Amplitude and wall-normal distance variation of small scales in turbulent boundary layers

PHYSICAL REVIEW FLUIDS

, 014606 (2022)

Amplitude and wall-normal distance variation of small scales

in turbulent boundary layers

Theresa Saxton-Fox

Adrián Lozano-Durán

and Beverley J. McKeon

Department of Mechanical and Civil Engineering, California Institute of Technology,

Pasadena, California 91125, USA

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139-3291, USA

(Received 7 October 2021; accepted 3 January 2022; published 20 January 2022)

The spatial organization of small scales around large-scale coherent structures in a

flat plate turbulent boundary layer is studied using a conditional-averaging technique

applied to experimental and computational data. The technique averages the small-scale

velocity conditioned on the projection coefficient between the instantaneous streamwise

velocity field and a model for large-scale velocity structures in the wake and logarithmic

regions. Two distinct scenarios are identified for the organization of the small scales:

amplitude variation, in which at a given wall-normal location the small-scale intensity

varies in amplitude across the streamwise extent of the large-scale structure, and height

variation, in which the small-scale velocity intensity remains nearly constant along a

curve that changed its wall-normal location across the streamwise extent of the large-scale

structure. Small scales that are energetic at the wall-normal location where the large-scale

structure is centered primarily show evidence of height variation, while small scales that are

energetic at wall-normal locations far from the center of the large-scale structure primarily

show evidence of amplitude variation. Connections can be drawn between the statistical

observations characterized by the amplitude modulation statistic and the structural picture

associated with vortex clusters.

DOI:

10.1103/PhysRevFluids.7.014606

I. INTRODUCTION

Coherent structures populate turbulent boundary layers and are central to their chaotic dynamics

[

]. These structures can be separated into large and small features through comparison to a

reference length scale. Using a reference length scale on the order of the boundary layer thickness,

the large-scale coherent structures are the very-large-scale motions (VLSMs) or superstructures

[

–

] in the logarithmic region and the large-scale motions (LSMs) in the wake region [

]. The

most commonly discussed small-scale coherent structures are the streamwise streaks [

]inthe

near-wall region and vortices (hairpin, cane, horseshoe, “typical eddies,” vortex clusters, etc.) in the

logarithmic and wake regions [

]. Very-large-scale motions and LSMs appear instantaneously

(without the mean subtracted) as bulges in the streamwise velocity field, with regions of uniform

momentum in their centers bounded by shear layers [

–

]. Small-scale vortices have been

observed in shear layers along the backs of low-speed streamwise velocity regions [

]. These

vortices may have a hairpin vortex structure: thin and long, arching over large scales in the flow

[

], though the persistence of hairpin vortices at high Reynolds number is a topic of continued

Present address: Department of Aerospace Engineering, University of Illinois at Urbana-Champaign,

Urbana, Illinois 61801, USA, tsaxtonf@illinois.edu

2469-990X/2022/7(1)/014606(18)

014606-1

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

FIG. 1. Schematic representations of (a) amplitude variation and (b) height variation. Large-scale fluctuat-

ing streamwise velocity structures are shown in red (fast) and blue (slow), while small scales are shown in gray

(dark indicating strong amplitude and light indicating weak amplitude).

debate [

]. The vortices may also have a more disorganized or chaotic structure at higher

Reynolds number [

The interaction of coherent structures enables the exchange of energy between scales and sustains

the turbulent state [

]. Preferential organization of one structure depending on the behavior of an-

other structure is one indication of interaction. Small-scale vortices at the edge of the boundary layer

and near the wall are organized around and influenced by large-scale velocity motions [

]. The

organization between large and small scales in turbulent boundary layers has been quantified by

filtering the velocity field into large and small scales and correlating the filtered signals, using data

from hot-wire measurements [

], large-eddy simulation [

], and direct numerical simulation [

In each case, the envelope and intensity of the small scales was observed to be strongly correlated

to the large-scale streamwise velocity signal with the sign of correlation changing from positive

near the wall to negative farther away. This relationship was found to be robust across Reynolds

numbers [

], geometries [

], pressure gradients [

], small-scale velocity components [

], and

wall permeability [

The observed correlation between large and small scales permits several interpretations of the

instantaneous organization of large and small coherent structures, including amplitude variation and

height variation. We focus here on these two instantaneous interpretations, which are illustrated in

Fig.

with flow moving from left to right. Large-scale fluctuating coherent structures of streamwise

velocity are shown in red (positive) and blue (negative) and small-scale coherent structures are

shown as circles with low amplitude (light gray) and high amplitude (dark gray). In the case of

amplitude variation [Fig.

1(a)

], multiple small-scale coherent structures, each at a fixed height,

change amplitude depending on the local behavior of the large-scale coherent structure. This is

the stress organization often inferred from the spectral picture offered by traditional conditional

averaging (see, e.g., [

]). In the case of height variation [Fig.

1(b)

], a single small-scale coherent

structure with a fixed amplitude changes height depending on the local behavior of the large-scale

coherent structure, while weak, incoherent small-scale activity is present throughout the flow. Height

variation is aligned with preferential vortex alignment on the back of low-speed structures (see, e.g.,

[

]) and with the ejection, sweep, and burst representation of the flow (see, e.g., [

]). The

amplitude variation and height variation representations lead to similar statistics but imply different

mechanisms driving the organization.

