Data-driven Turbulence Modeling Research Articles

A field inversion and symbolic regression enhanced Spalart–Allmaras model for airfoil stall prediction

A data-driven turbulence modeling method based on symbolic regression (SR) is proposed in this paper to enhance the prediction accuracy of the Spalart–Allmaras (SA) model for airfoil stall. Unlike traditional methods that rely on neural networks and lack physical interpretability, this paper utilizes SR to establish an analytic expression mapping local flow field variables to the SA model correction factor β. The training data are obtained through field inversion with the discrete adjoint method in the flow field of the S809 airfoil. Additionally, a relearning approach proposed in this paper is applied to the SR process to address the issue arising from the multi-solution nature of field inversion. The SA model embedded with β, referred to as the SA-SR model, can be integrated into computational fluid dynamics solvers with negligible computational cost. The generalization performance of the SA-SR model is tested under various conditions and airfoil types. The results indicate that the new model improves the predictive capability for airfoil stall without compromising the performance of the baseline SA model for attached flows.

Ensemble variational method with adaptive covariance inflation for learning neural network-based turbulence models

This work introduces an ensemble variational method with adaptive covariance inflation for learning nonlinear eddy viscosity turbulence models where the Reynolds stress anisotropy is represented with tensor-basis neural networks. The ensemble-based method has emerged as an important alternative to data-driven turbulence modeling due to its merit of non-derivativeness. However, the training accuracy of the ensemble method can be affected by the linearization assumption and sample collapse issue. Given these difficulties, we introduce the hybrid ensemble variational method, which inherits the merits of the ensemble method in non-derivativeness and the variational method in nonlinear analysis. Moreover, a covariance inflation scheme is proposed based on convergence states to alleviate the detrimental effects of sample collapse. The capability of the ensemble variational method in model learning is tested for flows in a square duct, flows over periodic hills, and flows around the S809 airfoil, with increasing complexity in the training data from direct observation to sparse indirect observation. Our results show that the ensemble variational method can learn relatively accurate neural network-based turbulence models in scenarios of small ensemble size and sample variances, compared to the ensemble Kalman method. It highlights the superiority of the ensemble variational method in practical applications, since small ensemble sizes can reduce computational costs, and small sample variance can ensure the training robustness by avoiding nonphysical samples of Reynolds stresses.

Data-driven turbulence modeling for fluid flow and heat transfer in peripheral subchannels of a rod bundle

This study presents a comparison of the performance of machine learning (ML) techniques, specifically multi-dimensional gene expression programming (MGEP), tensor basis neural network (TBNN), and also proposes a novel universally interpretable machine learning architecture to model the turbulent scalar flux (UIML-s) to enhance turbulence models for fluid flows at different Prandtl numbers in channels with complex shapes of walls in the channel cross section. In particular, peripheral subchannels of rod bundles are of primary interest. However, the accuracy of mean velocity and scalar distributions predicted by commonly used turbulence models still poses a challenge compared to data extracted from high-fidelity eddy-resolving numerical simulations, particularly for engineering applications involving complex geometry flows. In the present study, by utilizing an explicit algebraic expression for the nonlinear Reynolds-stress term obtained through both the evolutionary MGEP optimization and TBNN, the secondary flow structure has been adequately predicted in the cross-wise mean velocity distributions in the square duct and the rectangular channel with three longitudinal rods. This structure is also observed in the data from the concurrent runs performed by direct numerical simulation (DNS) but is completely absent in the results produced by a baseline Reynolds-averaged Navier–Stokes (RANS) closure, which employs the linear eddy viscosity model for the Reynolds stress tensor. Comparison of MGEP and TBNN has shown their nearly equal performance in a square duct flow; however, MGEP works better for the more complex geometry channel with three rods. Furthermore, based on the velocity field produced by the RANS-MGEP model, the ML modification of the gradient diffusion hypothesis, integrated into the aforementioned novel RANS-ML model called as UIML-s, significantly improves the mean scalar distributions in a flow with three bumps serving as a prototype for the peripheral subchannel of rod bundle. The normalized root mean squared error decreases from 13.5% to 7.6%, bringing the predicted distributions closer to the DNS data, particularly in the near-wall region. Another approach, MGEP-s, also yields the acceptable results, which are nearly identical to those from UIML-s. These findings highlight the potential of using data-driven calibration of turbulence models with nonlinear closures to enhance the predictability for RANS simulations of fluid flows, heat, and mass transfer in channels with complex geometry.

