[American Institute of Aeronautics and Astronautics 19th AIAA Computational Fluid Dynamics - San Antonio, Texas ()] 19th AIAA Computational Fluid Dynamics - Quantification of Uncertainties

American Institute of Aeronautics and Astronautics

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Quantification of Uncertainties in Compressible Flows with Complex Thermodynamic Behavior

1P. Cinnella 2P.M. Congedo 3L. Parussini, 4V. Pediroda Laboratoire SINUMEF,

Ecole Nationale Supérieure d’Arts et Métiers,

75013, Paris, France

Laboratoire des Ecoulements Géophysiques et Industriels,

INP Grenoble, 38041, Grenoble, France

Department of Mechanical Engineering,

University of Trieste, 34127, Trieste, Italy

A Tensorial-expanded Chaos Collocation method is used to quantify the effect of uncertainties on thermophysical properties of complex organic substances on the computed flow properties. Specifically, we investigate the effect of uncertainties introduced by several thermodynamic models on the numerical results provided by a computational fluid dynamics code for flows of molecularly complex gases close to saturation condition (dense gas flows). Most thermodynamic models of current use actually require data about thermophysical properties of the fluid, such as critical-point properties, which are typically affected by large uncertainties. Moreover, more accurate thermodynamic models typically require a larger number of input parameters, thus introducing new sources of uncertainty. The Tensorial-Expanded Chaos Collocation method investigated in this paper is used to perform both a priori and a posteriori tests on the output data generated by three popular thermodynamic models for dense gases with uncertain thermophysical input parameters. A priori tests check the sensitivity of each equation of state to uncertain input data via some reference thermodynamic outputs, such as the saturation curve and the critical isotherm. A posteriori tests investigate uncertainties introduced in the pressure field around an airfoil placed into a transonic dense gas stream.

I. Introduction In recent years, the scientific community has displayed increased interest in uncertainty analysis and management for computational fluid dynamics and its application to risk-based design methods. Precisely, uncertainty analysis addresses potential deficiencies in any phase or activity of the modeling process due to a lack of knowledge1, which are of stochastic nature. Sources of uncertainties may be intrinsically related to the physical modeling, to the boundary conditions, to discretization and solution errors and to geometrical errors due to machining tolerances. Research efforts have led to the development of several methods, which may be classified as non intrusive or statistical (e.g. the Monte Carlo method and the surface response method) or intrusive/non statistical (e.g. polynomial chaos methods). Among these methodologies increased interest has been developed into polynomial chaos methods because of the high degree of accuracy which may be easily attained by increasing the polynomial order. Specifically, their non-intrusive implementation has demonstrated great capabilities2,3. The Chaos Collocation approaches are simpler to implement then intrusive methods. A remarkable advantage of these approaches is that they may be coupled with a black-box deterministic solver, without any need to modify it. This means the non-intrusive methods are more versatile than intrusive methods since they may be applied to different problems just by changing the black-box solver.

In the present work we focus on the analysis of uncertainties related to thermodynamic modeling of real gas flows. Many problems in physics and engineering are characterized by thermodynamic conditions where the perfect gas approximation is no longer valid. In the past, a variety of studies have addressed real-gas flows with chemical reactions, such as hypersonic or combustion flows4. More recently, advances in computational fluid dynamics

1 Professor, AIAA Member, 151 bd de l’Hôpital, F-75013 Paris, France. E-mail: [email protected] 2 Post-Doc, Domaine Universitaire, B.P. 53, F-38041, Grenoble, France. E-mail: [email protected] 3 Post-Doc, via Valerio 10, I-34127, Trieste, Italy. E-mail: [email protected] 4 Assistant Professor, via Valerio 10, I-34127, Trieste, Italy. E-mail: [email protected]

