Self-similar hierarchies and attached eddies

PHYSICAL REVIEW FLUIDS

, 082601(R) (2019)

Rapid Communications

Beverley J. McKeon

Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA

(Received 14 May 2019; published 26 August 2019)

It is demonstrated that time-evolving coherent structures with features consistent with

Townsend’s attached eddies and the developments associated with the reconstruction of

flow statistics using the attached eddy hypothesis can be obtained from analysis of the

(linear) resolvent operator associated with the Navier-Stokes equations. A discrete number

of members of a single self-similar hierarchy, chosen by assuming a constant ratio of crit-

ical layer heights between neighboring members with overlapping wall-normal footprints,

produces a geometrically self-similar, fractal-like structure of the spatial distribution of

the velocity field and an associated signature of uniform momentum zones. The range

of convection velocities associated with the hierarchy gives rise to time evolution of the

velocity. The scaling of the streamwise wave number on a hierarchy can be reconciled

with Townsend’s distance-from-the-wall scaling for self-similar, attached eddies, at least

conceptually, by considering coherent spatial structures in the velocity field and thus

enforcing an exact equivalence between compared features. It has been shown previously

that both the linear and nonlinear terms in the resolvent framework have self-similar

representations in the logarithmic overlap region of the turbulent mean velocity profile.

The conditions under which self-sustaining self-similar behavior may be obtained from

self-similar hierarchical members in this region are elaborated and some similarities with

other results and theories associated with self-similarity in this region are identified.

DOI:

10.1103/PhysRevFluids.4.082601

I. SELF-SIMILARITY: STRUCTURAL AND SPECTRAL REPRESENTATIONS

Townsend’s [

] seminal distance-from-the-wall scaling has underpinned many theoretical and

observational descriptions of wall turbulence. The original argument posits that eddies with

diameters proportional to the distance from the wall should be required in order to obtain a

dissipation length scale also proportional to this distance in the equilibrium (or inertial) layer, such

that the motion of “the main eddies of the flow

...

is directly influenced by its presence” [

]. He

proposed using a self-similar function,

, to model the velocity field

(

) for an eddy

centered at (

) and superimposed on a uniform background flow,

(

)

[

(

−

)

(

−

)

(

−

)

]

(1)

These ideas have been elaborated effectively over several years into the attached eddy model

(AEM), most recently reviewed in Ref. [

], which deploys a (static, linear) superposition of

hierarchies of self-similar eddies and gives rise to a range of scaling results for the statistics of

the velocity field which are well borne out in experimental and numerical data, e.g., Refs. [

–

as well as a physical interpretation of the flow arrangement in the inertial region, e.g., Ref. [

In a discrete representation, the wall-normal eddy length scales (heights)

are related by a

mckeon@caltech.edu

2469-990X/2019/4(8)/082601(9)

082601-1

BEVERLEY J. MCKEON

geometric progression and the structures are randomly distributed over wall-parallel planes with

the number of structures belonging to hierarchy

described by

∼

−

. Knowledge of the precise

self-similar eddy form is not required to recover logarithmic scaling for the mean velocity profile

and wall-parallel fluctuation variances and a constant

variance, but is important for the precise

matching of higher-order statistics. Recent work [

] has shown that including a spatial exclusion

condition to prevent eddies of the same hierarchy overlapping each other leads to improved capture

of higher-order statistics. Perry and Marusic [

] noted that a wall-normal extent which scales with

distance from the wall can be obtained by two types of eddies: those which reach down to the

wall, i.e.,

, as well as those which do not,

, which they denoted as type-A and

type-B eddies, respectively. The inclusion of type-B eddies improved the accuracy of the AEM in

the wake region of the flow. Information on the eddy dynamics is not immediately available (but

could be determined using the Biot-Savart calculations). The predictive capabilities of the attached

eddy model are rightly celebrated, and the subject of significant current study.

Flores and Jiménez [

] and Hwang and Bengana [

] have identified self-sustaining turbulent

solutions to the Navier-Stokes equations in channel flow simulations with domains restricted to be

proportional to the distance from the wall, while recent work by Yang

et al.

