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

Karthik Duraisamy

TL;DR

This work extends Townsend's attached-eddy hypothesis by formulating an inverse problem to recover ideal single-eddy influence functions from DNS moments, then building a minimal Biot–Savart-consistent hairpin eddy template anchored to the inferred kernels. A spectral Influence kernel is introduced to map self-similar eddy footprints to the 1D energy spectrum, clarifying the origin of the $k_x^{-1}$ scaling and the log-layer behavior. The square hairpin with wall image reproduces mean and Reynolds-stress statistics across a range of Reynolds numbers, supporting the view that eddy morphology is secondary to kinematic influence functions once mean anchoring is fixed. The framework provides a transparent link between morphology, influence kernels, and spectra, offering a pathway to more sophisticated, yet parsimonious, models of wall turbulence and a direction for future inverse-design of eddy populations.

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 and 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 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.

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

TL;DR

This work extends Townsend's attached-eddy hypothesis by formulating an inverse problem to recover ideal single-eddy influence functions from DNS moments, then building a minimal Biot–Savart-consistent hairpin eddy template anchored to the inferred kernels. A spectral Influence kernel is introduced to map self-similar eddy footprints to the 1D energy spectrum, clarifying the origin of the scaling and the log-layer behavior. The square hairpin with wall image reproduces mean and Reynolds-stress statistics across a range of Reynolds numbers, supporting the view that eddy morphology is secondary to kinematic influence functions once mean anchoring is fixed. The framework provides a transparent link between morphology, influence kernels, and spectra, offering a pathway to more sophisticated, yet parsimonious, models of wall turbulence and a direction for future inverse-design of eddy populations.

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 and 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 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.
Paper Structure (19 sections, 43 equations, 12 figures, 1 table)

This paper contains 19 sections, 43 equations, 12 figures, 1 table.

Table of Contents

  1. Statistics of attached eddies
  2. Statistical assumptions and derivation of the Reynolds-stress formula
  3. Velocity superposition.
  4. Second moment and the cross-term issue.
  5. Reynolds stresses as covariances (cancellation of the cross term).
  6. Convergence of planar influence integrals
  7. Reynolds-stress kernels are less restrictive.
  8. Implications for Biot--Savart line-vortex eddies and the role of the image system.
  9. The ideal eddy contribution function
  10. Discrete formulation and identifiability
  11. A simplified influence-function model and the log law
  12. Predictions
  13. Physical and Statistical Insights
  14. Similarity structure and the Influence kernel
  15. From single-eddy spectra to the attached-eddy spectrum
  16. ...and 4 more sections

Figures (12)

  • Figure 1: Schematic of discrete representation of attached eddies with $n=21$ and $m=3$.
  • Figure 2: Optimal influence functions for $Re_\tau \approx 5200$ for the mean flow (left) and Reynolds stresses (right, with red=streamwise; green=spanwise; blue=wall-normal, and black=shear).
  • Figure 3: Reference (symbols) vs optimal attached eddy statistics for $Re_\tau \approx 5200$
  • Figure 4: A hypothetical model of the eddy influence function corresponding to the mean streamwise velocity (blue dashed lines) compared to the optimal influence function.
  • Figure 5: A prototypical hairpin-type eddy.
  • ...and 7 more figures