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Navier-Stokes is Wrong at Sub-Kolmogorov Scales and Why It Matters

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TUR - Mathematical aspects of turbulence: where do we stand?

It was pointed out by Robert Betchov in 1957 that scales of turbulence at and below the Kolmogorov dissipation length must be strongly affected by thermal noise, and his proposal has recently received confirmation from several numerical simulations. But does it matter?  Here we discuss some consequences, focussing especially on one example: turbulent high-Schmidt mixing. We argue that the classical predictions of Batchelor and Kraichnan for the viscous-diffusive range, although verified by numerical simulations of deterministic Navier-Stokes, are fundamentally altered by thermal noise. We employ an exact asymptotic method of Donev, Fai and vanden-Eijnden to account for the thermal noise effects at high-Schmidt numbers. Making also Kraichnan’s white-noise-in-time approximation for the turbulent velocity field, we solve analytically for the spectrum of the scalar concentration field. Interestingly, we find Batchelor’s prediction for the viscous-convective range is unaltered, despite violation of his basic assumptions. 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 power-law starting just below the Batchelor length. This spectrum corresponds to non-equilibrium giant concentration fluctuations, first experimentally observed in quiescent fluids by Vailati & Giglio in 1997. At higher wavenumbers, the concentration spectrum instead must go to a k2 equipartition spectrum due to equilibrium molecular fluctuations, a fact which raises conceptual questions about how to define “macroscopic gradients”. We discuss this issue in the context of the balance equations for scalar fluctuations and also for fluid kinetic energy itself.   Finally, we discuss general implications of these results for the main question of this entire program: “where do we stand on mathematical aspects of turbulence?”

This talk is part of the Isaac Newton Institute Seminar Series series.

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