A data-driven wall model for the prediction of
turbulent flow separation over periodic hills
Margaux Boxho ∗,1,2,3 , Koen Hillewaert2,1 , Grégoire Winckelmans3 , Michel Rasquin1 ,Thomas Toulorge1 and Sophie Mouriaux41

Cenaero, Rue des Frères Wright, 29, 6041 Gosselies, Belgium.
Aerospace and Mechanics Department (A&M), Université de Liège (ULiege), 4000 Liège, Belgium.
3
Institute of Mechanics, Materials and Civil Engineering (IMMC), Ecole Polytechnique de Louvain,
1348 Louvain-la-Neuve, Belgium.
4
Safran Tech, Rue des jeunes bois, Châteaufort, 7877 Magny-les-Hameaux, France.

2

Abstract
Aeronautic flows are characterized by turbulence, which consists of chaotic perturbations around a time-averaged flow field. Turbulence appears in a large variety oflength scales, ranging from unsteady flow features of the size of the aircraft, wing,
blade down to tiny whirls, which are many orders of magnitude smaller. Turbulence
has a profound impact on aerodynamic performance, but unfortunately, explicit
computation of all turbulent features remains intractable for the design and analysisof real-life geometries [1]. The smallest structures, requiring the largest computational effort, are found in the so-called boundary layer near the wall. This cost
can be avoided by modeling their time-averaged impact on the forces exchanged
between fluid and the wall. This saving, in turn, allows for the direct computationof the largest turbulent flow features away from the wall, which govern importantlarge-scale effects. The present work proposes the use of Deep Neural Networks(DNN) to link the wall shear stress components to volume data extracted at multiple wall-normal distances hwm and wall-parallel locations. The model focuseson separation since this phenomenon is currently not well-represented, whereas it
has a huge impact on aerodynamic performance and operating range. The model istrained using a high-fidelity database of the well-known two-dimensional periodic
hill flow. The conditions of this separated flow are such that it is still affordable tocompute all turbulent flow features directly, using Tier-1 modern supercomputers.

1

Introduction

Aeronautic flows are characterized by turbulence which consists of an irregular chaotic motion
superimposed upon the mean flow field. The turbulent flow features – also called whirls or eddies –occur over a large range of length scales. The largest eddies are formed by the flow around thegeometry. Big eddies are unstable and break down into smaller eddies, transferring almost all of theirkinetic energy. The newly formed smaller eddies then undergo the same process. This decomposition
occurs in a recursive manner such that kinetic energy is continuously transferred from larger toever-smaller eddies until finally, the viscous effects dominate where the kinetic energy is converted
into heat. The corresponding small scales are often many orders of magnitude smaller than thelargest; this ratio grows with the non-dimensional Reynolds number Re = uD/ν with u, D, and ν, arepresentative velocity, length scale, and viscosity, respectively. This process is commonly referred toas the energy cascade, introduced by Richardson and further refined by Kolmogorov. Turbulence∗

Corresponding author margaux.boxho@cenaero.be35th Conference on Neural Information Processing Systems (NeurIPS 2021), Sydney, Australia.

