Is the log-law a first principle result from Lie-group symmetry analysis?

The invariance method of Lie-groups in the theory of turbulence carries the high expectation of being a first principle method for generating statistical scaling laws. The purpose of this comment is to show that this expectation has not been met so far. In particular for wall-bounded turbulent flows, the prospects for success are not promising in view of the facts we will present herein. Although the invariance method of Lie-groups is able to generate statistical scaling laws for wall-bounded turbulent flows, like the log-law for example, these invariant results yet not only fail to fulfil the basic requirements for a first principle result, but also are strongly misleading. To note here is that not the functional structure of the log-law itself is misleading, but that its invariant Lie-group based derivation yielding this function is what is misleading. By revisiting the study of Oberlack (2001) [M. Oberlack, A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, pp. 299-328] we will demonstrate that all Lie group generated scaling laws derived therein do not convince as first principle solutions. Instead, a rigorous derivation reveals complete arbitrariness rather than uniqueness in the construction of invariant turbulent scaling laws. Important to note here is that the key results obtained in Oberlack (2001) are based on several technical errors, which all will be revealed, discussed and corrected. The reason and motivation why we put our focus solely on Oberlack (2001) is that it still marks the core study and central reference point when applying the method of Lie-groups to turbulence theory. Hence it is necessary to shed the correct light onto that study. Nevertheless, even if the method of Lie-groups in its full extent is applied and interpreted correctly, strong natural limits of this method within the theory of turbulence exist, which, as will be finally discussed, constitute insurmountable obstacles in the progress of achieving a significant breakthrough