Building-block-flow computational model for large-eddy simulation of external aerodynamic applications

Abstract

Computational fluid dynamics is an essential tool for accelerating the discovery and adoption of transformative designs across multiple engineering disciplines. Despite its many successes, no single approach consistently achieves high accuracy for all flow phenomena of interest, primarily due to limitations in the modeling assumptions. Here, we introduce a closure model for wall-modeled large-eddy simulation to address this challenge. The model, referred to as the Building-block Flow Model (BFM), rests on the premise that a finite collection of simple flows encapsulates the essential missing physics necessary to predict more complex scenarios. The BFM is designed to: (1) predict multiple flow regimes, (2) unify the closure model at solid boundaries and the rest of the flow, (3) ensure consistency with numerical schemes and gridding strategies by accounting for numerical errors, (4) be directly applicable to arbitrary complex geometries, and (5) be scalable to model additional flow physics in the future. The BFM is utilized to predict key quantities in five cases, including an aircraft in landing configuration, demonstrating similar or superior capabilities compared to previous state-of-the-art models. The design of BFM opens up new opportunities for developing closure models that can accurately represent various flow physics across different scenarios.

Arranz and colleagues introduce a closure model for computational fluid dynamics. Their approach is implemented using artificial neural networks. It predicts multiple flow conditions, is directly applicable to complex geometries, and ensures consistency with numerical schemes.

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 dollars¹. Virtual testing via computational fluid dynamics (CFD) might accelerate the process and alleviate costs². However, current CFD closure models do not comply with the stringent accuracy requirements demanded by the aerospace industry³. 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 savings⁴. 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 annually⁵. 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 States⁵. The implementation of novel turbulence modeling techniques also carries immense potential for mitigating environmental issues and reducing pollutant emissions⁶.

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 10¹³ to 10²⁰, 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) equations⁷. The approach has materialized in various forms, from pure RANS solutions to hybrid methods such as Detached Eddy Simulation and its variants⁸. 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. 1a. In these situations, RANS predictions tend to be inconsistent and unreliable, especially for geometries and conditions representative of the flight envelope of commercial airplanes⁹. 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 stall¹⁰. 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. 1b).

Fig. 1. Challenging flow phenomena and state of the art in computational fluid dynamics (CFD).

a 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. b Number of wind tunnel tests required during the design cycle as a function of the year. The horizontal axis also marks the introduction of widely used Reynolds-averaged Navier-Stokes (RANS) models. The graph is adapted from ref. ¹⁰¹. c Selected milestones in wall and subgrid-scale (SGS) modeling for wall-modeled (WM) large-eddy simulations (LES) of external aerodynamics. The text in blue highlights milestones in traditional (WM)LES; the text in red highlights machine-learning based milestones in (WM)LES. In (c) abbreviations stand for: law of the wall (LoW), thin boundary layer (TBL), scale similarity model (SSM), Dynamic Smagorinsky model (DSM), Wall-Adapting Local Eddy-viscosity (WALE), and artificial neural network (ANN).

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^11–13, 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 assumption¹⁴ (see Fig. 1c). Among the most prominent SGS models, we can cite the similarity model¹⁵, Vreman model¹⁶, dynamic Smagorinsky model¹⁷, deconvolution model¹⁸, and sigma model¹⁹, to name a few. A detailed account of SGS models can be found in ref. ²⁰. Regarding wall modeling, several strategies have been explored in the literature, and comprehensive reviews can be found in refs. ^8,21–24. 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^25–27, the full/simplified RANS equations^28–34, vortex-based models³⁵, or dynamic wall models^36,37.

In this context, machine learning (ML) has emerged as a powerful tool to enhance existing turbulence modeling approaches in the fluid community^38–43. 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 models⁴⁴. More recently, SGS models have been trained to predict the so-called perfect SGS terms using data from filtered direct numerical simulation (DNS)^45–54. Other approaches involve deriving SGS terms from optimal estimator theory⁵⁵ and deconvolution operators^56–60, as well as from reinforcement learning^61–63. 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. ⁶⁴. Other examples include spanwise rotating channel flows⁶⁵, flows over periodic hills⁶⁶, turbulent flows with separation⁶⁷, and boundary layer flows in the presence of shock-boundary layer interaction⁶⁸, with mixed results in a posteriori testing. Recent works have also leveraged semi-supervised learning to devise wall models⁶⁹, 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 LES⁴⁴, 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 here⁷⁰. The goal of this study is to introduce a 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.

