# Building-block flow model for computational fluids

Gonzalo Arranza,∗, Yuenong Linga, Sam Costaa, Konrad Gocb & Adrián Lozano-
Durána,∗
aDepartment of Aeronautics and Astronautics, Massachusetts Institute of
Technology, Cambridge, MA 02139
aThe Boeing Company, Everett, 98204, WA, USA
<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
###### Abstract

We introduce a closure model for computational fluid dynamics, referred to as
the Building-block Flow Model (BFM). The foundation of the model rests on the
premise that a finite collection of simple flows encapsulates the essential
physics necessary to predict more complex scenarios. The BFM is implemented
using artificial neural networks and introduces five unique advancements
within the framework of large-eddy simulation: (1) It is designed to predict
multiple flow regimes (wall turbulence under zero, favorable, adverse mean-
pressure-gradient, and separation); (2) It unifies the closure model at solid
boundaries (i.e., the wall model) and the rest of the flow (i.e., the subgrid-
scale model) into a single entity; (3) It ensures consistency with numerical
schemes and gridding strategy by accounting for numerical errors; (4) It is
directly applicable to arbitrary complex geometries; (5) It can be scaled up
to model additional flow physics in the future if needed (e.g., shockwaves and
laminar-to-turbulent transition). The BFM is utilized to predict key
quantities of interest in turbulent channel flows, a Gaussian bump, and an
aircraft in a landing configuration. In all cases, the BFM demonstrates
similar or superior capabilities in terms of accuracy and computational
efficiency compared to previous state-of-the-art closure models.

##  1 Introduction

The adoption of transformative low-emissions aero/hydro vehicle designs and
propulsion systems is significantly impeded by a major obstacle: the lengthy
and expensive experimental campaigns needed throughout the design cycle. These
campaigns can span years and cost billions of dollars [1]. Virtual testing via
computational fluid dynamics (CFD) might accelerate the process and alleviate
costs [2]. However, current CFD closure models do not comply with the
stringent accuracy requirements demanded by the aerospace industry [3]. These
limitations primarily stem from the challenging nature of turbulence, i.e.,
the chaotic and multiscale motion of flows, resulting in complex physical
phenomena.

Despite the challenges, novel modeling strategies hold the potential to unlock
substantial financial benefits worldwide, extending far beyond the aerospace
industry. The use of highly accurate CFD tools can facilitate the creation of
innovative vehicle designs that achieve up to 30% fuel savings [4]. This
breakthrough would result in substantial cost reductions, amounting to
hundreds of billions of dollars per year within the transportation sector
alone. For instance, the global ocean shipping industry consumes approximately
2 billion barrels of oil annually [5]. Similarly, the airline and trucking
industries consume approximately 1.5 billion barrels and 1.2 billion barrels
of oil per year, respectively. Additionally, a mere 5% reduction in
transportation drag is estimated to have an impact equivalent to doubling the
current wind energy production in the United States [5]. The implementation of
novel turbulence modeling techniques also carries immense potential for
mitigating environmental issues and reducing pollutant emissions [6].

In terms of prediction and control, the field of CFD is in the privileged
position of having a highly accurate framework to model flows: the
Navier–Stokes equations. However, the number of degrees of freedom involved in
real-world applications ranges between $10^{13}$ to $10^{20}$, making the
direct numerical simulation of all flow scales infeasible. Consequently, the
treatment of turbulence in industrial CFD has primarily relied on closure
models for the Reynolds-averaged Navier-Stokes (RANS) equations [7]. The
approach has materialized in various forms, from pure RANS solutions to hybrid
methods such as Detached Eddy Simulation and its variants [8]. Many RANS
models have been developed to overcome the limitations of their predecessors,
often by expanding and calibrating their coefficients to account for missing
physics. However, no practical model is universally applicable across the
broad range of flow regimes of interest to the industry. Examples of
challenging flow scenarios include separated flows, favorable/adverse pressure
gradient effects, shock waves, mean-flow three-dimensionality, and transition
to turbulence, among others, as illustrated in Fig. 1(left). In these
situations, RANS predictions tend to be inconsistent and unreliable,
especially for geometries and conditions representative of the flight envelope
of commercial airplanes [9]. Further CFD experience with aircraft at high
angles of attack has revealed that RANS-based solvers have difficulties
predicting maximum lift and the physical mechanisms for stall [10]. As a
result, the number of wind tunnel experiments required during the final stage
of the aircraft design cycle has remained at around ten for the past two
decades (Fig. 1 right).

Figure 1: (Left) Schematic of the different flow phenomena encountered over an
aircraft challenging current CFD methodologies. For compactness, all the
phenomena are overlaid in the schematic; however, not all of them occur
concurrently or even take place at all. (Right) Number of wind tunnel tests
required during the design cycle as a function of time. The horizontal axis
also marks the introduction of widely used RANS models. The graph is adapted
from ref.[11].