The schematic in Fig.

is broken up into three regions in the streamwise direction to examine

the local behavior. In Figs.

1(a)

and

1(b)

, in region I the large-scale streamwise velocity fluctuation

is positive and small scales are strongest near the wall; in region II, the large-scale transitions

from positive to negative and small scales are strongest at a middle height; and in region III, the

014606-2

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

large scale is negative and small scales are strongest away from the wall. The two explanations

differ in the spatial organization of the small-scale coherent structures and in the driving scale

organization mechanism that they suggest. Amplitude variation may point to an energy transfer

or stress-based mechanism, while height variation may point to a convective mechanism. The two

behaviors are visually distinguishable by the continuity of the small-scale intensity across regions

I–III. In amplitude variation [Fig.

1(a)

], a given small-scale coherent structure appears continuously

at a fixed height across the three regions. In height variation, a given small-scale coherent structure

appears continuously along a curve of changing height across regions I–III. The present work aims

to quantify the small-scale intensity across the streamwise extent of a large-scale coherent structure

to investigate whether amplitude or height variation is dominant.

The present study uses spatially resolved measurements and a different conditional averaging

technique to study the averaged small-scale velocity intensity across a continuous range of phases

of large-scale coherent structures. Both amplitude and height variation are observed with different

levels of importance depending on the type of large scale being averaged upon and the component

of the small-scale velocity intensity being considered. Spatial data are obtained using particle image

velocimetry and direct numerical simulation. A global conditional averaging technique, termed

conditional projection averaging, is used to correlate the large-scale signal and small-scale intensity.

The technique yields a smoothly varying, averaged large-scale field and a correlated smoothly

varying, averaged small-scale intensity field. The results of the averaging process are provided,

including the extent to which height and amplitude variation are observed in each component of the

velocity field. The sensitivity of the results to the details of the analysis technique are offered in the

Appendix.

II. SOURCES OF DATA

Turbulent boundary layer data were obtained from particle image velocimetry (PIV) measure-

ments in an experimental facility from Saxton-Fox [

] and direct numerical simulation (DNS) from

et al.

[

]. Data extracted from both sources were at similar Reynolds numbers, Re

3300

and Re

910 for the experiment and Re

3000 and Re

900 for the DNS. The experimental

data were collected in the Merrill wind tunnel at the California Institute of Technology, an incom-

pressible, recirculating facility. The turbulent boundary layer was developed in a 0

test

section with a flat plate that had a wire trip at the leading edge. The velocity was measured in the

streamwise–wall-normal plane using a double-pulsed YLF laser and a Photron Fastcam APX-RS

camera with a 17-mm Tamron macrolens. The measurement was recorded at a rate of 1.5 kHz with

a pulse separation of 35

s and yielded data with a field of view of 1

, where

refers to

the 99% boundary layer thickness. DaVis software from LaVision was used to process the data; a

double pass approach with windows of first 32 and then 16 pixels with 50% overlap was used to

yield a velocity field with 0

013

013 outer units or 14

5 inner units resolution per vector

[

For the computational data, the boundary layer was simulated from its initial laminar state

through a bypass transition to a fully canonical turbulent state [

]. In the present work, the

downstream portion of the simulation, where Re

≈

000, was analyzed. In this region, the span-

wise extent of the simulation was approximately 2

, using the local 99% boundary layer thickness.

Many streamwise–wall-normal planes were extracted from the DNS data at different spanwise

locations. These planes were compared directly to the PIV data and improved the convergence

of the analysis techniques, as only 30 temporal snapshots were available for the computational data,

compared to 5120 temporal snapshots for the experimental data. Converged statistics of the flow

were available for comparison to the experimental data just upstream of the region of interest, at

900, which are shown in Fig.

The notation adopted for the coordinate system was (

) (streamwise, wall-normal, span-

wise), with velocity components

(

) in the streamwise, wall-normal, and spanwise

directions. The mean velocity field of each data set

(

) was computed by performing a

014606-3

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

FIG. 2. Mean profile (blue dots) and rms (red dots) for the streamwise velocity from the PIV data at Re

3300 as a function of distance from the wall and mean profile (cyan stars) and rms (magenta stars) for the

streamwise velocity from the DNS data at Re

2900 [

]. The dashed line indicates the wall-normal distance

below which reflections corrupted the experimental data.

temporal average (for the experimental data) and combined temporal and spanwise average (for

the DNS data) of the velocity field. The fluctuating velocity about the mean velocity field was

defined as

(

)

(

)

−

(

). A characteristic mean velocity profile

(

)was

defined for the experimental data by performing a temporal average for the experimental data at

a streamwise location in the center of the field of view. The computational mean velocity profile

had been computed in the original computational campaign just upstream of the region of interest,

at Re

900. Similarly, the root mean square streamwise velocity fluctuation was calculated in

the center of the field of view for the experimental data (Fig.

