# Uncovering Optimal Attached Eddies in Wall-bounded Turbulence

Karthik Duraisamy
Department of Aerospace Engineering,
University of Michigan, Ann Arbor, MI 48109

###### Abstract

Townsend’s attached eddy hypothesis decomposes the logarithmic layer of high
Reynolds number turbulent boundary layers as a field of randomly distributed
self-similar eddies that are assumed to be attached to the wall and obey a
scaling relationship with the wall-distance. The attached eddy model has
emerged as an elegant and successful theory to explain the physics, structure
and statistics of the logarithmic layer in wall turbulence. Building on the
statistical framework of Woodcock & Marusic (2015), the present work details
quantitative results on the structure of the attached eddies and their impact
on velocity moment predictions. An inverse problem is posed to infer the ideal
eddy contribution function that yields precise first and second order velocity
moments obtained from direct numerical simulations. This ideal function is
further simplified and is used to guide the proposal of a hairpin-type
prototypical eddy which is shown to yield reasonable predictions of the
velocity moments. This hairpin-type structure is improved upon by a) solving a
constrained optimization problem to infer an idealized eddy shape, and b)
inferring the circulation distribution of a hairpin packet. The associated
forward and inverse modeling procedure is general enough to serve as a
foundation to model the flow beyond the log layer and the codes are open
sourced to the community to aid further research.

Characterizing and predicting the turbulent flow in the vicinity of a wall
continues to not only be a problem of scientific interest, but also one of
great practical relevance. Significant knowledge and insight has been gained
from the development of experimental diagnostics in the 1960s to the emergence
of direct numerical simulations in the 1980s, sophisticated measurements in
the 1990s, an emphasis on coherent structures in the 2000s and operator-based
dynamical models over the past decade. Against this backdrop of increasingly
sophisticated experimentation, computations and analyses, the attached eddy
hypothesis - originally proposed more than 50 years ago by Townsend (1976) \-
continues to serve as a simple, yet effective theory to qualitatively and
quantitatively describe structural and statistical aspects of turbulent
boundary layers, particularly focusing on the logarithmic region. The attached
eddy model (AEM) was given a firm mathematical footing by Perry & Chong
(1982), and further developed by Marusic and co-workers over the past 25 years
(Perry & Marušić, 1995; Marusic et al., 2013; Woodcock & Marusic, 2015; de
Silva et al., 2016b). Among other useful features, the AEM is able to explain
scaling behaviors of velocity moments, provide an explanation for uniform
momentum zones (de Silva et al., 2016a), and serve as a predictive model for
the von-Karman constant as a function of Reynolds number. A review of the
theory and developments of the AEM of wall turbulence can be found in Marusic
& Monty (2019). Over the past few years, theoretical and numerical analyses
(e.g. McKeon (2019); Lozano-Durán & Bae (2019)) have added further credibility
to this theory. Despite the success of the AEM, it is pertinent to remember
that it is fundamentally a statistical theory, and other hypotheses (e.g
Davidson et al. (2006); Davidson & Krogstad (2009)) can also be used to
explain the statistical characteristics of the log layer.

The attached eddy hypothesis is based on the principle that the physics and
statistical properties of the logarithmic layer can be explained by
considering geometrically self-similar eddies that extend from the wall. A key
assumption is that the length scale of each individual eddy follows a
probability distribution that is a function of the distance from the wall.
Hence, the term ‘attached’ alludes to the fact that every eddy can be assumed
to be randomly placed on the wall. The AEM is effectively an inviscid theory,
yet the range of scales is set by the Reynolds number.

A major advance was made by Woodcock & Marusic (2015) (henceforth W&M) who
established a rigorous statistical foundation for AEM. They provide a complete
derivation for all the velocity moments and demonstrate logarithmic scaling
relationships therein. They were also able to provide expressions for the
skewness and flatness of the wall-normal and spanwise fluctuations as a
function of the Reynolds number. While variants of the AEM continue to be
developed in the literature (e.g. Hwang & Eckhardt (2020)), we consider W&M as
the starting point of our exploration and exclusively consider zero pressure
gradient boundary layers. Particularly, the following questions are addressed:

  1. 1.

