Turbulent Boundary Layer Similarity Requirements for the Zagarola and Smits Parameters

A mathematical proof that identifies the parameters that lead to defect profile similarity in the 2D boundary layer is presented. Applied to the Zagarola and Smits similarity scaling parameters for turbulent boundary layers, the proof requires that the parameters satisfy two similarity conditions, one of which restricts their application to Clauser-style equilibrium datasets. This restrictive result contradicts the published literature, which claims that the Zagarola and Smits parameters work for most turbulent boundary layer flows including non-equilibrium flows. This new result adds to the existing theoretical and logical evidence that there are unresolved issues with the widespread applicability of the Zagarola and Smits parameters. In contrast, we tested the similarity scaling parameters identified in the defect velocity profile proof and found that they yield reasonable similar-like results for certain turbulent boundary layer equilibrium-type datasets, without the theoretical or logical issues encountered with the Zagarola and Smits parameters.