Results

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, ρ represents the flow density, u_i denotes the i-th velocity component, p signifies the pressure, and σ_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 $-\partial {\tau }_{ij}^{\mathrm{SGS}}/\partial x_j$, where ${\tau }_{ij}^{\mathrm{SGS}}$ 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, ${\tau }_{ij}^{\mathrm{SGS}}$ 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 addressed with an SGS model). An overview of the BFM 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^70,71. 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. The current model builds upon the previous wall model developed in ref. ⁷¹. One of the main conclusions of that study was that the primary limitation of the developed wall model stemmed from modeling errors due to the suboptimal performance of SGS models. The BFM formulation presented here addresses this issue by devising consistent SGS and wall models in a unified manner. A comparison of the BFM with ref. ⁷¹ can be found in the Methods section. In the following, we discuss the key components of the BFM: BBFs, model architecture, and training data.

Fig. 2. Schematic of the Building-block Flow Model (BFM).

The color over the surface of the aircraft represents the magnitude of the wall-shear stress (τ_w) normalized by the freestream velocity (U_∞) and density (ρ_∞). The tail of the aircraft is included in the model just for illustration purposes. The panel displays a cross-section of the grid close to the solid boundary, illustrating the three artificial neural networks (ANNs) that constitute the BFM, along with their respective inputs and outputs.

Building block flows

The BBFs offer simple representations of different flow regimes for which scale-resolving, high-fidelity DNS data or experimental data are 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 cases. Additionally, the computational affordability of the BBFs allows for the collection of a large number of high-fidelity data 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 a set of cases.

Fig. 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.

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 Bae⁷¹, is tasked with predicting the wall-shear stress at the solid boundaries (τ_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 tensor⁷². 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 ν_t. The inputs and output variables of the first ANN are non-dimensionalized using viscous scaling (i.e., the kinematic viscosity ν and the local grid size ∆). For the second and third ANNs, the variables are non-dimensionalized using semi-viscous scaling (based on ν 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.

Numerically-consistent training data

The training data is generated using a 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 gridding 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^73–76. This limitation is particularly important in external aerodynamics applications, where the typical grid sizes used in WMLES are orders of magnitude (10² to 10⁴) 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.

Validation cases

We evaluate the performance of the BFM in five cases: turbulent channel flows, a turbulent pipe flow, the separated flow over a bump, a simplified aircraft model and a realistic 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 Smagorinsky SGS model (DSM)^17,77 or Vreman SGS model (VRE)¹⁶ combined with the equilibrium wall model (EQ)³². 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.

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. Note that although this case is part of the training dataset, it is not guaranteed that BFM will perform well in a posteriori testing due to potential inconsistencies between an ANN-based model and the numerical schemes of the solver^42,78–80. Therefore, this case serves as a demonstration that our strategy to enforce numerical consistency is successful in actual WMLES across various Reynolds numbers and grid resolutions. We also use this case to show the robustness of BFM to grid resolutions finer than those used for training. The friction Reynolds numbers considered are Re_τ = u_τh/ν ≈ 2000 and 4200, where u_τ is the friction velocity, h is the channel half-height, and ν is the kinematic viscosity. The channel is driven by a streamwise mean pressure gradient, such that Re_c = U_ch/ν = 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: Δ = 0.05h, 0.1 and 0.2h, where the first was not included in the training dataset. The BFM accurately predicts the mean velocity profile (Fig. 4) and the mean wall shear stress, achieving an accuracy of 1% to 6% across the considered Reynolds numbers and grid resolutions.

Fig. 4. Mean velocity profile in a turbulent channel flow as a function of the wall-normal distance.

a Re_τ ≈ 2000 for grid sizes Δ ≈ 0.2h, and 0.1h. b Re_τ ≈ 4200 for grid sizes Δ ≈ 0.2h and 0.1h; and (c) Re_τ ≈ 4200 for Δ ≈ 0.05h. The dashed line is data from direct numerical simulations (DNS)¹⁰² and the blue symbols are the Building-block Flow Model (BFM). In (a) and (b) circles correspond to Δ ≈ 0.2h and triangles to Δ ≈ 0.1h.

