Single-eddy contributions to streamwise velocity variance in turbulent pipe flow. Part 1. The single-eddy intensity function

Abstract

Townsend's attached-eddy hypothesis explains the asymptotic statistics of wall turbulence through a superposition of self-similar attached eddies. At a given observation location, active portions contribute to momentum transfer, whereas inactive portions retain only wall-parallel fluctuations. Since this distinction depends on the observation height relative to the eddy height $y_l$, we extract single-eddy intensity functions from DNS of turbulent pipe flow at $Re_τ\simeq930$--$6000$. Spectral linear stochastic estimation and differencing isolate wall-coherent contributions at prescribed $y_l$, yielding the streamwise and wall-normal intensity functions $I_{uu}$ and $I_{vv}$. When expressed in $y^*=y/y_l$, both collapse across $y_l$ and share a peak at $y_a^*$, close to the von K'arm'an constant. Near $y_a^*$, both components remain significant, whereas for $y^*\ll y_a^*$, $I_{vv}$ diminishes while $I_{uu}$ remains finite, identifying active and inactive portions. Over the self-similar range, both peak intensities scale as $y_l^{-1}$, and $I_{uu}$ retains this scaling in both portions. Integrating the active portion yields an approximately constant contribution, while the inactive portion recovers the classical near-wall logarithmic variation. An outer logarithmic tendency with a slope close to the Townsend--Perry constant arises from integrating active contributions up to their outer cutoff. At the near-wall peak of $\langle uu\rangle^+$, the scaling steepens to $I_{uu}\sim y_l^{-(1+β)}$ with $β\simeq1/4$, yielding a finite asymptote with a $Re_τ^{-1/4}$ defect. The additional exponent is attributed to viscous attenuation of inactive footprints associated with a $y_l$-dependent dissipative scale. Wall-coherent motions also produce a broad outer shoulder, while a localized wall-incoherent residual may contribute to a secondary peak.