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Brief introduction to UQO VERVIEWUQ activities in hypersonic flowsOVER SOMEUQACTIVITIES IN HYPERSONIC FLOWSPietro Marco CongedoINRIA and AQUARIUS TeamApplied Modeling & Simulation SeriesNASA-Ames23rd October 2014
Brief introduction to UQUQ activities in hypersonic flowsO UTLINE1B RIEF2UQINTRODUCTION TOUQACTIVITIES IN HYPERSONIC FLOWSPM Congedo – Overview over some UQ activities in hypersonic flows1/52
Brief introduction to UQ1B RIEF2UQUQ activities in hypersonic flowsINTRODUCTION TOUQACTIVITIES IN HYPERSONIC FLOWSPM Congedo – Overview over some UQ activities in hypersonic flows2/52
Brief introduction to UQT HREE PILLARS FORUQ activities in hypersonic flowsPREDICTIVEC OMPUTATIONAL S CIENCEComputation cannot be truly predictive without the coupling to theory andexperiments. . .This coupling is precisely the verification and validation process! Verification: Is the computational method implemented correctly?Validation : Are we solving the right equations?PM Congedo – Overview over some UQ activities in hypersonic flows3/52
Brief introduction to UQUQ activities in hypersonic flowsHow experience can be taken into account in the numerical simulation ?ModelsBoundary Cond.Initial tainty Quantification is the end-to-end study of the reliability of scientificpredictionsPM Congedo – Overview over some UQ activities in hypersonic flows4/52
Brief introduction to UQF LUIDS MAKE THE UQUQ activities in hypersonic flowsCOMPLEX TO HANDLE CHALLENGING!Flows features unsteadiness, compressibility, discontinuities, turbulence, multiphasenature of the flow, and so on .Measurements in fluid mechanics devices delicate and expensiveComplexity of measuresSometimes scarce and inaccurate experimental data increased amounts of uncertaintiesComplex Numerical simulationThe variability of a CFD numerical solution should be estimated taking into accountevery possible source of uncertainty (operating point, modeling, )PM Congedo – Overview over some UQ activities in hypersonic flows5/52
Brief introduction to UQT HEUQ activities in hypersonic flowsMATHEMATICAL SETTING OF THE PROBLEMLet the output of interest u(x, t, ξ) be governed by the equation:L(x, t, ξ(ω); u(x, t, ξ(ω))) S(x, t, ξ(ω)),(1)Rnd ,where L (algebraic or differential operator) and S are on D T Ξ, x D with nd {1, 2, 3}, ξ(ω) {ξ1 (ω1 ), . . . , ξd (ωd )} Ξ with parameters space Ξ RdThe objective of uncertainty propagation is to find the probability distribution ofu(y, ξ) and its statistical moments µui (y) given byµui (y) Zu(y, ξ)i fξ (ξ)dξ.ΞHow compute this integral in an efficient way ?PM Congedo – Overview over some UQ activities in hypersonic flows6/52(2)
Brief introduction to UQ1B RIEF2UQUQ activities in hypersonic flowsINTRODUCTION TOUQACTIVITIES IN HYPERSONIC FLOWSPM Congedo – Overview over some UQ activities in hypersonic flows7/52
Brief introduction to UQM OTIVATION :UQ activities in hypersonic flowsPOST- FLIGHT ANALYSIS OF A SPACE MISSIONIt requires an accurate determination of the free-stream conditions for thetrajectoryThese quantities can be rebuilt from the heat flux and pressure measured on thespacecraftNASA Mars Science LaboratoryPM Congedo – Overview over some UQ activities in hypersonic flowsESA European eXPErimental Re-entry Testbed8/52