A highly accurate strategy for data-driven turbulence modeling

A highly accurate strategy for data-driven turbulence modeling

Progressive augmentation of turbulence models for flow separation by multi-case computational fluid dynamics driven surrogate optimization

In the field of data-driven turbulence modeling, the consistency of the a posteriori results and generalizability are the most critical aspects of new models. In this study, we combine a multi-case surrogate optimization technique with a progressive augmentation approach to enhance the performance of the popular k−ω shear stress transport (SST) turbulence model in the prediction of flow separation. We introduce a separation factor into the transport equation of a turbulent specific dissipation rate (ω) to correct the underestimation of the turbulent viscosity by the k−ω SST model in the case of flow separation for two-dimensional cases. The new model is optimized based on their performance on the training cases including periodic hills and curved backward-facing step flow. Simulation of the channel flow is likewise included in the optimization process to guarantee that the original performance of k−ω SST is preserved in the absence of separation. The new model is verified on multiple unseen cases with different Reynolds numbers and geometries. Results show a significant improvement in the prediction of the recirculation zone, velocity components, and distribution of the friction coefficient in both training and testing cases, where flow separation is expected. The performance of the new models on the test case with no separation shows that they preserve the successful performance of k−ω SST when flow separation is not expected.

Progressive augmentation of Reynolds stress tensor models for secondary flow prediction by computational fluid dynamics driven surrogate optimisation

Generalisability and the consistency of the a posteriori results are the most critical points of view regarding data-driven turbulence models. This study presents a progressive improvement of turbulence models using simulation-driven Bayesian optimisation with Kriging surrogates where the optimisation of the models is achieved by a multi-objective approach based on duct flow quantities. We aim for the augmentation of secondary-flow prediction capability in the linear eddy-viscosity model k−ω SST without violating its original performance on canonical cases e.g. channel flow. Progressively data-augmented explicit algebraic Reynolds stress models (PDA-EARSMs) for k−ω SST are obtained enabling the prediction of secondary flows that the standard model fails to predict. The new models are tested on channel flow cases guaranteeing that they preserve the successful performance of the original k−ω SST model. Subsequently, numerical verification is performed for various test cases. Regarding the generalisability of the new models, results of unseen test cases demonstrate a significant improvement in the prediction of secondary flows and streamwise velocity. These results highlight the potential of the progressive approach to enhance the performance of data-driven turbulence models for fluid flow simulation while preserving the robustness and stability of the solver.

Autoregressive transformers for data-driven spatiotemporal learning of turbulent flows

A convolutional encoder–decoder-based transformer model is proposed for autoregressively training on spatiotemporal data of turbulent flows. The prediction of future fluid flow fields is based on the previously predicted fluid flow field to ensure long-term predictions without diverging. A combination of convolutional neural networks and transformer architecture is utilized to handle both the spatial and temporal dimensions of the data. To assess the performance of the model, a priori assessments are conducted, and significant agreements are found with the ground truth data. The a posteriori predictions, which are generated after a considerable number of simulation steps, exhibit predicted variances. The autoregressive training and prediction of a posteriori states are deemed crucial steps toward the development of more complex data-driven turbulence models and simulations. The highly nonlinear and chaotic dynamics of turbulent flows can be handled by the proposed model, and accurate predictions over long time horizons can be generated. Overall, the potential of using deep learning techniques to improve the accuracy and efficiency of turbulence modeling and simulation is demonstrated by this approach. The proposed model can be further optimized and extended to incorporate additional physics and boundary conditions, paving the way for more realistic simulations of complex fluid dynamics.

