The Minimal Attached Eddy in Wall Turbulence: Statistical Foundations, Inverse Identification and Influence Kernels

Abstract

Townsend's attached eddy hypothesis models the logarithmic region of high Reynolds number wall turbulence as a random superposition of wall-attached, geometrically self-similar eddies whose sizes obey a scale-invariant population law. Building on the statistical framework of Woodcock & Marusic (2015), the present work (i) poses an inverse problem to infer the ideal single-eddy contribution (influence) functions for the mean velocity and Reynolds stresses from DNS moments, (ii) uses these inferred kernels to guide a minimal Biot--Savart-consistent hairpin-type eddy built from Rankine vortex rods together with an inviscid image system, and (iii) introduces and infers a spectral Influence kernel that maps a self-similar eddy footprint to its one-dimensional energy spectrum. The Influence-kernel viewpoint yields a transparent explanation for the emergence (and limitations) of the linear part of the energy spectrum, provides a clear scale-by-scale decomposition and helps rationalize why simple eddy templates can reproduce a broad set of log-layer statistics once the mean-flow anchoring is fixed. Exact closed-form expressions for the mean influence function and the Fourier-space streamwise velocity of a general straight-segment hairpin family with image are derived, revealing a clean mean-variance duality: the horizontal head determines the entire mean kernel $I_1$ while the inclined legs dominate the spectral energy $I_\phi$. This structural insight explains why the rectangular hairpin occupies a singular corner of the eddy design space and why replacing it without degrading either mean or spectral predictions is difficult. The model is further extended by allowing the eddy population density to vary with scale, yielding near-perfect predictions of mean velocity and streamwise variance across $Re_\tau = 6000$--$20000$.