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Quantifying uncertainties in a Venturi multiphaseconfiguration

Maria Giovanna Rodio, Pietro Marco Congedo

To cite this version:Maria Giovanna Rodio, Pietro Marco Congedo. Quantifying uncertainties in a Venturi multiphaseconfiguration. [Research Report] RR-8180, INRIA. 2012. �hal-00765009�

ISS

N02

49-6

399

ISR

NIN

RIA

/RR

--81

80--

FR

+E

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RESEARCHREPORT

N° 8180December 14, 2012

Project-Team Bacchus

Quantifying uncertaintiesin a Venturi multiphaseconfigurationMaria Giovanna Rodio, Pietro Marco Congedo

RESEARCH CENTREBORDEAUX – SUD-OUEST

351, Cours de la Libération

Bâtiment A 29

33405 Talence Cedex

Quantifying un ertainties in a Venturimultiphase on�gurationMaria Giovanna Rodio, Pietro Mar o CongedoProje t-Team Ba husResear h Report n° 8180 � De ember 14, 2012 � 21 pagesAbstra t: Modeling the omplex physi al stru tures of avitating �ows makes numeri al sim-ulation far to be predi tive, and still a hallenging issue. Understanding the role of physi al andparametri un ertainties in avitating �ows is of primary importan e in order to obtain reliablenumeri al solutions. In this paper, the impa t of various sour es of un ertainty on the predi tion of avitating �ows is analyzed by oupling a non-intrusive sto hasti method with a avitating CFDsolver. The proposed analysis is applied to a Venturi tube, where experimental data on erningvapor formation are available in literature. Numeri al solutions with their asso iated error barsare ompared to the experimental urves displaying a large sensitivity to the un ertainties of inletboundary onditions. Furthermore, this is on�rmed by omputing the hierar hy of most pre-dominant un ertainties by means of an ANOVA analysis. Finally, a simple algorithm is proposedin order to provide an optimized set of parameters for the avitation model, thus permitting toobtain a deterministi solution equal to the most probable one when onsidering physi al inletun ertainties.Key-words: avitation, un ertainty quanti� ation, Venturi tube, polynomial haos.

Quanti� ation des in ertitudes dans une on�gurationmultiphasique de type VenturiRésumé : La modélisation des stru tures physiques omplexes dans les é oulements avitantsdiminue la apa ité de prédi tion de la simulation numérique. Comprendre le r�le des in erti-tudes physiques et du modèle devient prioritaire pour obtenir une simulation numérique robuste.Dans e papier, l'impa t des di�érentes sour es d'in ertitudes est analysé en ouplant une méth-ode sto hastique non-intrusive ave un solveur CFD pour la avitation. L'analyse est appliquéeà une on�guration de type Venturi, où les données expérimentales sont disponibles en littéra-ture. Les solutions numériques ave les barres d'erreur asso iées sont omparées ave les ourbesexpérimentales. Un algorithme simple est proposé pour al uler un ensemble de paramètres op-timisé permettant d'obtenir une solution déterministe égale à la solution plus probable quandles in ertitudes physiques à l'entré sont onsidérées.Mots- lés : avitation, quanti� ation des in ertitudes, on�guration de type Venturi, haospolynomial.

Quantifying un ertainties in a Venturi multiphase on�guration 31 Introdu tionCavitation onsists in a lo al pressure drop below the vapor pressure at the liquid temperature,thus reating a phase hange and vapor bubbles formation. Their ollapse in high-pressure region an dramati ally lead to failure, erosion and other undesirable e�e ts. For this reason, there isa strong e�ort devoted to develop predi tive numeri al tools for avitating �ows in industrialappli ations. Unfortunately, an a urate des ription of intera tions between the vapour andliquid phases requires a urate physi al models and a way to take into a ount the dynami sof the interfa e. Moreover, multis ale e�e ts, turbulen e and thermodynami s should be also onsidered.Several numeri al approa hes have been proposed to reprodu e avitating �ows in externaland internal on�gurations. Prin ipally the models an be regrouped in two major ategories:interfa e models and two-phase models. In the �rst ase, the liquid and the vapor phase areseparated by an interfa e, then the systemati re onstru tion of interfa e and the appli abilityto omplex geometries are the most hallenging issues. Con erning two-phase models, the twophases are treated as a mixture. Di� ulties of these models are related to the mixture's propertiesestimation based on the liquid-vapor mixture ratios [1℄. Di�eren es between the various modelsin the se ond ategory mostly ome from the relation that de�nes the density �eld. For moredetails on erning the various modeling approa hes, Refs. [2, 3, 4℄ are strongly re ommended.Multiphase models are derived basing on onservation prin iples. By the way, model is typi allydependent on two types of parameters: �rst, on some physi al parameters, su h as for examplethe number of bubbles, that is not usually well measured; se ondly, on some empiri parameters,useful for �tting and alibration pro edures with respe t to the experimental data. Therefore,model parameters represent an important sour e of un ertainty. Moreover, it is not an easytask to well de�ne boundary and initial onditions, be ause of di� ulties en ountered in orderto ontrol a urately experiments in avitating �ows. As a result, onditions imposed for thesetting of a numeri al simulation, are a�e ted by a dramati randomness.A tually, the numeri al simulation of multiphase models is performed without onsideringthis set of un ertainties. The numeri al approa h remains an useful and fundamental tool but itis hard to prove its a ura y without performing a validation with respe t to the experiments,sin e there is no estimation of numeri al solution predi tivity. Finally, even if several models ofdi�erent omplexity exist in literature, no general onsensus exists on erning the a ura y orthe stability of a given model. Then, it is of primary importan e in avitating �ows to determinenot only a onverged numeri al solution but also a des ription of the variability of the solutionwith respe t to the known un ertainties, i.e. providing the statisti moments of the quantitiesof interest.In re ent years, the use of sto hasti methods applied to the numeri al simulation in �uidme hani s is being more and more di�used. Several methods have been proposed, allowing agood estimation of statisti properties with a redu ed omputational ost. One of the most usedmethods is based on Polynomial Chaos (PC) theory �rst introdu ed by Ghanem and Spanos[5℄ relying on Polynomial Chaos expansion of the random variables. Intrusive [5, 6, 7, 8℄ andnon-intrusive [9, 10, 11℄ formulations exist in literature, but the method used in this work is thenon-intrusive spe tral proje tion des ribed in [12, 13℄.Con erning multiphase �ows, a few papers exist treating un ertainty quanti� ation aspe ts.In 2000 and 2006 Li et al.[14℄ proposed a Markov sto hasti model to reprodu e the randombehavior of avitation bubble(s) near ompliant walls. In 2003, Fariborza et al.[15℄ proposed anempiri al model for the time-dis rete sto hasti nu leation of intergranular reep avities. Theyassumed nu leation to o ur randomly in time, with the temporal behavior being governed by aninhomogeneous Poisson pro ess. In 2007, Giannadakis et al.[16℄ des ribed the bubble breakup inRR n° 8180