Local turbulence generation using conditional generative adversarial networks toward Reynolds-averaged Navier–Stokes modeling

Data-driven turbulence modeling has been extensively studied in recent years. To date, only high-fidelity data from the mean flow field have been used for Reynolds-averaged Navier–Stokes (RANS) modeling, while the instantaneous turbulence fields from direct numerical simulation and large eddy simulation simulations have not been utilized. In this paper, a new framework is proposed to augment machine learning RANS modeling with features extracted from instantaneous turbulence flow data. A conditional generative model is trained to model the probability distribution of the local instantaneous turbulence field given local mean flow features. Then, the generative model is transferred to machine learning RANS modeling. The present work is mainly focused on generating a local instantaneous turbulence field using conditional generative adversarial networks (CGANs). Several GANs are trained first on the turbulence data from channel flow and periodic hill flow to generate complete one-dimensional and two-dimensional turbulence fields. Then, a CGAN is trained on the periodic hill flow data to generate local turbulence fields. Statistical analysis is performed on the generated samples from the GAN models. The first and second moments, the two-point correlation, and the energy spectra conform well to those of real turbulence. Finally, the information learned by the CGAN is used for machine learning RANS modeling by multitask learning, and the feasibility of the framework proposed in this paper is initially verified.

Wavelet Transforms and Machine Learning Methods for the Study of Turbulence

This article investigates the applications of wavelet transforms and machine learning methods in studying turbulent flows. The wavelet-based hierarchical eddy-capturing framework is built upon first principle physical models. Specifically, the coherent vortex simulation method is based on the Taylor hypothesis, which suggests that the energy cascade occurs through vortex stretching. In contrast, the adaptive wavelet collocation method relies on the Richardson hypothesis, where the self-amplification of the strain field and a hierarchical breakdown of large eddies drive the energy cascade. Wavelet transforms are computational learning architectures that propagate the input data across a sequence of linear operators to learn the underlying nonlinearity and coherent structure. Machine learning offers a wealth of data-driven algorithms that can heavily use statistical concepts to extract valuable insights into turbulent flows. Supervised machine learning needs “perfect” turbulent flow data to train data-driven turbulence models. The current advancement of artificial intelligence in turbulence modeling primarily focuses on accelerating turbulent flow simulations by learning the underlying coherence over a low-dimensional manifold. Physics-informed neural networks offer a fertile ground for augmenting first principle physics to automate specific learning tasks, e.g., via wavelet transforms. Besides machine learning, there is room for developing a common computational framework to provide a rich cross-fertilization between learning the data coherence and the first principles of multiscale physics.

Explainability analysis of neural network-based turbulence modeling for transonic axial compressor rotor flows

Model–consistent training has become trending for data-driven turbulence modeling since it can improve model generalizability and reduce data requirements by involving the Reynolds–averaged Navier–Stokes (RANS) equation during model learning. Neural networks are often used for the Reynolds stress representation due to their great expressive power, while they lack interpretability for the causal relationship between model inputs and outputs. Some post–hoc methods have been used to explain the neural network by indicating input feature importance. However, for the model–consistent training, the model explainability involves the analysis of both the neural network inputs and outputs. That is, the effects of model output on the RANS predictions should also be explained in addition to the input feature analysis. In this work, we investigate the explainability of the model–consistent learned model for the internal flow prediction of NASA Rotor 37 at its peak efficiency operating condition. The neural–network–based corrections for the Spalart–Allmaras turbulence model are learned from various experimental data based on the ensemble Kalman method. The learned model can noticeably improve the velocity prediction near the shroud. The explainability of the trained neural network is analyzed in terms of the model correction and the input feature importance. Specifically, the learned model correction increases the local turbulence production in the vortex breakdown region due to non–equilibrium effects, which capture the blockage effects near the shroud. Besides, the ratio of production to destruction and the helicity are shown to have relatively high importance for accurately predicting the compressor rotor flows based on the Shapley additive explanations method.

