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

A STUDY ON SPATIAL ORGANIZATION OF COHERENT STRUCTURES IN
CHANNEL FLOW USING UNSUPERVISED DEEP LEARNING

Mohammad Javad Sayyari
School of Mechanical Engineering
Pusan National University
Busan 46241, Republic of Korea
sayyari@pusan.ac.kr

Jinyul Hwang
School of Mechanical Engineering
Pusan National University
Busan 46241, Republic of Korea
jhwang@pusan.ac.kr

Kyung Chun Kim
School of Mechanical Engineering
Pusan National University
Busan 46241, Republic of Korea
kckim@pusan.ac.kr

ABSTRACT(Fukami et al., 2019) and unsupervised (Kim & Lee, 2020)
deep learning (DL) are applied to generate inflow for fully
developed turbulent channel flows. These studies along withothers have shown that the ML techniques have the potential
to learn the underlying physics of turbulent flows. However,
the ability of these methods for prediction of coherent structures has not been well examined yet. This study assesses thecapability of an unsupervised DL technique for predicting the
turbulence statistics as well as the spatial features of coherent
structures in a fully developed turbulent channel flow. For thispurpose, the training dataset is provided from a DNS of thechannel flow at Reτ ≈ 178. Then, the training dataset is fed
to an unsupervised DL network based on the generative adversarial network (GAN) to generate instantaneous flow fields.
Finally, the generated flow fields are compared to those of
DNS in terms of the turbulence statistics and coherent structures with the focus on the two-point correlation. The results
infer the potential of the unsupervised learning of turbulence
to be enhanced further and become a powerful tool in the turbulence researches. As the shape and behavior of the coherent structures are mainly preserved when they grow and travelin the streamwise direction in a boundary layer flow (Robinson 1991; Adrian et al. 2000), a DL network can be trained
at a low Reynolds number and be asked to generate instantaneous inflow boundary conditions at a high Reynolds number
in a turbulent boundary layer to reduce the computational costsof DNS related to the required large computational domain athigh Reynolds numbers. Recently, Kim & Lee (2020) showed
successful application of a deep unsupervised network to predict turbulent flow fields in a channel flow at a higher Reynolds
number (Reτ = 720) than the training dataset, but their study
is limited to the one-point and two-point turbulence statistics
without structural analysis.

This study aims to evaluate the capability of a GAN-based
unsupervised deep learning network for generating inflow turbulent fields in a channel with the focus on the spatial organization of the coherent structures. For this purpose, a DNS isconducted at Reτ ≈ 178 to collect the training dataset at thecross-stream plane of the channel at a fixed streamwise location. First, it is shown that the one-point and two-point turbulence statistics are well matched between the DNS and GAN
data. Then, we extracted intense streamwise velocity clusters from the generated flow fields and compared them withthose of the DNS in terms of their size distribution. Moreover, the two-point correlation coefficients computed from ourGAN data are compared with those of the DNS data. Interestingly, the GAN successfully predicts the structural characteristics hidden in the training data such as the relation between
the height and width of the identified clusters for negative,
positive, and superposed clusters, the mean spanwise distance
between the low-speed streaks, and the wall-normal variationof the conditional and unconditional length-scales in the wallnormal and spanwise directions. The successful performanceof the network in this study suggests its potential to be utilized
more in turbulence researches.

Introduction
As coherent structures play a critical role for understanding and controlling turbulent flows, a proposed method
for predicting turbulent flow fields should be able to capture them accurately. In this regard, the direct numericalsimulation (DNS) has shown promising results (Jiménez &Moin, 1991). However, the DNS is time-consuming especially at high Reynolds numbers. Alternatively, the computational costs of DNS related to a required large computational domain at high Reynolds numbers can be decreased byan accurate inflow turbulence generation method. To generate
the inlet boundary condition, several methods are summarized
by Wu (2017). Recently, Kim & Lee (2020) pointed out thedisadvantages of these methods, suggesting machine learning(ML) techniques as an alternative. In this regard, supervised

