Green's Functions at Zero Viscosity
H. M. Fried and J. Tessendorf
Journal of Mathematical Physics, vol 25, 1144-1154 (1984)
Fradkin-type propagator representations are written for solutions to Navier-Stokes and related equations, for arbitrary dimension D and arbitrary source geometry. In the limit of very small viscosity, velocity/vorticity solutions are given in terms of Cauchy position coordinates q of a particle advected by the velocity flow v, using a set of coupled equations for q and v. For localized point vortices in two dimensions, the vectors q become the time-dependent position coordinates of interacting vortices, and our equations reduce to those of the familiar, coupled vortex problem. The formalism is, however, able to discuss three-dimensional vortex motion, discrete or continuous, including the effects of vortex stretching. The mathematical structure of vortex stretching in a D-dimensional fluid without boundaries is conveniently described in terms of an SU(D) representation of these equations. Several simple examples are given in two dimensions, to anchor the method in the context of previoiusly known, exact solutions. In three dimensions, vortex stretching effects are approximated using a previous "strong coupling" technique of particle physics, enabling one to build a crude model of the intermittent growth of enstrophy, which may signal the onset of turbulence. For isotropic turbulence, the possibility of a singularity in the inviscid enstrophy af a finite time is related to the behavior of a single function characterizing the intermittency.