Feature importance in neural networks as a means of interpretation for data-driven turbulence models

This work aims at making the prediction process of neural network-based turbulence models more transparent. Due to its black-box ingredients, the model’s predictions cannot be anticipated. Therefore, this paper is concerned with the quantification of each feature’s importance for the prediction of trained and fixed NNs, which is one possible type of explanation for opaque models. Two conceptually different attribution methods, namely permutation feature importance and DeepSHAP, are chosen in order to assess global, regional and local feature importance. The neuralSST turbulence model, which serves as an example, will be investigated in greater detail. While the global importance scores provide a quick and reliable way to detect irrelevant features and may thus be used for feature selection, only the (semi-)local analysis provides meaningful and trustworthy interpretations of the model. In fact, the local importance scores suggest that hypotheses with a common high-level influence on the turbulence model, e.g. adjusting the net production of turbulent kinetic energy or the Reynolds stress anisotropy, are similarly affected by local mean flow structures such as attached boundary layers, free shear layers or recirculation zones.

Data-driven Reynolds-averaged turbulence modeling with generalizable non-linear correction and uncertainty quantification using Bayesian deep learning

The past few years have witnessed a renewed blossoming of data-driven turbulence models. Quantification of the concomitant modeling uncertainty, however, has mostly been omitted, and the generalization performance of the data-driven models is still facing great challenges when predicting complex flows with different flow physics not seen during training. A robust data-driven Reynolds-averaged turbulence model with uncertainty quantification and non-linear correction is proposed in this work with the Bayesian deep neural network. In this model, the Reynolds stress tensor is decomposed into linear and non-linear parts. The linear part is taken as the usual linear eddy viscosity model while the non-linear counterpart is learned by a Bayesian deep neural network. Independent tensor bases of invariants and tensors constituted by mean strain rate tensor and rotation rate tensor are embedded into the neural network to effectively consider key turbulence features in different flows. The proposed model is well validated through numerical simulations of four canonical flows that significantly deviate in geometrical configurations and/or Reynolds numbers from those in the training data. With the non-linear corrections of embedded invariants and tensors representing key features of turbulence, the proposed model not only improves the predictive capabilities of Reynolds-averaged turbulence models on the same mesh but also has better generalization performance when simulating complex turbulent flows with large scale separation. In addition, this model allows us to quantitatively demonstrate the confidence interval of the predicted flow quantities that are originated from the model itself.

Data-Driven RANS Turbulence Closures for Forced Convection Flow in Reactor Downcomer Geometry

Recent progress in data-driven turbulence modeling has shown its potential to enhance or replace traditional equation-based Reynolds-averaged Navier-Stokes (RANS) turbulence models. This work utilizes invariant neural network (NN) architectures to model Reynolds stresses and turbulent heat fluxes in forced convection flows (when the models can be decoupled). As the considered flow is statistically one dimensional, the invariant NN architecture for the Reynolds stress model reduces to the linear eddy viscosity model. To develop the data-driven models, direct numerical and RANS simulations in vertical planar channel geometry mimicking a part of the reactor downcomer are performed. Different conditions and fluids relevant to advanced reactors (sodium, lead, unitary-Prandtl-number fluid, and molten salt) constitute the training database. The models enabled accurate predictions of velocity and temperature, and compared to the baseline k − τ turbulence model with the simple gradient diffusion hypothesis, do not require tuning of the turbulent Prandtl number. The data-driven framework is implemented in the open-source graphics processing unit–accelerated spectral element solver nekRS and has shown the potential for future developments and consideration of more complex mixed convection flows.

