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Machine Learning-augmented Predictive Modeling of Turbulent Separated Flows over Airfoils

Anand Pratap Singh, Shivaji Medida, Karthik Duraisamy

arXiv:1608.03990v3cs.CE

TL;DR

Flow separation and related effects are poorly modeled, motivating data-driven turbulence-model augmentation. The paper uses inverse modeling and machine learning to reconstruct corrective model forms, which improve predictions for airfoil flows, including cases outside training data.

  • Problem

    Flow separation and related effects are poorly modeled, while the structural form of the turbulence model remains a modeling concern.

  • Method

    Inverse modeling extracts quantitative discrepancy information from limited data, and machine learning reconstructs it into corrective model forms embedded in a standard solver.

  • Results

    The data-assisted SA model significantly improves predictions of lift, drag, and stall onset angles, including airfoil shapes and conditions not used for training.

  • Takeaways & Limitations

    Embedding reconstructed corrective forms in a solver can achieve significantly improved predictions for turbulent separated airfoil flows.

Abstract

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A modeling paradigm is developed to augment predictive models of turbulence by effectively utilizing limited data generated from physical experiments. The key components of our approach involve inverse modeling to infer the spatial distribution of model discrepancies, and, machine learning to reconstruct discrepancy information from a large number of inverse problems into corrective model forms. We apply the methodology to turbulent flows over airfoils involving flow separation. Model augmentations are developed for the Spalart Allmaras (SA) model using adjoint-based full field inference on experimentally measured lift coefficient data. When these model forms are reconstructed using neural networks (NN) and embedded within a standard solver, we show that much improved predictions in lift can be obtained for geometries and flow conditions that were not used to train the model. The NN-augmented SA model also predicts surface pressures extremely well. Portability of this approach is demonstrated by confirming that predictive improvements are preserved when the augmentation is embedded in a different commercial finite-element solver. The broader vision is that by incorporating data that can reveal the form of the innate model discrepancy, the applicability of data-driven turbulence models can be extended to more general flows.

I. INTRODUCTION

Turbulence models rely on simplifying assumptions and calibration from canonical cases, limiting predictive accuracy for complex flows. The paper proposes combining inverse modeling and machine learning to reconstruct corrective model forms from limited experimental data.

  • Motivation: Simplifying assumptions about turbulence structure and calibration on canonical problems constrain practical model accuracy.Existing formulations balance physical intuition, empiricism, robustness, and computational cost.
  • Motivation: Improved computation, measurements, data availability, and machine learning motivate data-driven approaches to turbulence modeling.The paper positions these advances as opportunities to improve turbulence understanding and modeling.
  • Related work: Prior studies used data to infer parameters, quantify model discrepancies, perturb turbulence quantities, or learn corrective functional forms.These efforts span Bayesian inference, discrepancy modeling, neural networks, and evolutionary algorithms.
  • Approach: Field inversion and machine learning combine discrepancy inference, functional-form reconstruction, and predictive embedding in a three-step framework.The first two steps are offline training computations, while embedding the learned form is performed online during prediction.
  • Contribution: This work demonstrates that inverse modeling can extract quantitative information from very limited experimental data and machine learning can convert it into corrective model forms.The reconstructed forms are embedded in a standard solver to improve predictions.

II. PROBLEM AND APPROACH

Separated airfoil flows under strong adverse pressure gradients expose limitations of common RANS turbulence models, especially near stall. The paper addresses this problem with a data-driven framework that infers discrepancies from limited experimental data, learns corrective forms, and embeds them during prediction.