Computational setup
DNS
To collect the training dataset, a DNS of fully-developed
turbulent channel flow is performed using a fractional step
method (see Kim et al. 2002). The periodic boundary con-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
given in Table 2 if not mentioned. See Kim & Lee (2020) for
more details about the DL network.
Table 2: Hyperparameters of the network (KS is the ker-

dition is applied in the streamwise (x) and spanwise (z) directions while the no-slip boundary condition is imposed on thewalls. Also, the mesh is non-uniform in the wall-normal direction (y). Table 1 provides the parameters of the DNS data.
Here, h denotes the channel half-height. Throughout this paper, U, V , and W indicate the streamwise, wall-normal andspanwise velolcity components. The corresponding velocity
fluctuations are u, v, and w, respectively. The superscript +denotes the inner unit normalized by the friction velocity andthe viscous length scales. The angled brackets <> indicate theensemble average.

nel size)

Table 1: Parameters of the DNS data

(4π,2π)(192,256) 0.11 4.87 4.37 0.075

Deep unsupervised learning network
The original GAN, first introduced by Goodfellow et al.
(2014), consists of two main networks; namely, Generator (G)and Discriminator (D). The generator takes a random noise
vector (~Z) in the latent space, having dimension of LS, andproduces a fake image. Then, the fake and real images are
fed to the discriminator which gives a number that determines
the similarity between the two images (see Fig.1). The GAN
network can be formulated as (Goodfellow et al. 2014):

→
h 


− i− log 1 − DθD GθG Z
min max EX log DθD (X) +E→
Z

BS

lr

EP

KS

LS

100

1e-3

0.2

4

1e-3

120

3

256

Results and discussion
Instantaneous flow fields

θD

A sample of the generated instantaneous velocity fields
in the cross-stream plane is displayed in Fig.2. As illustrated, the spatial velocity fields are similar to the flow fields
of DNS qualitatively, showing that the network is capableof emulating the training dataset. We examine these spatial
fields statistically in terms of their structural characteristics.
Fig.3 shows the joint probability density function (JPDF) of
height and width of the identified structures in the instantaneous flow fields. Based on the physical geometry of the structures, their height and width are defined as ly = ymax − ymin
and lz = zmax − zmin , respectively. Here, ymin and ymax are theminimum and maximum wall-normal locations while zmin andzmax denote the minimum and maximum spanwise locationsof a given structure, respectively. As seen, the identified structures are widely distributed with different sizes. As shown, theGAN lines follow the same pattern as the DNS contour levels.
This agreement is also held when the structures are divided tothe positive and negative clusters based on Eq.4 (see Fig.3(b)and 3(c)). This figure shows that the size and the relation between the height and width of the identified structures in the
instantaneous flow fields are well learned by GAN.

(1)
where E denotes the expectation, and X, θG , and θD are thereal image, trainable parameters of G, and trainable parametersof D, respectively. The real image is a three-channel image that
includes the U + , V + , and W + information.
Gulrajani et al. (2017) suggested WGAN-GP which modifies the loss function of D as Eq.2:
h  ih2 i
LD = EX [−D (X)]+EXe D Xe +λD EX̂
∇X̂ D X̂ 2 − 1(2)→−where ||.||2 denotes the Euclidean norm, Xe = G( Z ) is the generated fake image, and X̂ = σ X + (1 − σ )Xe with σ being arandom number. In this study, the WGAN-GP is implemented
with the techniques proposed by Karras et al. (2017) such as
pixelwise normalization (pn). The loss function of the generator is also constrained as Eq.3(Wu et al. (2020)):e 22
LG,new = LG + λG ||S(X) − S(X)||aIn this study, the coherent structures are defined as theconnected points in a given instantaneous flow field that havestreamwise velocities less or larger than a specific value. The
connectivity rule is defined as the four orthogonal neighborsof each node in Cartesian coordinates. Thus, the positive andnegative coherent structures (clusters) are defined as (Hwang& Sung 2018; Lozano-Duran & Jiménez 2014):
Positive clusters: u > αurms(4)
Negative clusters: u < −αurms
Based on percolation analysis, we found α = 1 for boththe DNS and GAN data. Although not shown here, the GAN
and DNS results are well matched in the vicinity of the chosen
threshold value. This is consistent with the work of Schmekel
et al. (2022) who showed that the number and volume of clusters connected with the Reynolds stresses are preserved by a
DL convolutional neural network. It should be mentioned thatthe clusters having area of less than (30)2 in wall units arefiltered out in this study.

We used a total of 20,000 y–z planes from the DNS data
at a fixed streamwise location and resized them to have a resolution of 192 × 192. Then, they are divided into a numberof batches, having batch size of BS, and used as the training
dataset, or real images.

