PaperlayerView exact revision on arXiv

Source-linked AI summary

Data-Driven Modelling of the Reynolds Stress Tensor using Random Forests with Invariance

Mikael L. A. Kaandorp, Richard P. Dwight

arXiv:1810.08794v2physics.flu-dynphysics.comp-ph

TL;DR

RANS models are widely used but can have poor accuracy and predictive power, motivating data-driven closures built from reference data. The paper presents an invariant random-forest method and reports anisotropy predictions on par with TBNN and mean flows close to DNS and experimental datasets.

  • Problem

    RANS remains widely used for its computational tractability despite poor accuracy and predictive power in important flows, motivating data-driven closures based on reference datasets.

    Source: §1 · pp. 2–3

  • Method

    The Tensor Basis Random Forest predicts Reynolds-stress anisotropy from invariant mean-flow features using a tensor basis, with median tree predictions and solver stabilization.

    Source: §1 · p. 4, §2.2 · p. 10, §2.5 · p. 21

  • Results

    TBRF performs on par with TBNN, and its predicted mean flows closely match corresponding DNS and experimental datasets.

    Source: §4 · pp. 43–44

  • Takeaways & Limitations

    The results support using TBRF as a turbulence closure for the tested flow cases, including square duct and backward-facing-step flows.

    Source: §4 · pp. 43–44

    The solver’s γmax=0.8 choice is ad hoc and requires further work.

    Source: §2.7 · p. 26

Abstract

from arXiv · show

A novel machine learning algorithm is presented, serving as a data-driven turbulence modeling tool for Reynolds Averaged Navier-Stokes (RANS) simulations. This machine learning algorithm, called the Tensor Basis Random Forest (TBRF), is used to predict the Reynolds-stress anisotropy tensor, while guaranteeing Galilean invariance by making use of a tensor basis. By modifying a random forest algorithm to accept such a tensor basis, a robust, easy to implement, and easy to train algorithm is created. The algorithm is trained on several flow cases using DNS/LES data, and used to predict the Reynolds stress anisotropy tensor for new, unseen flows. The resulting predictions of turbulence anisotropy are used as a turbulence model within a custom RANS solver. Stabilization of this solver is necessary, and is achieved by a continuation method and a modified $k$-equation. Results are compared to the neural network approach of Ling et al. [J. Fluid Mech, 807(2016):155-166, (2016)]. Results show that the TBRF algorithm is able to accurately predict the anisotropy tensor for various flow cases, with realizable predictions close to the DNS/LES reference data. Corresponding mean flows for a square duct flow case and a backward facing step flow case show good agreement with DNS and experimental data-sets. Overall, these results are seen as a next step towards improved data-driven modelling of turbulence. This creates an opportunity to generate custom turbulence closures for specific classes of flows, limited only by the availability of LES/DNS data.

1. Introduction

RANS remains widely used despite accuracy and predictive-power weaknesses, while data-driven closures offer a way to use LES/DNS references for flow-specific models. This paper introduces an invariant random-forest closure and solver stabilization, addressing practical challenges in applying machine learning to turbulence modelling.

1. Introduction

  • RANS models remain the norm in most industrial applications because they are computationally tractable, despite poor accuracy and predictive power in important flows.Simple linear eddy-viscosity models remain much more widely used than nonlinear eddy-viscosity and Reynolds-stress transport models.Source: §1 · p. 2
  • Data-driven closures can use reference LES or DNS datasets to build models customized for particular flow classes.Supervised machine learning can distill large datasets into functional relationships.Source: §1 · p. 3
  • Practical data-derived closures must address sensitivity of mean flow to turbulence, nonlocal and nonequilibrium physics, invariance, smoothness, solver stability, and standard machine-learning challenges.The cited standard challenges include high-dimensional training data and overfitting.Source: §1 · pp. 3–4
  • The Tensor Basis Random Forest (TBRF) integrates a tensor basis into a modified random forest to guarantee Galilean invariance and significantly outperform a baseline RANS model for a specified class of flows.The solver is stabilized with a modified k-equation and relaxation against a Boussinesq stress tensor.Source: §1 · p. 4
  • Prior data-driven RANS work includes model calibration, structural approaches, inverse correction methods, and direct prediction using neural networks, random forests, or gene-expression programming.The authors identify frame invariance and solver stabilization as missing from prior random-forest methods and critical to robustness and practicality.Source: §1 · p. 5
  • The paper proceeds from TBRF methodology and datasets to anisotropy and flow-field results, then conclusions.The results section also proposes future work on uncertainty quantification.Source: §1 · p. 6

