Symmetries and their importance for statistical turbulence theory

Published in Mechanical Engineering Reviews, 2015 by Oberlack, Waclawczyk, Rosteck & Avsarkisov

The reader is confronted here with a collection of articles showing the results obtained in the group of M. Oberlack during the years 2010-2015. In the last three years, however, several (peer-reviewed) Comments and Notes have been published that do not confirm these results: https://zenodo.org/communities/turbsym.

Eventually all central results and conclusions by Oberlack et al. are based on the global methodological error that new statistical symmetries are produced that are not compatible to the deterministic description of Navier-Stokes turbulence. One of the latest investigations https://arxiv.org/abs/1606.08396 clearly reveals this fact again. Therein (viz. Section 5.2), particularly their results obtained in Sec.6.2 (pp.53-63) for a turbulent channel flow with constant wall-transpiration were revisited. As proven and demonstrated in detail, these results cannot be confirmed: For example, with the symmetries provided, a correct and to all orders consistent symmetry analysis shows that for the viscous case in the center region of the channel only a (featureless) linear invariant scaling law for the mean velocity field can be generated. In particular, the existence of an invariant logarithmic scaling law as presented by Oberlack et al. in Eq.(362) is not given. Nonetheless, even if we would allow to also include all nonphysical statistical symmetries, the invariant surface condition will only give at most a quadratic power law for the mean velocity field, and not a power-law of arbitrary exponent as claimed in Eq.(355). Hence, the apparent "successful validation" of the scaling laws to the DNS data, as shown e.g. in Figs.14-18 (pp.61-67), cannot be confirmed. In fact, the opposite is proven.

1. Methodology:

This section lists all parts of their symmetry analysis that contribute and forms the methodological error as a whole. This will be done in the order as they appear in the article. Details can be found in the links provided.

1.1. The functional symmetry analysis presented in Sec.2.7 (pp.15-22) is incomplete. It ignores the fact that the continuous coordinate $x$ (or in Fourier space the wavenumber $k$) constitute transformable quantities by themselves. As a consequence, for example, a certain class of scaling symmetries cannot be detected by this method (http://aip.scitation.org/doi/10.1063/1.4940357).

1.2. Even if formally the infinite hierarchy of MPC equations Eq.(195), or Eq.(202), is considered, this system remains to be unclosed due to its forward recursive form. Instead of true symmetry transformations, only the weaker form of equivalence transformations can be generated, for which, in a strict mathematical sense, the construction of invariant solutions is misleading and sometimes even ill defined if no further external information is provided for such unclosed systems (https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1508.06962, https://arxiv.org/abs/1511.00002).

1.3. All three "new statistical symmetries" Eqs.(232)-(234) are inconsistent and thus nonphysical. The latter two symmetries Eq.(233) and Eq.(234) are incompatible to the underlying deterministic description of Navier-Stokes turbulence in the way that they both violate the classical principle of cause and effect (https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1602.08039), while the former symmetry Eq.(232) does not respect the one-sided (non-symmetrical) connectedness of the MPC hierarchy, namely the fact that only the lower $n$-point correlation functions of $(n+1)$-order can be uniquely constructed from the higher $(n+1)$-point functions and not vice versa. 

Moreover the claim that "this group is not related to the classical translation group in usual $x$-space" (p.31) is incorrect. With a simple argument one can show that Eq.(232) forms a particular, yet inconsistent translation sub-group of $T_7\,$-$\,T_9$ in Eq.(181) if $\mathbf{k}_{(l)}$ is non-zero: 