Turbulent pipe flow

The second validation case is the turbulent flow in a pipe (see Fig. 5a). We use this case to evaluate the performance of BFM at Reynolds numbers much higher than those it was trained for. The friction Reynolds number is Re_τ = u_τR/ν = 39500, where R is the pipe radius, and the flow is incompressible. The pipe is assumed periodic in the streamwise direction with a length of 7.5R. As in the turbulent channel flow, the pipe is driven by a streamwise mean pressure gradient such that Re_c = U_cR/ν ≈ 1.27 million, with U_c representing the mean streamwise velocity at the pipe centerline. The mesh is spatially isotropic (see Fig. 5a) with a grid size of Δ = 0.1R, which yields roughly 30000 grid points.

Fig. 5. Turbulent pipe flow.

a Schematic and flow visualization of the pipe flow at Re_τ = 39500. The radial and axial slices depict the grid structure and the instantaneous streamwise velocity. b Mean velocity profile along the radial direction r from the wall (r/R = 0) to the pipe centerline (r/R = 1). Symbols correspond to Building-block Flow Model (BFM) (blue circles), Dynamic Smagorinsky model (DSM) (red squares), and Vreman model (VRE) (yellow triangles). The dashed line is experimental values⁸¹. The wall model used for DSM and VRE is the equilibrium wall model (EQ).

We evaluate the performance of BFM, DSM-EQ, and VRE-EQ using the same flow conditions and computational grid. The results are compared with experimental data⁸¹. Figure 5b displays the mean velocity profile, showing the improved agreement of BFM with the experimental results with respect to DSM-EQ and VRE-EQ. Similarly, BFM predicts the average wall shear stress within a 3% error compared to the experiment, whereas DSM-EQ and VRE-EQ overpredict the average wall shear stress by 22% and 13%, respectively.

Separated flow over a bump

The second validation case is a Gaussian bump designed by The Boeing Company and the University of Washington⁸², as shown in Fig. 6. The test consists of a three-dimensional tapered hump at a Reynolds number of 3.6 million and a Mach number of 0.17, based on the freestream quantities and the length of the bump along the spanwise direction, L. This test case has proven to be highly challenging for the RANS and LES community, with inaccurate prediction of the separation bubble and non-monotonic convergence of the solution with grid refinements^83,84 (see Methods section).

Fig. 6. Separated flow over a bump.

a Schematic of the Gaussian bump; (b) grid structure and (c) instantaneous streamwise velocity in the z/L = 0 plane. The velocity of the freestream is U_∞. Note that the figure does not show the whole domain in the streamwise direction. d Wall pressure coefficient $C_p=\frac{p-p_{\infty }}{\frac{1}{2}{\rho }_{\infty }{U}_{\infty }^2}$ and (e) wall friction coefficient $C_f=\frac{{\tau }_w}{\frac{1}{2}{\rho }_{\infty }{U}_{\infty }^2}$, along the z/L = 0 plane, where U_∞, p_∞ and ρ_∞ are the freestream velocity, pressure and density, respectively, and p and τ_w are averaged over time. f Wall pressure coefficient along the x/L = 0 plane. g Mean velocity profile at z/L = 0 at different streamwise locations. In (d–g) The solid lines denote Building-block Flow model (BFM) (blue), Dynamic Smagorinsky model (DSM) (red) and Vreman model (VRE) (yellow). White circles are experimental values^85,103. The wall model used for DSM and VRE is the equilibrium wall model (EQ).

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 data⁸⁵. Figure 6d, 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 presssure along the spanwise direction at the bump apex (x/L = 0), as shown in Fig. 6f, and the mean velocity profiles, as demonstrated in Fig. 6g. 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%. Similarly, the BFM accurately predicts the flow topology of the separation. This is illustrated in Fig. 7, which shows a reduced window (with limits z/L = ±0.13 and y/L≤ 0.075) downstream of the bump. The BFM captures the counter-rotating vortices that lift away from the surface⁸⁵, in contrast to VRE-EQ, which predicts a completely different flow topology. The results for DSM-EQ (not shown) are similar to those of VRE-EQ. 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 (Fig. 6e).

Fig. 7. Average flow over the bump.

The colormap corresponds to the mean streamwise velocity and the streamlines show the in-plane velocity at the plane x/L = 0.208. (a) experiments⁸⁵, (b) Building-block Flow Model (BFM) and (c) Vreman model with equilibrium wall-model (VRE-EQ). Note that axes are not to scale.

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 methodologies⁸⁶. 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 0.2.

The simulations are conducted using a grid with approximately 30 million control volumes. Four angles of attack are considered: α = 7.05^∘, 11.29^∘, 17.05^∘ and 19.57^∘. Figure 2 contains the model geometry, an inset of the mesh, and a visualization of the wall-shear stress predicted by BFM.