19th AIAA Computational Fluid Dynamics22 - 25 June 2009, San Antonio, Texas

AIAA 2009-3670

Copyright © 2009 by P.Cinnella, P.M. Congedo, L. Parussini, V. Pediroda. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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(CFD) as a fundamental design tool both in aerodynamics and in energetics have motivated the development of fluid dynamic solvers for non-reacting flows of gases operating in special thermodynamic conditions, characterized by a more accurate thermodynamic modeling. For instance, accurate numerical tools are required for the study of gases close to saturation conditions, or in the supercritical region, or in the two-phase regime5-13. In order to accurately describe the fluid thermodynamic response, several equations of state have been proposed in the literature, diversified according to the substance to be modeled. Equations of state base on theoretical and analytical criteria, such as the van der Waals, Redlich-Kwong, Peng-Robinson, Martin-Hou equations, and many other14, allow to model the thermodynamic behavior of a fluid for an extended range of operating conditions provided some thermodynamic inputs (e.g. critical temperature and pressure, acentric factor and other) are available for the substance of interest. However, such data are typically affected by more or less significant experimental errors. For instance, the determination of critical-point data is typically a quite delicate issue, because of the exasperated system sensitivity close to critical conditions and also because complex substances may decompose totally or just in part, at temperatures close to the critical one. In this case, critical point values just rely on estimations. Moreover, in order to increase the accuracy and/or the application range of a thermodynamic model, a larger number of expansion terms (for instance, virial expansion terms) is adopted. As a consequence a larger number of thermodynamic inputs is required to determine all of the model constants and, inevitably, new sources of uncertainty are introduced. Thus, one may arrive to the paradox that, increasing the complexity of the thermodynamic model to increase its (deterministic) accuracy, one finally ends up with larger modeling uncertainties because of the increased number of uncertain parameters. A possible alternative to the use of an analytical equation of state is interpolation on a database of thermodynamic properties. This avoids in part numerical complexities related to the use of analytical models, but requires some caution about interpolation errors in regions characterized by huge variations of thermodynamic properties. The computational load as well as the accuracy varies according to the chosen interpolation procedure. The thermodynamic database used for interpolations may both be generated by applying an analytical equation of state, which introduces the before-mentioned uncertainties in addition to interpolation errors, or directly made of experimental date, which are intrinsically affected by experimental uncertainties.

In the present work we adopt a chaos collocation method which allows dealing accurately and efficiently with systems characterized by many uncertain thermodynamic parameters. We analyze the sensitivity of several thermodynamic models of common use for the numerical simulation of dense gas flows to input parameter uncertainties both for the determination of some crucial thermodynamic information, such as the liquid/vapor coexistence curve, the critical isotherm and a sound-speed-related quantity known as the fundamental derivative of gas dynamics15 and for aerodynamic output data, such as pressure and force coefficients for dense gas flows past airfoils. The effect of different input distributions on the computed stochastic solution is also analyzed.

II. Flow solver and thermodynamic models The governing equations for two-dimensional compressible inviscid flow (2D Euler equations) completed by a

real-gas thermodynamic model are discretized using a cell-centered finite volume scheme for structured multi-block meshes of third-order accuracy, which allows computing flows governed by an arbitrary equation of state11. The scheme is constructed by correcting the dispersive error term of the second-order-accurate Jameson’s scheme16. The use of a scalar dissipation term simplifies the scheme implementation with highly complex equations of state and greatly reduces computational costs. Local time stepping, implicit residual smoothing and multigrid are used to efficiently drive the solution to the steady state. The accuracy properties of the numerical solver just described have been demonstrated in previous works11,17, and will not be discussed further.

In the present study, three well-known thermodynamic models are taken into account, which are described in the following. The first one is the simple Redlich-Kwong equation of state as modified by Soave18 (denoted hereafter as the RKS equation) the second one is the Peng-Robinson-Strijek-Vera (PRSV) cubic equation of state19, and the third one is the five virial expansion term Martin-Hou (MAH) equation20.

