coping with uncertainty in turbulent flow

8/9/2019 Coping With Uncertainty in Turbulent Flow

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Coping with Uncertainty in Turbulent Flow

Simulations

Pierre Sagaut

Institut Jean Le Rond dAlembert, Universite Pierre et Marie Curie - Paris 6,4 place Jussieu -case 162, F-75252 Paris cedex 5, [email protected]

1 Introduction

Real-life CFD applications involve a largenumber of choices that are to bemade in order to setup the computational configurations of the problem.These choices include the physical models and related arbitrary constants,boundary conditions, initial conditions as well as tuning parameters in thenumerical methods. In many cases, all physical and geometrical parameters

arenotexactlyknownandin someinstances arenot knownat all. Thislackofinformation raises the issue of taking into account uncertainties in the CFDprocess,modelingit and measuring the dependency/sensitivity of the resultswith respect tothese uncertainties. Consequently,this approachimplies thatinstead of seeking a single deterministic solution, we are now interested inrecovering a continuous description of the spaceof possible solutions spannedby uncertain parameters.

This article aims at providing a survey of recent progress made in thefield of uncertainty and error quantification and propagation in CFD, the

emphasis being puton Large-Eddy Simulation (LES) of turbulent flows. Allissues addressed below are also relevant to Reynolds-Averaged NumericalSimulations (RANS) and hybrid RANS/LES methods. Section 2 illustrateswhy,even in a simple academic turbulentflow,arbitraryparameters that ap-pearin turbulence models must be considered as uncertainparameters,sincetheir values are flow-dependent. The concept of robust model, i.e. whosesensitivity to the tuning of arbitrary parameters is minimal, is then pre-sentedin Section 3. The issueof representing the spaceof solutions spannedby possible variations of the computational setup parameters is then ad-dressed. Section 4firstillustrates the useof the generalizedPolynomialChaos(gPC) [Xiu andKarniadakis(2002), Xiu andKarniadakis(2003)]methodonan academic case and then exemplifies the use of the Kriging method[Krige(1951)] on an engineering problem.


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20 P. Sagaut

2 Inertial Range Consistent Subgrid/TurbulenceModels

In simulations, subgrid-scale models do not always yield their theoreticallyexpected solution,and thisisoftenexperiencedin the significant discrepancybetween a priori tests and a posteriori results. This is caused by simplifi-cations and assumptions related to the shape of the LES filter, the typeof turbulence at the filter cut-off, etc., in combination with the non-linearnature of the NavierStokes equations. We illustrate here this problem byconsidering the value of the adaptable model constant parameter C in theSmagorinsky subgrid viscosity model for large-eddy simulation. The well-known Lilly analysis [Lilly (1967)], which leads to C = C = 0.17 0.18,

was carriedout in a simplified asymptotic framework, which is not relevantin mostflowsofinterest,sinceit doesnot account for Reynolds effects, large-scale dynamics,shear. . . Usualtuning methodologyfor arbitraryparametersin turbulence/subgrid models relies on simple test case solutions: the modelparameters are adjusted so that a satisfactory solution is recovered. A verypopular test case is Decaying Homogeneous Isotropic Turbulence (DHIT),whichis characterized at the global level by the turbulent kinetic energy de-cay rate. The recovery of the observed decay ratevia LES or VLES methodis still a challengingissue. One of the main reasons for this is that the decay

rate is not universal, sinceit depends on several fine spectral featuresof thesolution, among which: the Reynolds number, the spectrum shape at verylarge scales and very small scales, cutoffeffects originating in the finite sizeof the wind tunnel/computational domain. Therefore, no universal value forC canbe expected,even inthe DHITcase. The exactvalueof theCconstantwas derived by [Meyers andSagaut(2006)] using Popes formulation for theturbulent kinetic energy spectrum E(k):

C(L/, ReL) =C

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C4/3

, (1)

where the spectrum shapeis defined as

E(k) =K02/3k5/3fL(kL)f(k), (2)

and the auxiliary function is defined as

(L/, ReL) =4

3

1

(L/)4/3

+0

x1/3G2(x/L)fL(x)f(xRe3/4L )dx, (3)

whereGis theLESfilter kernel,L theintegral lengthscale, theKolmogorovscale, theLES cutoffscale and C= 0.17-0.18the usual value. The filter-dependent parameter is computed as

=

4

3

+0

x1/3G2(x)dx

3/4. (4)


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Coping with Uncertainty in Turbulent Flow Simulations 21

