12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

ANALYSIS OF LARGE-SCALE/SMALL-SCALE INTERACTIONS IN
TURBULENT CHANNEL FLOW AT REτ = 5200 USING AUTO-ENCODER
COMBINED WITH MULTIVARIATE-PDF.

Agostini Lionel
Institut Pprime, CNRS / Université de Poitiers / ENSMA,
Poitiers, France
lionel.agostini@univ-poitiers.fr

Leschziner Michael
Department of Aeronautics,
Imperial College London,
London, UK

Cordier Laurent
Institut Pprime, CNRS / Université de Poitiers / ENSMA,
Poitiers, France

2 𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑖𝑛𝑔
𝑎𝑙𝑙 𝑦 +

ABSTRACTThe influence of especially√ energetic structures residingin the region around y+ ≈ 4. Reτ , are investigated in channel flow at Reτ ≈ 5200. The motivation is to shed light on
how the magnitude and length-scale properties of the smallscale near-wall turbulence are modulated by the outer structures, and whether this response is tied solely to the friction
velocity – i.e., whether the near-wall turbulence is universal
when scaled by the spatially and temporally varying largescale wall shear stress, rather than by its time-averaged value.
A unique aspect of the present framework is the use of an autoencoder algorithm to separate the extremely large full-volume
DNS fields into large-scale and small-scale motions. A second
original element is a formalism that derives the modulation,
conditional on large-scale fluctuations, from continuous statistical quantities represented as joint-multivariable probabilitydensity functions, thus obviating the need for any discrete representation or binning beyond that imposed by the discrete
DNS solution. A third novel aspect is the use of the lengthscale-wise derivative of the second-order structure function toquantify the modulation (increase or decrease) in the lengthscale, again conditional on the large-scale structures.

1

1

Input
1280 × 1280

𝑅𝑎𝑤 𝑓𝑖𝑒𝑙𝑑𝑠

𝐿𝑒𝑎𝑟𝑛𝑖𝑛𝑔

Output1280 × 1280

Bottleneck
20 × 20 × 2

𝑭𝒆𝒂𝒕𝒖𝒓𝒆𝒔 𝑫𝒆𝒕𝒆𝒄𝒕𝒊𝒐𝒏
𝑏𝑦 𝐴𝑢𝑡𝑜𝐸𝑛𝑐𝑜𝑑𝑒𝑟

Introduction

The impact of outer turbulent structures on the state of
the viscosity-affected near-wall layer is of considerable interest in the context of controlling wall friction. The key aspectsof this interaction are, first, the formation of large-scale fluctuations in the viscous and buffer layers, provoked by “footprinting” of outer scales in the log layer, and second, the attenuation or amplifications (“modulation”) of small-scale motions in this layer by the footprints. Importantly, both effects
rise with the Reynolds number, in tandem with the progressive rise in the√ streamwise turbulence energy around the location y+ ≈ 4. Reτ – i.e. y+ ≈ 280 at Reτ = 5200, the value
considered herein. Thus, as Re rises, the near-wall layer is increasingly perturbed by the strengthening footprints, which islikely to increasingly compromise the effectiveness of any control method designed to reduce the friction drag by damping
the small-scale turbulence in the viscosity-affected near-walllayer.

𝐿𝑎𝑟𝑔𝑒 𝑠𝑐𝑎𝑙𝑒𝑠
Figure 1. Schematic representation of the process of extracting the large-scale motions by using a multi-layer AutoEncoder. Patches on the plane y+ ≈ 280 are subdomains ofsize ∆x+ = 1280, ∆z+ = 1280 used for the learning process.

Agostini & Leschziner (2014, 2019a,b) have previously
examined a variety of aspects of the above large-scale/smallscale interactions by analysing the DNS data of Touber &Leschziner (2012) and Agostini & Leschziner (2014) for channel flow at Reτ = 1000, a value for which the interactions
are relatively weak. They did so by employing their own
spatially bi-dimensional “Empirical Mode Decomposition (BiEMD)” to separate the turbulence spectrum across many full-1

