NOTE: For a correctly rendered text of this review, see https://turbsym.blogspot.com/p/reviews.html

Anyone with experience in turbulence modelling immediately recognizes that this newly proposed turbulence model (Eqs.6-8,10) is a mathematical artefact without physical value.

In a nutshell, here's the construction strategy of their newly proposed modelling principle:

From the unclosed set of statistical equations, two arbitrary symmetries (Eqs.4-5) out of an infinite pool of possible symmetries are taken in order to modify existing turbulence models such that they now comply with these two particularly chosen symmetries. To achieve this, new mathematical field variables are introduced: The new velocity field Fraktur(U) and the new pressure field Fraktur(P). Hence, when also including any underlying DNS data to these newly created statistical models, we basically are dealing here with three different velocity fields (and three different pressure fields accordingly): The fine-grained instantaneous velocity U, the coarse-grained mean velocity <U> (denoted as U-bar in the paper), and then a new velocity field, called Fraktur(U), satisfying an own closed dynamical equation (Eq.6) defined in exactly the same form as the original Navier-Stokes equations, with the only difference that, instead of nu, a different viscosity parameter Fraktur(nu) is used. However, for what Fraktur(U) should actually stand for physically, or what fluid flow property it should actually represent, is not given nor discussed. The second point below provides such a discussion, resulting in a negative outcome.

The obvious points for criticism of this new modelling principle are:

1. Despite the fact that the two symmetries used are unphysical per se, in that they violate the classical principle of cause and effect (see e.g. https://doi.org/10.1103/PhysRevE.92.067001, https://arxiv.org/abs/1602.08039, https://arxiv.org/abs/1710.00669), their key modelling-argument in itself is circular: Two "new" symmetries are extracted from unclosed equations to then use them in order to find improved closing constraints for those same equations again. This effort is comparable to pull oneself up by one's own bootstraps.

1.1. The set of possible statistical symmetries is infinite, simply because from the outset the statistical equations themselves are unclosed. These equations, as well as their symmetries, need to be modelled empirically. To take any of such (unmodelled) symmetries, in order to exactly model those same equations again from which they originated, is of no physical value, since the modelling information in this situation has to come from the outside (experimental and/or numerical data) and cannot come from the inside, in particular not when the intended modelling-element, namely the set of symmetries, is itself unclosed and need to be modelled for its own use. To note in this regard is that also the set of symmetries in the functional Hopf-framework is infinite and therefore unclosed, alone already by the fact that the linear superposition principle exists and applies in this particular framework.

1.2. With modelling of symmetries, I mean to find which is the correct symmetry to take (if any) when the pool of symmetries to choose from is infinite. The aim is to avoid the high risk of choosing unphysical symmetries. On the contrary, a statistical symmetry can be regarded as correct or as approximately correct in the modelling sense if it's more or less consistent to the statistical data considered. However, for the two "new" statistical symmetries (Eqs.4-5) chosen here, this is definitely not the case, as they both fail to be consistent with the numerical data already for simple canonical cases as turbulent channel flow (see e.g. https://arxiv.org/abs/1412.3061, https://arxiv.org/abs/1606.08396), or turbulent jet flow (https://www.researchgate.net/publication/328080340). Excluding these two unphysical symmetries (Eqs.4-5) from the modelling process clearly shows that the matching to the DNS data improves by several orders of magnitude, which can be well explained from the fact that these two symmetries violate the classical principle of causality (for a summary of facts, see e.g. https://www.researchgate.net/publication/286732368).

1.3. Their motivation to choose from an infinite pool of possibilities exactly only those two symmetries Eq.4 and Eq.5, is based on the following overall wrong claim: "These statistical symmetries are closely connected with intermittency and non-Gaussianity of turbulence [6], and, further, they are part of the key building blocks for turbulent scaling laws (see [5] and references therein)" (p.29). Particularly their claim regarding the connection to intermittency and non-Gaussianity is definitely wrong and has been clearly refuted several times by now within different statistical frameworks (see e.g. https://doi.org/10.1103/PhysRevE.92.067001, https://www.researchgate.net/publication/311285232). Also from a pure phenomenological viewpoint, it's abundantly clear that intermittency is a symmetry-breaking phenomenon and not a symmetry-existing or symmetry-preserving one. Even if we would wrongly assume this to be the case, intermittency is definitely not described or featured by any global scaling symmetry, particularly not by the one given by Eq.5, which actually just mimics the standard scaling of a linear system when applied to any unclosed nonlinear system, namely in the way that it simply identifies all nonlinear terms just as error terms that only need to be corrected for by exploiting the unclosed terms. Obviously, such a symmetry is only a mathematical artefact of the unclosed system itself and therefore indeed unphysical, independent of the fact, of course, that this symmetry also violates the classical principle of cause and effect.

