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

NON-INTRUSIVE SENSING IN TURBULENT BOUNDARY LAYERS VIA
DEEP FULLY-CONVOLUTIONAL NEURAL NETWORKS

Luca Guastoni
FLOW, KTH Engineering Mechanics,
Swedish e-Science Research Centre (SeRC)
SE-100 44 Stockholm, Sweden
guastoni@mech.kth.se

Alejandro Güemes
Aerospace Engineering Research Group
Universidad Carlos III de Madrid
28911, Leganés, Spain
aguemes@ing.uc3m.es

Arivazhagan G. BalasubramanianFLOW, KTH Engineering Mechanics,
SE-100 44 Stockholm, Sweden
argb@kth.se

Andrea Ianiro
Aerospace Engineering Research Group
Universidad Carlos III de Madrid
28911, Leganés, Spainaianiro@ing.uc3m.es

Stefano Discetti
Philipp SchlatterAerospace Engineering Research Group
FLOW, KTH Engineering Mechanics,
Universidad Carlos III de Madrid
Swedish e-Science Research Centre (SeRC)
28911, Leganés, Spain
SE-100 44 Stockholm, Sweden
sdiscett@ing.uc3m.es
pschlatt@mech.kth.se

Hossein Azizpour
Ricardo Vinuesa
School Elect. Eng. and Comp. Sci., KTH
FLOW, KTH Engineering Mechanics,
Swedish e-Science Research Centre (SeRC) Swedish e-Science Research Centre (SeRC)
SE-100 44 Stockholm, Sweden
SE-100 44 Stockholm, Sweden
azizpour@kth.se
rvinuesa@mech.kth.se

ABSTRACT

can be designed either as an open-loop or closed-loop system.
In the first case, no information is obtained from the flow withmeasurements. This means that the control is prescribed independently from the current flow state. It can be a force, introduced with a periodic motion (Quadrio et al., 2009) or a deformation (Tomiyama & Fukagata, 2013) of the wall, or it canbe obtained through blowing and suction at the wall (Fahland
et al., 2021). In the closed-loop case, the drag is reduced bymodifying the velocity near the wall according to the changesof the velocity within the flow field (Choi et al., 1994). It ishighly desirable to monitor the flow with non-intrusive measurements. Hence, sensors are typically located at the wallin practical applications. One of the downsides of this approach is a poor reconstruction of the flow farther from thewall, a fact that hinders the possibility to design an accurate
control for coherent structures in that flow region. This highlights the need of a reliable estimation of the flow field that canbe performed, for instance, using extended proper orthogonal
decomposition (EPOD, Borée, 2003), which is equivalent tolinear stochastic estimation (LSE, Encinar & Jiménez, 2019),
or transfer functions (Sasaki et al., 2019). Recently, neuralnetwork models have shown excellent results in monitoring
the instantaneous state of the flow using quantities measured
at the wall in numerical simulations. For instance, convolu-

Flow-control techniques are extensively studied in fluid
mechanics, as a means to reduce energy losses related to friction, both in fully-developed and spatially-developing flows.
These techniques typically rely on closed-loop control systems that require an accurate representation of the state of theflow to compute the actuation. Such representation is generally difficult to obtain without perturbing the flow. For thisreason, in this work we propose a fully-convolutional neuralnetwork (FCN) model trained on direct-numerical-simulation(DNS) data to predict the instantaneous state of the flow at different wall-normal locations using quantities measured at thewall. Our model can take as input the heat-flux field at thewall from a passive scalar with Prandtl number Pr = ν/α = 6(where ν is the kinematic viscosity and α is the thermal diffusivity of the scalar quantity). The heat flux can be accurately
measured also in experimental settings, paving the way for the
implementation of a non-intrusive sensing of the flow in practical applications.

