Annual Review of Fluid Mechanics Attached Eddy Model of Wall … · 2018. 8. 8. · Much effort has...

FL51CH03_Marusic ARI 27 July 2018 10:38

Annual Review of Fluid Mechanics

Attached Eddy Model ofWall TurbulenceIvan Marusic and Jason P. MontyDepartment of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia;emails: [email protected], [email protected]

Annu. Rev. Fluid. Mech. 2019. 51:49–74 Keywords

wall turbulence, boundary layer, wall-bounded flow, turbulent shear flow

Abstract

Modeling wall turbulence remains a major challenge, as a sufficient physicalunderstanding of these flows is still lacking. In an effort to move towarda physics-based model, A.A. Townsend introduced the hypothesis that thedominant energy-containing motions in wall turbulence are due to largeeddies attached to the wall. From this simple hypothesis, the attached eddymodel evolved, which has proven to be highly effective in predicting velocitystatistics and providing a framework for interpreting the energy-containingflow physics at high Reynolds numbers. This review summarizes the hypoth-esis itself and the modeling attempts made thereafter, with a focus on thevalidity of the model’s assumptions and its limitations. Here, we review stud-ies on this topic, which have markedly increased in recent years, highlightingrefinements, extensions, and promising future directions for attached eddymodeling.

49

Review in Advance first posted on August 3, 2018. (Changes may still occur before final publication.)

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https://doi.org/10.1146/annurev-fluid-010518-040427

https://doi.org/10.1146/annurev-fluid-010518-040427


1. INTRODUCTION

Essentially, all models are wrong, but some are useful.—George E.P. Box

Advancing our understanding of wall turbulence requires high-fidelity data, in both space andtime, in concert with theory and modeling. While great advances have been made in recent yearsin generating large databases, there are relatively few models of wall-bounded turbulence, despitethe fact that this famously complex physical phenomenon has great importance in engineeringapplications and in nature. Progress in understanding wall turbulence has been relatively slow (incomparison to other problems of classical physics, such as solid mechanics or rigid body dynamics),which is arguably either a cause or effect of a lack of generally accepted or even openly debatedmodels. Models are simplifications, as George E.P. Box eloquently stated in the epigraph above;however, they are needed to provide predictions and conceptual frameworks to improve physicalunderstanding.

Much effort has gone into modeling small-scale vortex dynamics (see Pullin & Saffman 1998),necessitated by the demand for low-cost computer simulations of complex flows. These effortsevolved from the pioneering works of Kolmogorov and Taylor, where the assumption of isotropyfacilitates mathematical analysis. In the case of wall turbulence, however, nonisotropy of the flowis the norm, and considerable attention has been devoted to understanding the organized motionsthat retain their spatial coherence for a long time. These eddies, or coherent structures, have beenextensively reviewed and discussed in relation to their roles in the dynamics of the flow (Robinson1991, Adrian 2007, Jimenez 2012, McKeon 2017). In recent years, the focus of these studies hasmoved from the viscous-dominated near-wall region, where local production of turbulent kineticenergy is at a maximum, to the inertial-dominated logarithmic region, where the bulk productionof turbulent kinetic energy takes place at high Reynolds numbers ( Jimenez & Moser 2007, Marusicet al. 2010a, Smits et al. 2011). It is the modeling of this logarithmic region that is the focus ofthis review.

Quality wall turbulence data are becoming increasingly available; high-fidelity databases ofvelocity fields from wall turbulence simulations, as well as experiments with greater ranges of gov-erning parameters and boundary conditions, have expanded opportunities for model conceptionand validation. Specifically, modeling the logarithmic region improves when there is sufficientscale separation between viscous- and inertia-dominated motions, which occurs at high Reynoldsnumbers. The friction Reynolds number, Reτ = δUτ /ν, directly represents the ratio of the outerlarge length scale δ (the boundary layer thickness) to the viscous length scale ν/Uτ , where ν iskinematic viscosity and Uτ is the friction velocity. High–Reynolds number data, as well as newexperimental facilities and precision instrumentation, are now more available than ever due toincreasing computational power (Smits & Marusic 2013). Thus, it is timely that we summarizethe progress on what is arguably the most commonly cited model for the inertial region of wallturbulence: the attached eddy model (AEM).

As reviewed below, the AEM is essentially a conceptual model that allows one to consider thecomplex physics in terms of simple geometrically self-similar attached eddies that exist over a rangeof scales limited by the Reynolds number. Just as the logarithmic region grows with Reτ , so toodoes the range of attached eddy length scales. The model is useful because it can be related to keyprocesses and used for important applications. For example, de Giovanetti et al. (2016) showed thesalient role attached eddies play in skin friction generation. Mouri (2017) used attached eddies tostudy two-point correlations of the rate of momentum transfer and of the rate of energy dissipationin wall turbulence. Stevens et al. (2014) used the generalized laws for high-order statistics fromthe AEM to test large eddy simulations (LES) of wall turbulence used to characterize wind turbine

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z

δη

FLOW

U(z)

Figure 1Flow visualization of a turbulent boundary layer with the relevant length scales and a schematic of thetime-averaged mean velocity profile. Here, z is the distance normal to the wall, U is mean velocity, δ is theboundary layer thickness, and η is the Kolmogorov length, which scales with the smallest scales of motion.

power fluctuations and estimate probabilities of extreme events. In other work, Ahn et al. (2010)used attached eddies as a structure-based model for describing turbulent wall pressure fields, andSubbareddy et al. (2006) used the attached eddy framework to develop fast and physically realisticinitialization schemes for LES and detached eddy simulation of high-speed wall-bounded flows.The AEM is also ideally suited for describing and assessing atmospheric surface layer flows, owingto the extremely high Reynolds number (e.g., Katul & Vidakovic 1996, Hunt & Carlotti 2001,McNaughton 2004, Kunkel & Marusic 2006, Hutchins et al. 2012).

Importantly, the attached eddy framework enables nontrivial predictions of the energy-containing velocity statistics and its associated probability density function (PDF) of velocityfluctuations, thus providing valuable points of reference for turbulence theories. A recent paperby Hwang & Sung (2018) describes the AEM as providing a “unified theory for the asymptoticbehaviours of wall turbulence.” The AEM does, however, have distinct limitations and assump-tions, but before discussing those in Section 4, we first review its origins starting with Townsendand then consider its subsequent refinements.

2. TOWNSEND’S ATTACHED EDDY HYPOTHESIS

The AEM is based on the hypothesis proposed by Townsend (1951, 1961, 1976) that considerswall turbulence as a field of randomly distributed eddies (or coherent structures) with an importantproperty:

The velocity fields of the main eddies, regarded as persistent, organised flow patterns, extend to thewall and, in a sense, they are attached to the wall. (Townsend 1976, p. 152)

Townsend’s attached eddy hypothesis gained little traction over the ensuing decades after pub-lication, possibly due to the lack of high fidelity data at high Reynolds number and insufficientplanar/volumetric information to observe details of flow structures. However, over the past decadeor so, interest in the attached eddy hypothesis has greatly increased, with multiple groups pursuingvarious avenues of investigation based on attached eddy concepts.

