Symmetry analysis and invariant solutions of the multipoint infinite systems describing turbulence

Published in Journal of Physics, 2016 by Waclawczyk & Oberlack.

This publication is seriously flawed, both methodologically as well as technically. For all details, please see https://www.researchgate.net/publication/311285232.

Nevertheless, it is worth to briefly list and summarize the most important errors of this publication in the order as they appear in the text:

1. The two "new" additional symmetries (5) and (6) not only violate the separation constraint of the LMN equations, but also the causality principle of classical mechanics. For more details on this particular issue, please see https://doi.org/10.1103/PhysRevE.92.067001 and https://arxiv.org/abs/1602.08039.

2. The mentioned modification on p.3 "due to the presence of boundaries" breaks the symmetry (6) not only for the unbounded, but also of the bounded LMN equations for channel flow. The proof to this statement can be found in https://arxiv.org/abs/1412.6949 and https://www.researchgate.net/publication/287996194.

3. The transformation given above equation (7) is not a Lie-group. Hence the conclusion to (7) is not correct.

4. Function (9) is inconsistent in two aspects: (i) It is not invariant under the finite transformation given above (7). The problem here is that of point (3) just stated before, in that this transformation has been wrongly assumed by Oberlack et al. as a Lie-group having an infinitesimal representation. (ii) If the correct invariant transformation for this function is derived, it cannot be related anymore to the LMN symmetries.

5. On p.4 it is stated that $\gamma\approx 0.98$ is a reasonable result for $Re_\tau=150$. However, when their proposed "scaling law" as given in (10) is matched to DNS data for this Reynolds  number case, it shows that $\gamma\approx 0.98$ is way too high in order to be a reasonable result. In fact, throughout all low Reynolds number cases, the universal "scaling law" (10) offers a very poor predictive ability to forecast the complex, non-universal flow behaviour in a turbulent channel flow "with Re close to its critical value where the laminar-turbulent transition takes place" (p.3).

6. The functional solution $\Phi=1+C_1+C_2+\cdots$ on pp.6-7, with as given in (33) is not compatible to the Navier-Stokes-Hopf equation (14), if $F_{ij}\neq 0$. Compatibility is only obtained if $F_{ij}=0$, but this then leads to a zero correlation (34).

7. As it stands, result (34) is incorrect. It misses the complementary part proportional to $1/t^2$.

8. The result after (34) cannot be confirmed. A correct derivation compatible with the Navier-Stokes-Hopf equation reveals that this result must be zero when enforcing the assumptions of Oberlack et al.

9. The conclusions given on p.7 and 8 are not justified by the results when everything would be done correctly.