Figure 8a–c 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. The results are compared with experimental data⁸⁷. The three plots should be interpreted as a whole, as recent studies⁸⁸ have demonstrated that –due to the complexity of the case– accurate predictions of C_L and C_D may result from error cancellation during the integration of aerodynamic forces. This issue is less relevant in the pitching moment, which is sensitive to force distribution and hence less susceptible to error cancellation. Overall, BFM offers similar or improved accuracy in predicting the three aerodynamic coefficients, especially at high angles of attack during stall and post-stall phases. It is important to note that despite the accurate predictions of lift and drag by DSM-EQ and VRE-EQ at α = 7.05^∘, these are accompanied by a poorer prediction of the pitching moment compared to BFM.

Fig. 8. Aircraft in landing configuration.

a Lift coefficient, (b) drag polar and (c) pitching moment coefficient. The black lines denote experimental results, blue circles are for the Building-block Flow Model (BFM), red squares are for Dynamic Smagorinsky model (DSM) and yellow triangles for Vreman model (VRE). The wall model used for DSM and VRE is the equilibrium wall model (EQ). Pressure coefficient along the chord of the wing at (d) the 33% and (e) 55% spanwise section of the wing for α = 17.05^∘. The chord location is normalized by the sectional chord-length c_i. Empty circles represents experimental results, blue circles are for the BFM, red and yellow plus symbols are for DSM-EQ and VRE-EQ, respectively.

Inspection of the sectional pressure coefficient (C_p) along the wing, as depicted in Fig. 8d, e for two spanwise locations, reaffirms that BFM yields more accurate predictions compared to DSM-EQ and VRE-EQ. The improvements are more noticeable in the flaps (last 20% of the wing chord). 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.

Juncture flow in simplified aircraft

We evaluate the performance of the BFM using data from the NASA Juncture Flow Experiment⁹. This experiment involves a simplified aircraft configuration, as illustrated in Fig. 9, featuring flow separation at the juncture between the fuselage and the trailing edge of the wing⁹. The Reynolds number is calculated based on the fuselage length (L = 4839.2 mm) and the freestream velocity, yielding Re = 20.8 million. Additionally, the free-stream Mach number is M = 0.189, and the angle of attack is α = 5^∘. A cross-section of the grid, containing approximately 30 million points, is depicted in Fig. 9d. This analysis complements the CRM-HL case previously discussed, as the experimental campaign of the Juncture Flow contains mean velocity profiles at various locations, which were not available for the CRM-HL. Unlike integrated measurements such as lift and drag coefficients, mean velocity profiles are less susceptible to error cancellation, thus facilitating a more detailed evaluation of the models.

Fig. 9. Mean velocity profiles for the Juncture Flow in simplified aircraft.

The probes are located at (a) the upstream region of the fuselage (x = 1168.4 mm, z = 0 mm), (b) the wing-body junction (x = 2747.6 mm; y = 239.1 mm), and (c) the wing-body junction close to the trailing edge (x = 2922.6 mm; y = 239.1 mm). the symbols denote: Building-block Flow Model (BFM) (blue circles) and Vreman model with equilibrium wall-model (VRE-EQ) (yellow triangles). Closed symbols corresponds to U/U_∞ and open symbols to W/U_∞, where U_∞ is the freestream velocity. The lines correspond to experimental values⁹. δ is the local boundary layer thickness¹³. d NASA Juncture Flow model, indicating the approximate location of the probes. e Cross-section of the grid at the wing.

WMLES was conducted using both BFM and VRE-EQ. The results for the mean streamwise (U) and spanwise (W) velocity components are compared with experimental measurements at three different locations, as shown in Fig. 9a–c. The symbols in the figure represent the actual locations of the grid points. BFM yields accurate results in sections A and B, showing an improvement over VRE-EQ. BFM also improves predictions at section C. However, both models fail to capture the flow separation. This failure is not surprising, given that the thickness of the separation region is represented by only two grid points in the wall-normal direction, making accurate prediction extremely challenging. Despite this limitation, BFM still provides enhanced predictions of the mean velocity profile compared to VRE-EQ across the same grid resolution at all three locations. WMLES was also performed using DSM-EQ (not shown), which yielded slightly poorer predictions than VRE-EQ.

