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Is the log-law a first principle result from Lie-group invariance analysis?

The invariance method of Lie-groups in the theory of turbulence carries the high expectation of being a first principle method for generating statistical scaling laws. The purpose of this comment is to show that this expectation has not been met so far. In particular for wall-bounded turbulent flows, the prospects for success are not promising in view of the facts we will present herein. Although the invariance method of Lie-groups is able to generate statistical scaling laws for wall-bounded turbulent flows, like the log-law for example, these invariant results yet not only fail to fulfil the basic requirements for a first principle result, but also are strongly misleading. The reason is that not the functional structure of the log-law itself is misleading, but that its invariant Lie-group based derivation yielding this function is what is misleading. By revisiting the study of Oberlack (2001) we will demonstrate that all Lie-group generated scaling laws derived therein do not convince as first principle solutions. Instead, a rigorous derivation reveals complete arbitrariness rather than uniqueness in the construction of invariant turbulent scaling laws. Important to note here is that the key results obtained in Oberlack (2001) are based on several technical errors, which all will be revealed, discussed and corrected. The reason and motivation why we put our focus solely on Oberlack (2001) is that it still marks the core study and central reference point when applying the method of Lie-groups to turbulence theory. Hence it is necessary to shed the correct light onto that study. Nevertheless, even if the method of Lie-groups in its full extent is applied and interpreted correctly, strong natural limits of this method within the theory of turbulence exist, which, as will be finally discussed, constitute insurmountable obstacles in the progress of achieving a significant breakthrough.


Introduction
Predicting the mean temporal evolution and spatial structure of turbulent fluid flow still remains the Holy Grail of systematic turbulence research for more than a century now. Statistical scaling laws play a key role in this context. To date, aside from several inequalities (Constantin & Fefferman, 1994;Constantin et al., 1999), the only exact statistical scaling equality that has been rigorously derived from first principles, i.e. by only making sole use of the Navier-Stokes equations without any major assumptions or approximations, is, so far, Kolmogorov's 4/5-law for the third-order longitudinal structure function of the velocity field (Kolmogorov, 1941c;Frisch, 1995;Davidson, 2004). However its exact validity is only restricted to the asymptotic regime of infinite Reynolds number within the inertial range of an idealized unbounded flow, that of statistically homogeneous isotropic flow.
For wall-bounded flows, which inherently give raise to turbulent boundary layers, the situation is even more worse: All attempts to derive scaling laws from first principles were doomed to failure. Every derivation done so far necessarily involves such strong assumptions and approximations that the initially attempted linkage to the Navier-Stokes equations with its associated geometrical boundary conditions is lost. However, when withdrawing the claim of dealing with first principle derivations a vast functional range of different scaling laws exist.
Unfortunately this lack of mathematical rigor split the turbulence community into different schools each favoring their own beliefs in never-ending debates on holding the correct scaling behavior within fully developed turbulent boundary layers. In particular when regarding the issue of the functional form for the mean velocity distribution in the overlap region between the inner (near-wall) and outer regions. Even in the asymptotic limit of large Reynolds numbers, theoretical analysis does not predict a unique statistical law as how the mean velocity profile should scale within this thin layer from the wall. Two contrasting laws coexist in the literature: One is the classical log-law where its coefficients are believed to be universal, i.e. independent of the Reynolds number and first deduced by von Kármán (1930), while the second one is a power-law, where, opposed to universality, its coefficients are believed to depend weakly on the Reynolds number (Barenblatt, 1993;Barenblatt & Prostokishin, 1993;Cipra, 1996;George & Castillo, 1997;George, 2007;Barenblatt et al., 2014). Usually the log-law has been emphasized over the power-law, sometimes to the exclusion of the latter (Zagarola et al., 1997;Osterlund et al., 2000;Lindgren et al., 2004;Jiménez, 2012Jiménez, , 2013. However in recent years it shows that maybe also the opposite is a reasonable possibility, but nevertheless, the question which scaling law ultimately applies is definitely not resolved yet (Dallas et al., 2009;Marusic et al., 2010;Gad-el-Hak & Buschmann, 2011).