𝑦 + ≈ 280

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
units, each being 1/48th of the total data items in the plane,chosen randomly from 2D fields at y+ ≈ 280, where most energetic eddies correspond to large-scale motions – the AE will
only use the “library” of features learned at this wall-normal
location for extracting large-scale fluctuations produced by theouter-flow structures at any other wall-normal location. For the
training, subsets from seven DNS snapshots are used, and forvalidation subsets from three other snapshots are used. A validation set is required to ensure that the features learned by the
AE correspond to a valid statistical description of the externalflow structures. This means that not only the noise will befiltered out, but that also any singular events produced by the
structures are removed, which is a distinctive benefit of the
AE over other methods such as FFT- or EMD-based methodologies. The main advantage is that the AE, once trained, isable to identify information reflecting the true impact of external flow structures at any wall-normal location, and extract
them faithfully in isolation from spurious disturbances having
spectral length-scale features similar to the true LS structures,
which is not possible with low-pass filtering, wherein all information of a given length-scale spectrum is conserved whateverits source.
The success of this process is illustrated in Figs.2 and 3,
which show fields decomposition, spectra and streamwise
energy profiles for the large-scale and small-scale motions
respectively. Fig. 2 shows sub-domains of streamwisefluctuations fields across the wall-normal plane y+ ≈ 280 andy+ ≈ 12 covering the sub-domain x+ × z+ ≈ 16000 × 8000,
corresponding to 1/8 and 1/6 of the total computational-box
dimensions, respectively. Full streamwise-fluctuations fields
at wall-normal locations y+ ≈ 280 and y+ ≈ 12 are conveyed
by Figs.2(a) and 2(b), respectively. The large-scale motions
and the residual small-scale fields at y+ ≈ 12 are shown in
Figs. 2(c) and 2(d). Specific feature that deserve to be underlined are, first, the strong dominance of the large-scale component at the outer location; second, the presence of large-scale
footprints in the near-wall layer; third, the fine-grained smallscale features in the near-wall layer; and fourth the influenceof the large-scale footprints on the intensity of the small-scale
structures in the near-wall layer.
Fig.3(a) demonstrates the scale-separation process by
way of pre-multiplied energy-density spectra, in the y+ − λz+(spanwise wavelength) plane for the streamwise components.
The red contours identify the large scales, while the blue contours pertain to the small scales, the sum of the two being identified by the black contours. The corresponding energy profiles
in the wall-normal direction are given on the right-hand-side(r.h.s) plots. The l.h.s plot convey a well-defined scale separation, with the large scales confined to wavelength values
λz+ > 800, which compares to λz+ ≈ 100 at which the streamwise energy density peaks at y+ ≈ 12.

volume DNS realisations into small, medium and large-scalemodes and then constructing small-scale statistics, conditionalon large-scale skin-friction footprints to infer the mechanisms
by which small-scale amplification by positive footprints dominate asymmetrically over small-scale attenuation by negative
footprints, thus explaining the progressive rise in the damaging
perturbations of the near-wall layer by footprinting.
Although the statistical tools developed in previous studies can be applied, in principles, to data at higher Reynolds
numbers, the resource requirements posed by the EMD
quickly become untenable as the spectrum of scales, andthereby the size of the domain, widen rapidly with the
Reynolds number. This all but prevents the application of the
EMD to full-volume DNS data sets at Reτ = 5200, taken from
Lee & Moser (2015). To circumvent this problem, a novel approach is proposed herein and analysed in this study in whicha decoder-encoder deep-learning algorithm is used to separate
the scales across any 2d (x − z) plane into small-scale andlarge-scale fields. Once this is achieved, joint PDFs are constructed, as explained below, which allow the effects of thelarge-scale footprints on the small-scale near-wall layer to bestudied.
The present paper deals with four aspects :i the manner in which the decoder-encoder model is used
to separate the scales over full-volume DNS realisations;ii the type of conditional small-scale statistics derived fromthe separated fields.
iii the manner in which the statistics under (ii) yield insight
into the mechanisms by which the large scales affect thesmall scales;
iv the manner in which the decomposition under (i) allows
the “quasi-steady” hypothesis to be tested – -i.e. theproposition that the small-scale fields are universal whenscaled with the instantaneous and local large-scale wall
shear stress.

