For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity vorticity is constant an inner region separated from boundary layer. In this paper, by constructing higher order approximate solutions Navier-Stokes equations establishing validity layer expansion, we give rigorous proof exis...

A class of phenomenological Hopf equations describing mixing of a passive scalar by random flow close to the Batchelor limit ~i.e., advection by random strain and vorticity! is analyzed. In the Batchelor limit multipoint correlators of the scalar are constructed explicitly by exploiting the SL (N ,R) symmetry of the Hopf operator. Hopf equations close to this ‘‘integrable’’ limit are solved via...

We explore very fine scales of scalar dissipation in turbulent mixing, below Kolmogorov and around Batchelor scales, by performing direct numerical simulations at much finer grid resolution than was usually adopted in the past. We consider the resolution in terms of a local fluctuating Batchelor scale and study the effects on the tails of the probability density function and multifractal proper...

The maxima of the scalar dissipation rate in turbulence appear in the form of sheets and correspond to the potentially most intensive scalar mixing events. Their cross section extension determines a locally varying diffusion scale of the mixing process and extends the classical Batchelor picture of one mean diffusion scale. The distribution of the local diffusion scales is analyzed for differen...