Revisiting the mixing length scaling in pressure-gradient turbulent boundary layers via a symmetry approach
A century after Prandtl's mixing length hypothesis, full-profile scaling of the mixing length in pressure-gradient turbulent boundary layers (PG TBLs) remains debated, especially for adverse pressure gradients (APGs). This work presents a symmetry-based analytical model for the mixing length in equilibrium APG TBLs by extending the structural ensemble dynamics theory and coupling a two-layer total shear stress model. The framework unifies the inner layer, logarithmic region, half-power-law transition zone, and wake region with an invariant Karman constant. A critical Clauser parameter is identified, above which the logarithmic layer shrinks and transitions to the half-power-law scaling. The wake-region mixing-length parameter is analytically formulated, and the viscous sublayer and buffer layer thicknesses are determined self-consistently without ad hoc fitting. With only one finite-Reynolds-number correction parameter determined by the maximum shear stress, the model accurately predicts full profiles of mixing length, mean velocity, and Reynolds shear stress, validated against extensive numerical and experimental data. This work provides a unified, physically consistent framework for mixing-length scaling in PG TBLs and clarifies the transition mechanism from the log law to the half-power law under strong APG. It also enables assessment of the invariance of the logarithmic law and Karman constant using the full-profile scaling law of the mixing length.