Reconstructing self-similarity for improving wall skin-friction prediction under adverse pressure gradients

In the present study, we propose a novel skin-friction prediction formula based on a re-established self-similarity within the adverse-pressure-gradient (APG) turbulent boundary layer. The basic idea lies in introducing a novel velocity scale, which is derived mathematically and adapted physically from the linear total stress within the boundary layer. This scale assimilates concurrently and fundamentally the friction velocity, two distinct pressure-gradient velocity scales and the half-power law of the mean velocity in the intermediate region. Then, this scale formula is well validated across a comprehensive, multi-geometry database of APG flows over flat plates, curved plates, ramps and airfoils, which covers an unprecedented parameter range, with friction Reynolds number ranging from $10^2$ to $5\times 10^3$ and the Rotta–Clauser pressure-gradient parameter spanning from $10^{-1}$ to $10^2$ . Crucially, the proposed scale consistently recovers a classical logarithmic region across all tested APG conditions, thereby restoring the self-similar structures traditionally absent in strong or non-equilibrium pressure-gradient flows. Leveraging this reconstructed self-similarity, we further formulate a new, robust skin-friction prediction model which demonstrates predictive errors confined within $\pm 20\,\%$ for all the investigated non-equilibrium flow states.