Mathematical model for logarithmic scaling of velocity fluctuations in wall turbulence

TL;DR

This paper develops a mathematical model explaining the logarithmic scaling of velocity fluctuations in wall turbulence, linking it to characteristic velocities and the absence of characteristic heights, supported by random variable mathematics.

Contribution

It introduces a novel mathematical framework for understanding the logarithmic scaling in wall turbulence, connecting phenomenological models with rigorous probabilistic conditions.

Findings

Logarithmic functions describe velocity fluctuation moments in wall turbulence.

Necessary and sufficient conditions are derived using random variable mathematics.

The model aligns with characteristics of eddy-based phenomenological models.

Abstract

For wall turbulence, moments of velocity fluctuations are known to be logarithmic functions of the height from the wall. This logarithmic scaling is due to the existence of a characteristic velocity and to the nonexistence of any characteristic height in the range of the scaling. By using mathematics of random variables, we obtain its necessary and sufficient conditions. They are compared with characteristics of a phenomenological model of eddies attached to the wall and also with those of the logarithmic scaling of the mean velocity.