Discussion

Future opportunities of BFM for WMLES

The BFM has the potential to 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 report³. The BFM can also play a key role in Certification by Analysis, i.e., the development of simulation-based methods of compliance for airplane and engine certification¹. 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. The impact of excluding BBFs from the BFM training set is illustrated in the Methods section for the Gaussian Bump. 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 model⁷¹. 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.

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 grid resolutions considered here, the BFM matches or improves the predictions from 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 and grid resolutions to improve its robustness.

Methods

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

$$\frac{\partial \overline{\rho }}{\partial t}+\frac{\partial \overline{\rho }{\tilde{u}}_i}{\partial x_i}=0,$$ 2

$$\frac{\partial \overline{\rho }{\tilde{u}}_i}{\partial t}+\frac{\partial \overline{\rho }{\tilde{u}}_i{\tilde{u}}_j}{\partial x_j}=-\frac{\partial \overline{p}}{\partial x_i}+\frac{\partial {\tilde{\sigma }}_{ij}}{\partial x_j}-\frac{\partial {\tau }_{ij}^{\mathrm{SGS}}}{\partial x_j},$$ 3

$$C_v\frac{\partial \overline{\rho }\tilde{T}}{\partial t}+C_v\frac{\partial \overline{\rho }{\tilde{u}}_j\tilde{T}}{\partial x_j}=-\overline{p}\frac{\partial {\tilde{u}}_j}{\partial x_j}+{\tilde{\sigma }}_{ij}\frac{\partial {\tilde{u}}_j}{\partial x_i}$$ 4

$$+\frac{\partial }{\partial x_j}\left(\tilde{\kappa }\frac{\partial \tilde{T}}{\partial x_j}\right)-C_v\frac{\partial q_j^{\mathrm{SGS}}}{\partial x_j},$$ 5

where repeated indices imply summation, $\left(\overline{\cdot }\right)$ denotes filtered quantities, $\left(\tilde{\cdot }\right)$ is the Favre average, u_i is the i-th velocity component, ρ is the density, T is the temperature, κ is the thermal conductivity, C_v is the specific heat at constant volume, and σ_ij is the viscous stress tensor. The heat flux is computed as $q_j^{\mathrm{SGS}}=\left(\overline{\rho }{\nu }_t/\mathrm{Pr}\right)\partial \tilde{T}/\partial x_j$, where Pr is the turbulent Prandtl number and ν_t is the eddy viscosity. The applications considered in this work are low speed flow, and heat transfer and compressibility effects play no important role.

The simulations are conducted with the GPU version of the solver charLES developed by Cascade Technologies, Inc. (Cadence)⁸⁹. 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 model¹⁷ with the modification by Lilly⁷⁷ and the Vreman model¹⁶ 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 direction²⁷,

$$u_{\mid \mid }^{+}\left(y_{\perp }^{+}\right)=\left\{\begin{array}{cc}{y}_{\perp }^{+}+a_1{\left(y_{\perp }^{+}\right)}^2\hfill & \mathrm{for}{y}_{\perp }^{+}<23,\hfill \\ \frac{1}{\kappa }ln{y}_{\perp }^{+}+B\hfill & \mathrm{otherwise}\hfill \end{array}\right)$$ 6

where u_∣∣ is the model wall-parallel velocity at the second grid point off the wall, y_⊥ is the wall-normal direction to the surface, κ = 0.41 is the Kármán constant, B = 5.2 is the intercept constant, and a₁ is computed to ensure C¹ continuity. The superscript  + denotes inner units defined in terms of wall friction velocity and the kinematic viscosity.

BFM formulation

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

$${\tau }_{ij}^{\mathrm{SGS}}=-2{\nu }_t{\tilde{S}}_{ij},$$ 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 ($\tilde{\mathbf{S}}\equiv {\tilde{S}}_{ij}$) and rate-of-rotation tensors ($\tilde{\mathbf{R}}\equiv {\tilde{R}}_{ij}$)⁷²,

$${\nu }_t=f\left(I_1,I_2,I_3,I_4,I_5,\theta \right),$$ 8

where f represents a feedforward ANN, and θ denotes additional input variables, namely, ν, Δ and u_∥, where ν is the kinematic viscosity, Δ is the characteristic grid size and u_∥ 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

