# Structure and interpolation of the turbulent velocity profile in parallel flow

## TL;DR

This paper examines the classical turbulence velocity profile theory, identifies discrepancies with experimental data, and proposes a correction based on pressure gradient effects to improve the theoretical description.

## Contribution

It introduces an analytical correction to the classical turbulence profile theory accounting for pressure gradient effects, enhancing its accuracy.

## Key findings

- Discrepancies in the logarithmic layer can be explained by pressure gradient effects.
- A universal correction improves the match between theory and experiments.
- The correction enables better analytical interpolation of velocity profiles.

## Abstract

The classical scaling theory of turbulent parallel flow provides a framework for the description of the mean velocity profile through two functions of one variable, traditionally named law of the wall and law of the wake, and a universal logarithmic law characterized by von K\'arm\'an's constant. Despite its widespread adoption in research and in teaching, discrepancies between this theory and both experiments and numerical simulations have been repeatedly observed in the literature. Recently we have shown that in the logarithmic layer such discrepancies can be physically interpreted and analytically accounted for through an equally universal correction caused by the pressure gradient. This finding opens the way to a likewise improvement in the description of the law of the wall and of the law of the wake, an analytical interpolation of either of which is often useful for practical applications.