A Generalized Mixing Length Model with Adverse-Pressure-Gradient Effects

The adverse-pressure-gradient (APG) severely disrupts the inherent symmetry of the turbulent boundary layer (TBL). The universal physical behavior and scaling laws observed under zero-pressure-gradient (ZPG) conditions are significantly altered under APG, thereby challenging the predictive accuracy of turbulence models. To address this breakdown, we develop a generalized mixing length model by incorporating APG effects into a baseline ZPG framework. The model is directly constructed using APG-extended wall laws—superposing the log law and the half-power law for the mean velocity—and the logarithmic decay law for the total shear stress. Validated against ten databases spanning the entire APG region, various Reynolds numbers, and both equilibrium and non-equilibrium flows, the model demonstrates accurate predictions for both mixing length and mean velocity profiles. The key achievements are as follows: (i) a significant simplification of model parameters, wherein the inner model can be reduced to a parameterless scaling law, while outer parameters were reduced from five to two; and (ii) the unification of predictions across the entire APG range. This work demonstrates that consistent modeling of APG effects bridges slight and strong APGs, enabling a unified description of the APG region before separation and providing critical insights for turbulence modeling.