Mapping-based exact-integral formulation of skin-friction transformations for zero-pressure-gradient compressible turbulent boundary layers

Abstract

A long-standing route to efficient surface-drag prediction in zero-pressure-gradient compressible turbulent boundary layers is to map the skin-friction coefficient $C_f$ and momentum-thickness Reynolds number $Re_θ$ onto their `incompressible' counterparts. Reassessment against an extensive DNS database shows that existing formulations do not consistently recover the reference incompressible skin-friction behaviour, even when transformed data exhibit improved collapse. We define the mapped `incompressible' state as a constant-property counterpart of the physical compressible boundary layer and derive the transformation factors from prescribed mean-velocity and wall-normal-coordinate mappings. This definition-first approach links skin-friction scaling to the full-layer accuracy of the underlying velocity transformation and exposes inherited outer-layer errors. Van Driest's theory is recast in a finite-Re exact-integral form, with the classical vD I and II transformations recovered as leading-order asymptotic reductions. Their limitations at finite Reynolds numbers are quantified, and the historical success of vD II is traced to a fortuitous cancellation of truncation errors. The exact-integral formulation then yields modified transformations assessed through a priori scaling and standalone a posteriori prediction of $C_f$ from prescribed macroscopic and wall-thermal inputs. The VIPL-based modified transformation gives the best overall performance. Across $0.30 \leq M_\infty \leq 13.64$ and $-0.55 \leq \varTheta \leq 2.85$, its prediction errors remain below $11\%$, with a mean error of $3.07\%$. Overall, the analysis places skin-friction transformations on a mapping-based exact-integral footing, relating them directly to prescribed mean-flow mappings while avoiding the leading-order asymptotic truncations that limit classical van Driest theory at finite Reynolds numbers.