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

Karthik Duraisamy

arxiv: 2301.10905 · v5 · submitted 2023-01-26 · ⚛️ physics.flu-dyn · math-ph· math.MP

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

Karthik Duraisamy This is my paper

Pith reviewed 2026-05-24 09:49 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math-phmath.MP

keywords attached eddy hypothesiswall turbulencelogarithmic layerhairpin vortexinfluence kernelinverse identificationReynolds stressespopulation density

0 comments

The pith

Allowing the population density of wall-attached eddies to vary with scale produces near-perfect predictions of mean velocity and streamwise variance across Reynolds numbers 6000 to 20000.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Townsend's attached eddy hypothesis by posing an inverse problem that extracts single-eddy influence functions directly from DNS moments for mean velocity and stresses. These kernels guide construction of a minimal hairpin eddy assembled from Rankine vortex rods and an inviscid image system that satisfies the Biot-Savart relation. Closed-form expressions for the mean influence and Fourier-space velocity reveal that the horizontal head fixes the mean kernel while the inclined legs control the spectral energy. When the model further permits the eddy population density to change with scale, it reproduces observed log-layer statistics with high accuracy over the stated Reynolds-number range.

Core claim

The turbulence statistics in the log layer are treated as a linear superposition of geometrically self-similar wall-attached eddies whose sizes follow a scale-invariant population law. An inverse identification procedure yields the ideal eddy contribution kernels, which are then used to design a Biot-Savart-consistent minimal hairpin. Exact expressions demonstrate a mean-variance duality: the horizontal head sets the entire mean kernel while the legs dominate the spectral influence. Allowing the population density to vary with scale then produces near-perfect agreement with DNS mean velocity and streamwise variance for Re_τ = 6000--20000.

What carries the argument

The influence kernel that maps the footprint of a self-similar eddy to its additive contributions to mean velocity, Reynolds stresses, and the one-dimensional energy spectrum.

If this is right

The horizontal head of the hairpin alone determines the mean influence function.
The inclined legs dominate the contribution to the one-dimensional energy spectrum.
The rectangular hairpin occupies a singular position in the space of possible eddy shapes because it satisfies both mean and spectral requirements simultaneously.
Simple geometric eddy templates can reproduce a wide set of log-layer statistics once the mean-flow anchoring is fixed.
A scale-by-scale decomposition of statistics becomes available through the spectral influence kernel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same inverse-kernel procedure could be applied to experimental data to test whether the inferred minimal eddy template remains consistent across facilities.
Deriving the functional form of the scale-dependent population density from a dynamical argument rather than fitting would remove the remaining free function in the model.
The mean-variance duality identified for hairpins may extend to other coherent-structure families in different shear flows.

Load-bearing premise

Log-layer turbulence statistics arise as a linear superposition of geometrically self-similar wall-attached eddies whose sizes obey a scale-invariant population law.

What would settle it

DNS data at Re_τ = 15000 in which no choice of scale-dependent population density can simultaneously match both the measured mean velocity profile and the streamwise variance would falsify the claim.

Figures

Figures reproduced from arXiv: 2301.10905 by Karthik Duraisamy.

**Figure 2.** Figure 2: Optimal Influence functions for 𝑅𝑒𝜏 ≈ 5200 for the mean flow (left) and Reynolds stresses (right, with red=streamwise; green=spanwise; blue=wall-normal, and black=shear) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Reference (symbols) vs optimal attached eddy statistics for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: A hypothetical model of the eddy influence function corresponding to the mean streamwise [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: A prototypical hairpin-type eddy modeled using vortex line segments. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Velocity moments and eddy influence function. Lines: Optimal eddy influence function. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: Representation of attached eddy using 20 parameters. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Velocity moments and eddy influence function. Lines: Optimal eddy influence function. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: Prototypical eddies: Optimal attached eddy (left), Hairpin packet (right). [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: Velocity moments of optimal attached eddy packet (symbols) compared to DNS data. [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

read the original 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 $Re_\tau = 6000$--$20000$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The inverse kernels and closed-form hairpin expressions are the concrete new pieces, but the near-perfect fits rely on letting population density vary with scale, which adds flexibility that needs scrutiny for uniqueness.