Brief introduction to UQUQ activities in hypersonic flowsF LUSH A IR D ATA S YSTEM (FADS) ON E XPERTST measured densities can deviate up to 20% from the standard atmosphere modelRAFLEX instrumentation: FADS comprising a set of sensors flush mounted in the TPS to measure staticpressure (pressure taps) and heat flux (calorimeters)Shuttle-derived densities compared to the 1962 U.S. standardatmosphere [Talay et al. 1985]RAFLEX instrumentation on Expert Q1: What is the uncertainty on rebuilt free-stream conditions due tomeasurement uncertainties as well as of chemistry models? Q2: What should be the sensor accuracy for a prescribed uncertainty on freestream conditions?PM Congedo – Overview over some UQ activities in hypersonic flows9/52
Brief introduction to UQUQ activities in hypersonic flowsR EBUILDING METHODOLOGYKnowing that experimental data suffer from errors, the rebuilding methodologyneeds to integrate quantification of uncertaintiesEpistemic uncertainties on the chemistry models in the bulk and at the wall(surface catalysis) should also be taken into account Rebuilding the free stream conditions from the FADS data amounts to solving astochastic inverse problemPM Congedo – Overview over some UQ activities in hypersonic flows10/52
Brief introduction to UQVALIDATION ANDUQ activities in hypersonic flowsPREDICTIONValidationExperimental databases of ground facilities will be used to validate the simulation toolsand improve our understanding of physical phenomenaPredictionPrediction of the free stream properties in flight with their associated uncertaintyVKI Longshot M14 free piston tunnelPM Congedo – Overview over some UQ activities in hypersonic flowsVKI Plasmatron facility11/52
Brief introduction to UQUQ activities in hypersonic flowsStrong collaboration between INRIA (P.M. Congedo) and VKI (T. Magin) andseveral other peopleCharacterization of the free stream conditions in the VKI Longshot facility (BartVan Hove, J. Tryoen, M. Duvernet)Uncertainty Analysis of Carbon Ablation in the VKI Plasmatron (A. Turchi, F.Sanson, F. Panerai)Sensitivity analysis concerning the chemical reaction uncertainties during anatmospheric reentry (K. Tang, M. Panesi)Sensitivity analysis and characterization of the uncertain input data for theEXPERT vehicle (J. Tryoen, N. Villedieu)PM Congedo – Overview over some UQ activities in hypersonic flows12/52
Brief introduction to UQUQ activities in hypersonic flowsCharacterization of the free stream conditions in the VKI LongshotfacilityPM Congedo – Overview over some UQ activities in hypersonic flows12/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYMach14 Hypersonic Wind Tunnel with either nitrogen or carbon dioxideNo chemical thermo-reactive effectsRebuilding of free stream conditions: p1 , T1 , v1Epistemic uncertainty on Tv1PM Congedo – Overview over some UQ activities in hypersonic flows13/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYF ORWARD P ROBLEMInput dataFreestream pressure p1 , temperature T1 andshock speed v1Vibrational temperature Tv1Output dataStagnation-point pressure pst and heat fluxqstReservoir pressure p01D code :perfect gas equation of state in the reservoir / non-equilibrium flow model in the nozzle expansionKeyes’/Lemmon/Jacobsen/Sutherland viscosity models for freestream conditions/post-shock/stagnation pointFay-Riddel model to compute the stagnation heat fluxRankine-Hugoniot relations applied through the shock / flow assumed incompressible after the shock Sources of uncertaintiesUnknown: p1 U(190, 230), T1 U(45, 60) and v1 U(1500, 1800)Epistemic uniform uncertainty on Tv1 U(1000, 1500)PM Congedo – Overview over some UQ activities in hypersonic flows14/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYI NVERSE P ROBLEMMeasurementsStagnation-point pressure pst and heat fluxqstReservoir pressure p0UncertaintiesExperimental datavibrational temperatureValidation of a new inverse method for determining uncertainties on free streamconditions p1 , T1 and v1 from measurements Proposed approach relying on a Bayesian Inference settingPM Congedo – Overview over some UQ activities in hypersonic flows15/52
