2022

[1]HeeChang Lim

[1]School of Mechanical Engineering, Pusan National University, Busandaehak-
ro, Busan, 46241, Rep. of KOREA

2]Instituto Universitario de Matemática Pura y Aplicada, Universitat
Politècnica de València, València, 46022, Spain

3]FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm,
10044, Sweden

# A deep-learning approach for reconstructing 3D turbulent flows from 2D
observation data

Mustafa Z. Yousif<EMAIL_ADDRESS>Linqi Yu<EMAIL_ADDRESS>Sergio Hoyas<EMAIL_ADDRESS>Ricardo Vinuesa<EMAIL_ADDRESS><EMAIL_ADDRESS>*  [  [

###### Abstract

Turbulence is a complex phenomenon that has a chaotic nature with multiple
spatio-temporal scales, making predictions of turbulent flows a challenging
topic. Nowadays, an abundance of high-fidelity databases can be generated by
experimental measurements and numerical simulations, but obtaining such
accurate data in full-scale applications is currently not possible. This
motivates utilising deep learning on subsets of the available data to reduce
the required cost of reconstructing the full flow in such full-scale
applications. Here, we develop a generative-adversarial-network (GAN)-based
model to reconstruct the three-dimensional velocity fields from flow data
represented by a cross-plane of unpaired two-dimensional velocity
observations. The model could successfully reconstruct the flow fields with
accurate flow structures, statistics and spectra. The results indicate that
our model can be successfully utilised for reconstructing three-dimensional
flows from two-dimensional experimental measurements. Consequently, a
remarkable reduction of the experimental setup cost can be achieved.

###### keywords:

3D flow reconstruction, experiments, deep learning, turbulent flow, numerical
simulations

## Introduction

With the development of computational power and experimental tools, an
accurate description of various types of turbulent flows can be achieved. In
terms of numerical simulations, direct numerical simulation (DNS) can resolve
the flow structures from the large energy-containing scales to the smallest
scales in the dissipation range (Kolmogorov1941, ). In the area of
experimental fluid dynamics, significant technical advances have been achieved
through particle-image velocimetry (PIV) (Adrian1984, ), tomographic-PIV
(tomo-PIV) (Scarano2012, ) and four-dimensional time particle-tracking
velocimetry (4D-PTV) (Schanzetal2016, ). However, in both the numerical and
experimental approaches, substantial costs are required to accurately describe
the physics of the turbulent flows and these costs are proportional to the
Reynolds number. This nondimensional parameter is proportional to the size of
the problem, its characteristic velocity, and inversely proportional to the
fluid kinematic viscosity.

On the other hand, an enormous amount of data can be generated from
experimental and numerical studies. Turbulence is now a science that needs,
more than ever, new questions more than new data to solve those questions.
This motivates developing data-driven methods that can practically utilise the
data for addressing various turbulence-related problems. With the recent rapid
advances of machine-learning algorithms and the remarkable improvement in the
graphic-processing-unit (GPU) capabilities, machine learning has been applied
in various fields, including image processing, natural language processing,
robotics, weather forecasting, etc. In terms of fluid dynamics, deep-learning
algorithms have been effectively applied to tackle a wide range of problems
(Bruntonetal2020, ; Kutz2017, ; Vinuesa&Brunton2022, ), where deep learning is
a subset of machine learning in which neural networks with multiple layers are
utilised in the model (LeCunetal2015, ). Unlike the conventional linear
methods, deep-learning-based techniques can deal with complex non-linear
problems. This makes the deep-learning approach a good candidate to be applied
for various problems in turbulence, such as turbulence modelling
(Duraisamyetal2019, ; Gamahara&Hattori2017, ; Lingetal2016, ), prediction of
turbulent flows (Lee&You2019, ; Srinivasanetal2019, ; Yousifetal2022a, ),
reduced-order modelling (Nakamuraetal2021, ; Yousif&Lim2022, ), turbulent-flow
control (Fanetal2020, ; Han&Huang2020, ; Rabaultetal2019, ;
Vinuesa&Brunton2022, ) and non-intrusive sensing (Guastonietal2021, ;
Guemesetal2021, ). The reconstruction of turbulent flows from spatially-
limited data using deep-learning-based models has been recently a topic of
interest, considering the ability of several deep-learning models to map the
flow fields that are represented by spatially limited or low-resolution data
to high-resolution flow fields (Fukamietal2019, ; Fukamietal2021, ;
Guemesetal2021, ; Kimetal2021, ; Liuetal2020, ; Yousifetal2021, ;
Yousifetal2022b, ). The results obtained from the studies cited above indicate
that several deep-learning models have a remarkable potential to map turbulent
flow fields with spatially limited distribution to high fidelity flow fields
by making use of the available training data. Thus, such deep-learning models
possibly can reconstruct the missing regions in the flow fields by
compensating them with approximation functions represented by the trainable
parameters in the models.