2. Methods

The methods describe training a tensor-basis random forest on DNS/LES data to predict Reynolds-stress anisotropy from invariant mean-flow features and applying its predictions in a stabilized RANS solver, including stress blending and a modified k-equation.

2.1. Outline of framework for data-driven turbulence modeling

  • The framework learns anisotropy from DNS/LES statistics and RANS mean-flow features, then predicts for unseen cases and updates the mean flow with a modified RANS system.This is a corrective, single-prediction approach; the closed-loop alternative is untested, and the method is tested only for steady RANS cases, not unsteady flows.Source: §2.1 · pp. 7–9

2.2. Reynolds stress and realizability constraints

  • The predicted target is the dimensionless anisotropy tensor b, whose anisotropic stress component transports momentum; the Lumley triangle visualizes realizable states and highlights linear eddy-viscosity limitations.For two-dimensional linear eddy-viscosity RANS, predicted anisotropy invariants lie along the plane-strain line.Source: §2.2 · pp. 10–12

2.3. Invariance of tensor-valued functions

  • Frame invariance is enforced by expressing tensor-valued models through a finite integrity basis, using 47 invariants derived from mean-velocity, turbulent-kinetic-energy, and pressure gradients.For models based on strain and rotation alone, Pope’s integrity basis contains 10 tensors.Source: §2.3 · pp. 13, 15, 16

2.4. Tensor Basis Neural Network (TBNN)

  • The competing TBNN encodes Galilean invariance by combining a feed-forward network’s scalar coefficient outputs with tensor-basis inputs.Its inputs are five scalar invariants and ten basis tensors; this work uses the implementation and settings of Ling et al. (2016b).Source: §2.4 · pp. 16–17

2.5. Tensor Basis Random Forest (TBRF)

  • TBRF fits tensor-basis coefficients to DNS/LES anisotropy in decision-tree splits, then combines bagged trees using the median and spatially smooths predictions before solver use.A Gaussian filter with standard deviation 3 cell lengths is used; its width is an ad hoc choice for numerical stability.Source: §2.5 · pp. 19–21
  • Out-of-bag error provides a validation estimate for stopping tree training and optimizing forest size, while the TBRF is described as robust to hyperparameter choice and easy to train.The authors compare its training simplicity with neural networks, noting that it avoids selecting an optimization algorithm and its associated hyperparameters.Source: §2.5 · p. 22

2.6. Choice of input features

  • Because the five S,R invariants were insufficient for square-duct DNS anisotropy, the model uses invariants of S, R, and ∇k plus nine additional scalar features.The turbulent-kinetic-energy gradient is normalized using an antisymmetric tensor. FS1 uses S and R, FS2 adds Ak, and FS3 adds Wu et al. (2016) features; FS3 features are rotationally invariant, but some are not Galilean invariant.Source: §2.6 · pp. 22–23, Table 1 · p. 25

2.7. Propagation of the predicted anisotropy tensor

  • The solver converged with DNS-, TBNN-, and TBRF-derived anisotropy after under-relaxing predictions against Boussinesq stress and modifying k-equation production to match predicted stress.The blending parameter ramps from 0 to γmax = 0.8, an ad hoc limit set by increasing it until instability; lower γ improves stability but impairs mean-velocity accuracy.Source: §2.7 · pp. 24, 26, 27

3. Results

The results section compares TBRF predictions with a TBNN using the same feature sets across five flow cases, examining anisotropy predictions and resulting mean flows.

