Analytical solutions for rotating vortex arrays involving multiple vortex patches

Abstract

A continous two-parameter family of analytical solutions to the Euler equations are presented representing a class of steadily rotating vortex arrays involving N + 1 interacting vortex patches where N > 3 is an integer. The solutions consist of a central vortex patch surrounded by an N-fold symmetric alternating array of satellite point vortices and vortex patches. One of the parameters governs the size of the central patch, the other governs the size of the N satellite patches. In the limit where the areas of the satellite vortex patches tend to zero, the solutions degenerate to the exact solutions of Crowdy (J. Fluid Mech. vol. 469, 2002, p. 209). Limiting states are found in which cusps form only on the central patch, only on the satellite patches, or simultaneously on both central and satellite patches. Contour dynamics simulations are used to check the mathematical solutions and test their robustness. The linear stability of a class of 'point-vortex models' (in which the patches are replaced by point vortices) are also studied in order to examine the stability of the distributed-vorticity configurations to pure-displacement modes. On the other hand, a desingularization of all point vortices to Rankine vortices leads to a class of 'quasi-equilibria' consisting purely of interacting vortex patches close to hydrodynamic equilibrium.

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

“…3) tends to a probability measure µ V,t where t is some fixed positive number (quantum area). It turns out that µ V,t is the unique solution of a two-dimensional electrostatic equilibrium problem in the presence of the external potential V in the complex plane.…”

Initial stages of the interaction between uniform and pointwise vortices in an inviscid fluid

“…The approximate solutions at orders 0 (14) and 1 (21) and (22), or (23) and (24), are now compared with reference numerical solutions obtained by integrating the problem (12). However, before discussing this comparison, it is worthwhile to recall their mathematical and physical meanings.…”

“…Growing uniform vortices are inserted in a corotating vortex pair, until the Rankine vortex is reached. Other equilibria involving uniform and point vortices are found in [12], still starting from the streamfunction (1). A central uniform vortex is surrounded by an alternate distribution of pointwise and uniform vortices.…”

Family of propagating vortex-wave equilibria for the two-dimensional Euler equations

“…[7][8][9][10] Recently, Crowdy has been successful in using Schwarz functions to construct exact forms for various vortex equilibria. [11][12][13][14] Typically, these equilibria involve patches of constant vorticity accompanied by vortex singularities usually in the form of point vortices. The construction method relies on conformal mapping in the complex z plane, which gives an explicit formula for both the curve defining the patch ͑vortical͒ boundary and its Schwarz function S͑z͒.…”