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Development of a Validation and Uncertainty Quantification Framework for Closure Models in Multiphase CFD Solver

Yang Liu and Nam Dinh

Multi-Physics Model Validation WorkshopJune/28/2017

Multiphase Flow and boiling

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Multiphase Flow and boiling involves multi-scale phenomena with different physics

• Flow regime (cm)

• Bubble behavior, interfacial exchange (mm)

• Turbulence and nucleation (μm)

Eulerian-Eulerian two-fluid-model: closure dependent averaged conservative equations

Condensation&Evaporation

Turbulence

Interfacial forces

Condensation&Evaporation

Turbulence

Mass

Momentum

Energy

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Wall boiling Interfacial forcesTurbulence model

Active nucleation site density

Evaporation heat transfer

Single-phase convective

heat transfer

Quenching heat transfer

Near wall heat transfer and evaporation

Bubble induced turbulence

Wall function

Turbulent viscosity

Turbulent heat fluxTurbulent viscosity

Momentum exchange

Interfacialcondensation

Drag forceBubble size

Turbulence viscosity

force

Wall lubrication

force

Lift force

Bubble departure diameter

Bubble departure frequency

Condensation

U α h α

Uh

Nucleation

Bubble breakup

Bubble coalesce

Heat transfer partition

Closure Structure in MCFD Solver

• Wall boiling• Heat partitioning, nucleation

• Interfacial momentum

• Interfacial mass/heat transfer

• Bubble size

• Turbulence• Bubble induced turbulence

VUQ Framework for two-fluid model based solver

• For a solver that deal with Multi-physics & Multi-scale phenomena (e.g. MCFD or CTF), conservative variables are averaged and closure models are used for the missing information

• Empirical parameters exist in closures which are one of the major error/uncertainty source of the solver (another is numerical)

Purpose: Given scenario, code, closure model, and available database,• 1.What can we conclude on the uncertainty of the QoIs?• 2.Is there model form inconsistency between closures? • 3.What is the applicable space of the VUQ results? (how far can we extrapolate the VUQ work done

under condition A to an unknown condition?) • 4.What is the best option to improve the uncertainty? (which measurement can reduce uncertainty

mostly?)

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VUQ workflow

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MCFD platform and VUQ tool• boilEulerFoam based on OpenFOAM

– Original developer: Dr. A.Bui and Prof. N.Dinh– Major revision: C.Rollins and Prof. H.Luo (MAE, NCSU)– Selected Model implementation: Y.Liu and Prof.N.Dinh

• VUQ tool used in this workDAKOTADRAMPython scikit-learnQUESO

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VUQ Framework for Multi-physics / Multi-scale solver

Generalized workflow• State of art tools• Non-intrusive method• Flexibility for method/algorithm selection

Preparations• Data management• Validation metrics• Model form inconsistency evaluation

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Data management: NEKAMS: Store and manage VUQ database• NE-KAMS (Nuclear Energy – Knowledge base

for Advanced Modeling and Simulation) – Enable knowledge base centric process for

V&V, UQ and M&S activities

– Collect, document, qualify, structure, format, integrate and manage data and information in various forms and from various sources

Credit: Dr. W.Ren, ORNL

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NE-KAMSKnowledge

Base

Validation Experiment

V&V and UQ Guidelines

V&V and UQStandards and Requirements

V&V & UQ Assessments

Computation

M&SActivities

Data management ： Database example

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Data Source: Prof. Buongiorno group, MITData are automatically processed and stored in two scales

[F1] General information NoteSource Synthetic CRUD Test (MIT) Details can be found in[ref][F2] System configurationGeometry Vertical flow in rectangular

channelFluid materials water liquid/vaporHeater materials ITO sapphire heater with

synthetic CRUD[F3] Test programFlow conditions 500 kg/m2

Heat configurations 2um thick CRUD with10um diameter chimneyson a 45um pitch

Heat flux 1400 kW/m2

[F4] Data[D0] raw data IR counts distribution[D1] primary data temperature/heat flux

distributionUsed to current VUQ workin current practice

[D2] secondary data ensemble averagedtemperature, heat flux andnucleation information

Used to current VUQ workin current practice

[D3] ternary data Nucleation sites location/interaction etc.

Used for a more detailedmodeling approach

[F5] Data characteristicsApplicability boiling model VUQ for flow

boiling on low pressureQuality Good High resolution data with

designed surface

Validation Metrics: Evaluation of model uncertainty and model form inconsistency

• Confidence intervalsThere is a α% possibility that the true error between model and data are within the given interval

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�𝐄𝐄 − 𝑡𝑡 ⁄𝛼𝛼 2,𝑣𝑣 ⋅𝐬𝐬𝑛𝑛

, �𝐄𝐄 + 𝑡𝑡 ⁄𝛼𝛼 2,𝑣𝑣 ⋅𝐬𝐬𝑛𝑛

�𝐄𝐄 = �⃗�𝐲𝑚𝑚,𝑜𝑜𝑜𝑜𝑜𝑜 −1𝑛𝑛�𝑖𝑖=1

𝑛𝑛

�⃗�𝐲𝑒𝑒𝑖𝑖

𝐬𝐬 =1

𝑛𝑛 − 1�𝑖𝑖=1

𝑛𝑛

�⃗�𝐲𝑒𝑒𝑖𝑖 − �⃗�𝐲𝑒𝑒,𝑎𝑎𝑣𝑣𝑎𝑎2

⁄1 2

Experimental Data

Response

SimulationData

“Overlapping Coefficient”