read the letter

The paper's main advance is posing the attached eddy model as an inverse problem to recover single-eddy influence functions for the mean and spectrum directly from DNS moments, then using those to build a minimal Rankine-vortex hairpin with an image system and derive exact closed-form expressions. The mean-variance duality they identify, where the head sets the mean kernel and the legs set the spectral energy, is a clean structural observation that explains why certain simple shapes perform well once the mean anchoring is fixed. The spectral influence kernel is also new relative to the Woodcock and Marusic 2015 framework they cite, and the exact expressions for the straight-segment hairpin family are reproducible work that can be verified directly. These steps give a more transparent scale-by-scale view of how the log-layer statistics emerge from the superposition. The paper does well at showing why replacing the rectangular hairpin is difficult without losing either the mean or spectral match. The soft spot is the inverse step. Recovering the kernels amounts to a statistical deconvolution of the population from the moments, and the abstract gives no sign of regularization or tests for whether other eddy families could produce equivalent results. Once they allow the eddy population density to vary with scale, the predictions of mean velocity and streamwise variance become near-perfect across Re_tau 6000-20000, but that scale dependence functions as an extra fitted degree of freedom. This makes it harder to judge how much credit belongs to the specific eddy construction versus the added flexibility. The stress-test concern about stability and uniqueness therefore lands and should be addressed with explicit checks in the manuscript. This is aimed at people working on structural models and closures for wall turbulence. The derivations are specific enough to warrant referee time, even if the inverse procedure needs more robustness evidence.

Referee Report

3 major / 1 minor

Summary. The paper extends Townsend's attached eddy hypothesis by formulating an inverse problem to recover single-eddy influence kernels I1 (mean velocity) and Iφ (streamwise spectrum) from DNS moments, constructing a minimal Biot-Savart-consistent hairpin eddy from Rankine vortex rods with an inviscid image system, deriving closed-form expressions for the mean influence and Fourier-space velocity of straight-segment hairpins, establishing a mean-variance duality (head sets I1, legs set spectral energy), and showing that allowing the eddy population density to vary with scale produces near-perfect matches to mean velocity and streamwise variance for Re_τ = 6000–20000.

Significance. If the recovered kernels prove unique and stable, the work supplies a transparent structural account of why simple eddy templates can reproduce log-layer statistics once mean-flow anchoring is fixed, together with exact analytic expressions that could serve as benchmarks. The explicit duality between head and legs is a useful organizing principle. The closed-form derivations constitute a clear technical contribution.

major comments (3)

[inverse problem section / abstract] Abstract and § on inverse problem: the inverse recovery of I1 and Iφ from DNS moments is presented as the foundation for both the eddy construction and the subsequent predictions, yet no regularization, conditioning analysis, or uniqueness test is described. Because the mapping from population to moments can be many-to-one, the recovered kernels (and therefore the “minimal” eddy) may not be unique; this directly affects the load-bearing claim that the constructed hairpin is the appropriate template.
[abstract] Abstract: the near-perfect predictions of mean velocity and streamwise variance across Re_τ = 6000–20000 are obtained only after the eddy population density is allowed to vary with scale. This introduces an explicit free parameter that is adjusted to the data, so the reported agreement is not a parameter-free test of the attached-eddy superposition; the central claim therefore rests on a fitted degree of freedom rather than a predicted population law.
[duality / eddy construction] Abstract and duality paragraph: the mean-variance duality (horizontal head determines entire I1 while inclined legs dominate Iφ) is derived for a general straight-segment hairpin family. It is not shown whether the specific Rankine-vortex minimal eddy constructed in the paper satisfies the same clean separation or whether the duality survives the image-system and Biot-Savart constraints used to define the “minimal” template.

minor comments (1)

[abstract / methods] Notation for the spectral kernel Iφ should be introduced with an explicit definition (e.g., relation to the one-dimensional spectrum) at first use to avoid ambiguity with the mean kernel I1.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below and will revise the manuscript accordingly. The responses clarify the status of the inverse problem, the role of the population density, and the duality verification.

read point-by-point responses

Referee: [inverse problem section / abstract] Abstract and § on inverse problem: the inverse recovery of I1 and Iφ from DNS moments is presented as the foundation for both the eddy construction and the subsequent predictions, yet no regularization, conditioning analysis, or uniqueness test is described. Because the mapping from population to moments can be many-to-one, the recovered kernels (and therefore the “minimal” eddy) may not be unique; this directly affects the load-bearing claim that the constructed hairpin is the appropriate template.

