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This study highlights the importance of satisfying the eddy viscosity equivalence below the logarithmic layer, to deriving accurate compressibility transformations. First, we analyze the ability of known transformations to satisfy the eddy viscosity equivalence and show that the accuracy of these transformation is strongly dependent on this ability. Secondly, in a step-by-step manner, we devise new transformations that satisfy this hypothesis. An approach based on curve fitting of the incompressible Direct Numerical Simulation data for eddy viscosity profiles below the logarithmic layer provides an extremely accurate transformation. That motivates self-contained methods, making use of mixing length formulas, in the inner region. It is shown that the accuracy of existing transformations can be significantly improved by applying these ideas, below the logarithmic layer. Motivated by the effectiveness of the formulations derived from eddy viscosity equivalence, we introduce a new integral transformation based on Reynolds number equivalence between compressible and incompressible flow. This approach is based on defining a new compressible velocity scale, which affects the accuracy of transformations. Several choices for the velocity scale are tested, and in each attempt, it is shown that the eddy viscosity equivalence plays a very important role for the accuracy of compressibility transformations.

Recent (Bose \& Durbin, \textit{Phys. Rev. Fluids}, 1, 073602, 2016) direct numerical simulations (DNS) of adverse- and zero-pressure-gradient boundary layers beneath moderate levels of free stream turbulence ($Tu$ $\le 2\%$) revealed a \textit{mixed mode} transition regime, intermediate between orderly and bypass routes. In this regime, the amplitudes of the Klebanoff streaks and instability waves are similar, and potentially can interact. Three-dimensional visualizations of transitional eddies revealed a helical pattern, quite distinct from the sinuous and varicose forms seen in pure bypass transition. This raises the fundamental question of whether the helical pattern could be attributed to a previously unknown instability mode. Two-dimensional stability analyses are performed herein for base flows extracted from DNS flow fields. The three-dimensional structure of the eigenfunction of the most unstable mode indeed reveals a helical pattern. These instability modes are obtained significantly earlier than helical structures are seen in flow visualizations, even for innocuous base flow structures. It is inferred that the instability mode is the underlying cause of helical patterns that emerge in the transition region. The streak configuration leading to the formation of the helical mode instability is different from those leading to sinuous and varicose modes in previous studies of pure bypass transition. Thus, the mixed mode precursor is the distinctive cause.