Euler CFD vs Cornerstone of Turbulence Theory

The cornerstone of classical turbulence theory is that the rate of turbulent dissipation $\epsilon$ is independent of the Reynolds $Re =\frac{UL}{\nu}$ once big enough, where $U$ is typical large scale flow speed, $L$ typical large spatial scale and $\nu$ viscosity. The rationale is that turbulent dissipation occurs at the smallest spatial scale independent of the absolute size of the smallest scale. But the size of the turbulent dissipation varies with the large scale features of the flow. 

More precisely, a smallest Kolmogorov spatial scale $dx\sim \nu^{0.75}$ with velocity fluctuation $du\sim \nu^{0.25}$ results from $\frac{dudx}{\nu}\sim 1$ and $\nu(\frac{du}{dx})^2\sim 1$ with thus $du\sim dx^\frac{1}{3}$ expressing Hölder continuity of turbulent flow with exponent $\frac{1}{3}$ as the Onsager conjecture.

Euler CFD fits into this picture with turbulent dissipation from residual stabilisation of the form $C\frac{h}{\vert u\vert }R(u)^2$ with $R(u)$ the Euler equation residual and $h$ the mesh size with $C$ a stabilisation constant. Euler CFD can be seen as a best possible solution of the Euler equations in the balanced sense that $R(u)\sim h^{0.5}$ is small in a weak sense and also $hR(u)\sim h^{0.5}$ in a strong sense, with the same scaling of $du$ and $dx$ as above with $\nu = h$. The resort to best possible reflects that it is impossible to find physical solutions with residual $R(u)$ being small in a strong sense (because all exact solutions are unstable and thus nonphysical). 

Euler CFD shows to deliver mean values quantities such as drag and lift on bluff body flow which are independent of mesh size once small enough and stabilisation parameter.  This gives evidence of turbulent dissipation being independent of effective Reynolds number, that is evidence of the cornerstone of turbulence theory. But Euler CFD also delivers the size of the turbulent dissipation from case to case depending on large scale flow features. Euler CFD thus can be seen as a quantitative parameter-free turbulence model while classical turbulence theory is only qualitative. 

Recall from previous posts that observed independence of drag on Reynolds number once large enough directly connects to independence of turbulent dissipation. 

An overview of the support of the cornerstone is given here. For DNS support see this. 

Recall that the Taylor turbulent length scale $\sim Re^{-0.5}$ as the scale with significant turbulent dissipation, while the Kolmogorov smallest scale $\sim Re^{-0.75}$, which with 3 smallest scale modes per Taylor mode gives $Re^{0.25}\sim 3$, that is $Re\sim 100$ (in accordance with observation) as requirement for turbulence with Taylor microscale $\sim 10^{-1}$ resolved by mesh size $\sim 10^{-2}$. This indicates that Euler CFD with millions of mesh points can deliver drag and lift which do not change under mesh refinement (as observed). It connects to DNS for isotropic turbulence in a cube starting with a $32^3$ mesh with $128^3$ mesh reached by Kent in 1985 with $1024^3$ by Gotoh et al in 2000.

Recall that residual stabilisation focusses dissipation to smallest scales and thus fits into the Kolmogorov picture of constancy of turbulent dissipation for large enough Reynolds numbers.  The scale invariance of turbulence dissipation reflects that the Euler equations are scale invariant.