3. Results

  • TBRF predictions are compared with a TBNN using the same feature sets.The supplied passages describe the comparison but do not report its outcomes.Source: §3 · p. 27
  • The algorithms are trained and tested using data from five flow cases.The flow cases are discussed in Section 3.1.Source: §3 · p. 27
  • The results cover predictions of the anisotropy tensor and corresponding mean-flow predictions.Anisotropy predictions are presented in Section 3.2 and mean-flow predictions in Section 3.3.Source: §3 · p. 27

3.1. Flow cases

Five DNS or highly resolved LES flow cases provide training and test data, spanning separated flows and a square duct with turbulence-driven secondary motion.

3.1. Flow cases

  • Five flow cases with DNS or highly resolved LES data are used to train and test the algorithms.Their reference mean flows are shown in Figure 4, with velocity magnitude encoded by color and streamlines plotted.Source: §3.1 · p. 27, Figure 4 · p. 29
  • Periodic hills data cover five Reynolds numbers from 700 to 10595, based on inlet bulk velocity and hill height.The data are from Breuer et al. (2009).Source: §3.1 · p. 27
  • The converging-diverging channel DNS case has Re = 12600, based on channel half-height and maximum inlet velocity.The data are from Laval and Marquillie (2011).Source: §3.1 · p. 27
  • The curved backward-facing step uses LES at Re = 13700, while the backward-facing step uses DNS at Re = 5100.The Reynolds numbers are based on step height and center-channel inlet velocity for the curved step, and step height and free-stream velocity for the backward-facing step.Source: §3.1 · p. 28
  • The square-duct data span 16 cases from Re = 1100 to Re = 3500, and the duct exhibits turbulence-anisotropy-driven secondary motion.Figure 4 shows only the upper-right quadrant; the first four cases feature separation and reattachment, while the square duct isolates nonlinear-modeling effects because linear eddy-viscosity models do not capture its secondary motion.Source: §3.1 · p. 28, Figure 4 · p. 29

3.2. Anisotropy tensor predictions

Across the tested flows, the machine-learning methods generally reproduce anisotropy more accurately than baseline RANS, while square-duct results improve with additional features. Performance varies by flow region, and TBRF shows a localized shear-layer discrepancy in the backward-facing step.

3.2. Anisotropy tensor predictions

  • The evaluation covers four test cases, with training and prediction flows, sample counts, and usable feature counts specified for each.Hyperparameters were tuned for C1, C2, and C4; C3 used fully grown trees and all features.Source: §3.2 · p. 30, Table 2 · p. 31

3.2.1. Curved Backward Facing Step

  • For the curved backward-facing step, TBRF and TBNN give reasonable, relatively smooth predictions across tensor components and capture turbulence states better than RANS.RANS predicts mainly plane-strain turbulence and misses wall effects; both ML methods reproduce the LES state patterns, though TBRF shows noise and both underrepresent the shift toward axisymmetric expansion near the channel center.Source: §3.2.1 · pp. 32–34, Figure 5 · p. 32

3.2.2. Backward Facing Step

  • For the backward-facing step, both ML methods predict turbulence character qualitatively, with quantitative discrepancies; TBRF is more accurate near the wall at x/h = 4 and 6, while TBNN is slightly better at x/h = 10, 15, and 19.TBNN's later-section advantage comes with some unrealizable near-wall predictions; TBRF produced none in the reported studies, despite lacking an explicit realizability constraint. TBRF also errs toward the two-component boundary in the near-step shear layer.Source: §3.2.2 · p. 35, Figure 8 · p. 36

3.2.3. Square Duct

  • In the square duct, adding features improves both algorithms' anisotropy predictions: C4 better captures near-wall [b]11 and reduces component-magnitude errors relative to C3.Both ML methods underpredict [b]12 and [b]13 and overpredict [b]23, with these errors reduced in C4; RMSE is lower with more features for both algorithms. The five C3 features are largely redundant, limiting models based only on S and R.Source: §3.2.3 · pp. 36–37, Table 3 · p. 37

3.3. Anisotropy tensor propagation

The predicted anisotropy tensors are propagated through a stabilized solver to assess their resulting mean flows. The propagated fields compare favorably with reference data in both the square duct and backward-facing step.

