A unified approach for symmetries in plane parallel turbulent shear flows

Published in JFM, 2001 by M. Oberlack.

The key finding and conclusion of this publication is seriously misleading and flawed. Several technical mistakes were made in this study, including one analytical result that cannot even be reproduced. A detailed discussion can be found in https://arxiv.org/abs/1412.3069, and https://arxiv.org/abs/2202.04635 (see Appendix D). They shed the correct light onto the method of Lie-group symmetry analysis when applied to the theory of turbulence. Because, as with any other analytical method, also the powerful and appealing Lie-group symmetry method cannot solve or circumvent the closure problem of turbulence as it is misleadingly and continually claimed by M. Oberlack.

Although a complete list of all mistakes and their detailed discussion can be found in https://arxiv.org/abs/1412.3069, it may be of interest to summarize the key mistake here again: 

The symmetry determining equations (3.12a)-(3.12c) are incorrect in the way that they are too restrictive in not providing the full and complete set of Lie-point symmetries. Hence the misleading result of (3.24), which gives the wrong impression that in the presence of a symmetry-breaking velocity scale within the near wall region (i.e. for $a_1=a_4\neq 0$ and $b_1\neq 0$), only the log-law (3.29) can be obtained as a turbulent scaling law for the mean velocity profile when performing a systematic Lie-group symmetry analysis onto wall-bounded shear flows. Similar the misleading conclusion that if an external symmetry breaking length scale is present, then only an exponential scaling of the form (3.31) is possible, and if no external length or velocity scale will be present then only an algebraic scaling law of the form (3.27) will be visible in the flow. However, no such unique or privileged results are obtained when applying the Lie-group symmetry approach to wall-bounded shear flows correctly. Instead, a rigorous derivation reveals complete arbitrariness rather than uniqueness in the construction of invariant turbulent scaling laws. This just reflects the closure problem of turbulence, which also by the method of Lie-groups cannot be solved or bypassed: As the statistical equations are unclosed, so are their symmetries; and the problem persists even if the infinite hierarchy of statistical one- or multi-point correlations is formally considered.