). The data were normalized using

viscous units, with length scales normalized by

ν/

and velocities normalized by

, where

is the kinematic viscosity and

is the friction velocity. The friction velocity was estimated for

the experimental data using the Clauser method [

]. When compared to data from Wu

et al.

[

] at similar Reynolds numbers, the statistics of the experimental data were found to agree fairly

well above

≈

40 or equivalently

/δ

≈

05. The statistics were also separately compared to

those from DeGraaff and Eaton [

] (not shown here) and showed agreement above

≈

40 or

equivalently

/δ

≈

05. Below

≈

40, reflections from the laser impingement on the flat plate

caused inaccuracies in the velocity measurement.

Both experimental and computational data were used in the analysis that follows. The use of

experimental or computational data was not found to alter the trends observed in the results, and

therefore each result was shown using either experimental or computational data. The experimental

data had the benefit of a larger number of temporal snapshots available for analysis, which provided

additional confidence in the convergence of the results. The computational data had the benefits

of improved spatial resolution, improved accuracy very near the wall, a larger field of view, and

access to the spanwise component of velocity. The experimental data were used for streamwise and

wall-normal velocity results in the outer and logarithmic regions of the flow, while computational

data were used in the very-near-wall region and for the spanwise component of velocity. The number

of snapshots used for each result is provided in Table

. To test whether the difference in spatial

resolution between the two data sets affected the results, the smallest scales were filtered from the

data; the final amplitude of the results varied but the trends did not.

014606-4

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

III. METHODS

A. Decomposition of velocity into large and small scales

A Gaussian filter was used to decompose the turbulent data into large and small scales in the

and

directions [

]. An approximate cutoff frequency can be defined as the frequency where

the filtered signal is half of its maximum in the power spectrum, given by

√

2ln(2)

πσ

(1)

where

is the standard deviation of the Gaussian function. The values used for the standard

deviation in this study were

/δ

125

5], corresponding to a range of approximate cutoff

wavelengths

/δ

7]. These approximate cutoff wavelengths bounded the value used by

Mathis

et al.

[

/δ

1, and were within the range of filter sizes considered by Ganapathisub-

ramani

et al.

[

]. The final results were shown to be qualitatively insensitive to the filter size in the

Appendix.

The fluctuating large-scale streamwise velocity field

was defined by convolving the Gaussian

function with the fluctuating streamwise velocity field. This convolution was normalized by the

integral of the filter in the domain to minimize edge effects. The small scales were computed by

subtracting the fluctuating large-scale velocity field from the fluctuating velocity field

(

)

(

)

−

(

)

(2)

The equivalent procedure was used to define the large and small scales in the wall-normal and

spanwise velocity fields

, and

, respectively. The small-scale velocity intensity was

defined as the square of the small-scale velocity. A mean-added, large-scale streamwise velocity

field was defined as the sum of the fluctuating large-scale field with the mean

(

)

(

)

(

)

(3)

B. Model for large-scale coherent structures

In the present work, a spatially periodic model for the large-scale structures is used. Modeling

large-scale coherent structures as periodic in the streamwise and spanwise directions and in time is

supported by recent work characterizing the periodicity of large-scale velocity structures [

] and

the reported ubiquity of the VLSM frequency for amplitude modulation [

]. Periodic models have

a clearly defined phase with wavelengths and wave speeds that are known to be energetic in the flow

from experimental and computational results. Periodic models were previously shown to capture

instantaneously realistic flow phenomena including bulges, uniform momentum zones, and hairpin

vortices [

]. However, we acknowledge that instantaneous turbulent structures are not periodic

and that the parallel flow assumption required to use periodic structures in the streamwise direction

introduces limitations in the model.

Resolvent analysis of the Navier-Stokes equations was used to generate a template of large-scale

turbulent structures [

]. Following the parallel flow assumption, periodicity was assumed in the

streamwise direction and Fourier transforms were performed in the temporal, streamwise, and

spanwise variables. Given a mean velocity profile

(

) and values of the streamwise wave number

, spanwise wave number

, and temporal frequency

that are relevant for a given structure of

interest, the Navier-Stokes equations can be reformulated into an input-output system such that the

Fourier-transformed nonlinear term

is the input to a linear operator

H

and the Fourier-transformed

velocity and pressure fluctuations

are the output:

H

. By performing a singular value

decomposition on the linear resolvent operator, the most-amplified input and output structures can

be identified as the left and right singular vectors, respectively,

H

−

. We use a rank-1

approximation, such that the first left singular vector of the resolvent operator

(

)isusedasthe

representative wall-normal coherence for the structures at a specified wave number [

]. Finally, the

014606-5

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

TABLE I. Comparison between traditional conditional averaging and conditional projection averaging.

Parameter

Traditional

CPA

Location of data extracted

Single point

Regions of overlap with model

to evaluate condition

Comparison value

Single value

Local values of model

Comparison operation

Greater than or less than

Projection coefficient (greater than)

and maximization over phase

large-scale model is given by the real part of the sum of a left-propagating and right-propagating

wave

(

)

[

(

)

(

−

)

(

−

)

)]

(4)

which represents a downstream-propagating traveling wave where ̃

is an arbitrary phase.