Brief introduction to UQUQ activities in hypersonic flowsC HARACTERIZATION OFBAYESIANTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYINFERENCE FOR STOCHASTIC INVERSE METHODForward model : d F(m, Tv1 ), d (pst , qst , p0 ), m (p1 , T1 , v1 )“prior pdf set of experimental data posterior pdf fitting data”Bayes’ rulep(m d1 , . . . , dn ) Rp(d1 , . . . , dn m, Tv1 )pm (m)pTv1 (Tv1 )p(d1 , . . . , dn m, Tv1 )pm (m)pTv1 (Tv1 )dmdTv1Experimental data : {d1 , . . . , dn }Hypothesis: dj F(m, Tv1 ) ej , ej N (0, Γ), j 1, . . . , n,Γ diag(σp2st , σq2st , σp2 )0 p(d1 , . . . , dm m, Tv1 ) nYpdj (dj m, Tv1 ) j 1nYpe (dj F(m, Tv1 ), Γ)j 1prior pdf pm 1p1 [190,230] 1T1 [45,60] 1v1 [1500,1800]uniform pdf pTv1 1Tv1 [1000,1500]measurement errors σ obs (σpst , σqst , σp0 ) supposed knownPM Congedo – Overview over some UQ activities in hypersonic flows16/52
Brief introduction to UQINUQ activities in hypersonic flowsPRACTICESamples of the normalized posteriorp(m d1 , . . . , dn ) p(d1 , . . . , dn m, Tv1 )pm (m)pTv1 (Tv1 )obtained with Markov Chain Monte Carlo methodsLet (mk )Ksuch a sample (tb “burning” time),k tbE(f (m) d1 , . . . , dn ) KX1f (mk )K tb 1k tb computation of posterior statistics (mean, standard deviation, densities, errorbars, . . . )PM Congedo – Overview over some UQ activities in hypersonic flows17/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYN UMERICAL R ESULTSA noisy data vector {d1 , . . . , d10 } is generated solving the forward model for a true vector of input parameters(mtrue , Tvtrue ) (200, 50, 1621, 1245)true true perturbing the true output value dtrue (ptrue0 , pst , qst ) (Gaussian noise)PM Congedo – Overview over some UQ activities in hypersonic flows18/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYN UMERICAL R ESULTSA noisy data vector {d1 , . . . , d10 } is generated solving the forward model for a true vector of input parameters(mtrue , Tvtrue ) (200, 50, 1621, 1245)true true perturbing the true output value dtrue (ptrue0 , pst , qst ) (Gaussian noise)Various cases are considered :1a 5% observation error (σi /dtrue 0.05) with known Tv ,i2a 10% observation error (σi /dtrue 0.10) with known Tv , andi3a 10% observation error (σi /dtrue 0.10) with uncertainty on Tvi(Tv U (1000, 1500)).4Experimental observationsPM Congedo – Overview over some UQ activities in hypersonic flows18/52
Brief introduction to UQR ESULTS (5% AND 10% ERRORUQ activities in hypersonic flowsWITH KNOWNTv , 10%WITH UNKNOWNTv )Spread of supports of PDFbetween (i) and (ii) increase of measurement errorsAND between (ii) and (iii) adding uncertainty on thevibrational temperaturePropating PDFin a numericalcode and estimating outputvariabilityPM Congedo – Overview over some UQ activities in hypersonic flows19/52
Brief introduction to UQUQ activities in hypersonic flowsC HARACTERIZATION OFTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYN UMERICAL R ESULTS (1)meanstdstd/mean90% confidence intervalsp T Usp T Usp T Usp T Us 0.05σi /dtrueiTv known198.250.4416224.6320.651410.662.337 10 21.291 10 20.657 10 2[190.1,205.8][49.47,51.61][1605,1640] 0.10σi /dtrueiTv known200.948.8716208.5441.27119.874.523 10 22.601 10 21.226 10 2[185.2,213.8][46.60,51.03][1585,1653] 0.10σi /dtrueiTv unknown192.948.2516069.2601.52625.764.80 10 23.163 10 21.604 10 2[177.9,209.5][45.82,50.63][1562,1645]Uncertainty of p is the most important: std/mean nearly two and four timesmore with respect to that ones related to T , and Us , when Tv is known.Large variation of pressure mean (4%) observed when Tv is unknown or knownSmaller variations observed for mean of T (3.2%) and for mean of Us (1.60%)w.r.t. p when Tv is unknownFor a given observation, p most difficult parameter to predict, more than T Importance of free-stream pressurePM Congedo – Overview over some UQ activities in hypersonic flows20/52