This situation provokes a different perspective in deriving and interpreting statistical scaling laws for high-Reynolds-number turbulent boundary layer flows. Due to an overall lack of mathematical rigor the still-ongoing debate would greatly benefit from a twofold change in attitude: i) From an attitude which would regard all scaling laws as only being approximative models to the exact but unknown Navier-Stokes scaling behavior. Going with the insight of George Box that essentially "all models are wrong, but some are useful" (Box & Draper, 1987), the adequate question to ask is: How wrong do statistical scaling laws have to be to not be useful?
ii) From an attitude which shares the position of Paul Feyerabend in that "all methodologies, even the most obvious ones, have their limits" (Feyerabend, 1975). Nevertheless, in extending or in using new methodologies to construct a theoretical framework for generating turbulent scaling laws "anything goes" (Feyerabend, 1975), however only under the strong restriction, of course, that mathematical consistency is always ensured.
Exactly in this context we want to place our comment regarding the method of Lie-group invariance analysis when applied to the theory of turbulence. Although this method carries the high expectation of being a first principle method for generating turbulent scaling laws, we will show that it faces the same analytical problems as any other method when trying to generate such laws a priori. In other words, within the theory of turbulent flows the invariant method of Lie-groups needs to be reduced from its high expectation level of being a first principle method down to a method, which also, like any other analytical method in turbulence theory, is only capable to generate turbulent scaling laws a posteriori, i.e. on basis of physical experience by making use of assumptions and approximations. In this sense the invariant method of Lie-groups has the immanent property, too, to generate next to useful also non-useful turbulent scaling laws, which, if no physical or numerical experiment can be matched to them, need to be discarded.
The aim of our comment is twofold. At first to draw attention to the fact that the first-principle approach in Oberlack (2001), namely to generate statistical scaling laws for wall-bounded turbulent flows based on the "maximum symmetry principle", is heavily misleading. Moreover, our analysis will show that the results as they are derived in Oberlack (2001) cannot be reproduced. In one of the two necessary steps to generate the Lie symmetries in Oberlack (2001) we obtain a different intermediate result, leading thus to a diverse and more general symmetry in particular for the mean velocity field. Secondly, to clarify that although the methodology of Lie-group symmetry analysis (Ovsiannikov, 1982;Stephani, 1989;Fushchich et al., 1993;Olver, 1993;Ibragimov, 1994;Bluman & Kumei, 1996;Hydon, 2000;Cantwell, 2002) is a powerful and indispensable tool to generate non-trivial solutions for nonlinear differential equations, it nevertheless brings along strong natural limits when applied to the theory of turbulence in general. These limits need all to be recognized in order to correctly assess the feasibility of this method in future work.
This study is organized as follows: Section 2 revisits in detail the results given in Oberlack (2001). We will show that all statistical scaling laws derived in Oberlack (2001) are based on an incorrectly calculated invariance transformation, which misleadingly lead, for example, to the log-law as a specific result for the mean streamwise velocity profile. In contrast to these published and as we believe incorrect results, our newly recalculated and cross-validated invariance transformation does not allow for such a specification. Rather, it results in complete arbitrariness in the scaling behavior for the mean streamwise velocity profile, thus showing that the method of Lie-groups is unable to generate a functionally unique set of turbulent scaling laws from first principles. Section 3 then concludes by generally addressing the limits of the Lie-group methodology in turbulence research, in particular when generating statistical multi-point scaling laws beyond the one-point case. Next to the limits as statistical unclosedness and the presence of boundary conditions, the limits are further driven by the property of intermittency. The issue is that all multi-point functions are highly sensitive to intermittent effects, while a Lie-group based invariance analysis performed upon them, which standardly (up to some exceptions) only returns a finite dimensional Lie-group, is effectively unable to account for these effects. Yet, it should be clear that we do not criticize the method of Lie-groups itself, being a very useful mathematical tool indeed, when only applied to the right problems.

Phantom turbulent scaling laws as first principle solutions
As a representative example for a wall-bounded turbulent scaling law, we will for brevity herein only focus on the invariant Lie-group based derivation of the log-law for the mean streamwise velocity profile in the inertial region. In the conclusion of this section, however, we will briefly compare this log-law derivation also to its corresponding and equally valid power-law derivation.
At this point we already would like to emphasize that it is not the log-law itself to be criticized here, which, as we know, undoubtedly acts as a useful scaling law in the inertial region, but only its particular derivation and interpretation originating from the invariance method of Lie-groups. All the more so, as the claim exists of dealing here with a derivation from first principles (Oberlack, 2001;Oberlack & Rosteck, 2010).