4 Rodio & Congedolagrangian models using a sto hasti Monte-Carlo approximation. This study was oriented on theparti ular topi of avitation in the Diesel nozzle holes. In 2010, Mishra et al.[17℄ introdu ed amodel of avitation oupled to deterministi and sto hasti hemi al rea tions of solute hemi alspe ies.At our knowledge, onlyWil zynski [18℄ �rst and later Goel et al.[19℄ performed an un ertainties-based study on some hydrodynami avitation model parameters. In parti ular, Wil zynski [18℄applied a sto hasti model to apture the intera tion of turbulent pressure �eld on avitationnu lei population. Moreover, Goel et al.[19℄ performed a sensitivity analysis on several empir-i al parameters used typi ally in two-phase models. This study was done by means of a �nitedi�eren es method. In this ase, input data un ertainty hara terization is not required for thesensitivity analysis, that an be performed basing only on the mathemati al form of the model.The aim of this paper onsists in a systemati study for onsidering the probabilisti propertiesof the input parameters permitting to apture non-linearities in un ertainty propagation. Inparti ular, we present a sto hasti analysis, based on Polynomial Chaos method, by taking intoa ount avitation model parameters and physi al boundary onditions un ertainties. This isapplied to the numeri al simulation of a avitating �ow in a Venturi on�guration, that is onethe most-used on�guration for studying avitation. Then, it is of great interest to assess thepredi tivity of avitation model for this �ow. On e omputed the large varian e of the mostimportant �ow properties, the se ond ontribution of this paper is related to the formulation ofa simple algorithm for improving the predi tivity of the avitation model.This paper is organized as follows. Se tion 2 presents the governing equations for reprodu -ing avitation and the asso iated numeri al methods for solving the deterministi and sto hasti solutions. In Se tion 3, the Venturi on�guration is des ribed and the numeri al solution isveri�ed and validated with respe t to the experimental data. Moreover, the sour es of un er-tainties are introdu ed. Then, in Se tion 4, sto hasti analysis is presented, where the in�uen eof un ertainties is displayed in terms of mean, varian e for some quantities of interest. The mostpredominant un ertainties are determined with a ross-validation onsidering the whole set ofun ertainties or di�erent sour es of un ertainties separately. Finally, in Se tion 5, an inversemethod is brie�y des ribed permitting to provide new, more robust parameter for the avitationmodel in order to obtain more reliable numeri al solutions. Some on lusions and perspe tivesare drawn in se tion 6.2 Governing equations and Numeri al methods2.1 Governing equations and Numeri al Flow SolverThe numeri al simulations are performed by means of the ommer ially available ode FLUENT(release 12.0.16). For a basi two-phase avitation model, Fluent solves the set of Reynolds-averaged Navier-Stokes (RANS) equations governing the transport of mixture, oupled with a onventional turbulen e model. An impli it �nite volume s heme based on a segregated Pressure-Based Solver (SIMPLE algorithm for the pressure-velo ity oupling) is used, with a 2nd orderupwind di�erentiating s heme, ex ept for pressure and vapor fra tion where a standard and a1st order upwind s hemes are used, respe tively.First, let us introdu e the mass and momentum equations for the mixture as follows:∂ρ

∂t+

∂ρui

∂xj

= 0 (1)∂ (ρui)

∂t+

∂ (ρuiuj)

∂xj

= −

∂ (p)

∂xi

+∂

∂xj

[

(µ + µt)

(

∂ui

∂xj

+∂uj

∂xi

−

2

3δij

∂uk

∂xk

)]