The attached eddy hypothesis deserves careful consideration, as it stands alone as a conceptfrom which the further modeling derives. First, it strictly only applies to inviscid, asymptoticallyhigh–Reynolds number wall-bounded flows. The attached eddy hypothesis describes a propertyof the main eddies (inertial motions unaffected by viscosity) and does not mention small-scale

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motions that are affected by viscosity and dominate at low Reynolds number; vorticity is con-spicuously absent from the attached eddy hypothesis (this is addressed in Section 4.3). Second,Townsend’s statement implies that the inertial scales (also termed the energy-containing motions)that dominate the logarithmic region and beyond are large relative to the viscous length scale. Theupper physical bound of the logarithmic region is a fixed fraction of the boundary layer thickness,typically 0.15δ, and thus is independent of Reynolds number. For a given distance from the wall, z,in the logarithmic region, the inviscid energy-containing motions that contribute to the turbulentkinetic energy and mean shear (cross-stream vorticity) are the attached eddies that range from aheight of O(z) to a height of O(δ). Therefore, Townsend’s statement is that such motions mustfeel the presence of a wall that they are close to, relative to their size. However, the phrase “extendto the wall” is somewhat ambiguous, as the wall is infinitely far away from the flow at a given walldistance because the velocity is zero only below the continuum scale at the wall. The words “in asense,” therefore, mean that the velocity field is influenced by the wall but may not be physicallyconnected to it. Perhaps the simplest explanation is that the motions would behave differently ifthe wall were not present. The wall has a blocking effect on the energy-containing eddies from thewall up to their center, amplifying the wall-parallel motions and suppressing the wall-normal ones.

Finally, Townsend’s statement does not prescribe any details of the structure of the flow in termsof eddy shapes or organization (except that the velocity fields of the eddies should be somehoworganized and not random, the importance of which will be shown later). In other words, theattached eddy hypothesis is simply an observation about the flow. This observation has receivedsupport from numerous studies, as will be outlined below, and should be judged as such, regardlessof whether models derived from the observation are valid.

The attached eddy hypothesis allowed Townsend (1976) to make predictions of statistics in thelogarithmic region. This required the critical assumptions that (a) characteristic attached eddiesare self-similar, meaning that their energy density is constant and their entire geometry scales onlywith distance from the wall, and (b) eddies have a constant characteristic velocity scale. Accordingly,if we consider the velocity field at x = (x, y , z) from one representative eddy of height h, locatedat xe = (xe, ye, 0), and with a characteristic velocity scale Uo to be given by

Ui(x) = Uofi

(x − xe

h

), 1.

then using the above assumptions, Townsend (1976) deduced the components of the Reynoldsstress tensor by adding up the contributions of all eddies and multiplying by the probability thatan eddy of a given size exists:

uiuj

U2o

=∫ δ

δ1

PH(h)Ii j

( zh

)dh. 2.

Here, u1, u2, and u3 refer to the streamwise, spanwise, and wall-normal components of fluctuatingvelocity, respectively; the function

Iij

( zh

)=

∫ ∞

−∞fi

( xh

)f j

( xh

)d

( xh

)d

( yh

)3.

is the contribution from an individual eddy to the correlation of velocity (hereafter termed theeddy intensity function); PH(h) is the probability distribution function of eddies; and δ1 and δ arethe smallest and largest eddy length scales, respectively. It is important to note that the impositionof a no-penetration condition due to the wall means that f3 = z/h fn(x, y) (hence f3(0) = 0), whilethe no-slip condition is not imposed, such that the wall-parallel contributions ( f1, f2) are finite at

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the wall. This leads to the requirements that for small z/h,

I13 ∝ zh

, I33 ∝( z

h

)2, and

I11 and I22 approach nonzero values. 4.

Townsend then deduced the distribution of eddy sizes with wall distance necessary to produceinvariance of the inner-scaled Reynolds shear stress (−u1u3/Uτ

2 ≈ 1) with distance from the wall,as nominally observed in the logarithmic layer. This analysis leads to setting Uo = Uτ and havinga population density of eddies that varies inversely with size and hence with distance from thewall. With this, it follows that the other components of the Reynolds stress tensor are of the form

lu21/U2

τ = B1 − A1 log (z/δ),

u22/U2

τ = B2 − A2 log (z/δ), 5.

u23/U2

τ = B3.

Experiments at high Reynolds number (Etter et al. 2005, Kunkel & Marusic 2006, Nickels et al.2007, Hultmark et al. 2012, Marusic et al. 2013, Vincenti et al. 2013, Talluru et al. 2014) andrecent direct numerical simulations (DNS) (Pirozzoli & Bernardini 2013, Sillero et al. 2013, Lee& Moser 2015) give strong support for the formulations in Equation 5.

It is noted that in the above analysis, the eddy intensity functions are entirely dependent oneddy geometry; however, any eddy shape that satisfies the small-z/h conditions in Equation 4 willgive the same functional results. Thus, details of individual eddies are unimportant in predictingstatistical trends at very high Reynolds numbers.

3. ATTACHED EDDY MODEL

Townsend (1976) attempted to extend the attached eddy hypothesis toward a model of the flowusing a specific eddy geometry. For this, he chose streamwise vortices in the form of the double-cone roller as the candidate attached eddy, as illustrated in Figure 2. However, this did notsuccessfully predict the behavior of the streamwise velocity (it can be shown that the double-coneroller velocity field does not satisfy Equation 4), and this was the extent of Townsend’s modelingeffort (see Townsend 1976, p. 158, equation 5.8.5).

Although Townsend’s double-cone eddy did not prove successful, his attached eddy hypothesislaid the foundation for the construction of a more detailed kinematical model of wall turbulence,the first attempt of which was made by Perry & Chong (1982), hereafter termed PC82. In thissection, we review the key concepts outlined in PC82 and subsequent refinements, as well asextensions of the AEM that followed.

3.1. Perry & Chong’s (1982) Model

PC82 was strongly influenced by the concepts introduced by Theodorsen (1952) and flow vi-sualizations, particularly those of Head & Bandyopadhyay (1981) and Perry et al. (1981), whoadvocated hairpin-type vortices as the key underlying coherent structure of wall turbulence (seeFigure 3b). Thus, PC82 uses a vortex skeleton arising from the wall and the Biot–Savart lawto determine the inviscid velocity field around the representative eddy, termed the �-vortex(Figure 3a). Through a weighted sum of velocity contributions from discrete �-eddies of varyingsize (weighted by the inverse of their size), Perry & Chong (1982) derived the same statisticalbehaviors Townsend found from conceptual arguments. PC82 proposes a mechanism whereby



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FLOW

zy

x

Figure 2Townsend’s double-cone eddy. Adapted with permission from Townsend (1976).

the �-eddies originate and grow from the wall. However, it is important to note that the AEMdoes not require the attached eddies to originate from the wall.