occurs in many natural phenomena (smoke, waterfalls, winds, . . . ) and many important engineering
applications (wind loads on buildings, wind turbines, vehicles, aeronautics, . . . ). The comprehension
and characterization of turbulence are of great interest in the aeronautical sector for the design ofaircraft, wings, or jet engine components. Due to computational cost considerations, industrial
simulation tools rely on the Reynolds-Averaged Navier-Stokes (RANS) approach, which replaces
the explicit representation of turbulence by modeling its averaged impact on the mean flow. RANS
models frequently fail, due to their inherent modeling assumptions, to predict off-design operating
points where undesired flow patterns are significantly exacerbated and flow separation may occur. To
enable off-design performance prediction, the main challenge is to provide a computational approachwith fewer modeling assumptions while maintaining a reasonable computational cost. The directcomputation of all scales of turbulence up to the smallest is called Direct Numerical Simulations(DNS). DNS removes all reliance on modeling but remains far too computationally expensive except
for very simple geometries and flow conditions. The Large Eddy Simulation (LES) approach replaces
the explicit representation of the smallest scales by modeling their impact on the computed larger
scales. However, the saving that this brings may still not be enough to tackle real-life problems. One
can go one step further, considering that the smallest turbulent flow structures are found in a small
region – the boundary layer – near the wall. For a commercial aircraft wing, these flow featuresare 108 to 109 smaller than the wing section itself. Since Wall-resolved LES (wrLES) computes
these microscopic whirls in the boundary layer, it still requires prohibitively extensive computational
resources [1]. Wall modeled LES (wmLES) avoids this cost by modeling the impact of the turbulencein the boundary layer on the forces, and in particular, the wall shear stress exchanged between theflow and the wall. This approach allows economizing computational resources without compromising
resolution in the core flow. Literature is abundant concerning wall model proposals (see Larsson
et al. [2]): algebraic wall models, two-layer wall models, equilibrium and non-equilibrium wallmodels, wall models based on partial differential equations, . . . to name a few. Most wall models havevery restrictive model assumptions, which are not often met in practice. In particular, once the flow
separates or transitions, these hypotheses no longer hold.
The purpose of this work is to address the prediction of wall shear stress for separated flows using
deep neural networks (DNN). Given the difficulty of representing the complex behavior of theboundary layer in a large range of conditions, DNN offer unique opportunities due to their universal
approximation properties Hornik [3]. In literature, the application of DNN to wall modeling has
already been partially addressed. Yang et al. [4] attempt to develop a data-driven wall model for the
extrapolation at high Reynolds numbers. To improve the extrapolation capabilities, a priori physicalknowledge is incorporated into the model inputs. Zhou et al. [5] targeted separating turbulent flows
by considering the periodic hill problem. Although their a priori validation on different geometries
and bulk Reynolds gives encouraging results, the a posteriori validation on the nominal geometry at
Re = 10595 fails to predict the mean separation lines leading to an incorrect prediction of the mean
velocity profiles. Lozano-Durán and Bae [6] consider multiple flow configurations including corner
vortex flows and separation. They assumed that a flow is a non-linear combination of known buildingblock flows. Hence, their wall-flux-based wall model for LES using a self-critical machine-learning
approach was trained over DNS data, including flow over a flat plate, in a channel, in a turbulent duct,or separated flow at various Reynolds numbers. The obtained model can not predict the separation in
the NASA Juncture Flow, probably due to a lack of data. In our approach, we trained more advanced
architecture (such as a Gaussian Mixture Neural network) on an enlarged domain of dependency tobetter capture the complex relationship between instantaneous flow fields, geometrical parameters,
and wall shear stress.

2

The two-dimensional periodic hill

The data-driven model is trained using high resolution wrLES computations of the well-known doubly
periodic hill case (see Figure 3), which is described by Fröhlich et al. [7] and part of the well-known
ERCOFTAC Knowledge Base Wiki test cases [8]. This bi-periodic flow evolves between two walls
featuring a streamwise constriction, inducing turbulent separation and reattachment. The consideredbulk Reynolds number is Reb = 10595, whereby Reb = ubν h , with ub the bulk velocity measured
at the hill top, h the hill height and ν the kinematic viscosity. A constant pressure gradient drives
the flow whose magnitude is controlled to match the bulk Reynolds number following Benocci andPinelli [9]. wrLES results are obtained using a high-order compressible DG flow solver. The flow
simulation is performed at a low bulk Mach number Mb of 0.1, and statistics are accumulated over2

38 flow-through times (tc = ub /Lx , where Lx is the streamwise domain length), after evacuating
the numerical transient over 45 tc . The averaged results, listed in appendix A for the sake of brevity,
are compared to those obtained by Temmerman et al. [10] and Gloerfelt and Cinnella [11]. The flowpresents a precursory separation at the top of the hill, only present for compressible flows, a second
main separation from x/h ' 0.19 to x/h ' 4.21 and a third at the leeward base of the hill (seeFigure 4). Figure 4 shows the envelope of the time-varying friction coefficient. This graph, combinedwith the intermittency γ = t(τw <0) /ttot , highlights the variability of the separation location andconditions in time. Around the mean separation line (x/h = 0.19), the flow is 40% to 60% of thetime detached. According to this observation, the data-driven wall model is required to be correct
instantaneously in addition to being correct on average.