For the calculation of all caloric properties, the three equations are supplemented by the ideal gas contribution to the specific heat at constant volume, which is approximated here by a power law of the form:

( ) ( )n

ccvv T

TTcTc

= ∞∞ ,, (1)

where the c subscript denotes critical-point values, and the exponent n and the critical-point ideal specific heat

( ),v cc T∞ are material-dependent constants. The caloric equation of state associated to the chosen thermal equation of

state is completely determined via the compatibility relation:


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( ) ( )( ) ( ), 2

', ' ' ,

'r r

T

r vT

p de e w T w e c T dT T p

T

ρ

ρρ

ρρρ∞

∂ = = + − − ∂ ∫ ∫ (2)

where quantities with a prime subscript are auxiliary integration variables, and the subscript r indicates a reference state.

A. RKS equation The Redlich-Kwong equation is commonly considered as one of the best available two-parameter equations of

state. The original Redlich-Kwong equation of state, written in non-dimensional form is21:

( )0.5/RT a T

pv b v v b

= −− +

(3)

p being the fluid pressure, T the absolute temperature, v the specific volume, R the gas constant, and a and b two material-dependent parameters which may be related to the fluid thermodynamic properties at the liquid/vapor

critical point. The Soave modification consists in replacing the term 0.5/a T by a more general term ( )a T of the

form:

( ) ( )ca T a Tα=

with

2 20.42747 /c c ca R T p=

The c subscript denotes critical-point values. The α(T) function is a non dimensional factor depending on the

reduced temperature /r cT T T= and the substance acentric factor ω:

( ) ( ) 20.51 1 rT m Tα = + − (4)

with 20.480 1.57 0.176m ω ω= + − .

The covolume b may also be related to critical quantities:

0.08664 /c cb RT p=

The final result is an equation of state depending on four input parameters, i.e. the gas constant R, the critical-point temperature and pressure and the acentric factor. The last one may be derived from experiments or computed through the expression (see e.g. Ref. 14):

( ) ( )10

3 /log 1

7 1 /e c

ce c

T Tp

T Tω = −

− (5)

where Te is the normal boiling temperature. As a consequence, the RKS thermodynamic model, i.e. the RKS equation supplemented with the power law (1) for the ideal-gas specific heat, written in dimensional form depends at least on six material-dependent parameters. The number of free parameters may be reduced by adopting the adimensional form:

( )

20.42747

0.08664 0.08664c r c

rr c r r c

Z T Zp

v Z v v Z

α= −− +

(6)

Where the subscript r denotes reduced quantities (i.e. normalized with the critical-point values). The critical

compressibility factor ( ) ( ): /c c c cZ RT p v= is univocally determined by imposing that ( )1, 1 1r r rp v T= = = ,

which leads to the solution of a cubic equation for Zc with only one relevant critical root. Thus, the only free


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parameter in the thermal equation is the acentric factor ω. Similarly, the caloric relation (1) written in non-dimensional form is:

( ) ( ) ( ), , nv r v c

r

c T c TT

R R∞ ∞= (7)

In summary, the RKS model in non-dimensional form (6),(7) considered in the present work depends on three uncertain parameters, namely the acentric factor ω, the exponent n and the reduced ideal-gas constant-volume

specific heat at the critical temperature ( ),v cc T∞ .

B. PRSV equation Peng and Robinson22 proposed a cubic equation of state of the form:

2 22

RT ap

v b v bv b= −

− + − (8)

with

( ) ( )2 20.457235 / ; 0.077796 /c c c ca R T p T b RT pα= = .

The corrective factor α is expressed in the form (4) and the acentric factor is calculated via Similarly to the RKS model, the dimensional PRSV thermodynamic model given by equations (1),(4), and (8) depends on five material-dependent parameters. Here again, we recast the PRSV equation in non-dimensional form by using reduced variables:

2 22

c r rr

r r r r r r

Z T ap

v b v b v b= −

− + − (9)

with

( ) ( )20.457235 ; 0.077796r c r ca Z T b Zα= =

and Zc univocally determined by the solution of a cubic equation. Thus, the non-dimensional PRSV model (9),(7)

depends just only on the uncertain parameters ω, n and ( ),v cc T∞ .

