PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks

Nicholas Geneva, Nicholas Zabaras

arXiv:1807.02901v3physics.flu-dynphysics.comp-phstat.ML

TL;DR

Data-driven RANS models have often lacked uncertainty quantification, while their predictions can degrade on flows unlike their training data. This work uses an invariant Bayesian neural network to predict Reynolds-stress anisotropy and quantify uncertainty, providing confidence estimates and predictive bounds for flow quantities.

  • Problem

    Uncertainty quantification has largely been neglected in data-driven RANS models, despite prediction degradation on flows that differ from the training data.

    Source: §1 · p. 3

  • Method

    An invariant Bayesian deep neural network predicts the anisotropic Reynolds-stress tensor, and the framework propagates its uncertainty to quantities of interest.

    Source: §1 · p. 4, §6 · p. 38

  • Results

    The framework provides a means to assess model confidence and compute predictive bounds for flow quantities of interest.

    Source: §1 · p. 4, §6 · p. 39

  • Takeaways & Limitations

    The bounds and confidence estimates can help identify potentially low-accuracy regions that may warrant finer meshes or high-fidelity simulations.

    Source: §6 · p. 39

    The invariant neural network relies on local quantities, an assumption described as questionable for flows with strong inhomogeneity.

    Source: §3.1 · p. 11

Abstract

from arXiv · show

Data-driven methods for improving turbulence modeling in Reynolds-Averaged Navier-Stokes (RANS) simulations have gained significant interest in the computational fluid dynamics community. Modern machine learning algorithms have opened up a new area of black-box turbulence models allowing for the tuning of RANS simulations to increase their predictive accuracy. While several data-driven turbulence models have been reported, the quantification of the uncertainties introduced has mostly been neglected. Uncertainty quantification for such data-driven models is essential since their predictive capability rapidly declines as they are tested for flow physics that deviate from that in the training data. In this work, we propose a novel data-driven framework that not only improves RANS predictions but also provides probabilistic bounds for fluid quantities such as velocity and pressure. The uncertainties capture both model form uncertainty as well as epistemic uncertainty induced by the limited training data. An invariant Bayesian deep neural network is used to predict the anisotropic tensor component of the Reynolds stress. This model is trained using Stein variational gradient decent algorithm. The computed uncertainty on the Reynolds stress is propagated to the quantities of interest by vanilla Monte Carlo simulation. Results are presented for two test cases that differ geometrically from the training flows at several different Reynolds numbers. The prediction enhancement of the data-driven model is discussed as well as the associated probabilistic bounds for flow properties of interest. Ultimately this framework allows for a quantitative measurement of model confidence and uncertainty quantification for flows in which no high-fidelity observations or prior knowledge is available.

1. Introduction

RANS remains important for rapid engineering simulations, but Reynolds-stress modeling leaves substantial model-form uncertainty and can limit accuracy. This work proposes a Bayesian deep-learning framework that improves RANS predictions while quantifying uncertainty and testing transfer to flows with different geometries and fluid properties.

1. Introduction

  • RANS is widely used for rapid-turnaround engineering simulations, yet modeling the Reynolds stress can make its predictions highly inaccurate.LES and DNS remain outside the scope of practical engineering applications described here.Source: §1 · p. 2
  • Empirical calibration and transport-equation modeling create substantial model-form uncertainty, which remains a major source of uncertainty in RANS simulations.More complex models and region-specific model choices still rely heavily on empirical tuning.Source: §1 · pp. 2–3
  • Machine-learning turbulence models have improved RANS accuracy, but their uncertainty has largely been ignored as predictions degrade on flows unlike the training data.Data-driven methods also face the accuracy-versus-flow-versatility tension seen in traditional turbulence models.Source: §1 · pp. 3–4
  • Earlier uncertainty methods either tuned models using observations from a single flow or estimated confidence without providing probabilistic bounds for flow quantities.The cited alternatives respectively used high-fidelity observations or confidence measures based on Mahalanobis distance and kernel density estimation.Source: §1 · p. 4
  • The framework combines a Bayesian deep-learning model with uncertainty propagation to quantify model-form uncertainty and provide probabilistic bounds for quantities such as pressure and velocity.It is assessed on flows with geometries and fluid properties different from those in the training set.Source: §1 · p. 4

2. Problem Formulation

Time averaging in RANS creates a closure problem because the Reynolds stress must be approximated. The formulation focuses on the standard k−ϵ model and its assumptions, calibration, strengths, and limitations.