+ ρg (2)Inria

Quantifying un ertainties in a Venturi multiphase on�guration 5where i, j and k denote the axes oordinate, t is the time, ρ = αρv +(1− α) ρl is the mixturedensity where v and l indi ate the vapor and liquid phase, respe tively, and α is the vapor volumefra tion. The term u represents the mixture velo ity, where the slip velo ity is assumed to beequal to zero, p is the mixture pressure, g is the gravity a eleration, µm = αµv +(1− α)µl andµm,t are the mixture vis osity and turbulent mixture vis osity, respe tively. This last one is afun tion of both the turbulent kineti rate k and the turbulen e dissipation rate ǫ:

µm,t = ρCµ

k2

ǫ(3)where Cµ is a onstant (see Tab. 1).As a onsequen e, two additional transport equations for k and ǫ are needed:

∂ (ρk)

∂t+ ▽ ·

(

ρk~u)

= ▽ ·

(

µm,t

σk

▽k

)

+Gm,k − ρǫ (4)∂ (ρǫ)

∂t+ ▽

(

ρǫ~u)

= ▽ ·

(

µm,t

σǫ

▽ǫ

)

+ǫ

k(C1ǫGm,k − C2ǫρǫ) (5)where Gm,k = µm,t(▽~u + ▽~ut) : ▽~u represents the generation of turbulen e kineti energy,due to the mean velo ity gradients, C1ǫ and C2ǫ are two onstants, σk and σǫ are the turbulentPrandtl numbers for k and ǫ, respe tively.Two terms are assumed as negligible in Eq. (4) and (5): i) the term of turbulen e kineti energy generation (due to the buoyan y be ause of tiny liquid temperature variation); ii) the ontribution of the �u tuating dilatation in ompressible turbulen e to the overall dissipationrate, be ause the �ow is in ompressible.Constants values are provided for water in table 1 (see [20℄ for more details). The near-wallregion is modeled by a wall fun tion that links the vis osity-a�e ted region between the wall andthe fully-turbulent region. A named standard wall fun tion, based on the work of Launder andSpalding [21℄, is used in this paper.Table 1: Values of onstants for water

C1ε C2ε Cµ σk σε1.44 1.92 0.09 1.0 1.3The avitation is taken into a ount by using the S hnerr and Sauer model [22, 23℄, i.e. atransport equation with a sour e term for the vapor phase (v):∂αρv∂t

+∇ · (αρv~u) = me + mc =ρvρlρ

Dα

Dt, (6)where me and mc des ribe the mass transfer for evaporation and ondensation, respe tively.In order to lose the system, it is ne essary to introdu e a relation for the volume fra tion α. It an be de�ned as:

α =4

3πηR3

[

1 + 4

3πηR3

] , (7)where R is the bubble radius and η is the nu lei on entration per unit of pure liquid volume.In this work, η is assumed equal to a onstant, onsidering that no bubbles are reated ordestroyed. A simpli�ed Rayleigh-Plesset equation [24℄ is introdu ed in order to model the bubbleradius R:DR

Dt=

√

2

3

pb − p

ρl, (8)RR n° 8180

6 Rodio & Congedowhere D/Dt is the material derivative, pb is the bubble pressure, p is the pressure far from thebubble that is assumed to be equal to p.Finally, the omplete system is de�ned by eight equations (Eqs. (1), (2), (4), (5), (6), (7),(8)) with eight unknowns (ρ, u, p, pb, R, k, ǫ). Remark that the avitation model depends only onone parameter, i.e. the nu lei on entration per unit of pure liquid volume η.The quantities me and of mc related to the S hnerr and Sauer model [22, 23℄ an be obtainedby oupling Eqs. (6, 7, 8) as follows:me =

ρvρlρ

α(1 − α)3

R

√

2

3

pb − p

ρlwhen pb > p (9)and

mc =ρvρlρ

α(1− α)3

R

√

2

3

p− pbρl

when pb ≤ p. (10)2.2 Sto hasti MethodLet us onsider a sto hasti di�erential equation of the form:L (x, θ, φ) = f (x, θ) (11)where L is a non-linear spatial di�erential operator (for istan e, L is the steady Navier-Stokesoperator) depending on a random ve tor θ (whose dimension depends on the number of un ertainparameters in the problem) and f(x, θ) is a sour e term depending on the position ve tor x andon θ. The solution of the sto hasti equation (11) is the unknown dependent variable φ(x, θ), andis a fun tion of the spa e variable xǫRd and of θ. Under spe i� onditions, a sto hasti pro ess an be expressed as a spe tral expansion based on suitable orthogonal polynomials, with weightsasso iated with a orthogonal polynomials, with weights asso iated with a parti ular probabilitydensity fun tion. The �rst study in this �eld is the Wiener (1938) pro ess. The basi idea is toproje t the variables of the problem onto a sto hasti spa eproje t the variables of the problemonto a sto hasti spa e spanned by a omplete set of orthogonal polynomials Ψ that are fun tionsof random variables ξ (θ), where θ is a random event. For example, the unknown variable φ hasthe following spe tral representation:

φ (x, θ) =

∞∑

i=0

φi (x)Ψi (ξ (θ)) (12)In pra ti e, the series in Eq. (12) has to be trun ated to a �nite number of terms, here denotedwith N . The total number of terms of the series is determined by:N + 1 =