In the wake region of the boundary layer, where large-scale motions are energetically dominant

[

], the model was constructed using assumed wavelengths of

/δ

4 and

/δ

1, with a

convection velocity of

∞

(

ω/

)

∞

8. In the logarithmic layer, where VLSMs are en-

ergetically dominant, the model assumed wavelengths of

/δ

6 and

/δ

1, with a convection

velocity of

∞

(

ω/

)

∞

6. Modes derived from resolvent analysis have been shown to

be maximized at their critical layer [

], the height where their convection velocity matches the

local mean velocity. The convection velocity chosen for the large-scale motion model placed the

peak amplitude in the wake region, while the convection velocity for the very-large-scale motion

placed it in the logarithmic region.

C. Conditional projection averaging

We introduce the conditional projection averaging (CPA) method, which is developed to average

a field variable conditioned on the instantaneous streamwise location of a particular large-scale

coherent structure. Rather than averaging based on a condition evaluated at a single point, a global

condition is utilized through the projection between the spatially resolved instantaneous velocity

field and the spatial model of the large-scale structure. The projection between the velocity field

and the model acts to simultaneously extract data at the appropriate locations in the flow and

evaluate whether the velocity satisfies the chosen condition. This method can be used with structural

models that have a defined phase or for which a phase can be defined from the mode’s convection

in space and time. Resolvent analysis, dynamic mode decomposition, spectral proper orthogonal

decomposition, and space-time proper orthogonal decomposition all generate modes that could be

used for conditional projection averaging [

–

]. In this work the resolvent analysis described in

the preceding section is used to define the model. A summary of the similarities and differences of

CPA versus traditional conditional averaging is provided in Table

. The schematic representation

of the projection method is provided in Fig.

In traditional conditional averaging, one point in the flow at a time is generally selected to

evaluate a condition [

(

)]; in a flow with statistically homogeneous or slowly varying

streamwise and spanwise directions, this is equivalent to choosing a single height in the flow at

which to evaluate a condition. Multiple heights are often studied in sequence by systematically

changing the height of the conditional point in the flow. Global conditional averaging techniques

instead use information from many points in space and/or time to condition. Variable-interval time

averaging is an example of such a technique that uses local averages of turbulence quantities over a

region of time or space [

]. The CPA method used here is a global technique in which instantaneous

spatial data are projected onto a spatial model of a large-scale structure; the data that overlap with

heights where the model has nonzero amplitude are used to evaluate the condition, while data

that overlap with heights where the model has zero or near-zero amplitude are masked by the

014606-6

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

FIG. 3. Schematic of the projection method between (a) a visualization of the instantaneous fluctuating

streamwise velocity field and (b) an illustration of a large-scale model at four locations in its convection,

labeled as phases ̃

. The black box outlines the approximate size of a single PIV frame. The data are projected

onto the large-scale model at each model phase ̃

. The value of the projection coefficient is used to evaluate

the condition and average together data that compare favorably to the model at the same phase.

multiplication by zero and do not influence the resulting average. The models used in this study have

an approximately Gaussian distribution in the wall-normal coordinate with maxima located at their

critical layer, i.e., the height where the model’s convection velocity matches the local mean speed.

The choice of the model’s critical layer height (or equivalently convection speed) is analogous to

the choice of the wall-normal location in the flow at which the condition is evaluated in traditional

conditional averaging.

In traditional conditional averaging, a numeric value is selected with which to compare the

instantaneous data at a single point. Conditional projection averaging uses a spatial field rather

than pointwise information to evaluate the condition, so a singly valued condition is no longer

meaningful. Instead, the data at each spatial point are compared to the value of the model at the

same spatial point. The projection operation averages the result of these comparisons together, with

larger weight on the agreement in locations with larger model amplitude. In traditional conditional

averaging, one often looks at different numeric values for comparison, such as a negative versus

positive value. In CPA, one can vary the value of comparison by changing the phase of the periodic

model. A shift of

yields the binary positive-negative value switch of traditional conditional

averaging at any one location, but one can also consider finer phase shifts that change the weighting

of different streamwise locations in the computation of the projection coefficient.

In traditional conditional averaging, one determines whether the data satisfy the condition

through a greater than or less than operation [

(

) less than a single value]. In CPA, the

projection coefficient between the data and model is used to evaluate whether the data satisfy the

condition. Two separate operations are used to evaluate the projection coefficient. The first requires

that the projection coefficient is greater than some threshold value

, indicating a certain level

of average agreement between the data and the model. The second is a maximization condition,

which requires that the projection coefficient is larger at the model phase of interest than at any

other phase considered. This condition ensures that the instantaneous streamwise position of the

large scale in the data is similar to the streamwise position of the large scale in the model. To

evaluate the maximization condition, the projection between the data and the model is computed

at many model phases. The maximization condition was found to increase the robustness of the

method and is schematically demonstrated in Fig.