INTRODUCTION
One of the possible applications of active flow control inwall-bounded flows is to reduce the skin friction. The control

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
denotes viscous scaling, i.e. in terms of the friction velocity uτ
or the viscous length ℓ∗ = ν/uτ (where ν is the fluid kinematic
viscosity).
Note that the sampled fields include both the initial, transitional part of the flow and the final region affected by thefringe forcing. On the other hand, the neural-network models predict only a portion of the field. Depending on the size,
we can identify two different types of samples, as shown in figure 1: full domain (FD) samples have streamwise and spanwise
lengths of xs /δ0∗ = 600 and zs /δ0∗ = 50, respectively. Here δ0∗is the displacement thickness of the laminar boundary layer at
the inflow. The samples do not include the initial (x/δ0∗ < 200)
and the final region (800 < x/δ0∗ < 1000). When the streamwise length of the samples is reduced to xs /δ0∗ = 300, we referto them as half-domain (HD) samples. The grid points considered in the FD case are Nx,s × Nz,s = 1960 × 320, while for HD
they are 980 × 320.
In order to obtain a sufficiently large number of fields, fivedifferent realizations of the simulation are performed using
five different trip-forcing random seeds. The training dataset
includes 7,474 samples obtained from three of the five simulations. The validation dataset consists of 2,195 samples, takenfrom a separate DNS to avoid unwanted correlations with thetraining dataset. The test dataset is obtained from the remaining simulation, it includes 1,973 samples and the overall sampled time for testing is sufficient to obtain converged turbulence statistics.

tional neural networks (CNNs) have been used to correlate thetwo wall-shear-stress components to the instantaneous wallnormal heat flux (Kim & Lee, 2020). In that case both themeasurements and the predictions are at the same location, i.e.
at the wall. In this work, on the other hand, we consider input measurements at the wall and target flow-state estimation
above the wall. The chosen network architecture is a fullyconvolutional network (FCN), conceptually similar to the one
proposed by Guastoni et al. (2021). However, other architectures have been tested in the literature for the same task, e.g.
super-resolution generative adversarial networks (SR-GANs,
Güemes et al., 2021). Despite machine-learning-based control has been tested in experimental settings with promising
results (Gautier et al., 2015), it is difficult to acquire from experimental facilities the large datasets that are needed to train
neural networks models. For this reason, in this study weperform the training of the networks using the data obtained
from numerical flow simulations. Furthermore, the networks
are trained to predict the flow using measurements that can beobtained in experiments. Thanks to this choice, the trained
network models will be able to be fine-tuned on the experimental data once they are available. In future studies, we will
use the methods developed here in the water tunnel facility of
Universidad Carlos III de Madrid coupling time-resolved IR
thermography (Raiola et al., 2017) with tomographic Particle
Image Velocimetry (Discetti & Coletti, 2018), to assess thefeasibility of deploying this prediction framework in a practical application.

NEURAL-NETWORK MODEL
In this work, we consider several network architectures
for different types of predictions. Based on the quantities sampled at the wall that are provided as input to the neural network, three types of predictions are investigated, as summarized in table 1.

DATASET
The direct numerical simulation (DNS) from which themeasurements and the target output fields are sampled is performed using the pseudo-spectral code SIMSON (Chevalier
et al., 2007). While our previous work (Guastoni et al.,2021) focused on a fully-developed flow, namely a turbulent
open channel flow, in this work we simulate a zero-pressuregradient (ZPG) turbulent boundary layer (TBL). The inflow
condition for the velocity is a laminar profile. A random trip
forcing is applied to trigger the transition to a turbulent boundary layer. A fringe forcing is applied at the outflow in orderto achieve periodicity at the boundary, as requested by the solution method. A passive scalar θ representing the temperature of the fluid is also simulated. We consider Prandtl number Pr = 6 and impose an isothermal wall boundary conditionθ |y=0 = 0.
The choice of a spatially-developing flow implies anadditional degree of complexity in the predictions with respect to the previously-studied channel flow, since the friction Reynolds number Reτ (based on p
the boundary-layer thickness and the friction velocity uτ = τw /ρ, where τw is thewall-shear stress and ρ is the fluid density) increases with thestreamwise location x within the sampled fields. The highest considered Reτ is 396, which is similar to the maximum
value we expect to sample in our experimental setting. We
sample the wall-shear-stress components, as well as the wallpressure. Note that we considered a reference friction velocity at the middle of the computational domain, which implies
that the actual inner-scaled location that is actually sampled
slightly varies along the streamwise direction. However, thevariation is within ±0.1y+ . Furthermore, the flux of a passive
scalar ∂ θ /∂ y is sampled at the wall. The velocity-fluctuation
fields (whose streamwise, wall-normal and spanwise components are denoted u, v and w, respectively) are sampled at fourwall-normal locations: y+ = 15, 30, 50 and 100. Here the ‘+’

Table 1. Summary of inputs and outputs for different prediction types. The inputs are measured at the wall, the outputs are
sampled at a given wall-normal location.

