Alan R. Elcrat

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

Schwarz-Christoffel mapping of multiply connected domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.

On the propagation of sonic discontinuities in the unsteady flow of a perfect gas

Abstract The magnitude of discontinuities of the first derivatives of flow quantities in the unsteady flow of a perfect gas are shown to satisfy Riccatti equations along the orthogonal trajectories of sonic singular surfaces S(t) . These equations are greatly simplified if they are transformed to equations along the bicharacteristic curves in the characteristic manifold Σ = U S(t) . When this is done an explicit criteria for the decay or “blow up” of sonic discontinuities can be given.

Radial and circular slit maps of unbounded multiply connected circle domains

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

On finding a surface crack from boundary measurements

We prove the uniqueness of the determination of a surface crack from one special boundary measurement of an electrical or elastic field. Then we suggest and test a numerical algorithm for identification of a polygonal plane crack based on the Schwarz-Christoffel formula. The numerical experiments with cracks consisting of one or two intervals show the high stability and precision of this algorithm.

Classical free-streamline flow over a polygonal obstacle

In classical Kirchhoff flow, an ideal incompressible fluid flows past an obstacle and around a motionless wake bounded by free streamlines. Since 1869 it has been known that in principle, the two-dimensional Kirchhoff flow over a polygonal obstacle can be determined by constructing a conformal map onto a polygon in the log-hodograph plane. In practice, however, this idea has rarely been put to use except for very simple obstacles, because the conformal mapping problem has been too difficult. This paper presents a practical method for computing flows over arbitrary polygonal obstacles to high accuracy in a few seconds of computer time. We achieve this high speed and flexibility by working with a modified Schwarz-Christoffel integral that maps onto the flow region directly rather than onto the log-hodograph polygon. This integral and its associated parameter problem are treated numerically by methods developed earlier by Trefethen for standard Schwarz-Christoffel maps.

On the radial flow of a viscous fluid between porous disks

A boundary value problem which yields exact solutions of the Navier-Stokes equations for the flow between two infinite, coaxial, and permeable disks is studied here. This problem was introduced in [1], and various conclusions were drawn on the basis of approximations to the solution. In what follows the existence, uniqueness, and asymptotic behavior of a solution are rigorously established. These results are of interest in themselves in providing a complete discussion of an exact solution of the Navier-Stokes equations. On the other hand, the flow problem is closely related to the VON K.~RMAN problem of flow between rotating, coaxial disks, and, if suction or injection is allowed on the disks in that problem, the one studied here may be thought of as the special case in which the disks do not rotate. Many questions remain unanswered for the VON KARM,~N problem ([2], [3] give recent work on existence of a solution), and in particular nothing is known about uniqueness. The boundary value problem dealt with here is much simpler than the one which arises from the rotating disk flow problem, but it is of a similar type, and it may be hoped that these results will shed some light on that problem.

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.

Ideal jet flow in two dimensions

A jet is a stream of one fluid entering another at high speed. In the simplest classical model of jet flow, the geometry is two-dimensional, gravity and viscosity are ignored, the moving fluid is a liquid, and the stationary fluid is a gas whose influence is assumed negligible. The description of this idealized flow can be reduced to a problem of complex analysis, but, except for very simple nozzle geometries, that problem cannot be solved analytically. This paper presents an efficient procedure for solving the jet problem numerically in the case of an arbitrary polygonal nozzle.

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.