Townsend’s original work introduced the concept of the ‘eddy intensity
function’, later formalized by W&M as the ‘eddy contribution function’ defined
to be proportional to the average (in the streamwise and spanwise directions)
of the individual and pairwise velocity components. Velocity moments can then
be described by a weighted integral of the eddy contribution function over all
eddy sizes. While Townsend gave remarkably insightful descriptions of the
nature of this function (Figure 5.7, page 155 of Townsend (1976)), its precise
from is not typically discussed in the literature. We provide a quantitative
characterization of this function and solve an inverse problem to infer the
shape of this function to precisely match first and second order velocity
moments obtained from DNS data. A simple model of the eddy contribution
function is proposed and the resulting statistics is also described.

  2. 2.

Since the AEM invokes self-similarity, the mechanics of the velocity
fluctuations is governed by the prototypical eddy shape. W&M utilize an eddy
that has a complex shape (Figure 1 in W&M), presumably configured using
insight from DNS and/or PIV fields. It is also unclear whether the W&M eddy
satisfies the Navier-stokes equations. This work assumes extremely simple line
vortex-based eddies and uses them as building-blocks for quantitative
development.

  3. 3.

The sensitivity of the statistics and velocity moments to the eddy structure
is unknown. In this work, we solve a constrained optimization problem to
extract optimal eddy shapes. It is noted that Perry & Chong (1982) provide
valuable yet qualitative insight on the impact of simple eddy shapes. The role
of individual vs packets of eddies is also not well-quantified. In this work,
we attempt to characterize the impact of both of these prototypes on the
statistics of interest.

  4. 4.

The AEM is one of the most elegant and successful theories that explains the
physics, structure and statistics in turbulence. While the theory has always
provided valuable insight and W&M have provided a firm statistical foundation,
in the author’s opinion, the AEM remains somewhat esoteric to a sizeable
fraction of the fluid mechanics community. The author hopes that the building-
block nature of the present work, and the fact that the associated forward and
inverse modeling tools are available to the community
111https://github.com/CaslabUM/AttachedEddy will make this topic more
accessible, and serve as a starting point for further research.

## 1 Statistics of attached eddies

We begin with a short description of the attached eddy hypothesis and
modeling. This presentation generally follows W&M with a slightly different
pedagogy. Consider a collection of $n$ eddies (Figure 1) of length scale
$h_{e,i}$ that are placed on the wall at locations $\mathbf{x}_{e,i}$. Since
these are attached eddies, it is implicit that only the streamwise and
spanwise components of $\mathbf{x}_{e,i}$ are variable. Define
$\mathbf{h}_{e}\triangleq\\{h_{e,1},h_{e,2},..,h_{e,n}\\}$ and
$\mathbf{X}\triangleq\\{\mathbf{x}_{e,1},\mathbf{x}_{e,2},..,\mathbf{x}_{e,n}\\}$.
Assuming self-similarity and linearity, a quantity $q$ (e.g. spanwise
velocity) evaluated at a location $\mathbf{x}$ can be determined using the
superposition
$q(\mathbf{x},\mathbf{X}_{e},\mathbf{h}_{e},n)\triangleq\sum_{i=1}^{n}q\left(\frac{\mathbf{x}-\mathbf{x}_{e,i}}{h_{e,i}}\right).$

e

Figure 1: Schematic of discrete representation of attached eddies with $n=21$
and $m=3$.