Authors: We agree that a conditioning analysis and uniqueness tests are missing and constitute a genuine limitation. In the revision we will add a subsection that (i) specifies the linear system solved for the kernels, (ii) reports its condition number, (iii) describes any implicit regularization arising from the discretization, and (iv) presents numerical experiments that perturb the input moments and examine the stability of the recovered I1 and Iφ. While the mapping is not guaranteed to be injective, the kernels that emerge are the unique solution (within the chosen basis) that simultaneously satisfy the mean and spectral moments and permit a Biot-Savart-consistent eddy; we will document this explicitly. revision: yes
Referee: [abstract] Abstract: the near-perfect predictions of mean velocity and streamwise variance across Re_τ = 6000–20000 are obtained only after the eddy population density is allowed to vary with scale. This introduces an explicit free parameter that is adjusted to the data, so the reported agreement is not a parameter-free test of the attached-eddy superposition; the central claim therefore rests on a fitted degree of freedom rather than a predicted population law.

Authors: We acknowledge that allowing the population density to vary with scale introduces a calibrated function and that the quantitative agreement is therefore not a parameter-free test of the classical attached-eddy hypothesis. The kernels I1 and Iφ themselves are obtained from the inverse problem without reference to this density; the scale dependence is introduced only in the final superposition step to reach near-perfect fidelity. In the revision we will (i) state explicitly that the density is fitted, (ii) report the functional form used, and (iii) separate the claims that rest solely on the kernels from those that rely on the calibrated density. revision: yes
Referee: [duality / eddy construction] Abstract and duality paragraph: the mean-variance duality (horizontal head determines entire I1 while inclined legs dominate Iφ) is derived for a general straight-segment hairpin family. It is not shown whether the specific Rankine-vortex minimal eddy constructed in the paper satisfies the same clean separation or whether the duality survives the image-system and Biot-Savart constraints used to define the “minimal” template.

Authors: The analytic duality is obtained for the entire family of straight-segment hairpins that already incorporates the inviscid image system. The minimal eddy is assembled from Rankine rods whose geometry is chosen so that its induced velocity matches the recovered kernels while obeying the same image condition and Biot-Savart law. In the revision we will add a direct verification—by evaluating the head-only and leg-only contributions of the constructed eddy under the full Biot-Savart operator—that the clean separation between I1 (head) and Iφ (legs) is preserved for this specific template. revision: yes

Circularity Check

1 steps flagged

Scale-dependent population density fitted to DNS yields near-perfect match by construction

specific steps

fitted input called prediction [Abstract (final sentence)]
"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 $Re_τ = 6000$--$20000$."

The population density is permitted to vary with scale and is inferred from the same DNS moments that define the target statistics; the resulting near-perfect agreement is therefore achieved by construction of this additional fitted function rather than predicted from a fixed scale-invariant law.

full rationale

The paper's central claim of near-perfect predictions of mean velocity and streamwise variance rests on extending the model by allowing eddy population density to vary with scale and inferring that variation from DNS moments. This introduces a fitted degree of freedom, making the match equivalent to the input data rather than an independent derivation. The inverse identification of influence kernels from DNS is external data and not circular, and the cited Woodcock & Marusic (2015) framework has no author overlap. No self-citation chains, uniqueness theorems, or ansatzes smuggled via citation are load-bearing. The physical eddy construction and duality derivation appear independent of the target statistics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The work rests on the attached-eddy hypothesis and the assumption that DNS moments can be inverted to recover unique single-eddy kernels; the population density variation is introduced as an adjustable function.

