Component-wise outer similarity of turbulent boundary layers with strong adverse-pressure gradient

The adverse-pressure-gradient (APG) effect challenges the classical scaling of turbulent boundary layers (TBLs) over a flat plate, and it remains unclear whether outer-layer similarity holds across a wide range of APG strengths.We assemble a benchmark comprising the present direct numerical simulation of a suction–blowing-induced separation bubble and published profiles, spanning Clauser parameters ${\beta }_e=0.04$–$180$ and Reynolds numbers $R{e}_{\theta }=530$–$19000$. Across this broad $({\beta }_e,R{e}_{\theta })$ parameter range, existing APG normalizations do not simultaneously collapse the two mean-velocity defects and four Reynolds-stress components, leaving open whether outer similarity persists. Instead of defining new scalings, we normalize each quantity by its value at a common location $y_s$. Results are shown using $y_s=0.5\delta$, beyond the outer stress peaks. For each of the six quantities, the normalized profiles exhibit a common outer shape across the conditions, with collapse emerging for $y\mathrm{/}\delta \gtrsim 0.2$ for the velocity defects and $y\mathrm{/}\delta \gtrsim 0.4$ for the Reynolds stresses. The collapse-error analysis shows that outer self-similarity is retained over a broad range of $y_s$ within the outer region and is not restricted to $y_s=0.5\delta$. Furthermore, explicit expressions are provided for the normalized outer profiles. These results support component-wise outer similarity under strong APGs and provide compact references for APG turbulence-model assessment and conditional profile reconstruction.