(n+ p0)!

n! p0!(13)where n is the dimensionality of the un ertainty ve tor θ and p0 is the order of the polynomialexpansion. Substituting the polynomial haos expansion (12), into the sto hasti di�erentialequation (11) yields:

L

(

x, θ;N∑

i=0

φi (x) Ψi (ξ (θ))

)

= f (x, θ) (14)Equation (14) is solved through the weighted residual method. The ollo ation method is ob-tained by hoosing Dira -delta weighting fun tions. The oe� ients φi (x) are obtained usingInria

Quantifying un ertainties in a Venturi multiphase on�guration 7quadrature formulae based on tensor produ t of a 1D formula. Applying a ollo ation proje tionto equation (14), we obtain the solution of a deterministi problem for ea h ollo ation point.For further details, see Congedo et al.[13℄. In both ases, on e the haos polynomials and theasso iated φi oe� ients are omputed, the expe ted value and the varian e of the sto hasti solution φi (x, θ) are obtained from :EPC = φ0 (x) (15)

V arPC =

N∑

i=0

φ2

i (x)⟨

Ψ2⟩ (16)Another interesting property of PC expansion is to make easier sensitivity analysis based on theanalysis of varian e de omposition (ANOVA). ANOVA allows identifying the ontribution of agiven sto hasti parameter to the total varian e of an output quantity. It an be easily omputedby using some interesting properties of the previous development [12℄. Let us re all here that the ontribution to the varian e of a given random variables with index i, i.e. the �rst order Sobol'sindex, an be obtained by:

Si =

∑

α≥i φ2

i (x)⟨

Ψ2⟩

V arPC

(17)For more details, Ref. [12℄ is strongly re ommended.2.3 Coupling CFD and un ertainty quanti� ation toolNon-intrusive sto hasti method, presented in the previous se tion, allows redu ing the sto has-ti problem into a series of deterministi runs where spe i� values for parameters a�e ted byun ertainties are onsidered. Then, the CFD solver is not modi�ed and it remains ompletelyde oupled from the sto hasti ode. On e deterministi runs performed, they are used to om-pute statisti s of the solution by means of equations (15) and (16). Results presented here areobtained by onsidering various order of polynomial haos.3 Venturi on�gurationVenturi on�guration is one the most popular system for studying avitation from numeri al andexperimental point of view. In parti ular, the se tion hosen for this study has been designed toreprodu e avitating �ows developing on the blades of spa e turbopump.3.1 Case des ription and available experimental data in literatureWe have fo used our attention to the experimental results of the Venturi test se tion of theCREMHYG (Centre dâ��Essais de Ma hines Hydrauliques de Grenoble) [25℄. It is onstitutedof a pro�le with a onvergen e angle of 4.3 degree and a divergen e angle of 4 degree, equippedwith �ve probing holes to measure the lo al void ratio, instantaneous lo al speed and pressure.A s hemati representation of the tunnel and of probes position is reported in �gure 1, while adetailed des ription of the experimental devi e is given in [25℄ and in [26℄. The �uid used in thisexperiment is water (the physi al parameters are reported in table 2). The test ase operating onditions (summarized in table 3) yield an experimental avity length L between 70 mm and85 mm.RR n° 8180

8 Rodio & Congedo

Figure 1: S hemati representation of Venturi test se tionTable 2: Physi al parameters for waterT [K] ρv

[

kg/m3]

ρl[

kg/m3]

Pv [Pa]293 0.0173 998 2339Table 3: Test ase operating onditionsVinlet [m/s] Q

[

m3/s]

Pinlet [Pa] Poutlet [Pa]10.8 0.02375 36000 210003.2 Veri� ation and validation of the numeri al solutionFirst, a grid onvergen e study is performed following Roa heâ��s method [27℄ based on Ri hard-son extrapolation. This method allows estimating the real s heme's order of onvergen e that islower than theoreti al order of onvergen e of numeri al algorithm utilized. The order of on-vergen e p an be obtained handling three numeri al solutions omputed on grids of in reasingdensity, with onstant grid-re�nement ratio r :p = ln

[

f3 − f2f2 − f1

]

/ln (r) (18)where f is a solution fun tional and indi es 1 and 3 are referred to the �ner and the oarsergrid solution, respe tively. For the present omputations, we onsider H-type mesh, where the oarse, the medium and the �ne grid present, respe tively, 23541, 94164 and 376656 ells. Azoom near the Venturi throat for the �ne grid is reported in Fig. 2. The order of onvergen eInria

Quantifying un ertainties in a Venturi multiphase on�guration 9p is omputed in a point in the avitating region, between the thoat and the wall. For thepresent omputation, p is omputed on the �ow pressure and velo ity and, respe tively, we have al ulated a value of 1.5 and 1.9 against a theoreti al order of onvergen e of 2. The value ofp is important in order to hoose the grid that assures a good tradeo� between the solutiona ura y and the omputational ost. Moreover, we ompute the grid onvergen e index (GCI)on the �ner and medium grid. It represents an estimate of how far the numeri al solution is fromits asymptoti value. We ompute GCI values of 0.02% (0.055%) for stati pressure and 0.26%(0.99%) for the velo ity when a �ner (medium) grid is used, indi ating that the solution is wellwithin the asymptoti range. Moreover, in Fig. 3, results in terms of velo ity and void ratioare reported for the three grids at the station 1. It an be observed that medium and �ne gridsdisplay very similar results. Basing on these onsiderations, the medium grid has been hosenfor all the subsequent simulations.