C. MAH equation The comprehensive thermal equation of state of Martin and Hou20 is considered to provide a realistic description of

the gas behavior close to saturation conditions23,24 . It reads:

( )

( )∑= −

+−

=5

2ii

i

bv

Tf

bv

RTp (10)

where 115

c

c

RTb

p

β = −

with 20.533 31.883cZβ = − , and the functions fi (T) are of the form:

( ) c

Tk

Ti i i if T A BT C e

−= + +

with Tc the critical temperature. The gas-dependent coefficients Ai, Bi, Ci can be expressed in terms of the critical temperature and pressure, the critical compressibility factor, the Boyle temperature (which may be expressed as a function of the critical temperature) and one point on the vapor pressure curve. Globally, the MAH thermodynamic


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model (10),(1) requires the knowledge of seven material-dependent parameters, which reduce to six if the non-dimensional form is adopted (not reported for brevity).

III. Uncertainty quantification methods

D. Chaos Collocation method Let us consider the following stochastic differential equation:

( ) ( ).,;, θφθ xx fL = (11)

where L is a differential operator representative of space differentiation. L can be non linear and depends on the random parameters θ. The unknown dependent variable( )θφ ,x represents the solution and is a function of the space

variable dℜ∈x and of the random parameters θ;. Finally ( )θ,xf is a source term which depends on x and on the

random parameters θ. Under specific conditions, a stochastic process can be expressed as a spectral expansion based on suitable

orthogonal polynomials, with weights associated with a particular probability density function. The first study in this field is the Wiener process25. The basic idea is to project the variables of the problem onto a stochastic space spanned by a set of complete orthogonal polynomials Ψ that are functions of random variables ( )θξ , where θ is a

random event. For example, the variable φ has the following spectral finite dimensional representation:

( ) ( ) ( )( ).,,0

θφθφ ξxx ii i t Ψ=∑∞

= (12)

Equation (12) splits the random variable ( )θφ ,x into a deterministic part –the coefficient ( )xiφ –, and a stochastic

part –the polynomial chaos ( )( )θξiΨ –.

In practice, the series in Eq.(12) has to be truncated to a finite numbers of terms, here denoted with N. The number of total terms of the series is determined by:

( )∏=

+=+n

kkpN

1

11 (13)

where n is the dimensionality of the uncertainty vector θ and kp is the order of the expansion respect to the k-th

random variable. Substituting the Polynomial Chaos series (12), into the stochastic differential Eq.(11) we obtain:

( ) ( )( ) ( ).,;,0

θθψφθ xξxx fLN

i ii =

∑ =

(14)

The method of Weighted Residuals is adopted to solve this equation. The coefficients ( )xiφ are obtained

imposing the inner product of the residual with respect to a weight function is equal to zero. If we employ Dirac delta function as weight function we produce Collocation method. Using a collocation

projection on both sides of Eq.(14), we obtain:

( ) ( ), ; , 0, , .j j jL f j Nθ φ θ= =x x … (15)

This formulation leads to the solution of a linear system equivalent to solving a deterministic problem at each collocation point and is called Chaos Collocation26.

If in Eq.(12) the spectral representation employed is based on the tensorial product of one-dimensional orthogonal polynomials, the Chaos Collocation approach will be referred as the Tensorial-expanded Chaos Collocation method2,3.

In both cases, once the chaos polynomials and the associated φι coefficients have been determined, the expected value and the variance of the stochastic solution ( )θφ ,x are easily computed from:

)()( 0 xx φµφ ==PCE (16)

.)()( 2

1

22 Ψ== ∑ =

N

i iPCVar xx φσ φ (17)


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Property cp cT cZ eT n ( ), /v cc T R∞ ω

16.2 atm 630.2 K 0.2859 467 K 0.5255 78.37 0.4833 Tab. 1: Mean values adopted for a priori tests.

