machine learning uncertainty quantification in turbulence · machine learning uncertainty...

MACHINE LEARNING UNCERTAINTY QUANTIFICATION IN TURBULENCE

A. Da Ronch, M. Panzeri, J. Drofelnik, R. d’Ippolito

Aerodynamics and Flight MechanicsUniversity of Southampton, U.K.

22 November 2017DiPaRT 2017 Annual Meeting, Bristol, UK 1

MotivationCFD workflows contain considerable uncertainty, often not quantified

Generally, UQ performed for aleatory uncertainties (grids, BCs, etc)

BUT

Uncertainty in the closure coefficients is the dominant (and often neglected) source of error in RANS

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“The 𝛾 transition model has only been calibrated for classical boundary layer flows. Application to other types of wall-bounded flows is possible, but might require a modification of the underlying correlations.” [1]

“Determining the empirical correlations by numerical optimization, along with debugging the model, demands a very large amount of computations, and it is the hope that other researchers...” [2]

[1] Fluent user’s guide: https://www.sharcnet.ca/Software/Ansys/17.0/en-us/help/flu_th/flu_th_sec_turb_intermittency_over.html[2] N.N. Sorensen; 2009; “CFD Modelling of Laminar-turbulent Transition for Airfoils and Rotors Using the 𝛾 − 𝑅𝑒%&”; Wind Energy; 12; pp. 715-733

Aim & ObjectivesInvestigate UQ & sensitivity analysis of commonly used RANS model to uncertainty in the closure coefficients

This requires a large number of calculations, so:

1. Employ machine-learning approach for efficient & robust exploration of parameter space (10-dimensional) with minimal number of CFD runs

This allow us to:

2. Rank the contribution of each closure coefficient to model uncertainty

and, potentially:

3. Revisit (by calibration) the RANS model & explore the prediction capability for other test cases

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4

Outline

• Machine learning framework.

• CFD solver features.

• Validation of computational approach.

• Automation framework.

• Results.

• Conclusions.

Machine learning framework

Adaptive Design Of Experiments (ADOE) technique:

• Couple machine learning techniques to improve efficiency and quality of traditional DOE methods by adaptive selection of the design points

• Only relevant number of model evaluations are run (no over/under sampling)

• Maximizing the information associated with a given number of CFD simulation

• Based on high fidelity of aerodynamic loads results

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[3] A.DaRonch,M.Panzeri,M.A.AbdBari,R.d’Ippolito,M.Franciolini;2017;"AdaptiveDesignofExperimentsforEfficientandAccurateEstimationofAerodynamicLoads";AircraftEngineeringandAerospaceTechnology;89(4);doi:10.1108/AEAT-10-2016-0173


Adaptive Design Of Experiments (ADOE) vs traditional DOE techniques

Traditional DOE techniques: monolithic approach

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Design space

All design points are selected a priori

Simulation results +

Interpolation / LS fitting Response surface model



Adaptive DOE technique: iterative approach

Initialization

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Interpolation / LS fitting

Design space

A limited number of design points are selected

a prioriIntermediate Response

surface model




Iteration

8



Design space

Assign new pointsbased on information inferred

from the previous runsIntermediate Response

surface model




Finalization

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Design space

Assign new pointsbased on information inferred

from the previous runsFinal Response surface

model

CFD Solver FeaturesDLR-Tau: finite volume unstructured flow solver

Spatial discretization:

• Discretization of convective and diffusive fluxes is based on second order Roe’s flux difference splitting scheme.

• Venkatakrishnan’s flux limiter.

Numerical integration:

• Explicit time stepping.

• Fourth order Runge-Kutta scheme.

• To accelerate the convergence:• local time-stepping, • implicit residual smoothing,• full multigrid method.