free parameters (1)

scale-dependent eddy population density
Introduced to achieve near-perfect agreement with DNS mean and variance profiles

axioms (1)

domain assumption Townsend attached eddy hypothesis with scale-invariant population law (Woodcock & Marusic 2015)
Explicitly stated as the statistical foundation for the inverse problem

invented entities (1)

minimal Biot-Savart-consistent hairpin-type eddy built from Rankine vortex rods with inviscid image system no independent evidence
purpose: To realize the inferred influence kernels while satisfying Biot-Savart consistency
Constructed to match the inverse kernels; no independent evidence outside the model is provided

pith-pipeline@v0.9.0 · 5849 in / 1279 out tokens · 21431 ms · 2026-05-24T09:49:37.247557+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Townsend’s attached eddy hypothesis … random superposition of wall-attached, geometrically self-similar eddies whose sizes obey a scale-invariant population law … p(h) = 2/h³ … I₁(z/h) … inverse problem to infer … influence kernels … allowing the eddy population density to vary with scale
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

exact closed-form expressions for the mean influence function … mean-variance duality: horizontal head determines I₁ while inclined legs dominate I_φ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

work page
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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Physics of Fluids 21 (5), 055105

Davidson, PA & Krogstad, P- 2009 A simple model for the streamwise fluctuations in the log-law region of a boundary layer . Physics of Fluids 21 (5), 055105

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[4]

Journal of Fluid Mechanics 550 , 51--60

Davidson, PA , Nickels, TB & Krogstad, P- 2006 The logarithmic structure function law in wall-layer turbulence . Journal of Fluid Mechanics 550 , 51--60

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[5]

Journal of Fluid Mechanics 894

Hwang, Yongyun & Eckhardt, Bruno 2020 Attached eddy model revisited using a minimal quasi-linear approximation . Journal of Fluid Mechanics 894

work page 2020
[6]

Journal of fluid mechanics 774 , 395--415

Lee, Myoungkyu & Moser, Robert D 2015 Direct numerical simulation of turbulent channel flow up to . Journal of fluid mechanics 774 , 395--415

work page 2015
[7]

Journal of fluid mechanics 868 , 698--725

Lozano-Dur \'a n, Adri \'a n & Bae, Hyunji Jane 2019 Characteristic scales of townsend’s wall-attached eddies . Journal of fluid mechanics 868 , 698--725

work page 2019
[8]

Annual Review of Fluid Mechanics 51 , 49--74

Marusic, Ivan & Monty, Jason P 2019 Attached eddy model of wall turbulence . Annual Review of Fluid Mechanics 51 , 49--74

work page 2019
[9]

Journal of Fluid Mechanics 716

Marusic, Ivan , Monty, Jason P , Hultmark, Marcus & Smits, Alexander J 2013 On the logarithmic region in wall turbulence . Journal of Fluid Mechanics 716

work page 2013
[10]

Physical Review Fluids 4 (8), 082601

McKeon, Beverley J 2019 Self-similar hierarchies and attached eddies . Physical Review Fluids 4 (8), 082601

work page 2019
[11]

Journal of Fluid Mechanics 119 , 173--217

Perry, AE & Chong, MS 1982 On the mechanism of wall turbulence . Journal of Fluid Mechanics 119 , 173--217

work page 1982
[12]

Perry, AE & Maru s i \'c , Ivan 1995 A wall-wake model for the turbulence structure of boundary layers. part 1. extension of the attached eddy hypothesis . Journal of Fluid Mechanics 298 , 361--388

work page 1995
[13]

The Bell System Technical Journal 23 (3), 282--332

Rice, Stephen O 1944 Mathematical analysis of random noise . The Bell System Technical Journal 23 (3), 282--332

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[14]

Journal of Fluid Mechanics 786 , 309--331

de Silva, Charitha M , Hutchins, Nicholas & Marusic, Ivan 2016 a\/ Uniform momentum zones in turbulent boundary layers . Journal of Fluid Mechanics 786 , 309--331

work page 2016
[15]

Physical Review Fluids 1 (2), 022401

de Silva, Charitha M , Woodcock, James D , Hutchins, Nicholas & Marusic, Ivan 2016 b\/ Influence of spatial exclusion on the statistical behavior of attached eddies . Physical Review Fluids 1 (2), 022401

work page 2016
[16]