PC82 requires additional assumptions beyond Townsend’s attached eddy hypothesis because itis a discrete model: (a) The vorticity must be entirely contained within the legs of a representativeeddy such that the model remains inviscid; (b) the smallest and largest eddies must have heights of100 wall units and the boundary layer thickness, respectively; and (c) there must be hierarchies ofgeometrically similar eddies. Hierarchies are needed because the Biot–Savart law dictates that thevelocity field of a �-vortex will diminish in intensity in time as it convects its head away from the

FLOW

b

a

Ω

λλ

λkλk

0

Assumed rod geometry

ϕ

Ω

Rod of vorticity

Trailingvortexpairs

Saddle–nodecombination

Vortex filaments of platebeing wound into rods

hhh

Figure 3(a) Perry & Chong’s (1982) �-vortex, as inspired from (b) flow visualizations of hairpin vortices downstream of an oscillating trip in aboundary layer. Panel a adapted with permission from Perry & Chong (1982). Panel b adapted with permission from Perry et al. (1981).

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wall. In PC82, it is worth highlighting that a hierarchy is not a fixed eddy geometry but a group ofeddies with a certain representative geometry. Individual eddies within a hierarchy have a range ofgeometries during their lifetimes, provided that they begin at the hierarchy scale, h. In this way,the hierarchy concept is not only a physically plausible one in terms of eddy dynamics, but alsobridges the divide between discrete and continuous modeling because there is a continuous rangeof stages of eddy development within a discrete hierarchy.

Each hierarchy has a characteristic length that is a fixed fraction of the size of the next-smallesthierarchy (PC82 uses a factor of 2) to meet the attached eddy hypothesis conditions of similarityand inverse-scale population density. The discretized approach of PC82 is attractive because itprovides a framework to visualize the model and is amenable to numerical simulation. It alsoallows one to consider the physical mechanisms that might produce constituent elements of themodel, e.g., eddy geometry, spacing, creation, and growth. Illustrating this point, Figure 4a

a

b

Figure 4Cartoons of the geometric progression of scales. (a) Three scales of hierarchies of eddies (that areconstructed from one hierarchy and hence are geometrically self-similar) with a constant scale ratio betweenthem. That is, h2 = Ch1, h3 = C2h1, . . . , hn = Cn−1h1. (b) The assemblage of these hierarchies to representthe boundary layer with an inverse-scale probability distribution of scales, leading to a logarithmic profile forthe mean velocity and the streamwise and spanwise turbulence intensities, as per the attached eddy model.Adapted from sketches by A.E. Perry.



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k1z

105

104

103

102

101

100

10–1

10– 6 10– 4 10– 2 100 101

1

–1

– 53

No = 10

8

4

ϕuu(k1z)

Figure 5Perry & Chong’s (1982) calculation of one-dimensional u spectra for various numbers of hierarchies. Asignificant k−1

1 spectrum is not observed until 10 hierarchies of attached eddies are present. Figure adaptedwith permission from Perry & Chong (1982).

shows three isolated hierarchies of eddies, and Figure 4b shows how a simple assemblage of arange of hierarchies results in a picture that looks reasonably similar to a turbulent boundary layervisualization. The figure also illustrates the concept of the inverse-scale probability distributionfunction: Clearly there are many more small eddies than large, and smaller eddies exist closer tothe wall (because the eddies are attached and geometrically similar).

Since the PC82 model is based on representative velocity fields, any first- or second-order statis-tic can be predicted, including energy (power) spectra, with a k−1

1 region of the one-dimensional(1D) streamwise energy spectrum of the streamwise velocity, where k1 is the streamwise wavenumber, consistent with earlier predictions based on similarity arguments (Perry & Abell 1977).However, the number of hierarchies required to give a notable bandwidth of k−1

1 behavior is verylarge. PC82 shows that a model with 10 hierarchies exhibits only a limited range (Figure 5). If thesmallest eddy is of order 100 wall units, a geometric progression with 10 hierarchy scales meansthat the predicted Reynolds number required to observe even a small spectral region of k−1

1 is210 × 100, i.e., Reτ = O(100,000). This is a good example both of the limitations of using Fourieranalysis for a nonperiodic velocity field made up of discrete eddies and of the insights that can begained from having a simple conceptual model: A boundary layer containing purely self-similareddies spanning three orders of magnitude in scale exhibits less than a decade of k−1

1 scaling in thestreamwise velocity spectrum. Both the concept of self-similar eddies and the k−1

1 scaling of thespectra are contentious issues discussed further in Section 4.2.

3.2. Refinements

The discretized approach of PC82 has distinct advantages but also complicates the mathematics,and without computational power for numerical simulation, a continuous approach is preferable.Perry et al. (1986) expanded on PC82, showing that a continuous approach was possible andsimplified calculations substantially. In doing so, they provided a recipe for how to use the AEMof PC82 to predict statistics and analyzed the effect of various eddy shapes. It was shown that all

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eddy shapes give similar behavior for the first- and second-order statistics, with only differencesin the outer region. This is to be expected because individual eddy shape should not be a majorcontributor if the other essential characteristics (described above) relating eddies of different sizeare maintained. This is an advantage of the model, but it is also problematic because the detailedstructure of turbulence in the fully turbulent region remains elusive, and predictions beyond basicstatistics are dependent on eddy shape (e.g., energy spectra).

To date, the AEM has mainly been developed and refined based on canonical wall-boundedflows such as pipes, channels, and flat plate boundary layers (indeed, Townsend’s original hy-pothesis was written in his chapter on pipes and channels). Although these flows have differingouter boundary conditions, at asymptotically high Reynolds numbers, the effects of the outergeometry on the logarithmic region are expected to be negligible. At finite Reynolds numbers,what the effects of the outer boundary condition are when applying the AEM remains an openquestion.

The first attempts to quantitatively compute statistics based on specific geometries of a repre-sentative attached eddy were made by Perry & Marusic (1995) and Marusic & Perry (1995) in anattempt to reproduce results for adverse pressure gradient boundary layers. They showed goodagreement with Reynolds stress data by incorporating two types of attached eddy structures. Oneeddy type was needed for the regular logarithmic region, and the other accounted for the wakeregion, where the eddies did not extend all the way to the wall. In this way, their description wasin line with the classical two-part wall–wake formulation of Coles (1956). Further, by using themean momentum and continuity equations, Perry & Marusic (1995) showed that the AEM couldbe used in principle to reproduce the Reynolds stresses given the mean velocity profile and henceprovided the basis of a closure model. The challenge remained, however, that the details of therepresentative eddy geometry were not known a priori, and computed statistics relied on trial anderror calculations using different eddy geometries. For this reason, the AEM has never been fullyimplemented as a RANS (Reynolds-averaged Navier–Stokes)-based model, but it has continuedto shape thinking about wall turbulence. For example, Marusic (2001) showed that using a packetof hairpin vortices as the representative attached eddy produces calculated structure angles, two-point streamwise velocity correlations, and autocorrelations much closer to experimental results.This supports the findings of Adrian and coworkers (Zhou et al. 1999, Adrian et al. 2000) thatspatially coherent packets are a statistically significant structure for the streamwise Reynolds stressand transport processes in the logarithmic region of the flow. Similar findings for all componentsof the Reynolds stresses were recently reported by Baidya et al. (2017).