3

Data-driven wall-model

Formulation. The prediction of the flow-aligned and transverse wall shear stress (τw,ξ , τw,z ) can
be stated as a regression problem. Under the assumption of Gaussian distribution, N (µ, σ) of mean
µ and variance σ, where the approximation function f is parametrized using a neural network (NN)that contains no activation function on its last layer, and applying the maximum likelihood estimation,
we recover the mean squared error loss, where θ represents the neural network parameters (weightand bias), τw the vector of output labels, xi the input labels and d the set of i.i.d. training data sets:X2p(τw |x) = N τw ; µ = f (x; θ), σ 2 = 1 ⇒ arg minτwi − f (xi ; θ) .
(1)
θ

i ∈d
xi ,τw

Data preparation. The training set d is composed of input labels summarized in Table 1 and output
labels representing the shear stress components which will be applied as wall boundary conditions
for wmLES. Before training the NN, feature selection is used to identify pairs of input/output thatare highly correlated. Space-time correlations, illustrated in Appendix B (see Figure 5) are used
to this end. When the flow separates, local correlations are insignificant and thus information has
to be sought up- and down-stream. Therefore, it is necessary to enlarge the domain of dependence
to include up- and down-stream information in both space and time. Figure 1 illustrates wheredata is extracted relative to the position along the wall at which the shear stress is predicted. Threewall-normal positions and 21 wall-parallel positions are considered. Currently, only instantaneous
information is used for the training; future research will introduce time delays. The database iscomposed of Nξ × Nz × Nt pairs of input/outputs. Six instantaneous snapshots of the solution
are interpolated on a (Nξ , Nη , Nz ) = (180, 100, 10) probe grid per flow-through time tc , definedas tc = Lx /ub . The database is created over 35tc , of which 90% is used for training, and the rest
for testing. The validation database, used in Section 4, for a priori validation, covers 10tc where
snapshots are extracted at every wrLES time step, (∆t ub /h) = 5 · 10−3 .

Field

Table 1: Inputs of the data-driven wall model (h is the hill height)
Description Pre-scaling

Velocity
Pressure Gradients
Length scale

u
∇p
hwm

uξ /hwm , uη /hwm , uz /hwm
(hwm /h) (∂p/∂ξ), (hwm /h) (∂p/∂η), (hwm /h) (∂p/∂z)ln (hwm /y ? )According to Yang et al. [4], it is essential to introduce some physical knowledge to the input features
for a better generalization of the model. Following their guidelines and the work of Zhou et al.
[5], the inputs are pre-scaled using y ? (see Table 1). The length scale y ? is a near-wall scaling
compatible with the separation process inspired by Duprat et al. [12]. It is defined as, y ? = ν/uτ,p√√with uτ,p = (u2ν + u2p ) where uν = (νuk /hwm ) and up = |(ν/ρ)∂x p| 1/3 ; hwm is the wall-normal√
distance at which the field is interpolated and fed to the model, and uk = (u2ξ + u2z ) is the norm of
the wall-parallel velocity measured at that location.
Feedforward neural network. The present study uses two forked feedforward multilayer perceptron (MLP) neural networks MLP1 and MLP2 (Figure 6a), with the same structure, but trainedon different data sets. MLP1 is trained over the reference database (DB1), while MLP2 uses 123instantaneous snapshots over one tc , instead of 6. This second enriched database (DB2) aims to better
capture the intermittency. We also study the behavior of a Gaussian Mixture Neural Network (GNN)with K = 2 modes (Figure 6b), trained over DB2. These NN are further described in Appendix C.
The rectified linear unit (ReLU) is selected as the activation function to avoid the vanishing gradientproblem. After applying the physical scaling to each input/output pair, they are further normalizedusing a min-max scaling. Regarding the inputs, this scaling is applied to each field independently andnot to each component individually to conserve the relative amplitude of the components. For the
outputs, this scaling is applied per component because the spanwise wall shear stress is smaller than
the streamwise one (a common scaling would lead to a wrong prediction of the smaller component).
MLP1 and MLP2 are trained with the MSE loss function (Equation 1). The GNN is trained with adifferent loss (Equation 2), accounting for the mean and variance of each modes. The adam optimizerof Kingma and Ba [13], with its default parameters, is used to train the models. The training of MLP1
and MLP2 is stopped when the test loss starts to increase to avoid overfitting [14]. The NN models
are created and trained with the open-source library PyTorch [15].