2 Methodology
2.1 Decoder-encoder-based scale separation
As noted earlier, the EMD becomes untenable at thepresent Reynolds number, to which must be added some limitations regarding the possibly incomplete capture of largescale motions the size of which is comparable with the computational box size. Thus, an alternative data-driven methodology, referred to as ”Auto-Encoder” (AE), has been adopted
herein to separate the large-scale from small-scale motions.
An AE is an unsupervised machine-learning (ML) algorithm
that is trained to reconstruct a representation of its inputs froma data set that is highly reduced in volume and detail relative tothe full input (Agostini, 2020). As illustrated in Fig.1, an AE
has three parts: an encoder, a bottleneck (latent space), anda decoder. Depending on how its architecture is defined, an
AE can served different purposes, functioning as de-noisors(Vincent et al., 2010), anomaly detectors (Zhou & Paffenroth, 2017), and estimators (Guastoni et al., 2021), for example. In the present study, the goal is to use AE for extracting large-scale motions associated to the outer-flow structure
at any wall-normal location. By reducing the amount of information passing through the AE’s bottleneck, the AE must learn
the most significant features of the input so that the output image is as close as possible to the input image. The level ofcloseness is defined by the mean square of the reconstruction
error (MSE), steering the AE towards reconstructing the mostenergetic scales. Once the training (learning) is achieved –here done by feeding in sub-domains of size 1280 × 1280 wall

2.2

Conditional analysis of amplitude modulation based on multivariate PDFs

In recent papers (Agostini & Leschziner (2019a,b, 2021)),a novel approach to analysing any statistical flow property conditional on any value of the large-scale skin-friction fluctuation
is presented, bringing into focus the response of the turbulent
statistics to the large-scale motions. Thus, with the largescale/small-scale decomposition effected, conditional joint
PDFs of the form P(u1 , u2 , u3 ,C fLS ) are derived from the DNS
data, where C fLS is the “conditional variable” and u1 , u2 , u3
are velocity fluctuations in the x, y, z directions, respectively.
Conditional variables can then be extracted as exemplified by
the following expressions for the mean velocity and Reynolds

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
3.8e+00

8

3.4e+00

7

103

103 2.9e+00

102

102

6

z + (×103 )2.5e+00

4

y′

y+

5

2.1e+00
1.7e+003

1.3e+00

101

101
2

8.4e-01
4.2e-01

1

100
0
0

2

6

4

8

x + (×103 )

10

12

14

16

102

103

λz+

104

100 0.0e+00
02

4

u ′u ′ ′

6

8

10

8

Figure 3. Large-scale/small-scale decomposed field ofstreamwise fluctuations; (a) Premultiplied power spectra:(black) total fluctuations, (red) large-scale fluctuations and(blue) remaining fluctuations; (b) corresponding streamwiseenergy profiles.

7

z + (×103 )

6
5
4
3
2
1
0
0

2

4

6

2

4

6

2

4

6

8

10

12

14

16

8

10

12

14

16

8

10

12

14

16

x + (×103 )Other properties, such as conditional production rates,
follow likewise. The merit of these conditional values is thatthey bring to light the influence of large-scale fluctuations on
the statistical quantities considered at large positive and negative C fLS values without the obscuring influence, or weighting,of low-density levels in the PDF P(C fLS ), which is shown in
Fig.4. The PDF is seen to be mildly positively skewed, a properly also observed at lower Reynolds number at which the levelof skewness is markedly higher.

87

z + (×103 )

6
5
4
3
2
1
0
0

x + (×103 )87

z + (×103 )

6
5
4
3
2
1
0
0

x + (×103 )Figure 4. PDF of large-scale streamwise-velocity fluctuations derived from AE, dark blue and red lines correspond to
1% extreme events, light blue and orange lines correspond to
10% and green line to 50% (median of PDF).

Figure 2. Illustration of scale decomposition using the approach shown in Fig. 1 at over a portion of the x − z DNS
box (1/8th in streamwise direction and 1/6th in streamwise
direction); (a) and (b) full streamwise-fluctuations fields at
y+ ≈ 280 and y+ ≈ 13, respectively; (c) large-scale fluctuations fields (output of the auto-encoder) at y+ ≈ 13 and (d)small-scale fluctuations (total minus large-scale fluctuations)
at y+ ≈ 13.