Since this is an exact formulation of the Smagorinsky constant, it is seenthat this parameterisflow-dependent and filter-dependent. Therefore, it mustbe considered as an uncertain parameter in practical LES, since most of

these parameters remain unknown. It is worth noting that the usual valueis recoveredas an asymptotic limit(seeFig.1). The definition given aboveisverycomplex,and doesnot satisfythe basic constraintofnumerical viability.Meyers and Sagaut proposed an efficient approach to recover inertial-rangeconsistency, which makesit possible to mimic the exact behaviour using theasymptotic valueof theSmagorinsky model by remapping the total effectiveviscosity (defined as the sum of molecular and subgrid-induced dissipativeeffects) as follows:

eff =

2

Lilly+

2

(5)

This analysis and remapping approach was also applied to VariationalMultiscale variants of the Smagorinsky model. Other models including ad-ditional transport equations (DES, SAS, ) remain to be analysed from theinertial range consistency viewpoint. Let us also notice that the usual one-test-filter dynamic procedure fails in recovering the correct behaviour, asshownby Porte-Ageland co-workers[Porte-Agelet al. (2000)].

100

101

102

0

0.05

0.1

0.15

0.2

/

Cs

Fig. 1. Smagorinsky constant as a function of the ratio between cutoff length andKolmogorov scale. Solid line: exact value; dotted line: asymptotic value; dashedline: linear and quadratic remapping.


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22 P. Sagaut

3 The Concept of Robust Modelling

The existence of intrinsic uncertainties in LES/VLES modelling automati-

cally raises the question of model robustness, i.e. of the sensitivity of theresults to the tuning of the free parameters in the model. A robust modelwill naturallybe preferred for practicalpurposes,even ifit doesnotyield thebest possible results,sinceitwillprovide userswith resultsof constantlevel ofaccuracy. More sensitive modelswillbe disgardedin practice. Inthis section,we will propose some quality indicator, and show that robust models existand can be derivedin simple ways [Meyers et al. (2006)]. The quantificationof the sensitivity will be addressedin the next section.

An error measure must first be introduced in order to quantify the com-

mitted error. Following Meyers and coworkers, let us consider the followingerrorindicator

p(N, C) =

T0

kc0 kp(ELES(k, t) G2(k)EDNS(k, t))dk

2dtT

0

kc0 kpG2(k)EDNS(k, t))dk

2dt

. (6)

Fig. 2. Error for the Smagorinsky model as a function of the number of grid pointsN and the value of the Smagorinsky constant. Shadded regions denote optimalityregions for the mode constant, in which the committed error is within 20 percentsof the minimum possible error on the same grid. Symbols: square: p= -1; circle:p=0; right-pointing triangle: p=1; left-pointing triangle: p=2.


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Choosing p=-1,0 or 2, one can put the emphasis on the error committedon the integral lengthscale, the resolved kinetic energy and the resolved en-strophy, respectively. Since in practicalapplications one can be interestedin

predicting severalparameters at the same timewith satisfactoryaccuracy, itis convenient to introduce the following multiobjective error measure

(N, C) =

p=1,0,1,2

p(N, C)/p(N, C(p, N)

p=1,0,1,2

1/p(N, C(p, N)

, (7)which is expressed as an explicit function of the number of grid points Nand the subgrid model constant C. Here, C(p, N) denotes the value of he

constant whichyields the lowest error on p(N, C) at fixed N. A model willbe referred to as a robust model if, keeping the samevalue ofC, a constantlevel of accuracy is recovered when varyingN.Resultsobtainedin DHIT withthe usual Smagorinsky model are presented in Fig. 2, while those obtainedusing the inertial-range consistant model based on a quadratic remappingintroduced above are displayed in Fig. 3. It is seen that the former is notrobust, while thelatteris. This analysis can be extended to VLES, DES andSAS-type methods in a straightforward manner.

Fig. 3. Error map for the quadratically approximated inertial-range consistentSmagorinsky model. Same caption as in Fig. 2.


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4 Copping with Uncertainties: Response Surface

Several mathematical tools are available to compute the sensitivity of the

solution with respect to a parameterof the simulation. Localgradientof thesolution can be computed in several ways, including complex differentiation[Lu and Sagaut(2007)]. A limitation of this approachis that thelocalgradi-ent doesnot give access to the full spaceof spanned solutions, and thatit isa linearized analysis. The response surface approach is becoming more andmore popular to parameterize a full subspace, without relying on lineariza-tion. One example is the Kriging approach that is becoming increasinglyacknowledged(e.g. [Jouhaud et al. (2006)]). The present paperwill focusonresults obtained with a stochastic spectral method referred to as the gen-

eralized Polynomial Chaos (gPC) approximation [Lucor et al. (2007)]. ThegPC approach consists of discretizing the space spanned by the uncertainparameters using a pseudo-spectral method. This is a means of represent-ing second-order random fields parametrically through a finite setof randomvariables. The basis functions, which are orthogonal polynomials, should bechosen in accordance with the probability density function of the uncertainsolution of the problem to ensure an optimal convergence of the represen-tation. When this density is not known, one choice is to use basis functions