Recently, large-eddy simulation (LES) has gained momentum as a tool for both
scientific investigations and industrial applications. In LES, the large flow
motions containing most of the energy are directly resolved by the grid, while
the dissipative effect of the small scales is modeled through a subgrid-scale
(SGS) model. By additionally modeling the near-wall flow using a so-called
wall model, the grid-point requirements for wall-modeled LES (WMLES) scale at
most linearly with increasing Reynolds number [12, 13, 14], defined as the
ratio between the largest and smallest flow scales in the problem. Most widely
used SGS models were developed from the 1980s to the early 2000s and are based
on the eddy-viscosity assumption [15]. Among the most prominent SGS models, we
can cite the similarity model [16], Vreman model [17], dynamic Smagorinsky
model [18], deconvolution model [19], and sigma model [20], to name a few. A
detailed account of SGS models can be found in ref.[21]. Regarding wall
modeling, several strategies have been explored in the literature, and
comprehensive reviews can be found in refs. [22, 23, 8, 24, 25]. A widely
adopted method for wall modeling is the wall-flux approach. This approach
substitutes the no-slip and thermal boundary conditions at the wall with the
shear stress and heat flux values provided by the wall model. Popular examples
of these approaches include those based on the law of the wall [26, 27, 28],
the full/simplified RANS equations [29, 30, 31, 32, 33, 34, 35], vortex-based
models [36], or dynamic wall models [37, 38].

In this context, machine learning (ML) has emerged as a powerful tool to
enhance existing turbulence modeling approaches in the fluid community[39, 40,
41, 42, 43, 44]. Over the last two decades, there have been multiple efforts
devoted to the development of SGS and wall models using ML tools. Most ML-
based SGS models rely on supervised learning, which involves the discovery of
a function that maps an input to an output based on provided training input-
output pairs. Early approaches used artificial neural networks (ANNs) to
emulate and speed up conventional, yet computationally expensive, SGS
models[45]. More recently, SGS models have been trained to predict the so-
called perfect SGS terms using data from filtered direct numerical simulation
(DNS)[46, 47, 48]. Other approaches involve deriving SGS terms from optimal
estimator theory[49] and deconvolution operators[50, 51, 52], as well as from
reinforcement learning[53]. Similar ML methods have been employed to devise
wall models for LES via supervised learning. One of the initial attempts can
be found in ref. [54]. Other examples include spanwise rotating channel
flows[55], flows over periodic hills[56], turbulent flows with separation[57],
and boundary layer flows in the presence of shock-boundary layer
interaction[58], with mixed results in a-posteriori testing. Recent works have
also leveraged semi-supervised learning to devise wall models[59], although
these approaches are still limited to simple flow configurations.

Despite the progress made, ML-based SGS and wall models still face challenges
and have yet to serve as an effective solution for addressing long-standing
issues in turbulence modeling for CFD. Even two decades after the introduction
of the first ML-based SGS model for LES[45], there has been only a single
demonstration of an ML-based SGS model applied to a realistic aircraft
configuration. This demonstration was conducted using a prototype of the model
presented here[60]. The goal of this study is to introduce a novel, unified
SGS and wall model for WMLES, aiming to bridge the gap between our current
predictive capabilities and the demanding requirements of the aerospace
industry.

##  2 Results

###  2.1 The building-block flow model

The simulation of turbulent flows involves solving the equations for
conservation of momentum:

$\frac{\partial\rho u_{i}}{\partial t}+\frac{\partial\rho u_{i}u_{j}}{\partial
x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial\sigma_{ij}}{\partial
x_{j}},\ i=1,2,3,$ (1)

along with the conservation of mass, energy, and the equation of state for the
gas. Here, $\rho$ represents the flow density, $u_{i}$ denotes the $i$-th
velocity component, $p$ signifies the pressure, and $\sigma_{ij}$ stands for
the viscous stress tensor. Note that repeated indices in Eq. (1) imply
summation. The challenge arises in numerically solving the equations on a
computational grid with the ability to capture the smallest scales of the
problem. In most realistic scenarios, this task becomes computationally
intractable due to the hundreds of billions of grid points required.
Typically, affordable grids comprise of the order of hundred million points.
However, the latter grids fall short in capturing the smallest flow motions
that significantly contribute to the mean forces on the vehicle. The solution
lies in introducing a correction term to the right-hand side of Eq. (1),
denoted as $\partial\tau^{\mathrm{SGS}}_{ij}/\partial x_{j}$, where
$\tau^{\mathrm{SGS}}_{ij}$ is the SGS closure model. This corrective term
accounts for the physics of the small scales unresolved by the coarse grid.

