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

Abstract

We present a critical examination of the recent article by Wac lawczyk 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 Wac lawczyk 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 Wac lawczyk et al. (2014) is proposing.

A note on the notion "statistical symmetry"

“…for those new statistical symmetries can be constructed, they obviously violate the classical principle of causality as we fully elaborated in our Comment (Frewer et al, 2015(Frewer et al, , 2014a. This unphysical behavior can also be clearly seen when comparing to DNS data (Simens et al, 2009;Borrell et al, 2013); a detailed summary of this analysis is given e.g.…”

“…( 16) in Wac lawczyk et al (2014)), it is straightforward to show that it's already by definition impossible to construct any kind of variable transformation for the instantaneous (fluctuating) velocity field such that it can induce this symmetry on the mean (averaged) level, where all velocity correlation functions H {n} are just multiplied by the same global factor e as (see the proof e.g. in Appendix A of Frewer et al (2014a)). The same issue we face for the coarse-grained multi-point PDFs (Eq.…”

“…But since all new statistical symmetries of Wac lawczyk et al (2014), which were first determined and introduced in Oberlack & Rosteck (2010), are of a globally trivial form, it is analytically easy to show that no variable transformation of any kind on the fine-grained level in the sense of Q can be constructed at all. Every approach to construct such a transformation leads to a clear contradiction, as it was independently shown not only for the Lundgren-Monin-Novikov equations (see Frewer et al (2015Frewer et al ( , 2014a), but also for the Hopf equation (see Section 4.1 in Frewer et al (2014b)), as well as for the multi-point correlation equations (see Sections 4.2 & 4.3 in Frewer et al (2014b)). In particular for the new statistical scaling symmetry of the multi-point equations (Eq.…”

An example elucidating the mathematical situation in the statistical non-uniqueness problem of turbulence

“…For each order n in the hierarchy the total number of dynamical equations (2.13), along with the continuity constraint equations (2.17), is therefore always less than the total number of unknown functions. In total this just reflects the classical closure problem of turbulence for the moments which cannot be bypassed by simply establishing the formal connection (2.15) -for a more detailed discussion on this issue, please refer to Appendix A & C in Frewer et al (2014a) and Section 3 in Frewer et al (2014b).…”

“…For more details, see e.g. Section 2 in Frewer et al (2014a), Sections 3-5 in Frewer et al (2014b), and the references therein.…”