$$\begin{array}{ccc}\hfill I_1=\mathrm{tr}\left({\tilde{\mathbf{S}}}^2\right),\hfill & \hfill I_2=\mathrm{tr}\left({\tilde{\mathbf{R}}}^2\right),\hfill & \hfill I_3=\mathrm{tr}\left({\tilde{\mathbf{S}}}^3\right),\hfill \\ \hfill I_4=\mathrm{tr}\left({\tilde{\mathbf{S}}\tilde{\mathbf{R}}}^2\right),\hfill & \hfill I_5=\mathrm{tr}\left({\tilde{\mathbf{S}}}^2{\tilde{\mathbf{R}}}^2\right).\hfill & \hfill \hfill \end{array}$$ 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 rationale for dividing the model into three ANNs is twofold: 1) task division (wall model versus SGS stress model), and 2) compatibility constraints (between the wall model and the near-wall SGS stress model). For the former, the initial ANN is responsible for predicting the wall shear stress (acting as a wall model), while the other two ANNs predict the eddy viscosity ν_t. The second point relates to the ANN calculating ν_t at the control volumes adjacent to the wall. This ANN is not only tasked with obtaining ν_t for accurate mean flow predictions but also with ensuring compatibility with the wall model predictions, as the latter are tightly coupled with ν_t just above the boundary. A consequence of this coupling is that the output from the near-wall ANN scales differently from that of the ANN dedicated to the rest of the flow. Although it is feasible to train one single ANN for ν_t, it was found that a better-performing model could be trained by separating the ANN into two.

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 geometries⁵⁴. For the ANN responsible for predicting the wall-shear stress, the input and output quantities are non-dimensionalized using viscous scaling: ν and Δ. For the ANNs predicting the eddy viscosity, the input and output variables are non-dimensionalized using semi-viscous scaling: ${\left(\nu \sqrt{\tilde{\mathbf{S}}:\tilde{\mathbf{S}}}\right)}^{1/2}$ (where the symbol : denotes the Frobenius inner product) and Δ, where the former represents a local velocity scale akin to the wall friction velocity u_τ typically used in wall-bounded flows, and the use of Δ as a length scale allows for accommodating the effect of different grid resolutions.

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. The core region of the turbulent channel, where mean shear effects are weak, serves to model the SGS physics of isotropic turbulence. 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. ^90,91. Five cases with friction Reynolds numbers 550, 950, 2000, 4200 and 10000 are used. For the TPC flows, the data was obtained from ref. ⁷¹. The Reynolds numbers based on the mean pressure gradient are $R{e}_P=\sqrt{h^3\mathrm{d}P/\mathrm{d}x}/\nu =340,680$ and 962, and the Reynolds number based on the top wall velocity is Re_U = U_th/ν = 22360, where h is the channel half-height. The computational domain in the streamwise, wall-normal, and spanwise direction is L_x × L_y × L_z = 4πh × 2h × 2πh for channel flows and L_x × L_y × L_z = 2πh × 2h × πh for TPC cases.

To generate the training data, we perform WMLES simulations of the BBFs adjusting ν_t to match the mean DNS velocity profile and correct wall-shear stress using the E-WMLES approach⁷¹. The simulations are performed in charLES. The eddy viscosity is adjusted by finding a correcting factor k(y) to a base model ${\nu }_t^{\mathrm{base}}$ (in this case Vreman SGS model¹⁶),

$${\nu }_t\left(x,y,z,t\right)={\nu }_t^{\mathrm{base}}\left(x,y,z,t\right)k\left(y\right),$$ 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

$$\arg \underset{k\left(y\right)}{\min }\int \mid U^{\mathrm{DNS}}\left(y\right)-\tilde{U}\left(y\right){\mid }^2\mathrm{d}y$$ 11

where U^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 algorithm⁹² 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 ${\left({\nu }_t\right.∣}_w={\left(\left(\partial \tilde{u}/\partial y\right)\right.∣}_w^{-1}{\tau }_w/\rho -\nu$, following⁹³, where the subscript w denotes quantities at the wall, and τ_w is the mean wall-shear stress. The grid size for the WMLES cases with E-WMLES is Δ ≈ 0.1h and  ≈ 0.2h for the turbulent channel flows and Δ ≈ 0.1h for the TPC flows.