The attached eddy framework has also been used to develop scaling laws for the Reynoldsstresses beyond the logarithmic region that incorporate finite Reynolds number effects into Equa-tion 5. This includes the work of Marusic & Kunkel (2003) and Kunkel & Marusic (2006), andrecent refinements have been proposed by Vassilicos et al. (2015) and Laval et al. (2017). Agostini& Leschziner (2017) used a spectral-based analysis of DNS data to support an extended attachededdy description in the region below where the variance of streamwise velocity has a logarithmicprofile. The layer below the classical log region is often termed the mesolayer, and this is whereviscous effects become important for attached eddies. Moreover, it is the region where the near-wall presence of all scales of wall-attached eddies are felt and interactions across all these scalesare likely the strongest. Recent studies by Hwang (2016) and Cho et al. (2018) have focused onthese attached eddy interactions in the mesolayer and are related to other efforts that considerlarge-scale influence in the near-wall region (Bradshaw 1967, Marusic et al. 2010b). There isconsiderable scope for further work in extending the AEM beyond the log region, closer to thewall (see Agostini & Leschziner 2017) and up to the wake region, where effects such as outer flowgeometry, pressure gradients, and free-stream turbulence may be incorporated. Wall roughness



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may also be incorporated by determining a length scale based on the roughness geometry andusing that parameter as the scale of the smallest hierarchy of eddies in the AEM.

3.3. Beyond Second-Order Statistics

More recent extensions of the AEM have been motivated by Meneveau & Marusic (2013), whodeduced that if the variance of u1 follows a logarithmic profile (as per the AEM: Equation 5) thengeneralized logarithmic laws should also follow for high-order even moments, so long as the PDFof u1 is assumed to be Gaussian. Namely, 2p-order moments of u1 (where p is an integer) raised

to the power 1/p follow logarithmic behavior according to (u2p1 )1/p/U2

τ = Bp − Ap log (z/δ).Meneveau & Marusic (2013) showed experimental results to support this formulation, but theslopes (Ap ) were empirically found to differ from predictions that assume Gaussian statistics.Further work along a similar line was pursued by Mouri (2015).

These findings motivated Woodcock & Marusic (2015) to revisit the AEM to compute velocitystatistics beyond second-order moments and spectra. Their paper is important in that they pro-vided a revised mathematical framework for the AEM based on an extended form of Campbell’stheorem (Rice 1944), which they showed was applicable for velocity fields corresponding to a for-est of variously sized attached eddies that are randomly located on the wall. This enabled them toderive any moment of the velocity in the logarithmic region, including cross-correlations betweendifferent components of the velocity. They also showed that the previous assumption by Townsendand PC82 that PH(h) ∼ 1/h (see Equation 2 and the necessary requirement for the logarithmiclaws) follows directly as a consequence of the self-similarity of the attached eddies and thus is nota separate assumption. Woodcock & Marusic (2015) derived the results of Meneveau & Marusic(2013) and showed that they followed from the AEM without assuming a Gaussian velocity dis-tribution. de Silva et al. (2015b) applied an extension of this approach to structure functions andshowed that there are bridging relations between higher-order moments of velocity fluctuationsand structure functions and that these too are predicted from the AEM with corresponding strongexperimental support.

The higher-order statistics derived by Woodcock & Marusic (2015) and de Silva et al. (2015b)have also been obtained and extended by Yang et al. (2016a), who invoke the attached eddy hy-pothesis to express streamwise velocity fluctuations in wall turbulence as a hierarchical randomadditive process (HRAP). Due to its simplified structure, the HRAP formalism leads more directlyto predictions of the scaling behaviors for various turbulence statistics, including the logarithmicscaling of moments, structure functions, and correlation functions, as well as other nonobviouslogarithmic laws involving two-point statistics of velocity of various powers. Using a similar ap-proach derived from the AEM, Yang et al. (2017) made new predictions of the scaling of thevelocity structure function tensor for all velocity components and in all two-point displacementdirections, while Yang et al. (2016b,c) related moment-generating functions to the extended self-similarity hypothesis (Benzi et al. 1993). The resulting scaling behaviors that have been derivedhave now been successfully tested in a range of wall-bounded flows, including high–Reynoldsnumber Taylor–Couette flow (de Silva et al. 2017, Krug et al. 2017).

3.4. Are Attached Eddies Necessary?

Davidson & Krogstad (2009) and Davidson et al. (2006) have produced a simplified model thatdoes not assume that eddies are physically attached to the wall. In fact, their model does notrely on any physical characteristic of eddies other than an assumption that their kinetic energiesscale with distance from the wall. The results they obtain are the same as the AEM predictions

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for the streamwise velocity fluctuations, and their model also models the entire spectrum of wallturbulence in the logarithmic region, not only the large eddies. While the Davidson & Krogstad(2009) model highlights that the attached eddy hypothesis can be bypassed, this does not discountthat attached eddies are needed. Indeed, the AEM provides the mechanism that leads to theassumed kinetic energy scaling of Davidson & Krogstad (2009).

A related question is whether the wall itself is required to model wall turbulence, as posed byMizuno & Jimenez (2013) in their provocatively titled paper “Wall Turbulence Without Walls.”They show that the logarithmic region can be effectively modeled without considering the near-wall dynamics (i.e, without the wall). However, they also conclude that some knowledge of thewall is necessary in that wall-normal correlation lengths are required. While it may at first appearthat this contradicts the attached eddy hypothesis, in fact, it is subtly equivalent. The AEM beginsat the logarithmic region and does not require information about the near-wall dynamics. All thatis required is a length scale from the wall and the no-penetration boundary condition; these set awall-normal correlation length as required by Mizuno & Jimenez (2013).

4. LIMITATIONS AND KEY UNDERLYING ASSUMPTIONS

In this section, we highlight the key limitations and underlying assumptions of the AEM anddiscuss their consequences. Additionally, we again point out that the AEM was conceived as amodel for asymptotically high Reynolds number flows and therefore essentially is an inviscidmodel that does not account for turbulent enstrophy or dissipation. Further, the model providesa kinematic description for boundary layers and currently contains no dynamics, which limits itsuse for applications such as flow control. This may hopefully change going forward, for example,by combining the AEM with dynamical systems approaches (e.g., McKeon & Sharma 2010).

