måndag 25 april 2016

Reformulation of Clay Navier-Stokes Problem Needed 3

The official formulation of the Clay Navier-Stokes Problem does not include any reference to the Reynolds number $Re =\frac{UL}{\nu}$ with $U$ a typical flow speed, $L$ length scale, and $\nu$ viscosity scale, and thereby makes no distinction between laminar/smooth flow at small Reynolds numbers and turbulent/non-smooth flow at large Reynolds numbers.

Since no bound on the Reynolds number is given, it can only mean that the Reynolds number can have any size, in particular be arbitrarily large. By normalizing $U$ and $L$ to unity, the viscosity thus can be arbitrarily small (or normalizing  viscosity and length scale to unity and letting $U$ become large), which means that the Clay problem includes the incompressible Euler equations as the incompressible Navier-Stokes equations with vanishingly small viscosity.

This is precisely what the book Computational Turbulent Incompressible Flow is about! As a basic example from the book, let us consider flow around a sphere under vanishing viscosity depicted in the following pictures:

We see a distinct large-scale separation pattern developing consisting of 4 tubes of counter-rotating flow attaching to the rear of of the sphere, which are dissolved into turbulent flow further down-stream. We see that the length of the tubes increases with decreasing viscosity, which is consistent with Kolmogorov's conjecture that the total amount of turbulent dissipation stays roughly constant under vanishing viscosity (along with total drag), requiring the surfaces of intense dissipation of the 4 tube pattern to extend further downstream.

We thus discover a vanishing viscosity solution to the incompressible Euler equations, which is fundamentally different from the formal exact solution in the form of potential flow, which is symmetric in the flow direction with symmetric attachment and separation without the 4tube gross pattern, a formal exact solution which is unstable at separation and thus not a limit of Navier-Stokes solutions.

The official problem formulation does not fit well with this situation, since stability aspects are left out and the presence of vanishing viscosity solutions is hidden, and the Clay Navier-Stokes problem thus asks for a reformulation away from the deadlock of the present (meaningless) formulation.

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