  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.
The reason 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) 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.