Restricting our attention to a streamwise and spanwise square wall patch of
side $2L$ in which the eddies are assumed to be idependently and uniformly
distributed, and assuming that the range of eddy length scales follows a
probability density function $p(h)$, the expectation of $q$ in this region is
given by

$q(\mathbf{x},n)\triangleq\mathbb{E}_{\mathbf{X}_{e},\mathbf{h}_{e}}[q(\mathbf{x},\mathbf{X}_{e},\mathbf{h}_{e},n)]=\frac{1}{4L^{2}}\int_{h_{min}}^{h_{max}}\int_{-L}^{L}\int_{-L}^{L}\sum_{i=1}^{n}q\left(\frac{\mathbf{x}-\mathbf{x}_{e,i}}{h}\right)p(h)d\mathbf{x}_{e,i}dh.$
(1)

Homogenizing in the $x,y$ directions, assuming $L$ is large enough that each
eddy centered at the origin has a negligible induced contribution outside an
area $4L^{2}$ (Refer Campbell’s theorem Rice (1944) and appendix of W&M), it
can be shown that

$q(z,n)\triangleq\mathbb{E}_{x,y}[q(\mathbf{x},n)]\approx\frac{n}{4L^{2}}\int_{h_{min}}^{h_{max}}\int_{-L}^{L}\int_{-L}^{L}q\left(\frac{\mathbf{x}}{h}\right)p(h)dxdydh.$

Note that in the above equation, the entire field is written as a function of
one prototypical eddy, which is scaled by the probability density function of
the eddy sizes $p(h)$ and the eddy density $n/(4L^{2})$. If the mean eddy
density is $\beta$, then using Poisson’s law, the expected value (over all
numbers of eddies) is

$Q(z)\triangleq\mathbb{E}_{n}[q(z,n)]=\beta\int_{h_{min}}^{h_{max}}\int_{-L}^{L}\int_{-L}^{L}q\left(\frac{\mathbf{x}}{h}\right)p(h)dxdydh.$
(2)

Note that all the $q$’s defined above are random variables, yet $Q(z)$ is a
deterministic quantity. Now we are in a position to define the mean streamwise
velocity $U(z)$ as a superposition of eddies of various sizes $h$. This can be
written in terms of the induced velocity field $u_{1}(\cdot)$ of one
prototypical eddy.

$\displaystyle U(z)$
$\displaystyle\triangleq\beta\int_{h_{min}}^{h_{max}}\int_{-L}^{L}\int_{-L}^{L}u_{1}\left(\frac{\mathbf{x}}{h}\right)p(h)dxdydh+U_{\infty}$
$\displaystyle=\beta\int_{h_{min}}^{h_{max}}p(h)h^{2}\left[\int_{-L/h}^{L/h}\int_{-L/h}^{L/h}u_{1}\left(\frac{\mathbf{x}}{h}\right)d\left(\frac{x}{h}\right)d\left(\frac{y}{h}\right)\right]dh+U_{\infty}$
$\displaystyle=\beta\int_{h_{min}}^{h_{max}}p(h)h^{2}I_{1}\left(\frac{z}{h}\right)dh+U_{\infty},$
(3)

where the mean flow eddy contribution function is

$\displaystyle I_{1}\left(\frac{z}{h}\right)$
$\displaystyle\triangleq\int_{-L/h}^{L/h}\int_{-L/h}^{L/h}u_{1}\left(\frac{\mathbf{x}}{h}\right)d\left(\frac{x}{h}\right)d\left(\frac{y}{h}\right).$
(4)

Similarly, we can define the Reynolds stress tensor as

$\displaystyle R_{ij}(z)$
$\displaystyle\triangleq\beta\int_{h_{min}}^{h_{max}}p(h)I_{ij}\left(\frac{z}{h}\right)dh,$
$\displaystyle\textrm{ where }I_{ij}\left(\frac{z}{h}\right)$
$\displaystyle\triangleq\int_{-L/h}^{L/h}\int_{-L/h}^{L/h}u_{i}\left(\frac{\mathbf{x}}{h}\right)u_{j}\left(\frac{\mathbf{x}}{h}\right)d\left(\frac{x}{h}\right)d\left(\frac{y}{h}\right).$

Note the presence of an additional freestream velocity in the definition of
$U(z)$. This is required because we are working with induced velocity
fluctuations. The final piece we need is the probability distribution of the
eddy sizes. Using insight from Townsend (1976) and Perry & Chong (1982), W&M
propose that $p(h)\propto 1/h^{3}$ and thus
$p(h)=\frac{2/h^{3}}{1/h_{\min}^{2}-1/h_{\min}^{2}}$, with the note that
$h_{\max}=\delta$ (i.e.) the boundary layer thickness and $h_{\min}$ is set by
the friction Reynolds number $Re_{\tau}$. For notational simplicity, all
length and velocity scales should be assumed to be in wall units henceforth.