Currently there are two different and independent Lie-group based derivations existing in the literature to generate the log-law. The first and older derivation is based on the deterministic fluctuating Navier-Stokes equations (Oberlack, 2001), while the second and more recent derivation is based on the infinite statistical hierarchy of the multi-point velocity correlation functions (Oberlack & Rosteck, 2010). Although both derivations claim to generate the loglaw a priori from first principles, i.e. directly from the Navier-Stokes equations without any major assumptions and approximations, it is only the former derivation in Oberlack (2001) that possibly can claim this.
The reason is that the derivation in Oberlack & Rosteck (2010) is based on a statistical description in which the system of equations is obviously not closed † , that is, the invariance analysis in Oberlack & Rosteck (2010) is based on a system with more unknown functions than equations, while in Oberlack (2001) the corresponding invariance analysis at first sight is presumably based on an opposite system having more equations than unknowns functions, due to augmenting the unclosed fluctuating equations by the so-called "velocity product equations" (eq. (2.12), p. 302).
At first sight it thus seems that Oberlack (2001) has a decisive advantage over Oberlack & Rosteck (2010) when taking the perspective of an invariance analysis into which the "velocity product equations" are incorporated. Because, on the level of the fluctuating Navier-Stokes equations it seems that with this approach a natural invariant closure constraint has been achieved, since "the purpose of (2.12) [the velocity product equations] regarding the symmetry properties of plane shear flows is quite different" (p. 302) and "that (2.12) is crucial to find self-similar mean velocity profiles consistent with the second moment and all higher-order correlation equations" (p. 307), where "the major difference between the classical turbulence modelling approach and the present procedure is the treatment of equation (2.12)" (p. 309).
However, these statements are misleading. As we will demonstrate, this additional set of "velocity product equations" are completely redundant, not only from the perspective of the fluctuating equations themselves, but also from the perspective of any invariance analysis performed upon them. The immediate consequence is that, instead of the log-law, a completely indifferent result is obtained, being completely useless to gain any mathematical insight into the statistical solution manifold of the underlying dynamical process. Since this is also the case for all other velocity moments, it simply shows that for wall-bounded turbulent flows a first principle result based on the invariance method of Lie-groups is yet still missing. The obvious reason is that the closure problem of turbulence cannot be bypassed, also then not when utilizing the analytical invariance method of Lie-groups. Oberlack (2001) If we restrict to non-rotating flows, the transport equations which were analyzed for invariance in Oberlack (2001) are

Revisiting the results derived in
denoting the continuity (C ), the momentum (N i ) and the above-discussed "velocity product" (P ij ) equations for the fluctuating fields of the Reynolds decomposed Navier-Stokes equations. All overbared quantities denote the corresponding averaged fields. As the analysis is restricted to stationary parallel shear flows, the above set of fluctuating transport equations are augmented by the defining restrictions for the mean fields Note that in Oberlack (2001) (from eq. (2.7a-b), p. 301 onwards) not the original mean pressure fieldp =p(x 1 , x 2 ) is used, but instead the effective (lower dimensional) mean pressure field p * =p * (x 2 ) is used by decomposing the original field asp = −K · x 1 +p * , where K is the mean constant pressure gradient to drive the flow in the streamwise x 1 -direction. In order to allow for a possible Reynolds number dependence in the scaling law coefficients, the search in Oberlack (2001) to determine all invariant point-transformations for system (2.1)-(2.4) was hence extended to the class of equivalence transformations, in which the viscosity ν ∼ 1/Re is not treated as a parameter, but, next to the space-time coordinates, as an own independent variable.