[

] identifies families of

attached exact coherent structures. The statistics of these flows are consistent with the properties of

attached eddies, and thus appear to provide evidence that the latter are multiscale and self-sustaining

within the minimal unit construction introduced by Jiménez and Moin [

]. Further, these studies

give information on the dynamics of such flows, which strongly resemble those of the near-wall

cycle identified in the original study. The mean momentum balance analysis (MMB) of Klewicki

et al.

, e.g., Ref. [

], identifies self-similar scaling layers contributing to the mean Reynolds stress

and therefore the mean velocity profile.

These works provide physical space (structural) proposals for, or identification of, attached

eddies, with the numerical studies also documenting their spectral signature. Separate work related

to the linear dynamics of the Navier-Stokes equations (NSE) has identified self-similarity of the

input-output response. del Alamo and Jiménez [

] noted the self-similarity of transient growth

modes obtained from analysis of the NSE linearized relative to the turbulent mean velocity profile

and Hwang and Cossu [

] observed a geometric self-similarity of the most-amplified velocity

response to harmonic and stochastic input forcing in the logarithmic region. Both of these studies

used an eddy viscosity to provide closure of the equations. Here, we exploit the development of

McKeon and Sharma [

] in which the terms that are nonlinear in the fluctuations provide the

forcing to sustain the linear dynamics, obviating the need to use an eddy viscosity and leading

to a closed feedback loop (sketched in, e.g., Ref. [

], Fig.

). Moarref

et al.

[

–

]have

documented the self-similarity of both linear and nonlinear terms in the log region. Self-similar

hairpin vortex structures were also observed by Sharma

et al.

[

] using resolvent analysis, while

Vadarevu

et al.

[

] observed a self-similar, attached structure in the impulse response of the

eddy-viscosity-modified linearized Navier-Stokes operator.

The approach is outlined briefly below; the reader is referred to, e.g., Ref. [

], for complete

details. Consider the equations governing the instantaneous motion for a periodic turbulent chan-

nel flow under the Reynolds decomposition,

(

)

(

)

(

) and ̃

(

)

(

)

(

). Denoting the nonlinear term by

, and Fourier transforming in the homogenous

spatial directions and in time leads to the equation for a given wave number

frequency triplet

(

,ω

), and yields

−

(

∇

)

(

∇

)

∇

−

)

∇

(2)

Here ˆ

denotes a variable in the Fourier-transformed domain,

∇

[

,∂

∂

−

, and Re

/ν

, where

is the channel half-height,

is the friction velocity,

and

the kinematic viscosity. This can be written in terms of the resolvent operator

(

which can be constructed with knowledge only of the mean velocity profile, where we introduce

(

π/

,λ

π/

) for convenience and

ω/

. Here, the mean is one dimensional,

[

(

)

0]. The resolvent has been shown to be low rank for

combinations where wall

082601-2

SELF-SIMILAR HIERARCHIES AND ATTACHED EDDIES

FIG. 1. Self-similar structure on a resolvent hierarchy in channel flow at Re

15 000:

(

), with

(29

123) and

(

) defined as the largest scale on the hierarchy.

−

and

2. Here, all modes have the same peak amplitude, 0

(1). (a)–(d) Isocontours of instantaneous

streamwise fluctuation on an (

) plane,

(

)for

0 symmetric modes.

marked by black

dashed lines and green dashed boxes in (a) identify the regions extracted in (b)–(d). (e) Three-dimensional

isosurfaces of negative

(

)for

0 antisymmetric modes. (f), (g) Aggregated structures in a single

period of the largest scale, color coded by the appropriate scaling height

. Red, blue, and yellow:

respectively. (h) Instantaneous streamwise velocity on an (

) plane,

(

), for the fluctuations in

(a)–(d) added to the turbulent mean shown in (i). Note the aspect ratios in this figure.

