Growing vortex patches

Abstract

This paper demonstrates that two well-known equilibrium solutions of the Euler equations-the corotating point vortex pair and the Rankine vortex-are connected by a continuous branch of exact solutions. The central idea is to ''grow'' new vortex patches at two stagnation points that exist in the frame of reference of the corotating point vortex pair. This is done by generalizing a mathematical technique for constructing vortex equilibria first presented by Crowdy ͓D. G. Crowdy, ''A class of exact multipolar vortices,'' Phys. Fluids 11, 2556 ͑1999͔͒. The solutions exhibit several interesting features, including the merging of two separate vortex patches via the development of touching cusps. Numerical contour dynamics methods are used to verify the mathematical solutions and reveal them to be robust structures. The general issue of how simple vortex equilibria can be continued continuously to more complicated ones with very different vortical topologies is discussed. The solutions are examples of exact solutions of the Euler equations involving multiple interacting vortex patches.

The corotating hollow vortex pair: steady merger and break-up via a topological singularity

“…Those authors also start with a corotating point-vortex pair and show, by growing two small vortex patches at the two stagnation points of the associated flow field in the corotating frame, that this 2-point-vortex equilibrium is connected to the circular Rankine vortex through the continuous family of non-trivial rotating equilibria involving the steady merger of the two vortex patches which eventually touch. This continuous sequence of equilibria is shown in figure 2 of Crowdy & Marshall (2004).…”

Analytical solutions for rotating vortex arrays involving multiple vortex patches

“…The device of 'growing' vortex patches used here suggests possibilities for the future construction of new vortical equilibria based on combined pointvortex and vortex-patch models. We note here that the present authors Crowdy & Marshall (2004b) have explored the idea of 'growing' new vortex patches in a simpler analytical environment (which does not require the full machinery of automorphic functions used here) and shown that the simple co-rotating point-vortex pair is connected to the simple Rankine vortex solution by a continuous branch of exact solutions. By procedures such as (i) the desingularization of point vortices to uniform vortex patches (or vice versa); (ii) the growing of new point vortices at co-rotation points of existing equilibria as done by Aref & Vainchtein (1998); (iii) the growing of new vortex patches at co-rotation points of existing equilibria as done here; and (iv) the smooth continuation of touching vortex patches to a merged equilibrium as done by Cerretelli & Williamson (2003), it appears that even basic equilibria with simple vorticity distributions can be continously continued to more complicated ones with more elaborate vortical topology.…”

Stress fields around two pores in an elastic body: exact quadrature domain solutions

“…Indeed, Crowdy & Marshall [ 25 ] have shown that the very same class of mappings ( 3.1 ) explored here for these plane elasticity problems also provide analytical solutions of the two-dimensional Euler equations of fluid dynamics describing two rotating vortex patches. They also showed how that work can be extended to the case of any number of co-rotating vortex patches in equilibrium [ 26 ]; to do so, use is made of multiply connected quadrature domains and the function theory based on the Schottky–Klein prime function just described [ 24 , 23 ].…”