## 2 The ideal eddy contribution function and a practical model

We now address the following question: How should the ’perfect’ eddy
contribution function look like? Townsend and Perry & Chong have provided
insight into the behavior of the eddy distribution function, but the current
objective is to be precise. Towards this end, consider a discrete set of eddy
sizes
$\tilde{\mathbf{h}}=\\{h_{1},h_{2},h_{3},..,h_{m}\\}=\\{h_{\min},h_{\min}+\Delta
h,h_{\min}+2\Delta h,..,h_{\max}\\}$ and a similar discretization of the wall-
normal coordinate
$\tilde{\mathbf{z}}=\\{z_{1},z_{2},z_{3},..,z_{m}\\}=\\{h_{\min},h_{\min}+\Delta
h,h_{\min}+2\Delta h,..,h_{\max}\\}$. Then, the discretized velocity moments
are

$\displaystyle U(z_{r};\mathbf{d})$
$\displaystyle=\beta\sum_{l=1}^{m}p(h_{l})\left[\sum_{q}\sum_{p}u_{1}\left(\frac{{x_{p},y_{q},z_{r}}}{h_{l}};\mathbf{d}\right)\left(\frac{\Delta
x}{h_{l}}\right)\left(\frac{\Delta y}{h_{l}}\right)\right]\Delta h+U_{\infty}$
$\displaystyle R_{ij}(z_{r};\mathbf{d})$
$\displaystyle=\beta\sum_{l=1}^{m}p(h_{l})\left[\sum_{q}\sum_{p}u_{i}\left(\frac{{x_{p},y_{q},z_{r}}}{h_{l}};\mathbf{d}\right)u_{j}\left(\frac{{x_{p},y_{q},z_{r}}}{h_{l}};\mathbf{d}\right)\left(\frac{\Delta
x}{h_{l}}\right)\left(\frac{\Delta y}{h_{l}}\right)\right]\Delta h.$

This can be represented as a matrix vector product in the form
$\mathbf{U}(\tilde{\mathbf{z}})-U_{\infty}=\mathbf{A}\mathbf{b},$ where
$A_{ij}=I_{1}(z_{i}/h_{j})$ and $b_{i}=\beta p(h_{i})h_{i}^{2}\Delta h.$ We
would now like to extract these coefficients. Towards this end, we define the
eddy contribution function at selected locations, and interpolate for the
values in other locations in a piecewise linear fashion. In other words

$\mathbf{A}=\begin{bmatrix}I_{1}(1)&I_{1}(h_{1}/h_{2})&I_{1}(h_{1}/h_{3})&..&I_{1}(h_{1}/h_{m-1})&I_{1}(h_{1}/h_{m})\\\
I_{1}(h_{2}/h_{1})&I_{1}(1)&I_{1}(h_{2}/h_{3})&..&I_{1}(h_{2}/h_{m-1})&I_{1}(h_{2}/h_{m})\\\
..&..&..&..&..&..\\\
I_{1}(h_{m}/h_{1})&I_{1}(h_{m}/h_{2})&I_{1}(h_{m}/h_{3})&..&I_{1}(h_{m}/h_{m-1})&I_{1}(1)\end{bmatrix}$
$\triangleq\begin{bmatrix}c_{1}&c_{m+1}&c_{m+2}&..&c_{2m-2}&c_{2m-1}\\\
c_{2}&c_{1}&I_{1}(h_{2}/h_{3})&..&I_{1}(h_{2}/h_{m-1})&I_{1}(h_{2}/h_{m})\\\
..&..&..&..&..&..\\\
c_{m}&I_{1}(h_{m}/h_{2})&I_{1}(h_{m}/h_{3})&..&I_{1}(h_{m}/h_{m-1})&c_{1}\\\
\end{bmatrix}.$