  • Problem: Strong adverse pressure gradients are important for predicting airfoil separation, but common RANS models often mispredict stall onset at high angles of attack.These models tend to over-predict maximum lift and stall onset angle.
  • Problem: None of 11 tested eddy-viscosity turbulence models performed satisfactorily for airfoil flows near maximum lift conditions.The deficiency is attributed to assumptions and simplifications inherent in eddy-viscosity models.
  • Problem: Strong adverse pressure gradients disrupt equilibrium boundary-layer assumptions by altering outer-layer scaling while leaving viscous and log layers relatively unchanged.As angle of attack increases, the defect layer penetrates deeper into the boundary layer.
  • Results: The data-assisted SA model improved predictions of lift, drag, and stall onset, including for airfoil shapes and conditions outside the training set.The approach is demonstrated on turbulent separated flows over airfoils.
  • Approach: The framework uses limited experimental lift data to infer spatial discrepancy fields, transforms multiple inverse solutions into functions of local model variables, and embeds corrections during simulation.The workflow uses full-field inversion and machine learning before querying the learned mapping in prediction.

Discretization

The airfoil computations use a structured-grid finite-volume solver with specified high-order inviscid and second-order diffusive discretizations, implicit operators, and a discrete-adjoint gradient calculation for field inversion.

  • Solver and grid: The flow solver uses a cell-centered finite-volume formulation for compressible RANS equations on structured grids.Airfoil domains are discretized with a C-grid containing 291 wraparound and 111 wall-normal points.
  • Spatial discretization: Inviscid fluxes use third-order MUSCL discretization with Roe’s approximate Riemann solver, while diffusive terms use second-order central differencing.These choices define the spatial discretization used in the computations.
  • Implicit treatment: Implicit operators are constructed with a diagonalized alternating-direction implicit scheme.The D-ADI scheme is part of the solver’s implicit treatment.
  • Numerical checks: A grid-convergence study verified that numerical errors were sufficiently low not to obscure turbulence-modeling errors.The farfield boundaries are placed 35 chord lengths from the airfoil surface.
  • Field inversion: Field inversion requires gradients at every grid point, which are computed most effectively using a discrete adjoint approach.The required derivatives are detailed in the appendix.

III. FIELD INVERSION

The field-inversion framework infers spatially varying corrections to the SA turbulence model from experimental observations, using adjoint optimization to recover discrepancy fields. Lift-based inversion produces near-wall corrections consistent with pressure-based inversion and reveals reduced pre-separation turbulence production.

  • Motivation: The method targets inaccurate model functional forms rather than merely estimating parameters, aiming to construct generalizable corrections.The stated goal is to reformulate the model structure through corrective functional forms.
  • Correction formulation: A spatially varying multiplier β(x) modifies the SA production term and is equivalently an added source term δ(x) = (β(x) −1)P(x).Inferring the nondimensional β field is better conditioned because its initial value is unity.
  • Inverse problem: The inverse problem fits model outputs to experimental data while regularizing deviations of β from unity, then solves for β throughout the computational domain using adjoint optimization.The formulation supports lift or surface-pressure observations and is applied at every discrete location.
  • Observations: Because surface pressure is often unavailable experimentally, the study uses lift coefficient data to infer modeling information for airfoil flows.The framework is intended to assimilate a broader set of experimentally measured lift data.
  • Inverse results: The lift- and pressure-based objectives produce similar inverse solutions, with almost identical near-wall β(x) features and indistinguishable surface pressures.Discrepancies remain in the post-stall region, while the near-wall behavior is nearly unchanged.
  • Physical interpretation: The inferred correction reduces turbulence generation in the near-wall pre-separation region, causing earlier flow separation and altered wall shear stress.The inverse solution addresses excessive turbulence generation at high angles of attack associated with delayed separation and stall.

IV. MACHINE LEARNING

The machine-learning stage converts flow-specific inverse correction fields into feature-based corrective models. It learns β as a function of locally available mean-field features selected across representative inverse problems for predictive use.

  • Reconstruction objective: Each inverse solution provides an optimal correction field for one geometry and flow condition, but predictive modeling requires converting β(x) into reusable modeling knowledge.The reconstruction is based on inverse problems representative of deficiencies relevant to the predictive task.
  • Feature-based correction: The correction is reconstructed as β(x) ≈ β(η), where η contains features derived from mean-field variables available during prediction.The functional relationship is designed to apply across different problems using locally nondimensional quantities.

