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

INTERPRETATION AND PREDICTION FOR PRANDTL NUMBER EFFECTIN TURBULENT HEAT TRANSFER USING GENERATIVE ADVERSARIAL
NETWORKS

Hyojin Kim
Department of Mechanical Engineering
Yonsei University
50 Yonsei-ro, Seodaemun-gu, Seoul 03722
gywls072@yonsei.ac.kr

Junhyuk Kim
Department of Mechanical Engineering
Yonsei University
50 Yonsei-ro, Seodaemun-gu, Seoul 03722
junhyuk6@yonsei.ac.kr

Changhoon Lee
School of Mathematics and Computing
Yonsei University
50 Yonsei-ro, Seodaemun-gu, Seoul 03722
clee@yonsei.ac.kr

ABSTRACT

the shear stress and heat flux is observed. Many studies using direct numerical simulation (DNS) have been conducted
to investigate the effect of Prandtl number (Kasagi & Ohtsubo,
1993; Kawamura et al., 1998), but the observation of local heat
flux with Pr has not been sufficiently performed because it
is mostly limited to the conventional statistical approach. In
fact, the turbulent transport mechanism of heat and momentum near the wall occurs very locally and intermittently due tothe role of streamwise vortical structures. The dissimilarity between heat flux and streamwise shear stress is obviously foundin some regions, although there is a high correlation between
them (Abe et al., 2004).

We propose a interpretable deep learning model that embedded effect of physical parameter in turbulence. Turbulence
is a very complex flow, and the analysis of relationships between turbulent variables remains a fundamental challenge.
Recently, studies applying deep learning are being conductedin attempts to analyze turbulence. Deep learning can extract
physics features in data, which is turbulent analysis model tounderstand the physical relationship between variables in turbulent flows. In this study, we consider turbulent heat transfer to extract and analyze the effect of Prandtl number (Pr) in
the data. The deep learning model uses conditional generative
adversarial networks (cGAN) with decomposition algorithm.
Our model predicts surface heat flux with various Pr from wall
shear stresses in channel flow. The predicted surface heat flux
reflected the characteristics with respect to Pr well, and also
was statistically very similar to DNS. We analyzed the spatial
nonlinear relationship between wall shear stresses and surface
heat flux for Pr through gradient maps of trained our model.
Furthermore, for analysis of effect of Prandtl number, we observed decomposed field into universal and Pr-dependent features based on turbulent data sets. Through interpretation of
the deep learning model, it is possible to understand the physical interaction between variables, which can help to develop aturbulence model considering physics.

In this study, we focus on the revealing the complicate relation between the local heat flux and wall-shear stresses with
the Prandtl number-effect through deep learning (DL). Unlike the traditional approach, DL can find a nonlinear mapping
function between variables and predicts instantaneous fields
with high accuracy, showing excellent performance in many
turbulence problems (Kim & Lee, 2020; Kim et al., 2021;
Güemes et al., 2019, 2021). Although DL performed well in
discovering interrelationship between input and output in various turbulence problems, but there are still unresolved fundamental questions about how DL learns turbulence, what characteristics of turbulence DL learns, and which information isessential for prediction. To take a step forward, understanding
and interpretation of DL is research of substance. Recently,
there are a few attempts to interpret DL with embedded features of turbulence (Kim & Lee, 2019; Lu et al., 2020; Jagodinski et al., 2020), but it is still difficult to provide sufficientanswers to these fundamental questions. Therefore, we apply
deep learning to predict the surface heat flux with various Pr
from wall shear stress, and further investigate the physical phenomena between local heat flux and shear stress through theinterpretation of the trained model with the embedded of the
Prandtl number effect. The interpretable DL model could help
to provide an framework in which discover unknown physical phenomena in the data. In addition, interpretation of DL
would play a very important role in the development and per-INTRODUCTION
Turbulent heat transfer is an important phenomena observed in science and engineering fields, and there are the examples such as gas turbines, heat exchangers and nuclear reactors. The close analogy between temperature fluctuations andmomentum is well known, resulting in a high similarity between heat flux and shear stress at the wall (Antonia et al.,
1988; Kim & Moin, 1989; Kasagi et al., 1992). However, thedistribution of heat flux is highly depending on Prandtl number (Pr = ν/α; ν and α are kinematic viscosity and thermal
diffusivity, respectively.), and the strong nonlinearity between

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1.

