Equilibrium configurations of point vortices in doubly connected domains

Abstract

Point v ortex ows of a steady, t wo dimensional, inviscid, and incompressible uid are studied for doubly connected geometries. The Routh function is explicitly constructed, and equilibrium con gurations of vortices are found by determining critical points numerically. The numerical computations make use of an analogue of the Schwarz-Christo el transformation for doubly connected regions.

Algorithm 785

“…With the process outlined above, an inverse evaluation may still fail (a message will show so on the screen), but we believe this could happen only when inverse image sought is well merged to one of the prevertices. The reliability of this inversion process has been demonstrated in our applications (see Elcrat et al [1997]). …”

“…Daeppen's code, however, is not available in the public domain. Based on Daeppen's idea, which is an extension of Trefethen's treatment for computing Schwarz-Christoffel mapping for simply connected domains to doubly connected regions, we independently explored many details of implementing (1.1) and developed a robust, portable, well-structured, and well-documented Fortran 77 package, which has been well tested by numerous examples and some nontrivial applications (see Elcrat and Hu [1996], Elcrat et al [1997], andHu [1995]). Compared to the results reported by Daeppen [1988], we achieved, with our code, the better results in terms of accuracy, speed, and being user friendly for the same test problems.…”

Equilibrium vortex configurations in domains with boundary

“…Regarding these statements as guidelines, it is also noted that the term "corner" in (2) and (3) can in practice be extended to include rounded corners as well as strict corners. In each of the specific examples given below nondegeneracy of the minimum can be determined either analytically or numerically.…”

“…When there are no external boundaries such vortices have been studied both analytically and numerically by Saffman and Sheffield [9] when the airfoil is a flat plate and by Huang and Chow [5] in the case of Joukowski airfoils. Some computed examples when the airfoil is in a channel are given in [2].…”

“…8b). A computed example of a flow with a vortex attached to the wall of a channel when the obstacle is a flat plate is given in [2].…”