The Building-block Flow Model (BFM) provides the SGS tensor,
$\tau^{\mathrm{SGS}}_{ij}$ that tackles the challenge posed by missing scales
at both solid boundaries (traditionally addressed with a wall model) and in
the flow above these boundaries (traditionally modeled with an SGS model). An
overview of the model is shown in Fig. 2. The core modeling assumption is that
the physics of the missing scales in complex scenarios can be locally mapped
onto the small scales of simpler flows [60, 66]. Under this premise, it is
postulated that there is a finite set of simple flows, referred to as
building-block flows (BBFs), containing the essential flow physics necessary
to formulate generalizable SGS and wall models. In the following, we discuss
the key components of the BFM: BBFs, model architecture, and training data.

Figure 2: Schematic of the BFM applied to an aircraft in landing
configuration. The tail of the aircraft is included in the model just for
illustration purposes. The color over the surface of the aircraft represents
the magnitude of the wall-shear stress normalized by the freestream velocity
and density. The panel displays a cross-section of the grid close to the solid
boundary, illustrating the three ANNs that constitute the BFM, along with
their respective inputs and outputs.

###  2.2 Building block flows

The BBFs offer simple representations of different flow regimes for which
scale-resolving, high-fidelity DNS data is available. The configuration of the
BBFs entails an incompressible turbulent flow confined between two parallel
walls. Four types of BBFs are considered: wall-bounded turbulence under
separation, adverse, zero, and favorable mean pressure gradient. Examples of
these BBFs are illustrated in Fig. 3. The intentional exclusion of cases with
additional complexities, such as airfoils, wings, bumps, etc., is grounded in
the idea that the BBFs should encapsulate fundamental flow physics to predict
more intricate scenarios. Hence, the aim is to avoid case overfitting, e.g.,
correctly predicting the flow over a wing merely because the model was trained
on similar wings. Note that the largest scales of the flow are intended to be
resolved by the computational grid, whereas the BBFs represent the smaller
flow features. This justifies the geometric simplicity of the BBFs over more
complex geometries. Additionally, the computational affordability of the BBFs
allows for the collection of a large number of high-fidelity simulations for
training, which facilitates efficient exploration of multiple flow regimes.
This aspect is important as the BFM must learn the physical scaling of the
non-dimensional inputs and outputs that control the problem at hand, rather
than merely fitting to a set of cases.

Figure 3: Examples of building-block flows (from left to right): wall-bounded
turbulence under separation, adverse, zero, and favorable mean pressure
gradient. The training database comprises several cases for each building
block flow. Dark blue corresponds to positive streamwise velocity, and orange
corresponds to negative streamwise velocity. The solid lines and arrows
represent the mean velocity profile.

###  2.3 Model architecture

The BFM is implemented using three feedforward ANNs (see Fig. 2). The first
ANN, adapted from the ML-based wall model by Lozano-Durán and Bae [66], is
tasked with predicting the wall-shear stress at the solid boundaries
($\boldsymbol{\tau}_{w}$). The second ANN predicts the SGS stress tensor for
control volumes adjacent to the solid boundaries, while the third ANN does the
same for the remaining control volumes. The ANNs take as input the local
values of the invariants of the symmetric ($\tilde{\mathbf{S}}$) and
antisymmetric ($\tilde{\mathbf{R}}$) parts of the velocity gradient tensor
[63]. Additionally, the first and second ANNs incorporate information about
the wall-parallel velocity of the flow relative to the solid boundary
($u_{||}$). These inputs are chosen to ensure invariance under constant
translations and rotations of the reference frame, along with Galilean
invariance. The output of the ANNs is the value of the SGS stress tensor via
the eddy viscosity $\nu_{t}$. The inputs and output variables of the first ANN
are non-dimensionalized using viscous scaling (i.e., the kinematic viscosity
$\nu$ and the local grid size). For the second and third ANNs, the variables
are non-dimensionalized using semi-viscous scaling (based on $\nu$ and the
first invariant itself). The BFM is developed for GPU architectures using the
CUDA programming language, which is particularly efficient in evaluating ANNs
compared to the CPU counterparts.

###  2.4 Training data

The training data is generated using a novel method that incorporates an
‘exact-for-the-mean’ SGS/wall model. The approach, referred to as E-WMLES,
involves conducting WMLES with a control mechanism that identifies the
required SGS tensor to predict specific mean statistics of interest in the
flow. The statistics considered here are the mean velocity profiles and the
mean wall-shear stress, which are obtained from high-fidelity data of the
BBFs. The BBFs were simulated using E-WMLES in the same numerical flow solver
and grid strategies subsequently employed for implementing the BFM. The data
generated from these simulations were used to train the BFM, ensuring
consistency between the model and the numerical schemes of the flow solver.