4.1. Existence of Attached Eddies

The obvious primary assumption of the AEM is that wall turbulence involves attached eddies. Aquestion immediately follows: Do attached eddies actually exist or are they purely a statistical con-struct? This is perhaps where the seminal works of Townsend (1976) and Perry & Chong (1982)differ. Townsend’s formulation was clearly statistical with reference to only an average flow field,while PC82 invoked experimental evidence for a hairpin-type vortex structure as the candidaterepresentative attached eddy. Later refinements by Perry and coworkers adopted a more practicalview, where hairpin-type vortices do exist, but the candidate attached eddy to be used in the modelis an average of many realizations of instantaneous structures. Therefore, one is unlikely to detectthis structure for any given realization (just as one is unlikely to see a statistically average person).

The question of whether hairpin vortices exist in turbulent boundary layers has been a topic ofcontroversy for over 60 years, going back to the work of Theodorsen (1952). A number of studiesat Stanford during the 1970s (for example, Offen & Kline 1975) identified hairpin structuresas a likely explanation for transport mechanisms near the wall, but it was the work of Head& Bandyopadhyay (1981) that brought this topic to prominence, and it has been a source ofmuch debate and controversy ever since. Figure 6 shows example visualizations from numericalsimulations with opposing views. Figure 6a is from Wu & Moin (2009), who concluded that thereis a striking preponderance of hairpin vortical structures across the boundary layer, while Figure 6bis from Eitel-Amor et al. (2015), who concluded that there is no evidence of hairpin vorticesreaching into the wall region once the turbulent layer achieves a fully developed state.

Many other studies exist pertaining to the identification of eddies in wall turbulence, but we willnot review those here. This is in part because attached eddies are only statistically representative



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Figure 6Two example visualizations from numerical simulations with opposing views on the presence of hairpin-type vortices. Panel a is fromWu & Moin (2009), and panel b is from Eitel et al. (2015). Both panels adapted with permission.

structures and also because searching for candidate attached eddy shapes from flow visualizationsis highly problematic and potentially misleading. This is highlighted well if one considers thework of Jodai & Elsinga (2016), who showed compelling experimental evidence of hairpin auto-generation events in a turbulent boundary layer using time-resolved tomographic particle imagevelocimetry (PIV). Figure 7 shows three snapshots taken from a movie from Jodai & Elsinga(2016) over a period of less than 20 viscous time units in flow evolution. In the first snapshot, ahairpin vortex is identified, and the second and third shapshots show the same flow field at slightlylater times, where the field of view convects in the streamwise direction, allowing us to track theevolution of the individual structures. In the third snapshot, the same hairpin vortex structure fromthe first snapshot is identified, but in the second snapshot, no such hairpin structure is observed.The hairpin head is still present in the second snapshot, but it is not visible likely due to a strongdynamic event where interactions from neighboring structures lead to a drop in the local swirlingstrength below the chosen threshold level. This is more clearly seen in the original movie, but herewe wish to highlight how an incorrect conclusion can be made (as in the second snapshot) if oneconsiders only still images. The problem here also largely stems from the detection criterion used

–200 –100

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HairpinNo hairpin?

Figure 7Snapshots from a movie from Jodai & Elsinga (2016) using high-speed tomographic particle image velocimetry, showing flow evolutionin a turbulent boundary layer over a period of less than 20 viscous time units. A hairpin vortex is highlighted that becomes obscured inthe middle panel. Images adapted with permission from Jodai & Elsinga (2016).

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to identify the vortices. Unlike vorticity, a vortex has no clear (or at least agreed upon) definition.As in many studies, Jodai & Elsinga (2016) make the sensible decision to visualize the vortices usingan isosurface of swirling strength, here color-coded according to the local streamwise velocity inwall units. However, this definition and most others are based on measures related to the velocitygradient tensor. While velocity gradient structures are correlated with large-scale (energetic) flowstructures (see for example Christensen & Adrian 2001, Eisma et al. 2015), such structures aremore strongly correlated with enstrophy and therefore are often problematic when used as thebasis of a detection criterion for attached eddies.

The important point to keep in mind is that attached eddies account for the main energy-containing motions (Townsend 1976), which therefore include the mean-relative motions. Thatis, the AEM by its construction only accounts for the (inviscid) contributions to the mean ki-netic energy and the mean vorticity. The mean vorticity is the balance of the wildly fluctuatinginstantaneous vorticity, which is the result of similarly fluctuating small-scale velocity gradients,and this causes a certain disconnect between the mean vorticity and the velocity gradients. Inthe AEM, energy-containing eddies carry essentially no turbulent enstrophy. To highlight this,we show schematically in Figure 8 how real turbulence signals in the logarithmic region mightcompare to signals obtained from the AEM using discrete ensembles of attached eddy signals (this

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Figure 8Schematic comparing inner-scaled signals of streamwise velocity (U+) and spanwise vorticity (�+

y ) versus streamwise distance (x+) inthe logarithmic region from measurements and the attached eddy model. The attached eddy signals are attenuated, as they do notcontain small scales that are viscous dependent. Thus, they contain no fluctuating vorticity and contribute only to the mean vorticity.



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k1+ϕ+

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Figure 9Inner-scaled premultiplied spectra of streamwise velocity (k+

1 φ+uu) and spanwise vorticity (k+

1 φ+ωy ωy ) in the

logarithmic region for a turbulent boundary layer at Reτ = 10,000 (red lines), compared to computed spectrausing the attached eddy model (blue lines). The blue dashed line (attached eddy spanwise vorticity) isnominally zero—here, O(10−6) in magnitude.

is for illustrative purposes only). The attached eddy signals are attenuated, as they do not containsmall scales that are viscous dependent. This is reflected in the attached eddy signal that containsno fluctuating vorticity and contributes only to the mean vorticity. This is also highlighted, nowquantitatively, in Figure 9, where the premultiplied spectra of velocity and spanwise vorticity areshown in the log region for a turbulent boundary layer at Reτ = 10,000 compared to computedspectra using the AEM as described by Baidya et al. (2017). Again, we see that the contributionto the kinetic energy from the attached eddies does not account for the small-scale contributions(here, nominally for k+

1 > 10−3) and contains negligible enstrophy and dissipation energy.Jimenez (2012) has also highlighted the need to distinguish attached eddy contributions from

small-scale contributions, and there is a need for more effective detection criteria that are indepen-dent from the velocity gradient tensor. In line with this, Lozano-Duran & Jimenez (2014) definedstructures as connected sets of points in space; they distinguished two types of structures: vortexclusters, as discussed by del Alamo et al. (2006), which are surrogates for strong dissipation, andquadrant structures, which contribute to momentum transfer and are defined using a thresholdlevel of instantaneous pointwise tangential Reynolds stress. Such quadrant structures are relatedto the energy-containing attached eddies (see further discussion in Section 4.2).

