The Turbulent Dissipation Range is Not Described by (Deterministic) Navier-Stokes

Incompressible Navier-Stokes is generally believed to model low-Mach turbulent fluid flows down to almost the mean-free-path, ignoring effects of molecular noise. Estimating this noise using Landau-Lifschitz fluctuating hydrodynamics, we find that the turbulent energy spectrum is modified around the Kolmogorov scale, with expected exponential decay in the far dissipation range replaced by thermal equipartition, in agreement with predictions of Betchov 60 years ago. The far dissipation-range intermittency predicted by Kraichnan is replaced by Gaussian equilibrium statistics. We verify our arguments by simulations of a shell model of turbulence, finding also that inertial-range statistics are unaltered by thermal noise. Our results imply that turbulent processes which involve sub-Kolmogorov scale eddies, such as high-Schmidt mixing, combustion, condensation, etc. are not correctly modelled by deterministic Navier-Stokes. Instead, the Landau-Lifschitz stochastic equations must be employed, requiring fundamentally different numerical algorithms. New theoretical questions arise, because molecular viscosities are renormalized by thermal noise and become dependent upon numerical grid size. Novel experimental methods that can resolve sub-Kolmogorov scales are also urgently required.