082601-3

BEVERLEY J. MCKEON

turbulence is active, e.g., Ref. [

], thus a singular value decomposition of this operator using an

energy norm can be used to obtain an effective (rank-

) approximation of it, in terms of response

and forcing singular modes,

(

) and

(

), respectively, where a complex conjugate is denoted

∗

(

)

(

)

(

)

(

)

≈

∑

(

)

(

)

∗

(

)

(3)

Then the velocity and forcing can be approximated as a weighted sum of resolvent modes,

(

)

∑

(

)

(

)

(

)

(4)

(

)

∑

(

)

(

)

(5)

with the weights

(

) determined by projecting the true nonlinear forcing onto the forcing

modes,

∗

(

)

∑

∫∫

lij

(

)

(

)

∗

(

)

(6)

lij

(

)

=−

(

)

(

)

∫

∗

(

)

[

(

)

∇

(

)]

(7)

Here, the interaction coefficient

lij

(

) describes the projection of the forcing arising from

the interaction between two response modes at

onto the

th forcing mode at (

II. SELF-SIMILAR STRUCTURE ON RESOLVENT HIERARCHIES

Moarref

et al.

[

] identified the conditions for self-similarity of the resolvent and geometric self-

similarity of the singular functions and values to be the logarithmic variation of the mean velocity

in a region bound by

, localization of the response modes to this log region,

and a constraint on the aspect ratio,

1. We enforce these conditions and examine the

first resolvent modes. Self-similar hierarchies can then be defined as subsets

(

) of all mode

parameters

such that

(

)

{

(

)

(

)

,λ

(

)

(

)

max

{

,γ

(

/λ

)

}

(8)

where

denotes the wavelengths of an arbitrary reference mode in that hierarchy and

(

)

. All modes on a hierarchy are geometrically self-similar [

]. Wall-normal scaling with

a consequence of the localization of the response due to the existence of a critical layer, and is

consistent with attached eddy scaling. Similarly,

∼

, however,

∼

instead of simply

for

constant Reynolds number, i.e., the self-similarity of the resolvent yields a self-similar hierarchy

function

[

], which differs from

(

)

[

(

−

)

(

−

)

(

−

)

]

(9)

The former pertains to spectral scaling, while the AEM is developed from the spatial distribution

of self-similar, coherent physical (low-velocity) structures. Thus the coherent physical structure

082601-4

SELF-SIMILAR HIERARCHIES AND ATTACHED EDDIES

associated with members of a resolvent hierarchy represents a more faithful point of comparison to

attached eddies as proposed by Townsend. Figure

shows isosurfaces of the streamwise velocity

field associated with five members of hierarchy

with

√

(29

123) and

(

), at Re

15 000, i.e.,

(

)

∑

−

(

)

[

−

]

(10)

where

relates the

locations of the members of the hierarchy through

−

. For this

analysis,

2 to ensure overlap in

between the fluctuations associated with consecutive modes,

and for consistency with recent implementations of the AEM. For Figs.

1(a)

–

1(d)

[and Fig.

1(h)

each mode is symmetric about

0. The lowest member of the hierarchy is centered at the start of

inertial scaling of the mean profile,

√

[

], and the highest member is constrained

to have

4. The streamwise wavelengths are all in the spectral region in which Ref. [

]

observed wall scaling via coherence measurements. The mode amplitudes were selected to give

the same peak value, which is 0

(1). The exact values of the composite velocities in Fig.

are

dependent on the choice of magnitudes, however, the location of the isosurfaces of zero fluctuation

are only weakly so. Each mode convects with a different convection velocity,

(

), such

that the relative phases between modes—and therefore the zero isocontour locations—are functions

of time. The relative spanwise phase between modes on the hierarchy can be varied without altering

the following observations.

The velocity field associated with Eq. (

) reflects the self-similarity of the hierarchy in a visually

striking manner. Coherent wall-normal structure, associated with the aggregation of low-speed

regions arising from the different modes on the hierarchy and delineated by the isocontours of zero

fluctuation, can be clearly observed in the streamwise

wall-normal planes in Figs.

1(a)

–

1(d)

[also

in three dimensions in Figs.

1(e)

and

1(f)

]. While the (spatial) frequency of individual members

of the hierarchy is dictated by the wave-number scaling on a hierarchy, the coherent aggregated

structures require a different accounting. Two of the largest aggregated regions of negative velocity,

with local minima at

4 and footprints which reach down to the lowest

can be observed

in Fig.