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Validation – RAE 2822 aerofoil

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• Case 6 of AGARD Report AR 138

• C-grid: 369 x 65 cells

• Freestream Mach: 0.729

• Reynolds number: 6.5×100

• Prescribed 𝐶2: 0.743 ⟹ 𝛼 = 2.51∘

Automation framework

Simulation workflow implemented in Optimus PIDO platform

1. Integration and automation of the simulation tasks performed during each iteration of the DOE / optimization analyses• Input / output files management• Submission of jobs to launch the DLR-Tau flow solver on the HPC facility

Iridis4• Post-processing

2. Suite of optimization / DOE methods3. Surrogate modelling

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Mapping variables to input file(s)

CFD run

Extraction of lift / drag coefficients

Post-processing and re-ordering of pressure coefficients values

Evaluation of SSE between measured / simulated pressure coefficients

Workspace:- workflows - analyses (DOE,

optimizations)- response surface

models

Optimus GUI

Iridis-4 supercomputer

Results - Response Surface Model• 10 input parameters: 9 uncertain SA closure coefficients 1 and angle of attack,

𝑐;< – Calibrates the growth of 𝜈>𝑐;? – Ensures that the integral of 𝜈>

<@ABC can only increase𝜎 – Turbulent Prandtl number𝑐E? – Part of function g, which controls the slope of 𝑓E in destruction term𝑐EG– Part of the 𝑓E function in destruction term𝑐H< – Used in turbulent eddy viscosity calculation and production term𝜅 – Von Karman's constant𝑐>G, 𝑐>K – Part of 𝑓>? function in production and destruction terms

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Standardvalue

Lowerbound

Upperbound

𝜎 6.667 L 10M< 6.0000 L 10M< 1.400 L 10N𝜅 4.100 L 10M< 3.6000 L 10M< 4.200 L 10M<𝑐H< 7.100 L 10N 6.9000 L 10N 7.500 L 10N𝑐EG 2.000 L 10N 1.5000 L 10N 2.750 L 10N𝑐>G 1.200 L 10N 1.0000 L 10N 2.000 L 10N𝑐>K 5.000 L 10M< 3.0000 L 10M< 7.000 L 10M<𝑐;< 1.355 L 10M< 1.2893 L 10M< 1.400 L 10M<𝑐;? 6.220 L 10M< 6.0983 L 10M< 7.000 L 10M<𝑐E? 3.000 L 10M< 5.5000 L 10M? 3.525 L 10M<

Basic

Near-wallregion

Laminarregionandtripping

𝑐;<𝑐;?

𝜎

𝑐E?𝑐EG𝑐H<

𝜅𝑐>G𝑐>K

Results - Response Surface Model (cont’d)• System outputs consist of:

• Lift and drag coefficients, • 𝑆𝑆𝐸 between the experimental and numerical 𝐶R,• 𝑆𝑆𝐸 is used as the output quantity of interest.

• 4,538 deterministic CFD simulations were performed, using about 1,000 CPU hours.

• Generation of the response surface model:

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Initialization AdaptiveDOE Target Validation

25samplingpoints

• 1,025samplepoints(2<N + 1)

• Additional1,500CFDsimulations

• runningasetofdeterministicCFDsimulations.

• find the angle of attack matching 𝐶2 = 0.743

Parameter Sobol index 𝑘 0.775 𝑐𝑣1 0.111 𝜎 0.046 𝑐𝑏1 0.046 𝑐𝑤2 0.006 𝑐𝑏2 0.004 𝑐𝑡3 0.001 𝑐𝑤3 0.001 𝑐𝑡4 0.000

Results - Global Sensitivity Analysis

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• Relative contribution ofinput parameters to the totalvariability of 𝑆𝑆𝐸 quantifiedby relying on the variance-based Sobol indices.

• First order Sobol indicesestimated via Monte Carlointegration performed with10,000 random pointsevaluated on the surrogatemodel.

• Only few constants of SAturbulence model show highsensitivity.

Results - Calibration of Turbulence Model Coefficients

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Parameter Standardvalue

Calibratedvalue

𝑘 4.100 L 10M< 3.600 L 10M<

𝑐H< 7.100 L 10N 7.500 L 10N

𝜎 6.667 L 10M< 1.003 L 10N

𝑐;< 1.355 L 10M< 1.400 L 10M<

• Two-step approach used forcalibration:i. the identification of a global

optimum by a differentialevolution algorithm;

ii. gradient-based approach thatlaunches additional CFDsimulations using non-linearprogramming quadratic lineoptimisation scheme.