Brief introduction to UQC HARACTERIZATION OFA PPLICATIONUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYOF BAYESIAN - STRATEGY TO EXPERIMENTAL MEASUREMENTSSome experimental points chosen by considering the sensors acquisition in theLongshot facilityFive points are selected on the experimental curves, at different time-steps equalto 0.5, 5, 10, 15, 20 msUncertainties on error measurements are consideredUncertainty on the heat flux at the stagnation point is much more elevated withrespect to the pressure uncertaintyPM Congedo – Overview over some UQ activities in hypersonic flows21/52
Brief introduction to UQC HARACTERIZATION OFA PPLICATIONUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYOF BAYESIAN - STRATEGY TO EXPERIMENTAL MEASUREMENTSDifferent behavior with respect to the synthetic test-caseTemperature variability (in terms of standard deviation to mean ratio) is greater than thepressure one for each time-stepVelocity variability of the same order of magnitude than pressure, that is quite different withrespect to the previous resultsPM Congedo – Overview over some UQ activities in hypersonic flows22/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE FREE STREAM CONDITIONS IN THEVKI L ONGSHOT FACILITYC ONCLUSIONSetting up of a rigorous framework to take into account measurement and modeluncertainties in the resolution of the inverse problemDetailed probability characterization from MCMC samplingSome notions concerning the physics of the phenomenon are provided, i.e. thevariability of the free-stream pressureStagnation measurement error needed for a prescribed level of accuracy on thefree-stream conditionsThe importance of modelling some unknown variables, such as for example thevibrational temperature, can be assessed.PM Congedo – Overview over some UQ activities in hypersonic flows23/52
Brief introduction to UQUQ activities in hypersonic flowsUncertainty Analysis of Carbon Ablation in the VKI PlasmatronPM Congedo – Overview over some UQ activities in hypersonic flows23/52
Brief introduction to UQUQ activities in hypersonic flowsM OTIVATIONSTo understand the operational behavior of the TPS materialsTo study the gas/surface interaction physics occurring during reentryTo improve the prediction capacity and reduce the design marginsGalileo missionDestination: JupiterDate: 1989–2003.the best race car is the one that fallsapart right after the finish line.PM Congedo – Overview over some UQ activities in hypersonic flows24/52
Brief introduction to UQUQ activities in hypersonic flowsPlasmatron facility (The most powerful induction-coupled plasma wind tunnel in theworld)Role: performing reusable/ablative TPS testsGas: Air, N2 , CO2 , ArPower: 1.2 MW – most powerful ICP in the world –Heat-flux: up to 16 MW/m2 (superorbital re-entry)Pressure: 10 – 800 mbarPM Congedo – Overview over some UQ activities in hypersonic flows25/52
Brief introduction to UQUQ activities in hypersonic flowsPut the players togetherPM Congedo – Overview over some UQ activities in hypersonic flows26/52