1(a)

. The increasing “forking” corresponds to encompassing increasing numbers of periods

of the lower hierarchy members. Between these two large structures are two centered at the next

lowest

, i.e.,

2, and so on.

Three-dimensional isosurfaces of constant negative instantaneous velocity corresponding to

aggregated structures when mode phases are such that each mode is antisymmetric about

(for ease of visualization) are shown in Fig.

1(e)

. Figures

1(f)

and

1(g)

show an isosurface in one

period of the largest aggregated structure [

,λ

2] with different spatial regions color coded to

correspond to aggregated structures of different spatial scale, i.e., whose dimensions scale with each

, with only

1 : 3 shown for clarity. As most easily seen from below in Fig.

1(g)

, the total

number of structures of the three scales shown are

4, and

16. The smaller scales

also bear out the scaling relationship

(

)

(11)

These aggregations (henceforth simply called structures) are attached in the sense of Townsend

and the AEM in as much as their wall-normal dimension, or coherence, scales with distance from

the wall. The averaged numbers of structures in the streamwise and spanwise directions obey the

attached eddy scaling, but with a spatial exclusion which extends to eddies at different levels within

the hierarchy, i.e., different

, rather than simply for the same

which would be analogous

with the proposal of Ref. [

]. Equation (

) can be modified to give

(

)

[

(

−

)

(

−

)

]

(12)

082601-5

BEVERLEY J. MCKEON

subject to the spatial exclusion constraint. We note in passing the similarity of the aggregations to

the branching structure obtained for optimal thermal transport of a passive scalar by Ref. [

The summation of the mean profile and a single resolvent mode has been shown [

] to replicate

the recognizable bulge structure in isocontours of instantaneous velocity, commonly noted in

experimental and numerical observations, including the asymmetric (in

) ramp and cliff structures

which arise collocated with the half period of negative velocity fluctuation due to the profile of

wall-normal shear. Such features can also be observed in the composite fields of Fig.

1(h)

; the exact

geometry of the isosurfaces of velocity is dependent on both the hierarchy under consideration and

the mode weight relative to the local mean velocity [Fig.

1(i)

], i.e.,

(

)

(

)

(

)

Of note, this development identifies the convection velocity for each member of the structure hier-

archy, which increases with increasing distance from the wall,

, since

≈

(

)

/κ

Thus the images of Fig.

evolve in time in a simple way. Moving in the frame of the

th mode

on a hierarchy, i.e., convecting downstream at

, and considering the relative passing frequencies

of the modes immediately above and below on the hierarchy, i.e., ˇ

(

−

)

∓

/κ

, it can be shown that ˇ

−

=−

. Thus the boundaries of the aggregated

structures vary in time or, equivalently, with convection downstream, but in a predictable fashion.

This may have implications for the eddy shapes used in classical AEM formulations, in which

dynamic evolution is not present. It should also lead to some insight into the time evolution of

uniform momentum zones [Fig.

1(h)

III. REQUIREMENTS FOR SELF-SUSTAINING SELF-SIMILARITY

Besides the aspects of temporal evolution elucidated by the linear analysis above, insight into

nonlinear interactions and therefore the mechanisms by which turbulence self-sustains can also be

obtained, an analytical analog of the numerical solutions cited earlier, derived from the NSE and

the mean profile. A direct connection between the AEM and the self-sustaining minimal solutions

in the log region in the literature has been previously alluded to [

], but not demonstrated.

The linearity of the velocity decomposition in the resolvent framework means that hierarchies can

be linearly superposed, with the nonlinear interactions determining the wave numbers and weights

which are required for consistency with the nonlinear NSE. It is simple to show [

] that the

quadratic interaction between hierarchies is self-similar, meaning that if modes on a set of three

hierarchies are triadically consistent (resonant) at one location in the overlap region, they will also

be resonant at other wall-normal heights on the same hierarchies. Further, self-similar interaction

coefficients

lij

can be defined for the excitation of hierarchy

by hierarchies

over all

heights

where self-similarity is observed. These depend only on separation of the

location of interest and

the reference height

(or, equivalently,

and

) such that

∗

(

)

(

−

)

∑

∫∫

lij

(

−

)

(

)

∗

(

)

(13)

i.e., from Eq. (

lij

(

)

(

−

)

lij

(

−

)

(14)

Thus, for some range of spatial and temporal scales, it is possible that interactions between

hierarchies at one height are replicated in a scaled, self-similar way at other locations in the overlap

region, provided that the weights

(

) are also self-similar and that the influence of modes that

are not self-similar is minimal.

