Published in Journal of Physics A, 2017 by Waclawczyk, Grebenev & Oberlack.
This study faces three shortcomings:
(1) Their analysis misses a very
central aspect of the LMN equations which thus, opposite to their
claim, makes their Lie-group symmetry results incomplete.
(2) The
statements on the constraints regarding the infinite-dimensional
symmetry groups are misleading.
(3) The particular scaling symmetry
originating solely from the linearity of the LMN hierarchy is nonphysical
in that it violates the classical principle of cause and effect. This
fact was proven analytically in a rigorous manner in 2014:
https://doi.org/10.1103/PhysRevE.92.067001,
https://arxiv.org/abs/1412.6949,
https://arxiv.org/abs/1412.3061,
https://arxiv.org/abs/1602.08039.
Details
on (1): The LMN equations are always accompanied by at least five
physical constraints in order to guarantee their solutions to be
physical. These are the four well-known and so-called non-negativity,
normalization, coincidence and separation constraints, and a fifth one,
not so well-known, but an equally important constraint, the conditional
constraint first derived by Ievlev in 1970 (see relation 2.6 in https://link.springer.com/article/10.1007/BF01015004) and then later also discussed in Monin & Yaglom (see relation 19.139 in https://books.google.de/books?id=6xPEAgAAQBAJ).
However, for some unknown reason, the present symmetry analysis of
Oberlack et al. only considers the first two constraints, while the
three remaining constraints are not included. A cardinal mistake,
because when included into the symmetry finding process these
constraints will ultimately break certain symmetries. For example, as
has been clearly shown and discussed in Sec.II in https://doi.org/10.1103/PhysRevE.92.067001, the scaling symmetry $X=X_*+X_{**}$ (Eqs.47-48) in the present study is incompatible to the separation
constraint and thus violates one of the most intuitive physical
constraints of the LMN equations. In other words, the particular
transformation $X=X_*+X_{**}$ is not admitted as a symmetry by the physical LMN equations (which also
includes all constraints) — a vital result not obtained in the
present study, rendering their symmetry analysis thus as incomplete.
Details on (2): Not explicitly referred to as such, the infinite-dimensional symmetry group $X_{**}$ (Eq.48) is nothing else but the superposition principle of the unclosed
first order equation (Eq.3) of the linear LMN hierarchy. By definition,
a superposition symmetry is built up by translation operators in the
dependent variables with coefficients being the solutions of the
considered linear system of equations.
But for unclosed systems such a symmetry is of no value, since it is not
clear how to generate solutions without modelling the system. Hence,
their result that the coefficients $b^0_{(1)}$ and $b^0_{(2)}$ of the symmetry $X_{**}$ must be "solutions" of the unclosed Eq.3 is misleading.
Obviously, to guess a solution for the lowest order equation is not the
method of choice, since there is no guarantee a priori that, firstly,
this guess is consistent also for all higher order equations and,
secondly, that this guess in the end represents a physical solution
matching the DNS data. The chance is practically zero to analytically
guess such a correct solution. Hence, the symmetry $X_{**}$ is of no value for further investigations, and surely will also break
as soon as the truncated Eq.3 is modelled, since any appropriate model
will definitely be non-linear.
Details on (3): Even when ignoring the above fact (1), as has been done in the present study, the scaling symmetry $X=X_*+X_{**}$ (Eqs.47-48) is unphysical per se in violating the causality principle
of classical mechanics. This symmetry, which has its origin solely from
the linear structure of the LMN equations, clearly leads to wrong and
misleading conclusions in turbulence, in particular when used further to
generate statistical scaling laws. This fact has been proven
analytically in a rigorous manner and visualized several times by
comparing such generated scaling laws to DNS data for different flow
configurations, always with the clear result that a strong mismatch is
constantly observed. For more details, please see:
https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1606.08396, https://www.researchgate.net/publication/311285232, https://www.researchgate.net/publication/286732368.
Essentially, this scaling symmetry $X=X_*+X_{**}$ is a physically spurious symmetry simply due to the fact of not
providing in the performed symmetry analysis for the (coarse-grained)
LMN equations the necessary statistical link to the underlying
(fine-grained) Navier-Stokes equations. Because including this link will
break this symmetry (see Sec.I in https://doi.org/10.1103/PhysRevE.92.067001)
and thus will not give rise to such an unphysical symmetry in the first
place, showing again that in turbulence it is necessary to reveal all
information available and not to hide it in some benefiting way in order
to be able to announce some "progress" which, in the end, then turns
out to be misleading.
Conclusion: Considering the above facts (1)-(3), the only symmetries left for further analysis are the classical symmetries $X_1-X_{12}$ (Eqs.41-46) of the Navier-Stokes equations when formulated
statistically in the framework of the LMN equations. But this is not new
and also not a surprising result as this is to be expected.
The two "new" infinite-dimensional groups $X_*$ (Eq.47) and $X_{**}$ (Eq.48), however, are broken and thus not available anymore to serve
the unrealistic expectation that a "non-Gaussian solution for the tails
of pdf's could be derived based on the symmetries of the corresponding
LMN equations for velocity differences" (p.11). This expectation, which
in the present study is directly linked to the expectation in trying to
grasp the "phenomenon of internal intermittency of turbulence" (p.11)
with the help of symmetries, is unrealistic in so far as intermittency
has the well-known property to rather break than to restore symmetries,
not only on the fine grained (fluctuating) but also on the
coarse-grained (averaged) level. And since the available symmetries $X_1-X_{12}$ (excluding the broken symmetries $X_*$ and $X_{**}$ from this set due to the facts (1)-(3) discussed above) collectively
only form a finite and not an infinite-dimensional Lie-algebra, they are
more prone to be broken in a turbulent and intermittent flow, thus
favoring the situation of symmetry breaking, where inside this finite
algebra, obviously, a noncompact group as that of scale invariance is
again more prone to be broken than a compact group as that of rotation
invariance and so on.
Note: This comment refers to the "Accepted Manuscript" published online on 24 February 2017