Brief introduction to UQUQ activities in hypersonic flowsPut the players togetherPM Congedo – Overview over some UQ activities in hypersonic flows26/52
Brief introduction to UQUQ activities in hypersonic flowsPut the players togetherPLASMATRONExperimentalconditionsICP Rebuilding Code(boundary layer)qcwqcw(n) qcw(exp)Yesme, Te, yi,eγreac,wεwNoStagnation-line code(w/ ablative b.c.)yk,emc, Tw, qwPM Congedo – Overview over some UQ activities in hypersonic flows26/52
Brief introduction to UQUQ activities in hypersonic flowsUncertain inputs generate.uncertain outputs!!!STEP 1: BOUNDARY-LAYER CODESTEP 2: STAGNATION-LINE CODEvariableDynamic PressureStatic PressureCold Wall Heat FluxCold Wall alycityNitrogen/Oxygen ratioUniformUniformmeanerror ( )48 Pa8.0%20000 Pa0.3%2962 kW/m210.0%350 K10.0%range0.001–1(79/21) �0.50.8–0.95variableCs O CO2Cs O2 2COCs N CN3Cs C3N N N2TPS wall emissivityPM Congedo – Overview over some UQ activities in hypersonic flows27/52
Brief introduction to UQUQ activities in hypersonic flowsCoupled analysis: input uncertainty distributionsPM Congedo – Overview over some UQ activities in hypersonic flows28/52
Brief introduction to UQUQ activities in hypersonic flowsCoupled analysis w/ nitridationABLATION QOIvariablemeanvariancemass blowing ratetemperature0.029 [kg /m2 s]3.48e-52.17e 42661 [K] stoch nomεold ( )–28.4% 5.0%16.72%4.56%Wall mass blowing rateSob0.60.4error: 5.53%1mass flow ratedv/dyedgeTedgeO2N2NOONO oxidationO2 oxidationNitridationSublimationRecombinationWall emissivity0.80.6Sob0.8Wall temperatureerror: 20.17%10.40.20.200mass flow ratedv/dyedgeTedgeO2N2NOONO oxidationO2 oxidationNitridationSublimationRecombinationWall emissivityconsidering all the uncertainties slightly affect the error!PM Congedo – Overview over some UQ activities in hypersonic flows29/52
Brief introduction to UQUQ activities in hypersonic flowsCoupled analysis w/o nitridationABLATION QOIvariablemeanvariancemass blowing ratetemperature0.020 [kg /m2 s]1.94e-61.39e 42818 [K] stoch nomεold ( )–2.9% 0.8%1.15%1.80%Wall mass blowing rateSob0.60.4error: 4.18%1mass flow ratedv/dyedgeTedgeO2N2NOONO oxidationO2 oxidationSublimationRecombinationWall emissivity0.80.6Sob0.8Wall temperatureerror: 6.79%10.40.20.200mass flow ratedv/dyedgeTedgeO2N2NOONO oxidationO2 oxidationSublimationRecombinationWall emissivityRebuilding uncertainties affect the mass blowing rate!PM Congedo – Overview over some UQ activities in hypersonic flows30/52
Brief introduction to UQUQ activities in hypersonic flowsC ONCLUSIONSdecoupled analyisisStrong impact on the QOIs of a questionable phenomenon such as thesurface nitridation when consideredSmall variations of the QOIs uncertainties when nitridation is neglected:consequence of the analyzed ablation regimecoupled analyisisThe influence of the nitridation uncertainties remains the biggerMeasurement and model uncertainties from the rebuilding procedurecause the error to grow when nitridation is neglectedP ERSPECTIVESAssess more plausible ranges for the most influential parametersAnalyze different ablation regimesCompare the obtained results with the experimental measurementsPM Congedo – Overview over some UQ activities in hypersonic flows31/52
Brief introduction to UQUQ activities in hypersonic flowsSensitivity analysis concerning the chemical reaction uncertaintiesduring an atmospheric reentryPM Congedo – Overview over some UQ activities in hypersonic flows31/52