Further, the Reynolds-Orr equation for the energy budget associated with a single mode in a

hierarchy [

]gives

(

)

−

(

)

(

)

(15)

082601-6

SELF-SIMILAR HIERARCHIES AND ATTACHED EDDIES

with

(

)

=−

(

)

(

)

∫

{

(

)ˆ

(

)

∗

(

)

}

(16)

(

)

(

)

(

)

∫

[

(

)

∗

(

)

(

)

∗

(

)]

(17)

(

)

=−

(

)

(

)

∫

{

(

)

∗

(

)

}

(18)

On a self-similar hierarchy with self-similar weights, the mode scaling implies self-similarity of

(

) and

(

), while

(

) must also be self-similar if the forcing

(

) arises from

the interactions between continuous hierarchies [Eqs. (

), (

), and (

)—see Ref. [

] for further

discussion].

If the real velocity field can be modeled as an integral over hierarchies,

(

), then the scaling

results above admit the somewhat remarkable possibility of determining from the equations of

motion velocity fields, nonlinear interactions, and energy budgets which are all self-similar and

scale with distance from the wall, and closing the NSE using resolvent modes.

IV. SUMMARY AND CONCLUSIONS

Geometric self-similarity of the linear Navier-Stokes operator for turbulent flow has been ob-

served by several authors, however, the self-similar scaling (in spectral space) was in disagreement

with the full distance-from-the-wall scaling of Townsend and the AEM. It has been shown here

that this apparent conflict can be resolved by making an “apples-to-apples” structural comparison,

following the common AEM representation of a geometric progression (rather than a continuous

distribution) of eddy sizes. The aggregated structures associated with a geometrically self-similar

resolvent hierarchy shown in Fig.

obey the attached eddy scaling and reveal a stricter spatial

exclusion than previously posited through the AEM. However, these aggregations exist relative

to the turbulent mean profile, which is assumed (equally, acts as a constraint) in the resolvent

framework, whereas attached eddies are superposed on a uniform flow.

The resolvent formulation also reveals spatiotemporal information, i.e., dynamics, associated

with the self-similar resolvent hierarchies. Members of a hierarchy move with different convection

velocities, meaning that while the aggregations are self-similar at a given instant, their boundaries

evolve in time, in a self-similar way, with further implications for the direct comparison with the

AEM. The results of Refs. [

] concerning self-similarity of the nonlinear interactions between

resolvent hierarchies reveal the possibility of self-sustaining assemblies of hierarchies, with likely

connections to the self-similar, self-sustaining solutions obtained in minimal unit simulations in the

logarithmic layer.

The connection between the velocity response modes which are naturally most amplified in the

equations of motion, parametrized in spectral space, and empirical physical space reasonings going

back to Townsend seems to hold promise for improved modeling in both domains. We have focused

on the first resolvent modes herein, but the approach can be extended to consider higher-rank

resolvent approximations and to include the separate consideration of Orr-Sommerfield and Squire

contributions to the wall-normal vorticity, which was identified by Ref. [

] as an important step in

obtaining an efficient basis to represent real flows. The weights

(

) hold the key to nonlinear

closure of the resolvent framework; the work herein suggests that analytical progress to complement

data-driven resolvent approaches, e.g., Refs. [

], may be made. Connections between the

resolvent results, the AEM, the self-similar minimal unit and exact coherent solutions, and the mean

flow similarity of the MMB are the topic of ongoing work.

082601-7

BEVERLEY J. MCKEON

ACKNOWLEDGMENT

The support of ONR through Grant No. N00014-17-1-3022 is gratefully acknowledged.

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