θG

λG

Identification of the coherent structures++ ∆t +
Reτ (Lx /h, Lz /h) (Ny , Nz ) ∆y+min ∆ymax ∆z

178

λD

(3)

where S, LG and λG are the energy spectra, the loss functionof the original GAN and the strength of the constraint, respectively. The statistical constraint term converges to a low value
after enough number of epochs (EP). Hereafter, we call the
DL network used in this study as GAN for simplicity.
As seen in Fig.1, the generator and discriminator are both
networks which contain blocks of mainly convolution (conv)
and fully connected (fc) layers. The network utilizes the
Leaky-ReLU function with parameter a, and the ADAM optimizer with learning rate of lr. Additionally, the number ofreal images is increased at each iteration by mirror and spectral augmentations. The fixed parameters for the network are

Turbulence statistics
To validate our GAN results, one-point and two-point
turbulence statistics obtained from the generated flow fields
are compared with those of DNS. The mean statistics are
ensemble-averaged over 10,000 generated fields whereas thoseof DNS are averaged over the whole training dataset.
Fig.4(a) compares the mean streamwise velocity, nondimensionalized by the friction velocity, for the current results

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

Figure 1: The GAN model used in this study with the detailed architectures of the generator (G) and discriminator (D).
3

3

DNS

a)
2

2

U+
U+
20
15
10
5
0

1

0
30

1

2

3

4

5

6

y/h

y/h

GAN

b)

+

1

03
0

1

2

3

z/h
2

+

V+
V

0

1

2

y/h

3

4

5

6

y/h
f)

+
W+
W

5
2.5
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-2.5
-5

1

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2

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6

+

V
V+

1

0

1

2

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+
W+
W

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-2
-3

z/h

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z/h

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e)

4
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-3

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d)

y/h

y/h

c)

U
U+
20
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10
5
0

0

1

2

3

z/h

4

5

6

5
2.5
0
-2.5
-5

z/h

Figure 2: A sample of the generated velocity fields using the current GAN compared with DNS at Reτ ≈ 178.
a) 1.5b)0.02 0.2c)0.02 0.2

22

0.02 0.2

2

ly/h

1

0.5

0.5

1

1.5

Negativeclusters

Positiveclusters

All clusters
0.5

1

lz/h

lz/h

1.5

0.5

1

1.5

lz/h

Figure 3: JPDF of height and width of a) all the identified clusters b) positive clusters, and c) negative clusters. The graycontour shows the DNS data while the red line indicates the GAN data. The contour levels are logarithmically distributed.
and DNS along the outer coordinates. As seen, except for thesmall differences mainly in the core region, where GAN predicts slightly higher values, very good agreement is obtained
for the mean streamwise velocity profile. Also, as shown in
Fig.4(b), the turbulent intensity profiles (i.e. u+2
i,rms (y) , where idenotes the streamwise (i = 1), wall-normal (i = 2) and the
spanwise (i = 3) directions) are in a good agreement whencompared with those of DNS. However, GAN shows slightly
lower turbulence intensities for the near-wall peaks. As seen inthis figure, the GAN predicts lower values for the v and w components at all wall-normal locations which can be attributed tothe problem that the mean profiles of these velocity fluctuations show higher alternations around zero compared to thoseof DNS (not shown here). The same issue can be seen in the
results of Fukami et al. (2019), which is related to a lack ofzero-mean constraint in the network.
Fig.4(c) demonstrates premultiplied one-dimensional
spanwise spectra of the velocity fluctuations (Φu+ u+ =i

3i

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022kz Su+ u+ ) at two wall-normal locations; y/δ = 0.08 and 0.3,
i iwith kz = 2π/λz being the spanwise wavenumber corresponding to the spanwise wavelength λz . In the near-wall region, theenergy carried by large scales, which represent the superimposed large scales (or footprint), are well captured, and the premultiplied energy spectra for the near-wall and outer regions
follow the same trend over a wide range of scales when compared with the DNS data. More importantly, the location of thepeaks are very well captured in this figure, which shows that
the GAN is capable of predicting the spanwise length-scale ofthe most energetic motions at a given wall-normal location.
This result implies that the spatial organizations of near-wallstreaks as well as outer large-scale structures are well predictedin the generated flow fields. However, the peak values predicted by GAN are lower than those of DNS, especially for the
streamwise velocity.
To further explore the spatial organization of coherent
structures, the two-dimensional two-point correlation coefficients of the velocity fluctuations having spanwise distance of∆z are calculated as Eq.5:

to within the zone where the correlation curve intersects with
Ru+ u+ = 0.05 (Sillero et al. 2014) (see inset (a) of Fig.4(d)).
i i
The first and second intersection points are shown with a andb, respectively. The integral length-scales are computed only
for the lower half of the channel, due to the wall-normal symmetry of this figure with respect to the channel centerline, as
the plots at the left-hand side of Fig.5(a). This figure clearly
admits that the wall-normal ( j = 1) and spanwise ( j = 2) integral length-scales obtained from the GAN data follow the same
trend as the DNS data for all the velocity perturbations. In particular, the spanwise integral length-scales for all the velocity
components increase through out the boundary layer thickness,while the wall-normal integral length-scale of the streamwise
velocity increases sharper near the wall and decreases near thecenter of the channel, which is in agreement with other studies(Monty et al., 2007; Sillero et al., 2014). This means that thecoherent structures are persisted well even beyond the logarithmic layer in the channel flow (Lee & Sung, 2013), and the success of GAN to predict their length-scales provides evidence
that this unique characteristic is reflected in the generated flow
fields. However, the length-scales are underestimated by GAN
compared to the DNS values. To explore the GAN capability for learning the length-scales of specific events, we further compute the length-scales based on the correlations conditioned on the negative and positive clusters.
The conditional two-point correlation coefficients are defined as Eq.7 (Sillero et al., 2014):+< u+i (y, z,t)ui (yre f , z + ∆z,t) >(5)+ui,rms (y)u+i,rms (yre f )where yre f indicates the reference height. First, we simplify Eq.5 by focusing on the one-dimensional correlations at
y = yre f . Fig.4(d) shows the one-dimensional two-point correlation coefficients at two reference heights. As seen, the distant points are correctly uncorrelated in both DNS and GAN.
As the reference height is increased from the near-wall to thelogarithmic layer, the separation distance between the anticorrelated regions increases, indicating that the average spanwise length-scale of the structures are growing with height.
This is in agreement with other studies including Lee & Sung(2013) and Monty et al. (2007). This behavior is discussed
more in the next section. More importantly, the location of
the minimum two-point correlation coefficient for the streamwise velocity in the logarithmic region, which gives an estimate of the mean spanwise distance between the low-speed
streaks, is in good agreement with that of DNS for both wallnormal locations. As observed, the minimum location for thecorrelations of the spanwise and wall-normal velocity components are also in excellent agreement with DNS. These observations suggest the potential of the DL network to learn someunderlying physics hidden in the turbulent flow fields such as
the mean spanwise spacing between the low-speed streaks.
At y/h = 0.3, in particular, the positive correlations for the
streamwise velocity with spanwise separation of ∆z/h ≈ 0.5
are centered at the reference position and are flanked both sides
by negative values with a distance of approximately h (see inset (b) of Fig.4(d)). This reveals that the spatial organization of
the large-scale structures in the predicted flow fields are well
captured at this specific height.

Ru+ u+ (y, ∆z) =
i

i

+
< u+
i (y, z,t)ui (yre f , z + ∆z,t) > |C
(7)
+
i i
u+
i,rms (y)ui,rms (yre f )|C
where the subscript C defines the condition being considered;that is, C = P and C = N mean conditioned on the positive andnegative clusters formulated in Eq.4, respectively. Here, weonly constrained the reference point to be within the identified
clusters whereas the moving point is freely positioned on any
node in the computational domain. Note that at y = yre f and

Ru+ u+ ,C (y, ∆z) =u+i

To discuss the wall-normal variation of the two-point correlation coefficients, we track the integral length-scales at different reference heights. The integral length-scales are definedas Eq.6:Z ba