The corresponding infinitesimal invariance operator (Stephani, 1989;Olver, 1993;Ibragimov, 1994;Bluman & Kumei, 1996;Hydon, 2000;Cantwell, 2002) to generate these Lie-point equivalence transformations for the complete system (2.1)-(2.4) is thus given by the following scalar structure where Y is the prolongation in the infinitesimals for all derivatives appearing in system (2.1)-(2.3) including the constraints (2.4): The prolongation Y appears due to the already predetermined transformation rules for all relevant derivatives, as e.g. for u ′ i,t : , given by the yet unknown and still to be determined transformation rules for the corresponding dependent variables with respect to their independent variables. In other words, all components of Y in (2.6) can be explicitly expressed as functions of both the unknown infinitesimals for the independent variables ξ t , ξ x i , ξ ν , and dependent variables η u ′ i , η p ′ , ηū 1 , ηp * . The general and explicit formula for a prolongation of any differential order in terms of its independent and dependent infinitesimals can be found, for example, in the above cited literature. Now, according to Oberlack (2001), the desired equivalence transformations X (2.5) for the considered system (2.1)-(2.4) are determined by solving in respect to the constraints (2.4) the following corresponding invariance conditions (eqs. (3.12a-c), p. 305) In order to explicitly yield these invariant transformations, the calculation in Oberlack (2001) was performed in two steps. In the first step, by solving according to the flow assumptions (2.4) the invariance conditions (2.7) and (2.8), the resulting general equivalence transformation of subsystem (2.1)-(2.2) augmented by (2.4) got calculated in the infinitesimal form as (eq. (3.14), p. 307): This result of this first step can be obtained and confirmed e.g. by using the computer algebra system (CAS) based symmetry-software-packages GeM (Cheviakov, 2007), SADE (Filho & Figueiredo, 2011), or the latest DESOLV-II package (Vu et al., 2012). However, it should be noted that in first instance the latter DESOLV-II package of Vu et al. (2012) is not giving the invariance (2.10) in its general form. Instead it restricts the result (2.10) to a 2 (ν) = 0, g 1 (x 2 ,ū 1 ,p * , ν) = F (x 2 , ν). (2.11) In the following we will, for reasons of simplicity, only consider this special restriction (2.11), which, as a consequence, will only simplify but not change the general statement of this study. Hence, irrespective of this choice, whether to restrict (2.10) by (2.11) or not, the first step in Oberlack (2001) provides a reproducible result.
In clear contrast now to the second step, where the remaining invariance condition (2.9) for the system of the "velocity product equations" (2.3) got incorrectly recognized in Oberlack (2001) as a symmetry breaking condition within the already analyzed subsystem (2.7)-(2.8). This lead (amongst others) to the following wrong key restriction (eq. (3.18), p. 308): (2.12) However, our recalculation of this second step only shows complete redundancy in the invariant condition for the "velocity product equations" (2.3), rather than a symmetry-breaking mechanism in the invariances admitted by (2.1)-(2.2). That is to say, reduction (2.12) is not supported by our recalculation (see second part of Appendix A, and entire Appendix B). The essential error in Oberlack (2001) not to recognize this redundancy already resides in the idea to consider the invariance conditions of system (2.1)-(2.3) in the uncoupled infinitesimal form (2.7)-(2.9). This allowed to split the invariance analysis into two separate and independent steps. As was shown above, (2.7) and (2.8) were solved in a first step, followed by solving (2.9) in a second step using the results then from the first step. This reasoning, however, is misleading in that it's not complete. Since (2.1)-(2.3) represents a strongly coupled system, the corresponding invariance conditions may not be decoupled as expressed in (2.7)-(2.9), but must rather be solved as a coupled, or more precisely, as a combined functional vector, namely as XT n | Tn=0 = 0, with T n := (C , N i , P ij ), (2.13) in order to not only guarantee for consistency, but to also allow for the most general solution, i.e. to allow for the minimal restricted invariant solution manifold in that system. As a result, the coupled invariance condition (2.13) immediately shows that all restrictions coming from the "velocity product equations" (2.9) are essentially redundant over (2.7) and (2.8), and thus not symmetry-breaking as claimed in Oberlack (2001). In other words, condition (2.9) has no impact on evaluating (2.7) and (2.8). Because, splitting equation (2.13) into its relevant components XC | Tn=0 = 0, (2.14) the last condition (2.16), which includes (2.9), is now satisfied identically when evaluating its left-hand side (2.17) by using the condition (2.2) for the first two summands, since N i | Tn=0 = N i | N i =0 ≡ 0, and the condition (2.15), which includes (2.8), for the last two summands. Hence, the correct equivalence transformation for system (2.1)-(2.4) is thus either given by the unrestricted result (2.10), or with the restriction (2.11), but not if it's restricted by (2.12), which, according to Oberlack (2001), would also go along with two other wrongly enforced restrictions f 1 (t, ν) = b 1 (ν) t + b 2 (ν) and f 2 (t, ν) = b 3 (ν) (eq. (3.18), p. 308). To provide a better overview, Table 1 finally lists the resulting set of all infinitesimal generators. It explicitly shows and summarizes the substantial difference between the final results given in Oberlack (2001) and our on the restriction (2.11) based re-evaluation. In addition, Appendix A explicitly demonstrates that the key invariant transformation for the mean velocity field is definitely a to-the-restrictions (2.4) compatible symmetry for the equations (2.1)-(2.2), and, again, that all remaining equations (2.3) are redundant and thus obviously not symmetry breaking.