The unknown $I_{1}(\cdot)$ values are then interpolated 222For instance, if
$h_{\min}=186$ and $h_{\max}=5186$ and $m=11$, then
$I_{1}(h_{2}/h_{3})=0.42106c_{1}+0.57894c_{12}$ from the nodal locations
$\mathbf{c}$. Note that there are more unknowns ($2m$) than equations ($m$),
and so while it is possible to determine the unknowns that lead to perfect
match of a reference (DNS or experiment) velocity profiles, a choice has to be
made on the problem formulation. We choose to define the following least norm
problem:

$\\{\mathbf{c}_{\textrm{opt}},\beta_{\textrm{opt}}\\}=\min_{\mathbf{c},\beta}||\mathbf{U}_{ref}(\tilde{\mathbf{z}})-U_{\infty}-\mathbf{A}\mathbf{b}||_{2}^{2}.$

Figure 2 shows the optimal eddy contribution functions inferred from the
channel flow data (Lee & Moser, 2015) at friction Reynolds number
$Re_{\tau}\approx 5200$. Figure 3 confirms that the first and second velocity
moments are perfectly reproduced.

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
$Re_{\tau}\approx 5200$

Consider a collection of attached eddies that result in a simple function as
shown in Fig. 4.

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.

It will be shown below that it is indeed possible to construct an attached
eddy that corresponds to such a function. Using this influence function, one
can reconstruct the mean streamwise velocity as

$\displaystyle U(z)$
$\displaystyle=\frac{2\beta}{1/h_{\min}^{2}-1/h_{\max}^{2}}\int_{h_{min}}^{h_{max}}\frac{I_{1}\left({z}/{h}\right)}{h}dh+U_{\infty}$
$\displaystyle=\frac{2\beta}{1/h_{\min}^{2}-1/h_{\max}^{2}}\int_{z/h_{max}}^{z/h_{min}}\frac{I_{1}\left({z}/{h}\right)}{z/h}d(z/h)+U_{\infty}$
$\displaystyle\approx\frac{2\beta}{1/h_{\min}^{2}-1/h_{\max}^{2}}\left[\int_{z/h_{max}}^{1}\frac{-a_{0}}{z/h}d(z/h)+\int_{1}^{a_{1}/a_{2}}\frac{a_{1}-a_{2}z/h}{z/h}d(z/h)\right]+U_{\infty}$
$\displaystyle=\frac{2\beta}{1/h_{\min}^{2}-1/h_{\max}^{2}}\left[\int_{z/h_{max}}^{1}\frac{-a_{0}}{z/h}d(z/h)+\int_{1}^{a_{1}/a_{2}}\frac{a_{1}-a_{2}z/h}{z/h}d(z/h)\right]+U_{\infty}$
$\displaystyle=\frac{2\beta}{1/h_{\min}^{2}-1/h_{\max}^{2}}\left[a_{0}\log[z/h_{\max}]+a_{1}\log[a_{1}/a_{2}]-a_{1}+a_{2}\right]+U_{\infty}.$

It is thus clear that the Karman constant can be constructed as

$\kappa=\frac{1/h_{\min}^{2}-1/h_{\max}^{2}}{2a_{0}\beta}.$

Note that $W\&M$ use a Taylor series approximation on a generic eddy to derive
an alternate expression for $\kappa$.

