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

TL;DR

This paper develops a symmetry-based theory to predict velocity and stress profiles in equilibrium adverse pressure-gradient turbulent boundary layers, incorporating stress overshoot and invariant profiles at high pressure gradients.

Contribution

It introduces a novel symmetry approach to model total stress and velocity profiles in APG TBLs, including the effects of pressure-gradient induced stress overshoot and invariant ultimate states.

Findings

Accurately predicts the peak total stress location.

Derives invariant mean velocity profile at high pressure gradients.

Validates predictions with multiple DNS, LES, and experimental datasets.

Abstract

We propose a theory for predicting the mean velocity and Reynolds shear and normal stresses profiles in the wake region of equilibrium adverse pressure-gradient (PG, APG) turbulent boundary layers (TBLs). Firstly, we explore 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 overshoot and is proportional to the Clauser PG parameter $\beta$. The wall-normal location with peak stress is predicted. The total stress profiles with arbitrary $\beta$ are transformed into an invariant profile, which is the ultimate state of the total stress at infinite $\beta$. This transformation is equivalent to the outer scaling of the Reynolds shear stress recently-proposed by Wei & Knopp (JFM, 2023). The Reynolds normal stresses are predicted accordingly based on the similarity of the Reynolds shear and normal stresses in the wake region. Secondly, a defect power law is proposed for the stress and kinetic energy lengths in the wake region. Two critical parameters in the defect power law are identified to depend on $\beta$ and determine the length profiles. With the total stress and stress length models, the streamwise mean-velocity profile is predicted. Especially, an invariant mean velocity profile is derived, which describes the ultimate state of the mean velocity in the wake region at infinite $\beta$. This invariant profile is also equivalent to the outer scaling of Wei & Knopp. The theory also predicts the variation of the Coles' wake parameter $\Pi$ with $\beta$, in close agreement with the empirical relation that correlates hundreds of experimental data. The predictions are validated with five published DNS, LES, and experimental databases on the equilibrium APG TBLs.