The E-WMLES approach diverges from the standard practice within the community
of training SGS models using filtered or coarse-grained high-fidelity data.
Our preference for the E-WMLES approach arises from the known fact that the
SGS tensor in WMLES is not consistent with the filtered terms derived from the
Navier-Stokes equations [69, 70, 71, 72]. This limitation is particularly
significant in external aerodynamics applications, where the typical grid
sizes used in WMLES are orders of magnitude ($10^{2}$ to $10^{4}$) coarser
than the smallest length scale of the flow. In these situations, numerical
errors become comparable to modeling errors, and both must be accounted for by
the SGS and wall model to yield accurate predictions.

###  2.5 Validation cases

We validate the performance of the BFM in three cases: a turbulent channel
flow, the separated flow over a bump, and an aircraft in landing
configuration. The results are compared with simulations using established SGS
and wall models conducted using identical meshes as those for the BFM. The
models for comparison are the dynamic Smagorisky SGS model (DSM) [18, 62] or
Vreman SGS model (VRE) [17] combined with the equilibrium wall model (EQ)
[33]. The models are labeled as DSM-EQ and VRE-EQ, respectively. In the
validation cases below, the computational cost of conducting WMLES with BFM is
approximately 0.9 times that of DSM-EQ and 1.1 times that of VRE-EQ.

###  2.6 Turbulent channel flow

The first validation case is a turbulent channel flow, a canonical
configuration in which the flow is confined between two parallel walls, as
depicted by one of the BBFs in Fig. 3. This case is used to show the
robustness of the BFM across Reynolds numbers and grid resolutions. The
friction Reynolds numbers considered are Re${}_{\tau}=u_{\tau}h/\nu\approx
2000$ and $4200$, where $u_{\tau}$ is the friction velocity, $h$ is the
channel half-height, and $\nu$ is the kinematic viscosity. The channel is
driven by a streamwise pressure gradient, such that
Re${}_{c}=U_{c}h/\nu=48500$ and $112500$, respectively, with $U_{c}$
representing the mean streamwise velocity at the centerline of the channel.
Three spatially isotropic grid resolutions are tested: $\Delta=0.05h$, $0.1$
and $0.2h$. The BFM predicts the mean velocity profile (Fig. 4) and the mean
wall-shear stress within an accuracy of 1% to 6% for the Reτ and grid
resolutions considered.

Figure 4: The mean velocity profile in a turbulent channel flow as a function
of the wall-normal distance at (A) Re${}_{\tau}\approx 2000$ for grid sizes
$\Delta\approx 0.2h$, and $0.1h$; (B) Re${}_{\tau}\approx 4200$ for grid sizes
$\Delta\approx 0.2h$ and $0.1h$; and (C) Re${}_{\tau}\approx 4200$ for
$\Delta\approx 0.05h$. The dashed line is DNS data [73] and the blue symbols
are the BFM. In (A) and (B) circles correspond to $\Delta\approx 0.2h$ and
triangles to $\Delta\approx 0.1h$.

###  2.7 Separated flow over a bump

Figure 5: (A) Schematic of the Gaussian bump; (B) grid structure and (C)
instantaneous streamwise velocity in the $y/L=0$ plane. The velocity of the
freestream is $U_{\infty}$. Note that the figure does not show the whole
domain in the streamwise direction. (D) Wall pressure coefficient
$C_{p}=\frac{p-p_{\infty}}{\frac{1}{2}\rho_{\infty}U^{2}_{\infty}}$ and (E)
wall friction coefficient
$C_{f}=\frac{\tau_{w}}{\frac{1}{2}\rho_{\infty}U^{2}_{\infty}}$, along the
$y/L=0$ plane, where $U_{\infty}$, $p_{\infty}$ and $\rho_{\infty}$ are the
freestream velocity, pressure and density, respectively, and $p$ and
$\tau_{w}$ are averaged over time. (F) Mean velocity profile at $y/L=0$ at
different streamwise locations. In (D,E,F) The solid lines denote BFM (blue),
DSM-EQ (green) and VRE-EQ (orange). White circles are experimental values[74].

The second validation case is a Gaussian bump designed by The Boeing Company
and the University of Washington [75], as shown in Fig. 5. The test consists
of a three-dimensional tapered hump at a Reynolds number of $3.6\times 10^{6}$
and a Mach number of $0.17$, based on the freestream quantities and the length
of the bump $L$. This test case has proven to be highly challenging for the
CFD community, with inaccurate prediction of the separation bubble and non-
monotonic convergence of the solution with grid refinements[76, 77].

