Post on 11-Jul-2020
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 2 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 3 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 4 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 5 / 30
UQ Approach Overview
Outline
1 Introduction and OverviewMany Success StoriesRANS Problem
2 UQ Approach OverviewBayes’ TheoremStochastic models and UQ
3 Channel Flow ProblemModel DevelopmentUQ Results
4 Summary
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 6 / 30
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”.
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 7 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 8 / 30
UQ Approach Overview Stochastic models and UQ
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 9 / 30
UQ Approach Overview Stochastic models and UQ
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 10 / 30
UQ Approach Overview Stochastic models and UQ
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)
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 11 / 30
Channel Flow Problem
Outline
1 Introduction and OverviewMany Success StoriesRANS Problem
2 UQ Approach OverviewBayes’ TheoremStochastic models and UQ
3 Channel Flow ProblemModel DevelopmentUQ Results
4 Summary
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 12 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 13 / 30
Channel Flow Problem Model Development
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 14 / 30
Channel Flow Problem Model Development
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
Channel Flow Problem Model Development
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) =
`in for η < ηin`in + `out−`in
ηout−ηin (η − ηin) for ηin ≤ η ≤ ηout`out for η > ηout
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 16 / 30
Channel Flow Problem Model Development
Inhomogeneous Model Details
`(η) =
`in for η < ηin`in + `out−`in
ηout−ηin (η − ηin) for ηin ≤ η ≤ ηout`out 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:
`in = `+in/Reτ ηin = η+in/Reτ
Length scales (`+in, `out), blend points (η+in, ηout), and variance σ2 are
calibration parameters
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 17 / 30
Channel Flow Problem Model Development
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 18 / 30
Channel Flow Problem Model Development
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′),
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 19 / 30
Channel Flow Problem Model Development
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 20 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 21 / 30
Channel Flow Problem UQ Results
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 22 / 30
Channel Flow Problem UQ Results
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 23 / 30
Channel Flow Problem UQ Results
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 24 / 30
Channel Flow Problem UQ Results
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 25 / 30
Channel Flow Problem UQ Results
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 26 / 30
Summary
Outline
1 Introduction and OverviewMany Success StoriesRANS Problem
2 UQ Approach OverviewBayes’ TheoremStochastic models and UQ
3 Channel Flow ProblemModel DevelopmentUQ Results
4 Summary
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 27 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 28 / 30
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
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 29 / 30
Summary
Thank you
T. A. Oliver PSAAP V&V UQ Workshop August 17, 2011 30 / 30