# Wall-modeled large-eddy simulation based on building-block flows

Y. Ling111Department of Aeronautics and Astronautics, Massachusetts Institute
of Technology G. Arranz1 E. Williams1 K. Goc  K. Griffin  A. Lozano-Durán1

A unified subgrid-scale (SGS) and wall model for large-eddy simulation (LES)
is proposed by devising the flow as a collection of building blocks that
enables the prediction of the eddy viscosity. The core assumption of the model
is that simple canonical flows contain the essential physics to provide
accurate predictions of the SGS tensor in more complex flows. The model is
constructed to predict zero-pressure-gradient wall-bounded turbulence, adverse
pressure gradient effects, separation and laminar flow. The approach is
implemented using a Bayesian classifier, which identifies the contribution of
each building block in the flow, and a neural-network-based predictor, which
estimates the eddy viscosity based on the building-block units. The training
data are directly obtained from wall-modeled LES with an exact SGS/wall model
for the mean quantities to guarantee consistency with the numerical
discretization. The model is validated in canonical flows and the NASA High-
Lift Common Research Model and shown to improve the predictions with respect
to current modeling approaches.

##  1 Introduction

The ability to accurately simulate complex flows using computational fluid
dynamics (CFD) is crucial in many applications ranging from aircraft design to
drag reduction in pipelines (Smits & Marusic, 2013). In recent years, wall-
modeled large-eddy simulation (WMLES) has gained traction as an alternative to
traditional lower-fidelity models, such as Reynolds-averaged Navier-Stokes
(RANS)-based methods. Compared to RANS-based methods, WMLES is inherently
unsteady and better positioned to provide high accuracy in the presence of
non-equilibrium flows such as separation. Additionally, since WMLES only
resolves the large-scale motions far from the wall, its computational cost is
competitive compared to other CFD approaches (Choi & Moin, 2012; Yang &
Griffin, 2021). Recently, Goc et al. (2021) have shown that WMLES is close to
achieving the accuracy and turnaround times demanded by the aerospace
industry. While state-of-the-art WMLES performs satisfactorily in turbulent
boundary layers with sufficient grid resolution, its performance deteriorates
in the presence of less than 20 grid points per boundary layer thickness.
Unfortunately, the latter grid resolution is typical for external aerodynamics
applications (Lozano-Durán et al., 2022). In this report, we present a unified
SGS/wall model consistent with the numerical discretization and applicable to
complex geometries. The model is rooted in the idea that truly revolutionary
improvements in WMLES will encompass advancements in numerics, grid generation
and wall/SGS modeling.

The model is implemented using artificial neural networks (ANNs) within the
supervised learning paradigm, which has been extensively explored for modeling
in recent years. For example, SGS models have been trained using data from
filtered direct numerical simulation (DNS) (Gamahara & Hattori, 2017; Xie et
al., 2019). Examples of supervised learning for wall modeling include Yang et
al. (2019), Zhou et al. (2021), Zangeneh (2021) and Huang et al. (2019), to
name a few. Interestingly, most studies to date have trained the model with
data from higher-fidelity simulations, such as DNS or wall-resolved LES. As a
consequence, previous models ignored the nonnegligible errors arising from the
numerical discretization in actual WMLES. An exception is the work by Bae &
Koumoutsakos (2022) using reinforcement learning. Here, we take advantage of a
novel data preparation process to ensure numerical consistency between the
training data and the model deployed in the flow solver.

The report is organized as follows. Section 2 details the simulation setup,
numerical approach, model formulation, control scheme and training procedure.
Results for different test cases are presented in Section 3. These include
laminar channel flow, turbulent channel flow, turbulent Poiseuille-Couette
flow and the NASA High-Lift Common Research Model (CRM-HL). Finally,
concluding remarks are offered in Section 4.

##  2 Methodology

###  2.1 Model overview

The model presented is based on the hypothesis that complex flows can be
divided into subregions resembling canonical building-block flows. Following
the strategy proposed by Lozano-Durán & Bae (2020), we devise a model composed
of a classifier and a predictor. The classifier categorizes the flow into
different types of canonical flows, and the predictor provides the wall stress
and eddy viscosity based on the likelihood of each category. The model is
referred to as the BFM (building-block flow model). In this work, we consider
four types of building-block flows, viz., laminar channel flow, fully
developed turbulent channel flow, turbulent Poiseuille-Couette flow with
adverse pressure gradient and turbulent Poiseuille-Couette flow with
“separation”.

