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We propose a theory for predicting the mean velocity and Reynolds shear andnormal stresses profiles in the wake region of equilibrium adversepressure-gradient (PG, APG) turbulent boundary layers (TBLs). Firstly, weexplore the PG-induced dilation-symmetry-breaking of the total stress $\tau^+$to construct a modified defect power law for $\tau^+$. Crucially, a PG stress$P_0^+$ is identified, which quantifies the APG-induced total-stress overshootand is proportional to the Clauser PG parameter $\beta$. The wall-normallocation with peak stress is predicted. The total stress profiles witharbitrary $\beta$ are transformed into an invariant profile, which is theultimate state of the total stress at infinite $\beta$. This transformation isequivalent to the outer scaling of the Reynolds shear stress recently-proposedby Wei & Knopp (JFM, 2023). The Reynolds normal stresses are predictedaccordingly based on the similarity of the Reynolds shear and normal stressesin the wake region. Secondly, a defect power law is proposed for the stress andkinetic energy lengths in the wake region. Two critical parameters in thedefect power law are identified to depend on $\beta$ and determine the lengthprofiles. With the total stress and stress length models, the streamwisemean-velocity profile is predicted. Especially, an invariant mean velocityprofile is derived, which describes the ultimate state of the mean velocity inthe wake region at infinite $\beta$. This invariant profile is also equivalentto the outer scaling of Wei & Knopp. The theory also predicts the variation ofthe Coles' wake parameter $\Pi$ with $\beta$, in close agreement with theempirical relation that correlates hundreds of experimental data. Thepredictions are validated with five published DNS, LES, and experimentaldatabases on the equilibrium APG TBLs.

Quantifying equilibrium pressure-gradient turbulent boundary layers via a symmetry approach