The projection between the instantaneous fluctuating velocity

and a model ̃

at several model

phases ̃

is given in its most general form as

(

)

≡

(

)

(

)

(

)

(

)

(5)

where a dot product is performed in the numerator,

is one snapshot, and

is the projection

coefficient. Using the maximization condition, one defines the model phase that yields the highest

projection coefficient for a given snapshot,

(

)

arg max

(

)

∀

(6)

014606-7

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

FIG. 4. Projection coefficient

plotted against the model phase ̃

for a single snapshot of experimental

data.

This phase represents the approximated large-scale structure phase of the snapshot,

. The process

was repeated for each snapshot such that each snapshot was associated with one phase. The

threshold condition

(

,γ

(

)

(7)

enforces that the best projection coefficient has to be larger than a threshold value

for that

snapshot to be considered in any average based upon the approximated large-scale structure phase.

The two-dimensional projection between the data and the model was defined as

(

)

≡

(

)

(

)

(

)

(

)

(8)

where

is a single spanwise location in the data and

is a single spanwise location in the model.

For the experimental data, only a single spanwise location was available due to the planar nature

of the PIV system, while for the computational data, multiple spanwise locations were considered

in series. When analyzing the DNS data, a modification was leveraged to improve the efficiency

of the computation. The projection operation weights the data-to-model comparison largest at

locations where the model has the largest amplitude. The models used in this study have a roughly

Gaussian distribution in

, with a maximum at their center and a periodic distribution in

.To

improve efficiency, a one-dimensional projection was used, in which data from each snapshot at

each spanwise location

and at a single wall-normal location

was projected onto information

from the model at each phase ̃

and at a spanwise location

and wall-normal location

where

the model was maximized in space. The projection was computed using Eq. (

) with

substituted

for

, yielding a one-dimensional projection. The wall-normal location

for the comparison was

the critical layer height of the model. Both one-dimensional and two-dimensional projections were

tested on the experimental data; the results are shown in the Appendix to be fairly insensitive

to the dimensionality of the projection. Figure

shows an example of the variation of

with ̃

for ̃

,π/

π/

4] for a single snapshot of experimental data,

(

). The approximated

large-scale structure phase for the snapshot used in Fig.

was

(

)

π/

Once each snapshot was labeled with a large-scale phase, various flow quantities

from two-

dimensional planes with the same approximated large-scale phase

were averaged together,

(

,γ

)

∑

(

)

∀

s.t.

(

,γ

)

(

)

(

,γ

)

(9)

where

is the total number of planes considered for the average. For ease of visual inspection,

the averages are organized sequentially in the final results such that the phase continuously varies

across the horizontal axis. The number of frames used for the average at each phase condition for

each structure and type of data is provided in Table

014606-8

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

TABLE II. Number of planes of data used for the averaging in Figs.

–

. The number of distinct temporal

snapshots

versus total planes including distinct times and spanwise coordinates

are provided. The

number of two-dimensional planes per phase bin is also listed.

frames

π/

LSM, PIV

4385

481

691

533

399

495

730

650

406

LSM, DNS

8163

1180

1119

772

953

1271

891

976

1001

VLSM, PIV

5120

1055

885

384

359

861

915

314

347

VLSM, DNS

5529

748

556

587

854

764

557

642

821

IV. RESULTS

A. Instantaneous visualization of filtered velocity field

The filtered, instantaneous streamwise velocity field from the computational data set is visualized

at one snapshot over a small range of spanwise locations

/δ

3–0

8inFig.

[

]. The fluctu-

ating large-scale streamwise velocity field is shown in a streamwise–wall-normal plane in red and

blue contours, while the mean-added velocity field is shown at a single value

∞

85 in an

isosurface in green. The value of the

isosurface was chosen to match the convection speed of the

large-scale flow, which was calculated using a cross correlation between two consecutive snapshots.

This

isosurface is not collocated with the intermittent edge of the boundary layer. The small-scale

streamwise velocity field is shown using isosurfaces at two values

(

)

∞

=±

06 in

black and white.

The small-scale isosurfaces fully populate the space near the wall in Fig.

, where small-scale

fluctuations exceed the isocontour threshold. Farther from the wall, above

/δ

≈

25, the small-

scale fluctuations are sparse when the fluctuating large-scale streamwise velocity field is positive

and dense when the fluctuating large-scale streamwise velocity field is negative, consistent with a

sweep and ejection representation of the flow [

]. The threshold chosen to visualize the small-scale

field affected at what heights the isosurfaces appeared, but did not affect the qualitative behavior of

the results until the isosurfaces fully filled the space or restricted the isosurfaces to lie only below

/δ

≈

FIG. 5. Visualization of the instantaneous, filtered streamwise velocity field of a turbulent boundary layer

[

]. Red and blue contours represent the large-scale fluctuation

. The white and black isosurfaces represent

the small-scale fluctuation

at values of

∞

. The green isosurface represents an isosurface of the

large-scale laboratory-frame streamwise velocity field

at a value of 0

∞

, which is the approximated

convection velocity of the large scale in this snapshot.

014606-9

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

FIG. 6. Conditional projection average of the intermittency variable

(

) based on a projection onto a

model of a LSM. The isocontour

∞

is plotted for reference.

The rise and fall of the mean-added large-scale streamwise velocity isosurface (green in Fig.

