High Schmidt-Number Turbulent Advection and Giant Concentration Fluctuations

TL;DR

This paper analyzes how thermal noise influences high Schmidt-number turbulent mixing, revealing significant alterations in concentration spectra at small scales and providing exact solutions within a stochastic turbulence model.

Contribution

It introduces a stochastic model incorporating thermal noise into the Batchelor-Kraichnan theory, deriving exact solutions and revealing new phenomena in concentration fluctuations at microscopic scales.

Findings

Thermal noise renormalizes diffusivity in the viscous-convective range.

Giant concentration fluctuations with a $k^{-2}$ spectrum emerge below the Batchelor length.

At high wavenumbers, the spectrum transitions to a $k^2$ equipartition spectrum.

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. Using fluctuating hydrodynamics 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 and derive asymptotic high-Schmidt limiting equations for the concentration field, in which the thermal velocity fluctuations are exactly represented by a Gaussian random velocity which is white in time. We obtain the exact solution for the 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, we obtain a $k^{-2}$ power-law spectrum starting just below the Batchelor length. This spectrum corresponds to non-equilibrium giant concentration fluctuations, due to the imposed concentration variations advected by thermal velocity fluctuations which are experimentally well-observed in quiescent fluids. At higher wavenumbers, 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 novel applications of our methods to experimentally accessible laminar flows.