Proof: For $n$ different coordinate points $\mathbf{x}_{(n)}$, the usual $x$-space translation symmetry ($T_7\,$-$\,T_9$ in Eq.(181) for $\mathbf{f}$ constant) can be written as $\mathbf{x}^*_{(n)}=\mathbf{x}_{(n)}+\mathbf{c}_{(n)}$, for all $n\geq 0$, and, hence, since the relative coordinate is defined as $\mathbf{r}_{(l)}:=\mathbf{x}_{(l)}-\mathbf{x}_{(0)}$, this translation symmetry can be equivalently reformulated into the relative coordinate as $\mathbf{r}^*_{(l)}=\mathbf{r}_{(l)}+\mathbf{c}_{(l)}-\mathbf{c}_{(0)}$, for all $l\geq 1$. If we now choose $\mathbf{c}_{(0)}=\mathbf{0}$ and identify $\mathbf{c}_{(l)}=\mathbf{k}_{(l)}$, we obtain the (inconsistent) symmetry Eq.(232) from the usual (consistent) $x$-space translation symmetry $T_7\,$-$\,T_9$ in Eq.(181). From a different perspective, one now also sees why Eq.(232) is inconsistent if $\mathbf{k}_{(l)}$ is non-zero: Since the general relation $\mathbf{r}^*_{(l)}=\mathbf{r}_{(l)}+\mathbf{c}_{(l)}-\mathbf{c}_{(0)}$ must also hold in the limit of zero correlation length (i.e., in the limit $\mathbf{r}^*_{(l)}\rightarrow \mathbf{0}$ and $\mathbf{r}_{(l)}\rightarrow \mathbf{0}$) in order to ensure overall consistency in the MPC hierarchy, we get the vital relation $\mathbf{c}_{(l)}=\mathbf{c}_{(0)}$, where we thus see that all $\mathbf{c}_{(l)}$ cannot be chosen independently from $\mathbf{c}_{(0)}$. And since the global condition $\mathbf{c}_{(l)}=\mathbf{c}_{(0)}$ implies again the multi-point transformation $\mathbf{r}^*_{(l)}=\mathbf{r}_{(l)}$, we get the result that only for $\mathbf{k}_{(l)}=\mathbf{0}$, i.e., only for a broken symmetry Eq.(232), a consistent analysis can be ensured.

1.4. The two LMN symmetries Eq.(246) and Eq.(254) are nonphysical, in not only violating the classical principle of causality, but also violating the separation constraint naturally arising in the statistical LMN description of turbulence (https://doi.org/10.1103/PhysRevE.92.067001).

1.5. The two functional symmetries Eq.(290) and Eq.(293) for the characteristic functional $\Phi$ are nonphysical in the way that they both independently induce a conflict in the corresponding transformation rule for the probability density functional $P$ (https://arxiv.org/abs/1412.3061). Eventually this conflict just reflects again the violation of cause and effect. Moreover, the functional translation symmetry Eq.(293) may also violate the non-holonomic constraint $|\Phi|\leq 1$ of the Hopf equation if the translation functional $\Psi$ is not chosen properly, a further crucial issue not mentioned by Oberlack et al. Because, it is not clear at all how to choose $\Psi$ for this translation symmetry Eq.(293), where all space-time coordinates transform invariantly, such that the physical constraint $|\Phi^*([\mathbf{y}(\mathbf{x})],t)|=|\Phi([\mathbf{y}(\mathbf{x})],t)+\Psi([\mathbf{y}(\mathbf{x})])|\leq 1$ is satisfied for all times $t\geq 0$, in particular since $\Phi$ evolves independently from the time-independent functional $\Psi$. Important to note here is that in Waclawczyk & Oberlack (2013) (http://aip.scitation.org/doi/abs/10.1063/1.4812803) an alternative symmetry for Eq.(293) has been derived in order to induce Eq.(233). This derivation, however, is not performed on the basis of the full Navier-Stokes equations, but only on the basis of the inviscid Burgers equation. Anyway, the analysis in http://aip.scitation.org/doi/10.1063/1.4940357 and https://www.researchgate.net/publication/299368809 clearly shows that their argument is not valid: The alternative symmetry does not induce a translation invariance in the MPC equations. Because, as has been proven, this alternative symmetry leads to the unphysical result of globally zero multi-point velocity correlations.

2. Analyses:

This section deals with the question whether the analyses have been performed appropriately and rigorously. The brief answer is no. Here the details:

2.1. The adapted transformation Eq.(262) is not admitted as a symmetry transformation by the bounded LMN equations for plane channel flow as incorrectly claimed by Oberlack et al. (https://doi.org/10.1103/PhysRevE.92.067001, https://www.researchgate.net/publication/287996194).