)

follows a similar trend as the small-scale isosurfaces. The visual similarity between the height at

which the green isosurface is observed and the height where small scales are active suggests a

hypothesis that small-scale activity may be consistently present along lines of constant value of

the local large-scale flow. This hypothesis is consistent with previous observations that isocontours

of the full velocity field coincide with the locations of shear layers [

] and may be related to

the concept of critical layers, which amplify small-scale perturbations where the perturbation speed

matches the speed of the base flow or mean flow [

]. Because of this observation, the mean-added

large-scale velocity isocontour is visualized in addition to the fluctuating large-scale velocity field

in the results that follow.

B. Results for conditional projection averaging

An intermittency variable

was defined with the same definition used by Chauhan

et al.

[

]

such that its value was zero within the boundary layer and one outside of it, with a sharp interface

indicating the instantaneous height of the turbulent–non-turbulent interface (TNTI) for each frame

of PIV. The intermittency variable

(

) was averaged using the conditional projection averaging

technique with a LSM model, setting

using Eq. (

). Figure

shows the result of this averaging

process. The averaged streamwise velocity field

was also computed and an isocontour of

8 was overlaid, highlighting that this isocontour lies within the boundary layer height

on average.

The velocity fields and intensities averaged using the CPA method are shown in Figs.

and

Results conditioned on a projection with a LSM model in the wake region are shown in Fig.

while results conditioned on a projection with a VLSM model in the logarithmic region are shown

in Fig.

. Results averaged around the approximated phase of LSMs are shown with a linear wall-

normal scaling, while the results averaged around the approximated phase of VLSMs are shown

with a logarithmic wall-normal scaling. A single isocontour of the average streamwise velocity field

summed with the mean

(

)

(

)

(

)

is overlaid on each panel of Figs.

and

, with

∞

in Fig.

and

∞

in Fig.

, corresponding to the wave speed of the

LSM and VLSM models, respectively. The visualization of the isocontour was motivated by the

spatial relationship between the mean-added streamwise velocity field and the small-scale intensity

that was instantaneously observed in Fig.

. The filter sizes for the LSM- and VLSM-averaged

small-scale velocity fields were

and

125

, respectively. Experimental data were

used to compute the results of Figs.

7(a)–7(d)

8(b)

, and

8(c)

, while computational data were used

to compute the results of Figs.

7(e)

8(a)

8(d)

, and

8(e)

. Experimental data were used when possible

due to the larger number of temporal snapshots yielding better convergence, while computational

data were used for all averages of the spanwise velocity and to resolve near-wall behaviors.

The averaged unfiltered streamwise velocity fields [Figs.

7(a)

and

8(a)

] yielded periodic struc-

tures with a period and wall-normal coherence matching those of the model used for projection.

This behavior was expected and was plotted to provide a reference for the local large-scale behavior

when considering the small-scale velocity intensities. The small-scale velocity intensities averaged

014606-10

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

FIG. 7. (a) Streamwise velocity field and (b)–(e) small-scale velocity intensities averaged using projection

onto the LSM model. Conditional small-scale (b) and (d) streamwise, (c) wall-normal, and (e) spanwise

intensities are shown. Results are derived from (a)–(d) experimental and (e) numerical data.

FIG. 8. (a) Streamwise velocity field and (b)–(e) small-scale velocity intensities averaged using projection

onto the VLSM model. Conditional small-scale (b) and (d) streamwise, (c) wall-normal, and (e) spanwise

intensities are shown. Results are derived from (b) and (c) experimental and (a), (d), and (e) numerical data.

014606-11

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

on the phase of LSMs [Figs.

7(b)–7(e)

] were stronger farther from the wall in phases in which the

averaged large-scale streamwise velocity fluctuation

was negative and stronger nearer to the

wall in phases in which

was positive, consistent with previous studies [

]. All three velocity

intensities showed a clear dropoff in intensity at the averaged TNTI location. Within the internal

structure of the boundary layer, the streamwise and wall-normal small-scale intensities showed an

additional localization that was not observed in the spanwise small-scale intensity. The color bars

were chosen to highlight the wake region localization observed for the streamwise and wall-normal

intensities in Figs.

7(b)

and

7(c)

, while the color bars for the streamwise velocity in Fig.

7(d)

and

for the spanwise velocity in Fig.

7(e)

were chosen to highlight behavior in the logarithmic region.

The streamwise velocity intensity in the wake region of Fig.

7(b)

(

/δ

1) had a roughly

constant maximum amplitude in the wake region across the large-scale phases at a range of heights

/δ

6. The region of constant amplitude overlapped with the isocontour

∞

The wall-normal small-scale velocity intensity in Fig.

7(c)

was less peaked and less consistent in

maximum amplitude than the streamwise intensity but still showed a dropoff in intensity below the

isocontour

∞

8, rather than at the edge of the boundary layer, at heights that varied

with the large-scale phase. The streamwise velocity intensity [Fig.

7(d)

] and spanwise velocity

intensity [Fig.

