On Prandtl's turbulent flow expressions for arbitrary power-law and shear-stress exponents

This work overviews two forms of Prandtl's power-law representation for turbulent motions that have been frequently used throughout the literature. After clarifying the difference between these two power-law expressions and their sources, a unified formulation is provided that enables us to recover the solutions for all boundary-layer properties and wall-normal velocity for arbitrary power-law and shear-stress parameters. This is accomplished over a broad range of Reynolds numbers by leveraging closed-form relations that accompany Kármán's momentum-integral approach. The availability of a generalized formulation helps to streamline the process of determining the most suitable combinations of power-law exponents and shear-stress coefficients for a given application and Reynolds number. Our findings indicate that both power-law and shear-stress exponents have an appreciable bearing on boundary-layer characteristics. Specifically, decreasing the overall power-law exponent leads to a larger disturbance thickness coefficient but lower skin friction, displacement, and momentum thickness coefficients. Conversely, increasing the overall shear-stress exponent results in systematically larger boundary-layer coefficients. By connecting classical power-law models to modern turbulence analyses, this work may provide a flexible tool for evaluating boundary-layer properties with added degrees of freedom and controlled accuracy. The resulting framework offers a simple avenue to reconcile empirical observations with basic power-law representations, thereby reinforcing Prandtl's fundamental formulation.