Kenneth G. Miller

Some steady vortex flows past a circular cylinder

Steady vortex flows past a circular cylinder are obtained numerically as solutions of the partial differential equation Δψ = f(ψ), f(ψ) = ω(1 - H(ψ - α)), where H is the Heaviside function. Only symmetric solutions are considered so the flow may be thought of as that past a semicircular bump in a half-plane. The flow is transplanted by the complex logarithm to a semi-infinite strip. This strip is truncated at a finite height, a numerical boundary condition is used on the top, and the difference equations resulting from the five-point discretization for the Laplacian on a uniform grid are solved using Fourier methods and an iteration for the nonlinear equation. If the area of the vortex region is prescribed the magnitude of the vorticity ω is adjusted in an inner iteration to satisfy this area constraint. Three types of solutions are discussed: vortices attached to the cylinder, vortex patches standing off from the cylinder and strips of vorticity extending to infinity. Three families of each type of solution have been found. Equilibrium positions for point vortices, including the Foppl pair, are related to these families by continuation

Stability of vortices in equilibrium with a cylinder

The stability of steady inviscid vortex pairs in equilibrium with a circular cylinder is studied by discretizing equations derived from contour dynamics. There are two families of vortices, one with a pair of counter-rotating vortices standing behind the cylinder, which may be thought of as desingularizing the Foppl point vortices, and the other with the vortices standing directly above and below the cylinder. Vortices in the first family are found to be neutrally stable with respect to symmetric perturbations. When asymmetric perturbations are included, there is a single unstable mode and a single asymptotically stable mode. Vortices above and below the cylinder have two modes of instability, one symmetric and the other asymmetric, and likewise two asymptotically stable modes.

Variational formulas on Lipschitz domains

A rigorous treatment is given of variational formulas for solutions of certain Dirichlet problems for the Laplace operator on Lipschitz domains under interior variations. In particular we extend well-known variational formulas for the torsional rigidity and for capacity from the class of C1 domains to the class of Lipschitz domains. A motivation for this work comes from the use of variational methods in the study of Prandtl-Batchelor flows in fluid mechanics.

Some steady axisymmetric vortex flows past a sphere

Steady, inviscid, axisymmetric vortex flows past a sphere are obtained numerically as solutions of a partial differential equation for the stream function. The solutions found include vortex rings, bounded vortices attached to the sphere and infinite vortex tubes. Four families of attached vortices are described: vortex wakes behind the sphere, spherically annular vortices surrounding the spherical obstacle (which can be given analytically), bands of vorticity around the sphere and symmetric pairs of vortices fore and aft of the sphere. Each attached vortex leads to a one-parameter family of vortex rings, analogous to the connection between Hills spherical vortex and the vortex rings of Norbury

Stationary corner vortex configurations

Given a stable configuration of point vortices for steady two dimensional inviscid, incompressible fluid flow in a domainD, it is shown that there is another stable configuration of stationary point vortices inD with vortices near the original vortices plus additional vortices near any of the convex corners ofD. It follows that there are steady flows which have a finite sequence, of arbitrary length, of vortices of alternating sign descending into any convex corner ofD. Several computed examples are given.

Magnetic Relaxation in the Solar Corona

This is a mathematical study of the long-lived hydromagnetic structures produced in the tenuous solar corona by the turbulent, resistive relaxation of a magnetic field under the condition of extremely high electrical conductivity. The relaxation theory of Taylor, originally developed for a laboratory device, is extended to treat the open atmosphere where the relaxing field must interact with its surrounding fields. A boundary-value problem is posed for a two-dimensional model that idealizes the corona as the half Cartesian plane filled with a potential field (1) that is anchored to a rigid, perfectly conducting base and (2) that embeds a force-free magnetic field in the form of a flux-rope oriented horizontally and perpendicular to the Cartesian plane. The flux-rope has a free boundary, which is an unknown in the construction of a solution for this atmosphere. Pairs of magnetostatic solutions are constructed to represent the initial and final states of a flux-rope relaxation that conserve both the total magnetic helicity and total axial magnetic flux, using a numerical iterative method specially developed for this study. The collection of numerical solutions found provides an insight into the interplay among several hydromagnetic properties in the formation of long-lived coronal structures. In particular, the study shows (1) that the outward spread of reconnection between a relaxing flux-rope and its external field may be arrested at some outer magnetic flux surface within which a constant-α force-free field emerges as the minimum-energy state and (2) that this outward spread is complicated by an inward, partial collapse of the relaxing flux-rope produced by a loss of internal magnetic pressure.

Steady axisymmetric vortex flows with swirl and shear

A general procedure is presented for computing axisymmetric swirling vortices which are steady with respect to an inviscid flow that is either uniform at infinity or includes shear. We consider cases both with and without a spherical obstacle. Choices of numerical parameters are given which yield vortex rings with swirl, attached vortices with swirl analogous to spherical vortices found by Moffatt, tubes of vorticity extending to infinity and Beltrami flows. When there is a spherical obstacle we have found multiple solutions for each set of parameters. Flows are found by numerically solving the Bragg–Hawthorne equation using a non-Newton-based iterative procedure which is robust in its dependence on an initial guess. Steady axisymmetric vortices with swirl as solutions of the Euler equations are of interest for several reasons. Unlike axisymmetric flows without swirl the vortex stretching terms do not vanish and the helical-like streamlines inside the vortex are, in general, ergodic with associated mixing properties. Nevertheless these threedimensional flows can be obtained as solutions to elliptic boundary value problems in two variables. We obtain families of inviscid flows in which an axisymmetric vortex with swirl is embedded in an external flow. The external flow may be irrotational flow which is uniform at infinity or it may include shear. We will consider both flows past a sphere and flows in which the vortex is the only disturbance to the flow at infinity. The solutions obtained can be related to explicit solutions of classical interest such as Hill’s vortex and its generalization to flows with swirl found by Moffatt (1969) and Hicks (1899). A number of studies have indicated that under appropriate conditions inviscid

Computation of vortex flows past obstacles with circulation

Abstract Vortex boundaries are compared for vortex blobs in equilibrium with an irrotational flow past on obstacle. The ambient flow has circulation around the obstacle. The curves are varied among those which are star-shaped with respect to the equilibrium position of a stable point vortex and which bound an area fixed by giving the circulation of the vortex. The computations are based on a variational principle for energy as a functional of vorticity.

A trial-free-boundary method for computing Batchelor flows

The solutions of Batchelor flows in bounded domains are computed by a nonstandard trial-free-boundary method.