# Predicting unavailable parameters from existing velocity fields of turbulent
flows using a GAN-based model

Linqi Yu1,∗, Mustafa Z. Yousif1,2,∗
&Young-Woo Lee1, Xiaojue Zhu3, Meng Zhang1, Paraskovia Kolesova1
&Hee-Chang Lim1,†

1 School of Mechanical Engineering, Pusan National University
2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, Republic of Korea

2 German Engineering Research and Development Center, LSTME-Busan Branch

3 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3,
Göttingen 37077, Germany

∗ These authors contributed equally to this work.
† Corresponding author<EMAIL_ADDRESS>
###### Abstract

In this study, an efficient deep-learning model is developed to predict
unavailable parameters, e.g., streamwise velocity, temperature, and pressure
from available velocity components. This model, termed mapping generative
adversarial network (M-GAN), consists of a label information generator (LIG)
and an enhanced super-resolution generative adversarial network (ESRGAN). LIG
can generate label information helping the model to predict different
parameters. The GAN-based model receives the label information from LIG and
existing velocity data to generate the unavailable parameters. Two-dimensional
(2D) Rayleigh-Bénard flow and turbulent channel flow are used to evaluate the
performance of M-GAN. Firstly, M-GAN is trained and evaluated by 2D direct
numerical simulation (DNS) data of a Rayleigh-Bénard flow. From the results,
it can be shown that M-GAN can predict temperature distribution from the two-
dimensional velocities. Furthermore, DNS data of turbulent channel flow at two
different friction Reynolds numbers $Re_{\tau}$ = 180 and 550 are applied
simultaneously to train the M-GAN and examine its predicting ability for the
pressure fields and the streamwise velocity from the other two velocity
components. The instantaneous and statistical results of the predicted data
agree well with the DNS data, even for the flow at $Re_{\tau}$ = 395,
indicating that M-GAN can be trained to learn the mapping function of the
unknown fields with good interpolation capability.

##  1 Introduction

As a traditional topic in experimental and computational fluid dynamics (CFD),
turbulence has been investigated extensively to enhance the understanding of
its complicated characteristics and chaotic behavior over a hundred years.
Visualizing and statistically analyzing turbulent flows requires massive data
from flow fields. With various methods developed, high-fidelity turbulence
data can be efficiently generated for aerodynamic simulations and other
scientific applications. For instance, direct numerical simulation (DNS) has
been extensively applied in the CFD field to simulate a variety of turbulent
flows precisely at a specific range of Reynold numbers ($Re$), whose main
mechanism solves the Navier-Stokes equations numerically [1, 2, 3]. From the
perspective of experimental measurement, one of the most widely-used methods
is particle-image velocimetry (PIV), which plays a crucial role in the
experimental investigation of turbulent flow fields [4]. However, if an
ordinary two-dimensional (2D) PIV system is employed, only two velocity
components, e.g., wall-normal velocity ($v$) and spanwise velocity ($w$), can
be obtained but without streamwise velocity ($u$).

Note here that an unavailable parameter generally refers to a parameter that
cannot be measured directly using 2D measurement techniques such as PIV
measurement. For instance, if the normal section to the streamwise direction
of the flow is measured, the streamwise component cannot be obtained directly
and therefore is considered an unavailable parameter. In addition, pressure
($p$) and temperature ($T$) also cannot be obtained directly by those
measurements. These unavailable parameters are undoubtedly important for
turbulent flow analysis and engineering applications. Therefore, these three
velocity components are required to analyze the instantaneous flow structures
and statistics of turbulent flow [5, 6]. In addition, the fluctuating pressure
field over the bluff bodies under turbulent boundary layers causes severe
noise and vibration [7]. Moreover, the pressure fields within the turbulent
flow field are crucial to analyze the coupling mechanisms of
turbulence–acoustic [8] and flow control strategies for reducing noise [9].
Temperature is also crucial in turbulence fields, as its gradient drives the
heat transfer phenomenon. Additionally, temperature gradients in the near-wall
region significantly impact the overall performance of many engineering
systems [10, 11].

Currently, several methods are applied to obtain the aforementioned
unavailable parameters. Therefore, in general, acquiring the full components
of turbulent flow fields requires a high-performance PIV system such as a
tomographic PIV (Tomo PIV) [12], which realizes the acquisition of three-
dimensional (3D) turbulent flow with full velocity components. However, in
order to obtain the full velocity components, it would be necessary to incur
additional costs and employ complex processes to achieve the desired level of
precision. As for pressure measurement by PIV, Oudheusden [4] proposed a way
how to manipulate the instantaneous pressure field by combining the
experimental data with the governing equations, i.e., the Poisson equation. He
et al. [13] used data assimilation to determine the pressure of turbulent
velocity fields measured by PIV based on the unsteady adjoint formulation.
However, implementing these methods still needs the assistance of Tomo PIV. In
other words, the turbulent flow data from the 2D PIV system are insufficient
to solve the governing equation or apply data assimilation because of the lack
of one of the velocity components. Thermographic PIV (Thermo PIV) [14, 15] has
been developed to measure velocity fields and temperature based on
thermographic phosphor particles, which possess temperature-dependent
luminescence properties. Compared to the original PIV, Thermo PIV needs more
extra setup and equipment, which include an extra ultraviolet laser and two
extra cameras with suitable spectral filters. Nonetheless, it is difficult to
predict unavailable parameters like $u$, $p$, and $T$, based on the available
data ($v$ and $w$) obtained from an ordinary 2D PIV system. Therefore, this
work aims to find an innovative way to predict the unavailable parameters
mentioned earlier using existing flow fields.

