kinematic viscosity and magnetic diffusivity of the fluid. The response to f is then dominated by resonant contributions near the natural frequencies of the free undamped system. As h 0 grows in strength, these frequencies change, and the dynamo process is rendered less efficient. Ultimately the magnetic energy M (and also the kinetic energy E ) asymptote to steady values. Expressions for these values are obtained for the particular situation when ν [Lt ]λ and when the frequency ω 0 characteristic of the f -field is small compared with other relevant frequencies, notably ω and h 0 / l ; under these conditions, it is shown that \[ \frac{M}{E}\sim C\left(\frac{\Omega}{\omega_0}\right)^{\frac{1}{2}}\left(\frac{\nu}{\lambda}\right)^{\frac{1}{2}}\frac{L}{l}, \] where C is a number of order unity determined by the spectral properties of the f -field. The implications for the terrestrial dynamo are discussed.

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

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 asymptotic expansions is extended to a higher order in a small parameter =( = ) 1=2 , where is the kinematic viscosity of fluid and is the circulation. Alternatively, 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 . This is a natural extension of Fraenkel{Saman’s rst-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 innitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and ~ Rs(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius Rp(t) as Rs R0 +[ 2 log(4R0= p t )+1 :4743424]t=R0, ~ Rs R0 +2 :5902739t=R0, and Rp R0 +4 :5902739t=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 eld. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specied at the initial instant. This specication removes an indeterminacy of the rst-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.