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 $\textit{Re}_{b,i}$ 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 $\sqrt {2/C_{\!f,i}}\propto \ln (\textit{Re}_{b,i}\sqrt {C_{\!f,i}/2})$ . Moreover, it is quantitatively verified that the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$ , 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 $\textit{Re}_{b,i}$ 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.