Deep learning (DL) algorithms are rapidly developing and extensively used in
various fields [16, 17]. Recently, DL has been widely utilized in fluid
dynamics, benefiting from its capability of highly nonlinear mapping [18]. DL
is of great interest in the following problems about turbulent flows: temporal
flow data generation based on turbulent modeling [19, 20]; fluid flow
simulation [21]; reduced-order modeling [22, 23]; Prediction of turbulent flow
based on the information from previous temporal data [24, 25, 26]; super-
resolution reconstruction of turbulent flow [27, 28, 29, 30]. Another
important application of DL in fluid dynamics is mapping parameters of the
flow field, namely, mapping some parameters obtained from a fluid flow to the
other different parameters, which can be expressed as

$f:P_{a}\rightarrow P_{u},$ (1)

where $P_{a}$ is termed the set of available or already-existing parameters,
$P_{u}$ is the set of unavailable parameters, and $f$ is the mapping rule
taking the elements in $P_{a}$ to the one in $P_{u}$. The $f$ of note is the
complex non-linear functions hidden in deep learning neural network, which are
learned by the DL model through training. Guastoni et al. [31] applied
convolutional neural network models to map the wall-shear-stress components
and the wall pressure to the 2D instantaneous velocity-fluctuation fields at
different wall-normal locations in a turbulent open-channel flow. Jouybari et
al. [32] designed a multilayer perceptron type neural network to get the
mapping of various rough surface statistics to equivalent sand-grain height.
Lee et al. [33] proposed a transfer learning method based on empirical
correlations to predict the drag force on the statistics of rough surfaces.
Moreover, the mapping that takes local wall-shear stresses and wall pressure
fluctuations to local heat flux was proved to be possible by Kim and Lee [34].

Furthermore, generative adversarial networks (GAN) proposed by Goodfellow et
al. [35] have been used in various fields, e.g., the work of Shamsolmoali et
al. [36] for synthetic image generation. Other powerful network based on GAN
are super-resolution generative adversarial network (SRGAN) [37], and the
variant of SRGAN, i.e., enhanced SRGAN (ESRGAN) [38]. Recently, GAN-based DL
models have been rapidly implemented to solve turbulent flow problems,
particularly resolution reconstruction problems. Benefiting from deeper layers
and special loss functions (e.g., perceptual loss), SRGAN and ESRGAN show
better performance on the resolution reconstruction of flow fields than basic
other DL models [39]. Moreover, Yousif et al. [40] proposed a novel 2D3DGAN
based on the mechanism of ESRGAN, which could reconstruct the 3D turbulent
flow fields from 2D velocity fields. Regarding flow field parameters mapping,
Guemes et al. [41] evaluated the performance of SRGAN for reconstructing
turbulent-flow quantities from coarse wall measurements. The results showed
that the SRGAN could capture and rebuild the large-scale structures of the
flow even for the most complicated cases. Undoubtedly, the GAN-based DL models
also have the potential to tackle more turbulence-related problems.

In this article, we attempt to predict unavailable parameters ($P_{u}$) from
existing available parameters ($P_{a}$) using a deep learning model that can
help find a mapping rule between $P_{u}$ and $P_{a}$. A novel GAN-based model
is proposed, termed mapping-generative adversarial network (M-GAN), and is
applied to map available parameters to unavailable parameters. Two cases of
fluid flows are used to examine the performance of M-GAN. The 2D Rayleigh-
Bénard (RB) flow at Rayleigh number ($Ra$) =${10}^{8}$ is used as a
demonstration, where $u$ and $v$ are regarded as available parameters;
meanwhile, $T$ is the unavailable parameter needing to be predicted.
Furthermore, turbulent channel flow data at $Re_{\tau}$ = 180, 395, and 550,
are applied to test the M-GAN model sufficiently. 2D flow fields at
$Re_{\tau}$ =180 and 550 are the data for training and testing for M-GAN,
i.e., $v$, and $w$ are input data, and $u$ and $p$ are output data. Then
channel flow at $Re_{\tau}$ =395, which has never participated in the training
process, is used as an additional testing case to estimate the interpolation
ability of M-GAN. Note that a label information generator (LIG) is combined
with M-GAN to train and test the turbulent channel flow cases. The role of LIG
is to help M-GAN decide to output $u$ or $p$ when the DL model can only
receive the same information of $v$ and $w$. All the mentioned flow data are
generated using DNS.