The model architecture is summarized in Figure 1. The model inputs are the
invariants of rate-of-strain and rate-of-rotation tensors ($\bar{S}_{ij}$ and
$\bar{R}_{ij}$, respectively), which are defined as

$\begin{array}[]{ll}I_{1}=\operatorname{tr}\left(\mathbf{\bar{S}}^{2}\right),&I_{2}=\operatorname{tr}\left(\mathbf{\bar{R}}^{2}\right),\\\
I_{3}=\operatorname{tr}\left(\mathbf{\bar{S}}^{3}\right),&I_{4}=\operatorname{tr}\left(\mathbf{\bar{S}\bar{R}}{}^{2}\right),\\\
I_{5}=\operatorname{tr}\left(\mathbf{\bar{S}}^{2}\mathbf{\bar{R}}^{2}\right),&I_{6}=\operatorname{tr}\left(\mathbf{\bar{S}}^{2}\mathbf{\bar{R}}^{2}\mathbf{\bar{S}\bar{R}}\right),\end{array}$
(1)

where $\bar{S}_{ij}$ and $\bar{R}_{ij}$ are

$\displaystyle\bar{S}_{ij}=\frac{1}{2}\left(\frac{\partial\bar{u}_{i}}{\partial
x_{j}}+\frac{\partial\bar{u}_{j}}{\partial x_{i}}\right),\
\bar{R}_{ij}=\frac{1}{2}\left(\frac{\partial\bar{u}_{i}}{\partial
x_{j}}-\frac{\partial\bar{u}_{j}}{\partial x_{i}}\right).$ (2)

The anisotropic component of the resolved shear stress is given by the eddy-
viscosity model

$\bar{\tau}_{ij}^{d}=-2\nu_{t}\bar{S}_{ij},$ (3)

and the eddy viscosity is assumed to be a function of the invariants (Lund &
Novikov, 1993)

$\nu_{t}=f(I_{1},I_{2},I_{3},I_{4},I_{5},I_{6},\theta),$ (4)

where $f$ represents an ANN, and $\theta$ denotes additional input variables,
namely, $\nu$, $\Delta$ and $u_{\parallel}$, where $\nu$ is the kinematic
viscosity, $\Delta=\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}$ is the
characteristic grid size and $u_{\parallel}$ is the magnitude of the wall-
parallel velocity measured with respect to the wall. The latter is only used
for the ANN close to the wall. The mapping between inputs and output is
learned from data generated from controlled WMLES simulations, which are
detailed in Section 2.2.

Figure 1: Schematic of the building-block flow model. The panel shows the
classifier-predictor structure and the ANN architecture. The flow state refers
to $(I_{1},...,I_{6},\nu,\Delta,u_{||})$ and different ANNs are used for the
inner and outer layer. The bottom of the figure depicts the building-block
flows considered (from left to right): turbulent channel flow, Poiseuille-
Couette flow with mild adverse pressure gradient, Poiseuille-Couette flow with
separation and laminar Poiseuille flow.

The input and output variables of the ANN are given in nondimensional form.
This is crucial for the generalizability of the model. The
nondimensionalization of the input and output features is done using
parameters that are local in both time and space to guarantee the
applicability of the model to complex geometries. Two types of ANNs are
considered. For grid points at the wall and the first grid points off the
wall, the input and output quantities are nondimensionalized using viscous
scaling ($\nu$ and $\Delta$). For the rest of the points, the input and output
variables are nondimensionalized using semi-viscous scaling
[$(\sqrt{2\bar{S}_{ij}\bar{S}_{ij}}\nu)^{1/2}$ and $\Delta$].

The simulations for canonical cases are performed using the solver by Bae &
Lozano-Durán (2021) (i.e., the incompressible, staggered, second-order
accurate central finite difference method). The CRM-HL simulation is carried
out in charLES, a finite-volume, compressible flow solver. Readers are
referred to Goc et al. (2021) for more details about charLES.

###  2.2 Building blocks and data preparation

Figure 1 also illustrates the collection of building-block flows. The
Poiseuille flow is used to generally represent laminar flows. The turbulent
channel flow models the regime where turbulence is fully developed without
significant mean-pressure gradient effects. In both cases, the bottom and top
walls are static. In the turbulent Poiseuille-Couette flows, the top wall
moves at a constant speed ($u_{t}$) in the streamwise direction, and an
adverse pressure gradient is applied in the direction opposed to the top wall
velocity. The adverse pressure gradient ranges from mild to strong, so that
the flow separates (i.e., zero wall stress) on the bottom wall.