Brief introduction to UQUQ activities in hypersonic flowsS ENSITIVITY ANALYSIS CONCERNING THE CHEMICAL REACTION UNCERTAINTIES DURINGAN ATMOSPHERIC REENTRYUnderstanding kinetic and radiative processes is crucialduring the reentryPhysico-chemical models are complex and prone to manyuncertaintiesRates for kinetic and energy relaxation are usually the mostuncertain since they are difficult to messure experimentally Q1: Which elementary reactions are important in chemicalmechanisms? Q2: What is the influence of the uncertainty of their ratecoefficients on heat flux prediction?PM Congedo – Overview over some UQ activities in hypersonic flows32/52
Brief introduction to UQUQ activities in hypersonic flowsS ENSITIVITY ANALYSIS CONCERNING THE CHEMICAL REACTION UNCERTAINTIES DURINGAN ATMOSPHERIC REENTRYR EACTIONSFOR WHICH THE RATE COEFFICIENTS ARE CONSIDERED UNCERTAINQuantity of interest: Radiative heat flux at a distance corresponding to thestand-off distance for the ERC capsulePM Congedo – Overview over some UQ activities in hypersonic flows33/52
Brief introduction to UQUQ activities in hypersonic flowsS ENSITIVITY ANALYSIS CONCERNING THE CHEMICAL REACTION UNCERTAINTIES DURINGAN ATMOSPHERIC REENTRYANOVAANALYSIS OF THEH EAT F LUXCoefficient of variation 0.099%Uncertain excitation reactions for the two species only involve the groundelectronic level and the metastable levelsThe radiating levels are high-lying energy levels above the metastable states.How computing efficiently the ranking of most predominant uncertainties ?PM Congedo – Overview over some UQ activities in hypersonic flows34/52
Brief introduction to UQUQ activities in hypersonic flowsS ENSITIVITY ANALYSIS CONCERNING THE CHEMICAL REACTION UNCERTAINTIES DURINGAN ATMOSPHERIC REENTRYAnchored ANOVA analysis (IJNME 2014, K. Tang, P.M. Congedo) applied to 50uncertainties problemCompared to standard ANOVA, anchored ANOVA, with arbitrary anchor point,loses orthogonality if employing the same measure Reduced number of deterministic computationsCovariance decomposition of output variance adaptivitySome variables can be inactive for certain order of interactionsThe active dimension is determined by means of a thresholdPM Congedo – Overview over some UQ activities in hypersonic flows35/52
Brief introduction to UQUQ activities in hypersonic flowsS ENSITIVITY ANALYSIS CONCERNING THE CHEMICAL REACTION UNCERTAINTIES DURINGAN ATMOSPHERIC REENTRYA NCHORED ANOVAANALYSIS(IJNME 2014, K. TANG, P.M. C ONGEDO )0.35APPLIED TO50UNCERTAINTIES PROBLEMVar-contriPercentage100%0.380%Sorted variances0.2560%0.20.1540%0.120%0.0500%7 8 3 9 5 11 10 12 26 4 29 28 33 32 1 35 36 39 30 34 40 2 6 15 37 14 27 31 42 25 44 38 46 13 41 49 48 50 45 43 47 19 18 20 16 17 23 24 21 22UncertaintiesPM Congedo – Overview over some UQ activities in hypersonic flows36/52
Brief introduction to UQUQ activities in hypersonic flowsS ENSITIVITY ANALYSIS CONCERNING THE CHEMICAL REACTION UNCERTAINTIES DURINGAN ATMOSPHERIC REENTRYM AJORREACTIONS AMONG50REACTIONS FOR WHICH THE RATE COEFFICIENTS ARE CONSIDERED UNCERTAIN Low order interactions of input variables have the main impact on QOI The anchored ANOVA analysis allow us to retrieve the reactions that are significantto the QOIPM Congedo – Overview over some UQ activities in hypersonic flows37/52
Brief introduction to UQUQ activities in hypersonic flowsSensitivity analysis and characterization of the uncertain input datafor the EXPERT vehiclePM Congedo – Overview over some UQ activities in hypersonic flows37/52