Parameter Optimisationrun1 Optimisationrun2 Optimisationrun3 OptimalDOEpoint𝑘 3.600 L 10M< 3.600 L 10M< 3.600 L 10M< 3.610 L 10M<𝑐H< 7.500 L 10N 7.500 L 10N 7.500 L 10N 7.431 L 10N𝜎 9.970 L 10M< 1.003 L 10N 1.009 L 10N 1.163 L 10N𝑐;< 1.400 L 10M< 1.400 L 10M< 1.400 L 10M< 1.380 L 10M<𝑐E? 3.000 L 10M< 3.000 L 10M< 3.000 L 10M< 3.260 L 10M<𝑐;? 6.220 L 10M< 6.220 L 10M< 6.220 L 10M< 6.260 L 10M<𝑐>G 1.200 L 10N 1.200 L 10N 1.200 L 10N 1.963 L 10N𝑐EG 2.000 L 10N 2.000 L 10N 2.000 L 10N 1.815 L 10N𝑐>K 5.000 L 10M< 5.000 L 10M< 5.000 L 10M< 3.040 L 10M<

𝑆𝑆𝐸 1.40 L 10M< 1.40 L 10M< 1.40 L 10M< 1.39 L 10M<

Exp Data Coarse grid Medium grid Standard SA Calibrated SA Standard SA Calibrated SA 𝛼 [°] 2.29 2.51 2.37 2.39 2.27 CD [counts] 127 150 137 142 124 Cm −0.095 −0.091 −0.096 −0.092 −0.096

Results - Improved Prediction AccuracyRAE 2822 CASE 6:

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M= 𝟎. 𝟕𝟐𝟗, 𝑹𝒆 = 𝟔. 𝟓 L 𝟏𝟎𝟔 , 𝑪𝑵 = 𝟎. 𝟕𝟒𝟑 M= 𝟎. 𝟕𝟐𝟗 , 𝑹𝒆 = 𝟔. 𝟓 L 𝟏𝟎𝟔 ,𝜶 = 𝟐. 𝟓𝟏𝒅𝒆𝒈


Results - Improved Prediction Accuracy (cont’d)RAE 2822 CASE 6, medium grid:

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C-grid: 865 x 161 cellsx/c

C p

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1Exp DataDLR-Tau (standard)DLR-Tau (calibrated)

M= 𝟎. 𝟕𝟐𝟗, 𝑹𝒆 = 𝟔. 𝟓 L 𝟏𝟎𝟔 , 𝑪𝑵 = 𝟎. 𝟕𝟒𝟑


Results - Improved Prediction Accuracy (cont’d)RAE 2822 CASE 9:

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M= 𝟎. 𝟕𝟑, 𝑹𝒆 = 𝟔. 𝟓 L 𝟏𝟎𝟔 , 𝑪𝑵 = 𝟎. 𝟖𝟎𝟑 M= 𝟎. 𝟕𝟑, 𝑹𝒆 = 𝟔. 𝟓 L 𝟏𝟎𝟔 , 𝑪𝑵 = 𝟎. 𝟖𝟎𝟑

Results – ONERA M6 wing

• Favourable agreement with experimental data is achieved with calibrated SA model.

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Results – ONERA M6 wing (cont’d)

• Enlarged view highlights favourable performance of calibrated SA model near shock.

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Conclusions• Uncertainty in turbulence closure coefficients is the dominant source of error in

RANS simulations.• Only selected closure coefficients have large impact on uncertainty of output

quantities of interest.• It was found that calibrated SA turbulence model slightly outperforms standard

version for transonic, wall-bounded flows around the RAE 2822 aerofoil.• The expected prediction accuracy holds for more complex transonic flows

around the ONERA M6 wing, as well as across different flow solvers.

Acknowledgements:• EPSRC, grant number EP/P006795/1• NOESIS Solutions• IRIDIS support team at the University of Southampton

All data are available at:https://doi.org/10.5258/SOTON/D0263

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THANKYOUFORYOURATTENTIONFormoreinformationcontact:

DrJernejDrofelnik:[email protected]:[email protected]

Results - Improved Prediction AccuracyRAE 2822 test case:

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x/c

Cp

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1Exp DataDLR-Tau (standard)DLR-Tau (calibrated)

x/c

Cp

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1Exp DataSU2 (standard)SU2 (calibrated)

• Prediction accuracy of the DLR-Tau flow solution has significantlyimproved, SSE is reduced from 0.206 to 0.137.

• The improvement in prediction accuracy when using SU2 is virtuallyidentical to that observed in DLR-Tau, showing the generality of theabove conclusions across different flow solvers.

machine learning uncertainty quantification in turbulence · machine learning uncertainty...

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