A still open problem of Chaos Collocation approach is the difficulty to select collocation points, since the choice is not unique when multi dimensional uncertainties are considered. However, this ambiguity disappears if a tensorial-expanded representation is adopted: in this case the collocation points are the Gauss quadrature points of the polynomial with order 1+kp in each dimension.

For the present computations, we retain and assess the Tensorial-expanded Chaos Collocation method.

IV. Preliminary results In this section we present some preliminary results about the effect of uncertainties i) on the thermodynamic data

provided by a given model and ii) on the results provided by a computational fluid dynamic solver for real gas flows using such uncertain models.

E. Uncertain state diagrams We first perform a priori tests on the sensitivity of the RKS, PRSV and MAH models to uncertain input

parameters. To measure such sensitivity, we plot uncertain state diagrams in the Amagat (pressure/volume) plane and compute error bars on the liquid/vapor coexistence curve, the critical isotherm and two particular curves of crucial importance for dense gas flows (see e.g. Ref. 17 and references cited therein): these are isolines of the fundamental derivative of gas dynamics15:

3 2

2 2: ,

2s

v p

a v

∂Γ = ∂

where 2

s

pa v

v

∂ = − ∂ is the sound speed and the subscript s denotes derivatives taken at constant entropy. The

curve of the p-v plane 0Γ = is referred-to as the transition line27 and the thermodynamic region included between this curve and the coexistence curve, where Γ<0, is called the inversion zone27. Fluids which exhibit a negative Γ region in their single-phase vapor state are called the Bethe-Zel’dovich-Thompson fluids (BZT) and are theoretically predicted to give rise to non-classical compression and expansion waves in the transonic and supersonic regime. In the past, BZT flows have been extensively studied in the past (see Ref. 28 for a review) because of their possible advantages as working fluids in energy-conversion cycles. Another important thermodynamic region for dense gas flows is the region such that 1Γ < . In such conditions, the fluid sound speed displays a reversed behavior compared to classical perfect-gas-like fluids for which Γ>1 everywhere: a decreases during isentropic compressions, thus leading to a non-monotone behavior of the Mach number with pressure. Because of the above considerations, we also analyze the effect of thermodynamic uncertainties on the shape and location of the Γ=0 and Γ=1 isolines of the p-v plane, on the size (area) of the inversion zone in the non-dimensional

p-v plane, denoted as ( )0A Γ < , and on the minimal value of Γ on the critical isotherm, i.e.:

( )min, min ,cT cv

T vΓ = Γ .

The results presented in the following have been obtained by Tensorial-expanded Chaos Collocation method with expansion polynomial order 3. For all of the thermodynamic models used for this study, we supposed Gaussian

distributions of the uncertain input parameters, of the form ,30

µµ ℵ

where µ is the mean value of the parameter,

taken from the literature and/or producer data, and the standard deviation is supposed to be about 3% of the mean value. This value is roughly representative of the estimated experimental uncertainty on the input data. The mean values used for the three thermodynamic models under analysis and reported in table 1 and are taken from the thermophysical properties the heavy fluorocarbon PP10 (C13F22) as reported in Ref. 24. Figure 1 shows the uncertain state diagrams and a close-up of the inversion zone for PP10, as predicted by the RKS, the PRSV and the MAH model. For all of the models, the mean stochastic solution is almost superposed to the