The optimization process for a given case is as follows: 1) WMLES simulation is performed with a fixed τ_w –equal to the correct value from DNS simulations, ${\tau }_w^{\mathrm{DNS}}$– 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

$$\frac{\mid U^{\mathrm{DNS}}\left(y\right)-\tilde{U}\left(y\right)\mid }{U^{\mathrm{DNS}}\left(y\right)}<0.03$$

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

$$\frac{\mid U^{\mathrm{DNS}}\left(y\right)-\tilde{U}\left(y\right)\mid }{U_t}<0.02.$$

Other approaches have been proposed in the literature to enforce compatibility between models and numerical schemes, most of them in the context of the RANS equations^42,94. Advances in model-consistent approaches for LES are more limited due to the chaotic nature of the system and the inherent inconsistency between the filtered Navier-Stokes and the LES equations. Noteworthy examples include reinforcement learning^61,69,95, filter-consistent formulations^75,76, and optimal LES formulations^71,96,97.

Gaussian bump: details of the computational set-up

The computational domain and set-up is as in ref. ⁸⁴. The domain is a rectangular prism that extends from  − L to 1.5L in the streamwise direction (with respect to the bump apex),  ± 0.5L in the spanwise direction, and from 0L to 0.5L in the vertical direction. The lateral and top boundaries are free-slip, a constant uniform inflow is imposed at the inlet, and the non-reflecting characteristic boundary condition with constant pressure is applied at the outlet.

For the results presented in Fig. 6, the grid has three levels of isotropic refinement. Each refinement level has roughly 10 control volumes along the wall-normal direction and the average size of each level is twice the size of the previous level (see Fig. 6). The grid size of the layer closer to the wall is the smallest with ${\mathrm{\Delta }}_{\min }\approx 0.026h$, where h = 0.0838L is the bump height. The number of control volumes is 8.7 million. This resolution yields of the order of 5 points per boundary layer at the bump apex. The simulations have been performed with a varying time step to ensure that the Courant–Friedrichs–Lewy number is less than 2.

CRM-HL: details of the computational set-up

We follow the computational set-up from ref. ⁹⁸. A semi-span aircraft geometry is simulated in a hemi-sphere of radius 1000c_a, where c_a is the mean aerodynamic chord. The symmetry plane is treated with free-slip and no penetration boundary conditions. A uniform plug flow is used at the front half of the hemisphere. A non-reflecting boundary condition with specified freestream pressure is imposed at the rear half of the hemisphere. Several grid refinements are considered, as illustrated in Fig. 2, being the coarsest and the smallest grid elements ${\mathrm{\Delta }}_{\max }\approx c_a$ and ${\mathrm{\Delta }}_{\min }\approx 2\times 10^{-3}{c}_a$, respectively. This leads to a total number of grid points is 30 million and the number of grid points per boundary layer thickness ranges from zero at the leading edge of the wing to twenty at the trailing edge. The reader is referred to⁹⁸ for more details about the numerical set-up and gridding strategy.

Juncture flow: details of the computational set-up

We follow the computational set-up from ref. ¹³. The aircraft is centered in a rectangular prism whose sides are about 5 times the fuselage length, L, in the three directions. A uniform plug flow is imposed at the bottom and front boundaries; a non-reflecting boundary condition is used at the top and rear boundaries; and the lateral sides are modeled as free-slip. The mesh is constructed using a Voronoi diagram with a background grid size of Δ ≈ 0.04L, and several layers of refinement around the aircraft that leads to minimum grid size over the aircraft’s surface of ${\mathrm{\Delta }}_{\min }\approx 4\times 10^{-4}L$. The reader is referred to ref. ¹³ for more details about the numerical set-up and gridding strategy.

Effect of the grid size

The current version of BFM has been trained on coarse grids with 5 to 10 points per boundary layer. The behavior of BFM when extrapolating on finer grids is shown in Fig. 10a, b for the separated flow over the bump. We consider a finer grid with a minimum grid size (${\mathrm{\Delta }}_{\min }\approx 0.015h$) that is roughly half the one presented in the Results, leading to a mesh with 29 million control volumes. The prediction of BFM becomes slightly less accurate upstream the bump, with lower C_f and  − C_p predictions at the bump apex; however the prediction downstream the apex are consistent with the experiments. We note that VRE-EQ and DSM-EQ still fails to predict the separation.

Fig. 10. Convergence study of the Building-block Flow Model (BFM) and comparison with other models.

a, c, e Wall pressure and (b, d, f) wall friction coefficients along the z/L = 0 plane. In (a, b), the results are obtained in a grid with ${\mathrm{\Delta }}_{\min }\approx 0.015h$. Legend as in Fig. 6. In (c, d), the lines correspond to Dynamic Smagorinsky model (DSM) (red) and Vreman model (VRE) (yellow) in a grid with ${\mathrm{\Delta }}_{\min }\approx 0.003h$ ( ≈ 450 million control volumes)⁸⁴. The wall model used for DSM and VRE is the equilibrium wall model (EQ). The symbols are Spallart-Allmaras (S-A) (squares), (S-A)-RC-QRC2000 (triangles). The dashed blue line is BFM from Fig. 6. In (e, f), the lines correspond to BFM (blue) and BFM trained without the building blocks accounting for adverse and favorable mean pressure gradient effects (green). White circles are experimental values⁸⁵.