Brief introduction to UQC HARACTERIZATION OFF ORWARDUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEPROBLEMInput dataFreestream pressure p and Mach numberM Catalytic recombination coefficient γ and gasreaction rate coefficientskr , r {1, . . . , nR }Output dataStagnation-point pressure pst and heat fluxqstCOSMIC code [Barbante 2001]: Axisymmetric NS eqs, high temperaturereacting flows with gas/surface interactionMUTATION library: 5 species air (N, O, N2 , O2 , NO) Park 2001 chemicalmechanism, assuming thermal eq. Sources of uncertaintiesUnknown: p U(16.3, 24.3) and M U(13.7, 17.3)Arrhenius gas reaction rate coefficients: log10 (Ar ) N (µr,0 , σr2 )Epistemic uniform uncertainty on γ U(0.001, 0.002)PM Congedo – Overview over some UQ activities in hypersonic flows38/52
Brief introduction to UQC HARACTERIZATION OFI NVERSEUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEPROBLEMFADS measurementsStagnation-point pressure pst and heat fluxqstUncertaintiesExperimental data chemical modelparametersInvestigation of one point of the trajectory of the EXPERT vehicle wherechemical non-equilibrium effects are important(p 20.3, T 245.5, M 15.5) Objective: determine uncertainties on freestream conditions p and M fromthe FADS dataPM Congedo – Overview over some UQ activities in hypersonic flows39/52
Brief introduction to UQC HARACTERIZATION OFN UMERICALUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEDIFFICULTIESLarge number of uncertaintiesTime consuming CFD computation( 1h for one deterministic calculation!)Definition of efficient numerical techniques for Uncertainty Quantification (UQ)PM Congedo – Overview over some UQ activities in hypersonic flows40/52
Brief introduction to UQC HARACTERIZATION OFP ROPOSED1UQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEAPPROACHStudy of the forward problemprior uniform uncertainty on p and M use of NISP methods to compute PC expansions of pst and qstsensitivity analysis from PC expansionscompute PC metamodels with uncertainties that have the most impact2Resolution of the stochastic inverse problemBayesian Inference methodsuse of PC metamodels to accelerate the Bayesian algorithmPM Congedo – Overview over some UQ activities in hypersonic flows41/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLED ETERMINISTIC C ODECOSMIC [Barbante 01]Pressure and temperature iso-contours of the flow around EXPERT obtained withinput data mean valuesPM Congedo – Overview over some UQ activities in hypersonic flows42/52
Brief introduction to UQUQ activities in hypersonic flowsVery short digression on PCPM Congedo – Overview over some UQ activities in hypersonic flows42/52
Brief introduction to UQUQ activities in hypersonic flowsP OLYNOMIAL C HAOS (PC)EXPANSIONS[Wiener 38; Cameron & Martin 47; Ghanem & Spanos 91]Every QOI u can be expanded in a convergent series of the formu(ξ) uPC (ξ) PXuα Ψα (ξ),α 0P (nξ No)!/nξ !No!, No: expansion degree{Ψα }α 0,.,P polynomial functions orthogonal w.r.t pξcorrespondence between pξ and {Ψα }{uα }α 0,.,P : deterministic spectral coefficientsDetermination of {uα } by a non-intrusive spectral method (NISP)uα kΨα k 2Zu(ξ)Ψα (ξ) kΨα k 2nXu(x, t, ξi )Ψα (ξi )ωii 1(ξi , ωi ) quadrature formulae points and weigths deterministic code used as a blackboxPM Congedo – Overview over some UQ activities in hypersonic flows43/52
Brief introduction to UQUQ activities in hypersonic flowsC HARACTERIZATION OFTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEU NCERTAINTY Q UANTIFICATIONPC forward expansions obtained from NISP methodsPpPCst (p , M , γ, k1 , k2 , k3 , k4 ) Pα (pst )α Ψα (p , M , γ, k1 , k2 , k3 , k4 )PCqst (p , M , γ, k1 , k2 , k3 , k4 ) α (qst )α Ψα (p , M , γ, k1 , k2 , k3 , k4 )Estimation of means, variances, and sensitivity informationµσ2No 26.49 · 1031.36 · 106pstNo 36.49 · 1031.37 · 106No 46.49 · 1031.39 · 106No 22.75 · 1059.73 · 109qstNo 32.75 · 1052.01 · 1010No 42.75 · 1056.18 · 1010TABLE : Means (µ) and variances (σ2 ) of pst and qst for No 2, 3, 4Coefficients of variation (σ/µ) : 18% for pst , 52% for qst !PM Congedo – Overview over some UQ activities in hypersonic flows44/52
