2026 Zhu Compressible Turbulent Boundary Layer Skin Friction

A long-standing route to efficient surface-drag prediction in zero-pressure-gradient compressible turbulent boundary layers is to map the skin-friction coefficient Cf and momentum-thickness Reynolds number Reθ onto their ‘incompressible’ counterparts, for which established skin-friction scalings apply. Reassessment against an extensive direct numerical simulation database shows, however, that existing formulations do not consistently recover the reference incompressible skin-friction behaviour, even when transformed compressible data exhibit improved collapse. We therefore define the mapped ‘incompressible’ state as a constant-property counterpart of the physical compressible boundary layer and derive the associated transformation factors from prescribed mean-velocity and wall-normal-coordinate mappings. This definition-first approach treats the resulting skin-friction scaling to the full-layer accuracy of the underlying velocity transformation and exposes inherited outer-layer mapping errors. On this basis, van Driest’s skin-friction theory is recast within a finite-Reynolds-number exact-integral formulation, with the classical van Driest I and II transformations emerging as leading-order asymptotic reductions. We further demonstrate that these reductions are not uniformly reliable over practical parameter ranges, and the historical success of the classical van Driest II transformation is traced to a fortuitous cancellation of finite-Reynolds-number truncation errors. The exact-integral formulation is then used to construct modified skin-friction transformations, which are assessed through both a priori scaling and standalone a posteriori prediction of Cf from prescribed macroscopic flow and wall-thermal inputs.