Comparison with RANS and WMLES

Figure 10c, d display the C_p and C_f values obtained using DSM-EQ and VRE-EQ on grids with 452 million control volumes (${\mathrm{\Delta }}_{\min }\approx 0.003h$), as reported in ref. ⁸⁴. The figure also includes results from RANS simulations using the Spalart-Allmaras (S-A) model⁹⁹ and the (S-A)-RC-QCR2000¹⁰⁰, performed by Iyer & Malik⁸³ on a grid with 21 million cells (simulating only the half span), which is effectively five times the grid size used by BFM. While all simulations show convergence upstream of the bump, VRE-EQ and S-A fail to capture the flow separation; the (S-A)-RC-QCR2000 predicts separation but inaccurately in terms of C_p within the bubble and C_f downstream. Only DSM-EQ with 452 million control volumes follows closely with the experimental results. Comparable accuracy was achieved by BFM on a grid with fewer than 9 million control volumes, which contains 52 times fewer control volumes.

Effect of the building-blocks flows on the BFM accuracy

We demonstrate that excluding BBFs modeling favorable/adverse pressure gradients from the training set significantly diminishes the performance of BFM on the Gaussian bump, as shown in Fig. 10e, f. The grid resolution matches that of Fig. 6. The green line represents a version of BFM with the SGS trained solely on BBFs derived from turbulent channel flows, without considering BBFs that account for the effects of favorable/adverse pressure gradients. In the absence of these specific BBFs, the model fails to accurately capture flow separation. These results illustrate how the BFM framework can systematically incorporate additional flow physics by integrating additional BBFs, a capability not explicitly demonstrated by previous ANN-based SGS models.

Effect of improved subgrid-scale model

We demonstrate the improvements in performance achieved by integrating the SGS and wall models. To this end, we compare our results with those using the BFM ANN-based wall model combined with a traditional SGS model. We consider three cases: a turbulent pipe, a Gaussian bump, and CRM. The results for the pipe and CRM, reproduced from ref. ⁷¹, employ DSM as the SGS model and a more comprehensive version of the wall model described here, trained with additional building blocks. Figure 11a depicts the mean velocity profile in the turbulent pipe, showing that the BFM provides improved predictions and corrects the discrepancies in the mean velocity profile near the wall compared to using DSM as the SGS model. Figure 11b presents a comparison between the BFM and DSM combined with the ANN currently responsible for τ_w (the bottom ANN in Fig. 2) over the Gaussian bump. The enhancements made by the BFM’s SGS-model components (top and middle ANNs in Fig. 2) in capturing flow separation are clearly visible. Finally, Fig. 11c–e display the integrated forces and moments on the CRM at attack angles of 7^∘ and 19^∘ as noted in ref. ⁷¹, with BFM yielding more accurate predictions of lift and moment coefficients at 7^∘. At higher angles, both models perform comparably. Overall, the BFM approach, which combines the SGS and wall model into one entity, enhances the performance of WMLES compared to solely developing ANN-based wall models.

Fig. 11. Effect of the subgrid scale model in the Building-block Flow Model (BFM).

a Mean velocity profile along the radial direction in the turbulent Pipe. b Friction coefficient along the y/L = 0 plane in the Gaussian bump. c Lift coefficient, (d) drag polar and (e) pitching moment coefficient for NASA Common Research Model High-Lift (CRM-HL). In (a, c, d, e) blue circles correspond to BFM and green triangles correspond to results from ref. ⁷¹. In (b) blue line correspond to BFM and green line to Dynamic Smagorinsky model (DSM) with the wall model from BFM as shown in the bottom ANN of Fig. 2.

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 set-up.

Peer review

Peer review information

Communications Engineering thanks the anonymous, reviewers for their contribution to the peer review of this work. Primary Handling Editors: Anastasiia Vasylchenkova.

Data availability

The data supporting the findings of this study are available from the corresponding authors.

Code availability

Access to the code is available from the corresponding authors upon reasonable request.

Competing interests

The authors declare no competing interests.