Generalisation of the Prandtl relation for the skin-friction law to compressible turbulent channel and pipe flows

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\!f,i and the bulk Reynolds number Reb,i 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\!f,i∝ ln (Reb,i√ C\!f,i/2) . Moreover, it is quantitatively verified that the transformed C\!f,i and Reb,i , 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\!f,i and Reb,i 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.