The remainder of this paper is written as follows. Section II presents the
generation of flow data using DNS. The design of M-GAN is introduced in the
methodology part (section III), where the training and testing procedures are
also included. The results, including instantaneous contours and turbulence
statistics, are plotted and discussed in section IV. Finally, section V
summarizes the article and proposes potential future research based on this
study.

##  2 Data generation

###  2.1 2D Rayleigh-Bénard flow

In the Rayleigh-Bénard flow case, the governing equations with
incompressibility condition can be expressed as

$\frac{\partial{\bf u}}{\partial t}+{\bf u}\cdot\nabla{\bf
u}=-\frac{1}{\rho}\nabla p+{\nu}{\nabla^{2}}{\bf u}+{\bf F_{b}},$ (2)

$\nabla\cdot{\bf u}=0,$ (3)

As for the temperature field, an advection-diffusion equation is applied as

${\frac{\partial{T}}{\partial t}+{\bf
u}\cdot\nabla{T}}={{\kappa}{\nabla^{2}}{T}},$ (4)

where ${\bf u}$, $p$, $\rho$, $T$, $\nu$, $\kappa$, and $t$ represent velocity
vector, pressure, density, temperature, kinematic viscosity, thermal
diffusivity, and time, respectively. The Boussinesq approximation is applied
for this flow and the body force ${\bf F}_{b}$ is taken to only depend
linearly on the temperature and to be in the direction of gravity. Besides,
the possible dependencies of density, viscosity, and thermal diffusivity on
temperature are ignored, so these parameters are considered constant.

The simulation is carried out by performing DNS using a well-validated second-
order finite-difference code [42, 11]. One important control parameter of
Rayleigh-Bénard flow is the Rayleigh number, i.e., $Ra$=$\alpha g\Delta
L^{3}/(\nu\kappa)$, where $\alpha$ is the thermal expansion coefficient, $g$
is the acceleration of gravity, and $\Delta$ is the temperature difference
between the upper and bottom surface with a depth $L$. The no-slip and
constant temperature boundary conditions are applied for the bottom and top
plates, and periodic boundary condition is assigned to the horizontal
direction. The details of this case are listed in Table 1. More information on
the simulation of Rayleigh-Bénard flow can be found in these papers [11, 43].

| $Ra$ |  $N_{x}\times N_{y}$ |  $\mathrm{\Gamma}$ |  $Pr$ |  $Nu$ |  $\Delta t^{+}$ |
---|---|---|---|---|---|---|---
| $10^{8}$ |  $512\times 256$ |  $2$ |  $1$ |  $26.1$ |  $0.1$ |

Table 1: Simulation parameters of Rayleigh-Bénard flow. $N_{x}$ and $N_{y}$
are the grid resolution in the horizontal and vertical directions.
$\mathrm{\Gamma}$ is the aspect ratio, $\mathrm{\Gamma}$=$W/L$, where $W$ and
$L$ are the width and depth of the domain. $Pr$ and $Nu$ are the Prandtl
number and Nusselt number. $\Delta t^{+}$ is the dimensionless time step of
the simulation. The superscript $``+"$ represents that the quantity is
nondimensionalized by $u_{\tau}$ and $\nu$.

In this case, 9000 snapshots are used as training data, while 3000 snapshots
are used to test the trained model. To save the computational expense, all the
data are interpolated to reduce the resolution from 512$\times$256 to
256$\times$128.

###  2.2 Turbulent channel flow

In the turbulent channel flow case, the momentum equation for an
incompressible viscous fluid is expressed as

$\frac{\partial{\bf u}}{\partial t}+{\bf u}\cdot\nabla{\bf
u}=-\frac{1}{\rho}\nabla p+{\nu}{\nabla^{2}}{\bf u}.$ (5)

The open-source CFD finite-volume code OpenFOAM-5.0x is applied to perform the
DNS calculation. As mentioned earlier, the current study has used three
different friction Reynolds numbers, i.e., $Re_{\tau}$ = $u_{\tau}\delta/\nu$
= 180, 395, and 550, where $u_{\tau}$ is the friction velocity, and $\delta$
is half of the channel height. The detailed simulation parameters of each
$Re_{\tau}$ are listed in Table 2. The periodic boundary condition is used in
the streamwise ($x$) and spanwise ($z$) directions. The no-slip boundary
condition is applied to the upper and lower walls of the channel. The obtained
turbulence statistics have been validated by comparing them with the results
from Kim et al.[2] and Moser et al.[3].