Type

Inputs

OutputsI∂ u/∂ y, ∂ w/∂ y, pu, v, wII∂ θ /∂ y, ∂ w/∂ y, pu, v, wIII∂ θ /∂ yu, v, w

In the first problem, a neural-network model is trained
to predict the velocity-fluctuation fields farther from the wallusing the streamwise and spanwise wall-shear-stress components, as well as the wall-pressure fields. The predictionsof the first problem are denoted as type I. These predictions
use the same inputs and outputs as those of our previous
work (Guastoni et al., 2021). In the second problem thestreamwise wall-shear stress is substituted with the heat flux
field corresponding to the passive scalar at Pr = 6. We refer tothese predictions as type-II predictions. Finally, a third problem is considered, using only the heat-flux field at Pr = 6 as
input (type III). The latter type aims to reproduce our experimental setting, in which we will be able to measure only thewall heat-flux field.

2

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

z/δ0∗

50

0.3

25
0

0

200

400

Transitional

Figure 1.

x/δ0∗

600

800

1000

−0.3

FringeRepresentation of full domain (FD) and half domain (HD) in a sampled streamwise velocity-fluctuation field at y+ = 30.

All the trained models are fully-convolutional neural networks, meaning that the input information is processed bya sequence of convolutional layers, but there are no fullyconnected layers at the end, as it happens for convolutional
neural networks, that are typically employed for classification
tasks on the entire input. The inputs of the FCN model are normalized with the mean and standard deviation computed on the
training samples. The velocity-fluctuation fields predicted by
the FCN are scaled with the ratio of the corresponding rootmean-squared (RMS) values and the streamwise RMS value,
following Guastoni et al. (2021). The scaled output quantities
are indicated with b•.
FCNs allow an accurate reconstruction of the flow fields
thanks to their capability to identify simple features and tocombine them into progressively more complex ones. The
FCN used as reference (Guastoni et al., 2021) is relatively
shallow (i.e. few convolutional layers), with a high numberof kernels per layer. On the other hand, the network architectures tested in this work have a higher number of layerswith fewer kernels per layer. These modifications are designed
to enhance the compositional capabilities of the model without increasing its GPU-memory footprint and computationaltraining cost. Note that the output of each convolutional layer
is slightly smaller than the input, depending on the size of the
convolutional kernel (Dumoulin & Visin, 2016). When a very
high number of layers is used, the output can become significantly smaller than the input. In our work, the size of theoutput is kept constant by modifying the size of the input fieldaccording to the architecture. This is realized by sampling alarger area in the streamwise direction and by padding periodically the field in the spanwise direction. Different models
with a varying number of layers and trainable parameters aretrained in order to identify the best combination of these network architecture parameters. Such comparison is performedon type-III predictions at y+ = 30, in an effort to optimize the
network performance for its experimental use. The FCN istrained using the Adam (Kingma & Ba, 2015) stochastic algorithm to minimize the mean-squared error (MSE) of the predictions with respect to the turbulent fields sampled from the
DNS:
Nx,s

bDNS ) =L(buFCN ; uNz,sFigure 2. MSE in the predictions as a function of the numberof layers of the network for type-III predictions. The filledsquare marker represents the network model used in Guastoni
et al. (2021), while the filled circle is the model proposed inthis work. These two latter models are compared in all thesubsequent analysis.

an intermediate output, in an effort to drive the internal flow
representation of the FCN towards physically-meaningful andinterpretable quantities.

RESULTS AND DISCUSSION
The quality of the network predictions is assessed using
the MSE with respect to the corresponding DNS fields andthe turbulence statistics accuracy. The pre-multiplied twodimensional power-spectral densities are also computed, toassess the amount of energy reconstructed for the different
scales. The comparison of the different network architectures
is based on the MSE because it is the metric used to optimize
the network performance during training. While increasing
the number of trainable parameters in the network does nothave a clear effect on the MSE, figure 2 highlights the correlation between the number layers and the network performancefor type-III predictions. The higher the number of layers, thelower the corresponding mean-squared error in the predictions.