In this study, we present a deep-learning-based method to reconstruct three-
dimensional (3D) turbulent flows from two-dimensional (2D) data, in such a way
that mimics the reconstruction of 3D turbulent flows from 2D PIV measurements.
In contrast with the proposed studies in the literature, which are based on
simple approaches such as assumptions of frozen velocity (Bruckeretal2012, ;
Brucker1995, ; Zhangetal2008, ) and frozen turbulence via Taylor hypothesis
(Ganapathisubramanietal2008, ), exploiting homogeneity in the flow
(Chandramoulietal2019, ) and proper-orthogonal decomposition (POD)
(Braudetal2004, ; Hamdietal2018, ), we apply a deep-learning-based approach to
map unpaired intersected 2D turbulent flow sections to 3D flow fields. We
propose a generative-adversarial-network (GAN)-based model, 2D3DGAN, to
reconstruct three-dimensional turbulent flows. Unlike the traditional
convolutional-neural-networks (CNNs)-based models, GAN-based models
(Goodfellowetal2014, ) have shown the ability to capture high-frequency data
in detail, and remarkable accuracy in terms of image transformation and super-
resolution problems (Ledigetal2017, ; Mirza&Osindero2014, ; Wangetal2018, ;
Zhuetal2017, ). In a typical GAN, two networks, namely the generator ($G$) and
the discriminator ($D$), compete with each other. Here, $G$ generates
artificial samples similar to the real ones, whereas $D$ distinguishes the
artificial samples from the real ones. The goal of the training process is to
make $G$ generate artificial samples that are difficult to distinguish using
$D$. We utilise a combination of 2D and 3D CNNs to build the 2D3DGAN, which
can make use of both the supervised deep-learning method and the adversarial
networks, $i.e.$ $G$ and $D$ networks. We remark that the 2D3DGAN is robust to
increasing Reynolds number and the complexity of the flow.

## Results

Building 2D3DGAN architecture. The process of training GAN can be expressed as
a min-max two-player game with a value function $V(D,G)$ such that:

$\begin{split}\begin{subarray}{c}\rm min\\\
{G}\end{subarray}~{}\begin{subarray}{c}\rm max\\\
{D}\end{subarray}~{}V(D,G)=\mathbb{E}_{x_{r}\sim P_{\rm data}(x_{r})}[{\rm
log}D(x_{r})]+\mathbb{E}_{z\sim P_{z}(z)}[{\rm log}(1-D(G(z)))],\end{split}$
(1)

where $x_{r}$ represents real data and $P_{\rm data}(x_{r})$ is its
distribution. Note that $\mathbb{E}$ represents the operation of calculating
the average of all the data in the training mini-batch. In the second term of
the right-hand side of equation (1), $z$ is a random vector used as an input
to $G$, whereas $D(x_{r})$ represents the probability that the data is real
and not artificial. The output from $G$, $i.e.$ $G(z)$, is expected to
generate data that is similar to the real one, such that the value of
$D(G(z))$ is close to 1. On the other hand, in $D$, $D(x_{r})$ returns a value
close to 1, whereas $D(G(z))$ returns a value close to 0. Thus, $G$ is trained
in a direction that minimizes $V(D,G)$, and $D$ is trained in a direction that
maximizes $V(D,G)$. Additional details are provided in the Methods section.

The proposed 2D3DGAN is inspired by the works of Wang et al. (Wangetal2018, )
and Yousif et al. (Yousifetal2021, ), where the input to the network is data
that contains limited information about the flow instead of the random vector
$z$ in equation (1). While the models of (Wangetal2018, ; Yousifetal2021, )
are designed to reconstruct 2D high-resolution data from spatially limited 2D
data, our 2D3DGAN is designed to reconstruct 3D flow fields from 2D data. As
shown in figure 1, the input to the 2D3DGAN is data represented by a cross-
plane of unpaired intersected 2D flow observation planes, $i.e.$ each plane
contains data of two velocity components, and the data are collected at a
period that is different from the other plane. Hence, the velocity fields in
the planes are at different instants.

In this study, two cases are used to evaluate the performance of the proposed
model: a turbulent channel flow at two different friction Reynolds number,
$Re_{\tau}=$180 and 500, and the flow around a finite wall-mounted square
cylinder with aspect ratio, $AR=$4, at a Reynolds number based on the free
stream velocity and the cylinder width $d$ of $Re_{d}=500$.

Let us take the case of turbulent channel flow in figure 1 as a demonstration
of matching the unpaired data before feeding it to the 2D3DGAN. The data from
the two sections are synthetically unpaired such that no instantaneous
velocity of each section is found at the same time as the instantaneous
velocity in the other section. After that, the flow data of the ($x-y$)
section is matched with the data of ($y-z$) section by utilising the square of
$L_{2}$ norm error for the intersection line of the wall-normal velocity such
that:

$v_{(y-z)}^{*}w_{(y-z)}^{*}=\begin{subarray}{c}{\rm argmin}\\\
v_{(y-z)},w_{(y-z)}\end{subarray}\left(\left\|v_{(x-y)}^{y}-v_{(y-z)}^{y}(t)\right\|_{2}^{2}\right),$
(2)

where $v$ and $w$ are the wall-normal and spanwise instantaneous velocity
components. Here, $v_{(x-y)}^{y}$ represents the wall-normal velocity in the
$(x-y)$ plane, and $v_{(y-z)}^{y}(t)$ represents the wall-normal velocity in
the $(y-z)$ plane as a function of time, $t$. The superscript ‘$*$’ indicates
the matched velocity data using equation (2). This procedure mimics the
matching of two data sets from two planar PIV experiments conducted each for a
different plane. The first plane is the observation plane (which is here
represented by the ($x-y$) plane) with velocity components $(u,v)$, where $u$
is the instantaneous streamwise velocity component, and the second one is the
$(y-z)$ plane with velocity components $(v,w)$. Note that the 3D label data
used for training the model are at the same instants of the observation plane.
A similar approach is followed for the case of flow over a finite wall-mounted
square cylinder with the observation plane being the central $(x-y)$ plane
$(z/d=0)$. The $(x-z)$ plane at $y/d=2$ is used to provide the matched
velocity data with the data from the observation plane.

Figure 1: Procedure of reconstructing three-dimensional turbulent flows using
the 2D3DGAN. The domain from each simulation contains planar data of the flow
obtained from a cross-plane. Note that the flow data of the two planes in the
cross plane are entirely unpaired, $i.e.$ the velocity fields in the planes
are at different instants. The superscript ‘$*$’ indicates the velocity
components that are matched with the velocity data from the observation plane
by applying the procedure explained in equation (2).

Instantaneous flow reconstruction. First, we examine the ability of the model
to reconstruct the 3D instantaneous velocity fields. Figure 2(a) shows the
reconstructed instantaneous velocity fields of the case of turbulent channel
flow $(u^{+},v^{+},w^{+})$ at the $(x-z)$ plane, located at $y^{+}=16.78$ and
46.95, for the flow at $Re_{\tau}=180$ and $500$, respectively. As can be seen
in the figure, the velocity fields are successfully reconstructed by the model
with commendable precision even though the $(x-z)$ plane is not introduced to
the model during the training process. Note that the superscript ‘$+$’
indicates that the quantity is scaled in inner units using the fluid
viscosity, $\nu$ and the friction velocity, $u_{\tau}$. The flow structure is
investigated in figure 2(b) by utilising the $Q$-criterion for vortex
identification (lozanoetal2014, ). As shown in the figure, the reconstructed
instantaneous velocity fields reveal a vortical structure $(Q^{+})$ that is
similar to that obtained from the DNS data, indicating that the model could
successfully reproduce the flow fields with high accuracy.

Figure 2(c) shows the reconstructed instantaneous velocity fields for the case
of flow around a finite wall-mounted square cylinder $(u\textfractionsolidus
U_{\infty},v\textfractionsolidus U_{\infty},w\textfractionsolidus U_{\infty})$
at $y/d=3$, where $U_{\infty}$ is the free stream velocity. As shown in the
figure, all the three velocity fields are in good agreement with the DNS data.
Note that the velocity fields along the height of the cylinder show very good
agreement with the DNS data even for this region, $i.e.$ near the free end of
the cylinder, which is not seen by the model as input during the training
process. This indicates that the model can reconstruct the complex three-
dimensional turbulent flow around the cylinder within all the regions.
Furthermore, the vortical structure obtained from the reconstructed
instantaneous velocity fields is generally consistent with the results from
DNS as shown in figure 2(d).

Finally, the accuracy of the reconstruction is examined via the $L_{2}$-norm
relative error:

$\epsilon=\frac{1}{J}\sum_{j=1}^{J}\frac{\left\|{\alpha}_{j}^{\rm
DNS}-{\alpha}_{j}^{\rm REC}\right\|_{2}}{\left\|{\alpha}_{j}^{\rm
DNS}\right\|_{2}},$ (3)

where $\alpha_{j}^{\rm DNS}$ and $\alpha_{j}^{\rm REC}$ represent the ground
truth (DNS) and the reconstructed instantaneous velocity components,
respectively, and $J$ represents the number of the test snapshots. As can be
observed from table 1, for the case of turbulent channel flow, no significant
increase in the error values is observed in the channel when increasing
$Re_{\tau}$ from 180 to 500, indicating the robustness of the model to
increasing Reynolds number. Also, for the case of flow around a finite wall-
mounted square cylinder, the error values are acceptable as compared to the
error values of the turbulent channel, a fact that further supports the
ability of the model to reconstruct the velocity fields of complex flows. Note
that the errors in $u$ are low, while the errors in $v$ and $w$ are
comparatively higher. This is explained by the fact that in these flows the
main physics is driven by the streamwise component, which is the main focus of
the deep learning model when performing the predictions. It is worth noting
here that in GAN-based models, the mapping of high frequency fluctuations in
the data is related to the adversarial loss. In other words, in GAN-based
models, synthetic data is generated for the high frequency fluctuations. This
is the main difference between GAN-based models and traditional CNNs, where
the results are usually blurry with few flow details that can be predicted.
Also, it is important to note that, despite the deviations in $v$ and $w$
error values, the main flow features are very well reproduced (as observed in
figure 2), and as discussed from the statistical and spectral perspectives
next.