| $Re_{\tau}$ |  $L_{x}\times L_{y}\times L_{z}$ |  $N_{x}\times N_{y}\times N_{z}$ |  $\Delta x^{+}$ |  $\Delta z^{+}$ |  $\Delta y_{w}^{+}$ |  $\Delta y_{c}^{+}$ |  $\Delta t^{+}$ |
---|---|---|---|---|---|---|---|---|---
| $180$ |  $4\pi\delta\times 2\delta\times 2\pi\delta$ |  $256\times 128\times 256$ |  $8.831$ |  $4.415$ |  $0.63$ |  $4.68$ |  $0.113$ |
| $395$ |  $4\pi\delta\times 2\delta\times 2\pi\delta$ |  $385\times 257\times 385$ |  $12.553$ |  $6.277$ |  $0.541$ |  $5.115$ |  $0.023$ |
| $550$ |  $4\pi\delta\times 2\delta\times 2\pi\delta$ |  $512\times 336\times 512$ |  $13.492$ |  $6.746$ |  $0.401$ |  $5.995$ |  $0.030$ |

Table 2: Simulations parameters of two turbulent channel flows. $L$ and $N$
are the domain dimension and the number of grids. The superscript $``+"$
represents that the quantity is nondimensionalized by $u_{\tau}$ and $\nu$.
$\Delta y_{w}^{+}$ and $\Delta y_{c}^{+}$ are the spacing near the wall and at
the center of the channel.

We use the pressure implicit split operator algorithm to solve the coupled
pressure momentum system. We discretize the convective fluxes using a second-
order accurate linear upwind scheme. Additionally, all the discretization
schemes employed in the simulations, including convective fluxes, have second-
order accuracy. We maintain the maximum Courant–Friedrichs–Lewy (CFL) number
below 1 during the simulations to ensure stability.

Training and testing data of a $y-z$ plane from the 3D channel flow domain are
collected for both $Re_{\tau}$ = 180 and 550. 10,000 snapshots of flow
velocity and pressure fields are obtained. The data of the flow at $Re_{\tau}$
=395 and 550 are interpolated to match the grid size of the DNS data of the
flow at $Re_{\tau}$ = 180, which is 128 and 256 in $y$ and $z$ directions,
respectively. By applying this data processing technique, we can ensure that
the flow data at all three Reynolds numbers are suitable inputs for the model.

Figure 1: Architecture of M-GAN: (a) generator and (b) discriminator.

##  3 Methodology

###  3.1 M-GAN

GAN, developed by Goodfellow et al. [35], has been successfully applied to
tackle the image generation problem. Subsequently, based on the traditional
GAN, Ledig et al. [37] and Wang et al. [38] proposed SRGAN and ESRGAN,
respectively. SRGAN and ESRGAN have excellent performance in recovering high-
resolution images from multiple images. In the turbulence field, SRGAN and
ESRGAN have proven their capability to reconstruct the super-resolution data
of turbulent flows [28, 29, 30]. Considering the previous great mapping
ability of ESRGAN for resolution reconstruction, this study intends to develop
the M-GAN to map some flow field parameters to other parameters.

Like the traditional GAN, M-GAN comprises two parts, i.e., generator ($G$) and
discriminator ($D$). As shown in Fig. 1 (a), $G$ mainly contains deep
convolutional neural networks (CNNs)-based multi-layers termed residual in
residual dense blocks (RRDBs) [44]. The sub-elements of RRDBs are dense blocks
(DBs) consisting of convolutional and leaky ReLU activation function layers.
Skip connection is applied in RRDBs and DBs to avoid vanishing gradient. In
addition, $G$ comprises a convolution-pooling part (CPP) following the input
and a multiscale part (MSP) preceding the output. The CPP module includes
several convolutional and max pooling layers that extract input data
information and compress the data size. Conversely, the MSP module uses
upsampling layers and convolutional layers with various-sized filters to
extract information from the output of RRDBs and construct the output data.
More details of CPP and MSP are listed in Table 3 and Table 4, respectively.
Besides, the label information generator (LIG) is added to the input of $G$.
LIG is used in specific situations, such as $v$ and $w$ are the input, and $u$
or $p$ are the output. Note that LIG can provide information to $G$ for
determining which parameters to output. Table 5 shows the detailed structure
of LIG. The label is passed into a dense layer to generate label information.
After reshaping, the label information is concatenated to $p_{a}$ and input
into the next process. Notably, the use of LIG is optional. Sometimes LIG can
be neglected when the outputting only includes one parameter in $P_{u}$. Fig.
1 (b) shows the detailed structure of $D$. The Major component of $D$ is CBL,
including three different layers, i.e., convolutional, batch normalization,
and LReLU activation function layers.

Table 3: CCP structure.

