The minimal attached eddy in wall turbulence: statistical foundations, inverse identification and influence kernels

Townsend’s attached-eddy hypothesis models the logarithmic region of high-Reynolds-number wall turbulence as a random superposition of wall-attached, self-similar eddies whose sizes obey a scale-invariant population law. Building on the statistical framework of Woodcock & Marusic (2015 Phys. Fluids , vol. 27, no. 1, p. 015104), 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 direct numerical simulations, (ii) uses these inferred kernels to guide a minimal Biot–Savart-consistent hairpin-type eddy built from Rankine vortex rods 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 an explanation for the emergence 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. 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 mean-variance duality: for a rectangular hairpin, the horizontal head determines the entire mean kernel while the inclined legs dominate the spectral energy. It is shown that the rectangular hairpin yields a precise plateau in the influence function, resulting in a logarithmic mean velocity profile and desirable properties for the streamwise fluctuations. The model is further extended by calibrating the eddy population density as a function of scale, yielding accurate descriptions of mean velocity and streamwise variance across italic Re Subscript tau Baseline equals 6000 minus 20 000 Re τ = 6000 − 20 000 $\textit{Re}_{\tau }=6000{-}20\,000$ .