Figure 1: Generation of the database combining a single output (red dot circled) with a set of inputs(dotted red rectangle) spreading along the wall-normal and wall-parallel directions.

4

Preliminary results and a priori study

We first compare the distribution of the reference and predicted friction using boxplots (Figure 8) atthree locations. The graphs confirm the observations of Figure 4. Near the separation (Figure 8a), thewide range of wall shear stress values is not captured by MLP1 and MLP2, while GNN recovers themean and variance of the distribution. MLP1 and MLP2 are probably not able to retrieve the richnessof the distribution due to the focus on regression of the mean in Equation 1. At the reattachment(Figure 8b), similar observations are made. However, the GNN slightly overestimates the variance.
In the converging region (Figure 8c), the MLP1 captures better the mean, while the GNN is betterat representing the variance; perhaps a third mode could improve its performance. Secondly, themean shear stress is compared along the bottom wall; this average is computed over 10tc . As in
Zhou et al. [5], we define its relative error as E = τ̂ w − τw / (|τw |max ), with . the time averaging
operator. Figure 2 shows the spatial evolution and the corresponding relative error. GNN captures theaverage separation well, although it does not understand the pre-separation, generating a relative errorof 10% at x/h ' 0. MLP1 and MLP2 miss the mean separation line and directly assume that theflow separates at the hilltop. Regarding the recirculation bubble, there is an underestimation of theshear stress between x/h ' 2 and x/h ' 3. This inaccurate prediction is less important because theflow in the bubble does not directly drive the flow. The models capture the mean reattachment point
well. Moreover, GNN correctly reproduces the reattachment between x/h = 6 and x/h = 7 witha relative error of less than 1%. The tiny separation at the hill foot is only predicted by MLP2. In
the converging part, none capture the friction peak. MLP1 behaves best with only 25% relative error.
GNN is the worst with 50% of relative error. Nonetheless, it predicts perfectly the wall shear stress
up to x/h = 8.5 while MLP1 and MLP2 find a higher relative error at the windward face of the hill.
4

Figure 2: Comparison of the wall shear stress predicted by the data-driven wall models (MLP1, MLP2
and GNN) to the reference obtained by wrLES. Predicted values (top) and relative error (bottom).

5

Conclusion and future work

This work constitutes a first step towards a data-driven wall model for massively separated flows.
The use of advanced correlations led to the selection of relevant input/output pairs and windowing
information for the training of three DNNs. The a priori validation demonstrates satisfactory results
for most of the bottom wall, between x/h ' 1 and x/h ' 8 for all models. GNN appears to correctly
capture the mean separation while missing the friction peak. We expect that the use of more modes
could improve this. We can furthermore question the spatial richness of the database, in particularat the hilltop where a quite complex flow is found. These points will be further investigated. We
conclude that Gaussian Mixture Neural networks can be a great asset for the development of wallmodels applied to separation. In the future, we will include additional geometrical information suchas the radius of curvature which is another parameter influencing separation. Furthermore a priori
physical relations should be added to improve generalization. Our approach will include multiple
streamwise and wall-normal positions. These positions are not yet encoded in the neural network,
due to its architecture. Therefore, self-attention layers would be integrated as well.