2.1. Governing Equations

  • The RANS momentum equation requires closure by approximating the Reynolds-stress term.This closure challenge follows from time averaging the Navier–Stokes equations.Source: §2.1 · p. 5
  • The Boussinesq approach represents Reynolds stress through a form analogous to the stress–strain relation of a Newtonian fluid, using turbulent kinetic energy and viscosity.The passage identifies the fluctuation velocity relative to its time-averaged value.Source: §2.1 · p. 5
  • The Boussinesq eddy-viscosity form is used for the simplifications it provides when substituted into RANS, not because it is accurate.Source: §2.1 · p. 5

2.2. RANS Turbulence Models

  • The study uses the standard k−ϵ closure for fully turbulent, incompressible flow, expressing effective viscosity through turbulent kinetic energy and dissipation.The k−ϵ model is described as a commonly used RANS closure.Source: §2.2 · p. 6
  • The dissipation transport equation is an empirical fit, and the model’s five tunable constants are taken from Launder et al.’s data fitting over turbulent flows.The stated values are Cµ = 0.09, Cϵ1 = 1.44, Cϵ2 = 1.92, σk = 1.0, and σϵ = 1.3.Source: §2.2 · p. 6
  • The k−ϵ model is robust, efficient, and broadly valid for fully turbulent flows, but can be inaccurate for complex flows, separation, and sharp pressure gradients.The converged simulations using the stated parameters serve as the baseline RANS simulations.Source: §2.2 · p. 6

3. Data-Driven Framework

The framework learns a direct mapping from baseline RANS information to improved Reynolds-stress fields, then propagates model uncertainty into flow quantities. It combines an invariant Bayesian neural network with stochastic RANS simulations to estimate uncertainty bounds.

3. Data-Driven Framework

  • Training uses baseline RANS and high-fidelity LES solutions from a small library of flows to learn direct predictions of the Reynolds-stress field.The framework can also be applied to tuned coefficients or correction terms; this work models the stress tensor directly. LES data can vary with mesh resolution and subgrid-scale model, potentially introducing physical inconsistencies.Source: §3 · pp. 7–8

3.1. Invariant Neural Network

  • The invariant network learns coefficients of tensor basis functions from five invariants to produce the anisotropic stress tensor, preserving Galilean and coordinate-transform invariance.The representation assumes the mapping between RANS and LES physics can be expressed through those invariants; local inputs also leave spatial correlations unmodeled, a concern for strongly inhomogeneous flows.Source: §3.1 · pp. 9–11

3.2. Bayesian Neural Network

  • Stein variational gradient descent represents the Bayesian network posterior with particles, enabling predictive uncertainty estimates that include epistemic uncertainty.The network uses Gaussian weight priors and a Gamma-distributed precision; output-wise noise represents uncertainty that cannot be reduced with more training data.Source: §3.2 · pp. 12, 15

3.3. Uncertainty Quantification with SDD-RANS

  • SDD-RANS samples posterior network particles to generate Reynolds-stress fields, holds each field fixed in a forward RANS simulation, and uses Monte Carlo statistics to quantify uncertainty in flow quantities.A single network predicts the stress field over the domain to retain spatial smoothness; explicit stress representation and noisy predictions can raise convergence concerns, particularly at higher Reynolds numbers.Source: §3.3 · pp. 15–17

3.4. Framework Implementation

  • The implementation trains an invariant model with SVGD particles, predicts a Reynolds-stress field for each particle on a new flow, and runs constrained forward simulations to obtain probabilistic bounds.Training combines baseline RANS and high-fidelity data from different flows; each forward simulation fixes the predicted stress field while updating the remaining state variables.Source: §3.4 · pp. 17–18

4. Numerical Implementation and Training

The section describes the CFD setup and neural-network training choices, then explains how the predicted Reynolds-stress field is incorporated into RANS simulations with wall treatment.

4.1. CFD Methods

  • OpenFOAM supplied the CFD simulations: simpleFoam produced baseline RANS solutions, while pimpleFoam with a damped Smagorinsky model produced LES data.Both domains used second-order discretization, and simulations ran below a CFL number of 0.3.Source: §4.1 · p. 18, Table 1 · p. 19, Table 2 · p. 19

4.2. Machine Learning Implementation

  • Training used Leaky Rectifier activations and tapered hidden layers; points were resampled every 10 epochs, while mixed-flow mini-batches and normalized inputs supported training.The training-point subset was evenly distributed across flows; invariant inputs were sigmoid-rescaled and tensor basis functions were L2-normalized.Source: §4.2 · pp. 20, 22
  • The five-hidden-layer network was selected because networks above six layers significantly over-fit validation data, while the six-layer network showed noisy MSPE.The tested architectures had similar MNLL behavior, but differed in converged MSPE.Source: §4.2 · pp. 20–21, Figure 3 · p. 21
  • Validation and test performance converged quickly with minimal over-fitting, and the training run took about 3.0 hours on one NVIDIA P100 GPU.The test MSPE used 1,000 randomly selected points from each test flow.Source: §4.2 · p. 22, Figure 4 · p. 23
  • The learned mixing coefficients identified physical flow regions and maintained symmetries in both training flows, with minor over-fitting ahead of the square cylinder.The square-cylinder visualization is at bulk Reynolds number 5000 and used no domain reflections to impose symmetry.Source: §4.2 · p. 23, Figure 5 · p. 24, Figure 6 · p. 24