This change in the result is decisive as it now leads to a completely different picture in Oberlack (2001) when generating invariant scaling laws for the mean streamwise velocity profileū 1 =ū 1 (x 2 ) in the limit of infinite Reynolds number (ν → 0). Considering the derivation of the log-law, for which, according to Oberlack (2001), the friction velocity breaks the scaling symmetry of the mean velocity profile as a 1 = a 4 = 0, the corresponding invariant surface condition is then no longer given as leading to (eq. (3.29), p. 312) Instead one obtains the more general and correct invariant surface condition as the function F (x 2 ) is not closer specified, except for only being at least once integrable. Hence the derivation of the invariant log-law (2.19) is completely misleading in Oberlack (2001) as the correct equivalence transformation of the system (2.1)-(2.4) does not allow for such a unique functional specification. It rather induces the generation of any desirable function for the mean velocity profile as given in (2.21), making this result thus essentially useless. The same holds true not only for the log-law, but for all invariant scaling laws which were derived in Oberlack (2001), including, for example, the exponential scaling law in the wake region of turbulent boundary layer flow which were later extensively utilized in the papers by Khujadze & Oberlack (2004) and Günther & Oberlack (2005). Considering, for example, the following statements made in Oberlack (2001) such as that "In the case of the logarithmic law of the wall, the scaling with the distance from the wall arises as a result of the analysis and has not been assumed in the derivation." (p. 299), or "... important to note that group theoretical arguments very much guide the finding ... where the mean velocity profiles are applicable." (p. 306), or "This is an assumption in the classical derivation [von-Kármán-derivation] of the log law of the wall but is a result of the present analysis [Oberlack-(2001)-derivation]." (p. 312), or "The theory is fully algorithmic and no intuition is needed to find a self-similar mean velocity profile." (p. 321), we have to conclude, in retrospect, that all these statements are not correct and are even misleading if the correct result (2.21) is not being put forward. That is, without considering the generalized result (2.21), the study in Oberlack (2001) basically gives a wrong impression for using the method of Lie groups in turbulence theory. Because, when performing the analysis of Lie-groups for turbulent flows correctly, in the end one still has to make strong assumptions and to use more than a strong intuition in order to get from a general expression as (2.21) the correct statistical scaling law which the underlying and corresponding physical or numerical experiment is proposing. For example, Oberlack (2001) claims that under certain conditions which are assumed to be valid close to the wall, either the algebraic-law "(3.27)" (eq. (3.27), p. 312) or the loglaw "(3.29)" (eq. (3.29), p. 312) emerges as a functionally unique result when employing the method of Lie-groups, namely by adjusting only a finite set of group parameters towards compatible values. However, the re-evaluated and correct result for the algebraic-law conditions (a 1 = a 4 = 0 and F (x 2 ) = 0) and the correct result (2.20)-(2.21) for the log-law conditions (a 1 = a 4 = 0 and F (x 2 ) = 0) both show that this is not the case. The reason is that (2.22) as well as (2.23) each form a completely indifferent result: Any arbitrary (integrable) function F (x 2 ) can be chosen to represent after its integration the law of the wall. That means, from the perspective of Liegroup theory, neither the algebraic-law nor the log-law is a special or privileged scaling law; any desirable function can be derived from this theory, e.g. choosing F (x 2 ) = const. in (2.22) or (2.23) gives of course the algebraic-law "(3.27)" or the log-law "(3.29)" respectively. But this choice for F (x 2 ) is not privileged, one can also choose e.g. a complicated hypergeometric function, then leading to a law-of-the-wall which behaves not like a pure power-law or a loglaw, but as an integrated hypergeometric function, thus ultimately having infinitely many different but equally privileged laws of the wall. Hence, neither the power-law "(3.27)" nor the log-law "(3.29)" is a first-principle result from Lie-group theory. In Oberlack (2001) both these laws were just derived under wrong conditions, in that a mathematical correct theory has been unfortunately misapplied. In this regard the reader should note that all these problems initially already exist in the earlier publication Oberlack (1999) which only then got generalized in Oberlack (2001).