We evaluate the performance of the BFM in predicting the mean velocity
profiles, and the mean pressure and friction at the wall (characterized by the
non-dimensional coefficients $C_{p}$ and $C_{f}$, respectively). The
simulation consists of roughly 9 million control volumes. Similar WMLESes are
conducted using DSM-EQ and VRE-EQ. The numerical findings are compared with
experimental data[74]. Figure 5(D,E) illustrates the evolution of $C_{p}$ and
$C_{f}$ along the centerline of the bump ($x/L$). Notably, the BFM yields more
accurate predictions of the $C_{p}$ and $C_{f}$ evolution in the separation
region ($x/L>0.1$) compared to DSM-EQ and VRE-EQ, which even fail to predict
flow separation. The improved predictions by the BFM also extend to the mean
velocity profiles, as demonstrated in Fig. 5F. The BFM predicts mean velocity
with up to 3% accuracy, whereas both DSM-EQ and VRE-EQ yield a similar result
that overestimate the mean velocities by more than 30%. Nevertheless, there is
still room for improvement upstream of the bump ($x/L<0$), where all models
exhibit low accuracy in the prediction of $C_{f}$.

###  2.8 Aircraft in landing configuration

The third validation case is an aircraft in landing configuration,
specifically the NASA Common Research Model High-Lift (CRM-HL). This case
serves as the gold standard used by the aerospace community to assess the
capabilities of different CFD methodologies [78]. The CRM-HL is a
geometrically realistic aircraft that includes the bracketry associated with
deployed flaps and slats, as well as a flow-through nacelle mounted on the
underside of the wing (Fig. 2). The Reynolds number of the aircraft is 5.49
million based on the mean aerodynamic chord and freestream velocity, and the
freestream Mach number is 0.20.

The simulations are conducted using approximately 30 million control volumes.
Two angles of attack are considered: $\alpha=7.05^{\circ}$ and
$19.57^{\circ}$. Figure 2 contains the model geometry, an inset of the mesh,
and a visualization of the wall-shear stress predicted by BFM.

Figure 6 displays the forces acting on the aircraft, characterized by the lift
($C_{L}$), drag ($C_{D}$), and pitching moment ($C_{M}$) coefficients for BFM,
DSM-EQ, and VRE-EQ, compared with experimental data [79]. For
$\alpha=7.05^{\circ}$, DSM-EQ and VRE-EQ predict values of $C_{L}$ and $C_{D}$
close to experimental results. However, further analysis has shown that this
accurate prediction is coincidental and due to error cancellation during the
integration of total forces [80]. This is corroborated by the poorer
predictions of the pitching moment by DSM-EQ and VRE-EQ. The case of
$\alpha=19.57^{\circ}$ corresponds to the angle of attack of the maximum lift
coefficient. DSM-EQ and VRE-EQ underpredict $C_{L}$, whereas the BFM offers
improved predictions. Inspection of the sectional pressure coefficient, shown
in Fig. 7, indicates that the BFM maintains the flow attached over a longer
section of the wing compared to DSM-EQ and VRE-EQ, explaining the enhanced
performance of the BFM. Overall, BFM yields improved predictions for the three
coefficients compared to traditional models. The results are still far from
satisfactory, especially for the pitching moment, and there is clear potential
for further improvements. Nonetheless, it is worth noting that BFM has never
‘seen’ an aircraft-like flow or been trained in a case resembling an airfoil
or a wing. This demonstrates the ability of the BFM methodology to offer
predictions that go beyond simple regression.

Figure 6: (A) Lift coefficient, (B) drag polar and (C) pitching moment
coefficient for the CRM-HL. The black lines denote experimental results, blue
circles are for the BFM, squares are for DSM-EQ and orange squared for VRE-EQ.
Figure 7: Pressure coefficient along the chord of the wing at the 82% spanwise
section of the wing for $\alpha=19.57^{\circ}$. The chord location is
normalized by the sectional chord-length $c_{\text{ref}}$. White circles
represents experimental results, blue circles are for the BFM, orange and
green squares are for DSM-EQ and VRE-EQ, respectively.

##  3 Discussion

###  3.1 Opportunities of BFM for CFD

The BFM has the potential to significantly advance the resolution of key Grand
Challenges in the aerospace industry. These challenges include simulating an
aircraft across its entire flight envelope and addressing off-design turbofan
engine transients –both highlighted in the NASA CFD Vision 2030 report [3].
The BFM can also play a significant role in Certification by Analysis, i.e.,
the development of simulation-based methods of compliance for airplane and
engine certification [1]. This marks a long-awaited milestone in the aerospace
industry, with the prospect of Certification by Analysis estimated to result
in substantial cost savings, amounting to billions of dollars in the design
process.

A distinctive advantage of the BFM, which might contribute to addressing the
aforementioned challenges compared to previous models, is its ability to
accommodate various flow regimes through the BBF database. The current version
of BFM includes separated flow and the effects of zero, adverse, and favorable
mean pressure gradients. However, the process of incorporating additional flow
physics is scalable by expanding the building block database, a capability
absent in current SGS/wall models. Future versions of BFM could involve BBFs
accounting for laminar flow, compressibility effects, shock waves, wall
roughness, and the laminar-to-turbulent transition. Efforts to include these
new physics are already underway in our group.