Architecture of conditional generative adversarial networks (cGAN) with decomposition algorithm.

Deep learning model

formance improvement of itself and for providing the guidanceof DL construction (e.g., hyper parameter optimization).

Conditional generative adversarial network (cGAN) is
used for prediction and interpretation of surface heat flux withvarious Pr. In figure 1, cGAN consists of two networks, generator (G) and discriminator (D), and is trained through competitive learning of these two networks. The input data is used
for the discriminator as conditioning, and this constraint allows the generator to generate output image dependent on the
input one. Finally, we can obtain a generator that generates afake image similar to the real image while being dependent
on the input data. This process can be described as a min/max
problem, and the loss function used for training is as follows:

METHODOLOGY
Data collection
To collect datasets for training of the DL model, we performed DNS of turbulent channel flow with passive temperature. The mean flow in streamwise direction is driven by a constant pressure gradient. The constant temperature and no-slip
conditions are imposed at both walls, and periodic boundary
conditions are used in the horizontal directions. The governing equations are the continuity equation, the incompressible
Navier–Stokes equation and the energy equation, as follows:

∂ ui
= 0,
∂ xi
∂ ui
∂ ui
∂p
1 ∂ 2 ui
+uj
=−
+
,
∂t
∂xj
∂ xi Reτ ∂ x j ∂ x j
∂T
∂T
1
∂ 2T
+uj
=
,
∂t
∂xj
PrReτ ∂ x j ∂ x j

min max Ey∼PY [log D(y|x)] + Ex∼PX [log(1 − D(G(x)|x))], (4)G

D

In this study, x is the streamwise and spanwise wall shear stressand Prandtl number, and y is the surface heat flux for Pr. Additionally, for in detail analysis of the Pr-effect, we applied adecomposition algorithm to cGAN that can separate surface
heat flux into universal features qU
w and Pr-dependent features
qPw . For the extraction of the Pr-effect, the generator is divided
into a universal generator GU and the parameter effect generator GP . GU generates universal features that have common
characteristics of surface heat flux for all Pr from wall shear
stresses. GP extracts Pr dependent features with properties for
the Pr-effect from the wall shear stress and Prandtl number. FiP
nally, the surface heat flux (qw = qU
w + qw ) can be obtained as
the sum of the decomposed features, and the total loss function
for training of the model is as follows.

(1)
(2)
(3)

The equations are nondimensionalized by channel half widthδ , friction velocity uτ and temperature difference ∆T between top and bottom walls. x1 (x), x2 (y) and x3 (z) denotestreamwise, wall-normal and spanwise direction, respectively.
The corresponding velocity components are u1 (u), u2 (v) andu3 (w), respectively. The dimensionless parameters are friction
Reynolds number(Reτ = uτ δ /ν), which is fixed at 180 andmolecular Prandtl number, ranging from 0.1 to 7.
Spatial discretization in the horizontal direction and thewall-normal direction used a pseudo-spectral method using Fourier expansion and a central difference scheme, respectively. The second-order Adams–Bashforth and Crank–
Nicolson schemes were applied for the temporal integrationof the nonlinear and viscous terms, respectively. The domain
size is 4πδ × 2δ × 2πδ for all Pr, and the number of grid
points for Pr = 0.1 − 1 and Pr = 2 − 7 is 128 × 129 × 128
and 192 × 129 × 192, respectively.
The collected data are wall-shear stresses and surface heat
flux of various Pr, and are divided into training, validation, andtesting data. The testing data are sufficiently decorrelated fromthe training data.