In summary, the open problem is that turbulence theory should predict a certain function for the one-dimensional mean velocity profileū 1 (x 2 ), which in our opinion, has not been achieved yet. For example, choosing the sophisticated method of Lie-groups to construct e.g. a scaling law in the vicinity of the wall, will only give the useless result (2.22) or (2.23), which in each case is just an alternative yet more complicated representation for the unknown onedimensional functionū 1 (x 2 ). One thus gained nothing in using Lie-groups, one just shifted the problem from one unknown functionū 1 (x 2 ) to another unknown function F (x 2 ). However, in Oberlack (2001) the result for the law-of-the-wall is that F (x 2 ) must be a constant function in the wall-normal coordinate x 2 , which definitely is an assumption, and thus forms a contradiction to all relevant statements made in Oberlack (2001).
In this sense Lie-group theory offers no answer, nor does it give any prediction a priori in how turbulence should scale. This failure simply reflects the classical closure problem of turbulence, which, also with Lie-group theory, cannot be bypassed. However, using this method to nevertheless get such an answer would be the same as guessing it, and if one knows what to expect a posteriori then, of course, one can manually arrange everything backwards, and pretend that theory is predicting this result. But such an approach has nothing to do with science.
This concludes the section on deriving a specific scaling law when using the invariance method of Lie-groups. It showed that although this method carries the high expectation of being a first principle method in turbulence theory, a convincing result is yet still missing in the literature to date -and remains very questionable if such a result is achievable at all.
While the invariance analysis just presented is set in the broader context of an analysis on Lie-groups in general, we would like to close the comment by pointing out all natural limits this method faces when using it within the theory of turbulence.

The limits of Lie-group methodology in the theory of turbulence
Regarding the limits in using the invariance theory of Lie-groups for differential equations in turbulence research, the main fact should be faced that this analytical approach experiences effectively the same closure-problem issues as the corresponding statistical transport equations for the correlation moments of the flow fields. Changing to a possible nonlocal integral and formally closed framework, e.g. to the higher statistical level of probability densities in order to allow for a formally complete and fully determined statistical description, as in the framework of the infinite Lundgren-Monin-Novikov chain of equations (Lundgren, 1967;Friedrich et al., 2012) or the functional Hopf equation (Hopf, 1952;McComb, 1990), will only render every systematic invariance analysis nearly infeasible. In particular for wall-bounded flows, the existence of boundary conditions and the usage of a nonlocal integral framework are both obstructive for an analysis based on Lie-groups, as they both favor the mechanism of symmetry breaking. Thus the main possibilities of the Lie-group methodology are already exhausted at this point.
A further serious limit for the methodology of Lie-groups in turbulence, in particular as standardly used since the key paper by Oberlack (2001), e.g. as in Oberlack & Günther (2003) (2013); Avsarkisov et al. (2014) and Wac lawczyk et al. (2014), is in the direction to consistently aim at generating useful statistical scaling laws for all higher order moments of a certain n-point correlation function in the inertial range. Even when applying a Lie-group based invariance analysis for a complex spatiotemporal (spatially nonlocal and temporally chaotic) system correctly, the result is standardly only given by a finite dimensional Lie-group, particularly in elements as scaling or translation of flow fields, i.e. the result is in general (up to some exceptions) repeatedly only given by invariant transformations of global nature, and predominantly is due to the non-integrability of these systems. The consequence are global invariant scaling functions, which are not only incapable of locating the domain where they should apply, but also in which in a naive way all higher moments are iteratively connected by constant global group parameters. It is therefore obvious that an extremely complex and omnipresent spatiotemporal property as that of intermittency cannot be captured in this restrictive manner. An invariantly constructed statistical set of global scaling laws simply cannot account for the complicated intermittent behavior which is constantly observed even in the inertial range of constant energy flux.