x axis [m]

yax

is[m

]

-0.04 -0.02 0 0.02 0.04 0.06 0.08-0.04

-0.02

0

0.02

0.04

0.06

0.08

Figure 2: Zoom of the �ne grid near the Venturi throatIn Fig. 4, the deterministi void fra tion and velo ity pro�les obtained with the S hnerr modelat three stations, are plotted with the best solution shown in [25℄ and with the experimentaldata. There is a good agreement between experimental and deterministi results obtained withthe S hnerr Model in terms of void ratio at station 1 and 2 when y > 0.0035 m. On the opposite,di�eren es between experiments and omputation are observed when y < 0.0035 m, displaying amaximal error of 200% at x = 0 m as in Barre et al.. At station 3, the deterministi solution doesnot shown void fra tion (alpha=0) that orrespond an error of 100% at y=0.003 m, even if thesolution is better than the estimation in Barre et al.. On the ontrary, for the velo ity, we anobserve a good agreement between experimental and deterministi results (S hneer Model) atstation 2 and 3, better than the omputation of Barre et al. that, on the opposite, shows a betteragreement at station 1. The Fig. 5 represents the mean wall pressure longitudinal evolution.The numeri al predi tion �ts with measurements in the �rst part of divergent (0 < x < 0.07) aswell as in the part where the experimental results indi ate a more rapid re ompression pro ess.RR n° 8180

10 Rodio & Congedo

Vx [m/s]

yax

is[m

]

2 4 6 8 10 12 14

0

0.01

0.02

0.03

0.04

0.05

Coarser_GridMedium_GridFiner_Grid

(a) Station 1 Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

Coarser_GridMedium_GridFiner_Grid

(b) Station 1Figure 3: Velo ity pro�les and void fra tions at the station 1 for the three grids used in the mesh onvergen e study3.3 Sour es of un ertaintyIn Barre [1℄, physi al measurements are provided with their estimated errors at the inlet �ow andat the �ve stations where probes are present. These estimations have been used in this paperin order to ompare experimental error bars with those one omputed by means of sto hasti simulations. Then, the following experimental errors have been taken into a ount: ±0.25%for the inlet �ow rate, ±0.05 [bar] for the inlet pressure, ±19 [Pa] for the pressure at di�erentstations, ±15% for the void ratio at di�erent stations.As already explained in 2.1, avitation model is a�e ted globally by one un ertainty on n.Moreover, experimental un ertainties on inlet onditions (pressure and velo ity) have been takeninto a ount. Then, globally, three un ertainties are onsidered in the sto hasti simulation. Themean values, maximal variations and pdf type for ea h parameter are summarized in next table.Seeing that an a urate estimation of probability density fun tion for the physi al measures, usedas input parameters in the numeri al simulation, is not available, we used sistemati ally uniformpdf. This hoi e represents a robust safety strategy in order to analyse un ertainty propagationof physi al un ertainties. Con erning modelling un ertainty, this epistemi (i.e. due to a la k ofknowledge) variable is treated again as a uniform pdf, that is one of the possible options when onsidering this kind of un ertainty.Table 4: Mean values, maximal variation and probability density fun tion (PDF) for model andoperating onditions un ertainties Mean values Max variations PDFn De oupled/Coupled 1010 ±105 Uniformp De oupled/Coupled 36000 [Pa] ±5000 [Pa] Uniformv De oupled/Coupled 10.8

[

m3/s]