Figure 2: Instantaneous velocity fields and flow structures. (a) $(x-z)$ Plane
instantaneous velocity fields of the turbulent channel flow case; from left to
right: streamwise velocity, wall-normal velocity and spanwise velocity; the
planes are located at $y^{+}=16.78$ and 46.95, for the flow at $Re_{\tau}=180$
and $500$, respectively. (b) Isosurfaces of the instantaneous flow structure
for the turbulent channel flow case; $Q^{+}=0.006$ and 0.002 for the flow at
$Re_{\tau}=$ 180 and 500, respectively. (c) Instantaneous velocity fields for
the case of flow over a finite wall-mounted square cylinder at the $(x-z)$
plane, located at $y/d=3$; from left to right: streamwise velocity, wall-
normal velocity and spanwise velocity. (d) Isosurfaces of the instantaneous
flow structure for the case of flow around a finite wall-mounted square
cylinder; $Q(d/U_{\infty})^{2}=0.0068$. Note that REC denotes reconstructed
field.

Spatial distribution of the flow. In order to examine the ability of the
proposed model to reconstruct the velocity fields with accurate spatial
distribution, the probability density function of each velocity component, pdf
is calculated for both cases. As shown in figure 3(a), the pdf plots of the
reconstructed velocity components for the case of turbulent channel flow show
a remarkable agreement with those obtained from the DNS data, for all the
regions along $y^{+}$. Also, for the case of flow over a finite wall-mounted
square cylinder (figure 3(b)), we can see that the pdf plots are consistent
with the DNS data for all the three velocity components. Here, the results
indicate the ability of the model to reconstruct the instantaneous velocity
fields with accurate spatial distributions.

Furthermore, the ability of the model to reproduce the spectral content of the
flow in the turbulent channel is investigated by utilising the premultiplied
streamwise and spanwise power-spectral densities, $i.e.$
$k_{x}\mathit{\Phi}_{\alpha\alpha}$ and $k_{z}\mathit{\Phi}_{\alpha\alpha}$,
where $k_{x}$ and $k_{z}$ are the streamwise and spanwise wavenumbers,
respectively, and $\mathit{\Phi}_{\alpha\alpha}$ represents the corresponding
wavenumber spectrum of each velocity component, $\alpha$. Figure 3(c) shows
$k_{x}^{+}\mathit{\Phi}_{\alpha\alpha}^{+}$ and
$k_{z}^{+}\mathit{\Phi}_{\alpha\alpha}^{+}$ as a function of $y^{+}$ and the
corresponding inner-scaled wavelengths, $\lambda_{x}^{+}$ and
$\lambda_{z}^{+}$. The spectra of the reconstructed velocity components are
generally in agreement with the results obtained from DNS, with a slight
deviation at high-wavenumber regions. These results further support the
performance of the proposed model to properly represent the spatial
distribution of the velocity fields.

Figure 3: Probability density functions of the velocity components. (a) pdf of the turbulent channel flow case as a function of $y^{+}$; from left to right: streamwise velocity, wall-normal velocity and spanwise velocity. The shaded contours represent the results from the DNS data and the lines represent the results from the reconstructed data. The contour levels are in the range of 20%-80% of the maximum pdf with an increment of 20%. Results for the flow at $Re_{\tau}=$ 180\. (b) pdf of the velocity components for the case of flow over a finite wall-mounted square cylinder; from left to right: streamwise velocity, wall-normal velocity and spanwise velocity. Note that REC denotes reconstructed field. (c) Premultiplied streamwise (top) and spanwise (bottom) power-spectral density of the velocity components from the turbulent channel flow case, as a function of $y^{+}$ and $\lambda^{+}$; from left to right: streamwise velocity, wall-normal velocity and spanwise velocity. The shaded contours represent the results from the DNS data and the lines represent the results from the reconstruction. The contour levels are in the range of 10%-90% of the maximum $k_{x}^{+}\mathit{\Phi}_{\alpha\alpha}^{+}$ and $k_{z}^{+}\mathit{\Phi}_{\alpha\alpha}^{+}$, with increments of 10%. Results for the flow at $Re_{\tau}=$180. Table 1: $L_{2}$-norm relative error of the reconstructed velocity fields. Case |  $\epsilon(u)$ |  $\epsilon(v)$ |  $\epsilon(w)$
---|---|---|---
Turbulent channel flow $Re_{\tau}=180$ | 0.041 |  0.577 |  0.471
Turbulent channel flow $Re_{\tau}=500$ | 0.044 |  0.589 |  0.515
Flow around a finite wall-mounted square cylinder | 0.068 |  0.743 |  0.649

