The Turbulent 'Mixing' Layer as a Problem in the Non-equilibrium Statistical Mechanics of a Vortex Gas

Abstract

The objective of this paper is to unravel any relations that may exist between turbulent shear flows and statistical mechanics, through a detailed numerical investigation in the simplest case where both can be well defined. The shear flow considered for the purpose is the 2D temporal mixing layer, which is a time-dependent flow that is statistically homogeneous in the streamwise direction (x) and evolves from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain with period L. The connections to statistical mechanics are explored by revisiting, via extensive computer simulations, an appropriate initial value problem for a finite but large collection of (N) point vortices of same strength (\gamma) and sign constituting a 'vortex gas'. Such connections may be expected to be meaningful as hydrodynamics, since the flow associated with the vortex gas is known to provide weak solutions of the Euler equation. Over ten different initial conditions classes are investigated using simulations involving up to 10^4 vortices, with ensemble averages evaluated over up to 10^3 realizations and integration over 10^4 L/(\Delta U) (where \Delta U is the velocity differential across the layer, given by N\gamma/L). (see PDF for complete abstract)