2bDNS (i, j)|uFCN (i, j) − u∑i=1 ∑ j=1 |b
Nx,s Nz,sThe network architecture trained in Guastoni et al. (2021)
and the deepest network in this work have roughly the same
number of trainable parameters, however the latter networkshows a prediction error that is about 50% lower than the former. This result suggests a higher importance of the compositional capabilities of the network over its capacity. Despiteachieving the best performance in the comparison, our deepest network was not selected for the subsequent analysis primarily because of the low ratio between the output and input
field size. In particular, in an experimental setting, we would, (1)where boldface indicates the vectors containing the three velocity components and | • | represents the L2 norm. We refer to the error in the individual components using L(•) for
brevity. For type-III predictions, an additional auxiliary loss
function is also considered: streamwise, spanwise wall-shear
stress and wall-pressure field are predicted by the network as

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
not be able to increase the size of the input fields as done inthis numerical investigation, hence a deeper network would
inevitably result in a smaller output field in which fewer turbulent features are represented. In the remaining of this work,
we present the results obtained with the second best architecture, which has about half of the layers than our deepest one,allowing to maintain a more acceptable output/input size ratio,
while providing a comparable performance in terms of MSE.
Note that this network has a higher number of trainable parameters, having a higher number of kernels per layer than the
deepest network trained.
Given the network architecture, the predictions on the
full-domain and on the half-domain datasets are compared.
One of the advantages of the FCN is that the architecture does
not depend on the size of the input. Either datasets can thenbe used to train the neural-network model. Despite providing
more information per sample during training, the predictionsof the model optimized on the full-domain dataset are less accurate than the ones on the half-domain dataset. This can be
explained by considering that the boundary layer is a spatiallydeveloping flow. This means that each sample contain a range
of Reynolds numbers that need to be predicted. The smaller
the range, the more accurate the predictions. This result is encouraging for the future experimental applications, since theinput data that can be obtained are limited to a small interrogation window (e.g. obtained from particle-image velocimetry),
with a small Reynolds-number range. Because of these observations, the prediction results in the subsequent part of thepaper will be only related to the half-domain dataset.
Once the network architecture and the dataset are chosen, the model is trained three times with different random initialization in order to verify the consistency of the stochastic
optimizations. The reported results show the average performance of the three models. When three inputs are considered(type-I and type-II predictions), the present network is able toreconstruct the non-linear relation between input and output
fields with a higher accuracy than the FCN proposed in Guastoni et al. (2021), as shown in figure 3 (type-I predictions).
The improvement is consistent across the entire range of investigated wall-normal locations. On the other hand, the performance degrades in a similar way as we move farther awayfrom the wall. The accuracy of the present network at y+ = 50
is comparable to that of the FCN in Guastoni et al. (2021) at
y+ = 30. At y+ = 100, the MSE in the wall-normal and spanwise directions is similar for both architectures. The accuracy
improvement is even more pronounced when considering thepredicted turbulent statistics. In short, the error is lower at ally+ locations and even at y+ = 100, the current FCN performs
substantially better than its predecessor.
Type-II predictions represent an intermediate step towards the flow estimation using only the heat flux. Whenpassing from type I to type II, the error is higher for all thevelocity components, however the streamwise component is
the most affected, as the heat flux at the wall is less correlated to the velocity-fluctuations away from the wall than thestreamwise wall-shear stress. The percentual increment of the
MSE due to the added difficulty of the predictions is similar
for both networks, note however that the predictions from the
FCN proposed in this study were significantly more accurate
than the FCN in Guastoni et al. (2021). For this reason, thecurrent FCN is found to perform better across the entire rangeof wall-normal locations, even for type-II predictions.
Type-III predictions are performed at two wall-normal
distances: y+ = 15 and 30. The resulting MSE is about threetimes higher when compared to type-II predictions, indicat-

u
v
w

0.8

MSE

0.6

0.4

0.2

0.0
15

30

50

100
+
ytarget

FCN proposed here
Guastoni et al. (2021)

60

ERMS [%]

50
40
30
20
10
0
15

30

50

100
+
ytargetFigure 3. MSE (top) and turbulence-statistics error (bottom)
obtained in type-I predictions with respect to target fields at
different wall-normal locations. The error for each velocity
component is normalized with the square of the corresponding
fluctuations intensity.

ing that information in the spanwise wall-shear stress and wallpressure has an important role in the reconstruction of the
fields away from the wall. Note that making the use of a deeper
FCN necessary to achieve satisfactory predictions of this type,as shown by the error comparison with Guastoni et al. (2021)
in table 2. With the use of the auxiliary loss the MSE is only
Table 2. Error comparison in Type-III predictions at y+ =30 using the network model from Guastoni et al. (2021) andthe one proposed here. The variance of the statistical error iscomputed across the different training runs.