Type of layers | Shape
---|---
Input ($P_{a}$) | (128, 256, 2)
Conv2D.(3, 3) | (128, 256, 56)
MaxPooling(2, 2) | (64, 128, 56)
Conv2D.(3, 3) | (64, 128, 56)
MaxPooling(2, 2) | (32, 64, 56)
concat(Output of last layer and LIG) | (32, 64, 64)

Table 4: MSP architecture.

First branch | Second branch | Third branch
---|---|---
Conv2D.(3, 3) | Conv2D.(5, 5) | Conv2D.(7, 7)
UpSampling(2, 2) | UpSampling(2, 2) | UpSampling(2, 2)
Conv2D.(3, 3) | Conv2D.(5, 5) | Conv2D.(7, 7)
LeakyReLU | LeakyReLU | LeakyReLU
UpSampling(2, 2) | UpSampling(2, 2) | UpSampling(2, 2)
Conv2D.(3, 3) | Conv2D.(5, 5) | Conv2D.(7, 7)
LeakyReLU | LeakyReLU | LeakyReLU
Add (first branch, second branch, third branch)

Table 5: LIG structure.

Type of layers | Shape
---|---
Input (label inf.) | (1)
Dense | (32×64×8)
Reshape | (32, 64, 8)

The overall algorithm is described as follows. First, $P_{a}$ and label
information (here, we assume to have the label information) are input to $G$.
The generated unavailable parameters ($p_{u}^{{}^{\prime}}$) are obtained from
the output of $G$. Then, both $P_{u}$ (real one) and $P_{u}^{{}^{\prime}}$
(generated one) are fed into $D$. The non-transformed discriminator value
$D_{Ra}$ is calculated from the output of $D$, which is formulated as

$D_{Ra}(P_{u},P_{u}^{{}^{\prime}})=\sigma\left(C\left(P_{u}\right)-\mathbb{E}_{P_{u}^{{}^{\prime}}}\left[C(P_{u}^{{}^{\prime}})\right]\right),$
(6)

$D_{Ra}(P_{u}^{{}^{\prime}},P_{u})=\sigma\left(C\left(P_{u}^{{}^{\prime}}\right)-\mathbb{E}_{P_{u}}\left[C(P_{u})\right]\right),$
(7)

where $\sigma$ is the sigmoid function, and $\mathbb{E}$ is the average
calculating operator. Similar to ESRGAN, $D$ is designed to predict the
probability that $P_{u}$ is relatively more realistic than
$P_{u}^{{}^{\prime}}$. Using equations 6 and 7, the discriminator loss
$L_{D}^{Ra}$ and adversarial loss $L_{G}^{Ra}$ are calculated. The formulation
of the discriminator loss ($\mathcal{L}_{D}$) is expressed as

$\mathcal{L}_{D}=L_{D}^{Ra}=-\mathbb{E}_{P_{u}}\left[{\rm
log}(D_{Ra}(P_{u},P_{u}^{{}^{\prime}}))\right]-\mathbb{E}_{P_{u}^{{}^{\prime}}}\left[{\rm
log}(1-D_{Ra}(P_{u}^{{}^{\prime}},P_{u}))\right].$ (8)

Besides, the adversarial loss is expressed as

$L_{G}^{Ra}=-\mathbb{E}_{P_{u}}\left[{\rm
log}(1-D_{Ra}(P_{u},P_{u}^{{}^{\prime}}))\right]-\mathbb{E}_{P_{u}^{{}^{\prime}}}\left[{\rm
log}(D_{Ra}(P_{u}^{{}^{\prime}},P_{u}))\right].$ (9)

As mentioned before, $D$ predicts the probability that the $P_{u}$ is
relatively more realistic than the $P_{u}^{{}^{\prime}}$ in the training
process. When $P_{u}$ is more realistic than $P_{u}^{{}^{\prime}}$,
$D_{Ra}(P_{u},P_{u}^{{}^{\prime}})$ tends to be 1 and
$D_{Ra}(P_{u}^{{}^{\prime}},P_{u})$ tends to be 0. Thus, $L_{D}^{Ra}$ will
decrease and tend to be 0. Inversely, $G$ plays a role in generating more
realistic $P_{u}^{{}^{\prime}}$ tending to be similar to $P_{u}$. In this
situation, $D_{Ra}(P_{u},P_{u}^{{}^{\prime}})$ will decrease from 1 to 0, and
$D_{Ra}(P_{u}^{{}^{\prime}},P_{u})$ will increase from 0 to 1, which makes
$L_{G}^{Ra}$ tend to be 0. The above description is the adversarial process
between $G$ and $D$. In this process, $G$ and $D$ compete with each other.
Meanwhile, they are also promoted mutually.

Figure 2: Architecture of the AE-FE.