Finally, the machine-learning nature of the BFM also unlocks other
opportunities for uncertainty quantification and adaptive grid refinement.
These might be accomplished by computing a confidence score based on the
similarity between the actual input data and the BFM training data. This
feature has already been tested in a preliminary version of the BFM, which
incorporates only a wall model[66]. The work has demonstrated that a
confidence score can be determined by the distance of the input data to the
closest sample in the training set. If the input data appears unusual, the
model assigns a low confidence score, indicating that the flow is unfamiliar
and does not align with any knowledge from the database. Regions with low
confidence scores can be used to assist in automatic grid refinement in
subsequent simulations, to identify the need for additional building blocks,
and for uncertainty quantification.

###  3.2 Conclusions

We have introduced the Building-Block Flow Model (BFM) for large-eddy
simulation. The model is designed to address challenges encountered in CFD
within the aerospace industry, specifically the demand for accurate and robust
solutions at an affordable computational cost. The core assumption of the BFM
is that the subgrid-scale physics of complex flows can be effectively
represented by the physics of simpler canonical flows. Implemented using three
feedforward ANNs for GPU architectures, the BFM is applicable to arbitrarily
complex geometries. Unlike previous models, the training data for the BFM is
derived from controlled WMLES with an ‘exact’ model for the mean quantities of
interest. This approach ensures consistency with the numerical discretization
and grid structure of the solver. We have shown that, at the resolutions
considered here, the BFM offers improvements in predictions over conventional
SGS and wall models in both simple and complex scenarios.

Truly revolutionary improvements in WMLES will encompass advancements in
numerics, grid generation, and wall/SGS modeling. The BFM addresses all these
aspects by devising SGS/wall models consistent with the numerics and the grid.
The modularity of the BFM opens up new opportunities for developing SGS/wall
models that have broad applicability across different scenarios. In essence,
the BFM offers a single model that can accurately represent various flow
physics, eliminating the need for multiple specialized models for specific
flow types. To enhance model performance, we will continue the training of
future versions of the BFM with additional building blocks to account for a
richer collection of flow physics.

#### Acknowledgments

This work was supported by the National Science Foundation under grant
#2317254 and by an Early Career Faculty grant from NASA’s Space Technology
Research Grants Program (grant #80NSSC23K1498). S. C was supported by The
Boeing Company. G. A. was partially supported by the NNSA Predictive Science
Academic Alliance Program (PSAAP; grant DE-NA0003993). The authors acknowledge
the support of the CTR Summer Program 2022 at Stanford University. The authors
also acknowledge the Massachusetts Institute of Technology, SuperCloud, and
Lincoln Laboratory Supercomputing Center for providing HPC resources that have
contributed to the research results reported here.

#### Author contributions

A.L.D. devised the idea and also wrote the manuscript, G.A. developed the
model and wrote the manuscript, Y.L. developed and trained preliminary
versions of the BFM, S.C. ported the BFM to GPU architecture, K.G. run the
simulations for conventional WMLES and provided guidance on test cases and
their numerical setup.

#### Competing interests

The authors declare no competing interests.

##  Appendix A Methods

###  A.1 Numerical solver and conventional models

The BFM provides the closure model for the SGS stress tensor
$\tau_{ij}^{\mathrm{SGS}}$ for the compressible LES equations

$\displaystyle\frac{\partial\bar{\rho}}{\partial
t}+\frac{\partial\bar{\rho}\tilde{u}_{i}}{\partial x_{i}}$ $\displaystyle=$
$\displaystyle 0,$ (2)
$\displaystyle\frac{\partial\bar{\rho}\tilde{u}_{i}}{\partial
t}+\frac{\partial\bar{\rho}\tilde{u}_{i}\tilde{u}_{j}}{\partial x_{j}}$
$\displaystyle=$ $\displaystyle-\frac{\partial\bar{p}}{\partial
x_{i}}+\frac{\partial\tilde{\sigma}_{ij}}{\partial
x_{j}}-\frac{\partial\tau^{\mathrm{SGS}}_{ij}}{\ \partial x_{j}},$ (3)
$\displaystyle C_{v}\frac{\partial\bar{\rho}\tilde{T}}{\partial
t}+C_{v}\frac{\partial\bar{\rho}\tilde{u}_{j}\tilde{T}}{\partial x_{j}}$
$\displaystyle=$ $\displaystyle-\bar{p}\frac{\partial\tilde{u}_{j}}{\partial
x_{j}}+\tilde{\sigma}_{ij}\frac{\partial\tilde{u}_{j}}{\partial x_{i}}$ (4)
$\displaystyle+$ $\displaystyle\frac{\partial}{\partial
x_{j}}\left(\tilde{\kappa}\frac{\partial\tilde{T}}{\partial
x_{j}}\right)-C_{v}\frac{\partial q^{\mathrm{SGS}}_{j}}{\partial x_{j}},$ (5)