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deterministic solution obtained by taking input parameter equal to the mean values of the distribution. Moreover, the simpler RKS and PRSV models provide similar mean curves and display similar (modest) sensitivity to the input uncertainties. In fact, error bars on the results are of the same order of the supposed uncertainty range for input quantities. The MAH model predicts quite different mean curves with respect to the previous models. These are expected to be closer to experimental data; however, they are also affected by greater error bars, which become particularly large for the Γ isolines. This behavior explains the extreme difficulty of obtaining reliable experimental data for dense gas flows based on preliminary numerical data provided by the MAH model (see e.g. Ref. 24). Table 2 reports results about mean values of the inversion zone size and minimum Γ value on the critical isotherm obtained for the three models, as well as the corresponding standard deviations. These values are very close to the deterministic ones for each model, which confirms the preceding considerations on the stochastic state diagrams. Note however that for all the considered models the standard deviation for these quite sensitive parameters is non negligible. Also note that the MAH model predicts a much larger inversion zone compared to the RKS and PRSV models. This is in contrast with older studies which suggest that the MAH model is too conservative with respect to negative values of Γ29.

MODEL ( )( )0Aµ Γ < ( )( )0Aσ Γ < ( )det

0A Γ < ( )min, cTµ Γ ( )min, cTσ Γ min, detcTΓ

RSK 0.008705 0.0026784 0.008664 0.14184 0.015223 0.14195 PRSV 0.006409 0.0013447 0.006388 0.17016 0.014538 0.17026 MAH 0.032742 0.0033641 0.028427 -0.029655 0.13656 -0.030085

Tab. 2: Means and standard variations for the area of the inversion zone and the minimum ΓΓΓΓ value on the critical isotherm.

F. Stochastic simulations of dense gas flows over an airfoil As a second step, a posteriori tests have been performed to measure the effect of thermodynamic uncertainties on

the computed results for the steady transonic inviscid flow of a dense gas over a NACA0012 airfoil at 0.95M∞ =

and 0° angle of attack. The thermodynamic conditions of the free-stream are taken fixed (no uncertainty) and

correspond to 0.985rp = and 0.622rρ = . Computation are performed using a half C-grid composed by 100×32

cells with mean height of the first cell closet o the wall equal to 0.001 chords and outer boundary located at 10 chords. This grid represents a reasonable compromise between accuracy and computational cost, given the number of CFD runs required by the stochastic solver. Some considerations about the effect of the computational grid on the quality of the stochastic simulations are reported later in this Section. The results have been obtained by applying the Tensorial-Expanded Chaos Collocation method with expansion polynomial order equal to 3. A convergence study showed that this choice ensures sufficient independency of the stochastic solutions from the expansion polynomial order. The uncertain input parameters in the thermodynamic models are assumed to be normally distributed, with means and standard deviations listed in Table 3. The standard deviations of the input distributions are about 1.5% of the corresponding mean values. This choice is motivated by the fact that, for the chosen thermodynamic free-stream conditions, large variations of the input parameters drive the flow into the humid vapor region. Since the flow solver is not equipped with two-phase models, a small value of the standard deviation was chosen to avoid that, for some of the collocation points, two-phase flow may develop. This is of course a limitation of the proposed method, which will be investigated in future research. The reference quantities used to investigate the sensitivity of the computed flow-field to the uncertain thermodynamic model are the drag coefficient and the pressure coefficient. Four series of results are considered in the following: the first and the second one have been obtained by means of the RKS and PRSV models with three uncertain parameters (namely, ω, ( ), /v cc T R∞ and n). The third and fourth series have been obtained via the

multiparameter MAH model and differ between them for the number of uncertain parameters included in the simulations. In one case, just three model parameters are assumed to vary, namely, the critical pressure and temperature and ( ), /v cc T R∞ . In the second one, all of the free parameters in the MAH model are allowed to vary

according to Gaussian distributions.

cp cT cZ eT n ( ), /v cc T R∞

( )16.15,0.27ℵ ( )597.1,9.95ℵ ( )0.294,0.005ℵ ( )462.6,7.7ℵ ( )0.528,0.009ℵ ( )597.1,9.95ℵ

Tab. 3 : Distributions of the uncertain parameters used for a posteriori tests. MAH model.


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Fig.1 : Top: uncertain state diagrams. Bottom: close up of the inversion zone.

