Bayesian Uncertainty Quantification Applied to RANS ...roystgnr/stanford_rans.pdfBayesian Uncertainty Quantiﬁcation Applied to RANS Turbulence Models Todd A. Oliver The University

PECOSPredictive Engineering and Computational Sciences

Bayesian Uncertainty Quantification Applied to RANSTurbulence Models

Todd A. Oliver

The University of Texas at Austin

August 17, 2011

T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 1 / 30

Outline

1 Introduction and OverviewMany Success StoriesRANS Problem

2 UQ Approach OverviewBayes’ TheoremStochastic models and UQ

3 Channel Flow ProblemModel DevelopmentUQ Results

4 Summary


Introduction and Overview Many Success Stories

UQ and Validation Hierarchy

Coupled Subsystems& Coupled Submodels

Fewer, more difficult experiments

Isolated Components & Sub Models

Many "Simple" Experiments

Increasing Complexity

& Cost

Full SystemRare

Experiments

Prediction of Quantity of Interest

Full System Validation

Coupled Calibration& Validation

Component Calibration& Validation

Successes so far include:• Sensitivity analysis and forward UQ for full system

• Inverse and forward UQ for many component models


Introduction and Overview Many Success Stories

Other Success Stories

Full System SimulationSensitivity analysis and forward UQ (R. Stogner et al.)

Component Models and Other Exercises• Chemical kinetics (K. Miki, S.H. Cheung, C. Simmons)

• EAST Shock tube analysis (M. Panesi, K. Miki, S. Prudhomme)

• Thermocouple calibration (P. Bauman, J. Jagodzinski)

• Surface reaction efficiency (O. Sahni, R. Upadhyay)

• Optimal experimental design (G. Terejanu et al.)

UQ SoftwareQUESO (E. Prudencio) enables inverse UQ in all modeling domains


Introduction and Overview RANS Problem

RANS Introduction

Motivation• Majority of engineering simulations of turbulent flows use

Reynolds-averaged Navier-Stokes (RANS) models

• RANS models well-known to be imperfect and unreliable• Uncertainty due to uncertain parameters

I Model parameters are not constants of nature• Uncertainty due to model inadequacy

I Closure models are not physical laws

• Must quantify effects of these uncertainties on model predictions

Approach• Formulate stochastic models to represent uncertainty

• Use Bayesian probabilistic approach to calibrate and compare models


UQ Approach Overview

Outline




4 Summary


UQ Approach Overview Bayes’ Theorem

The Bayesian Approach

P (B|A) =P (B ∩A)

P (A)

P (A|B) =P (A ∩B)

P (B)

P (B|A) =P (A|B)P (B)

P (A)

Let A = data, B = parameters

Then P (A|B) = the modelThomas Bayes

posterior knowledge =likelihood of data · prior knowledge

probability of data

“Theories have to be judged in terms of their probabilities in light of the evidence”.


UQ Approach Overview Stochastic models and UQ

What constitutes the model?

Physics• Mathematical representation of physical phenomena of interest

• At macroscale, usually deterministic (e.g., RANS)

Experimental UncertaintyModel for uncertainty introduced by imperfections in observation process

Model UncertaintyModel for uncertainty introduced by imperfections in physical model

Prior InformationAny relevant information not encoded in above models



Model⇒ Likelihood

p(D|θ) =

∫p(D|Dtrue, θ)︸︷︷︸

Experimental uncertainty

p(Dtrue|θ)︸︷︷︸Prediction model

dDtrue

p(Dtrue|θ) =

∫p(Dtrue|Dphys, θ)︸︷︷︸

Model uncertainty

p(Dphys|θ)︸︷︷︸Physical model

dDphys

• Physics + model uncertainty = Prediction model

• Prediction model + experimental uncertainty = Likelihood

• Different/further decomposition possible depending on availableinformation

• Models coupled with prior form a stochastic model class



UQ Using Stochastic Model Classes

Processes• Single Model Class, M

I Calibration: p(θ|D) ∝ p(θ) p(D|θ)I Prediction: p(q|D) =

∫p(q|θ,D) p(θ|D) dθ

I Experimental design• Multiple Model Classes,M = {M1, . . . ,MN}

I Calibration, prediction, and experimental design with each model classI Model comparison/selection: P (Mi|D,M) ∝ P (Mi|M) p(D|Mi)I Prediction averaging: p(q|D,M) =

∑i p(q|D,Mi)P (Mi|D,M)

Software used at UT• DAKOTA: Forward propagation• QUESO: Calibration, model comparison

