User:ZaRaT/sandbox

Bounded Flows
Work by Barenblatt and others has shown that besides the logarithmic law of the wall — the limit for infinite Reynolds numbers — there exist power-law solutions, which are dependent on the Reynolds number. In 1996, Cipra submitted experimental evidence in support of these power-law descriptions. This evidence itself has not been fully accepted by other experts.

In 2001, Oberlack claimed to have derived both the logarithmic law of the wall, as well as power laws, directly from the Reynolds-averaged Navier–Stokes equations, exploiting the symmetries in a Lie group approach. All scaling laws derived in should represent the leading order solutions of a corresponding averaged Navier-Stokes boundary value problem for sufficiently high Reynolds numbers, and hence should all be valid in spatial regimes where the influence of the boundaries can be ignored. This derivation from first principles however lacks mathematical rigor. It is based on several critical assumptions, mainly to justify the use of the pure temporal scaling symmetry of the inviscid Euler equations, a symmetry not admitted by the viscous Navier-Stokes equations, not even to leading order in any strict perturbative sense for small viscosities. The well-known challenging issues related to the vanishing viscosity limit in transiting from the viscous Navier-Stokes to the inviscid Euler equations, particularly for bounded flows, also resides in every Lie group analysis for variable transformations, in that the limit is singularly unstable for symmetry transformations and non-unique for equivalence transformations. These crucial issues were not addressed in the derivation, making this derivation from first principles thus highly questionable. The difference here between a symmetry and an equivalence transformation is that the former defines viscosity as an arbitrary but fixed parameter, while the latter contrarily defines viscosity as an own additional Lie group variable next to all coordinates and flow variables. Eventually an ultimate test for these scaling laws allegedly emerging from first principles would be to show if the proposed Lie group methodology allows for a consistent unified treatment in agreement with experiment or DNS data when higher-order statistical quantities are included, in particular when involving the quantity of dissipation with its anomalous behavior in the vanishing viscosity limit. Otherwise the assumptions made in order to derive these proposed wall scaling laws in distorts the conclusion of dealing with a rigorous derivation from first principles.