Exceptions can be found of course in lower dimensional systems, as e.g. in the Burgers equation. However, the nonlinear Burgers equation is in itself a very specific case as it's connected to the linear heat equation via the Cole-Hopf transformation (see e.g. Kevorkian & Cole (1996)). It therefore not only allows for an infinite dimensional Lie-group due to the linear superposition principle, but also to formally express the general solution of the Burgers equation in a closed form, which straightforwardly then also extends to any statistical description of the Burgers equation, as e.g. to the statistical functional Hopf approach -a result somehow or other not recognized by Wac lawczyk & Oberlack (2013) as several formal particular solutions of the Burgers-Hopf equation get calculated although the general solution has been already formulated in Hosokawa & Yamamoto (1970).
Although the origin of intermittency is still object of research, debates and conjectures there is nevertheless strong evidence that intermittent behavior in a complex system as found in all turbulent flows is closely related to the continuous dynamical breaking of symmetries induced by the system itself (Tsinober, 2013;Saint-Michel et al., 2013;Castellani, 2003;Kurths & Pikovsky, 1995;Frisch, 1985). This breaking mechanism however acts such that when the system finally reaches a statistically fully developed state the broken symmetries are not globally restored in a statistical sense (Frisch, 1995;Biferale et al., 2003). For example in the inertial range of turbulence the results clearly show that the flow cannot be globally invariant under scaling, neither in a deterministic nor in a statistical sense. Inertial range intermittency when measured with the longitudinal multi-point structure functions show a clear lack of global statistical self-similarity. This breakdown of global scale invariance in the structure functions is expressed by an anomalous or multifractal scaling behavior. The general rule is, the higher the order of the structure functions, the larger is the departure from global statistical self-similarity, that is, inertial-range intermittency becomes more important for higher-order moments and becomes even more pronounced the larger the Reynolds number gets (Falkovich & Sreenivasan, 2006). But in this regime intermittency not only breaks global statistical self-similarity, also global statistical isotropy may break, however then in a more weaker sense. The reason is that a noncompact group as that of scale invariance is more prone to be broken than a compact group as that of rotation invariance (Frisch, 1983). A statistical anisotropy is measured by comparing the scaling exponents between the longitudinal and the transversal structure functions, which turn out to be different even for an initially prepared homogeneous isotropic flow (Biferale & Procaccia, 2005;Biferale et al., 2008;Benzi et al., 2010). A recent theoretical investigation could show that this difference does not seem to depend on the Reynolds number, i.e. that the weak but existent breaking of isotropy in the inertial range is essentially not based on a finite Reynolds number effect (Grauer et al., 2012). Hence there is a strong indication that next to global self-similarity also global isotropy may not be statistically restored in the limit of infinite Reynolds numbers, as was originally and contrarily postulated in both the K41-theory (return to self-similarity and isotropy, Kolmogorov 1941) and in the later refined K62-theory (only return to isotropy, Kolmogorov 1962).
It is important to mention here that the dynamical process of symmetry-breaking inside such an intermittent-evolving system does not remove the property of scaling per se. Quite the contrary, it rather allows for fundamentally new and more complex scaling laws, which all unfold on a local level without being necessarily connected to an underlying global symmetry or global invariance principle (Frisch, 1991;Benzi et al., 1993;Sreenivasan, 1991;L'vov & Procaccia, 2000;Fujisaka & Grossmann, 2001;Chakraborty et al., 2010).
With this knowledge at hand, it is thus highly questionable if reasonable results can be gained at all when applying the method of Lie-groups onto any set of multi-point equations in the sense of Oberlack et al., as it standardly only leads to finite dimensional transformation groups involving constant (globally valid) group parameters. A result, which, with reasonable certainty, is unable to produce the correct scaling behavior of the turbulent velocity structure functions, all the more so, as with each increase in their order the intermittent behavior consecutively becomes more pronounced. Anyhow, ignoring this insight by continuously using these finite dimensional (global) scaling Lie-groups would in fact only give the same wrong results as a direct dimensional analysis would do.