7(e)

] were plotted with color bars that highlighted their behavior in the logarithmic

region, while still being averaged on the LSM model in the wake region. Their behavior in the

logarithmic region shows change in the maximum amplitude over the large-scale phases for both

velocity components. The height at which the maximum occurs appears fairly constant across the

large-scale phases.

The analysis was performed again with a projection on a VLSM model in the logarithmic

region. The streamwise velocity in the logarithmic region, shown in Fig.

8(b)

, showed variation

in amplitude as well as a change in the height where that maximum occurred. The maximum

height approximately followed the isocontour

∞

6. The wall-normal small-scale velocity

intensity [Fig.

8(c)

] was most energetic in the wake region at a fairly constant height across the

VLSM phases. Variation in the maximum amplitude was observed, stronger in the presence of a

negative VLSM fluctuation and weaker in the phases of a positive VLSM fluctuation. The color

bar for the streamwise small-scale velocity intensity was shifted to highlight the near-wall cycle

in Fig.

8(d)

. The maximum amplitude of the streamwise small-scale velocity intensity varied over

the VLSM phases and remained at a fairly constant wall-normal location. The spanwise velocity

intensity [Fig.

8(e)

] had fairly constant maximum amplitude across the VLSM phases. The height

at which the maximum occurred varied across the phases and coincided with the height of the

∞

6 isocontour that was overlaid (the assumed wave speed of the VLSM model).

V. DISCUSSION AND CONCLUSIONS

Both height and amplitude variation effects were observed in the small-scale velocity intensities

averaged using the conditional projection averaging technique. The height and amplitude variation

trends depended on the component of the small-scale velocity field being averaged, the wall-normal

region where the small-scale activity was studied, and the type of large-scale structure used for

projection. In the wake,

and

both showed some evidence of height variation when the projection

was performed with the LSM model: Both intensity fields showed localization and a change in the

wall-normal location of the maximum amplitude over the phases that was not exclusively tied to the

location of the TNTI [Figs.

7(b)

and

7(c)

]; however, when

was averaged using the VLSM model

for projection in Fig.

8(c)

, amplitude variation was observed with little height variation. In the

logarithmic region,

and

showed evidence of amplitude variation when the projection applied

used the LSM model in the wake, but both showed evidence of height variation when the projection

applied used the VLSM model in the logarithmic region. Further,

in the near-wall cycle region

showed evidence of amplitude variation when the projection applied used the VLSM model in the

logarithmic region.

014606-12

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

FIG. 9. Interpretation of the height and amplitude variation behavior observed. Large-scale fluctuations are

shown in red and blue with a large-scale velocity isosurface drawn in gray curves. Small-scale features in the

same region of the flow as the large scales, represented with orange circles, follow the large-scale isosurface.

Small scales in other regions of the flow, represented with green ovals, change amplitude depending on the

local shear stress imposed by the large-scale fluctuations.

A hypothesis to explain these various observations is provided in Fig.

. Small scales that were

active and strong in the same region of the flow as the large-scale structure used for the projection

often showed evidence of height variation (

and

in the wake when averaged on LSMs and

and

in the logarithmic region when averaged on VLSMs). Small scales that were active and

strong in a different region of the flow as the large-scale structure used for the condition projection

often showed evidence of amplitude variation (

and

in the logarithmic region when averaged

on LSMs,

in the wake when averaged on VLSMs, and

in the near-wall cycle region when

averaged on VLSMs). We hypothesize that small scales that are physically near the central height

of a large-scale coherent structure experience height variation as a result of that structures’ presence

(through participation in packets and/or bursts and sweeps), while small scales that physically

occupy a different space than the large-scale coherent structure of interest instead experience a

change in the stress that they experience [

]. A faster (slower) structure above the small scale

yields a stronger (weaker) local shear stress for that small scale, while a faster (slower) structure

below the small scale yields a weaker (stronger) local shear stress for that small scale. In Fig.

red and blue objects represent the large-scale feature, orange represents the small scales in the same

region of the flow as the large scale, and green represents the small scales in other regions of the

flow. Darker (lighter) green indicates stronger (weaker) small scales. A black isosurface of the large

scale including the mean is also provided, and the orange small scales are shown following that

isosurface.

The height variation behavior observed here is consistent with the preferred spatial organization

of vortical structure observed by, e.g., Adrian [

] and the vortical fissures of Klewicki and Hirschi

[

], as well as the burst and sweep perspective of Kline

et al.

[

]. The stress-based amplitude

modulation behavior observed here is consistent with the observations of Baars

et al.

[

]. In this

paper, we connect the statistical observations characterized by the amplitude modulation statistic

and the structural picture associated with vortex clusters and hairpin packets. Investigations of causal

and dynamic interactions between the large- and small-scale motions are reserved for future work.

Previously, no significant differences were observed in the correlations between large-scale

streamwise velocity signals and the envelope of different small-scale velocity components

, and

[

]. However, in the present work, the height versus amplitude variation trend seems dependent

on the component of the small-scale velocity being considered. The reason for this difference may

be that the present study did not normalize by the average small-scale intensity locally. It was

important in this study to observe the intensity level normalized by a global parameter, in order to

track the maximum value of the intensity across the height of the boundary layer. By retaining the

strengths of different components of the velocity field at different heights in the flow, the strongest

014606-13

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

small-scale structures were highlighted and continuity of the amplitude of the small-scale structures

was observed.

