The method of Lie symmetry groups is a successful tool to either model dynamical rules that should admit a certain given set of symmetries, or to provide insight into the structure of the solution space for a given closed set of dynamical equations, including the possibility to even allow for their full integration.

The equations of turbulence, however, are different, both conceptually and practically. These equations are mathematically unclosed and need to be modelled empirically. Therefore, from the unclosed and unmodelled theory itself, caution has to be exercised when extracting new symmetry-based information from it.

For Navier-Stokes turbulence, no breakthrough with symmetries has been achieved yet and is still in a very immature state. Up to date, all systematic results to predict the statistical scaling behaviour of turbulent flows with Lie-group symmetries, are either not rigorous to convince or are not correct to be adopted.

Hence, this community welcomes all results, positive as well as negative ones, on its way to understand the need for using the method of Lie symmetry groups in turbulence.

For a more detailed description of this community, the challenges and the insurmountable difficulties it faces, even if the method of Lie symmetry groups is applied and interpreted correctly to the theory of turbulence, please follow the link "Read more" below.

The method of Lie symmetry groups is a successful tool to either model dynamical rules that should admit a certain given set of symmetries, or to provide deep insight into the structure of the solution space for a given but closed set of dynamical equations, including the possibility to even allow for their full integration. The (statistical) equations of turbulence, however, are different, both conceptually and practically. These equations are mathematically unclosed and need to be modelled empirically. Hence, caution has to be exercised when extracting new (statistical) symmetries from the unclosed and unmodelled theory itself, not to run into any circular arguments. For example, to derive new symmetries from unclosed equations to then use them in order to close those same equations again, is such a circular argument. Also to explore the solution structure of unclosed equations with new symmetries only admitted by those unclosed equations, turns out to be inconclusive, not only because the obtained symmetries are unclosed by themselves, but also once the equations are closed by an empirical model these new symmetries will most likely get broken.

For Navier-Stokes turbulence, no breakthrough with symmetries has been achieved yet. Up to date, all systematic results to predict the statistical scaling behaviour of turbulent flows with Lie-group symmetries, are either not rigorous to convince or are not correct to be adopted. In the former case, the Lie-group symmetry results are standardly based on strong low-order assumptions which typically turn out to be incompatible to associated higher-order relations in showing an increasing mismatch to empirical results the higher the statistical order gets, while in the latter case, the Lie-group symmetry results are already inconsistent from the outset in violating certain immutable constraints already on the lowest statistical order. One reason for this prominent failure and the missing breakthrough is that the classical Navier-Stokes theory does not allow for a local space symmetry, in strong contrast, for example, to the theory of quantum fields, which is based on such a symmetry, the local gauge symmetry, which successfully predicts the unknown functional structure of the interacting fields between the various elementary particles.

To unravel the complexity of Navier-Stokes turbulence, not only the unclosed statistical equations but also their defining deterministic equations, the instantaneous Navier-Stokes equations themselves, should be considered and taken into account in any modelling and solution finding process. In particular, since the deterministic equations naturally define a causal structure on the statistically induced equations, that is, since the deterministic Navier-Stokes equation implies its statistical equations and not opposite, a strict principle of cause and effect is formulated by this asymmetric relation which should be respected and not violated during any process of modelling and quantifying statistical analysis.

Also, not the instantaneous and deterministic symmetry, but rather the subsequent statistical symmetry breakdown is what ultimately produces diversity and complexity in nature. Turbulence is the result of such successive symmetry breakdowns, a state, where in particular dynamical symmetries are not recovered, also not in a statistical and universal sense! An eminent example is, e.g., the anomalous scaling behaviour of an isotropic turbulent flow, which significantly deviates from the statistical scaling symmetry of the governing Navier-Stokes equation.

Using in turbulence the method of Lie symmetry groups, Landau's comment should be recalled, to show here again the importance to follow the causal structure defined by the deterministic equations onto its induced set of statistical equations, which by themselves cannot be statistically universal due to exhibiting and featuring scale dependence: "It may be thought that the possibility exists in principle of obtaining a universal formula, applicable to any turbulent flow, which should give [the velocity correlation functions] for all distances that are small compared with [the integral length scale]. In fact, however, there can be no such formula... [since in this case] the result of averaging cannot be universal" (Fluid Mechanics, Landau and Lifshitz, Chapter 34).