A. Features

The feature design replaces dimensional SA variables with locally nondimensional quantities, while neural networks learn the resulting correction relationship efficiently from inverse-problem data. Feature selection uses validation SSE, and feed-forward networks map inputs through hidden layers to one output.

  • Feature rationale: The baseline SA source terms depend on local viscosity, working-variable, vorticity, and wall-distance quantities, but these dimensional variables are unsuitable as direct machine-learning inputs.Dynamically similar flows can have different numerical values for dimensional quantities.
  • Feature construction: Inputs are rescaled into locally nondimensional quantities representative of turbulence, including χ = ˆν/ν.The locally nondimensionalized SA source terms depend only on normalized vorticity and χ.
  • Model selection: The machine-learning alternatives include Gaussian-process regression and artificial neural networks, with neural networks selected for evaluation cost independent of training-data size.Input selection is assessed using validation-set sum squared error.
  • Neural network architecture: A feed-forward neural network forms linear combinations of inputs, applies nonlinear activations through hidden layers, and produces the correction at its output layer.The illustrated architecture has three inputs, two hidden layers, and one output.
  • Training and evaluation: Training uses error back-propagation to determine network weights, after which prediction cost depends on hidden-node count rather than training-data volume.The implementation typically uses about 100 nodes and sigmoid activation across three layers.

V. RESULTS

The framework trains neural-network augmentations for the SA turbulence model from inverse inferences over airfoil data. Across tested airfoils and conditions, the augmented model improves lift, drag, and surface-pressure predictions while remaining grid converged.

  • Test cases: The study evaluates the framework on three wind-turbine airfoils with varying thickness: S805, S809, and S814.These airfoils are used for neural-network training and testing.
  • Method: Full-field inversion uses experimentally available lift coefficients across angles of attack and Reynolds numbers, followed by neural-network reconstruction of model corrections.For each airfoil and condition, inversion is based on lift; one S809 case is also compared with pressure-based inversion.
  • Lift and drag: Significant stall-prediction improvement is obtained in lift, including at Re = 3×10^6, which was not included in training.The augmented model predicts drag rise at lower angles of attack than baseline SA, matching the observed qualitative trend.
  • Generalization: The augmented model performs equally well for S805 and S809, airfoil shapes not used in training.This indicates predictive improvements beyond the training geometry.
  • Surface pressure: The NN-augmented model provides considerable improvements in surface-pressure distributions across the reported airfoil cases.The comparisons include S809, S805, and S814 pressure predictions.
  • Numerical verification: Both the baseline SA and NN-augmented SA solutions are sufficiently grid converged at the resolutions used.The comparison covers pressure and skin-friction solutions on different spatial resolutions.

B. Predictive variability

The authors assess whether neural-network augmentations preserve baseline behavior and how predictions vary with training data. The augmented models remain more accurate across cases, but their quality depends on training-set selection and the ensemble test is not formal uncertainty quantification.

  • Baseline-preserving behavior: The augmented model is intended not to affect problems where the original SA model is already accurate.The results report no deterioration in low-angle-of-attack regions.
  • Predictive variability: All NN-augmented models are more accurate than the base SA model across the cases examined.None diverges from the base SA model at α = 0°.
  • Uncertainty: The ensemble comparison is useful for testing variability but does not constitute formal uncertainty quantification.A more formal uncertainty-quantification approach is identified as a separate need.
  • Scope boundary: Training-set variability can warn users when models operate far from the conditions in which they were trained.The paper links significant variability in predictions with this warning.
  • Training-data sensitivity: Predictions vary with the selection of training data, as illustrated by the S809 surface-pressure comparisons.The main predictive model uses data-set P, while other data-set combinations are used to study variability.

C. Portability and Convergence

The NN-augmented SA model was transferred from ADTURNS to AcuSolve, preserving predictive improvements while retaining baseline-like convergence and low computational overhead.

