Logarithmic distribution of mean velocity and turbulent kinetic energy in a pipe flow

TL;DR

This paper develops a Lie-group based similarity theory that predicts universal logarithmic profiles for mean velocity and turbulent kinetic energy in turbulent pipe flow, validated by extensive experimental data and explaining recent observations.

Contribution

It introduces a new theoretical framework using Lie-group analysis to derive universal logarithmic profiles and a spatial invariant for turbulent pipe flow.

Findings

Universal Karman constant approximately 0.45.

High-accuracy prediction of mean velocity profiles up to Re=40 million.

Validation of the theory with experimental data and explanation of recent kinetic energy observations.

Abstract

A Lie-group based similarity theory is developed for both momentum and energy distributions in a turbulent pipe flow, leading to asymptotic logarithmic profiles of mean velocity and turbulent kinetic energy. Both channel and pipe data over a wide range of Re yield 0.45 to be the universal Karman constant. A new spatial invariant characterizing outer dynamics is discovered and validated by reliable experimental data. The theory predicts the mean velocity profile (MVP) with 99% accuracy for high Re experimental data (up to 40 millions), and offers a quantitative explanation for recent observation of logarithmic kinetic energy distribution by Hullmak et al. (Phys. Rev. Lett. 108, 094501).