Thermal Noise Effects in High-Schmidt Turbulent Mixing*

We study thermal noise effects on the Batchelor-Kraichnan theory of turbulent high-Schmidt mixing in the viscous-diffusion range at sub-Kolmogorov scales. Fluctuating Navier-Stokes equations for a binary fluid at low Mach numbers is linearized around the turbulent velocity. The latter is modelled a la Kraichnan as a Gaussian random field with spatially-constant strain, white-noise in time. We derive high-Schmidt equations for the concentration field in which thermal velocity fluctuations are given exactly also by a white-noise model. Classical predictions in the viscous-convective range are unchanged, but in the viscous-diffusive range, just below the Batchelor length, expected exponential-decay scalar spectra are replaced by k-2 power-law due to non-equilibrium “giant concentration fluctuations''. Such fluctuations are experimentally well observed in quiescent fluids, where imposed concentration variations are advected by thermal velocity fluctuations. At higher wavenumbers, our scalar spectrum goes to k2 equipartition spectrum due to equilibrium scalar fluctuations. We discuss detailed predictions for specific binary mixtures and other possible effects of thermal noise in sub-Kolmogorov scales, including turbulent combustion.