Now, we answer the question whether it is possible to recreate the above
hypothetical influence function using an attached eddy. Consider a square
hairpin (Figure 5) with unit circulation as the attached eddy, along with its
image across the $z=0$ plane to enforce no-penetration. It is remarked that
this eddy has a square structure, yet the leg of the eddy aligned with the
wall is canceled by an equal and opposite image vortex pair, and is thus
consistent with the kinematics. Using the Biot-savart law to compute the
induced velocity $u_{1}$ in Eqn. 4, Figure 6 shows that such a simple attached
eddy can reproduce the mean streamwise velocity extremely accurately. Further,
the second moments are also seen to be reasonably well-predicted as shown in
Figure 6.

Figure 5: A prototypical hairpin-type eddy modeled using vortex line segments.

Figure 6: Velocity moments and eddy influence function. Lines: Optimal eddy
influence function. Symbols: Square hairpin at $45^{\circ}$. For the right
figures, colors follow Figure 2.

## 3 The optimal attached eddy

To assess whether the above attached eddy shape can be improved to yield more
accurate statistics, we pose a constrained optimization problem. The
prototypical attached eddy is now represented using 20 parameters as shown in
Fig. 7. It is noted that the shape is symmetric across the $y=0$ plane, and a
mirror image is used across the $z=0$ plane to satisfy inviscid boundary
conditions.

Figure 7: Representation of attached eddy using 20 parameters.

The following optimization problem is solved

$\displaystyle\mathbf{d}_{\textrm{opt}}$
$\displaystyle=\min_{\mathbf{d},\beta}(||\mathbf{U}_{ref}(\tilde{\mathbf{z}})-\mathbf{U}(\tilde{\mathbf{z}};\mathbf{d})||_{2}^{2}+||\mathbf{R}_{11,ref}(\tilde{\mathbf{z}})-\mathbf{R}_{11}(\tilde{\mathbf{z}};\mathbf{d})||_{2}^{2}+||\mathbf{R}_{22,ref}(\tilde{\mathbf{z}})-\mathbf{R}_{22}(\tilde{\mathbf{z}};\mathbf{d})||_{2}^{2}$
$\displaystyle+||\mathbf{R}_{33,ref}(\tilde{\mathbf{z}})-\mathbf{R}_{33}(\tilde{\mathbf{z}};\mathbf{d})||_{2}^{2}+||\mathbf{R}_{13,ref}(\tilde{\mathbf{z}})-\mathbf{R}_{13}(\tilde{\mathbf{z}};\mathbf{d})||_{2}^{2})$
$\displaystyle\textrm{s.t. }\mathbf{C}\mathbf{d}\leq\mathbf{f}\textrm{ and
}\mathbf{d}_{l}\leq\mathbf{d}\leq\mathbf{d}_{u},$

where the constraints $\mathbf{C}$ and $\mathbf{f}$ are designed to ensure
that $d_{20}>d_{17}>d_{14}>d_{11}>d_{8}>d_{5}>d_{2}>0$, and
$1>d_{19}>d_{16}>d_{13}>d_{10}>d_{7}>d_{4}>0$. The bounds
$\mathbf{d}_{l},\mathbf{d}_{u}$ ensure that the x and y extents of the
prototypical eddy is bounded. Sequential quadratic programming is used to
solve the above constrained optimization problem. The results of the
optimization is shown in Figures 8, 9, and improvement is noticeable over the
simpler eddy (Figure 6) .

Figure 8: Velocity moments and eddy influence function. Lines: Optimal eddy
influence function. Symbols: Optimal attached eddy. For the right figures,
colors follow Figure 2.

Figure 9: Prototypical eddies: Optimal attached eddy (left), Hairpin packet
(right).

Figure 10: Velocity moments of optimal attached eddy packet (symbols) compared
to DNS data. For the right figures, colors follow Figure 2.

