Anisotropy analysis of a non-equilibrium turbulent boundary layer

Direct numerical simulation data of a non-equilibrium turbulent boundary layer evolving under adverse and then favorable pressure gradients (IJHFF (2025), 116; JFM (2024), 997:A75.) is analyzed to determine the state of turbulence in various regions of the flow. This database allows us to study the upstream history effects, a phenomenon that challenges turbulence models. The fundamental state of turbulence anisotropy is investigated using barycentric maps. Three distinct streamwise positions with different upstream histories are chosen: a small-defect APG (H=1.60, Re\textsubscript{$\theta$}= 5335, Re\textsubscript{$\tau$}=1010), a large-defect APG (H=2.78, Re\textsubscript{$\theta$}=11727, Re\textsubscript{$\tau$}=850), and a small-defect FPG (H=1.60, Re\textsubscript{$\theta$}=12287, Re\textsubscript{$\tau$}=2156) with a long non-equilibrium APG history. Near the wall, the dominance of $u^2$ over $v^2$ and $w^2$ weakens as the mean velocity defect increases. Further from the wall, the trajectory on the map deviates significantly, shifting toward a state resembling axisymmetric contraction, where $u^2$ and $w^2$ dominate $v^2$ more strongly than in small defect TBLs. In the subsequent FPG region, while the near-wall turbulence responds to the local favorable pressure gradient and resembles the small-defect APG case, the outer layer retains a distinct "memory" of the upstream APG. The result provides clear evidence that the inner and outer layers respond to changing pressure gradients at different rates, a challenge for turbulence models.