2.3
stresses:ui |C fLS =Z+∞ Z+∞ Z+∞ui−∞ −∞ −∞

P(u1 , u2 , u3 ,C fLS )du1 du2 du3P(C fLS )u′i u′j |C fLS = (ui − ui |C fLS )(u j − u j |C fLS )|C fLS= ui u j |C fLS − ui |C fLS u j |C fLS

Conditional analysis of length-scale modulation based on structure function

The second-order structure function is proposed herein asa basis for investigating length-scale modulation. Given thefields of velocity u(x, z) across any y-plane, and the large-scale
skin-friction C fLS (x, z), the latter being wall footprint of theouter-large scale structures shifted forwards in streamwise direction by ∆x = ∆y tan θ , with θ ≈ 11o , the starting point isthe derivation of the PDFs from the DNS data set in whichsu,δ z = u(x, z + δ z) − u(x, z) is the instantaneous difference between velocity fluctuations separated by a given (i.e. chosen)
value δ z (the subscript y is omitted henceforth).
The nth order structure function (Sn ) for a given δ z can

(1)

(2)

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
is substantially favoured by the fact that large-scale fluctuations are highly correlated in y+ , subject to a spatial streamwise lag, as is discussed by Hutchins & Marusic (2007) andseveral other authors. This has also been found to be the casein the present data set. Hence, here, the assumption has been
invoked that uLS is represented by C fLS , subject to the spatiallag ∆x+ = ∆y+ tan(θ ), with the angle arising from the twopoint correlation maps. A consequence of this lag, and thus
the gradient in uLS , is a degree of ambiguity as regards thederivative of S(δ z), which is conditional of uLS – or rather its
C fLS equivalent. This is a problem similar to that discussed
earlier in relation to spanwise variations in uLS . However, thelag between two neighbouring y+ planes is small and the effecton the statistics is thus negligible.

be derived from:Sun (δ z) = ⟨|u(x, z + δ z) − u(x, z)|n ⟩ =Zsnu,δ z P(su,δ z )dsu,δ z(3)
The representation of the structure function via its PDF is
important in the context of, and is consistent with, the strategy
of using the multi-variable joint PDF strategy, as expressed byequation (1).
Here, the focus is on the second-order structure function,
or rather its incremental derivative with respect to δ z, usedas the surrogate of the scale-wise distribution of the energy.
As demonstrated by Davidson et al. (2006a,b) and Agostini& Leschziner (2017) , there is a close relationship between theenergy spectra and the derivative of the structure function. The
preference for the structure function is rooted in the fact that it
is a purely data-driven method, the results of which do not depend upon a projection basis and on how the hyper parameters
are tuned, as is the case for FFT (window size, window function, etc.) and wavelet-based analyses (see De Moortel et al.
(2004)). Moreover, as the conditional statistics are derived
from the joint pdf of the structure function, the conditional
wavelength can be determined without any specific constrainson the length of the input signal, as this latter can continuously
vary with the value of C fLS . Another advantage of the structure
function is that it allows the contribution of the largest scales
resolved by the DNS to be represented much more accuratelythat via the spectra, which suffer from limitation imposed by
the FFT when the sample of scales is sparse and the domain
size limited. As will emerge below, length-scale shifts in thisderivative will be used to identify length-scale modulation conditional on C fLS .
By using the multi-variable PDF approach, the variationof the structure function and its derivative conditional on C fLS
can be investigated in the same sense as previously undertaken
in relation to amplitude modulation, by way of:Sun (δ z) =ZZsnu,δ z P(su,δ z ,C fLS )dsu,δ z dC fLS(4)

P(su,δ z ,C fLS )dsu,δ zP(C fLS )(5)