In addition to adversarial loss, the loss function of G ($\mathcal{L}_{G}$)
contains some extra parts, i.e., pixel loss ($L_{pixel}$), perceptual loss
($L_{perceptual}$). $L_{pixel}$ is the pixel-based error, which is calculated
by comparing the value difference between $P_{u}$ and $P_{u}^{{}^{\prime}}$.
Different from $L_{pixel}$, $L_{perceptual}$ is the error between the features
of $P_{u}$ and $P_{u}^{{}^{\prime}}$, which are extracted by the feature
extractor (FE). Regarding the original ESRGAN, VGG19 [45], a very deep
convolutional network, is used as FE. However, VGG19 is designed to process
image data with three channels, i.e., RGB. In other words, VGG19 can only
receive the flow data with three components like $u$, $v$, and $w$. In this
study, $P_{u}$ and $P_{u}^{{}^{\prime}}$ only contains one component, so
applying VGG19 to extract features is unsuitable. This study develops an
autoencoder-based DL algorithm as a FE (AE-FE) to replace VGG19. FE is a pre-
trained model using $P_{u}$ data. As shown in Fig. 2, after the $P_{u}$ are
input the AE-FE, various features data with different sizes are output from
several convolutional layers. The above extra loss terms are computed using
the Mean-squared error (MSE). The total loss function of the $G$ is expressed
as

$\mathcal{L}_{G}=L_{G}^{Ra}+\lambda L_{pixel}+L_{perceptual},$ (10)

where, $\lambda$ is the coefficient used to balance the magnitude of various
loss terms, which is set to be 1000.

This study uses the open-source library TensorFlow 2.3.0 to implement the DL
model. The customized sample Python code for the proposed M-GAN is available
on the web page (https://fluids.pusan.ac.kr/fluids/65416/subview.do).

###  3.2 Training and testing of M-GAN

Figure 3: Schematic of the 2D Rayleigh-Bénard flow case.

For the training process of the 2D Rayleigh-Bénard flow case, the input data,
i.e., $P_{a}$ are velocity components $u$ and $v$. Meanwhile, the target data,
i.e., $P_{u}$ is temperature $T$. Firstly, $u$ and $v$ are input into $G$ to
output $P_{u}^{{}^{\prime}}$, with a batch size of 16. Then, generator loss is
calculated using $P_{u}$ and $P_{u}^{{}^{\prime}}$ based on equation 10. The
discriminator loss is calculated by equation 8 through $P_{u}$ and
$P_{u}^{{}^{\prime}}$. After computing the losses, the optimization algorithm
updates the weights of both $G$ and $D$ to minimize losses during the training
period. The adaptive moment estimation (Adam) algorithm is used in this study
for optimization [46]. In addition, the target data, $P_{u}$, is normalized
using the min-max normalization function with a range of 0 to 1 to enhance the
training performance. After training, the $u$ and $v$ values in the test
dataset are fed into trained M-GAN to generate predicted $T$
($P_{u}^{{}^{\prime}}$), which is depicted in Fig. 3.

Figure 4: Schematic of the Channel flow case.

In the training process of the turbulent channel flow case, the input data
comprise wall-normal velocity $v$ and spanwise velocity $w$, while the target
data consist of the streamwise velocity $u$ and pressure $p$. For predicting
$u$, a label of 0 is inputted into LIG, whereas a label of 1 is used for
predicting $p$. In contrast to the 2D Rayleigh-Bénard flow case, we train the
M-GAN using both channel flow datasets at $Re_{\tau}$ = 180 and 550 together.
The training details and methods are similar to the 2D Rayleigh-Bénard flow
case mentioned above, so the detailed explanation is neglected here. As shown
in Fig. 4, we input $v$ and $w$ from channel flow and the corresponding label
into the trained M-GAN to predict the corresponding $u$ or $p$.

Here, the channel flow data at $Re_{\tau}$ = 395 mentioned above never attend
the training process of M-GAN, and they are used to evaluate the interpolation
ability of M-GAN. The $v$ and $w$ from channel flow at $Re_{\tau}$ = 395 and
label are inputted into the M-GAN trained by channel flows at $Re_{\tau}$ =
180 and 550 to predict the $u$ or $p$. Here, if the M-GAN trained by channel
flows at $Re_{\tau}$ = 180 and 550 could also predict the unavailable
parameters of channel flow at $Re_{\tau}$ = 395 from its $v$ and $w$, it could
be proved that the proposed M-GAN has good interpolation ability. In other
words, the M-GAN might be appropriate for all the channel flows at a range of
Reynolds numbers ($Re_{\tau}$ = 180 to 550) to predict their unavailable
parameters. The verification of the instantaneous and statistical results from
all the above cases will be presented in due course.