Figure 2 shows the iso-contours of the mean pressure coefficient and its standard variation for the four cases under study. The deterministic distributions obtained by setting the model parameters to the mean values of the chosen distributions are also reported for reference. For the simple RKS and PRSV models, the mean solution provided by the chaos collocation method is very similar to the deterministic one. As it could be expected, the


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strongest sensitivity of the solution is located around the shock wave. However, the output distribution is quite symmetric and the mean location of the shock wave almost coincides with the deterministic one. This is confirmed by inspection of Figure 3, which shows the wall distributions of the pressure coefficient, with error bars corresponding to µ ±σ. Note that for the simple models the standard variation of the pressure coefficient is almost everywhere negligible, except in the shock region, where it becomes up to 40% of the mean. This shows, on the one hand, that the nonlinear effects in the shock regions strongly amplify the uncertainties on the input parameters; on the other, these variations result in a non negligible dispersion of the computed drag coefficient. Figure 4 shows the computed probability density function for the drag coefficient, cd. This has been obtained via Monte-Carlo sampling over 500 points of the output distribution reconstructed via equation (9). Almost Gaussian distributions are observed for both the RKS and the PRSV models. The RKS cd distribution exhibits a higher mean than the PRSV one, which is consistent with the fact that the first model predicts a smaller inversion zone and consequently more reduced dense gas effects and higher drag. On the other hand, such a model is also slightly less sensitive than the PRSV one to input uncertainties, and the standard variation of the results is lower (see Table. 4 for numerical values).

Figure 2 also displays results for the comprehensive MAH model. Several remarks are in order. First of all, both the deterministic and the mean stochastic solutions provided by this model strongly differ from those given by the simpler ones. This is due to the fact that, for the chosen fluid properties and far-field thermodynamic conditions, the MAH model predicts the flow to evolve essentially within or in the immediate neighboring of the inversion zone. In such conditions, the flow physics is dramatically affected by dense gas effects. Namely, compression shock waves tend to disintegrate and the flow field is almost smooth. On the other hand, the location and extent of the inversion zone are extremely sensitive to the model parameters, as demonstrated by the preceding a priori tests. As a consequence, the aerodynamic field displays a much greater sensitivity to the input parameter than in the previous computations with the simpler models. Note that the solution computed by considering only three well-chosen uncertain parameters is quite close to that computed by allowing all the model parameters to vary. This demonstrates that the dispersion of the computed results may be quantified with reasonable accuracy using a reduced number of uncertain parameters, that is, with a substantial reduction in the global computational cost. The computed probability density functions for the drag coefficient obtained in the two cases are also shown in figure 4. In this case the results display significant non-Gaussian interactions, with more dispersion of the data on the high-cd side of the curve. High-drag values are associated to the presence of a stronger shock wave in the solution, i.e. closer-to-ideal gas behavior. The drag coefficient is affected by a very large uncertainty, with standard deviation which is larger than the mean value for both the case with 3 and the case with 6 uncertain model parameters. In the fist case, however, the mean drag coefficient is about 20% lower than in the second one, in spite of the relatively similar mean pressure coefficient distribution at the wall. This indicates that integrated values are more sensitive than local ones, essentially because of large uncertainties on the shock location. Finally, remark that the means are higher than the deterministic drag coefficient, since small changes in the model parameter produce large variations in the size of the inversion zone, with significant probability for the flow to evolve outside it. To investigate how different distributions of the input uncertain parameters affect the stochastic solution, computations where performed using the MAH model with three uncertain parameters (reduced pressure, temperature and specific heat). The mean values of the distribution are the same of Tab. 3, the variation interval being equal to ±3σ. The resulting mean solution, shown in Fig. 5, is quite similar to that obtained for Gaussian input. The main differences are observed in the shock region, where a steeper mean shock profile is produced, compared with the results of Figs 2 and 3. Moreover, the standard deviation of the solution is globally slightly lower, with large variation more localized in the shock region than in the previous case. The computed mean drag coefficient is close to the previous one, but slightly lower, due to the higher weight given to input parameters leading to a large BZT region. Precisely, the mean drag coefficient is equal to 0.3031×10-2. The standard variation of the drag coefficient, in turn, is definitely lower, consistently with the results of Fig. 5.b, and equal to 0.1560×10-2, even if it remains high in absolute terms (about 50% of the mean value). These results confirm the exasperate sensitivity of the MAH equation to the uncertainties of any nature on the input parameters, and put in question the choice of this equation as a reliable design tool for dense gas flows.

