Consistent outer scaling and analysis of adverse pressure gradient turbulent boundary layers

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect U_e − U, an outwards shift of the peak value of the Reynolds shear stress −〈uv〉 and an appearance of the outer peak value of the Reynolds normal stress 〈uu〉. Here U_e is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling U_zs = U_e(δ^∗/δ) and ensure that the original properties of the Zagarola–Smits scaling are maintained as Re → ∞. Here δ^∗ is the displacement thickness and δ is the boundary layer thickness. Based on the established correlation between U_e − U and −〈uv〉, the scaling is extended to −〈uv〉. Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter β = (δ^∗/ρu²_τ)(dP_e/dx) of these databases spans two orders of magnitude. Here P_e is the boundary layer edge pressure, u_τ is the friction velocity and ρ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the −〈uv〉 peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.