±0.025% Uniform Inria

Quantifying un ertainties in a Venturi multiphase on�guration 114 Sto hasti analysis4.1 General strategyThe analysis of the ontribution of the three un ertainties to the varian e of the outputs ofinterest (vapor fra tion, velo ity and pressure) is �rst performed following a de oupled analysis:a single sour e of un ertainty is taken into a ount, the other sour e being held onstant equalto the respe tive mean values. This analysis is arried out with several orders of PC, usinga full tensorization for the hoi e of ollo ation points. The L2 norm for the mean and thevarian e of pressure are omputed when un ertainties on the input and on the model are takeninto a ount (the 5th order of polynomial haos is taken as referen e). Convergen e urves arereported in �gure 6. As it an be seen, the sto hasti solutions are well onverged for an orderequal to 3. Results obtained with this de oupled analysis are resumed in �gure 7, where radialvapor distributions with error bars at di�erent se tions are reported. Ex ept at the station 1,un ertainty propagation due to the inlet un ertainties is mu h more strong than that one relatedto model un ertainty. Let us fo us on the inlet un ertainties results (�gure 7(a, ,e)): experimentalbars are well inside the numeri al error bars at station 5. At station 3, this happens only fory > 0.003. At station 1, numeri al error bars are more tiny than the experimental ones.In a se ond step, all the sour es of un ertainty (modelling and inlet onditions) are simul-taneously taken into a ount in order to assess possible intera tions between the model andthe operating onditions that might ontribute to the varian e of void fra tion. De oupled and oupled analysis are used in order to ross-validate statisti estimations. Convergen e urvesfor the oupled analysis are reported in �gure 6. Also in this ase, sto hasti solution are well- onverged for a polynomial haos order equal to 3. Mean and varian e ontours in the �ow�eldare omputed for vapor fra tion, for velo ity and pressure, that are reported in �gures 8, 9 and10, respe tively. For the three variables, varian e is maximal near the wall. Finally, in �gure11, radial vapor (a, ,e) and velo ity (b,d,f) distributions with error bars at di�erent se tions arereported. The same qualitative on lusions derived from the de oupled un ertainties are on-�rmed in this ase: at stations 3 and 5, experimental error bars are in luded in the numeri albars, while at station 1, larger di�eren es are observed between the numeri al and experimental urves.5 Setting of optimized parameters for the avitation modelWhen experimental data are not available, the use of UQ te hniques ould be of great interest inorder to set up some empiri al parameters, usually treated like epistemi un ertainties. Insteadof running every time a sto hasti problem, the omputation of an optimized empiri al param-eter ould allow to obtain a solution very similar to the most probable one by running only adeterministi simulation.Let us fo us on our ase of study. The avitation model is a�e ted by an epistemi un ertaintyon n, while aleatory un ertainties ara terize the inputs. It ould be useful to optimize the valueof n permitting to obtain in a deterministi framework, the most probable (in a sto hasti sense)solution. This ould be a general approa h for optimizing epistemi un ertainties basing on thealeatory un ertainties, when experiments are not available.Remark that the optimized value do not permit a-priori to reprodu e better experimentalresults but the idea is to give the most probable solution basing on the hosen multiphase model.In this way, when omparing with experiments, some more de�nitive on lusions an be drawnin terms of predi tivity of the model without onditionating the results making some arbitrary alibration of the model. This is faster with respe t to a bayesian alibration and do not requireRR n° 8180

12 Rodio & Congedoexperimental data.The obje tive of the study is to �nd optimal model parameters ensuring that the asso iatedsimulation be equal to the most probable solution when onsidering un ertainties on inlet on-ditions. Using the notations introdu ed in the previous se tion, this problem to solve an bemathemati ally expressed as:minn

∑

j

∥

∥µuj−Qj(n)

∥

∥ (19)where µuiis omputed with respe t to the inlet un ertainties. The quantity Q denotes the �eldvalue obtained in the ell j when the model parameter n is used, and ‖ξ‖ is a L2-norm.The problem de�ned by Eq. (19) is a parti ular optimization problem. The output of interestis the vapor fra tion. An optimized value of n equal to 1010.1309 is obtained, permitting to obtainthe most probable solution using a deterministi approa h with an error of 0.1%.6 Con lusion and Future WorkIn this paper, we performed a sto hasti analysis of a avitating �ow evolving in a Venturi on�guration. Main results of this analysis are the following:� The ross-validation between de oupled and oupled analysis and ANOVA results displaysimilar qualitative behaviors in terms of the omputation of most predominant un ertain-ties.� A third order of the polynomial haos expansion is su� ient to attain onvergen e over allthe �ow�eld for the pressure and the vapor fra tion.� Experimental un ertainties on inlet boundary onditions are predominant with respe t tothe model-un ertainty. The meaning of this analysis is twofold: �rst, the hoi e of the modelseems to be less important when inlet un ertainties are strong; se ondly the predi tivity ofthe numeri al simulation of avitating �ows seems questionable sin e the large variation ofthe vapor fra tion.Finally, we proposed a very simple algorithm for using the sto hasti solution in order to getsome optimized parameter related to the epistemi un ertainty. The idea is to use this optimizedparameter for obtaining the most probable solution only by running a deterministi simulation.Di�erent on�gurations will be investigated in a future work.

Inria

Quantifying un ertainties in a Venturi multiphase on�guration 13

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 1

0

0.0005

0.001

0.0015

0.002

EXPBarre et al. [24]Schneer_Model

(a) Station 1 Vx [m/s]y

axis

[m]

0 2 4 6 8 10 12

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

EXPBarre et al. [24]Schnerr_Model

(b) Station 1

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 1

0

0.002

0.004

0.006

0.008

EXPBarre et al. [24]Schneer_Model

( ) Station 3 Vx [m/s]

yax

is[m

]

0 2 4 6 8 10 120

0.001

0.002

0.003

0.004

0.005

0.006

EXPBarre et al. [24]Schnerr_Model

(d) Station 3

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8

0

0.002

0.004

0.006

0.008

0.01EXPBarre et al. [24]Schneer_Model

(e) Station 5 Vx [m/s]

yax

is[m

]

-2 0 2 4 6 8 10 120

0.002

0.004

0.006

0.008

EXPBarre et al. [24]Schnerr_Model

(f) Station 5Figure 4: Radial vapor (a, ,e) and velo ity (b,d,f) distributions. Comparison with results shownin [25℄ and experimental measurements at di�erent se tions.RR n° 8180

14 Rodio & Congedo

x axis[m]

(P-P

v)/P

v

0 0.1 0.2

0

5

10

15

EXPBarre et al. [24]Schnerr_Model

Figure 5: Longitudinal mean wall pressure evolution

Polynomial Chaos Order

L2(µ

(p))