##  4 Results and discussion

###  4.1 2D Rayleigh-Bénard flow

The capability of M-GAN is examined in this sub-section for the case of 2D
Rayleigh-Bénard flow. Fig. 5 shows the predicted instantaneous temperature
field from the mapping of velocity components using M-GAN. The defined
parameters: $t^{+}$ is the dimensionless time, and $T^{+}$is the dimensionless
temperature calculated from $T^{+}$ = $T/T_{\tau}$, where $T_{\tau}$ =
$-\kappa\partial_{y}\langle T\rangle_{x,t}|_{y=0}/u_{\tau}$ is a
characteristic temperature scale like the $u_{\tau}$ for the velocity [47].
From these instantaneous temperature contours, we can find that all the
detailed features of predicted results using M-GAN are consistent with the
corresponding DNS data, even if the intense large-scale rolls (LSRs) keep
moving along x direction with time going.

Figure 5: Predicted instantaneous temperature field of the 2D Rayleigh-Bénard
flow case. Figure 6: Statistical results of the 2D Rayleigh-Bénard flow case:
(a) average temperature profile; (b) probability density function of
temperature field.

Moreover, the mapping ability of M-GAN is further validated by plotting and
comparing the statistical results from the predicted and DNS data. The average
temperature profile along the wall unit $y^{+}$ and the probability density
function (PDF) are plotted in Fig. 6. All the predicted results show
commendable agreement with the results obtained from DNS data indicating that
M-GAN can map the available parameters $u$ and $v$ to unavailable parameter
$T$ of the 2D Rayleigh-Bénard flow.

Although the results show that M-GAN can work well for the RB flow case, the
RB flow case still cannot sufficiently verify the mapping ability of M-GAs,
because Rayleigh-Bénard flow does not have very complex and chaotic behavior
at a low Rayleigh number ($Ra$=$10^{8}$). Thus, the next sub-section will
illustrate a more comprehensive verification using turbulent channel flow
data.

###  4.2 Turbulent channel flow

Figure 7: Predicted instantaneous streamwise velocity and pressure fields of
the turbulent channel flow at $Re_{\tau}$ = 180.

In this sub-section, the mapping capability of M-GAN for taking the available
parameters $v$ and $w$ to unavailable parameters $u$ and $p$ is examined. Fig.
7 and Fig. 8 show the predicted instantaneous streamwise velocity and pressure
fields of the turbulent channel flow at $Re_{\tau}$ = 180 and 550. The
predicted results of both cases agree well with related the DNS results, where
most fluctuating features of predicted results are consistent with the DNS
data. Through observation by comparing the results based on different
parameters and $Re_{\tau}$, we can find that the results of channel flow at
$Re_{\tau}$ = 180 have more accuracy as compared to the results from the
results of the flow at $Re_{\tau}$ = 550. Besides, when comparing the results
from the same channel flow case, the predicted $u$ field is better than the
results from $p$. Fig. 9 also indicates this observation, where root-mean-
square (RMS) velocity and pressure fluctuations ($u_{rms}^{+}$ and
$p_{rms}^{+}$) are plotted. The reason for that could be attributed to the
fact that pressure fields are more chaotic than streamwise velocity fields,
and channel flow at $Re_{\tau}$ = 550 is more complicated than the one at
$Re_{\tau}$ = 180.

Figure 8: Predicted instantaneous streamwise velocity and pressure fields of
the turbulent channel flow at $Re_{\tau}$ = 550. Figure 9: Root-mean-square
profiles of streamwise velocity and pressure fluctuations: (a) and (c) are
$u_{rms}^{+}$ and $p_{rms}^{+}$ of the channel flow at $Re_{\tau}$ = 180; (b)
and (d) are $u_{rms}^{+}$ and $p_{rms}^{+}$ of the channel flow at $Re_{\tau}$
= 550. Figure 10: Probability density functions of the streamwise velocity and
pressure fields of channel flows: (a) and (c) are PDF($u$) and PDF($p$) of the
channel flow at $Re_{\tau}$ = 180; (b) and (d) are PDF($u$) and PDF($p$) of
the channel flow at $Re_{\tau}$ = 550. The isoline levels are in the range of
10% – 90% of the maximum PDF with an increment of 20%.

Figure 10 shows the probability density function (PDF) plots of the $u$ and
$p$ fields. Note that a larger probability means that the parameter magnitude
appears more frequently and dominates the flow field to a greater extent. The
PDF of predicted velocity and pressure fields can basically match the
reference results. However, the results obtained from channel flow at
$Re_{\tau}$ = 550 show less accuracy than the results from channel flow at
$Re_{\tau}$ = 180. At the same time, the deviation shown in PDF($p$) is much
larger than the one in PDF($u$), which is consistent with the previous
discussion in the first paragraph of this subsection.