Turbulence statistics. To examine the capability of the proposed model to
reproduce the turbulence statistics, the first and second-order turbulence
statistics of the reconstructed velocity fields are calculated and shown in
figure 4. As can be observed in figure 4(a), for the case of turbulent channel
flow at $Re_{\tau}=180$, the profile of the mean streamwise velocity $(U^{+})$
and the root-mean-square (rms) profiles of the velocity components $(u_{\rm
rms}^{+},v_{\rm rms}^{+},w_{\rm rms}^{+})$ are in excellent agreement with the
results obtained from DNS. The Reynolds stress profile
$(-\overline{u^{\prime}v^{\prime}}^{+})$ exhibits a relatively good agreement
with the results obtained from DNS, with a slight deviation in the range
between $y^{+}$ = 50 and 125. The fluctuating streamwise vorticity
$(\omega_{x,{\rm rms}}^{+})$ profile is also in agreement with the results
obtained from DNS. Here the results indicate that the model can reproduce the
turbulence statistics of the flow with remarkable accuracy. The turbulence
statistics for the flow at $Re_{\tau}=500$ indicate that the reconstructed
data are generally in good agreement with the DNS. Nonetheless, the Reynolds
shear-stress profile exhibits a slight underprediction for a region that
starts from the maximum shear stress and continues along $y^{+}$. As can be
seen in figure 4(b), for the case of the flow over a wall-mounted square
cylinder, the streamwise, as well as the spanwise profile of the mean
streamwise velocity $(\overline{u}\textfractionsolidus U_{\infty})$ exhibit an
excellent agreement with the results obtained from the DNS for the examined
elevations along the cylinder height. The comparison of the spanwise profile
of the Reynolds shear stress
$(\overline{u^{\prime}w^{\prime}}\textfractionsolidus{U_{\infty}^{2}})$ also
reveals generally good agreement with the results obtained from the DNS.
However, it exhibits a slight deviation at the examined wall-normal location,
$i.e.$ $y/d=3$. This might be attributed to the increase in the complexity of
the flow as the wall-normal distance increases (Bourgeoisetal2011, ; Saha2013,
; Yousif&Lim2021, ; Lazpitaetal2022, ).

Figure 4: Turbulence statistics. (a) Turbulence statistics of the turbulent
channel flow case; from top to bottom: mean streamwise velocity profile, root-
mean-square profiles of the velocity components, Reynolds shear-stress stress
profile and root-mean-square profile of the streamwise vorticity. (b)
Turbulence statistics of the flow around a finite wall-mounted square cylinder
case; from top to bottom: streamwise profile of mean streamwise velocity at
$y/d=1$, streamwise profile of mean streamwise velocity at $y/d=3$, spanwise
profile of mean streamwise velocity at $y/d=3$ and spanwise profile of
Reynolds stress at $y/d=3$. Note that REC denotes reconstructed field.

## Discussion and conclusions

This study presents a deep-learning-based method to reconstruct 3D turbulent
flows from 2D data. A combination of 2D and 3D CNNs was utilised to build a
model based on GAN, named 2D3DGAN. The model is designed to reconstruct 3D
turbulent flows from a cross-plane of unpaired data (the data of each plane in
the cross-plane is collected at a different time interval), such that it can
be utilised to reconstruct 3D turbulent velocity fields from two-dimensional
PIV measurements.

The reconstructed instantaneous velocity fields and the 3D flow structures
show an excellent agreement with the DNS results for both the cases used to
test the model, $i.e.$ turbulent channel flow and the flow around a finite
wall-mounted square cylinder, even for the regions where the data is not
introduced to the model during the training process. The error analysis
reveals that the model is robust to increasing Reynolds number and the
complexity of the flow. The model also successfully reproduces the turbulence
statistics with very good accuracy. Furthermore, the flow spectra for the case
of turbulent channel flow also reveal a commendable agreement with the result
obtained from the DNS data, indicating a great ability of the model to
maintain a realistic spatial behaviour of the velocity fields.

This study demonstrates for the first time that GAN-based models can be
successfully used for reconstructing 3D turbulent flows from 2D data. This
approach opens the doors to discovering new data-driven methods that can
reconstruct the 3D turbulent flows from 2D experimental measurements, which
can result in a remarkable cost reduction in complexity of the experimental
setup required to achieve accurate 3D experimental measurements.

## Methods

Building the 2D3DGAN architecture. The architecture of 2D3DGAN is shown in
figure 5(a). Here, $G$ consists of two stages. First, two sections of 2D data
are introduced to $G$ and passed through a 2D CNN and two stages of reshaping
with a dense layer between them. After that, the 3D data are passed through a
deep 3D CNN represented by residual in residual dense blocks (RRDBs) and
multi-scale part (MSP). The MSP, which consists of three parallel 3D
convolutional sub-models with different kernel sizes, is applied to the data
features extracted by the RRDBs. More details regarding MSP can be found in
Yousif et al. (Yousifetal2021, ; Yousifetal2022b, ). The outputs of the three
sub-models are summed and passed through a final 3D convolutional layer to
generate an artificial 3D data ($3D_{a}$). The artificial and real data are
fed to $D$ and passed through a series of 3D convolutional, batch
normalization, and leaky-ReLU layers. As a final step, the data are passed
through a 3D convolutional layer. The non-transformed discriminator outputs
using the real and artificial data, $i.e.$ $C(3D_{r})$ and $C(3D_{a})$, are
used to calculate the relativistic average discriminator value ($D_{Ra}$)
(Jolicoeur-Martineau2018, ):