2 2.5 3 3.5 4

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

INLETMODELCOUPLED

Polynomial Chaos Order

L2(σ

2 (p))

2 2.5 3 3.5 4

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

INLETMODELCOUPLED

Figure 6: Convergen e urves for the mean (left) and the varian e (right) of the pressure in the�ow�eld. A L2 norm is used. Inria

Quantifying un ertainties in a Venturi multiphase on�guration 15

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 1

0

0.0005

0.001

0.0015DeterministicMeanEXP

(a) Station 1 Void Ratioy

axis

[m]

0 0.2 0.4 0.6 0.8 1

0

0.0005

0.001

0.0015 DeterministicMeanEXP

(b) Station 1

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

DeterministicMeanEXP

( ) Station 3 Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 1

0

0.002

0.004

0.006DeterministicMeanEXP

(d) Station 3

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.80

0.002

0.004

0.006

0.008DeterministicMeanEXP

(e) Station 5 Void Ratio

yax

is[m

]

0 0.05 0.1

0

0.002

0.004

0.006

0.008

DeterministicMeanEXP

(f) Station 5Figure 7: Radial vapor distributions with error bars at di�erent se tions when de oupled un er-tainties are onsidered: un ertainties on inlet onditions (a, ,e) and on the model (b,d,f). Theresults are ompared with experimental measurementsRR n° 8180

16 Rodio & Congedo

x axis (m)

yax

is(m

)

0 0.05 0.1

0

0.02

0.04

0.450.40.350.30.250.20.150.10.05

Standard Deviation of Vaoid Ratio

x axis (m)

yax

is(m

)

0 0.05 0.1

0

0.02

0.04

0.950.850.750.650.550.450.350.250.150.05

Mean of Void Ratio

Figure 8: Contour of mean (top) and varian e (bottom) of the vapor fra tion in the �ow�eld.The whole set of un ertainties is taken into a ount.Inria

Quantifying un ertainties in a Venturi multiphase on�guration 17

x axis (m)

yax

is(m

)

0 0.1 0.2 0.3-0.1

-0.05

0

0.05

0.112.5211.5610.69.648.687.726.765.84.843.882.921.961

Mean of Velocity

x axis (m)

yax

is(m

)

0 0.1 0.2 0.3

-0.05

0

0.05

0.1 2.942112.626322.310531.994741.678951.363161.047370.7315790.4157890.1

Standard Deviation of Velocity

Figure 9: Contour of mean (top) and varian e (bottom) of the velo ity in the �ow�eld. Thewhole set of un ertainties is taken into a ount.RR n° 8180

18 Rodio & Congedo

x axis (m)

yax

is(m

)

0 0.1 0.2 0.3

-0.05

0

0.05

0.11400012555.611111.19666.678222.226777.785333.333888.892444.441000

Standard Deviation of Pressure

x axis (m)

yax

is(m

)

0 0.1 0.2 0.3-0.1

-0.05

0

0.05

0.1

56000520004800044000400003600032000280002400020000160001200080004000

Mean of Pressure

Figure 10: Contour of mean (top) and varian e (bottom) of the pressure in the �ow�eld. Thewhole set of un ertainties is taken into a ount.Inria

Quantifying un ertainties in a Venturi multiphase on�guration 19

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 1

0

0.0005

0.001

0.0015

DeterministicMeanEXP

(a) Station 1 Vx [m/s]y

axis

[m]

0 2 4 6 8 10

0

0.0005

0.001

0.0015DeterministicMeanEXP

(b) Station 1

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8 1

0

0.002

0.004

0.006DeterministicMeanEXP

( ) Station 3 Vx [m/s]

yax

is[m

]

0 5 10 150

0.002

0.004

0.006DeterministicMeanEXP

(d) Station 3

Void Ratio

yax

is[m

]

0 0.2 0.4 0.6 0.8

0

0.002

0.004

0.006

0.008

0.01DeterministicMeanEXP

(e) Station 5 Vx [m/s]

yax

is[m

]

-5 0 5 10

0

0.002

0.004

0.006

0.008

DeterministicMeanEXP

(f) Station 5Figure 11: Radial vapor (a, ,e) and velo ity (b,d,f) distributions with error bars at di�erentse tions. The whole set of in ertainties is taken into a ount. The results are ompared withexperimental measurementsRR n° 8180

