Girimaji APS 2011 UQ Turbulence

Error bounds on engineering turbulence models: A framework for uncertainty quantification (UQ)

Sharath S. Girimaji

Department of Aerospace EngineeringTexas A&M University

2011 Annual meeting of American Physical Society – Division of Fluid Dynamics Baltimore, Maryland

November 20, 2011

Motivation

Someday CFD will replace experiments as the main design tool

Are we there yet? Not really

Why not? Turbulence modeling, numerics etc.

Which is more important? Modeling errors (IMHO)

Current modeling status Making lines pass through symbols,

Current engineering models not predictive

but post-dictive

For engineering turbulence models to become predictive tools:

1.Improve fidelity

2. Uncertainty quantification

UQ in turbulence context

• How do you assess error/uncertainty of a chaotic system?

• Is the concept of UQ meaningful in turbulence computations?

• Is standard UQ terminology/framework applicable to turbulence?

Best we can do: Ignorance management UQ

What should we aim for: In the absence of data

• Catastrophic failure detection

• Physics-based error/uncertainty estimates

Or at least: as in Hurricane trajectory predictions

• Multi-valued prediction envelop; not a single-valued prediction

• Multiple calculations w/different models

Further Questions

Is Reynolds stress a metric of error in mean flow prediction?• Yes: In a well-conducted experiment

• No: In a good numerical implementation of a bad model

• No: In a bad numerical implementation of a good model

• May be: In a good numerical implementation of a good model

Therefore,

1)Mean flow UQ is possible only if Reynolds stress is reliable

2)UQ of Reynolds stress is necessary

Objectives

Develop a framework for UQ of Reynolds stress Rij:

• Identify general sources of uncertainties in Rij

• Characterize and categorize general sources of uncertainties

• Develop guidelines/procedures for quantifying source errors

Provide some direction for Rij UQ in• Empirical/Semi-empirical closure• Higher-order physics based closure

Types of uncertainty – turbulence context

General Types of uncertaintyAleatoric/statistical uncertainty: • From sources outside model parameters• Statistical analysis to quantify uncertainty

• Bayesian approach possible

Epistemic /systematic uncertainty: • From sources within – but not accounted for due to various reasons• Physics-based analysis needed to quantify uncertainty

• Must be based on turbulence physics

Generally, both aleatoric and epistemic occur together and difficult to separate

Sources of Uncertainty

Model Calculations: Mean and moments are not chaotic

Numerical errors: aleatoric/epistemic – standard V&V may apply

Closure errors: Type of error depends upon the level of

modeling

•Empirical models large aleatoric errors

•Physics-based models more epistemic errors

Empirical closure models

Empirical models: Zero equation closures

•Most errors aleatoric; no basis for establishing epistemic error

•Model coefficient sensitivity can be established

•For a given flow, best-fit model coefficient can be found

• Bayesian Analysis

•No basis for assessing error in a new family of flows

Semi-empirical models: Standard 1 & 2 equation models

•For K & possible to distinguish aleatoric and epistemic

•Error in constitutive relation can be estimated using ARSM

Incomplete UQ of CGiven the constitutive model ij = -C Sij

Epistemic uncertainty can estimated using the ARSM closure model.

ARSM reduction

RANSLESDNS

2-eqn. ARSM

Averaging Invariance

Application

7-eqn. SMC Multi-point/global

stability effects

Linear Pressure

Effects: RDT

Nonlinear pressure effects

Spectral and dissipative processes

2-eqn. PANS

Navier-Stokes Equations

Reynolds stress closure models

Any attempt at UQ must start at RSCM level• Turbulent transport Gradient-Diffusion model• Remaining closures Piece-wise homogeneous

All closure models based on incomplete information

Critical `game-changing’ closure• Inhomogeneity Missing global stability physics•Rapid Pressure-strain correlation (r) Missing information

Less critical Closures• Slow Pressure-strain correlation (s) Missing information•Dissipation () Missing details of cascade

Global Instabilities

Flow instabilities are the biggest reason of RSCM error• Multi-point phenomena not amenable to one-point closure

• Large-scale inviscid instabilities more dynamically important

• Small-scale viscous instabilities interesting but unimportant

• Dissipative and not dynamically important

Must resolve large-scale inviscid instabilities

rational UQ unlikely

What else can be done:

Possibility of instability can be forecast and be prepared

for all consequences

Global Instability/Coherent Structure Forecast

Hypothetical velocity amplitude vector

Ominous sign 1: Large variation in velocity gradients

Global Instability/Coherent Structure Forecast

Ominous sign 2: Vanishing 2nd derivative

UQ Rapid Pressure Strain Corr.Traditional RPSC development

•Much theoretical/analytical work to improve RPSC fidelity•Seeking single-valued closure model – despite multi-valued possibility

• = f(R, A, ) simplified to = f(R, A)

•In pursuit of an improbable single-valued closure with limited basis• Conflicting closure requirements (realizability vs. linearity)• Unholy compromises made • Much knowledge of physics discarded

Change in model development paradigm• Multivalued closure model: For a given R and A,

• min = f(R, A) based on worst case • probable = f(R, A) based on most probable • max = f(R, A) based on best case

Redirect modeling effort to explore the range of closure space and identify most probable

Closed streamline flows : Modal behavior.

Most unstable mode

Most stable mode

Most unstable mode

Most stable mode

Closed streamline flows : envelope.

Hypothetical velocity amplitude vector

Envelope of probable behavior

Some observations on UQ for turbulence closures

Epistemic UQ possible only at RSCM level

Locality and incomplete basis are main reasons for epistemic uncertainty

Success of UQ hinges on how well we can fill-in incomplete information

Critical Uncertainty 1: Global stability effectsVirtually impossible to fill-in at one-point closure level

No option but resolve

Critical Uncertainty 2: Rapid-pressure strain correlation

Possible to place bounds on missing information Mijkl tensor

Other RSCM uncertainties are less critical and can be managedUQ for RANS-LES hybrids can be addressed within this framework

Further observations

Future investments: Improved closure fidelity Vs. UQ?

• Point of diminishing returns for improving fidelity

• More practical gains by redirecting theory/analysis

toward uncertainty quantification

CFDers must resolve the phenomena they cannot model,Should model the phenomena physics allows,And have the wisdom to know the difference.