To date, all characteristic eddies are purely imagined structures that have velocity fields sat-isfying the AEM criteria only; these velocity fields do not satisfy the Navier–Stokes equations.Ideally, finding the geometry of the representative attached eddy would come from the Navier–Stokes equations. This remains the topic of future work, but some studies along these lines havealready been conducted. For example, using analysis and numerical simulations, Farano et al.(2015) showed that hairpin vortex structures can be the outcome of a nonlinear optimal growthprocess, similar to how streaky structures in the viscous region can be the result of a linear optimal

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growth mechanism. In a later study using an analysis of the distribution of the most energeticwavelengths in the wall-normal direction for the inner and outer optimal disturbances, Faranoet al. (2017) concluded that the optimal structures are likely attached eddies, as they scale with theirdistance from the wall. Another intriguing possibility is that the representative attached eddy isrelated to an invariant solution of the Navier–Stokes equations. This has essentially been proposedby Y. Hwang et al. (2016) and Cossu & Hwang (2017), who considered filtered Navier–Stokesequations and reported that there is a family of self-sustaining motions with behaviors consistentwith attached eddies. Driven by high–Reynolds number experimental data, one could use Navier-Stokes-based, control systems approaches, such as that of Zare et al. (2017), to uncover candidateeddies that would better model the turbulent flow.

4.2. Self-Similarity

The assumption of self-similar attached eddies is a strong one, as it demands that the main eddiesare all of similar geometry, that the geometry scales with distance from the wall, and that the eddies’energy density is constant. Because self-similarity is so important to a number of mathematicalconstructs used to describe aspects of wall turbulence, including the AEM, there have been manystudies to test the assumption using a variety of statistical tools. A brief review of such studies isincluded here to provide confidence in this key assumption.

4.2.1. Two-point measurements. Linear growth of eddy geometry with distance from the wallis a necessary (but not sufficient) condition for self-similarity. Various measures of eddies’ spanwisewidths have shown a linear increase with wall distance in the logarithmic region for boundary layers(Tomkins & Adrian 2003, Hutchins et al. 2005), channels (del Alamo et al. 2006, Monty et al.2007), and pipes (Bailey et al. 2008). Tomkins & Adrian (2003) found that individual eddies didnot grow self-similarly in time, which is consistent with the notion of hierarchies; hierarchies areself-similar, but individual eddies within a hierarchy are all at different stages of development andare not themselves self-similar (see PC82).

Baars et al. (2017) also reported evidence in support of self-similarity from an analysis of two-point measurements in the wall-normal direction over a change in Reynolds number of threeorders of magnitude. Baars et al. (2017) used spectral coherence analysis to define structuresthat are attached to the wall and showed that these structures are self-similar with a constantstreamwise/wall-normal aspect ratio of ∼14.

4.2.2. Volumetric data analysis. From simulations or experiments, 3D data have also beenextensively employed to test the self-similarity of structures. Many studies set out to look for orfind hairpin-type eddies as the dominant contributors to the kinetic energy in the logarithmicregion; however, del Alamo et al. (2006) used the discriminant criterion of Chong et al. (1990)to identify clusters of small-scale vortices from DNS data. It turned out that many of thesewere large, attached structures that are kinematically identical to packets of hairpin vortices. Thevortex clusters were shown to be self-similar in size owing to their large-scale velocity structureeven though the small-scale vortices that formed each cluster were not self-similar. As mentionedin Section 4.1, Lozano-Duran & Jimenez (2014) extended this work and focused on quadrantstructures, which contribute to momentum transfer and thus can be interpreted as surrogates forattached eddies. Lozano-Duran & Jimenez (2014) found that these eddies become large, attachto the wall, extend across the logarithmic layer, and are geometrically and temporally self-similar,with lifetimes proportional to their size (or distance from the wall). These structures also explainthe previously observed symmetry between sweeps and ejections (Lozano-Duran et al. 2012).



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Hwang & Sung (2018) investigated so-called u clusters using a criterion based on streamwisevelocity. They chose this criterion based on their prior studies of how negative-u1 regions areassociated with the net Reynolds shear force ( J. Hwang et al. 2016a) and of how their role ininteractions relates to momentum transport ( J. Hwang et al. 2016b). They concluded that thesestructures are indeed attached to the wall and self-similar. Consistent with the AEM, they alsoconcluded that these structures lead to multiple uniform momentum zones (UMZs) and to thelogarithmic variation of the streamwise turbulence intensity, with the structures’ population den-sity scaling inversely with their height. These UMZs are consistent with the findings of Meinhart& Adrian (1995), who first developed the concept, and Adrian et al. (2000), who referred to thesezones as “nested hierarchies.”

One of the most convincing studies using experimental data in support of the self-similarityassumption and the employment of a roughly hairpin-type vortex as a characteristic eddy comesfrom the conditional averaging analysis of Dennis & Nickels (2011). Using high-fidelity, time-resolved stereo-PIV data, they examined the swirling strength field conditionally averaged ona positive spanwise swirl event at a specified wall distance. The key result of their analysis isshown in Figure 10, and the authors made the following observations: “the averaged hairpins. . . are very similar in their shape throughout the height of the boundary layer. There is minimalvariation in angle and the size of the average hairpin scales simply with the height it is found inthe boundary layer” (p. 212). Clearly there is evidence for self-similar scaling in wall turbulenceover some range of scales. However, we still need to know what fraction of energy is attributed

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Figure 10Isosurfaces of conditionally averaged swirling fields resembling packets of hairpin-type vortices. Adaptedwith permission from Dennis & Nickels (2011).

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to self-similar attached eddies as Reynolds number becomes very large. The key question is, Willthe self-similar scales dominate the energy spectrum as Reynolds number becomes very large, asoriginally anticipated in the development of the AEM? This will help elucidate the relationshipbetween A1 in Equation 5 and k−1

1 scaling of the 1D spectra.

4.2.3. Energy methods. Fourier analysis is a frequently used tool in turbulent flows due tothe large range of scales of motion present and the need for understanding the contributions ofindividual scales, as evidenced by this review. The advent of high-fidelity simulations and experi-mental techniques (i.e., PIV) provide opportunities for advanced analysis such as multidirectionaland multicomponent energy spectra, proper orthogonal decomposition (POD), and velocity gra-dient tensor interrogation. These tools are needed because of the complexity of the flow and theconsequent difficulty in interpreting statistics involving averaging over two or more dimensions.

del Alamo et al. (2004) conducted one of the first studies to show high-resolution 2D energyspectra in a channel flow. Their simulations were at relatively low Reynolds number, and they foundno evidence of self-similarity in the inertial scales in the fully turbulent region. That is, their spectradid not exhibit the expected constant aspect ratio across the range of scales of the energy-containingmotions. However, a recent paper by Chandran et al. (2017) reported novel experimental results for2D spectra of the streamwise velocity at high Reynolds number in a turbulent boundary layer andshowed that as Reynolds number increases, self-similarity is approached in the large scales (eddieslonger than ∼14 times the wall distance they are centered at). Approximately constant energydensity of a large range of scales was also demonstrated for the first time. The key observation hereis that self-similar eddies only dominate the energy spectrum at extremely high Reynolds number(recall that the AEM is designed to be most accurate at asymptotically high Reynolds number).