  • Portability: The NN-augmented SA model was implemented in AcuSolve, a commercial unstructured finite-element solver, using locally non-dimensional flow features for portability.AcuSolve uses dimensional RANS equations, making locally non-dimensional features essential for transferring the neural-network model.
  • Portability: NN augmentation significantly improved lift and drag predictions for the S809 airfoil across three Reynolds numbers.Its effectiveness was comparable to that observed in the ADTURNS solver framework.
  • Portability: Baseline solutions from ADTURNS and AcuSolve are not expected to be identical because their SA implementations use different rotation and curvature corrections.AcuSolve includes these corrections, whereas ADTURNS does not.
  • Convergence: Baseline and NN-augmented SA models showed comparable convergence characteristics in AcuSolve.The initial condition for all runs was uniform free-stream.
  • Convergence: Less than 10% additional compute time was required for passing features to the ANN and obtaining model corrections.The overhead was measured relative to the baseline calculation.

VI. SUMMARY AND PERSPECTIVES

The study uses full-field inversion and machine learning to augment the SA model for separated airfoil flows. The resulting model improves predictions beyond its training cases while preserving accuracy, convergence, and cross-solver portability, although broader generalization remains open.

  • Summary: A data-driven framework combining full-field inversion and machine learning was used to develop turbulence-model augmentations.The procedure infers spatial model discrepancy fields and reconstructs them as neural-network corrections embedded during solver iterations.
  • Perspectives: The present work demonstrates potential for data-driven modeling of turbulent separated airfoil flows, while more development is needed for general settings and formal uncertainty quantification.The authors identify training, data, baseline-model, and extrapolation uncertainties as requiring further treatment.
  • Summary: The data-assisted SA model significantly improved lift, drag, and stall-onset predictions for untrained airfoil shapes and flow conditions.Improvements were confirmed for cases excluded from the training set.
  • Summary: No accuracy deterioration was observed at low angles of attack where the original model was accurate.This preserves performance in regimes already represented adequately by the baseline model.
  • Summary: Lift-only inference also produced considerable improvements in surface-pressure distributions, supporting that the augmentation did not overfit the lift data.The authors connect the pressure improvements with confidence that the model improves predictions for physically appropriate reasons.
  • Summary: Predictions moved closer to experiments across all training-set ensembles despite expected variability between training sets.The ensemble was used to assess sensitivity to training data.
  • Summary: Predictive improvements were preserved when the augmentation was transferred from a structured finite-volume solver to AcuSolve.The reported convergence-cost overhead was minimal.
  • Perspectives: Future augmentations should preserve accurate baseline regions, solver convergence, realizability, and invariance, especially during extrapolation.The paper also identifies broader feature sets and alternative machine-learning methods as areas for exploration.

APPENDIX A : DISCRETE ADJOINT METHOD FOR FIELD INVERSION

The appendix formulates field inversion as a high-dimensional gradient-based optimization and explains its discrete adjoint implementation. It also illustrates how experimentally informed discrepancy fields improve predictions for separated flows.

  • Discrete adjoint method: Adjoint gradients make optimization feasible when the number of control variables equals the number of control volumes.The optimization uses a gradient-based limited-memory BFGS method, with automatic differentiation and pseudo-time stepping for solution procedures.
  • Discrete adjoint method: The adjoint derivative uses the governing equations and adjoint variables to compute the objective gradient with respect to the field parameters.The adjoint variable corresponding to the SA working variable is denoted by ψ˜ν.
  • Separated-flow application: Using lift data to generate inverse fields was also confirmed to produce good surface-pressure predictions for airfoil flows.A separation-bubble prediction was reported as 15% more accurate than the baseline, with pressure data identified as valuable for inferring discrepancy.
  • Separated-flow application: For a smooth-hump separated-flow benchmark, matching wall pressure over 0.5 ≤ x/c ≤ 1.5 also improved Reynolds-stress predictions.The inverse objective used surface pressure only, while the Reynolds-stress improvement appeared as an additional field outcome.
  • Spalart–Allmaras model: The framework uses the one-equation Spalart–Allmaras model, whose working variable is modified eddy viscosity and whose transport equation contains production and destruction terms.The model also defines vorticity-related and wall-damping functions through the stated model relations and constants.
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