20 Rodio & CongedoReferen es[1℄ A. Hosangadi and V. Ahuja. A numeri al study of avitation in ryogeni �uids, part ii:New unsteady model for dense loud formation. In Pro eedings of the 6th InternationalSymposium on Cavitation, Wageningen, The Netherlands, 2006. Pro eedings of the 6thInternational Symposium on Cavitation.[2℄ J. P. Fran and C. Pellone. Analysis of thermal e�e ts in a avitating indu er using rayleighequation. Journal of Fluids Engineering, 129:974�983, 2007.[3℄ E. Gon alves and R. F. Patella. Numeri al simulation of avitating �ows with homogeneousmodels. Computers & Fluids, 38:1682�1696, 2009.[4℄ Y. Utturkar, J. Wu, G. Wang, and W. Shyy. Re ent progress in modeling of ryogeni avitation for liquid ro ket propulsion. Prog. Aerospa e S i., 41(7):558�608, 2005.[5℄ R. G. Ghanem and S. D. Spanos. Sto hasti Finite Elements : a Spe tral Approa h. SpringerVerlag, 1991.[6℄ D. Xiu and G. E. Karniadakis. The wiener-askey polynomial haos for sto hasti di�erentialequations. Journal of S ien e Computing, 26, 2002.[7℄ I. Babuska, R. Tempone, and G.E. Zouraris. Galerkin �nite elements approximation ofsto hasti �nite elements. SIAM J. Numer. Anal., 42:800�825, 2004.[8℄ P.G. Constantine, D.F. Glei h, and G. Ia arino. Spe tral methods for parameterized matrixequations. SIAM J. Matrix Anal. A., 31:2681�2699, 2010.[9℄ O.P. Le Maitre, O.M. Knio, H.N. Najm, and R.G. Ghanem. A sto hasti proje tion methodfor �uid �ow, i: Basi formulation. J. Comput. Phys., 173:481�511, 2001.[10℄ O.P. Le Maitre, M.T. Reagan, H.N. Najm, R.G. Ghanem, and O.M. Knio. A sto hasti proje tion method for �uid �ow, . ii. random pro ess. J. Comput. Phys., 181:9�44, 2002.[11℄ M.T. Reagan, H.N. Najm, R.G. Ghanem, and O.M. Knio. Un ertainty quanti� ationin rea ting-�ow simulations through non-intrusive spe tral proje tion. Combust. Flame,132:545�555, 2003.[12℄ Thierry Crestaux, Olivier Le Maître, and Jean-Mar Martinez. Polynomial haos expansionfor sensitivity analysis. Reliability Engineering & System Safety, 94(7):1161�1172, 2009.[13℄ P. M. Congedo, C. Corre, and J.-M. Martinez. Shape optimization of an airfoil in a bzt �owwith multiple-sour e un ertainties. Computer Methods in Applied Me hani s and Engineer-ing, 200(1):216�232, 2011.[14℄ S. Li, Z. G. Zuo, and S. C. Li. Sto hasti study of avitation bubbles near boundary wall.Journal of Hydrodynami s, 18(3):487�491, 2006.[15℄ S. J. Fariborza, D. G. Harlowa, and T. J. Delpha. Intergranular reep avitation withtime-dis rete sto hasti nu leation. A ta Metallurgi a, 34(7):1433�1441, 1986.[16℄ E. Giannadakis, D. Papoulias, M. Gavaises, C. Ar oumanis, C. Soteriou, and W. Tang.Evaluation of the predi tive apability of diesel nozzle avitation models. In Pro eedings ofSAE International Congress, Detroit, 2007. Inria

Quantifying un ertainties in a Venturi multiphase on�guration 21[17℄ S. K. Mishra, K. Sudib, P. A. Deymier, K. Muralidharan, G. Frantziskonis, S. Pannala, andS. Simunovi . Modeling the oupling of rea tion kineti s and hydrodynami s in a ollapsing avity. Ultrasoni s Sono hemistry, 17(1):258�265, 2010.[18℄ L. Wil zynski. Sto hasti modeling of avitation phenomena in turbulent �ow. Advan es inFluid Me hani s III, 29:1�10, 2000.[19℄ T. Goel, S. Thakur, R. T. Haftka, W. Shyy, and J. Zhao. Surrogate model-based strategyfor ryogeni avitation model validation and sensitivity evaluation. Int. J. Numer. Meth.Fluids, 58:969�1007, 2008.[20℄ B. E. Launder and D. B. Spalding. Le tures in Mathemati al Models of Turbulen e. A ademi Press, London, 1972.[21℄ B. E. Launder and D. B. Spalding. The numeri al omputation of turbulent �ows. ComputerMethods in Applied Me hani s and Engineering, 3:269�289, 1974.[22℄ G. H. S hnerr and J. Sauer. Physi al and numeri al modeling of unsteady avitation dy-nami s. In Fourth International Conferen e on Multiphase Flow, New Orleans, USA, 2001.Fourth International Conferen e on Multiphase Flow.[23℄ W. Yuan, J. Sauer, and G. H. S hnerr. Modeling and omputation of unsteady avitation�ows in inje tion nozzles. Me anique & Industries, 2:383�394, 2001.[24℄ C. E. Brennen. Cavitation and Bubble Dynami . Oxford University Press, 1995.[25℄ S. Barre, J. Rolland, G. Boitel, E. Gon alves, and R. Fortes-Patella. Experiments andmodelling of avitating ï¬�ows in venturi: atta hed sheet avitation. European Journal ofMe hani s B/Fluids, 28(1):444�464, 2009.[26℄ B. Stutz and J.-L. Reboud. Two-phase �ow stru ture of sheet avitation. Physi s of Fluids,9(3678):1�12, 1997.[27℄ P. J. Roa he. Veri� ation and Validation in Computational S ien e and Engineering. Her-mosa, 1998.

RR n° 8180

RESEARCH CENTREBORDEAUX – SUD-OUEST

351, Cours de la Libération

Bâtiment A 29

33405 Talence Cedex

PublisherInriaDomaine de Voluceau - RocquencourtBP 105 - 78153 Le Chesnay Cedexinria.fr

ISSN 0249-6399