where repeated indices imply summation, $(\bar{\cdot})$ denotes filtered
quantities, $(\tilde{\cdot})$ is the Favre average, $u_{i}$ is the $i$-th
velocity component, $\rho$ is the density, $T$ is the temperature, $\kappa$ is
the thermal conductivity, $C_{v}$ is the specific heat at constant volume, and
$\sigma_{ij}$ is the viscous stress tensor. Note that repeated indices imply
summation. The heat flux is computed as
$q^{\mathrm{SGS}}_{j}=(\bar{\rho}\nu_{t}/\text{Pr})\partial\tilde{T}/\partial
x_{j}$, where Pr is the turbulent Prandtl number and $\nu_{t}$ is the eddy
viscosity. The applications considered in this work are low speed flow, and
heat transfer and compressibility effects play no significant role.

The simulations are conducted with the solver GPU version of charLES developed
by Cascade Technologies, Inc. (Cadence) [61]. The code integrates the
compressible LES equations using a kinetic-energy conserving, second-order
accurate, finite volume method. The numerical discretization relies on a flux
formulation which is approximately entropy preserving in the inviscid limit,
thereby limiting the amount of numerical dissipation added into the
calculation. The time integration is performed with a third-order Runge-Kutta
explicit method. The ANNs of the BFM are evaluated using the GPU capabilities
of the solver.

The mesh generation follows a Voronoi hexagonal close-packed point-seeding
method, which automatically builds locally isotropic meshes for arbitrarily
complex geometries. First, the water-tight surface geometry is provided to
describe the computational domain. Second, the coarsest grid resolution in the
domain is set to uniformly seeded points. Additional refinement levels are
specified in the vicinity of the walls if needed. Thirty iterations of Lloyd’s
algorithm are undertaken to smooth the transition between layers with
different grid resolutions.

The dynamic Smagorinsky model [18] with the modification by Lilly [62] and the
Vreman model [17] are considered as SGS models. For the wall, we use an
equilibrium wall model. The no-slip boundary condition at the wall is replaced
by a wall-stress boundary condition. The walls are assumed adiabatic and the
wall stress is obtained by an algebraic equilibrium wall model derived from
the integration of the one-dimensional stress model along the wall-normal
direction [28],

$u_{||}^{+}(y_{\perp}^{+})=\begin{cases}y_{\perp}^{+}+a_{1}(y_{\perp}^{+})^{2}\text{\,
\, \, for $y_{\perp}^{+}<23$},\\\ \frac{1}{\kappa}\ln{y_{\perp}^{+}}+B\text{\,
\, \, \, otherwise}\end{cases}$ (6)

where $u_{||}$ is the model wall-parallel velocity at the second grid point
off the wall, $y_{\perp}$ is the wall-normal direction to the surface,
$\kappa=0.41$ is the Kármán constant, $B=5.2$ is the intercept constant, and
$a_{1}$ is computed to ensure $C^{1}$ continuity. The superscript $+$ denotes
inner units defined in terms of wall friction velocity and the kinematic
viscosity.

###  A.2 BFM formulation

The anisotropic component of the SGS stress tensor is given by the eddy-
viscosity model

$\tau^{\text{SGS}}_{ij}=-2\nu_{t}\tilde{S}_{ij},$ (7)

where $\tilde{S}_{ij}$ is the rate-of-strain tensor. The eddy viscosity is a
function of the first five invariants ($I_{k}$) of rate-of-strain
($\mathbf{\tilde{S}}\equiv\tilde{S}_{ij}$) and rate-of-rotation tensors
($\mathbf{\tilde{R}}\equiv\tilde{R}_{ij}$) [63],

$\nu_{t}=f(I_{1},I_{2},I_{3},I_{4},I_{5},\theta),$ (8)

where $f$ represents a feedforward ANN, and $\theta$ denotes additional input
variables, namely, $\nu$, $\Delta$ and $u_{\parallel}$, where $\nu$ is the
kinematic viscosity, $\Delta$ 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 by the ANNs acting on the control
volumes adjacent to the walls. The invariants are defined as

$\begin{array}[]{ccccc}I_{1}=\operatorname{tr}\left(\mathbf{\tilde{S}}^{2}\right),&I_{2}=\operatorname{tr}\left(\mathbf{\tilde{R}}^{2}\right),&I_{3}=\operatorname{tr}\left(\mathbf{\tilde{S}}^{3}\right),\\\
I_{4}=\operatorname{tr}\left(\mathbf{\tilde{S}\tilde{R}}{}^{2}\right),&I_{5}=\operatorname{tr}\left(\mathbf{\tilde{S}}^{2}\mathbf{\tilde{R}}^{2}\right).\end{array}$
(9)

