This book is based on conclusions from several flawed publications on Lie-group symmetry analysis. These are:

1. "Symmetries and invariant solutions of turbulent flows and their implications for turbulence modelling", in Oberlack, M., Busse, F.H.: Theories of Turbulence. Springer, New York (2002)
2. "Symmetrie, Invarianz und Selbstähnlichkeit in der Turbulenz", Oberlack, M.: Shaker Verlag, Aachen (2000)
3. "Symmetries and their importance for statistical turbulence theory", Oberlack, M., Waclawczyk, M., Rosteck, A., Avsarkisov, V.: Bull. JSME 2, 1-72 (2015)
4. "Lie symmetry analysis of the Hopf functional-differential equation", Janocha, D.D., Waclawcyk, M., Oberlack, M.: Symmetry 7, 1536–1566 (2015)

All four publications above, which are cited and relied on by Kollmann several times in Secs.2,9 & Appx.F, are seriously in error, not only technically but also methodologically.

In the following we will briefly go through Kollmann's incorrect and misleading quotes, which will then be corrected.

(a) "Detailed accounts and critical analysis of the various closure assumptions and a wealth of applications can be found in the literature, for instance, ... Oberlack [30] (symmetry analysis and its application to turbulence modelling), ... are excellent sources." [p.7]

Correction: Key results on turbulent scaling laws in Oberlack [30] cannot be reproduced. Oberlack [30], which partly is an identical copy of the flawed JFM publication "A unified approach for symmetries in plane parallel turbulent shear flows", Oberlack, M., JFM 427, 299-328 (2001), makes several wrong claims, for example, the most prominent one is that for plane parallel turbulent shear flow the log-law can be uniquely derived by solely using only a Lie-group symmetry analysis (along with the single assumption that the friction velocity is symmetry breaking). A correct and thorough symmetry investigation, however, shows that in reality the opposite is true: The Lie-group method itself cannot place any restrictions on solution functions for turbulence scaling. The reason is that the corresponding statistical equations are unclosed and therefore also their set of possible symmetries. Ultimately, every conceivable symmetry can be generated from the unclosed equations, and thus also every conceivable invariant scaling law. In other words, the closure problem of turbulence cannot be bypassed just by making sole use of the Lie-group method, as wrongly and thus misleadingly claimed by Oberlack. In fact, the main problem in Oberlack [30] is that the Lie-group symmetry method was not correctly and fully applied therein.

For more details on this issue, please see: https://arxiv.org/abs/1412.3069

(b) "Among all possible transformations, the class of transformations that do not change the form of the equations play a special role; they are called symmetry transformations or simply symmetries of the system, Frisch [16], Sect. 2.2, Oberlack [17], Oberlack et al. [18]." [p.22]

Correction: In this context, only the Fisch [16] citation is correct, since Frisch discusses symmetries (and its corresponding statistical averages, see e.g. Secs.6,8) only in connection with the closed instantaneous Navier-Stokes equations and not detached from it, as Oberlack does it, by always considering only the statistically induced unclosed equations. In fact, the invariant statistical transformations obtained by Oberlack are only equivalence and not true symmetry transformations, simply because his analysis is permanently centered around an unclosed system. In contrast to a true symmetry transformation, which maps a solution of a specific (closed) equation to a new solution of the same equation, an equivalence transform acts in a weaker sense in that it only maps an (unclosed) equation to a new (unclosed) equation of the same class. Hence, even for all the possible infinite invariant transformations that still can be obtained in Oberlack [17,18] when only applying a complete and correct Lie-group invariance analysis to it, we cannot expect any information about the solution structure of the unclosed statistical equations as long as they are not modelled. Hence, without modelling these unclosed equations, an a priori prediction as how turbulence scales is and will not be possible. Only a posteriori, as a so-called curve-fitting method, if the data is already available, the Lie-group symmetry method can sometimes be helpful to approximately determine the desired invariant scaling of the underlying data via an iterative trial-and-error procedure.

To note is that since the unclosed statistical equations involve complete arbitrariness in choosing a particular invariant transformation from an infinite set, one has to be aware of the problem that in this process also non-physical invariant transformations might get chosen, which then, obviously, cannot be matched to any given data anymore. Hence, it is important to have a physical guideline as how to make a physically sound choice. One such guideline is the classical principle of cause and effect, namely that any statistical invariance must have a cause in the underlying instantaneous Navier-Stokes equations, where the cause itself, however, need not to be an invariant. This robust guideline is permanently ignored by Oberlack et al., with the result that in each of their publications since 2010 non-physical invariant scaling laws get constantly proposed. See e.g. their latest correction, which had to be forced by the journal: https://doi.org/10.1017/jfm.2019.985.

For more details on this combined issue, please see:

https://arxiv.org/abs/1412.3061

https://arxiv.org/abs/1602.08039

https://arxiv.org/abs/1710.00669

https://hal.archives-ouvertes.fr/hal-01888353v2

The following articles also refute Oberlack et al. [18] directly:

https://doi.org/10.1103/PhysRevE.92.067001,
and its supplement: https://doi.org/10.13140/rg.2.1.1238.2803

https://doi.org/10.1063/1.4940357,
and its supplement: https://doi.org/10.13140/rg.2.1.2780.0721/1

https://arxiv.org/abs/1606.08396

https://doi.org/10.13140/rg.2.2.35698.76480

(c) "Janocha et al. [18] started the symmetry analysis of the Hopf fde with the aid of Lie transformation group theory." [p.137], or "Janocha et al. [9] contain a detailed analysis of the transformation groups of the fde obtained above for the pure IVP of the Burgers pde." [p.611]

Correction: Janocha et al. is methodologically and technically so seriously flawed that nowhere throughout that study their findings and conclusions can be relied upon. Their "new" method simply does not capture the invariance property of functional-differential equations correctly. As a result, symmetries get generated therein which in reality do not exist. See e.g. their partial correction in https://doi.org/10.1007/978-3-319-29130-7_2, which refers to the overall correction: https://doi.org/10.3390/sym8040023, and its supplement: https://doi.org/10.13140/rg.2.1.4794.4081

In this regard, please also see:

https://zenodo.org/record/1204432

https://zenodo.org/record/1204426

(d) "Oberlack et al. [18] present a detailed explanation of the Lie group approach for systems of differential equations, which is the source for the following development for incompressible fluids and the spatial description." [p.23]

Correction: To introduce and to comprehend the method of Lie-group symmetry analysis, there are endless far better books and articles out there that can be cited than the seriously flawed and misleading paper [18] by Oberlack et al., which, except for the introductory Secs.2.1-2.6, cannot be relied upon, in particular not their "new" symmetry method of Lie-groups for functional differential equations, starting from Sec.2.7 onwards through to the very end of the article.

For details, see again:

https://doi.org/10.3390/sym8040023,
and its supplement: https://doi.org/10.13140/rg.2.1.4794.4081

https://doi.org/10.13140/rg.2.2.35698.76480

Overall Conclusion: This book should be red-flagged since it relies in certain sections on studies that are clearly in error.