Complete eddy self-similarity in turbulent pipe flow

TL;DR

This study provides experimental evidence for the existence of geometrically self-similar eddies in turbulent pipe flow, supporting Townsend's attached eddy hypothesis and revealing full three-dimensional self-similarity across specific size ranges.

Contribution

The paper demonstrates the presence of fully three-dimensional geometrically self-similar eddies in turbulent pipe flow through detailed PIV measurements and analysis, confirming theoretical predictions.

Findings

Eddy structures exhibit geometric self-similarity in the (r, θ)-plane for spanwise scales from 1.03 to 0.175 R.

Two-point correlations show self-similarity in the streamwise direction for scales from 0.88 to 0.203 R.

Velocity magnitude also exhibits self-similarity, with velocity scale depending on eddy size.

Abstract

For wall-bounded turbulent flows, Townsend's attached eddy hypothesis proposes that the logarithmic layer is populated by a set of energetic and geometrically self-similar eddies. These eddies scale with a single length scale, their distance to the wall, while their velocity scale remains constant across their size range. To investigate the existence of such structures in fully developed turbulent pipe flow, stereoscopic particle image velocimetry measurements were performed in two parallel cross-sectional planes, spaced apart by a varying distance from 0 to 9.97$R$, for $Re_\tau = 1310$, 2430 and 3810. The instantaneous turbulence structures are sorted by width using an azimuthal Fourier decomposition, allowing us to create a set of average eddy velocity profiles by performing an azimuthal alignment process. The resulting eddy profiles exhibit geometric self-similar behavior in the $(r,\theta)$-plane for eddies with spanwise length scales ($\lambda_\theta/R$) spanning from 1.03 to 0.175. The streamwise similarity is further investigated using two-point correlations between the two planes, where the structures exhibit a self-similar behaviour with length scales ($\lambda_\theta/R$) ranging from approximately $0.88$ to $0.203$. The candidate structures thereby establish full three-dimensional geometrically self-similarity for structures with a volumetric ratio of $1:80$. Beside the geometric similarity, the velocity magnitude also exhibit self-similarity within these ranges. However, the velocity scale depends on eddy size, and follow the trends based on the scaling arguments proposed by \cite{Perry1986}.