The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow

TL;DR

This paper extends spectral models of turbulent pipe flow to better predict the variation of turbulence intensity and energy peaks in the intermediate layer, aligning with high Reynolds number experimental data.

Contribution

It introduces a new wavenumber range to the spectral model to account for the outer peak of turbulent kinetic energy and the mean flow gradient behavior.

Findings

The extended model matches high Reynolds number data for outer peak energy.

The mean flow gradient has an outer peak at the same location as turbulent kinetic energy.

The model predicts a logarithmic decay of the mean flow gradient beyond the outer peak.

Abstract

The spectral model of Perry, Henbest & Chong (1986) predicts that the integral length-scale varies very slowly with distance to the wall in the intermediate layer. The only way for the integral length scale's variation to be more realistic while keeping with the Townsend-Perry attached eddy spectrum is to add a new wavenumber range to the model at wavenumbers smaller than that spectrum. This necessary addition can also account for the high Reynolds number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high Reynolds number data by Hultmark, Vallikivi, Bailey & Smits (2012, 2013). The finding of Dallas, Vassilicos & Hewitt (2009) that it is the eddy turnover time and not the mean flow gradient which scales with distance to the wall and skin friction velocity in the intermediate layer implies, when combined with Townsend's (1976) production-dissipation balance, that the mean flow gradient has an outer peak at the same location as the turbulent kinetic energy. This is seen in the data of Hultmark, Vallikivi, Bailey & Smits (2012, 2013). The same approach also predicts that the mean flow gradient has a logarithmic decay at distances to the wall larger than the position of the outer peak, a qualitative prediction which the aforementioned data also support.