I Metropolis-Hastings, DRAM, Adaptive Multi-Level Sampling



Summary: Four Stage Bayesian Framework

Stochastic Model DevelopmentGenerate extension of physical model to enable probabilistic analysis

• Closure parameters viewed as random variables

• Stochastic representations of model and experimental errors

CalibrationBayesian update for parameters: p(θ|D) ∝ p(θ)L(θ;D)

PredictionForward propagation of uncertainty using stochastic model

Model ComparisonBayesian update for plausibility: P (Mj |D,M) ∝ P (Mj |M)E(Mj ;D)


Channel Flow Problem

Outline




4 Summary


Channel Flow Problem Model Development

Application Overview

Models• Four competing RANS turbulence models

I Baldwin-Lomax; Spalart-Allmaras; Chien, low Re k-ε; Durbin’s v2-f• Four competing model uncertainty representations

I Three velocity-based with different spatial correlation assumptionsI One Reynolds stress-based

• Sixteen total stochastic models

Calibration Data and Prediction QoI• Fully-developed, incompressible channel flow

• Calibrate using DNS data for Reτ ≈ 1000, 2000 (Reτ ≡ uτδ/ν)

• Predict centerline velocity at Reτ = 5000



Physical Modeling ApproachRANS• Decompose flow into mean and fluctuating parts: u = u+ u′.

• Average the Navier-Stokes equations. For channel flow,

− d

dη

(1

Reτ

du+

dη+ τ

)= 1,

where u+ = u/uτ , τ+ = u′v′/u2τ , u2

τ = ν dudy , η = y/δ.

• Model Reynolds stress using eddy viscosity τ ≈ νt dudy• Make up or choose a turbulence model for νt (Baldwin-Lomax,

Spalart-Allmaras, k-ε, k-ω, ...)

Key PointModel inadequacy introduced by closure model—i.e., combination of eddyviscosity assumption and turbulence model



Stochastic Models: Basic IdeasGoalCreate model that produces distribution over mean velocity fields

• Incorporate information from RANS solution

• Model the fact that RANS solution represents incomplete knowledge

Simple Example

〈u〉(y; θ, α) = u(y; θ) + ε(y;α)

• u is the RANS mean velocity

• ε is a random field representing uncertainty due to RANS infidelity

• 〈u〉 is stochastic prediction of true mean velocity

Issues• Where to introduce uncertainty model representing model inadequacy

• Details of that model (e.g., distribution for ε, dependence on scenario)T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 15 / 30


Velocity-Based Stochastic Models

Multiplicative Gaussian Error

〈u〉+(η; θ, α) = (1 + ε(η;α))u+(η; θ)

where ε is a zero-mean Gaussian random field.

Covariance Structures• Independent: cov(ε(η), ε(η′)) = σ2δ(η − η′)

• Correlated (homogeneous): cov(ε(η), ε(η′)) = σ2 exp(− 1

2(η−η′)2`2

)• Correlated (inhomogeneous):

cov(ε(η), ε(η′))〉 = σ2(

2`(η)`(η′)`2(η)+`2(η′)

)1/2exp

(− (η−η′)2

`2(η)+`2(η′)

)where

`(η) =

ìn for η < ηinìn + òut−ìn

ηout−ηin (η − ηin) for ηin ≤ η ≤ ηoutòut for η > ηout



Inhomogeneous Model Details

`(η) =

ìn for η < ηinìn + òut−ìn

ηout−ηin (η − ηin) for ηin ≤ η ≤ ηoutòut for η > ηout

• All length scales non-dimensionalized by channel height

• Inner lengths scale with viscous length, not channel height

• Rewrite inner variables using viscous scales:

ìn = `+in/Reτ ηin = η+in/Reτ

Length scales (`+in, òut), blend points (η+in, ηout), and variance σ2 are

calibration parameters



Covariance Models

Homogeneous

0 0.2 0.4 0.6 0.8 10.97

0.98

0.99

1

1.01

1.02

1.03

η

ε

Inhomogeneous

0 0.2 0.4 0.6 0.8 10.97

0.98

0.99

1

1.01

1.02

1.03

η

ε

Inhomogenous covariance enables better representation of two-layerstructure of channel flow



