In this work, the classical Prandtl relation for the skin-friction law of incompressible turbulent channel and pipe flows is generalised to compressible cases. Specifically, based on the law of the wall and asymptotic analysis, a skin-friction transformation is proposed to map the skin-friction law of compressible turbulent channel and pipe flows to the classical Prandtl relation. It has been theoretically proven that the skin-friction coefficient C\!₅, ₈ and the bulk Reynolds number Re₁, ₈ for compressible turbulent channel and pipe flows, where the subscript i denotes the transformed quantity obtained from the proposed skin-friction transformation, adhere to the Prandtl relation, expressed as 2/C\!₅, ₈ (Re₁, ₈ C\!₅, ₈/2). Moreover, it is quantitatively verified that the transformed C\!₅, ₈ and Re₁, ₈, obtained from direct numerical simulations (DNS) of compressible turbulent channel flows with bulk Mach numbers ranging from 0. 2 to 4, and friction Reynolds numbers from 200 to 2000, elegantly collapse into the Prandtl relation for the incompressible skin-friction law. Additionally, the transformed C\!₅, ₈ and Re₁, ₈ from DNS of compressible turbulent pipe flows, with bulk Mach numbers ranging from 1. 5 to 3, and friction Reynolds numbers from 200 to 1000, are also unified with the Prandtl relation for the incompressible skin-friction law.
Zhao et al. (Mon,) studied this question.