Guastoni et al.

Current FCN

0.592

0.271

0.638

0.335

L(w)/w2RMS

0.850

0.330

ERMS (u) [%]36.82 ± 0.67

11.87 ± 0.49

ERMS (v) [%]39.48 ± 0.92

14.14 ± 0.93

ERMS (w) [%]57.26 ± 1.24

14.04 ± 0.74

L(u)/u2RMS
L(v)/v2RMS

about 5% lower, while the predicted turbulence statistics are
20% better than when the auxiliary loss function is not used.
A sample type-III prediction at y+ = 30 using the auxiliary loss
function is shown in figure 4. From this figure, it is possible
to observe that the FCN is able to reconstruct the large-scalefeatures of the flow in all three velocity components startingfrom the heat-flux field only. The smaller-features reconstruc-4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ture for Computing (SNIC) at PDC. S.D. and A.I were partially supported by the project ARTURO, ref. PID2019109717RB-I00/AEI/10.13039/501100011033, funded by the
Spanish State Research Agency.

tion is less accurate, in particular the maximum positive andnegative fluctuations are typically underestimated. This is related to the use of the mean-squared error as loss function forthe optimization.
A more comprehensive overview of the predicted energy
at the different scales is provided by the spectra, shown in figure 5. We compare the spectra of type-II and type-III predictions at y+ = 30. The amount of reconstructed energy islower in type-III predictions than in type-II. Furthermore, it ispossible to observe that eliminating the spanwise wall-shear
stress and wall pressure has a higher impact on the prediction of the shorter wavelengths, both in the streamwise andspanwise direction. The accuracy reduction is more evident
in the pre-multiplied wall-normal and spanwise spectra. Thisis expected, as the wall-pressure is well correlated with thewall-normal component of the velocity and the spanwise wallshear stress helps to improve the prediction of the corresponding velocity-fluctuation component.

REFERENCES
Borée, J. 2003 Extended proper orthogonal decomposition: atool to analyse correlated events in turbulent flows. Exp.
Fluids 35 (2), 188–192.
Chevalier, M., Schlatter, P, Lundbladh, A. & Henningson,
D. S. 2007 SIMSON : a pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep.. KTH Royal Institute of Technology, Stockholm.
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control
for drag reduction in wall-bounded flows. J. Fluid Mech.
262, 75–110.
Discetti, S. & Coletti, F. 2018 Volumetric velocimetry for fluid
flows. Meas. Sci. Technol. 29 (4), 042001.
Dumoulin, V. & Visin, F. 2016 A guide to convolution arithmetic for deep learning. Preprint arXiv:1603.07285 .
Encinar, M. P. & Jiménez, J. 2019 Logarithmic-layer turbulence: A view from the wall. Phys. Rev. Fluids 4, 114603.
Fahland, G., Stroh, A., Frohnapfel, B., Atzori, M., Vinuesa,
R., Schlatter, P. & Gatti, D. 2021 Investigation of blowing
and suction for turbulent flow control on airfoils. AIAA J.
59 (11), 4422–4436.
Gautier, N., Aider, J.-L., Duriez, T., Noack, B. R., Segond,
M. & Abel, M. 2015 Closed-loop separation control using
machine learning. J. Fluid Mech. 770, 442–457.
Guastoni, L., Güemes, A., Ianiro, A., Discetti, S., Schlatter, P.,
Azizpour, H. & Vinuesa, R. 2021 Convolutional-networkmodels to predict wall-bounded turbulence from wall quantities. J. Fluid Mech. 928, A27.
Güemes, A., Discetti, S., Ianiro, A., Sirmacek, B., Azizpour,H. & Vinuesa, R. 2021 From coarse wall measurements toturbulent velocity fields through deep learning. Phys. Fluids
33 (7), 075121.
He, K., Zhang, X., Ren, S. & Sun, J. 2016 Deep residual learning for image recognition. In 2016 IEEE Conf. Comp. Vision
and Pattern Recogn. (CVPR), pp. 770–778.
Kim, J. & Lee, C. 2020 Prediction of turbulent heat transfer
using convolutional neural networks. J. Fluid Mech. 882,
A18.
Kingma, D. P. & Ba, J 2015 Adam: A method for stochastic
optimization. In 3rd Int. Conf. on Learning Representations,
ICLR 2015, San Diego, CA, USA, May 7-9, 2015.
Quadrio, M., Ricco, P. & Viotti, C. 2009 Streamwise-travellingwaves of spanwise wall velocity for turbulent drag reduction. J. Fluid Mech. 627, 161–178.
Raiola, M., Greco, C. S., Contino, M., Discetti, S. & Ianiro,
A. 2017 Towards enabling time-resolved measurements of
turbulent convective heat transfer maps with ir thermography and a heated thin foil. Int. J. Heat Mass Trans. 108,
199–209.
Ronneberger, O., Fischer, P. & Brox, T. 2015 U-net: Convolutional networks for biomedical image segmentation. pp.
234–241. Springer.
Sasaki, K., Vinuesa, R., Cavalieri, A. V. G., Schlatter, P. &Henningson, D. S. 2019 Transfer functions for flow predictions in wall-bounded turbulence. J. Fluid Mech. 864,
708–745.
Tomiyama, N. & Fukagata, K. 2013 DNS of drag reduction ina turbulent channel flow using spanwise traveling wave-likewall deformation. Phys. Fluids 25 (10), 105115.