3.3. Anisotropy tensor propagation

  • The study propagates predicted anisotropy tensors for the square duct and backward-facing step using the stabilized solver described in Section 2.7.For the square duct, DNS-derived anisotropy is also propagated as a best-case reference for separating propagation errors from anisotropy-prediction errors.Source: §3.3 · p. 37, §3.3.1 · p. 39

3.3.1. Square Duct

  • In the square duct, propagated TBRF anisotropy gives mean-velocity profiles closer to DNS than either nonlinear eddy-viscosity model.The predicted profiles preserve key features and appropriate amplitudes; their additional errors relative to propagation using DNS anisotropy are similar in magnitude to the propagation errors themselves.Source: §3.3.1 · pp. 39–40, Figure 10 · p. 40

3.3.2. Backward Facing Step

  • For the backward-facing step, propagated TBRF anisotropy improves the predicted recirculation-region size and reattachment location over baseline RANS.The propagated-field reattachment point is 6.32, compared with the experimental 6.0 ± 0.15; skin-friction results mostly fall within the experimental ±0.0005cf error bounds.Source: §3.3.2 · pp. 40–41, Table 4 · p. 41

4. Conclusions

The TBRF predicts Reynolds-stress anisotropy with Galilean invariance and robust, realizable predictions. Propagated predictions produce mean flows close to DNS and experimental data, while broader flow coverage and uncertainty quantification remain future work.

4. Conclusions

  • The tensor basis ensures Galilean invariance, and taking the median of decision-tree predictions yields robust anisotropy predictions.The method was trained on invariant features from RANS flow fields and DNS or highly resolved LES anisotropy data.Source: §4 · pp. 41, 43
  • TBRF performs on par with TBNN and is relatively easy to implement and train, with out-of-bag samples providing validation error estimates.Its remaining hyperparameters are described as robust under standard settings.Source: §4 · p. 43
  • A custom solver was introduced to propagate the predicted anisotropy, and the solver’s numerical stability is improved by the approach.The conclusion also reports propagation results for the square duct and backward-facing step.Source: §4 · p. 43
  • Propagated predictions for the square duct and backward-facing step show close agreement with DNS and experimental data.The backward-facing-step streamlines and skin-friction coefficient are compared with DNS and experimental references.Source: Figure 11 · p. 42, Figure 12 · p. 42, §4 · p. 44
  • Tree variance may support future uncertainty quantification, but meaningful prediction bounds require further investigation.The authors observed that individual trees often have high variance when their prediction is relatively poor.Source: §4 · p. 44
  • The study considers idealized canonical flows; performance on more complex cases, higher-Reynolds-number extrapolation, and unsteady RANS remain open questions.The authors identify highly detached flows as one example for further investigation.Source: §4 · p. 44

Appendix: TBRF implementation details

The tensor-basis decision tree recursively splits feature space, fitting basis coefficients in each resulting bin to DNS anisotropy. The random forest’s median aggregation and standard settings help produce robust predictions.

Appendix: TBRF implementation details

  • Each split greedily minimizes the mismatch between the DNS anisotropy response and the best constant coefficient fit in the two bins.The split objective uses the Frobenius norm, chosen for invariance to unitary transformations, including rotations.Source: Appendix: TBRF implementation details · pp. 54–55
  • The tree recursively applies splits until reaching a specified maximum depth or minimum number of samples per bin.The CART starting point uses response averages within the left and right bins.Source: Appendix: TBRF implementation details · p. 54
  • Training cost is dominated by sorting and scales as O(N log^2 N), with the procedure described as fast and independent of a starting guess.The cost is stated in terms of the number of data values N.Source: Appendix: TBRF implementation details · p. 56
  • The method predicts coefficients multiplying tensor basis elements rather than directly predicting the final anisotropy, allowing predictions outside the range of training outcomes.Pruning and regularization reduced irregular predictions, but the median across trees was more successful than the mean at reducing outlier sensitivity.Source: Appendix: TBRF implementation details · p. 57
  • Standard settings produce anisotropy predictions relatively insensitive to hyperparameters compared with the tuned TBRF.The example uses fully grown trees, all features for splits, and Γ = 1 × 10^-15; Figure 13 illustrates robustness.Source: Appendix: TBRF implementation details · p. 58, Figure 13 · p. 58

Related papers

Loading 1810.08794v2…