Overlapping coefficient

Model form inconsistency evaluationTotal Data Model Integration

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'sin 1,2,inconsistency TDMI gleε

∞∝ −Y Y

Divide-and-Conquer Approach

Model inconsistency in MCFD solver mainly stems from

• Potential conflict of assumptions between different closures

• Divide tightly coupled phenomena and treated them independently

Case Study I: interfacial momentum closure

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Expression Model

Drag Schiller-Naumman

Lift Tomiyama

Wall lubrication Antal

Turbulent dispersion Gosman

Virtual mass Rusche

Bubble size Anglart

3( )

4D Da b

s

C

Dρ α= − − −a b a bM U U U U

( ) ( )La L bC ρ α= − × ∇ ×a b aM U U U

21( )

2(y )WL

a WL b S rwC D xρ α= − ⋅a bM U U n

3

4

tTD bDa bt

s bt

C

D Pr

υρ α

σ= − − ∇a bM U U

1( )

2VM ba b vm

D D

Dt DtCρ α= − −aU U

M

ref,1 sub,2 ref,2 sub,1 sub

sub,1 sub,2

(T ) (T )subs

D T D TD

T T

− + −=

−

Case Study II： Wall heat transfer closures

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Kurul & Podowski(1991) : With different closure options (Version A & B) Shaver & Podowski(2014)

Surrogate construction

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CMFD solver Surrogate

Sampling +

Gaussian Process

I.C.B.C.

Closure parameter Closure parameter

QoIs QoIsI.C.B.C.

fixed fixed

Surrogate accuracy evaluation by cross validation

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QoI MaximumRMS

MaximumABS

Voidfraction 5.98e-3 3.98e-3

Gasvelocity 2.27e-3 1.70e-3

Relativevelocity 2.31e-3 1.66e-3

Low pressure adiabatic flow High pressure subcooled boiling flow

QoI Maximum RMS

Maximum ABS

Void fraction 2.37e-3 1.48e-3

Gas velocity 2.41e-3 1.54e-3

Relative velocity 5.81e-4 3.82e-4

Liquid Temperat

ure4.58e-2 2.70e-2

Global Sensitivity Analysis: Morris Measure and Sobol indices

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Interfacial momentum terms: interfacial forces and bubble size

Wall lubrication hasinfluence on all regions

Parameters have similarsensitivity in all regionsadiabatic flow, but havedifferent behavior inboiling flow

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Wall Boiling Model

8.55E-01Cwall3.70E-01C32.34E-02C2

-7.39E-03C1

-1.85E-03Prt

-8.06E-04vonKarman-6.12E-04yPlusSL

Comparison with Sobol indices

Global Sensitivity Analysis: Morris Measure and Sobol indices

Parameter Selection

• Reason: parameter identifiability issue– For complex non-linear model,

there exists different combinations of parameters that fit the data equally well

– Thus the inverse Bayesian can be performed only on a subset of parameters without identifiability issue

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Parameter Selection: ad hoc approach

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• Check parameter identifiability among most important parameters

• Randomly get rid of one if identified

• Include parameter with intermediate importance one at a time, and check

• Do not include that one if identified

• Directly get rid of not important one

Inverse Bayesian Inference using MCMC: Gen-I model, version B

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Check of convergence: Burn in pattern and autocorrelationJoint sampling of parameters

Bubble Effective area factor

Bubble diameter constant

Turbulent convective constant

One experiment, averaged over heater surface

Validation metrics example• One experiment, averaged over heater surface

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ConstructingConfidence intervals

Model form inconsistency identification:

One experiment, Distribution along heated wall

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Indication of model form inconsistency

Gen-I model, version A Shaver model

Issue identified

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Universal ‘optimal’ parameter estimates do not exist

Inference with datasets on all conditions simultaneously

Extrapolation can lead to large error

Inference with datasets on one condition, then apply parameter distribution to other conditions

A possible solution: gain knowledge from multiple validation results

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Test: infer posterior distribution through

interpolation and extrapolation

Desired future work• With many units, the desired data needs / best available model/parameter can be obtained and aid

decision making

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1 2 3

1 1 1

1 2 3

...

......

... ... ... ...

A A AB B BC C C

Closure options

Phenomena

IET .1

IET .2

SET .1SET .2

SET .3 SET .4

………. Micro-scale

Meso-scale

Macro-scaleValidation database Unit. A

Validation database Unit. B

……

Summary: Current Achievement

Purpose: Given scenario, code, closure model, and available database,

• 1.What can we conclude on the uncertainty of the QoIs? (Answered)• 2.Is there model form inconsistency between closures? (Partially Answered)• 3.What is the applicable range of the VUQ results? (can we extend the VUQ work done

under condition A to B) (Partially Answered)

• 4.What is the best option to improve the uncertainty?

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Summary: Current limitation

• Depends on multiple datasets with uncertainty known– High fidelity data with thorough uncertainty analysis is desired– Acquire more data sources from literature/high fidelity simulation– Evaluating “missing” uncertainty information

• Parameter identifiability not fully resolved, the parameter selection depends on a lot trial and error– Trying state-of-art mathematical methods, e.g. active subspace

• Simultaneous measurement of multiple physics is essential for a comprehensive evaluation of all closures (wall heat transfer behavior, near wall flow and bubble dynamics, bulk flow, etc.)– However, those kind of measurements are currently not available

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