High Schmidt-number turbulent advection and giant concentration fluctuations

Abstract

We consider the effects of thermal noise on the Batchelor-Kraichnan theory of high Schmidt-number mixing in the viscous dissipation range of turbulent flows at sub-Kolmogorov scales. Starting with the nonlinear Landau-Lifshitz fluctuating hydrodynamic equations for a binary fluid mixture at low Mach numbers, we justify linearization around the deterministic Navier-Stokes solution in the dissipation range. For the latter solution we adopt the standard Kraichnan model, a Gaussian random velocity with spatially constant strain but white noise in time. Then, following prior work of Donev, Fai, and Vanden-Eijnden [J. Stat. Mech.: Theory Exp. (2014) P04004], we derive asymptotic high Schmidt limiting equations for the concentration field, in which the thermal velocity fluctuations are exactly represented by a Gaussian random velocity that is likewise white in time. We obtain the exact solution for concentration spectrum in this high Schmidt limiting model, showing that the Batchelor prediction in the viscous-convective range is unaltered. Thermal noise dramatically renormalizes the bare diffusivity in this range, but the effect is the same as in laminar flow and thus hidden phenomenologically. However, in the viscous-diffusive range at scales below the Batchelor length (typically micron scales) the predictions based on deterministic Navier-Stokes equations are drastically altered by thermal noise. Whereas the classical theories predict rapidly decaying spectra in the viscous-diffusive range, either Gaussian or exponential, we obtain a ${k}^{\ensuremath{-}2}$ power-law spectrum over a couple of decades starting just below the Batchelor length. This spectrum corresponds to nonequilibrium giant concentration fluctuations, which are due to the imposed concentration variations being advected by thermal velocity fluctuations and which are experimentally well observed in quiescent fluids. At higher wave numbers, the concentration spectrum instead goes to a ${k}^{2}$ equipartition spectrum due to equilibrium molecular fluctuations. We work out detailed predictions for water-glycerol and water-fluorescein mixtures. Finally, we discuss broad implications for turbulent flows and applications of our methods to experimentally accessible laminar flows.