Ltotal = LcGAN + λ1 Lmse + λ2 LPr ,

(5)

with
N

Lmse =

1 p∑ (Gi (x, Pr) − yi )2 ,N p i=1

(6)

N

LPr =

1 p P
∑ (Gi (x, GU (x), Pr))2 ,N p i=1(7)

where the first and second terms on the right-hand side in equation (5) are the cGAN loss and mean squared loss (MSE), respectively. The last term is the physical parameter loss, which212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2.

Instantaneous surface heat flux for Pr, which was not used in training, obtained from wall shear stresses using cGAN.

can decompose the surface heat flux into universal and Prdependent features. λ1 and λ2 are 10 and 200, respectively.
In equation (6) and (7), N p is the number of inputs, and G(x)denotes the predicted surface heat flux through the generator.
GP (x) and GU (x) are a Pr-dependent feature and the universal feature, respectively. In the training process, the parametersof the generator (G) are trained in the direction of minimizing
Ltotal , and the parameters of the discriminator (D) are trainedin the direction of maximizing LcGAN .

For
qualitatively
evaluationofthetrainedmodel, CNN was considered as a comparativemodel.D First, we
E examined the correlation coefficient
00DL(R = qwDNS qwDL /(σ (qDNS
w )σ (qw )); qw is surface heat
flux, and σ is standard deviation) between generated heat
flux and that of DNS in table 1. Both cGAN and CNN
were able to predict accurately at approximately 0.9 for all
P except Pr = 7. Although the surface heat flux predicted
through cGAN had a relatively slightly lower correlation
coefficient compared to CNN, cGAN was able to generate a
DNS-like surface heat flux. Additionally, probability density
function(P.d.f.) of surface heat flux predicted through CNN
and cGAN are presented in figure 3. For trained Pr in
figure3(a), CNN generated surface heat flux for all Pr that wasslightly underestimated than that of DNS, but the heat flux
predicted through cGAN was very similar to the distributionof DNS. Furthermore, we tested our model for surface heat
flux for untrained Prandtl number in figure3(b), which is
not considered for learning. Similar to the results for the
trained Prandtl number, the heat flux generated by CNN isunderpredicted than the distribution of DNS. On the other
hand, cGAN was able to predict similar to the results of DNS
for strong heat flux. These results mean that our model cangenerate a heat flux that reflects the physical characteristics of
the Prandtl number well. Although cGAN has slightly lower
prediction accuracy than CNN in point-by-point statistics,
we consider only cGAN with more accurate predictions forstrong local heat fluxes, which are more important in turbulent
heat transfer.

RESULTS AND DISCUSSION
Prediction of surface heat flux with Pr
We apply cGAN with decomposition algorithm for prediction of surface heat flux with Pr from streamwise wall shear
stress, spanwise wall shear stres and Prandtl number in turbulent channel flow. In figure 2, we qualitatively evaluated ourmodel by considering for the surface heat flux with untrained
Pr, which was not used in training. Our model generated surface heat flux field that was slightly overpredicted than one of
DNS for the lowest Pr = 0.1, but the predicted field is similar
to that of DNS. The surface heat flow for Pr = 1 and 7 predicted through cGAN well have the characteristics of the heatflow observed in DNS.

Table 1. Correlation coefficient between target data (DNS
data) and surface heat flux with Pr predicted by the deep learning model.

trained Pr

Untrained Pr

cGAN

CNN

0.2

0.890

0.909

0.71

0.968

0.973

2

0.975

0.979

5

0.922

0.931

0.1

0.802

0.822

0.4

0.918

0.937

1

0.967

0.977

3

0.945

0.958

7

0.876

0.897

Interpretation of Pr-effectFor interpretation of the trained model, we analyzed theunderlying physical properties between wall shear stresses andsurface heat flux with respect to Pr through gradient maps obtained from the trained model. For reliable analysis, we focused on trained Pr, where our model made more accurate
predictions in the previous section. The sensitivity analysis
through gradient map can understand the relation between input and output, which has been applied to image classification
and regression problems (Simonyan et al., 2013; Kim & Lee,2019). The gradient map is defined as follows.