Ru+ u+ (y, ∆z)dr j
i

i

i

i,rms

while the unconditional one reaches Ru+ u+ = 1 (see Fig.5(b)).
i iThe length-scale of the conditional correlation is defined as the
distance between the two intersection points of the correlation
curve and the line Ru+ u+ ,C = 0.3 (Lee & Sung, 2013; Sillero
i i
et al., 2014). The wall-normal and spanwise length-scales of
the conditional correlation for the streamwise velocity component (i = 1) are shown at right-hand side of Fig.5(a) as afunction of yre f /h. Although not shown here, this figure also
has an almost symmetric behavior with respect to the centerline of the channel, and thus is only plotted for the upper halfof the channel. Surprisingly, trends of the GAN results for the
positive and negative clusters are in a good agreement whencompared to those of the DNS. However, the lengths obtained
from the GAN data are shorter, especially for the wall-normal
length-scale near the channel center. Fig.5(a) confirms that the
GAN not only learns the characteristic length-scales of the superposed structures at different wall-normal locations, but alsopreserves the length-scales of their constitutive events such as
the positive and negative clusters in a deeper level. Similar to
our results, Lee & Sung (2013) reported the same conditional
analysis for the streamwise two-point correlation and showedthat the spanwise length-scales for the positive and negative
events increases from near the wall to y/δ = 0.5 and 0.6, respectively, where δ is the boundary layer thickness in theirstudy, beyond which the curves start to decay. These locations
are in excellent agreement with our GAN and DNS results.
Note that the streamwise turbulent intensities averaged
separately over the positive and negative structures are in a

Wall-normal variation of the two-point correlation coefficientΓi, j =

(yre f )|C
re f )∆z = 0, the conditional correlation gives Ru+ u+ ,C = ui,rms
+
(y

(6)

where the wall-normal and spanwise directions of integration
correspond to j = 1 and j = 2, respectively, so that r1 = y
and r2 = ∆z. The integration is performed on the lines passing through the reference point (y = yre f or ∆z = 0) limited

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

b)

15

6

u+2
i,rms

<U+>

5
GAN
DNS

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i=1,DNS
i=1,GANi=3,DNS
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i=2,GAN

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d)

c)

i i

Φu+u+

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i=1,DNS (y/h=0.08)
i=1,GAN (y/h=0.08)
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i=2,DNS (y/h=0.3)
i=2,GAN (y/h=0.3)
i=2,DNS (y/h=0.08)
i=2,GAN (y/h=0.08)
i=3,DNS (y/h=0.3)
i=3,GAN (y/h=0.3)
i=3,DNS (y/h=0.08)
i=3,GAN (y/h=0.08)

Ru+iu+ii=1,GAN (y/h=0.08)
i=1,GAN (y/h=0.3)
i=2,GAN (y/h=0.08)
i=2,GAN (y/h=0.3)
i=3,GAN (y/h=0.08)
i=3,GAN (y/h=0.3)
i=1,DNS (y/h=0.08)
i=1,DNS (y/h=0.3)
i=2,DNS (y/h=0.08)
i=2,DNS (y/h=0.3)
i=3,DNS (y/h=0.08)
i=3,DNS (y/h=0.3)

5

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y/h

0.5

a)
0

a

b

-0.1
-1

-0.5

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0.5

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b)

0.2

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∆z=h
-0.2
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R=0.05

0.1

10-1

100

101

-3

-2

-1

λz

0

1

∆z/h

2

3

Figure 4: One-point and two-point statistics obtained from the GAN data compared with the DNS data: a) Mean stream-

wise velocity profile, b) Turbulence intensities, c) Pre-multiplied spanwise energy spectrum obtained at y/h = 0.08 andy/h = 0.3, and d) One-dimensional spanwise two-point correlation coefficient obtained at y/h = 0.08 and y/h = 0.3. In
the inset (a) of figure (d), the dashed line shows Ru+ u+ = 0.05 while a and b points indicate the intersections of this line
i iwith the curve of Ru+ u+ at y/h = 0.3.

Γi, j/h0.8
0.6

i=1,j=1,DNS
i=2,j=1,DNS
i=3,j=1,DNS
i=1,j=2,DNS
i=2,j=2,DNS
i=3,j=2,DNS
i=1,j=1,GAN
i=2,j=1,GAN
i=3,j=1,GAN
i=1,j=2,GAN
i=2,j=2,GAN
i=3,j=2,GAN

0.6
0.4
j=1, P, DNS
j=2, P, DNS
j=1, N, DNS
j=2, N, DNS
j=1, P, GAN
j=2, P, GAN
j=1, N, GAN
j=2, N, GAN

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15
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DNS, N
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GAN, P
GAN, N
GAN, Unconditioned

0.8

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u+2rms1

li=1, j/h

a)