Sun (δ z)|C fLS =Z

snu,δ z3

Amplitude modulation

Small-scale amplitude modulation by large-scale outer
structures in wall-bounded flow has been studied extensively
over the past decade or two, both experimentally and computationally, the latter by probing DNS data. Previous computational studies have been restricted to Reτ < 1000, in whichcase, the outer structures are relatively weak and the relevantinteractions rather tenuous. Here, amplitude modulation isconsidered at the much higher Reynolds number Reτ = 5200.
An example is given in Fig.5(a,b), in which the conditional response of the streamwise stress to the large-scale skinfriction C fLS is shown (the profiles are cuts at the C fLS shownin the Fig.5(a), corresponding values of C fLS /C f are given
in Fig.4). While this figure conveys a number of interesting
features deserving discussion - not least the negative correlation between the outer large-scale fluctuations and the smallscale fluctuations in the outer region - the key feature to highlight here is that positive and negative footprints are associated, respectively, with strong positive modulation and damping. Fig.6 demonstrates that the modulation is driven by distortions in the conditional mean profiles, which then translate into
corresponding variations in the stress-production rates, shownin Fig.7. Specifically, the production of the streamwise energy
increases substantially at the wall for positive footprints anddecreases for negative ones. The reverse occurs in the outer regions, although the strain increments are considerably weaker
than in the inner layer, and this reversal ties up with the negative correlations between the stresses in the outer and inner
regions, mentioned above by reference to Fig.5. These results
illustrate the more general observation that the mechanisms bywhich the footprints provoke the modulation of near-wall turbulence within the near-wall layer concur broadly with thosederived at the much lower Reynolds number at Reτ = 1000
presented by Agostini & Leschziner (2019a) .
The present processing methodology also allows light
to be shed on the question of whether the turbulent stresses
as well as production rates are universal when scaled withthe local, instantaneous, large-scale wall shear stress – i.e.,
whether the Quasi-Steady Hypothesis (QSH) is satisfied(Chernyshenko (2021)). An example is shown in Fig.8, again
by reference to the streamwise stress. When the streamwise
stress and the wall-normal distance are normalised by the local large-scale shear stress, instead of its mean value (as inFig.5), the wall-normal profiles of the conditional streamwise
stress approximately collapse on one another, at least up toy+ ≈ 80. The implication is that the QSH is valid only inthe region not much thicker than the buffer layer. The region
well above the buffer layer is, in contrast, increasingly populated by intermediate-size structures with a lifetime longer

With the joint PDF, P(su,δ z ,C fLS ), derived separately foreach and every computational y place and for a set of values δ z,
the second-order structure function conditional on C fLS can bederived using equation (5) with n = 2. Alongside, the conditional structure function Su2 (δ z)|C fLS , its premultiplied derivadS2 (δ z)utive δ z dδ|can be computed at every wall-normal locaz C fLS
tion. It is this derivative on which attention focuses primarily.
Again, the reason is that the variation of this variable across δ z
is closely connected to the energy-density spectrum Φuu (λz ),with λz connected to the separation δ z, as is discussed at lengthin Agostini & Leschziner (2017).
The objective here is to examine statistics that are conditional on a single reference field. This reference field is chosen to be C fLS . This allows questions to be asked about thevalidity of the quasi-steady concept which hypothesizes thatscaling the wall-normal small-scale turbulence properties withthe local large-scale wall shear stress τw,LS results in identicalstatistics, i.e. statistics that do not vary with C fLS . However,a question that needs to be addressed is whether the condition
C fLS is equivalent to the condition uLS (y+ ). This equivalence412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
than the small-scale structures, and this detracts from applicability of the QSH. Consistently, the production rate for the
streamwise stress, not included herein, also complies with the
QSH in the near-wall region, but not beyond (a more extensive
discussion of the pertinent mechanisms can be find in Agostini& Leschziner (2019b)).