The ANNs are fully-connected feedforward networks. The first ANN, tasked with
predicting the wall-shear stress, consists of 6 layers with 40 neurons per
layer. The second near-wall ANN consists of 10 layers with 12 neurons per each
layer; and the outer-layer ANN consists of 10 layers with 16 neurons per
layer. The non-dimensionalization of the input and output features is attained
by using parameters that are local in both time and space to guarantee the
applicability of the model to complex geometries. For the ANN responsible for
predicting the wall-shear stress, the input and output quantities are non-
dimensionalized using viscous scaling: $\nu$ and $\Delta$. For the ANNs
predicting the eddy viscosity, the input and output variables are non-
dimensionalized using semi-viscous scaling:
$(\nu\sqrt{\mathbf{\tilde{S}}:\mathbf{\tilde{S}}})^{1/2}$ and $\Delta$.

###  A.3 Training data preparation

The mean velocity profiles and wall-shear stress are extracted from DNS data
of the BBFs. The turbulent channel flows are used to model the regime where
turbulence is fully developed without significant mean-pressure gradient
effects. Separation, adverse and favorable mean pressure gradient effects are
learned from turbulent Poiseuille-Couette (TPC) flows. In these cases, the top
wall moves at a constant speed ($U_{t}$) in the streamwise direction, and a
mean pressure gradient is applied in the direction opposed to the top wall
velocity. The pressure gradient ranges from mild to strong, so that the flow
“separates” (i.e., zero wall shear stress) on the bottom wall. Favorable mean
pressure gradient effects are obtained from the upper wall of the TPC cases.

For turbulent channel flows, the mean DNS quantities are obtained from the
database in refs.[64, 65]. Five cases with friction Reynolds numbers 550, 950,
2000, 4200 and 10000 are used. For the TPC flows, the data was obtained from
ref.[66]. The Reynolds numbers based on the mean pressure gradient are
$\mathit{Re_{P}}={\sqrt{h^{3}\mathrm{d}P/\mathrm{d}x}}/\nu=340,680$ and $962$,
and the Reynolds number based on the top wall velocity is
$\mathit{Re_{U}}=U_{t}h/\nu=22,360$, where $h$ is the channel half-height. The
computational domain in the streamwise, wall-normal, and spanwise direction 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 TPC cases.

To generate the training data, we perform WMLES simulations of the BBFs
adjusting $\nu_{t}$ to match the mean DNS velocity profile and correct wall-
shear stress using the E-WMLES approach[66]. The simulations are performed in
charLES. The eddy viscosity is adjusted by finding a correcting factor $k(y)$
to a base model $\nu^{\text{base}}_{t}$ (in this case Vreman SGS model [17]),

$\quad\nu_{t}(x,y,z,t)=\nu^{\text{base}}_{t}(x,y,z,t)k(y),$ (10)

where $x$, $y$, and $z$ are the streamwise, wall-normal, and spanwise
directions, respectively, of the BBFs. The correcting factor is in turn
computed by solving the optimization problem

$\arg\min_{k(y)}\quad\int|U^{\text{DNS}}(y)-\tilde{U}(y)|^{2}\mathrm{d}y$ (11)

where $U^{\text{DNS}}$ is the mean velocity profile from DNS and $\tilde{U}$
is the mean velocity profile from WMLES obtained using the eddy viscosity in
(10). The free Conjugate-Gradient algorithm and the Bayesian Global
optimization algorithm [67] are used to minimize (11) for the turbulent
channel and the TPC flows, respectively. 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
$\left.\nu_{t}\right|_{w}=\left.\left(\partial\tilde{u}/\partial
y\right)\right|_{w}^{-1}\tau_{w}/\rho-\nu$, following [68], where the
subscript $w$ denotes quantities at the wall, and $\tau_{w}$ is the mean wall-
shear stress. The grid size for the WMLES cases with E-WMLES is $\Delta\approx
0.1h$ and $\approx 0.2h$ for the turbulent channel flows and $\Delta\approx
0.1h$ for the TPC flows.

The optimization process for a given case is as follows: 1) WMLES simulation
is performed with a fixed $\tau_{w}$ –equal to the correct value from DNS
simulations, $\tau_{w}^{\text{DNS}}$– and with an initial random $k(y)$; 2)
the simulation is run until the statistical steady state is reached; 3) the
integral in (11) is evaluated, and 4) a new guess of $k(y)$ is provided by the
optimizer. This approach is continued until the condition

$\frac{|U^{\text{DNS}}(y)-\tilde{U}(y)|}{U^{\text{DNS}}(y)}<0.03$

at each $y$ location for the turbulent channel flow cases is satisfied. For
the TPC cases, this condition was too stringent, since velocities are close to
$0$ near the wall, and we relaxed the condition to

$\frac{|U^{\text{DNS}}(y)-\tilde{U}(y)|}{U_{t}}<0.02.$

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