An example from turbulence how not to use the invariant function method of Lie-group symmetries

TL;DR

This paper critiques the misuse of the invariant function method of Lie-group symmetries in turbulence modeling, emphasizing that it should not be applied to unclosed systems like the statistical Navier-Stokes equations, and refutes recent claims of solutions.

Contribution

It provides a detailed critique showing the incorrect application of the invariant method in turbulence research and clarifies the limitations of Lie-group symmetries for unclosed systems.

Findings

The invariant solution method is misapplied in turbulence modeling.

Claims of deriving turbulence solutions from first principles are incorrect.

Misapplication leads to wrong conclusions about turbulence behavior.

Abstract

The recent Reply by Oberlack et al. [Phys. Rev. Lett. 130, 069403 (2023)] fails to rebut the critique that a mathematical solution method has been misapplied in their original work. On a point-by-point basis we prove that all arguments put forward in that Reply are incorrect. Therefore, the fact that the invariant solution method of Lie-group symmetries should not be used for unclosed systems in the same way as for closed systems still holds true. Ignoring this fact only leads to wrong conclusions. Claims such as having derived solutions of the statistical Navier-Stokes equations from first principles, or having found a measure for the intermittent behaviour of turbulence or particularly the true scaling in wall-bounded turbulent shear flows, are incorrect.