2.2. The functional analysis presented in physical coordinate space (pp.36-37) is flawed. In particular, the derivation of the functional Galilean invariance Eqs.(271)-(286) is incorrect. Due to the Comment http://dx.doi.org/10.3390/sym8040023, this mistake has now been acknowledged in http://www.springer.com/in/book/9783319291291 (pp.13-14). In this regard, see also the additional Comment on this issue: https://www.researchgate.net/publication/301553065.

2.3. The interpretation and justification of the translation symmetry Eq.(233) that "Kraichnan (1965) recognised this first" and that "he observed the first element of the infinite set of symmetries in (233)" (p.31) is incorrect and constitutes a misinterpretation of Kraichnan's idea to random Galilean invariance. This misconception has been revealed in https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.067001.

2.4. It has not been recognized by Oberlack et al. that their ad-hoc introduced $Z_{zij}$-symmetries Eqs.(298)-(301), or Eq.(353), actually are particular specifications of a symmetry reflecting the linear superposition principle; a generic symmetry which ultimately is of no predictive value, because, as already correctly noted by them before, it "cannot directly be adopted for the practical derivation of group invariant solutions" (p.31). The problem is that through the use of the superposition principle within the unclosed MPC system, their Lie-group symmetry approach degenerates down to a non-predictive incremental trial-and-error method, which is not in the sense of the inventor to forecast turbulent scaling laws from a "first principle" theory which is "fully algorithmic" and where "no intuition is needed" (https://doi.org/10.1017/S0022112000002408, p.321). For a more detailed discussion on the subtle and problematic issue of using the linear-superposition symmetry within unclosed equations, see https://arxiv.org/abs/1606.08396.

3. References:

This section deals with the question whether the shown results were referenced correctly. The brief answer is no. Here the reason:

The "new" statistical scaling symmetry Eq.(234) for the two-point correlation (TPC) equations was first derived and mentioned in 2004 (https://doi.org/10.1007/s00162-004-0149-x), and was only later generalized in 2010 (http://dx.doi.org/10.3934/dcdss.2010.3.451) for the infinite hierarchy of multi-point correlation (MPC) equations. It is not correct that the importance of the results obtained in Khujadze & Oberlack (2004) "was not observed therein", nor that "it was stated that they may be mathematical artifacts of the averaging process and probably physically irrelevant" (p.30). Instead, this at that time so-called third scaling symmetry has been expressly and knowingly applied to various turbulent flow cases over the years long before Oberlack & Rosteck (2010) appeared. In particular the case of fractal-generated turbulence provided a "successful" application of the third scaling symmetry according to M. Oberlack at that time around 2008. Hence, although G. Khujadze now distances himself from his own discovery because of knowing better by now that this third scaling symmetry is nonphysical, the initial finding of the scaling symmetry Eq.(234) is not correctly referenced and the work of G. Khujadze not adequately acknowledged by M. Oberlack. His statement that the scaling symmetry Eq.(234) "was first presented and its key importance for turbulence recognised in Oberlack & Rosteck (2010)" (p.30) is incorrect and fails to do justice to the fact that G. Khujadze laid the basis for the now more generalized and as "new" pretended statistical scaling symmetry Eq.(234). To prove this statement, a few links are provided below, which clearly show that during the time 2004-2009 this third scaling symmetry was strongly promoted as a physical and useful symmetry and not as mathematical artifact as now claimed by M. Oberlack in retrospect for unclear reasons. Note that the conference video in the third link is particularly worthwhile to study this issue:

2005: 

http://onlinelibrary.wiley.com/doi/10.1002/pamm.200510259/pdf 
2007:

http://www.tsfp-conference.org/proceedings/2007/69-new-scaling-laws.pdf 

2008:

https://www.newton.ac.uk/seminar/14423 (Video)

https://www.newton.ac.uk/files/seminar/20081001143015001-151993.pdf
http://link.springer.com/chapter/10.1007%2F978-1-4020-6472-2_39
http://link.springer.com/chapter/10.1007/978-3-642-02225-8_3 
2009:

http://www.tsfp-conference.org/proceedings/2009/vol_1-17.pdf

4. Conclusions:

This section deals with the question whether the interpretations and conclusions are justified by the results. The clear answer is no. Here the details:

4.1. It is not true that "the new statistical symmetries mirror key properties of turbulence such as intermittency and non-gaussianity" (p.1). In fact, as was rigorously proven, their conclusion is based not only on a methodological error but also on several technical errors (https://arxiv.org/abs/1412.6949, https://doi.org/10.1103/PhysRevE.92.067001, https://www.researchgate.net/publication/287996194).

