tisdag 12 februari 2019

Kolmogorov/Onsager: Turbulent Velocity 1/3 Hölder Continuous

Let me here recall the derivation by a scaling argument of the law of Kolmogorov/Onsager stating that fully developed turbulent velocities are Hölder continuous with exponent 1/3.

If $dx$ is smallest scale in space and $du$ the corresponding variation of velocity u, then we have with $\nu >0$ the (small) viscosity:
  • $\nu (du/dx)^2 \sim 1$ (finite rate of turbulent dissipation)
  • $\frac{du\times dx}{\nu}\sim 1 $ (Reynolds number on smallest scale $\sim 1$).
We solve to get $dx\sim \nu^{\frac{3}{4}}$ and $du\sim \nu^{\frac{1}{4}}$ and so $du\sim dx^{\frac{1}{3}}$ showing Hölder continuity 1/3.

The idea is that the flow will by instability develop smaller and smaller structures until the local Reynolds number becomes so small ($\approx 1000$) that this cascade stops on a smallest scale generating the bulk of the turbulent dissipation.

We see that velocity gradients $\frac{du}{dx}\sim \nu^{-\frac{1}{2}}$ are large, since $\nu$ is small, and so velocities are non-smooth.

The official formulation of the Clay Navier-Stokes Prize Problem by Fefferman asks about existence of smooth solutions. By the above argument this question cannot have a positive answer and so the question does not serve well as a Prize Problem.

A pure mathematician may counter this argument by claiming that a velocity with very large gradients still can be smooth, just with very large derivatives. And so even a turbulent solution of the Navier-Stokes equations can be viewed to be smooth, just with very large derivatives, and so asking for existence of smooth solutions in fact can be meaningful and so the Prize Problem in fact is meaningful. I think this means twisting the logic and terminology, which is not in the spirit of meaningful mathematics, pure and applied.

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