$D_{Ra}(3D_{r},3D_{a})=\sigma(C\left(3D_{r})\right)-\mathbb{E}_{3D_{a}}\left[C(3D_{a})\right],$
(4)

$D_{Ra}(3D_{a},3D_{r})=\sigma(C\left(3D_{a})\right)-\mathbb{E}_{3D_{r}}\left[C(3D_{r})\right],$
(5)

where $\sigma$ is the sigmoid function. In Equations (4) and (5), $D_{Ra}$
represents the probability that the output from $D$ using the real 3D data is
relatively more realistic than the output using the generated 3D data.

The discriminator loss is then defined as:

$L_{D}^{Ra}=-\mathbb{E}_{3D_{r}}\left[{\rm
log}(D_{Ra}(3D_{r},3D_{a}))\right]-\mathbb{E}_{3D_{a}}\left[{\rm
log}(1-D_{Ra}(3D_{a},3D_{r}))\right].$ (6)

The adversarial loss for the generator can be expressed in a symmetrical form
as:

$L_{G}^{Ra}=-\mathbb{E}_{3D_{r}}\left[{\rm
log}(1-D_{Ra}(3D_{r},3D_{a}))\right]-\mathbb{E}_{3D_{a}}\left[{\rm
log}(D_{Ra}(3D_{a},3D_{r}))\right].$ (7)

The total loss function of $G$ is defined as:

$\mathcal{L}_{G}=\beta_{1}L_{G}^{Ra}+\beta_{2}L_{\rm voxel}+\beta_{3}L_{\rm
perceptual}+L_{\rm continuity}+L_{\rm momentum},$ (8)

where $L_{\rm voxel}$ is the error calculated based on the voxel (volume
pixel) difference between the generated data and the ground truth data.
$L_{\rm perceptual}$ represents the difference in the extracted features of
the real and the generated data. In this study, we use a 3D convolutional
auto-encoder (3DCAE) to extract the features from the data obtained from the
generator and the ground truth data as shown in figure 5(b). Note that $L_{\rm
continuity}$ and $L_{\rm momentum}$ represent the error of the continuity and
momentum equations. In the loss function, $\beta_{1}$, $\beta_{2}$, and
$\beta_{3}$ are the coefficients used to balance the loss terms, and their
values are set to 10, 1000 and 2000, respectively. The square of the $L_{2}$
norm error is used to calculate all the loss terms of the generator except
$L_{G}^{Ra}$, such that:

$L_{\rm voxel}=\frac{1}{M}\sum_{m=1}^{M}\left\|3D_{r}-3D_{a}\right\|_{2}^{2},$
(9)

$L_{\rm
perceptual}=\frac{1}{M}\sum_{m=1}^{M}\left\|\mathscr{F}_{FE}(3D_{r})-\mathscr{F}_{FE}(3D_{a})\right\|_{2}^{2},$
(10)

$L_{\rm continuity}=\frac{1}{M}\sum_{m=1}^{M}\left\|\left(\frac{\partial
u_{r}}{\partial x}+\frac{\partial v_{r}}{\partial y}+\frac{\partial
w_{r}}{\partial z}\right)-\left(\frac{\partial u_{a}}{\partial
x}+\frac{\partial v_{a}}{\partial y}+\frac{\partial w_{a}}{\partial
z}\right)\right\|_{2}^{2},$ (11)

$L_{\rm momentum}=L_{\rm momentum}^{u}+L_{\rm momentum}^{v}+L_{\rm
momentum}^{w},$ (12)

$L_{\rm momentum}^{u}=\frac{1}{M}\sum_{m=1}^{M}\left\|\left(\frac{\partial
u_{r}}{\partial t}+u_{r}\frac{\partial u_{r}}{\partial x}+v_{r}\frac{\partial
u_{r}}{\partial y}+w_{r}\frac{\partial u_{r}}{\partial
z}\right)-\left(\frac{\partial u_{a}}{\partial t}+u_{a}\frac{\partial
u_{a}}{\partial x}+v_{a}\frac{\partial u_{a}}{\partial y}+w_{a}\frac{\partial
u_{a}}{\partial z}\right)\right\|_{2}^{2},$ (13)

$L_{\rm momentum}^{v}=\frac{1}{M}\sum_{m=1}^{M}\left\|\left(\frac{\partial
v_{r}}{\partial t}+u_{r}\frac{\partial v_{r}}{\partial x}+v_{r}\frac{\partial
v_{r}}{\partial y}+w_{r}\frac{\partial v_{r}}{\partial
z}\right)-\left(\frac{\partial v_{a}}{\partial t}+u_{a}\frac{\partial
v_{a}}{\partial x}+v_{a}\frac{\partial v_{a}}{\partial y}+w_{a}\frac{\partial
v_{a}}{\partial z}\right)\right\|_{2}^{2},$ (14)