10

5
0.2
0
2

0
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yref/h

0.5

1

1.5

2

y/h

Figure 5: a) Wall-normal variation of the streamwise and wall-normal integral length-scales for all the velocity components(on the left), and length-scales for the streamwise velocity component corresponding to the line Ru+ u+ ,C = 0.3 obtained
based on the correlations conditioned on the positive (P) and negative clusters (N) (on the right). b) Conditional andunconditional turbulent intensities for the streamwise velocity component. The dash-dot line is the channel centerline.
very good agreement with the DNS results as given in Fig.5(b).
This figure reveals that the turbulence intensities carried by
the negative clusters are higher than those of the positive clusters which in turn are higher than the unconditional intensities
in a large portion of the channel height. Also, the near-wallpeak locations of the negative clusters, accurately predicted by
GAN, occur at a larger distance from the wall compared to theother two curves.

which is in agreement with the integral length-scales presented
in Fig.5(a). Also, the distance between the anti-correlated regions (dashed lines) increases with the wall distance. These regions are located symmetrically at the left and right sides of theu and v correlations whereas the negative lobes of the w component are positioned above and below the reference point,suggesting the existence of a quasi-streamwise roll mode. In
all cases, these regions disappear completely from the spanwise direction in the DNS data away from the wall. The findings discussed above are in agreement with the cross-stream
analysis of Sillero et al. (2014) for a channel flow. Interestingly, these features are well preserved in the results obtained
from the generated flow fields for a large portion of the flow
thickness as shown with the thin lines in Fig.6. The GAN datashows almost identical correlation maps for the transversal velocities at all wall-normal locations as presented in Fig.6(b)and 6(c). While the correlations of the u fluctuations show
similar patterns near the wall between the GAN and DNS data,

A better understanding of the correlations dimensions canbe achieved by Fig.6 which shows the 2D shape of the correlations at different wall-normal locations, covering from near
the wall to the logarithmic and outer regions. The thick lines
in this figure indicate the DNS data while the GAN results
are shown with the thin lines. As depicted in Figs.6(a–c),
the size of the objects for the three velocity fluctuations increases with the wall-normal location. In addition, the objectsof u, v, and w fluctuations get closer to circular, elliptical, andsquarish shape, respectively, as they move away from the wall,5

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

y/h

1.5
1

0.5
0

-1

0

1

-1

∆z/h

a) Ru+u+

0

∆z/h

1

b) Rv+v+

-1

0

1

∆z/h

c) Rw+w+

Figure 6: Two-dimensional two-point correlation coefficients of the a) streamwise, b) wall-normal, and c) spanwise velocity components at different reference points shown with colored circles (yre f /h = 0.08, 0.30, 1.01). The thick lines
indicate the DNS data while the thin lines are for the GAN data. At each wall-normal location, the contour level with
R = 0.1 is shown with a solid line while the dashed line indicates R = −0.1. The dash-dot line indicates the spanwise
symmetry.
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they deviate near the edge of the boundary layer. In fact, thefluctuations surrounding the reference point in the GAN data
maintain their coherence in a larger wavy pattern compared tothat of the DNS with some negatively correlated regions which
are not seen in the DNS correlation. As discussed by Sillero
et al. (2014), flux of the velocity covariance in the plane normal to the velocity component is equal to zero theoretically. In
our study, this integral is roughly of order 10−6 and 10−2 forthe DNS and GAN data shown in Fig.6(a), respectively. As
a result, the positive and negative correlations are strictly distributed differently in the DNS and GAN data, but as shown inthis study, the important characteristics are still present in the
GAN data.

Summary and conclusions
In this study, we successfully trained a GAN-based network with the cross-stream DNS data of a turbulent channel
flow to generate similar flow fields that resemble those of the
DNS both qualitatively and quantitatively. In particular, our results reveal that not only the flow field patterns, one-point andtwo-point statistics are quite matched between the DNS and
GAN data, but also the intense velocity coherent structures obtained from the generated flow fields follow the same behavior
as the DNS results in terms of the relation between their height
and width. Comparing the two-point correlation coefficientsfor the DNS and GAN data in the cross-stream plane, it turned
out that the wall-normal variation of the length-scales obtained
from the conditional and unconditional correlations are wellreflected in the GAN data. Overall, a satisfactory agreement isobtained between the DNS and GAN flow fields in this study,
with a general underprediction in the GAN results. The evidences gathered in this study clearly show the potential of theunsupervised deep learning network to learn structural features
hidden in the training dataset.

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