The training data is generated using WMLES with an exact-for-the-mean eddy-
viscosity and wall model (denoted by ESGS). This enables the model to account
for the numerical errors of the flow solver. The ESGS model is based on the
anisotropic minimium-dissipation (AMD) SGS model (Rozema et al., 2015)
combined with a controller that iteratively adjusts the AMD eddy viscosity to
match the DNS mean velocity profile. A Dirichlet non-slip boundary condition
is applied at the walls and the correct wall-shear stress is enforced by
augmenting the eddy viscosity at the walls such that
$\left.\nu_{t}\right|_{w}=\left.\left(\partial\bar{u}/\partial
y\right)\right|_{w}^{-1}\tau_{w}/\rho-\nu$, following Bae & Lozano-Durán
(2021). For turbulent channel flows, the mean DNS quantities are obtained from
the database by Jimenéz and coworkers (Del Alamo et al., 2004; Hoyas &
Jiménez, 2006; Lozano-Durán & Jiménez, 2014). Four cases with friction
Reynolds numbers 550, 950, 2000 and 4200 are used. For the turbulent
Poiseuille-Couette flows, our in-house code is used to generate DNS data. Two
cases (labeled as PC-0 and PC-2) are considered, which correspond to
separation and adverse pressure gradient, respectively. The Reynolds numbers
based on the adverse pressure gradient are
$\mathit{Re_{P}}={\sqrt{h^{3}\mathrm{d}P/\mathrm{d}x}}/\nu=$ 680 and 340, and
the Reynolds number based on the top wall velocity is
$\mathit{Re_{U}}=u_{t}h/\nu=22,360$. The computational domain is $L_{x}\times
L_{y}\times L_{z}=4\pi h\times 2h\times 2\pi h$ for channel flows and
$L_{x}\times L_{y}\times L_{z}=2\pi h\times 2h\times\pi h$ for PC-0 and PC-2,
where $h$ is the channel half-height. The grid size for the WMLES cases with
ESGS is $\Delta\approx 0.2h$.

###  2.3 Predictor and classifier

The architecture of the ANNs is a fully connected feedforward neural network
with 6 hidden layers and 40 neurons per layer. The model comprises different
ANNs. The first ANN is used to predict the eddy viscosity far from the wall,
where viscous effects are negligible (i.e., outer layer ANN). To account for
near-wall viscous effects, two ANNs are used to predict $\nu_{t}$ at the wall
and at the first grid point off the wall. The input features to the outer
layer ANN are $(I_{1},...,I_{6},\nu,\Delta)$. For inner layer ANNs, the input
features are identical to the outer layer ANN in addition to $u_{||}$. The
ANNs were trained using stochastic gradient descent by randomly dividing
training data into two groups, the training set (80% of the data) and test set
(20% of the data). For laminar flows, the eddy viscosity is set to zero.

The classification of the flow at a given point is done in two steps. First,
the flow is classified as turbulent or laminar. If the flow is turbulent, it
is further classified as zero-pressure-gradient wall turbulence (ZPG),
adverse-pressure-gradient wall turbulence (APG), or separated turbulent flow.
A Naive Bayes (NB) classifier is employed whose inputs are the
nondimensionalized invariants (together with $u_{\parallel}$ normalized with
$\nu/\Delta$ for the second classification). For the turbulent/laminar
classifier, the invariants are normalized with $\nu$ and $\Delta$, whereas for
the second classification, $\sqrt{2\bar{S}_{ij}\bar{S}_{ij}}$, is used. The
output of the classifier is a label that is fed to the predictor, as shown in
Figure 1.

The classifier is trained with the same cases as the predictor in addition to
synthetic data generated for laminar Poiseuille flow at
$\mathit{Re_{\tau}}=[5-150]$. The classifier is applied to all grid points to
discern between turbulent and laminar flow. The second classification is only
applied to the points in contact with the wall. The classifier shows 100%
accuracy in the first classification. For the turbulent classification, Figure
2(a) shows the confusion matrix. The numbers in the cells indicate the
percentage of samples that are classified as a given class, showing that flows
are predicted with $>75\%$ accuracy.

Figure 2: (a) Confusion matrix of the classifier for the first grid point off
wall. C stands for ZPG flow, A for APG flow and S for separated flow. (b) Mean
streamwise velocity profile for laminar channel flow ($U_{c}$ is the
centerline velocity).