Differentiable physics-enabled closure modeling for Burgers’ turbulence

Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with machine learning to develop closure models for Burgers’ turbulence. We consider the one-dimensional Burgers system as a prototypical test problem for modeling the unresolved terms in advection-dominated turbulence problems. We train a series of models that incorporate varying degrees of physical assumptions on an a posteriori loss function to test the efficacy of models across a range of system parameters, including viscosity, time, and grid resolution. We find that constraining models with inductive biases in the form of partial differential equations that contain known physics or existing closure approaches produces highly data-efficient, accurate, and generalizable models, outperforming state-of-the-art baselines. Addition of structure in the form of physics information also brings a level of interpretability to the models, potentially offering a stepping stone to the future of closure modeling.

Toward More General Turbulence Models via Multicase Computational-Fluid-Dynamics-Driven Training

The accuracy of machine-learned turbulence models often diminishes when applied to flow cases outside the training data set. In an effort to improve the predictive accuracy of data-driven models for an expanded set of cases, an extension of a computational fluid dynamics (CFD)-driven training framework consisting of three key steps is proposed. Firstly, a list of candidate flow-related parameters is selected to supplement Pope’s general tensor basis hypothesis. Secondly, modeling an additional production term may benefit the overall predictions in certain situations. Finally, the Reynolds-averaged Navier–Stokes (RANS) evaluations of candidate models are performed on several different flows simultaneously during the model training iterations. Five free-shear and five wall-bounded flow cases are chosen to train or test data-driven turbulence models. It is shown that the machine-learned models from the present multicase CFD-driven framework can significantly improve the predictive accuracy for the test cases where the baseline RANS results showed significant error from the ground truth. Meanwhile, for cases in which the baseline produced good results, the new models do not perform worse. Further analysis shows that the new models can adapt to opposite trends of turbulent diffusion required for the different cases with a common correction. Moreover, the trained models can be simplified and still achieve similar improvement as the whole expressions.

Explaining the physics of transfer learning in data-driven turbulence modeling.

Transfer learning (TL), which enables neural networks (NNs) to generalize out-of-distribution via targeted re-training, is becoming a powerful tool in scientific machine learning (ML) applications such as weather/climate prediction and turbulence modeling. Effective TL requires knowing (1) how to re-train NNs? and (2) what physics are learned during TL? Here, we present novel analyses and a framework addressing (1)-(2) for a broad range of multi-scale, nonlinear, dynamical systems. Our approach combines spectral (e.g. Fourier) analyses of such systems with spectral analyses of convolutional NNs, revealing physical connections between the systems and what the NN learns (a combination of low-, high-, band-pass filters and Gabor filters). Integrating these analyses, we introduce a general framework that identifies the best re-training procedure for a given problem based on physics and NN theory. As test case, we explain the physics of TL in subgrid-scale modeling of several setups of 2D turbulence. Furthermore, these analyses show that in these cases, the shallowest convolution layers are the best to re-train, which is consistent with our physics-guided framework but is against the common wisdom guiding TL in the ML literature. Our work provides a new avenue for optimal and explainable TL, and a step toward fully explainable NNs, for wide-ranging applications in science and engineering, such as climate change modeling.

Deep reinforcement learning for turbulence modeling in large eddy simulations

Over the last years, supervised learning (SL) has established itself as the state-of-the-art for data-driven turbulence modeling. In the SL paradigm, models are trained based on a dataset, which is typically computed a priori from a high-fidelity solution by applying the respective filter function, which separates the resolved and the unresolved flow scales. For implicitly filtered large eddy simulation (LES), this approach is infeasible, since here, the employed discretization itself acts as an implicit filter function. As a consequence, the exact filter form is generally not known and thus, the corresponding closure terms cannot be computed even if the full solution is available. The reinforcement learning (RL) paradigm can be used to avoid this inconsistency by training not on a previously obtained training dataset, but instead by interacting directly with the dynamical LES environment itself. This allows to incorporate the potentially complex implicit LES filter into the training process by design. In this work, we apply a reinforcement learning framework to find an optimal eddy-viscosity for implicitly filtered large eddy simulations of forced homogeneous isotropic turbulence. For this, we formulate the task of turbulence modeling as an RL task with a policy network based on convolutional neural networks that adapts the eddy-viscosity in LES dynamically in space and time based on the local flow state only. We demonstrate that the trained models can provide long-term stable simulations and that they outperform established analytical models in terms of accuracy. In addition, the models generalize well to other resolutions and discretizations. We thus demonstrate that RL can provide a framework for consistent, accurate and stable turbulence modeling especially for implicitly filtered LES.