Sk (i, j) =3

∂ N(I)∂ I k (i, j)(8)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4. Average of gradient maps for streamwise wall
shear stress with respect to surface heat flux for Pr obtained
through cGAN.
Figure 3. Probability density function (P.d.f.) of surface heat
flux for (a) trained Pr(= 0.2, 0.71, 2, 5) and (b) untrained Pr(=
0.1, 0.4, 1, 3, 7) obtained through deep learning models.

where Si,k j is gradient map for input I k (i, j) with respect to output N(I). and i and j are indices in x- and z-direction. respectively. k is the input variable.
First, we investigate nonlinear relationship between
streamwise wall shear stress and local heat flux, and the average gradients map between them for Pr are presented in figure 4. The positive peak moves from upstream to downstream
with increasing Pr. It is relevant with that the temperaturefield is more dominated by large scale motion with decreasing Pr, which increases the convection velocity of temperature (Kowalewski et al., 2003). For the average of gradientmaps for spanwise wall shear stress in figure 5, the signifiτ̃τ̃w,zcant patterns are skew symmetricity(Si,w,z
j = −Si,− j ) for all Pr.
The case of Pr = 0.2, 0.71 indicates that the local heat flux canbe enhanced when the vortex pair is downstream or vortices
cross diagonally at the center point, even though the streamwise wall shear stress is weak. The gradient for Pr = 5 is relatively weak, meaning that the surface heat flux at high Pr isweakly influenced by the spanwise wall shear stress comparedwith other Prandtl numbers. In short, we identified the nonlinear relationship between wall shear stresses and local heat flux
for considering the Pr effect.
Next, for analysis of noticeable physical properties, we
confirmed relationship between the decomposed features, universal and Pr-dependent features, and the surface heat flux obtained through the trained model. The two-point correlation
between them are presented in figure 6. The universal feature
had a high correlation coefficient with the surface heat flux for
all Pr at −50 < rx+ < 50. This indicates that universal featuresreflect the common characteristics of surface heat flux regardless of Prandtl number. For relation between surface heat flux
and Pr-dependent feature in figure6(b), when Pr > 1, the highFigure 5. Average of gradient maps for spanwise wall shear
stress with respect to surface heat flux for Pr obtained throughcGAN.

correlation coefficient between them is at rx+ < 0. On the other
hand, a high correlation is observed at rx+ > 0 when Pr < 1.
This result suggests that as the Prandtl number decreases, ourmodel can extract features that well reflect the physical properties shown under the effect of large scale motion. These results
imply that our model can automatically separate the surface
heat flux with respect to Prandtl number without additional
knowledge in the training process, and the decomposed features well reflect the influence on Prandtl number.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(MSIP) (2017R1E1A1A03070282, 2022R1A2C2005538).

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Figure 6. Two-point correlation along streamwise direction(a) between surface heat flux and universal features and (b)surface heat flux and Pr-dependent features.

CONCLUSION
We developed cGAN with a new decomposition algorithm to predict turbulent heat transfer with Pr. The model
was able to generate well not only for the Prandtle numbers
considered in the training process, but also for the thermal flux
for the Prandtl numbers not used in the training. Our model
quantitatively generated a more accurate surface heat flux for
all Pr than that of comparative model, and the predicted heat
flux very similar to the distribution of DNS.
For the interpretation of the trained model, we investigate
gradient maps for wall shear stresses with respect to surface
heat flux. Through the gradient map, we confirmed the nonlinear correlation between input and output well for Prandtl
number, and our model well reflected the effect observed in
the surface heat flux with Pr. Furthermore, our model can extract the effect of physical parameter based on turbulence data
without detailed knowledge, allowing analysis of the physicalparameter effect and the relationship between turbulence variables. Our framework would be extended to analyze effect ofother physical parameters such as Reynolds number, discoverphysical phenomena in data, and understand deep learning of
turbulent flows

Acknowledgments
This work was supported by National Research Foundation of Korea (NRF) grants funded by the Korean government5