Appendix A. Explicit symmetry † validation without CAS
This section consists of two parts, where the result of the second part will depend on the result of the first part. In the first part we will demonstrate that the key transformation for the mean velocity fieldū 1 (which is taken as a sub-group from the Lie-group transformation on the right-hand side of Table 1) is definitely admitted as a symmetry transformation of the equations (2.1)-(2.2) and that it's compatible to all restrictions (2.4). Then, in the second part, we will easily see that at this stage the remaining "velocity product equations" (2.3) are just redundant to the already transformed momentum equations (2.2) and thus not symmetry breaking as contrarily claimed in Oberlack (2001). Both parts will be performed without using a computer algebra system (CAS).

Part I.
The key sub-group transformation we want to consider is obtained if we put all group functions to zero (on the right-hand side of Table 1), except F = 0, which, for simplicity, we want to restrict only to a pure x 2 -dependence. In non-infinitesimal form the transformation is thus given as the following arbitrary x 2 -dependent translation of the mean and fluctuating streamwise velocity fields Now, when inserting this transformation into the continuity equation (2.1) we obtain the invariant result which is also the case for the momentum equations (2.2) k † Note that since the considered system of equations (2.1)-(2.4) is underdetermined (unclosed) all admitted invariances only act as equivalence transformations and not as symmetry transformations (Meleshko, 1996;Ibragimov, 2004;Bila, 2011;Chirkunov, 2012;Frewer et al., 2014). Although in this section as well as in the next section we will designate these invariant transformations as symmetries, we have to keep in mind that in a strict sense they are not symmetries but only equivalences.
and for the restriction equations (2.4), which, after employing transformation (A.1), stay unchanged as well Hence, the (2.4)-restricted equations C (2.1) and N i (2.2) admit transformation T (A.1) as a symmetry.
Part II. In order to complete the validation procedure of Part I, we also have to check if the "velocity product equations" P ij (2.3) admit T (A.1) as a symmetry transformation. But these "equations" are redundant to the momentum equations N i (2.2). They all show a degenerate behavior when trying to independently transform them from the underlying momentum equations N i . Consider e.g. the first diagonal component If we would treat (A.5) as an own and from N 1 independent equation, we have to globally conclude that either u ′ 1 must be zero or that N 1 must be zero in order to satisfy equation (A.5). But, since the former choice u ′ 1 = 0 has to be excluded, obviously, equation (A.5) can only be equivalent (necessarily and sufficiently) to the latter choice i.e. the "velocity product equation" P 11 is redundant and thus mathematically equivalent to the momentum equation N 1 . Therefore, since the latter equation stays invariant under transformation T (A.1), the former equation will stay invariant too. To be explicit, let's apply transformation T (A.1) to equation (A.5) Now, sinceÑ 1 = 0 (as shown in (A.3)), and since by construction F = 0 (the initial assumption), we obtain the equivalent relatioñ which therefore will turn equation (A.7) into the (redundant and necessary) invariant form The conclusion that P 11 admits T (A.1) as a symmetry transformation due to that N 1 admits it (and vice versa), can also be readily verified when using any symmetry-determining computer algebra package (see Appendix B): The symmetry results stay completely unaffected when either including or excluding the "velocity product equation" P 11 (2.3) in a corresponding symmetry analysis next to the continuity equation C (2.1), the momentum equations N i (2.2) and all constraint equations (2.4). Hence, relation (A.9) is a fully redundant invariance of the actual system (A.2)-(A.4) and thus clearly not symmetry breaking. This conclusion, of course, also applies to all other components of P ij (2.3) -the procedure is exactly the same as given just before for P 11 .

Appendix B. Explicit symmetry validation with CAS
By using the Maple-based symmetry-packages GeM (Cheviakov, 2007), SADE (Filho & Figueiredo, 2011) and DESOLV-II (Vu et al., 2012), we will demonstrate that irrespective of whether the "velocity product equations" P ij (2.3) are excluded from or included into the analysis, the final result will always stay unchanged in all three symmetry-determining algorithms. In the case of GeM and SADE this result is given by (2.10), while for DESOLV-II it is further restricted by (2.11).
Hence, all three CAS results explicitly show that the "velocity product equations" P ij (2.3) are not only redundant from the perspective of the fluctuating Navier-Stokes equations themselves (2.1)-(2.2), but also redundant from the perspective of a symmetry analysis performed upon them (2.14)-(2.15). The symmetry breaking mechanism as claimed in Oberlack (2001), which should solely arise from P ij (2.3), is thus not supported.