4.3. Constrained R-S Simulation

  • The solver holds the sampled Reynolds-stress field constant in the momentum equation, using baseline RANS TKE for boundary treatment and a blending function near walls.The blending function transitions from baseline RANS stress near the wall to the data-driven prediction in the bulk flow; the selected tuning parameter was 0.05.Source: §4.3 · pp. 24–25

5. Numerical Results

The study tests SDD-RANS on backward-step and wall-mounted-cube flows at three Reynolds numbers, using geometries that differ from the training flows. Results show improvements in selected flow regions, while predictive uncertainty varies with location and Reynolds number.

5.1. Backwards Step

  • For the backward step, SDD-RANS improves anisotropic-term shape predictions for a11 and a33 at Re=500 and 2500, but underpredicts a22 and has small variance.Predictions are consistently upstream of LES; at Re=5000, a11 and a33 favor one region rather than the two seen at lower Reynolds numbers.Source: §5.1 · p. 26, Figure 8 · p. 27, Figure 9 · p. 27
  • For the backward step, SDD-RANS predicts the second recirculation region at Re=500 but offers little improvement at Re=2500 and 5000 beyond x/h ≤7.5.At the higher Reynolds numbers, it also misses the upper recirculation region present in LES.Source: §5.1 · pp. 28–30, Figure 11 · p. 29, Figure 12 · p. 30

5.2. Wall Mounted Cube

  • For the cube, SDD-RANS improves Reynolds-stress predictions near the front edge, while predictive variance is concentrated behind the obstacle.The bulk region has little variability, where baseline RANS is accurate.Source: §5.2 · p. 32, Figure 14 · p. 33
  • In the cube’s recirculation region, SDD-RANS mean velocity shows no marginal improvement over baseline RANS, although its larger variance often encloses part of the LES solution.The authors suggest this uncertainty may indicate flow physics absent from the training data.Source: §5.2 · p. 33, Figure 15 · p. 34
  • For the wall-mounted cube, SDD-RANS captures detached flow above the cube that baseline RANS misses, with more accurate recirculation predictions at the lower Reynolds number.Velocity-sample variance rises with Reynolds number in the recirculation region but remains small in the bulk flow.Source: §5.2 · pp. 34–35, Figure 16 · p. 35, Figure 17 · p. 35
  • On the cube’s y = 0.5h plane, SDD-RANS predicts detached side flow, though some asymmetry remains.Source: §5.2 · p. 35, Figure 18 · p. 36, Figure 19 · p. 37
  • Cube pressure predictions improve on baseline RANS, correcting an unphysical pressure drop at the leading edge; uncertainty nearly captures LES values across the faces.Upstream pressure variance is smaller in the laminar region.Source: §5.2 · p. 37, Figure 20 · p. 37, Figure 21 · p. 38

6. Conclusions

The conclusion presents an invariant Bayesian neural-network and Monte Carlo framework for propagating Reynolds-stress uncertainty into flow quantities. It also discusses challenges in predicting unseen flows and potential uses of the resulting uncertainty bounds.

6. Conclusions

  • The framework uses an invariant Bayesian neural network to model anisotropic Reynolds-stress distributions and Monte Carlo simulation to propagate uncertainty to flow quantities.It was investigated on two flows to assess generalization.Source: §6 · p. 38
  • The invariant network was difficult to train and predict satisfactorily on unseen flows; the authors suspect five inputs provide insufficient information.They identify finding important local and non-local variables as a critical research direction.Source: §6 · p. 38
  • Non-local information may benefit Reynolds-stress modeling, although it is difficult to implement on non-uniform grids; generalization remains an open problem.The authors note convolutional neural networks as one way to apply non-local modeling.Source: §6 · p. 39
  • The framework could assess predictive confidence, locate potentially less accurate regions for finer meshes or high-fidelity simulations, and provide bounds when training data are limited.It takes more computation than deterministic data-driven approaches but remains low-cost relative to high-fidelity simulations.Source: §6 · p. 39
  • Bayesian predictions could support experimental-design studies of how training-data quantity and flow selection affect predictions; detailed epistemic-uncertainty analysis remains beneficial.Source: §6 · p. 39
Loading 1807.02901v3…