The organization of the small scales that occupy the same space as the large-scale feature of

interest along a specific isocontour of the large-scale feature is not conclusively shown in this work,

but is suggested by it. Such a localization may suggest that small scales sit at a constant height

in some local definition of the large-scale flow, rather than at a constant height relative to a mean.

Critical layer mechanisms are responsible for the amplification of structures at a specific height

relative to a base flow or a mean [

]. Continued work is suggested to explore whether critical layer

mechanisms could be responsible for the amplification of small scales along specific isocontours of

some local “large-scale flow” in fully developed turbulent boundary layers.

ACKNOWLEDGMENTS

This work was made possible through United States Air Force Grants No. FA9550-12-1-0060

and No. FA9550-16-1-0361 and through a National Defense Science and Engineering Graduate

Fellowship. The authors would like to thank Xiaohua Wu for providing access to and support using

his computational data. The support of the Center for Turbulence Research at Stanford Univer-

sity and useful conversations there with Kevin Rosenberg and Aaron Towne are also gratefully

acknowledged.

APPENDIX: SENSITIVITY OF RESULTS

The sensitivity of the results to the details of the projection and averaging technique were

examined. Parameters that could have an effect on the results and were examined were the use of

a one-dimensional versus two-dimensional projection, the size of the filter that defined large versus

small scales, and the wave speed of the model used for projection. The sensitivity of the results to

each of these parameters was explored for the small-scale streamwise velocity intensity averaged

on the LSMs in the wake region as a somewhat concise demonstration of the method’s sensitivity.

1. Two-dimensional vs one-dimensional model

The results were not highly sensitive to the use of a one-dimensional versus a two-dimensional

model of the large-scale coherent structure for the projection. The small-scale streamwise velocity

intensity averaged on the approximated phase of a LSM, computed using a one-dimensional

projection, is shown in Fig.

10(a)

and can be compared to Fig.

7(b)

. Due to the low sensitivity

and the improved speed of the method using the one-dimensional projection, the other sensitivity

results also used the one-dimensional projection.

2. Filter width

The small-scale streamwise velocity intensity defined using different sizes of a Gaussian filter are

shown in Fig.

. The definition of the small scales

was modified by changing the definition of

the standard deviation of the Gaussian filter

. The filter width was quartered to 0

125

[Fig.

10(b)

]

and doubled to 1

[Fig.

10(c)

]. The color bar was adjusted to enable the visualization of each result.

The size of the filter was observed to affect the magnitude of the intensity of the small scales, but not

to alter the spatial organization of the intensity. The effect of the spatial resolution was also studied

by filtering the smallest scales and was also found to affect the magnitude of the intensity of the

small scales, but not to alter the qualitative behavior.

3. Projection model wave speed

The wave speed of the model used for projection was varied in order to adjust the height where

projection was performed. The height of projection was the location where the mean velocity profile

014606-14

AMPLITUDE AND WALL-NORMAL DISTANCE VARIATION ...

FIG. 10. Averaged streamwise small-scale velocity intensities are shown in the streamwise–wall-normal

plane using the experimental data and the one-dimensional conditional projection averaging approach using a

LSM model. The small scales are defined using a standard deviation of (a) 0

,(b)0

125

,and(c)1

for the

Gaussian filter.

matched the model wave speed:

(

)

where

was changed to 0

∞

and 0

∞

(results in

the study used 0

∞

and 0

∞

With the change of the projection height, the structures in the averaged streamwise velocity

field [Figs.

11(a)

and

11(b)

] shifted towards and away from the wall, respectively. In each panel

the isocontour corresponding to

of the large scale is shown as a solid black curve.

Figure

demonstrates that the streamwise small-scale velocity intensity was sensitive to the height

where the condition projection took place. In the

∞

case [Fig.

11(c)

], amplitude variation

was observed more strongly in the small-scale streamwise velocity intensity in the wake region,

while height variation was observed in the logarithmic region. This may have reflected that the

phases of the VLSMs were being averaged as much as or more than the phase of the LSMSs,

yielding height variation in the logarithmic region around the VLSMs, rather than in the wake

FIG. 11. Phase-averaged PIV data with an assumed large-scale convection velocity of (a) and (c) 0

∞

and (b) and (d) 0

∞

. Conditional projection averages are shown of (a) and (b) the streamwise fluctuating

velocity field and (c) and (d) the streamwise small-scale velocity intensity. The isocontour

is shown

by a solid black line in each panel.

014606-15

SAXTON-FOX, LOZANO-DURÁN, AND MCKEON

region around the LSMs. This result suggests that the height of projection has a leading-order effect

on which large-scale coherent structure is averaged upon. In the

∞

case [Fig.

11(d)

], some

height variation was observed but was much less localized or strong than was observed for the

original results in Fig.

. The localization of the small-scale streamwise velocity intensity that was

still observed was not isolated along the curve

∞

, but instead showed a height profile

associated more with the location where

∞

would pass.

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