##  3 Results

###  3.1 Laminar channel flow

The BFM is first tested in a laminar channel flow with fixed centerline
velocity. The simulation is started from a random flow field and the data are
collected after transients. As shown in Figure 2(b), the BFM produces the same
solution as the analytical one by correctly classifying all the points as
laminar flow. Regarding wall-shear stress, the analytical solution corresponds
to a friction Reynolds number $\mathit{Re_{\tau}}=19.64$, and the prediction
of BFM is $\mathit{Re_{\tau}}=19.62$.

###  3.2 Turbulent channel flow

The BFM is next tested in turbulent channel flows. Two grid sizes
($\Delta=0.2h$ and $\Delta=0.1h$) are considered to assess the
generalizability of the model. The results of BFM are compared with the
Dynamic Smargorinsky (DSM) and Vreman SGS models using the ODE-based
equilibrium wall model (EQWM) by Larsson et al. (2016). The mean velocity
profiles are shown in Figure 3, and Table 1 summarizes the friction Reynolds
number predicted by WMLES. BFM exhibits the best performance in the coarse
grid. The predictions of BFM deteriorate when the grid size is refined as
shown in Figure 4; however, they are still comparable to those from
traditional models. In this preliminary version of BFM, the model was trained
only at one grid resolution. Better grid-convergence behavior is expected in
future versions of BFM trained at various grid sizes.

Figure 3: Mean streamwise velocity for turbulent channel flow using a coarse
grid ($\Delta=0.2h$). (a) $\mathit{Re_{\tau}}\approx 550$, (b)
$\mathit{Re_{\tau}}\approx 950$, (c) $\mathit{Re_{\tau}}\approx 2000$ and (d)
$\mathit{Re_{\tau}}\approx 4200$. Figure 4: Mean streamwise velocity for
turbulent channel flow using a medium grid ($\Delta=0.1h$). (a)
$\mathit{Re_{\tau}}\approx 550$, (b) $\mathit{Re_{\tau}}\approx 950$, (c)
$\mathit{Re_{\tau}}\approx 2000$ and (d) $\mathit{Re_{\tau}}\approx 4200$.

Case | $Re_{\tau}$ | DSM-EQWM | Vreman-EQWM | BFM | Trained | $Re_{P}$
---|---|---|---|---|---|---
550-C | 547 | 587(7.3%) | 578(5.7%) | 540(1.3%) |  | N/A
950-C | 943 | 1034(9.7%) | 1002(6.3%) | 929(1.5%) |  | N/A
2000-C | 2003 | 2228(11.2%) | 2157(7.7%) | 1976(1.3%) |  | N/A
4200-C | 4179 | 4641(11.1%) | 4474(7.1%) | 4179(0.0%) |  | N/A
550-M | 547 | 552(0.9%) | 545(0.4%) | 537(1.8%) |  | N/A
950-M | 943 | 973(3.2%) | 944(0.1%) | 906(3.9%) |  | N/A
2000-M | 2003 | 2137(6.7%) | 2050(2.3%) | 1988(0.7%) |  | N/A
4200-M | 4179 | 4502(7.7%) | 4286(2.6%) | 4006(4.1%) |  | N/A
PC0 | 7 | 76(1014%) | 68(871%) | 33(371%) |  | 680
PC1 | 194 | 164(15.5%) | 169(12.9%) | 172(11.3%) |  | 480
PC2 | 264 | 225(14.8%) | 221(16.3%) | 227(14.0%) |  | 340

Table 1: Friction Reynolds numbers predicted by different SGS and wall models.
$\mathit{Re_{\tau}}$ are the benchmark values from DNS. The “Trained” column
shows whether the case has been used for training. Cases ended with M were
simulated with a finer grid ($\Delta=0.1h$). Numbers in parentheses are the
relative errors defined by
$\left|(\mathit{Re_{\tau,\text{pred}}}-\mathit{Re_{\tau}})/\mathit{Re_{\tau}}\right|$.
$\mathit{Re_{P}}$ is the Reynolds number defined by the adverse pressure
gradient applied.

###  3.3 Turbulent Poiseuille-Couette flow

Three cases are considered: PC0 (separation), PC1 (stronger APG) and PC2 (mild
APG). The mean velocity profiles and wall-shear stress predictions are shown
in Figure 5 and Table 1, respectively. As shown in Table 1, BFM gives the best
wall-stress predictions for all cases. For PC0, BFM is capable of capturing
separation better than other models. For PC2 and PC1, the improvement is
marginal. As shown in Figure 5, BFM also outperforms or matches the accuracy
of other models in terms of the prediction of the mean velocity profile except
for PC1. The classifier was found to be responsible for hindering the overall
performance in this case. Assuming a perfect classification for PC1, the
results by BFM outperform other models, as shown in Figure 5. A more accurate
classifier is anticipated to help the model achieve better mean profile and
wall-shear stress predictions.