$L_{\rm momentum}^{w}=\frac{1}{M}\sum_{m=1}^{M}\left\|\left(\frac{\partial
w_{r}}{\partial t}+u_{r}\frac{\partial w_{r}}{\partial x}+v_{r}\frac{\partial
w_{r}}{\partial y}+w_{r}\frac{\partial w_{r}}{\partial
z}\right)-\left(\frac{\partial w_{a}}{\partial t}+u_{a}\frac{\partial
w_{a}}{\partial x}+v_{a}\frac{\partial w_{a}}{\partial y}+w_{a}\frac{\partial
w_{a}}{\partial z}\right)\right\|_{2}^{2},$ (15)

where $M$ represents the number of snapshots in the training mini-batch, which
is fixed in this study to 8, and $\mathscr{F}_{FE}$ in equation (10)
represents the function of the feature extractor, $i.e.$ the 3DCAE.

Figure 5: Architecture of 2D3DGAN and 3DCAE. (a) Schematic of 2D3DGAN showing
the generator network (left) and the discriminator network (right). The
generator network receives a dataset represented by two planes in the flow,
the first one is the observation plane, and the second one is the plane that
is matched with the observation plane by applying the procedure explained in
equation (2), which is indicated by the superscript ‘$*$’. The output from the
generator network, $i.e.$ the artificial 3D flow data is fed to the
discriminator network, and the latter tries to distinguish if the data is
artificial or true. (b) Schematic of the feature extractor (3DCAE). The main
features of the flow fields are extracted using three layers in the encoder
part of the 3DCAE.

Overview of the training setup. The 2D3DGAN is trained separately with the
data of turbulent channel flow at
$Re_{\tau}=u_{\tau}\delta\textfractionsolidus\nu=180$ and 500, and with the
data of flow around a finite wall-mounted square cylinder with $AR=$4, and at
$Re_{d}=U_{\infty}d\textfractionsolidus\nu=500$, where, $\delta$ is the
channel half-width. For the case of turbulent channel flow, the transfer-
learning technique (Guastonietal2021, ; Yousifetal2021, ; Yousifetal2022a, )
is used to train the model for the flow at $Re_{\tau}=500$, in order to
further reduce the computational cost (represented by the training time) and
the required training data. For both cases, 80% of the collected data are used
as training set and the rest, $i.e.$ 20%, are used as testing set. The open-
source library TensorFlow 2.4.0 (Abadietal2016, ) is used for the
implementation of the 2D3DGAN. The adaptive moment estimation (Adam)
optimization algorithm (Kingma&Ba2017, ) is used to update the weights of the
model. The learning rate is initially set to $10^{-4}$ and progressively
decreased during the training process to finally reach $1.25\times 10^{-5}$.
The number of trainable parameters in the model is approximately 5.7 million
for the generator network and 3.74 million for the discriminator. The training
of the model using a machine with a single NVIDIA TITAN RTX GPU for the case
of turbulent channel flow requires approximately 48 hours for $Re_{\tau}=180$
and for $Re_{\tau}=500$, with the aid of transfer learning, requires
approximately 30 hours. The training of the case of flow over a wall-mounted
square cylinder requires approximately 40 hours.

Direct numerical simulation of turbulent channel flow. DNS calculations of a
fully-developed incompressible turbulent channel flow at friction Reynolds
numbers $Re_{\tau}=180$ and 500 are performed to generate the training and
testing datasets. The incompressible Navier–stokes equations have been solved
using the LISO code (Hoyas&Jimenez2006, ), similar to the one described by
Lluesma-Rodriguez et al. (Lluesmaetal2021, ). This code has successfully been
employed to run some of the largest simulations of wall-bounded turbulent
flows (alcantara2021, ; Hoyasetal2022, ; Oberlacketal2022, ). Briefly, the
code uses the same strategy as that described by Kim et al. (kimetal1987, ),
but uses a seven-point compact-finite difference scheme in the $y$ direction
with fourth-order consistency and extended spectral-like resolution (Lele1992,
). The temporal discretization is a third-order semi-implicit Runge–Kutta
scheme (Spalartetal1991, ). The wall-normal grid spacing is adjusted to keep
the resolution to $\Delta y=1.5\eta$, $i.e.$ approximately constant in terms
of the local isotropic Kolmogorov scale
$\eta=\left(\nu^{3}/\varepsilon\right)^{0.25}$, where $\varepsilon$ is the
isotropic dissipation of turbulent kinetic energy.

The dimensions of the computational domain for both simulations are set to
$8\pi\delta,2\delta$ and $3\pi\delta$ in the streamwise, wall-normal and
spanwise directions, respectively. A uniform distribution of the grid points
is used in the streamwise and spanwise directions and a non-uniform
distribution is used in the non-homogeneous wall-normal direction. For the
flow at $Re_{\tau}=180$, the grid spacings in the streamwise ($\Delta x^{+}$)
and spanwise directions ($\Delta z^{+}$) are 8.55 and 4.27, respectively,
while for the flow at $Re_{\tau}=$ 500, $\Delta x^{+}=8.33$ and $\Delta
z^{+}=4.16$. The grid spacing near the wall in the wall-normal direction is
$\Delta y_{w}^{+}=0.53$ and $0.74$ for the flow at $Re_{\tau}=$ 180 and 500,
respectively. The simulation time step ($\Delta t^{+}$) is set to 0.07 and
0.09 for the flow at $Re_{\tau}=$ 180 and 500, respectively. The flow is
periodic in the streamwise and spanwise directions, whereas the no-slip
condition is applied to the channel walls. A total of 1,000 consecutive
snapshots are collected for each of the two simulations.

