A critical examination of the statistical symmetries admitted by the Lundgren-Monin-Novikov hierarchy of unconfined turbulence

TL;DR

This paper critically examines proposed new statistical symmetries in turbulence theory, demonstrating they are unphysical and asserting that only classical symmetries are valid within the Lundgren-Monin-Novikov hierarchy, correcting prior flawed claims.

Contribution

It provides a rigorous critique of recent proposed symmetries in turbulence, establishing that no new physical symmetries exist beyond the classical ones in the LMN framework.

Findings

Proposed symmetries violate causality principles.

Existing symmetries are the only physically consistent ones.

Previous claims of new symmetries are flawed and incorrect.

Abstract

We present a critical examination of the recent article by Waclawczyk et al. (2014) which proposes two new statistical symmetries in the classical theory for turbulent hydrodynamic flows. We first show that both symmetries are unphysical in that they induce inconsistencies due to violating the principle of causality. In addition, they must get broken in order to be consistent with all physical constraints naturally arising in the statistical Lundgren-Monin-Novikov (LMN) description of turbulence. As a result, we state that besides the well-known classical symmetries of the LMN equations no new statistical symmetries exist. Yet, aside from this particular issue, the article by Waclawczyk et al. (2014) is flawed in more than one respect, ranging from an incomplete proof, to a self-contradicting statement up to an incorrect claim. All these aspects will be listed, discussed and corrected, thus obtaining a completely opposite conclusion in our study than the article by Waclawczyk et al. (2014) is proposing.