At this juncture, we remark that the AEM is a statistical and non-dynamic
model, and thus prototypical eddies cannot be considered to be physical
structures. However, from a conceptual and aesthetic standpoint, the eddy
shape in Figure 9 may be unsatisfactory. Indeed, analyses and observations
from experiments and numerical simulations have suggested the following
features of dynamically-important eddies: a) Hairpin-type structures are
present the form of eddy packets (rather than individual eddies ), and b) A
preferential alignment of approximately $45^{\circ}$ to the wall. Simply
stacking the hairpins of Figure 5 with equal circulation strengths did not
yield meaningful results, and thus the circulation strength of the individual
hairpins in the packet (Figure 9) were optimized and found to follow a roughly
quadratic distribution with hairpin size. It is noted that - in contrast to
the above square hairpins, single / stacked lambda vortices did not yield
accurate predictions of the statistics.

We conclude by stating that there are several directions for future work,
including a) a more complete consideration of statistics such as the structure
function, high-order moments, and the energy spectrum; b) assessment of the
impact of the prototypical eddy in explaining physical characteristics of the
flow; c) use of more sophisticated topological optimization and richer
parametrizations of the eddies; and d) extension to other scenarios including
boundary layers with pressure gradients and in combination with detached
eddies. Further we note that opportunities exist in modeling flow regions
beyond the log layer. In fact, the current inference approach and predictions
already include the outer layer.

Acknowledgement This research was supported by NASA grant # 80NSSC18M0149
(Technical monitor: Dr. Gary Coleman).

Declaration of interests. The author reports no conflict of interest.

## References

  * Davidson & Krogstad (2009) Davidson, PA & Krogstad, P-Å 2009 A simple model for the streamwise fluctuations in the log-law region of a boundary layer. Physics of Fluids 21 (5), 055105.
  * Davidson et al. (2006) Davidson, PA, Nickels, TB & Krogstad, P-Å 2006 The logarithmic structure function law in wall-layer turbulence. Journal of Fluid Mechanics 550, 51–60.
  * Hwang & Eckhardt (2020) Hwang, Yongyun & Eckhardt, Bruno 2020 Attached eddy model revisited using a minimal quasi-linear approximation. Journal of Fluid Mechanics 894.
  * Lee & Moser (2015) Lee, Myoungkyu & Moser, Robert D 2015 Direct numerical simulation of turbulent channel flow up to. Journal of fluid mechanics 774, 395–415.
  * Lozano-Durán & Bae (2019) Lozano-Durán, Adrián & Bae, Hyunji Jane 2019 Characteristic scales of townsend’s wall-attached eddies. Journal of fluid mechanics 868, 698–725.
  * Marusic & Monty (2019) Marusic, Ivan & Monty, Jason P 2019 Attached eddy model of wall turbulence. Annual Review of Fluid Mechanics 51, 49–74.
  * Marusic et al. (2013) Marusic, Ivan, Monty, Jason P, Hultmark, Marcus & Smits, Alexander J 2013 On the logarithmic region in wall turbulence. Journal of Fluid Mechanics 716.
  * McKeon (2019) McKeon, Beverley J 2019 Self-similar hierarchies and attached eddies. Physical Review Fluids 4 (8), 082601.
  * Perry & Chong (1982) Perry, AE & Chong, MS 1982 On the mechanism of wall turbulence. Journal of Fluid Mechanics 119, 173–217.
  * Perry & Marušić (1995) Perry, AE & Marušić, 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.
  * Rice (1944) Rice, Stephen O 1944 Mathematical analysis of random noise. The Bell System Technical Journal 23 (3), 282–332.
  * de Silva et al. (2016a) de Silva, Charitha M, Hutchins, Nicholas & Marusic, Ivan 2016a Uniform momentum zones in turbulent boundary layers. Journal of Fluid Mechanics 786, 309–331.
  * de Silva et al. (2016b) de Silva, Charitha M, Woodcock, James D, Hutchins, Nicholas & Marusic, Ivan 2016b Influence of spatial exclusion on the statistical behavior of attached eddies. Physical Review Fluids 1 (2), 022401.
  * Townsend (1976) Townsend, AAR 1976 The structure of turbulent shear flow. Cambridge university press.
  * Woodcock & Marusic (2015) Woodcock, JD & Marusic, I 2015 The statistical behaviour of attached eddies. Physics of Fluids 27 (1), 015104\.