Cambridge university press

Townsend, AAR 1976 The structure of turbulent shear flow\/ . Cambridge university press

work page 1976
[17]

Physics of Fluids 27 (1), 015104

Woodcock, JD & Marusic, I 2015 The statistical behaviour of attached eddies . Physics of Fluids 27 (1), 015104

work page 2015

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

work page

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page

[3] [3]

Physics of Fluids 21 (5), 055105

Davidson, PA & Krogstad, P- 2009 A simple model for the streamwise fluctuations in the log-law region of a boundary layer . Physics of Fluids 21 (5), 055105

work page 2009

[4] [4]

Journal of Fluid Mechanics 550 , 51--60

Davidson, PA , Nickels, TB & Krogstad, P- 2006 The logarithmic structure function law in wall-layer turbulence . Journal of Fluid Mechanics 550 , 51--60

work page 2006

[5] [5]

Journal of Fluid Mechanics 894

Hwang, Yongyun & Eckhardt, Bruno 2020 Attached eddy model revisited using a minimal quasi-linear approximation . Journal of Fluid Mechanics 894

work page 2020

[6] [6]

Journal of fluid mechanics 774 , 395--415

Lee, Myoungkyu & Moser, Robert D 2015 Direct numerical simulation of turbulent channel flow up to . Journal of fluid mechanics 774 , 395--415

work page 2015

[7] [7]

Journal of fluid mechanics 868 , 698--725

Lozano-Dur \'a n, Adri \'a n & Bae, Hyunji Jane 2019 Characteristic scales of townsend’s wall-attached eddies . Journal of fluid mechanics 868 , 698--725

work page 2019

[8] [8]

Annual Review of Fluid Mechanics 51 , 49--74

Marusic, Ivan & Monty, Jason P 2019 Attached eddy model of wall turbulence . Annual Review of Fluid Mechanics 51 , 49--74

work page 2019

[9] [9]

Journal of Fluid Mechanics 716

Marusic, Ivan , Monty, Jason P , Hultmark, Marcus & Smits, Alexander J 2013 On the logarithmic region in wall turbulence . Journal of Fluid Mechanics 716

work page 2013

[10] [10]

Physical Review Fluids 4 (8), 082601

McKeon, Beverley J 2019 Self-similar hierarchies and attached eddies . Physical Review Fluids 4 (8), 082601

work page 2019

[11] [11]

Journal of Fluid Mechanics 119 , 173--217

Perry, AE & Chong, MS 1982 On the mechanism of wall turbulence . Journal of Fluid Mechanics 119 , 173--217

work page 1982

[12] [12]

Perry, AE & Maru s i \'c , Ivan 1995 A wall-wake model for the turbulence structure of boundary layers. part 1. extension of the attached eddy hypothesis . Journal of Fluid Mechanics 298 , 361--388

work page 1995

[13] [13]

The Bell System Technical Journal 23 (3), 282--332

Rice, Stephen O 1944 Mathematical analysis of random noise . The Bell System Technical Journal 23 (3), 282--332

work page 1944

[14] [14]

Journal of Fluid Mechanics 786 , 309--331

de Silva, Charitha M , Hutchins, Nicholas & Marusic, Ivan 2016 a\/ Uniform momentum zones in turbulent boundary layers . Journal of Fluid Mechanics 786 , 309--331

work page 2016

[15] [15]

Physical Review Fluids 1 (2), 022401

de Silva, Charitha M , Woodcock, James D , Hutchins, Nicholas & Marusic, Ivan 2016 b\/ Influence of spatial exclusion on the statistical behavior of attached eddies . Physical Review Fluids 1 (2), 022401

work page 2016

[16] [16]

Cambridge university press

Townsend, AAR 1976 The structure of turbulent shear flow\/ . Cambridge university press

work page 1976

[17] [17]

Physics of Fluids 27 (1), 015104

Woodcock, JD & Marusic, I 2015 The statistical behaviour of attached eddies . Physics of Fluids 27 (1), 015104

work page 2015