Acknowledgments and Disclosure of Funding
The present research benefited from computational resources made available on the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles, infrastructure funded by the Walloon Region under the
grant agreement n°1117545. The funding of M. Boxho by SafranTech is gratefully acknowledged.

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[2] Johan Larsson, Soshi Kawai, Julien Bodart, and Ivan Bermejo-Moreno. Large Eddy Simulation
with modeled wall-stress: recent progress and future directions. Mechanical Engineering
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[3] K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks,
4(2):251–257, 1991.
[4] X. I. A. Yang, S. Zafar, J.-X. Wang, and H. Xiao. Predictive large-eddy-simulation wallmodeling via physics-informed neural networks. Phys. Rev. Fluids, 4(3):034602, 2019-03-15.
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ISSN 2469-990X. doi: 10.1103/PhysRevFluids.4.034602. URL https://link.aps.org/
doi/10.1103/PhysRevFluids.4.034602.
[5] Zhideng Zhou, Guowei He, and Xiaolei Yang. Wall model based on neural networks for
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[6] Adrián Lozano-Durán and Hyunji Jane Bae. Self-critical machine-learning wall-modeled LES
for external aerodynamics. arXiv:2012.10005 [physics], 2021. URL http://arxiv.org/
abs/2012.10005.
[7] Jochen Fröhlich, Christopher. P. Mellen, Wolfgang Rodi, Lionel Temmerman, and Michael A.
Leschziner. Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech., 526:19–66, 2005.
[8] Underlying Flow Regime case UFR 3-30: 2D Periodic Hill Flow, 2013. URL https://kbwiki.ercoftac.org/w/index.php/Abstr:2D_Periodic_Hill_Flow.
[9] C. Benocci and A. Pinelli. The role of the forcing term in the large-eddy simulation ofequilibrium channel flow. Engineering turbulence modeling and experiments, page 287–96,
1990.
[10] L. Temmerman, M. Leschziner, C. Mellen, and J. Fröhlich. Investigation of subgrid-scale models
and wall-function approximations in large eddy simulation of separated flow in a channel withstreamwise periodic constrictions. Int J Heat Fluid Flow, 24(2):157–180, 2003.
[11] X. Gloerfelt and P. Cinnella. Investigation of the flow dynamics in a channel constricted byperiodic hills. In 45th AIAA Fluid Dynamics Conference. AIAA, 2015.
[12] C. Duprat, G. Balarac, O. Métais, P. M. Congedo, and O. Brugière. A wall-layer model forlarge-eddy simulations of turbulent flows with/out pressure gradient. Physics of Fluids, 23(1):015101, 2011-01. ISSN 1070-6631, 1089-7666. doi: 10.1063/1.3529358. URL http://aip.scitation.org/doi/10.1063/1.3529358.
[13] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization.
arXiv:1412.6980 [cs], 2017. URL http://arxiv.org/abs/1412.6980.
[14] Stuart Geman, Elie Bienenstock, and René Doursat. Neural networks and the bias/variancedilemma. Neural Computation, 4(1):1–58, 1992. ISSN 0899-7667. doi: 10.1162/neco.1992.4.1.
1. URL https://www.mitpressjournals.org/doi/10.1162/neco.1992.4.1.1.
[15] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan,
Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas
Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy,
Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, highperformance deep learning library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'AlchéBuc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32,
pages 8024–8035. Curran Associates, Inc., 2019. URL http://papers.neurips.cc/paper/
9015-pytorch-an-imperative-style-high-performance-deep-learning-library.
pdf.
[16] Dominic Edelmann, Tamás F. Móri, and Gábor J. Székely. On relationships between the Pearson
and the distance correlation coefficients. Statistics & Probability Letters, 169:108960, 2021.
[17] Gábor J. Székely, Maria L. Rizzo, and Nail K. Bakirov. Measuring and testing dependence bycorrelation of distances. Ann. Statist., 35(6):2769–2794, 2007.