Figure 11: Spanwise energy spectra of the predicted streamwise velocity and
pressure from channel flow at $Re_{\tau}$ = 180. Figure 12: Spanwise energy
spectra of the predicted streamwise velocity and pressure from channel flow at
$Re_{\tau}$ = 550

In Fig. 11 and Fig. 12, the spanwise energy spectrum of $u$ and $p$
($E_{uu}^{+}(k_{z}^{+})$ and $E_{pp}^{+}(k_{z}^{+})$) at various wall
distances are plotted to evaluate the ability of M-GAN to predict the $u$ and
$p$ fields with realistic behavior. In these two figures, the results of
$E_{uu}^{+}(k_{z}^{+})$ have more accuracy than the results of
$E_{pp}^{+}(k_{z}^{+})$). Besides, with the $y^{+}$ increasing, the deviation
of the energy spectrum also increases. This consistent with previously
discussed instantaneous contours and RMS results. Another interesting thing
observed from the energy spectrum results is that in the flow at $Re_{\tau}$ =
180, the predicted energy in high wavenumbers is higher than the DNS results.
On the contrary, the predicted energy in high wavenumbers is mostly lower than
the DNS results for the channel flow case at $Re_{\tau}$ = 550.

With the above discussion, we can observe that the M-GAN can predict the
unavailable parameters from the corresponding available data based on flow
data used for training. However, the interpolation ability of the model still
needs to be further investigated, In other words, whether the model can work
in the channel flow case at a Reynolds number between $Re_{\tau}$ = 180 and
550 or not. Regarding this issue, a channel flow at $Re_{\tau}$ = 395, which
never joins the training, is used for the interpolation ability test. Same as
the previous operation, $v$ and $w$ are regarded as available data to predict
$u$ and $p$ using pre-trained M-GAN trained by the data of channel flows at
$Re_{\tau}$ = 180 and 550.

Fig. 13 shows the predicted instantaneous streamwise and pressure fields for
the flow at $Re_{\tau}$ = 395 . Moreover, Fig. 14 shows the statistical
results of RMS velocity and pressure fluctuations. Although there are larger
errors than in previous results, especially the $p$ field cannot be predicted
well, the results still are acceptable. Generally, most of the features of
flow fields can be constructed and $u_{rms}^{+}$ has a relatively good
agreement with the DNS result. This indicates that the proposed M-GAN has a
good interpolation ability to work successfully in a case different from the
training data set. In other words, M-GAN can learn the mapping function
between $P_{a}$ and $P_{u}$.

Figure 13: Predicted instantaneous streamwise velocity (a) and pressure (b)
fields of the turbulent channel flow at $Re_{\tau}$ = 395. Figure 14: Root-
mean-square profile of the streamwise velocity (a) and pressure (b)
fluctuations of the turbulent channel flow at $Re_{\tau}$ = 395.

##  5 Conclusions

We proposed an efficient deep learning-based model, namely, M-GAN to predict
unavailable flow parameters from existing ones. Two cases of classical fluid
flows were used to validate the performance and interpolation ability of
M-GAN.

First, the 2D Rayleigh-Bénard flow was used as a demonstrating case to show
the capability of M-GAN for mapping available parameters $u$ and $v$ to
unavailable parameters $T$. In this case, the results indicated that the M-GAN
could indeed predict the temperature field of the 2D Rayleigh-Bénard flow from
its velocity field.

Then, two turbulent channel flows at $Re_{\tau}$ = 180 and 550 were used
together to train M-GAN and validate the predicting performance of M-GAN. In
this case, wall-normal velocity $v$ and spanwise velocity $w$ were regarded as
available data and passed into M-GAN to predict unavailable parameters that
are streamwise velocity $u$ and pressure $p$. Here, LIG was applied to help
M-GAN decide which parameter ($u$ or $p$) should be output when the input
layer received the same $v$ and $w$. The prediction’s instant contours and
turbulence statistics were exhibited and compared to DNS data. The results
illustrated that M-GAN successfully predicted $u$ and $p$ from the
corresponding $v$ and $w$ in both turbulent channel flow cases. However, the
pressure field and the flow fields of the channel flow at $Re_{\tau}$ = 550
were relatively more difficult to be predicted because of their more chaotic
turbulence characteristics. Finally, turbulent channel flow at $Re_{\tau}$ =
395, which was never used to train the M-GAN, was employed to test the
interpolation ability of M-GAN. As the results showed, the pre-trained M-GAN,
which was based on the training data set of channel flows at $Re_{\tau}$ = 180
and 550, also could predict the unavailable parameters $u$ and $p$ from $v$
and $w$ from channel flow data at $Re_{\tau}$ = 395. Thus, it was proved that
M-GAN learned the mapping law from $v$ and $w$ to $u$ and $p$ of the channel
flows at a range of Reynolds numbers ($Re_{\tau}$ = 180 to 550). In other
words, M-GAN showed a good interpolation ability for a specific range of
Reynolds numbers. In conclusion, this article has shown that M-GAN can predict
unavailable flow parameters from existing ones, i.e., velocity fields.
Besides, with good interpolation ability, M-GAN can work, even for turbulent
flows with a range of Reynolds number.

## 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-2022-CRE-0282).

## Data Availability

The data that supports the findings of this study are available within this
article.

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