Log-linear law of the mean streamwise velocity in turbulent boundary layers with moderate adverse pressure gradients

TL;DR

This paper introduces a new log-linear law for the mean velocity profile in turbulent boundary layers with moderate adverse pressure gradients, improving upon the classical log law for such conditions.

Contribution

The study derives and validates a log-linear velocity law for APG turbulent boundary layers, extending the classical log law applicable to ZPG conditions.

Findings

The log-linear law fits experimental data better than the classical log law under APGs.

The law reduces to the classical log law as the pressure gradient parameter approaches zero.

Validation covers a range of pressure-gradient parameters from 0.73 to 9.0.

Abstract

An essential feature of canonical zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs) is that the mean streamwise velocity exhibits a logarithmic dependence on the wall-normal distance, known as the log law. In this study, we demonstrate that this conventional log law is not suitable for turbulent boundary layers subjected to pressure gradients (PGs). Instead, a log--linear law is theoretically derived for TBLs under moderate adverse pressure gradients (APGs), based on the total shear-stress balance and a rescaled eddy-viscosity model. The validity of the proposed log--linear law is assessed using available datasets of incompressible APG TBLs with the Clauser pressure-gradient parameter $\beta$ ranging from 0.73 to 9.0. Compared with the conventional log law, the present log--linear formulation shows significantly improved agreement with the measured mean velocity profiles. In the limiting case of $\beta \to 0$, the proposed law naturally recovers the classical log law.