1.0e+01

103

103

8.8e+00

7.4e+00

6.0e+00

102

τ+

y+

102

4.7e+00

101

101

3.4e+00

100

−0.4

−0.2

0.0

CfLS /Cf

0.2

0.4

100 0

2.0e+00

0.6

4

10

5

uu|CfLS /uτ2

Fig.9 shows six maps of the premultiplied derivative of
the second-order structure function in the δ z+ − y+ plane at
the six conditional values of C fLS shown in the pd f of figure
4. Attention is first drawn the choice of the abscissa: 4δ z+ .
The choice of this multiplier is rooted in the correspondence
between the derivative of the structure function and the spectra, an equivalence demonstrated and discussed in Agostini &Leschziner (2017). In the latter, the abscissa is conventionally
the wave length λz+ . Multiplying δ z+ by 4 is observed to shift
the peak in the map at y+ = 12 and C fLS = 0 to the value to4δ z+ ≈ 100, corresponding to λz+ ≈ 100 – i.e., the accepted
level of the spanwise separation distance between streaks asderived from two-point correlations.
There are three observations that can now be made byreference to Fig.9 9. First, as C fLS increases, there is a progressive shift of the maximum towards the wall. This process and the mechanisms responsible have been discussed indetail in Agostini & Leschziner (2019a). In essence, large
positive C fLS levels cause a steepening of the strain close to
the wall, thus generating increased small-scale turbulence andthinning the viscous sublayer. More interesting in the present
context is that the increase in C fLS is also accompanied bya progressive reduction in the length-scale maximum from4δ z+ ≈ 100 to around 65, which may be interpreted as indicating the length-scale modulation. The third feature deserved to be highlighted is the progressive widening of thescale spectra around the buffer layer for increasing C fLS . Thiswidening appears to be due primarily to a shift of the range ofshort length scales towards lower values, a process especially
noticeable when tracking the left-most edge of the spectra aty+ ≈ 10. In contrast, the longer scales around the buffer layer,
at 4δ z+ ≈ 200, experience no discernible shift (the downwardpointing tongues at 4δ z+ > 1000 signify the footprinting process).
To examine whether the QSH applies to the length-scale
modulation, a map of contours of the wave length corresponding to the maximum value of the derivative of the structure
function at different y+ values and conditional on C fLS is derived. This is shown in Fig.10(a). The map demonstrates thatthe wave length of the maximum derivative location at any
given y+ is decreasing, as already observed by reference toFig.9. For example, at y+ ≈ 13, the wave length in Fig.10(a)is seen to decline from 100 (green shade) to around 70 (darkblue shade). Next, the map in Fig. 10(a) is re-scaled withthe wall-normal distance and the wave length normalised withthe large-scale wall shear stress and δ z+,LS = δ z+ × uτ,LS /uτ .
This yields the map given in Fig.10(b), which permits two major observations to be made. First, the contours are broadly
horizontal in the positive range of C fLS , implying that the QSH
applies to length-scale modulations of the fluctuations within
the near-wall layer, y+ < 40. Second, the hypothesis is notsatisfied for negative large-scale fluctuations. As the lengthscale modulation is here considered within the confines of thenear-wall layer, up to y+ of approximately 40, the lack of ofadherence of length-scale fluctuations with the QSH at negative C fLS is inconsistent with the behaviour observed in respect

Figure 5. Conditional response of the streamwise stress to
the large-scale skin friction C fLS , scaled with average uτ . Profile colours correspond to locations in the pdf in Fig.4

103

102

102

y+

y+

103

101

100 0

101

10

20

1000.030

U + |CfLS

1.0

0.5

1.5

dU + |CfLS /dy +Figure 6. Profiles of conditional streamwise velocity and its
wall-normal derivative at the locations marked in the pdf in
Fig.4.

16.4

103

103

14.0

y+

9.2
6.8

101

102

y+

11.6

102

101

4.4

100 −0.5

0.0

2.0

0.5

CfLS /Cf

100 0

10

5

+
y + × (−2uv + dU
dy + |CfLS )Figure 7. Conditional response of the production of thestreamwise energy component to the large-scale skin friction,
scaled with average uτ . Profile colours correspond to locationsin the pdf in Fig.4

1.0e+01

103
8.8e+00

103

7.4e+00

6.0e+00

4.7e+00

101

102

τLS+

yLS+

102

101

3.4e+00

100

−0.4

−0.2

0.0

CfLS /Cf

0.2

0.4

0.6

2.0e+00

100 0

5

uu|CfLS /uτ2, LS

Length-scale modulation

10

Figure 8. Conditional response of the streamwise stress to
the large-scale skin friction C fLS , scaled with the local largescale uτ,LS . Profile colours correspond to locations in the pdf
in Fig.45

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

103

8.5
103
7.6
6.6
102
5.7
4.8
3.8 1
10
2.9
1.9
0
1.0
1010
0

103

103

8.5
7.6
6.6
5.7
4.8
3.8
2.9
1.9
1.0

103

8.5
7.6
6.6
5.7
4.8
3.8
2.9
1.9
1.0

y+

y+

101
0
1010
0

101

4δz +

102

103

4δz +

102

y+

y+

102

101

101
100

100

101

4δz +

102

103

4δz +

102

y+

y+

102

101

101
0
1010
0

101

4δz +

102

101

4δz +

102

40

0.50

130.0
120.0
110.0
100.0
90.0
80.0
70.0
60.0
50.0

0.50

130.0
120.0
110.0
100.0
90.0
80.0
70.0
60.0
50.0

30
20

y+

103

8.5
103
7.6
6.6
102
5.7
4.8
3.8 1
10
2.9
1.9
0
1.0
1010
0

102

8.5
7.6
6.6
5.7
4.8
3.8
2.9
1.9
1.0

10
−0.50 −0.25

0.00

0.25

CfLS /Cf

40
30
20

yLS+

103

8.5 3
10
7.6
6.6
5.7
102
4.8
3.8 1
10
2.9
1.9
0
1.0
1010
0

103

10
−0.50 −0.25

Figure 10.

