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An adverse-pressure-gradient turbulent boundary layer with nearly-constant $\boldsymbol{β\simeq 1.4}$ up to $\boldsymbol{Re_θ \simeq 8,700}$

In this study, a new well-resolved large-eddy-simulation (LES) of an incompressible near-equilibrium adverse-pressure-gradient (APG) turbulent boundary layer (TBL) over a flat plate is presented. In this simulation, we have established a near-equilibrium APG over a wide Reynolds-number range. In this so-called region of interest (ROI), the Clauser--Rotta pressure-gradient parameter $β$ exhibits an approximately constant value of around 1.4, and the Reynolds number based on momentum thickness reaches $Re_θ=8700$. To the authors' knowledge, this is to date the highest $Re_θ$ achieved for a near-equilibrium APG TBL under an approximately constant moderate APG. We evaluated the self-similarity of the outer region using two scalings, namely the Zagarola--Smits and an alternative one based on edge velocity and displacement thickness. Our results reveal that outer-layer similarity is achieved, and the viscous scaling collapses the near-wall region of the mean flow in agreement with classical theory. Spectral analysis reveals that the APG displaces some small-scale energy from the near-wall to the outer region, an effect observed for all the components of the Reynolds-stress tensor, which becomes more evident at higher Reynolds numbers. Generally, the effects of the APG are more noticeable at lower Reynolds numbers. For instance, the outer peak of turbulent-kinetic-energy (TKE) production is less prominent at higher $Re$. While the scale separation increases with $Re$ in zero-pressure-gradient (ZPG) TBLs, this effect becomes accentuated by the APG. Despite the reduction of the outer TKE production at higher Reynolds numbers, the mechanisms of energization of large scales are still present.