With multiple components of velocity available, the Reynolds shear stress cospectrum can becalculated, and this has also been used to demonstrate self-similarity by examining the cospectrumscaling (Hoyas & Jimenez 2006, Baidya et al. 2017). Using relatively higher Reynolds numberDNS, Hoyas & Jimenez (2006) used a novel technique whereby they averaged the spectra overlogarithmically spaced spectral bands. They convincingly demonstrated that energetic structureshave characteristic widths that increase linearly with wall distance. Baidya et al. (2017) conductedhigh–Reynolds number experiments with ×-wires, allowing the analysis of the 1D spectrum of u1

and u3. They observed mixed scaling of the streamwise velocity but pure wall-distance scaling ofthe wall-normal component and the Reynolds shear stress cospectrum.

POD is a useful technique not constrained by harmonic functions that also provides informationabout the energy over the inhomogeneous wall-normal direction, which is obviously critical toassessing the veracity of the AEM. Hellstrom et al. (2016) exploited snapshot PIV over the crosssection of a turbulent pipe flow at moderate Reynolds numbers (Reτ ≈ 1,000–3,000) and showedthat the size of the most energetic modes scales with distance from the wall. Interestingly, theyfound the POD modes to be self-similar over a decade in range of spanwise (azimuthal) wavenumbers, even at moderate Reynolds numbers.

4.2.4. Structure functions. Evidence for and against the k−11 scaling of the 1D streamwise energy

spectra can be found in the literature (see Perry et al. 1986, Morrison et al. 2004, Nickels et al.2005, Vallikivi et al. 2015). Overall, there is unconvincing evidence of k−1

1 at laboratory-scaleReynolds numbers, especially in pipe flows. However, Chung et al. (2015) postulated that theazimuthal crowding of large-scale eddies in pipes due to converging geometry (see Monty 2005)could be a reason for the observed lack of k−1

1 scaling.Recall that 1D spectra are the integrated contributions across all spanwise wavelengths (not

only the energetic self-similar wavelengths found by Hellstrom et al. 2016) and contain the mixed



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contribution of a range of eddies that may or may not be attached. Chandran et al. (2017) usedtheir 2D spectra to highlight this issue and predicted that an appreciable range of k−1

1 scalingwill only be observed for friction Reynolds numbers above 60,000, which is above most availablelaboratory data.

The known limitations of 1D spectra analysis led Davidson et al. (2006) to propose 1D structurefunctions, which are the spatial analogs of the Fourier spectrum, as better tools for analyzing self-similarity. Davidson et al. (2006) showed that dominance of self-similar eddies should result inlogarithmic dependence on the separation, r , of the second-order structure function and providedfurther convincing support from experimental data at high Reynolds number. de Silva et al. (2017)and Yang et al. (2017) then showed that higher-order structure functions (of arbitrary even order)also show logarithmic dependence in support of the self-similarity of attached eddies in wallturbulence.

4.2.5. Navier-Stokes-based analysis. Busse (1970) was one the first to consider attached eddieswith the equations of motion. He used variational methods to conclude that the maximum mo-mentum transport to the wall requires setting a distribution of the energy-containing motions inthe form of attached eddies, as described by Townsend (1956). Encouragingly, in recent years,more analysis methods have emerged that directly use the Navier–Stokes equations to study wallturbulence. For example, Klewicki and colleagues (Klewicki et al. 2009, Klewicki 2013) analyzedthe averaged Navier–Stokes equations and showed that a self-similar hierarchical structure isrequired for invariant solutions associated with the leading-order mean dynamics.

Other studies have involved linear analysis. del Alamo & Jimenez (2006) used the linearizedNavier–Stokes equations together with an eddy viscosity for channel flows and found the lineartransient growth modes to be self similar, which is consistent with the notion of attached eddies,although no direct connection was drawn. Using essentially the same model, Hwang & Cossu(2010) showed that the self-similarity extends also to harmonic forcing (resolvent modes) andstochastic response modes. Later work by Hwang (2015) used filtered LES to simulate energy-containing motions only at a given spanwise length scale in the logarithmic region. These motionswere found to be self-sustaining (Hwang & Bengana 2016) at each of the spanwise length scales, andHwang (2015) showed them to be self-similar with respect to the given spanwise length. Hwangconcluded that the statistical and dynamical features of a family of these extracted self-sustainingmotions are consistent with Townsend’s attached eddies.

Other notable studies based on the Navier–Stokes equations include those of McKeon andcolleagues (McKeon & Sharma 2010, Moarref et al. 2013, Sharma & McKeon 2013, McKeon2017), who performed a systems approach analysis of the Navier–Stokes equations, in which thenonlinear advective term is interpreted as forcing the linear Navier–Stokes operator. By lookingat the resolvent of this linear operator, they extracted modes for wall turbulence and demonstratedthat these modes are nominally self-similar for a logarithmic mean velocity profile. While it is notyet clear exactly how these modes relate to the attached eddies, this line of investigation seemspromising for future studies.

4.3. Unresolved Scales

Models of complex dynamical systems are designed to capture only the essential physics; theAEM only attempts to capture the main energy-containing eddies in the fully turbulent region.Therefore, the model does not attempt to capture any small-scale motions below the logarithmicregion, which may be important for control applications, or any motions in the wake region,where entrainment/mixing problems would benefit. In the fully turbulent region at high-enough

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Outer flow scaling Kolmogorov scaling

A1

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Type-CeddiesType-Ceddies

Area =A1 log(z/δ)

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k1z ϕ+uu (k1z)

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Figure 11Sketch of contributions from different eddy types to inner flow–scaled premultiplied u-spectra, k1zφ+

uu(k1z), with scaling and overlapregions following Perry et al. (1986). Adapted with permission from Marusic & Perry (1995).

Reynolds number, we expect that the modeled eddies will be the assumed self-similar structuresdescribed above and that such eddies will carry most of the energy. At commonly studied Reynoldsnumbers, this may not be true and the unresolved energy is likely significant.

Figure 11, taken from Marusic & Perry (1995), is a schematic illustration of the resolved andunresolved scales in the turbulent wall region. Only the Type-A eddies referenced in the figure areattached and self-similar, and only these are part of the AEM. The Type-B eddies referenced in thefigure are physically detached eddies that were expected to scale with the boundary layer thicknessand hence are invariant with Reynolds number. They were thought to be eddies centered in thewake region, far above the logarithmic region. Recent studies revealing superstructures and very-large-scale motions (see Smits et al. 2011) have shed new light on Type-B eddies, and it appearsthat they are weakly dependent on Reynolds number (Hutchins & Marusic 2007, Lee et al. 2014).Further research into the behavior of Type-B eddies at large Reynolds number is required tofully understand the extent of the unresolved energy due to Type-B eddies. It also remains to bedetermined how much of the superstructure-scale energy is due to attached structures, i.e., howmuch contributes to the Type-A and Type-B eddies (since they overlap in spectral space).

