Turbulence & Symmetries

Seriously Flawed, including Mathematical Errors & Conclusions that do not match Results The following JFM publication by Oberlack (2001) is seriously flawed: A unified approach for symmetries in plane parallel turbulent shear flows https://doi.org/10.1017/S0022112000002408 Details and corrections are given in the Comment  https://arxiv.org/abs/1412.3069 Note that JFM has the obscure policy not to publish any Comments on any of their articles.

https://doi.org/10.5281/zenodo.1101197 A detailed theoretical investigation is given which demonstrates that a recently proposed set of statistical symmetries is physically void. Although they are mathematically admitted as unique symmetry transformations by the underlying statistical Navier-Stokes equations up to the functional level of the Hopf equation, by closer inspection, however, they lead to physical inconsistencies and erroneous conclusions in the theory of turbulence. These new statistical symmetries are thus misleading in so far as they form within an unmodelled theory a set of analytical results which at the same time lacks physical consistency. Our investigation will expose this inconsistency on different levels of statistical description, where on each level we will gain new insights for their non-physical transformation behavior. With a view to generate invariant turbulent scaling laws, the consequences will be finally discussed when trying to analytically exploit su...

https://arxiv.org/abs/1412.3069 Essence of this Comment: Since the statistical equations in turbulence are unclosed, so are their symmetries. Hence, when performing any invariant analysis on such an unclosed system of equations, it will predominately result into a non-unique set of invariant solutions involving arbitrary functions for which from the outset it is not clear how to specify them.  The closure problem of turbulence cannot be circumvented, or “somewhat bypassed” in the sense of Oberlack  ➽ , by just employing the method of Lie-group invariant analysis alone, since the arbitrariness in the results just gets shifted from one function to another. Hence, without modelling these unclosed equations, an a priori prediction from Lie-group theory alone as how turbulence scales is and will not be possible. Only a posteriori, by anticipating what to expect from numerical or experimental data, the adequate invariant scalings can be generated through an iterative tri...

https://doi.org/10.5281/zenodo.1116165 The above link is a collection of examinations showing that the closure problem of turbulence cannot be "somewhat bypassed" by formally considering the infinite set of correlation equations and its admitted set of symmetries, as misleadingly claimed by the group of Oberlack et al.  ➽

https://doi.org/10.5281/zenodo.1101170 The above link is a collection of comments continuing a central comment by in Symmetry (2016):  Comments on Janocha et al.: Lie Symmetry Analysis of the Hopf Functional-Differential Equation

https://doi.org/10.5281/zenodo.1101136 The above link is a collection of comments continuing a central comment in J.Math.Phys. (2016):  Comment on “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation”

https://doi.org/10.5281/zenodo.1098507 The above link is a collection of comments continuing a central comment in Phys.Rev.E (2015):  Comment on “Statistical symmetries of the Lundgren-Monin-Novikov hierarchy” The latest comments on this issue are given in  https://arxiv.org/abs/1710.00669 https://www.researchgate.net/publication/311285232