Reynolds Stress-Based Stochastic Models

Motivation• Structure of the RANS equations is not uncertain

• Only the closure (i.e., Reynolds stress tensor field) is uncertain

Additive Model• 〈u′v′〉+(η;θ,α) = T+(η;θ)− ε(η;α) where T+ obtained by solving

RANS+turbulence model

• Find 〈u〉 by forward propagation through mean momentum

− d

dη

(1

Reτ

d〈u〉+

dη+ 〈u′v′〉+

)= 1,

• ε chosen to be zero-mean Gaussian random field with

cov(ε(η), ε(η′)) = kin(η, η′) + kout(η, η′),



Reynolds Stress-Based Stochastic Models

Reynolds Stress

0 0.2 0.4 0.6 0.8 1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

η

ε

Velocity

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

η∆

u

Large Reynolds stress uncertainty in inner layer


Channel Flow Problem UQ Results

Results Overview

• Calibration:I Joint posterior PDFs for model parameters for each model class

• Model comparison:I Posterior plausibility for each stochastic model classI Examine joint and conditional plausibilities

• QoI Prediction:I Compare predictions (PDF for QoI) of each model class



Sample Parameter Posterior PDFs

Spalart-Allmaras

0.5 1 1.50

2

4

6

κ

p(κ

)

0.5 1 1.50.5

1

1.5

cv1

κ

0.5 1 1.50

1

2

3

4

5

cv1

p(c

v1)

Chien k-ε

0.5 1 1.5 2 2.50

1

2

3

σk

p(σ

k)

0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

σε

σk

0.5 1 1.5 2 2.50

1

2

3

σε

p(σ

ε)

Parameter joint posterior PDFs computed using Adaptive Multi-LevelAlgorithm implemented in QUESO Library



Joint Model Plausibility

Baldwin Spalart Chien DurbinIndependent ≈ 0 ≈ 0 ≈ 0 ≈ 0

Homogeneous ≈ 0 ≈ 0 ≈ 0 ≈ 0Inhomogenous ≈ 0 ≈ 0 0.995 3.24× 10−3

Reynolds Stress ≈ 0 1.36× 10−3 ≈ 0 ≈ 0

Chien k-ε coupled with inhomogeneous velocity uncertainty modelstrongly preferred



Uncertainty Model Plausibility

Conditioned on turbulence model, which uncertainty model is preferred?

Uncertainty Model Baldwin Spalart Chien DurbinIndependent ≈ 0 ≈ 0 ≈ 0 ≈ 0

Homogeneous ≈ 0 ≈ 0 ≈ 0 ≈ 0Inhomogenous ≈ 1 6.69× 10−3 ≈ 1 0.9998

Reynolds Stress ≈ 0 0.993 ≈ 0 1.86× 10−5

Observations• Data prefers the inhomogenous correlation structure for all models

• Makes sense given the two-layer structure of the mean velocity profile



Turbulence Model Plausibility

Conditioned on uncertainty model, which turbulence model is preferred?

Turb Model Indep Homog Inhomog Rey StressBaldwin ≈ 1 0.779 ≈ 0 ≈ 0Spalart ≈ 0 ≈ 0 1.01× 10−5 0.99995Chien ≈ 0 ≈ 0 0.996 1.01× 10−5

Durbin ≈ 0 0.221 3.33× 10−3 4.11× 10−5

ObservationPreferred turbulence model depends on uncertainty model



QoI Predictions

Uncertainty Model Comparison

24 24.5 25 25.5 26 26.5 27 27.5 280

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

q = <u>+(1)

p(q

)

IND

SE

VLSE

ARSM

Turbulence Model Comparison

24 24.5 25 25.5 26 26.5 27 27.5 280

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

q = <u>+(1)

p(q

)

BL

SA

Chien

v2−f

Observations• Different stochastic model extensions lead to significantly different

uncertainty predictions

• With same stochastic extension, turbulence models similar for this QoI


Summary

Outline




4 Summary


Summary

Summary

PECOS: Many success stories• Sensitivity analysis and forward UQ for full system

• Inverse UQ, forward UQ, and model comparison for componentmodels

• QUESO library: Algorithm research and enabling technology

RANS Application• Applied Bayesian framework for calibration and comparison of models

to popular RANS models

• Stochastic modeling of model inadequacy crucial to both modelcomparison and prediction

• Chien k-ε model preferred by the data over competitors in this case


Summary

Ongoing Work

• Continued stochastic model developmentI Often critical to conclusionsI Build what is known into the model

• Apply Bayesian UQ ideas for coupled modelsI Philosophy on parameter updatingI Surrogate QoIs and QoI-aware model comparison

• Prediction validationI More than comparing model output to available dataI Given model output and data, what can be said regarding prediction?I Different procedures for different situations⇒ No silver bullet


Summary

Thank you


Documents

Bayesian Uncertainty Quantification Applied to RANS ...roystgnr/stanford_rans.pdfBayesian Uncertainty Quantiﬁcation Applied to RANS Turbulence Models Todd A. Oliver The University