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SUMMARY:Navier-Stokes is Wrong at Sub-Kolmogorov Scales and Why It Matters
  - Gregory Eyink (Johns Hopkins University)
DTSTART:20220414T150000Z
DTEND:20220414T160000Z
UID:TALK172289@talks.cam.ac.uk
DESCRIPTION:It was pointed out by Robert Betchov in 1957 that scales of tu
 rbulence at and below the&nbsp\;Kolmogorov dissipation length must be stro
 ngly affected by thermal noise\, and his proposal&nbsp\;has recently recei
 ved confirmation from several numerical simulations. But does it matter?&n
 bsp\;\nHere we discuss some consequences\, focussing especially on one exa
 mple: turbulent&nbsp\;high-Schmidt mixing. We argue that the classical pre
 dictions of Batchelor and Kraichnan for the viscous-diffusive range\, alth
 ough verified by numerical simulations of deterministic Navier-Stokes\, ar
 e fundamentally altered by thermal noise. We employ an exact asymptotic&nb
 sp\;method of Donev\, Fai and vanden-Eijnden to account for the thermal no
 ise effects at high-Schmidt&nbsp\;numbers. Making also Kraichnan&rsquo\;s 
 white-noise-in-time approximation for the turbulent velocity&nbsp\;field\,
  we solve analytically for the spectrum of the scalar concentration field.
  Interestingly\,&nbsp\;we find Batchelor&rsquo\;s prediction for the visco
 us-convective range is unaltered\, despite violation of&nbsp\;his basic as
 sumptions. Thermal noise dramatically renormalizes the bare diffusivity in
  this range\,&nbsp\;but the effect is the same as in laminar flow and thus
  hidden phenomenologically. However\,&nbsp\;in the viscous-diffusive range
  at scales below the Batchelor length (typically micron scales) the&nbsp\;
 predictions based on deterministic Navier-Stokes equations are drastically
  altered by thermal noise.&nbsp\;Whereas the classical theories predict ra
 pidly decaying spectra in the viscous-diffusive range\,&nbsp\;we obtain a 
 k^{&minus\;2} power-law starting just below the Batchelor length. This spe
 ctrum corresponds&nbsp\;to non-equilibrium giant concentration fluctuation
 s\, first experimentally observed in quiescent fluids by Vailati & Giglio 
 in 1997. At higher wavenumbers\, the concentration spectrum instead must g
 o to a k^2 equipartition spectrum due to equilibrium molecular fluctuation
 s\, a fact which raises&nbsp\;conceptual questions about how to define &ld
 quo\;macroscopic gradients&rdquo\;. We discuss this issue in the&nbsp\;con
 text of the balance equations for scalar fluctuations and also for fluid k
 inetic energy itself. &nbsp\;\nFinally\, we discuss general implications o
 f these results for the main question of this entire&nbsp\;program: "where
  do we stand on mathematical aspects of turbulence?&rdquo\;
LOCATION:Seminar Room 2\, Newton Institute
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