4.2. Their view that "these new symmetries have important consequences for our understanding of turbulent scaling laws" (p.1) cannot be shared, because, as already mentioned before, these "new" statistical symmetries as proposed by Oberlack et al. are, in every respect, unphysical in being not only incompatible to the underlying deterministic description of Navier-Stokes turbulence, but also in violating several physical constraints which naturally go along with the dynamical equations when considering any higher-level statistical descriptions of turbulence, as the LMN or the Hopf functional theory. In all three statistical approaches to turbulence, i.e., in the MPC, LMN and Hopf approach, the mathematically correctly admitted new statistical symmetries turn out to be unphysical by closer inspection (https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1412.6949, https://arxiv.org/abs/1606.08396).

4.3. Their conclusion that "the symmetries form the essential foundation to construct exact solutions" (p.1) has to be treated with caution. Because, for unclosed systems as the MPC equations, one can always construct infinitely many invariant and functionally independent solutions which are all nonphysical, i.e., which all are not reflected in the data if a physical experiment or a DNS is performed. In other words, for unclosed systems which are particularly defined forward recursively as the MPC equations, a particular (mostly nonphysical) "exact invariant solution" is not privileged among the infinitely many other independent and exact invariant solutions which can be always straightforwardly generated for such systems. Hence, the word "solution" is thus misleading if no further external information is provided for such unclosed systems (https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1508.06962, https://arxiv.org/abs/1511.00002, https://arxiv.org/abs/1609.08155, https://arxiv.org/abs/1412.3069).

4.4. It is not true that "even new scaling have been forecasted from these symmetries and in turn validated by DNS" (p.1). Because, when performing a correct and overall consistent Lie-point symmetry analysis on the infinite hierarchy of MPC equations, it shows that every true and serious forecasting of turbulent scaling will fail simply due to the fact that Lie-group symmetry theory cannot circumvent or bypass the closure problem of turbulence (https://arxiv.org/abs/1412.3069, https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1606.08396, https://arxiv.org/abs/1609.08155).

4.5. It is not true that some of the "new statistical symmetries... have been employed even in very early turbulence models" (p.68). Despite the fact that this sentence is an oxymoron (because how can a symmetry be "new" when it already has been used in the past), these "new statistical symmetries" proposed by Oberlack et al. have never been used or applied in science anywhere before. For example, their claim that Kraichnan back in the 1960's already observed the "new statistical translation symmetry" Eq.(233) which "he named random Galilean invariance" (p.31) is incorrect and, as already said, constitutes a severe misinterpretation of Kraichnan's idea to random Galilean invariance (https://doi.org/10.1103/PhysRevE.92.067001, https://arxiv.org/abs/1412.6949, https://www.researchgate.net/publication/287996939).

4.6. One can fully agree with their following statement that it will be "impossible to make turbulence models consistent with some of the [new] symmetries such as the new scaling symmetry (234)" (p.68). Because, any turbulence model that will admit e.g. Eq.(234) as a symmetry, has to be a linear model, but which then, of course, cannot be a reasonable model for turbulence anymore.

4.7. Finally, to address the open question that "so far completeness of all admitted symmetries of the MPC equation has not been shown... [which] appears to be necessary not only from a theoretical point of view but rather essential to generate scaling for all higher moments" (p.68) will only lead to a superfluous task, because, as already said, arbitrary invariances can be constructed when considering an unclosed infinite hierarchy of equations as the MPC equations considered by Oberlack et al.; therefore completeness in this very sense can never be obtained in such infinite systems (https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1508.06962, https://arxiv.org/abs/1511.00002).