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

## TL;DR

This paper introduces a symmetry-based framework to analyze and predict non-equilibrium pressure-gradient turbulent boundary layers, revealing universal scaling laws and flow structures across various aerodynamic conditions.

## Contribution

It develops a Lie-group-informed formalism to quantify non-equilibrium effects and derive a universal multilayer defect scaling law for turbulent boundary layers under pressure gradients.

## Key findings

- Universal multilayer defect scaling law for TSS evolution.
- Identification of boundary-layer decoupling during abrupt PG transitions.
- Validation across diverse aerodynamic systems.

## Abstract

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 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 unified characterization of stress evolution dynamics, providing new physics-based parameterizations that could promote machine learning of complex wall-bounded turbulent flows.