Figure 5: Mean streamwise velocity for Poiseuille-Couette flow. Three cases,
PC0 (separation), PC1 (stronger APG) and PC2 (mild APG), are overlaid on the
same plots.

###  3.4 CRM-HL

Finally, BFM is tested in the CRM-HL as representative of realistic aircraft
in high-lift configuration (Lacy & Sclafani, 2016). We focus on free air
simulations. The reader is referred to Goc et al. (2022) for a detailed
description of the physical and numerical setup.

We perform simulations for two angles of attack, $\alpha=7.05^{\circ}$ and
$19.57^{\circ}$, and compare the lift, drag and pitching moment coefficients
with experimental data and simulations from Goc et al. (2022) using DSM with
EQWM (labeled as DSM-EQWM). The grid has 40 million elements and is the same
as in Goc et al. (2022). A slice of the grid is depicted in Figure 6. The
simulations are conducted in charLES with a version of BFM that assumes ZPG.
Future versions of BFM will be retrained using WMLES data from charLES to
guarantee accuracy in the classification and consistency with the numerical
discretization.

Figure 6: Geometry of the CRM-HL model and a cross-sectional view of the grid
over the wing surface.

Figure 7 displays the lift, drag and pitching moment coefficients. For
$\alpha=7.05^{\circ}$, DSM-EQWM correctly predicts $C_{L}$ and $C_{D}$.
However, further analysis has shown that this accurate prediction is
coincidental and due to error cancellation when integrating the total forces.
This is corroborated by the poor predictions of pitching moment by DSM-EQWM.
However, BFM provides more consistent results. The case of
$\alpha=19.57^{\circ}$ corresponds to the angle of attack of maximum lift
coefficient. DSM-EQWM underpredicts the $C_{L}$, whereas BFM accurately
predicts $C_{L}$. The drag coefficient is overpredicted by both models, but
the pitching moment computed using BFM is closer to the experimental values,
suggesting that BFM provides a more accurate distribution of the forces over
the wing. Inspection of the sectional pressure coefficient (Figure 8)
indicates that BFM maintains the flow attached over a longer section of the
wing compared to DSM-EQWM, which explains the enhanced performance of BFM.

Figure 7: (a) Lift, (b) drag and (c) pitching moment coefficients for CRM-HL.
The black lines denote experimental results, squares are for DSM-EQWM and
circles are for BFM. Figure 8: Pressure coefficient at the 82% spanwise
section of the wing for $\alpha=19.57^{\circ}$. Black $\times$ represents
experimental results, orange squares are for DSM-EQWM and blue circles are for
BFM.

##  4 Conclusions

We have presented a unified SGS/wall model for large-eddy simulation by
devising the flow as a collection of building blocks whose information enables
the prediction of the SGS stress tensor. The model is referred to as the
building-block flow model (BFM) and is grounded in the idea that truly
revolutionary improvements in WMLES will encompass advancements in machine-
learning-based modeling consistent with the numerics and grid generation.

The first version of BFM discussed here is constructed to predict wall-
attached turbulence, adverse-pressure-gradient turbulence, separation and
laminar flow. The model is trained using WMLES with an exact-for-the-mean
SGS/wall model that guarantees consistency with the numerical schemes and the
gridding strategy. The model is applicable to complex geometries and is
implemented using artificial neural networks.

The BFM has been validated in canonical flows. These include laminar
Poiseuille flow, turbulent channel flows, and turbulent Poiseuille-Couette
flows mimicking adverse-pressure-gradient effects and separation. The
performance of BFM in complex scenarios is evaluated in the NASA Common
Research Model High-Lift (CRM-HL). We have shown that BFM outperforms
traditional SGS/wall models in the canonical flows and the CRM-HL. The only
exceptions are the coincidentally accurate predictions by the equilibrium wall
model due to error cancellation. To the best of our knowledge, this is the
first demonstration of a successful SGS/wall model implemented via artificial
neural networks applicable to realistic aircraft configurations.

### Acknowledgments

The authors acknowledge the MIT SuperCloud and Lincoln Laboratory
Supercomputing Center for providing HPC resources that have contributed to the
research results reported within this paper.

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