Sparse Bayesian Learning of Explicit Algebraic Reynolds-Stress models for turbulent separated flows

A novel Sparse Bayesian Learning (SBL) framework is introduced for generating stochastic Explicit Algebraic Reynolds Stress (EARSM) closures for the Reynolds-Averaged Navier–Stokes (RANS) equations from high-fidelity data. Building on the recently proposed SpaRTA (Sparse Regression of Turbulent Stress Anisotropy) algorithm of Schmelzer et al. (2020), corrections to an underlying Linear-Eddy-Viscosity Models (LEVM) (namely, the k−ω SST model) are formulated as physically-interpretable, frame-invariant tensor polynomials and built from a redundant dictionary of candidate functions. The SBL-SpaRTA algorithm yields a sparse model structure (which favors interpretability and reduces overfitting), while endowing model coefficients with a measure of uncertainty, namely, posterior probability distributions. The framework is used to learn customized stochastic closure models for three separated flow configurations, characterized by different geometries but similar Reynolds number. The resulting stochastic models are then propagated through a CFD solver for all three configurations by means of probabilistic chaos collocation (Loeven et al., 2007). SBL-SpaRTA predictions of velocity profiles and friction coefficient distributions outperform those of the baseline LEVM, for training as well as for test cases. Furthermore, the prediction uncertainty intervals encompass reasonably well the reference data and tend to become large in regions of large discrepancy between RANS and high-fidelity predictions, thus warning the user about model reliability. Finally, a global sensitivity analysis of the stochastic models is carried out, illustrating the role of the dominant corrective terms and providing insights for future improvement of data-driven turbulence models.

Data-Driven Transition Models for Aeronautical Flows with a High-Order Numerical Method

Over the past years, there has been innovative ideas about data-driven turbulence modeling proposed by scholars all over the world. This paper is a continuity of these significant efforts, with the aim of offering a better representation for turbulence physics. Previous works mainly focus on turbulence viscosity or Reynolds stress, while there are few works for turbulence transition. In our work, two mapping functions between average flow parameters and transition intermittency, a virtual physical quantity describing the amount of turbulence at a given position, are refactored, respectively, with neuron networks and random forests. These two functions are then coupled with the Spalart–Allmaras (SA) model to reconstitute two models for transition prediction. To demonstrate that these two coupled models provide improved prediction accuracy on transition compared with previous SA models, we conduct test cases all under a high-order weighted compact nonlinear scheme (WCNS). The prediction results of both coupled models significantly improved the capture of natural transitions occurring in the flows. Furthermore, the interpolation generalisation and extrapolation generalisation abilities of the coupled models are also demonstrated in this paper. The results emphasize the potential for machine learning as a supplementary in turbulence transition modeling.

Ensemble Kalman method for learning turbulence models from indirect observation data

In this work, we propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. Data-driven turbulence models have emerged as a promising alternative to traditional models for providing closure mapping from the mean velocities to Reynolds stresses. Most data-driven models in this category need full-field Reynolds stress data for training, which not only places stringent demand on the data generation but also makes the trained model ill-conditioned and lacks robustness. This difficulty can be alleviated by incorporating the Reynolds-averaged Navier–Stokes (RANS) solver in the training process. However, this would necessitate developing adjoint solvers of the RANS model, which requires extra effort in code development and maintenance. Given this difficulty, we present an ensemble Kalman method with an adaptive step size to train a neural-network-based turbulence model by using indirect observation data. To our knowledge, this is the first such attempt in turbulence modelling. The ensemble method is first verified on the flow in a square duct, where it correctly learns the underlying turbulence models from velocity data. Then the generalizability of the learned model is evaluated on a family of separated flows over periodic hills. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes.