Applicability tests for empirical formulas: Analysis of turbulent flows and general comments

Direct and differential-invariant methods for processing experimental data are used to analyze Nikuradse’s results for a turbulent flow of a viscous liquid in round smooth tubes for various Reynolds numbers. It is found that direct testing ratios do not make it possible to draw a correct inference about the applicability of the Prandtl logarithmic profile. The explanation of this paradoxical result is given. It is shown that although direct applicability tests for empirical formulas do not generally ensure the necessary conditions of their applicability, they make it possible to draw subtle qualitative inferences that cannot be reached using the least-squares method.