Fig. 4. Uncertainty error bars on the 643 grid resolved turbulent kinetic energyspectrum computed using a Smagorinsky model with uncertain constant: envelopeof possible solutions


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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

0

2

4

6

8

10

12

14

16

18

20

22

24

0

500

1000

1500

2000

2500

3000

k

E(k)

PDF

Fig. 5. pdf of the turbulent kinetic energy E(k) in DHIT with uncertain Smagorin-sky constant following a betadistribution

Fig. 6. Flow configuration for the hot jet exhaust of an aeronautical engine coolingsystem

that areoptimal for the representation of the randominputs to the problem.Using this approach, a continuous reconstruction of the spaceof possible solu-tionsisobtained using a restricted setof usual LESrealizations. The analysisof decaying turbulence with an uncertain Smagorinsky model has been per-formed. We thus treat theSmagorinskyconstant as anuncertain input tothestochastic problem, and its probability distribution is assumed. In order to


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26 P. Sagaut

V/U0

Z/D

-0.05 0 0.05 0.10

0.5

1

1.5

2

2.5

3

W/U0

Z/D

0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

W/U0

Z/D

-0.2 -0.1 0 0.10

0.5

1

1.5

2

2.5

3

U/U0

Z/D

0.6 0.7 0.8 0.9 1 1.10

0.5

1

1.5

2

2.5

3 X/D=8

V/U0

Z/D

-0.1 0 0.1 0.2

0.5

1

1.5

2

2.5

U/U0

Z/D

-0.5 0 0.5 1 1.0

0.5

1

1.5

2

2.5

3 X/D=1

Maximum and minimum values

Exp 50037

Standard LES

Fig. 7. Mean velocity response based on the Kriging approach

simplify the system, we make the additionalassumption thanCs is a randomvariable, which means thatits statisticalproperties do not dependon spatialor temporal dimensions. The induced uncertainty on the computed energyspectrum is illustrated in Fig. 4. Here, the distribution of the Smagorinskyconstantis uniformand theLES computational domain is discretizedwith a643 grid.

Looking at the results, itis observed that all scales do not respond to theuncertainty in the same way, i.e. the level of sensitivity is scale-dependent.Another striking featureis the existenceof a modewhichis almostinsensitive

to variations in the Smagorinsky model constant. This featurewas observedfor smaller grid sizes as well. The scale-by-scale sensitivity is better illus-trated in Fig. 5, which displays the pdfof the value ofE(k) for all k. Here,the distribution of the Smagorinsky constant is a beta(4, 4) distribution andtheLES computational domain is discretizedwith a483 grid. It is seen thatboth the amplitude and the shapeof the pdf distribution are scale-dependent


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X

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Y=0

Exp 50037

Maximum and m

Standard LES

Y=0

Y

0 0.020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X/D=8

Y

0 0.02 0.04 0.060

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X/D=1

Fig. 8. Cooling effectiveness response based on the Kriging approach

functions,showing the great complexity of theLES solution response toran-domvariability in theSmagorinsky constant.

We now illustrate the use of the response surface approach on a prac-tical engineering problem, namely the separated flow at the exhaust ofthe cooling system in a aeronautical engine (see figure 6). Most numericalmethods used for practical engineering purposes involve artificial dissipa-tion. However, itis knownthatnumericaldissipationand subgrid/turbulencemodel induced dissipation are in competition. They must be tuned in anad hoc manner to adapt to the case in order to recover the best possi-ble results (e.g. [Garnier et al.(2001), Ciardi et al. (2005)]). In the followingexample [Jouhaud et al. (2008)], it is chosen to retain both the Smagorin-sky constant CS and the artificial fourth-order dissipation parameter smu4which appearsin Jamesons scheme as uncertain optimization parameters. Avariability range of 30% around the standard values Cs = 0.18 andsmu4 = 0.01 is considered. The sensitivity of the mean flow with respect

to these two parameters is illustrated in Figs. 7 and 8, which compare thestandard LES solution (i.e. the LES solution computed using the standardparameters), experimental data, and the extreme values retrieved from theresponse surface built for the mean flowsolutionat everygrid point. The sen-sitivity of the solution is directly related to the differences between the two


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28 P. Sagaut

extrema profiles. It is observed that the robustness of the solution dependson both the spatial location and the physical variable under consideration.

