On the Lie symmetries of characteristic function hierarchy in compressible turbulence
Published in European Journal of Applied Mathematics, 2022 by Praturi, Plümacher & Oberlack.
All problems that one encounters in the incompressible case translate here one-to-one to the compressible case: A symmetry analysis of an unclosed system with an infinite hierarchy is performed again, leading to two nonphysical "symmetries" (4.17) and (4.18), where the latter one is incorrectly elevated again to a symmetry that should reflect external intermittency in turbulent flows. But, exactly as in the incompressible case, these two "symmetries" are just mathematical artefacts of the unclosed equations and not reflected by experiment or numerical simulation. They both fail to model turbulence or to derive turbulent scaling laws that can be matched to relevant data.
To say that "All of the symmetry groups derived have been instrumental in obtaining scaling laws for various canonical flows and have been verified against direct numerical simulations and/or experimental data [1,17,18,19,11]" is not true: [1] and [11] are refuted in https://arxiv.org/abs/2202.04635 (see Sec.D and Sec.3, respectively), [17] was self-refuted in the forced Corrigendum https://doi.org/10.1017/jfm.2019.985, in [18] the nonphysical "intermittency symmetry" is not used since it had to be put to zero, and [19] does not show any turbulence-relevant scaling, but only the trivial instantaneous scaling that can be trivially predicted once the scaling of the lowest moments is known. All this information is not revealed to the reader. By the way, the "new scaling laws" in [18] are not new at all. They already have been derived earlier and can be found in http://arks.princeton.edu/ark:/88435/dsp015t34sm99d. For more details see: https://arxiv.org/abs/2205.15652.
Then, not to include the separation constraint into the symmetry analysis is clearly a mistake, as explained in https://doi.org/10.1103/PhysRevE.92.067001. Ignoring or violating this constraint will inevitably lead to nonphysical invariant transformations, as (4.18), that will map physical solutions to nonphysical ones, regardless of the fact that this invariance, like (4.17), additionally also violates the classical principle of cause and effect (for a detailed proof, see https://arxiv.org/abs/1710.00669). To silence the separation constraint by saying "We consider a arbitrarily large domain that is not infinite in extent" [p.11] surely does not help, because in practice an arbitrarily large domain is already practically infinite as far as an experiment or a numerical simulation is concerned. That means, for example, when performing a numerical simulation in a sufficiently large computational box, then the separation constraint should be fulfilled in this domain up to the numerical accuracy chosen, and the larger the box, the better. For the two invariant transformations (4.17) and (4.18), however, this is not the case, regardless of how large the computational box is chosen. In particular, this is because (4.18) violates this constraint independent of the domain size with a term of constant amplitude, while (4.17) violates it for all flow configurations that allow for non-zero fields at the (infinite far) boundary of the domain.
To note is that the cited paper [21], on which the current paper heavily relies on, is methodologically and technically flawed, as was first proven in https://doi.org/10.1103/PhysRevE.92.067001, and then further clarified in http://dx.doi.org/10.13140/RG.2.1.1238.2803, https://arxiv.org/abs/1602.08039, http://dx.doi.org/10.13140/RG.2.2.35698.76480, and finally in https://arxiv.org/abs/1710.00669. Therefore, the statement in the present paper, that (4.18) is a relevant symmetry and a measure for intermittent behavior of turbulence, is not true.
For an up-to-date presentation of this overall topic, see https://arxiv.org/abs/2202.04635.