Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach

A closer look at predicting turbulence statistics of arbitrary moments when based on a non-modelled symmetry approach

“…The present review will unequivocally demonstrate that no breakthrough or any progress in [1] has been made "to overcome the problem related to the closure problem of turbulence" [p. 5]. Instead, the paper [1] is highly misleading in that it shows near-perfect matching results to analytically derived turbulent scaling laws, which are claimed to be true solutions of the statistical Navier-Stokes equations obtained from first principles alone, without making any prior assumptions or using any modelling approaches. Furthermore, two so-called 'symmetries' are emphasized and claimed to be a measure of the intermittent and non-Gaussian behaviour of turbulence.…”

“…Said differently, once the mean field has been fitted to data, the scaling behaviour of the full-field correlations as shown in Figs. 1-3 in [1] is trivial to predict.…”

“…First of all, the scaling behaviour shown in Figs. 1-3 in [1] is clearly dominated by the mean flow and says nothing about the turbulent part. Due to the overwhelming dominance of the mean flow in the two flow regions considered, the n-th full-field correlation in the streamwise direction U n 1 is close to equal to the n-th power of the mean field itself, U n 1 , i.e., the near-perfect fitting results all reveal only the trivial aspect that U n 1 ≈ U n 1 .…”

“…Already more than a century ago, Osborne Reynolds was among the first to have realized that it is necessary to separate the fluctuating part from the mean flow when trying to analyze and understand the intricate nature of turbulence. Notwithstanding this, the authors of [1] "depart here from the usual approach" [p. 2] and instead of analyzing and showing the scaling behaviour of the turbulence-relevant fluctuation correlations, they only show the trivial scaling relation of the full-field correlations. By doing so, they claim to have finally found the prediction rule for the scaling of turbulence in two regions of turbulent channel flow, the inertial and center region.…”

“…The celebrated scaling of the fluctuation correlations of the velocity field, however, was shown in an earlier publication, but had to be corrected in [3], with the result that their new scaling theory is no better and partially even worse than the well-known classical one. Now, the authors' general argument 1 which is supposed to justify their approach of analyzing only the full-field correlations is that once the scaling relation of these is determined, the corresponding scaling relation of the fluctuation correlations can be retrieved and derived from it by just using the bijective Reynolds decomposition, which, for the streamwise components as studied in [1], is given iteratively by…”