CONCLUSIONSIn this work, we assessed the prediction capabilities ofa fully-convolutional network (FCN) using DNS data sampled from a turbulent boundary layer flow, with a maximum
Reynolds number of Reτ = 396. We optimize the architecture of the FCN in order to minimize the MSE in the predictions, while maintaining a satisfactory output/input ratioof the field size. The resulting network yields type-III predictions at y+ = 30 with an error that is 50% lower than thatof the previously-studied architectures (Guastoni et al., 2021).
A higher number of layers determines a larger receptive field
for the network, and it enhances its compositional capabilities. The prediction accuracy is proven to be more sensitive tothis parameter than the network capacity (i.e. the number oftrainable parameters). The use of alternative, yet similar network architectures (e.g. ResNet by He et al. (2016) or UNetby Ronneberger et al. (2015)) was only partially explored andfor this reason not reported, preventing a more comprehensive
analysis of the available network architectures. The use of skip
connections, present in the aforementioned architectures, canpotentially improve the network performance further, but thisassessment is left for future work. The architectural improvements described here are essential to achieve a satisfactory velocity field reconstruction, given the additional difficulties related to the choice of a spatially-evolving flow, and the use to
input quantities that are less informative than the ones used in
the previous studies.
After assessing the prediction capabilities of the FCN
using DNS data, future work will be devoted to testing theneural-network model in an experimental setting. In this case,the resolution of the measured quantities is typically lower
than that of DNS samples. To address this issue, the FCN
model should be first trained using DNS data aptly modified
to mimic experimental data and then using samples taken directly from a wind tunnel. Testing the trained networks on this
synthetic experimental data represents the next step towards
the network model deployment in an experiment.

ACKNOWLEDGEMENTS
The authors acknowledge the funding provided by the
Swedish e-Science Research Centre (SeRC), the G. Gustafsson Foundation and the Knut and Alice Wallenberg (KAW)
Foundation. The numerical simulations were carried out
on resources provided by the Swedish National Infrastruc-5

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

103

103

102

102

102

101

λ+
z

103

λ+
z

λ+
z

Figure 4. Sample result for type-III prediction at y+ = 30, obtained using the proposed FCN with auxiliary loss functions. The firstrow corresponds to the DNS input heat flux for Pr = 6, normalized with the mean and standard deviation computed on the training
samples. The second and third rows show the streamwise DNS velocity-fluctuation field and the corresponding prediction obtained from
FCN, respectively. Similarly, the fourth and fifth rows represent the wall-normal velocity fluctuation of the target and predicted fields.
Finally, the sixth and seventh rows show the spanwise velocity fluctuation component of the target and predicted fields, respectively.
The velocity-fluctuation fields are scaled by the respective RMS quantities.

101
102

103

Type II
Type III

101
102

103

λ+
x

λ+
x

102

103
λ+
x

Figure 5. Pre-multiplied two-dimensional power-spectral densities. The three rows represent kz kx φuu (left), kz kx φvv (center), kz kx φww(right). The contour levels contain 10%, 50% and 90% of the maximum DNS power-spectral density. Shaded contours refer to thereference data, while contour lines refer to type-II (blue) and type-III (orange) predictions, respectively.

6