Revisiting the theory of van Driest: a general scaling law for the skin-friction coefficient of high-speed turbulent boundary layers

The skin-friction coefficient is a dimensionless quantity defined by the wall shear stress exerted on an object moving in a fluid, and it decreases as the Reynolds number increases for wall-bounded turbulent flows over a flat plate. In this work, a novel transformation, based on physical and asymptotic analyses, is proposed to map the skin-friction relation of high-speed turbulent boundary layers (TBLs) for air described by the ideal gas law to the incompressible skin-friction relation. Through this proposed approach, it has been confirmed theoretically that the transformed skin-friction coefficient $C_{f,i}$ , and the transformed momentum-thickness Reynolds number $Re_{\theta ,i}$ for compressible TBLs with and without heat transfer, follow a general scaling law that aligns precisely with the incompressible skin-friction scaling law, expressed as $ (2/C_{f,i} )^{1/2}\propto \ln Re_{\theta ,i}$ . Furthermore, the reliability of the skin-friction scaling law is validated by compressible TBLs with free-stream Mach number ranging from $0.5$ to $14$ , friction Reynolds number ranging from $100$ to $2400$ , and the wall-to-recovery temperature ratio ranging from $0.15$ to $1.9$ . In all of these data, $ (2/C_{f,i} )^{1/2}$ and $\ln Re_{\theta ,i}$ based on the present theory collapse to the incompressible relation, with a squared Pearson correlation coefficient reaching an impressive value $0.99$ , significantly exceeding $0.85$ and $0.86$ based on the established van Driest II and the Spalding–Chi transformations, respectively.