2. From a pure mathematical viewpoint, the introduction and necessity of the new velocity field Fraktur(U) and pressure field Fraktur(P) is clear. But not from a physical viewpoint! What exactly is the physical meaning, for example, of Fraktur(U)? Is it a coarse-grained variable operating on the same or similar time- and length-scales as the mean velocity field <U>? Surely, Fraktur(U) cannot operate on the same scales as the instantaneous velocity U, otherwise the authors would not talk of a "closed set of equations" (p.31). So, on what time- and length-scales does the new field Fraktur(U) then operate on? This important issue is not discussed in their paper. Had they done so, they would have immediately recognized that either their new field Fraktur(U) cannot be physically realized, or that their new model is still unclosed in that dynamical equations for Fraktur(U) are still required to universally close the whole set of equations. Let's briefly discuss these two only possible outcomes.

2.1. Let's imagine for a minute Fraktur(U) is a fully coarse-grained velocity variable operating on the same or similar time- and length-scales as the mean velocity field <U>. Then, consistently, we have the obvious relation: <Fraktur(U)>=Fraktur(U). That means, Eq.6 is a closed equation when averaging it according to <,>. In other words, the new field Fraktur(U) need not to be modelled, it can be directly determined from Eq.6 when supplementing it with appropriate initial and boundary conditions (ICs and BCs). Of course, these ICs must be coarse-grained too, that is, they must be of the same or similar length scale as those for the mean field <U>. So, <U> and Fraktur(U) must be physically visible on the same or similar length scale. For example, let's take channel flow: The existence of the mean field <U> is clear, but what flow quantity is Fraktur(U)? In particular, according to Eq.6, it even should evolve exactly to the original Navier-Stokes equations only with a different viscosity Fraktur(nu), where, of course, under the fixed BCs considered, Fraktur(nu) has to be considerably larger in value than nu (Fraktur(nu)>>nu). This will avoid triggering any of the well-known instability properties of the Navier-Stokes equations so that Eq.6 for Fraktur(U) can indeed be regarded as a closed equation when averaging it according to <,>. Hence, we should have next to <U> a second independent "mean" flow, namely Fraktur(U), but which, so far as I know, has not been reported or seen in any experiment or numerical simulation. It's clear, such a velocity field as Fraktur(U), evolving on the same or similar scale as <U>, is physically fictitious and does not exist!

2.2. Now, let's imagine Fraktur(U) is a velocity field that operates on time- and length-scales which lie somewhere between the fine-grained instantaneous velocity field U and the coarse-grained mean velocity <U>. Such a graining can be constructed by filtering U to a certain space-time scale (similar as in the LES methodology): Fraktur(U):=int(U*f,dt,dx), where f is the filter function defining the space-time resolution. But now, why should this medium-grained velocity field Fraktur(U) evolve exactly according to the pre-defined closed Eq.6? Fraktur(U) is definitely a flow quantity that needs to be modelled in its own respect now, namely all those fine non-resolved scales that were removed or suppressed by the chosen filter function! Hence, Eq.6 has to be corrected to an unclosed equation involving new stresses that act on Fraktur(U). It are these medium-grained stresses that need to be modelled, and they are certainly different to the coarse-grained Reynolds stresses.

Conclusion: The proposed model (Eqs.6-8,10) as it stands is methodologically flawed on three counts:

(1) The key modelling principle is based on a circular argument, in that unmodelled symmetries from unclosed equations get extracted and used in order to close these same equations again.

(2) The particular model is based on two unphysical symmetries that violate the classical principle of cause and effect.

(3) Since the authors claim that system Eqs.6-8,10 is closed, the new velocity and pressure field, Fraktur(U) and Fraktur(P), must operate on the same or similar time- and length-scales as the mean velocity and pressure field <U> and <P>. However, as discussed, such additional fields are physically fictitious and do not exist!