Quantifying non-equilibrium pressure-gradient turbulent boundary layers through a symmetry-based framework

This study establishes a symmetry-based framework to quantify non-equilibrium processes in complex pressure-gradient (PG) turbulent boundary layers (TBLs), using a Lie-group-informed dilation-symmetry-breaking formalism. We derive a universal multilayer defect scaling law for the evolution of total shear stress (TSS). The law shows that gradually varying adverse pressure gradients (APGs) break the dilation symmetry in the two-layer defect scaling of equilibrium TSS, leading to three-layer TSS structures. For abrupt PG transitions, we identify boundary-layer decoupling into (1) an equilibrium internal boundary layer and (2) a history-dependent outer flow, arising from disparate adaptation timescales. The framework introduces a unified velocity scale u** mapping non-equilibrium TSS to canonical zero-PG scaling. Validation spans different aerodynamic systems, including developing APG on airfoils, APG-to-favorable-PG transition on a Gaussian bump, and favorable PG to rapidly amplifying APG in a converging-diverging channel. The work enables improved prediction of non-equilibrium PG TBL behavior through a unified characterization of stress evolution dynamics, providing new physics-based parameterizations that could promote machine learning of complex wall-bounded turbulent flows.