Direct numerical simulation of flow around a finite wall-mounted square
cylinder. In the second case, we consider the flow around a finite wall-
mounted square cylinder with $AR=$4, at $Re_{d}=500$. The spectral-element-
method (SEM)-based open-source code Nek5000 developed by Fischer et al.
(Fischeretal2022, ) is used to perform the DNS. In the SEM (Patera1984, ), the
computational domain is decomposed into elements, and the solution is
expressed in terms of Lagrange interpolants of order $N$ within those
elements. The location of the nodes inside the elements follows the
Gauss–Lobatto–Legendre (GLL) distribution, whereas there is an isoparametric
mapping for the shape of the elements and there are no restrictions regarding
the position of the elements in the domain. This means that this method allows
the flexibility to compute complex geometries, while still preserving the
characteristics of a high-order spectral method. In the present study, the
velocity field is expressed in terms of Lagrange interpolants of order $N=5$,
and order $N-2=3$ is considered for the pressure field. The nonlinear terms
are treated explicitly by third-order extrapolation (EXT3), whereas the
viscous terms are treated implicitly by a third-order backward differentiation
scheme (BDF3). The no-slip boundary condition is applied for the cylinder
walls and the ground, while periodic boundary conditions are used in the
spanwise direction. At the top of the domain, we impose a constant streamwise
velocity, a zero spanwise velocity and zero stress in $y$. Furthermore, the
input is a laminar boundary layer, and the output is the stabilized boundary
condition by Dong et al. (Dongetal2014, ). The dimensions of the simulation
domain are $(L_{x},L_{y},L_{z})/d=(60,12,12)$ in $x$, $y$ and $z$,
respectively, and the total number of grid points is around 20 million. We
collect a total of 10,000 snapshots with a time setp among snapshots of
$\Delta tU_{\infty}/d=0.02$.

Data preparation and pre-processing. To reduce the computational cost of
training the model and to increase the training and testing data, each
simulation domain in the case of turbulent channel flow is divided into 12
identical sub-domains having a size of $2\pi\delta$, $2\delta$ and
$\pi\delta$. Furthermore, the grid size of each sub-domain is interpolated and
reduced to $64\times 48\times 48$ grid points with a uniform distribution in
the $x$ and $z$ directions and a non-uniform distribution in the $y$
direction. For the case of flow around a wall-mounted square cylinder, the
size of the domain that is used for training and testing the model is set to
$10d\times 7.6d\times 8d$, and the data are interpolated and reduced into a
uniform grid distribution of $48\times 48\times 48$ points. The input data to
the model are normalised using the min-max normalisation to obtain values
between 0 and 1.

## Data availability

All the data analysed in this paper were produced with the in-house and open-
source softwares described in the code-availability statement. Reference data
and the scripts used to produce the data figures, as well as instructions to
train and test the 2D3DGAN are available on the following web page:

(https://fluids.pusan.ac.kr/fluids/65416/subview.do).

## Code availability

The DNS calculations of the turbulent channel flow case are performed using
in-house code LISO, contact SH for the code availability. The DNS calculations
of the case of flow over a finite wall-mounted square cylinder are performed
using the open-source code Nek5000: (https://nek5000.mcs.anl.gov/). The open-
source library TensorFlow 2.4.0 is used for the implementation of the 2D3DGAN:
(https://www.tensorflow.org/).

## Acknowledgments

This work was supported by ‘Human Resources Program in Energy Technology’ of
the Korea Institute of Energy Technology Evaluation and Planning (KETEP),
granted financial resource from the Ministry of Trade, Industry & Energy,
Republic of Korea (no. 20214000000140). In addition, this work was supported
by the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIP) (no. 2019R1I1A3A01058576). This work was also supported by
the National Supercomputing Center with supercomputing resources including
technical support (KSC-2021-CRE-0244). R.V. acknowledges the financial support
from the ERC Grant No. “2021-CoG-101043998, DEEPCONTROL”. SH was funded by
Contract No. PID2021-128676OB-I00 of Ministerio de Ciencia, innovación y
Universidades/ FEDER.

## Author contributions

M.Z.Y. and H.L. designed the research. M.Z.Y. and L.Y. wrote the code of the
deep-learning model with input from R.V. R.V. and S.H. performed the
simulations and provided the data. M.Z.Y., L.Y. and H.L. wrote the manuscript.
R.V. and S.H. reviewed and edited the manuscript. H.L. supervised the project.

## Competing interests

The authors declare no competing interests.

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