Type-C eddies include Kolmogorov scales and smaller detached eddies that might be remnantsof eddies once attached earlier in their lifetimes. Marusic & Perry (1995) had limited insight intothe structure of such eddies and suggested that a better knowledge of the characteristic attachededdies (Type-A) is required because these set the unknown boundary, as shown in Figure 11. Itis plausible that some of the Type-C energy is related to intense small-scale velocity fluctuationsin the high-vorticity regions of the eddies that make up the Type-A eddies. As discussed above,del Alamo et al. (2006) have shown that although the large-scale turbulent structures may be self-similar, the small-scale vorticity that surrounds the large structures is not. Here, we describe two



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different scales of motion, but at low Reynolds number, the scales overlap significantly, as shown inFigure 9; even at Reτ = 10,000, there is still considerable overlap in wave number space betweenthe energy-containing and the viscous-dependant scales. However, as Reynolds number increases,these viscous-scaled motions decrease in importance to the logarithmic region, and again, theseare not intended to be modeled by the AEM.

It is possible that a significant fraction of Type-C eddy energy is due to either detachedremnants of attached eddies or eddies whose geometry is self-similar even though their velocitysignatures are incoherent with the wall. If so, Type-C eddies will not separate in scale from Type-Aeddies as Reynolds number increases. Moreover, contributions from Type-C eddies may contain asignificant fraction of the overall energy (as they do at Reτ = 10,000, shown in Figure 9). There isa need to better understand the nature and origins of Type-C eddies in the context of the AEM, aswell as a broader need for research into which scales should be modeled as attached. It is likely thatthere are motions that are energetically important that are detached and not able to be modeledwithin the attached eddy framework, and it is also possible that there are detached eddies that areself-similar and follow wall scaling (these are eddies whose geometry scales with distance fromthe wall, but their influence on the velocity field is local). The contributions from attached/self-similar, attached/non-self-similar, detached/self-similar, and detached/non-self-similar eddies asa function of Reynolds number needs to be measured. Correlation-based methods should be usefulin this regard, as they have the potential to separate out these different motions.

4.4. Random Distribution over the Plane of the Wall

The final key assumptions of the AEM we highlight here are that the attached eddies are assumedto occur randomly in space and that they are independent of each other. These assumptions arehighly desirable mathematically, as they enable the HRAP framework of Yang et al. (2016a) toproceed and allow one to treat the problem using theories such as Campbell’s theorem (Rice1944), where Poisson point processes can be invoked (Woodcock & Marusic 2015). We knowthat these assumptions are not strictly true, but the success of the predicted formulations for thevarious velocity statistics suggests that they provide a reasonable approximation to leading order.Moreover, testing the veracity of these assumptions is problematic, and currently we must rely onindirect evidence.

One indirect method for testing whether random spatial distributions of noninteracting at-tached eddies is reasonable is to compare the results from continuous integral approaches thatrely on Campbell’s theorem (Perry et al. 1986, Woodcock & Marusic 2015) to those from discretemethods where brute force calculations are made for hierarchies of differently sized self-similarattached eddies, producing volumetric databases of instantaneous velocities for multiple realiza-tions. Such an approach was used by de Silva et al. (2015a) to show that attached eddies (where therepresentative attached eddy was a packet of hairpin vortices) could produce instantaneous flowfields resembling UMZs in turbulent boundary layers, consistent with the findings of Adrian et al.(2000). The results of the discrete attached eddy computations showed striking agreement withthe empirical observation that the number of UMZs increased with the logarithm of the Reynoldsnumber.

However, other indirect evidence shows clear limitations of the AEM assumptions. For exam-ple, the AEM predicts the flatness (kurtosis) for all velocity components to be invariably greaterthan 3 (i.e., super-Gaussian behavior), while experimental results show sub-Gaussian behaviorfor the streamwise component of velocity (Meneveau & Marusic 2013, Woodcock & Marusic2015). de Silva et al. (2016) considered this issue by performing discrete attached eddy computa-tions where the locations of the eddies over the wall were freely prescribed (different from a pure

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random scattering). As mentioned above, adopting the Poisson point process to describe the eddylocations has mathematical advantages, but it has the obvious problem of treating eddies as pointparticles and thus makes no consideration for the locations of neighboring eddies. By mandatinga minimum distance between any two eddies of the same height, de Silva et al. (2016) were able todemonstrate that this spatial exclusion produces predictions in better agreement with experimen-tal observations, namely, sub-Gaussian behavior for the streamwise component of velocity andsuper-Gaussian behavior for the other velocity components. These findings suggest that spatialexclusion between the main energy-containing eddies is likely to play an important role in the lawsthat govern their spatial arrangement, which is likely to be more disperse than a Poisson pointprocess.

SUMMARY POINTS

1. Townsend’s attached eddy hypothesis is a statement about the scaling of the dominant co-herent motions in the logarithmic region. Essentially, the hypothesis is only that velocityfields of those eddies scale with distance from the wall.

2. The attached eddy model (AEM) evolved Townsend’s hypothesis by introducing theconcept of self-similar representative eddies, with current models standing out from otherconceptual models owing to the mathematical rigor of the methodology and predictions.

3. The AEM provides powerful and accurate predictions of the scaling behaviors for variousturbulence statistics, including the logarithmic scaling of moments of the streamwise andspanwise components of velocity, structure functions, and correlation functions, as wellas other nonobvious logarithmic laws involving two-point statistics of velocity of variouspowers.

4. The AEM has proven to be a highly effective predictive tool and framework for consid-ering the flow physics, but all models have limitations. The most important limitationof the current AEM is that accuracy is limited to very high Reynolds number flows andthat only the attached eddies in the spectrum are modeled.

5. Studies of attached eddies have markedly and continuously increased over the past decade,pointing to a future of more accurate predictions of the kinematics and dynamics of high–Reynolds number turbulence.

FUTURE ISSUES

1. Are attached structures, having self-similar behavior, the dominant features in wall tur-bulence at very high Reynolds number?

2. The extent to which eddies in the flow are attached needs to be ascertained. While it isknown that there are attached and detached eddies, it is not yet clear what percentage ofwhich scales are attached and detached or how these different eddies scale.

3. The AEM could be extended to include a greater range of scales and extend nearer toor farther from the wall, either by extending the model itself or combining with othermodels.



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4. Current characteristic eddies used in the AEM are imagined structures whose veloc-ity fields do not satisfy the Navier–Stokes equations. Finding candidate eddies that aresolutions to the Navier–Stokes would be extremely valuable.

5. Moving from the current AEM, which is purely kinematic, to a model incorporatingdynamics is essential for flow control applications.

DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS

The authors wish to thank Spencer Zimmerman and Rio Baidya for providing the data used inFigures 8 and 9, respectively, and Woutijn Baars, Daniel Chung, David Dennis, Gerrit Elsinga,Nick Hutchins, Yongyun Hwang, Simon Illingworth, John Elsnab, and Hyung Jin Sung for helpfuldiscussions and feedback on the manuscript. The financial support of the Australian ResearchCouncil is gratefully acknowledged.

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