6

Checklist
The checklist follows the references. Please read the checklist guidelines carefully for information on
how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or
[N/A] . You are strongly encouraged to include a justification to your answer, either by referencing
the appropriate section of your paper or providing a brief inline description. Please do not modify thequestions and only use the provided macros for your answers. Note that the Checklist section does
not count towards the page limit. In your paper, please delete this instructions block and only keep
the Checklist section heading above along with the questions/answers below.
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes] The work scope is the prediction of separation and theenhancement of current wall models using DNN.
(b) Did you describe the limitations of your work? [Yes] see section 4(c) Did you discuss any potential negative societal impacts of your work? [No] This work
is to low TRL to discuss such societal impacts.
(d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes] No potential negative societal impacts has been detected, this work has
for future goal to better understand aero-engine and to be able to reduce their carbon
impact.
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A] Not applicable
here.
(b) Did you include complete proofs of all theoretical results? [N/A] Not applicable here.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See section 2(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [Yes] See section C(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] No, we run numerical simulation whose averaged resultscompared well with the literature.
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [Yes] See section 3
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] Data are generated
with Argo-DG developed at Cenaero (see section 2)(b) Did you mention the license of the assets? [Yes] PyTorch is used as an open-source
library and is cited in the reference [15](c) Did you include any new assets either in the supplemental material or as a URL? [N/A](d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [No] The database is not linked to "people".
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content? [No] The database does not contain such feature, weonly deal with instantaneous value of the wall shear stress.
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable? [N/A] Not applicable here.
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable? [N/A] Not applicable here.
(c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation? [N/A] Not applicable here.
7

A

APPENDIX: VALIDATION

Our results obtained on the two-dimensional periodic hill at Reb = 10595 are compared withthose obtained by Temmerman et al. [10] and by Gloerfelt and Cinnella [11]. Figure3 shows thevertical profiles of the time-averaged streamwise velocity u/ub and the Reynolds shear stress u0 u0 /u2b .
Considering the average velocity profiles and Reynolds stresses, good agreement with the references
is observed.
3.0

2.5

2.5

2.0

2.0

y/h

y/h

3.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0
0

1

2

3

4

u/ub

5

6

7

8

9

0

1

2

3

4

5

u u /ub
′

′

6

7

8

9

2

Figure 3: Averaged velocity profile u/ub (top) and stress component u0 u0 /u2b (bottom) of the 2D
Periodic Hill flow at Reb = 10595, compared with ERCOFTAC database (star red curve), X. Gloerflt
and P. Cinnela at M = 0.1 (diamond blue curve).

Figure 4: Friction coefficient Cf measured on the bottom surface of the periodic hill (circle black
curve), compared with ERCOFTAC database (star red curve), X. Gloerflt and P. Cinnela at M = 0.1(diamond blue curve). The graph depicts also the friction coefficient envelop (gray area), intermittency
γ measurement (dash-dotted line) and mean separation and reattachment locations (green squares).
p
The friction coefficient defined as u2τ /(0.5ρu2b ) where uτ = τw /ρw is the friction velocity, isshown in Figure 4. There is good agreement with the results obtained by Gloerfelt and Cinnella[11] at M = 0.1. A precursory separation is visible at x/h ' 0. According to Gloerfelt andCinnella [11], this pre-separation arises only when using compressible flow solvers. The mean
separation line is located at x/h ' 0.19. In the separation bubble extending to x/h ' 4.21, the flow
decelerates and then re-accelerates due to the change in curvature. We note two divergences: (i) the
deceleration/acceleration process is slightly shifted downstream, and (ii) the secondary separation
bubble at the bottom of the hill is predicted to occur before x/h = 7.0. We suspect an alternative
definition of the geometry and, more specifically, at the junction between the flat part of the bottom
and the hill. Our DG solution captures well the mean reattachment location. The recovery region
extends from x/h ' 4.2 to x/h ' 7.0, with a secondary recirculation bubble at x/h ' 7.2, followed
by a strong acceleration on ascent.