Brief introduction to UQUQ activities in hypersonic flowsC HARACTERIZATION OFTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEU NCERTAINTY Q UANTIFICATIONp M γO2 N2 2O N2O2 O 2O ONO O N O ONO N N O NS1ST,1S2ST,2S3ST,3S4ST,4S5ST,5S6ST,6S7ST,7pst3.99 · 10 14.07 · 10 15.87 · 10 15.99 · 10 16.76 · 10 42.30 · 10 34.54 · 10 61.65 · 10 43.90 · 10 61.17 · 10 42.21 · 10 47.56 · 10 46.58 · 10 42.28 · 10 3qst1.07 · 10 26.18 · 10 22.52 · 10 17.14 · 10 19.40 · 10 23.03 · 10 19.31 · 10 42.70 · 10 26.35 · 10 42.23 · 10 23.01 · 10 21.02 · 10 18.86 · 10 23.00 · 10 1TABLE : Sobol first order (Si ) and total order indices (ST,i ) for No 3PM Congedo – Overview over some UQ activities in hypersonic flows45/52
Brief introduction to UQC HARACTERIZATION OFBAYESIANUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEINFERENCE FOR THE STOCHASTIC INVERSE PROBLEMForward model : d F(m, c), d (pst , qst ), m (p , M ),c (γ, k1 , k2 , k3 , k4 )“prior pdf set of experimental data posterior pdf fitting data”Bayes’ rulemeasurement errors parameters σ st (σpst , σqst ) supposed knownmetamodel pPCst convergent, negligible dependence on chemical parameters use of pPCst (p , M , γ0 , k1,0 , k2,0 , k3,0 , k4,0 ) as a reduced PC metamodel inthe Bayesian settingPM Congedo – Overview over some UQ activities in hypersonic flows46/52
Brief introduction to UQC HARACTERIZATION OFBAYESIANUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEINFERENCE FOR THE STOCHASTIC INVERSE PROBLEMmetamodel pPCst convergent, negligible dependence on chemical parameters use of pPCst (p , M , γ0 , k1,0 , k2,0 , k3,0 , k4,0 ) as a reduced PC metamodel inthe Bayesian setting (chemical parameters fixed at their mean values)metamodel qPCst non-convergent, important dependence on all parameters can not be used as a metamodel in the Bayesian settingidea : solve first the stochastic inverse problem considering only {p1st , . . . , pnst }and using pPCst in order to modify prior distribution of (p , M )PM Congedo – Overview over some UQ activities in hypersonic flows47/52
Brief introduction to UQUQ activities in hypersonic flowsC HARACTERIZATION OFTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEN UMERICAL R ESULTSp 40.0201716.516M 15.51514.51413.5F IGURE : Joint posterior density relying on pPCstPrior distribution of (p , M ) modified through Bayesian algorithm usingmetamodel for pstA set of couples (p , M ) plausible w.r.t measurements of pstPM Congedo – Overview over some UQ activities in hypersonic flows48/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE UNCERTAIN INPUT DATA FOR THEEXPERT VEHICLEN UMERICAL R ESULTSIf M known: M 15, uniform prior distribution p [16.3, 24.3] pseudo-measurements of pst posterior distribution for p µ 20.35, σ2 0.48, 90% confidence interval [18.97, 21.71]posterior approx. Gaussian because pst roughly linear w.r.t. (p , M )It is not the case when considering qst measurements!PM Congedo – Overview over some UQ activities in hypersonic flows49/52
Brief introduction to UQC HARACTERIZATION OFUQ activities in hypersonic flowsTHE UNCERTAIN INPUT D
priorpdf p m 1 p 1 [190 ,230 ]1 T1 45 60 1 v1 1500 1800 uniform pdf p T v1 1 T v1 [1000,1500] measurement errors σ obs (σ pst,σqst,σ 0) supposed known PM Congedo - Overview over some UQ activities in hypersonic flows 16/52
