Generalized Helmholtz–Kirchhoff Model for Two-Dimensional Distributed Vortex Motion

Abstract

The two-dimensional Navier–Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz–Kirchhoff model for the evolution of point vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the representation of the flow as a system of interacting Gaussian (i.e., Oseen) vortices, which previous experimental work has shown to be an accurate approximation to many important physical flows [P. Meunier, S. Le Dizes, and T. Leweke, C. R. Phys., 6 (2005), pp. 431–450].