B

APPENDIX: CORRELATION MAPS

Before training any machine learning models, it is important to analyse the database to select relevant
input/output pairs. Feature selection can improve model performances and reduces the computational
cost of modeling. Space-time correlations (Pearson [16] and distance proposed by Székely et al.
[17]) are adopted to perform feature selection for the present problem. While Pearson correlations
only detect linear relations between data, distance correlations can extract both linear and non-linear
8

relationships. An example of such correlation map is shown on Figure 5. These correlation maps
expose the relation between the instantaneous wall shear stress τw,ξ and the instantaneous wallparallel velocity component uξ . The main observation is the domain of dependence shift in thepositive δξ/h with a maximum value of 0.45 (for Pearson correlation). We conclude that separation
drives the behavior of the wall shear stress at x/h ' 0.05.

Figure 5: Pearson (top) and distance correlation (bottom) between τw,ξ and uξ at (ξ, η)/h ' (0, 0.1).

C

APPENDIX: NEURAL NETWORK ARCHITECTURE

Description of MLP1 and MLP2. The neural network used for the present study is a forked
fully-connected feedforward neural network with a common set of parameters (blue rectangle) and aset of target parameters (red rectangles), dedicated to each task. There are two distinct tasks (green
rectangle): (1) the prediction of the streamwise wall shear stress τw,ξ and (2) the prediction of the
spanwise wall shear stress τw,z . The common neural network is composed of seven hidden layerswith 400, 200, 200, 200, 100, 100, 50 neurons. Each target neural network is composed of one hidden
layer of 50 neurons. The inputs are clearly described in Table 1. The training is stopped once the test
loss (gray curve on Figure 7) starts to increase to avoid overfitting. The neural network model is thus
saved after 150 epochs (red star). Moreover, after 150 epochs, the success rate measured based on therelative error is above 98% for both outputs.
Common task parameters

k = 1, . . . , KθµkInputs

Outputs

x

NN

σk

N

pk

Σ

p

πk
y
Target parameters of the task

(a)

(b)

Figure 6: Schematic of the architecture of (a) a feedforward neural network with common parameters
and target parameters for the prediction of the wall shear stress components; (b) a Gaussian Mixture
Neural Network with K modes.
Description of GNN. This NN is depicted in Figure 6b where the distribution of τw,ξ is predicted.
This NN is trained via a different loss function accounting for the mean and variance of the K modes.
9For the present study, only two modes are considered (K = 2). The loss function is:

!
K
X
X
(yi − µk (xi ))2
1
√
exp −
L=
log
,
2σk2 (xi )
2πσk (xi )xi ,yi ∈d

(2)

k=1

where µk , σk and πk are respectively the mean, the variance and the mixture coefficient of modek. The network contains four hidden layers with respectively 400, 200, 100, and 20 neurons each.
The inputs are those described in Table 1 while it tries to predict the distribution of τw,ξ only. τw,z
distribution is not inferred here. The model is evaluated after 100 epochs (Figure 7).

Figure 7: Evolution of the losses with training epochs for both the training and testing datasets: MLP1
(left), MLP2 (middle) and GNN (right) with red stars being the epoch at with models are evaluated.
Section 4 discusses the validation of the model prediction against wrLES results in an a priori study.
As a first step, model predictions are plotted in boxplot (see Figure 8) to have a more in-depth viewof their distribution. The analysis of Figure 8 is already done in the results section.

(a)
(b)
(c)
Figure 8: Boxplot comparing the model prediction (1:MLP1, 2:MLP2 and 3:GNN) and the groundtruth distribution (4) for three different locations along the wall: (a) ξ/h ' 0, (b) ξ/h ' 4.5 and (c)ξ/h ' 8.

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