The figures reveal the existence of twocases. In the first case, the exper-

imental data lie within the two extrema, meaning that it might be possibleto tune up the parameters to match the experimental results. This will ob-viously lead to a very accurate LES solution at the considered location. Inthe second case, the experimental data areoutside the response surface: theexact solution doesnot belong to the spaceof solutions spanned by theLESmethod on the considered grid.

5 Concluding Remarks

The present paper aimed at presenting recent results dealingwith unresolvedscale modelling for unsteadysimulation of turbulentflows. The emphasiswasput on LES, but proposed analysis tools can be extended to hybrid RANS-LES method in a straightforward manner. In a first step, the concept ofinertial range consistency was introduced. Let us notice that this non-trivialissueisof directinterest for practicalapplication,sinceitwas shownthat theusual value of the Smagorinsky constant can be used if and only ifL/ >20 30 and that / > 100, i.e. for coarse grids in flows such that L/ >

2000 3000. Since similar criteria may be defined for hybrid approaches,one may wonder if model tuning on test cases with nearly-infinite Reynoldsnumberisof practical interest forVLES modelswhichinvolve severaltuningparameters.

A second step is to design robust models, i.e. models which will leadto very good results even if some parameters of the simulation (grid res-olution, ) are changed. Using the error map approach and the conceptof inertial-range consistency, it will be shown that robust, nearly-optimalsubgrid models can be designed, which satisfy the three basic modelling

constraints: 1/ physical constistency 2/ robustness 3/ numerical viability[Meyers and Sagaut(2006), Meyers et al. (2006)]. In a third step, a deeperinsight in the solution sensitivity was gained using the gPC approach. Onemay wonder ifDHIT is a relevant test case for model validation, since it isa very simple turbulent flow. Let us first remark that theoretical analysisneeds relevant, unambiguous test cases. A last pointis therefore the reliabil-ity of somefamous test cases forLES-like model validation,such as the planechannel flow. Itwas recentlyshownthat the error associatedwith coarse-gridplane channel DNS exhibits a complex non-linear behavior, which can leadto misleading interpretations of the results in [Meyers andSagaut(2007)].These authors show that coarse-grid DNS may, in some cases, lead to ex-act prediction of usual test parameters such as skin friction, mean centrelinevelocity and peakof streamwise turbulence intensity.


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The last point dealt with the representation of the uncertainty in thesolution which originates in the lack of knowledge of complex flows. Theresponse surface approachisobserved tobe anefficient tool for that purpose.

Response surfaces can be used toinvestigate the sensitivity of the solution,but also to draw error maps atlowcostis some reference data are available.

Acknowledgements. The author would like to warmly acknowledge thefruitful and enlightning collaborations which lead to the results presentedhere. The error map approach and the robust modelling theory were devel-opped in close collaboration with Dr. Johann Meyers. The Kriging-basedresults were obatined with Dr. Jean-Christophe Jouhaud. Dr. Didier Lucorwas the main contributor to the development of the gPC-based researches

presented here.

References

[Ciardi et al. (2005)] Ciardi, M., Sagaut, P., Klein, M., Dawes, W.N.: A dynamicfinite-volume scheme for large-eddy simulation on unstructured grids. Journalof Computational Physics 210, 632655 (2005)

[Garnier et al.(2001)] Garnier, E., Sagaut, P., Deville, M.: A class of explicitENO filters with application to unsteady flows. Journal of Computational

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proach for CFD/wind tunnel data comparison. Journal of Fluids Engineer-ing 128(4), 847855 (2006)

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[Krige (1951)] Krige, D.G.: A statistical approach to some basic mine valuationsproblems on the Witwatersrand. Journal of Chemical, Metallurgy and MiningSociety of South Africa 52, 119139 (1951)

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[Meyers and Sagaut (2006)] Meyers, J., Sagaut, P.: On the model coefficient for the

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[Meyers et al. (2006)] Meyers, J., Sagaut, P., Geurts, B.J.: Optimal model param-eters for multi-objective large-eddy simulations. Physics of Fluids 18, 095103(2006)


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[Meyers and Sagaut (2007)] Meyers, J., Sagaut, P.: Is plane channel flow a friendlytest-case for the testing of LES subgrid scale models? Physics of Fluids 19,048105 (2007)

[Porte-Agel et al. (2000)] Porte-Agel, F., Meneveau, C., Parlange, M.: A scale-dependent dynamic model for large-eddy simulation: application to a neutralatmospheric boundary layer. Journal of Fluid Mechanics 415, 261284 (2000)

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coping with uncertainty in turbulent flow

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