0.00

0.25

CfLS /Cf

Maps of conditional 4δ z+ |C fLS in the y+ − C fLS
dS2 (δ z)uplane derived from δ z dδ|+ (for streamwise fluctuaz C fLS
tions); (a) nominal field scaled with mean wall shear stress;(b) field in (a) rescaled with large-scale wall shear stress todemonstrate adherence to the quasi-steady hypothesis.

dS2 (δ z)u
Figure 9. Maps of δ z dδ|+ in the y+ − δ z+ plane conz C fLS
ditional on the six C fLS shown in figure Fig.4; (a)-(f) arranged
in order of rising C fLS values.

fined interpretation of the logarithmic structure function law
in wall layer turbulence. Physics of Fluids 18 (6), 5112.
Davidson, PA, Nickels, TB & Krogstad, P-Å 2006b The logarithmic structure function law in wall-layer turbulence.
Journal of Fluid Mechanics 550, 51 –60.
De Moortel, I., Munday, SA & Hood, A.W. 2004 Wavelet analysis: the effect of varying basic wavelet parameters. Solar
Physics 222 (2), 203–228.
Guastoni, L., Güemes, A., Ianiro, A., Discetti, S., Schlatter, Ph., Azizpour, H. & Vinuesa, R. 2021 Convolutionalnetwork models to predict wall-bounded turbulence fromwall quantities. Journal of Fluid Mechanics 928.
Hutchins, N. & Marusic, I. 2007 Large-scale influences innear-wall turbulence. Philosophical Transactions of theRoyal Society of London A: Mathematical, Physical andEngineering Sciences 365 (1852), 647–664.
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of
turbulent channel flow up to Reτ ≈ 5200. Journal of Fluid
Mechanics 774, 395 –415.
Touber, E. & Leschziner, M. A. 2012 Near-wall streak modification by spanwise oscillatory wall motion and dragreduction mechanisms. Journal of Fluid Mechanics 693,
150 –200.
Vincent, Pascal, Larochelle, Hugo, Lajoie, Isabelle, Bengio,
Yoshua, Manzagol, Pierre-Antoine & Bottou, Léon 2010
Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion.
Journal of machine learning research 11 (12).
Zhou, Chong & Paffenroth, Randy C 2017 Anomaly detection
with robust deep autoencoders. In Proceedings of the 23rd
ACM SIGKDD international conference on knowledge discovery and data mining, pp. 665–674.

of the amplitude modulation. An interpretation of this inconsistency is given in a more extensive paper evolving at the timeof writing.

REFERENCES
Agostini, L. 2020 Exploration and prediction of fluid dynamical systems using auto-encoder technology. Physics of Fluids 32 (6), 067103.
Agostini, L. & Leschziner, M. 2014 On the influence of outerlarge-scale structures on near-wall turbulence in channel
flow. Physics of Fluids 26 (7), 075107.
Agostini, L. & Leschziner, M. 2017 Spectral analysis of nearwall turbulence in channel flow at Reτ = 4200 with emphasis on the attached-eddy hypothesis. Phys. Rev. Fluids 2,014603.
Agostini, L. & Leschziner, M. 2019a The connection between
the spectrum of turbulent scales and the skin-friction statistics in channel flow at Reτ ≈ 1000. Journal of Fluid Mechanics 871, 22–51.
Agostini, L. & Leschziner, M. 2019b On the departure ofnear-wall turbulence from the quasi-steady state. Journal of
Fluid Mechanics 871.
Agostini, L. & Leschziner, M. 2021 Statistical analysis ofouter large-scale/inner-layer interactions in channel flow
subjected to oscillatory drag-reducing wall motion using a multiple-variable joint-probability-density function
methodology. Journal of Fluid Mechanics 923.
Chernyshenko, S. 2021 Extension of qsqh theory of scale interaction in near-wall turbulence to all velocity components.
Journal of Fluid Mechanics 916, A52.
Davidson, PA, Krogstad, P-A, Nickels, TB et al. 2006a A re-6