Model ( )dcµ ( )dcσ ( )detdc

RKS 0.6214×10-1 0.7854×10-3 0.6214×10-1 PRSV 0.5151×10-1 0.1157×10-2 0.5150×10-1 MAH, 3 unc. 0.3846×10-2 0.3523×10-2 MAH, 6 unc. 0.4899×10-2 0.6091×10-2

0.2587×10-2

Tab. 4 : Mean and standard deviations of the computed drag coefficient for different models.


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We conclude this Section with some remarks about the effect of the computational grid on the computed stochastic results. Stochastic simulations using non intrusive methods require several separate deterministic computations, whose number becomes larger and larger when the number of uncertain parameters is increased. This reduces the possibility of using very fine computational meshes, because of the limited turn-around time allocated to the stochastic simulation. Figure 6 shows the effect of increased grid density on the stochastic and deterministic results computed by using the MAH model. The results have been computed using a coarser and a finer grid composed by 50x16 and 200x64 cells, respectively. The stochastic results take into account only three uncertain input parameters. Grid convergence of the mean stochastic solution (Fig. 6a) is slightly slower than that of the deterministic one for this very sensitive case characterized by significant dense gas effects, namely in the shock region. Nevertheless, convergence of the solution standard variation is quite satisfactory and the medium grid solution provides already a reasonable representation of the solution expectancy and variance. Moreover, solution accuracy may be increased with modest computational over-cost, by applying Richardson extrapolation to the results obtained on two coarse grids of increasing density. This technique has been successfully applied in Ref. 30 to improve the solution accuracy for flow optimization problems by using genetic algorithms.

V. Conclusions A Tensorial-Expanded Chaos Collocation method has been applied to the quantification and analysis of

thermodynamic uncertainties in real gas flows past airfoils. A priori test showed that more complex models, used to achieve a more accurate description of the fluid thermodynamic response are also more sensitive to uncertainties on the fluid thermophysical properties, because of the larger number of input parameters they require. Uncertainties are particularly critical when dense gas or non-classical gas dynamic effects are to be quantified. A posteriori tests performed for a transonic inviscid dense gas flow over a symmetric airfoil show that the largest error bars on the solution are achieved in the region close to the shock, because of the highly nonlinear behavior of the flow-field. For simple models like the Soave-Redlich-Kwong (RKS) or the Peng-Robinson-Strijek-Vera (PRSV), the mean stochastic solution is almost superposed to the deterministic one, and the output distribution is almost a Gaussian with standard deviation being within roughly 10% of the mean, at worst, i.e. approximately the same order of the input dispersion. For the multiparameter Martin-Hou (MAH) model, on the other hand, the computed results are much more sensitive to model parameter uncertainties. This behavior remains qualitatively similar even for a different choice of the input distributions. The output distribution for the drag coefficient exhibit significant non-Gaussian interactions, the worst expected results being close to those predicted by using simple models. This puts in question the opportunity of adopting the MAH model for the design of dense gas flows, since it is likely to provide too optimistic estimates of the computed drag coefficient. The PRSV model seems to offer a reasonable alternative both in terms of accuracy and computational cost.

References 1Guide for the verification and validation of computational fluid dynamics simulations. AIAA Guide G-077-1998,

1998] 2 L Parussini, V Pediroda, FictitiousDomain with Least-Squares Spectral Element Method to Explore Geometric

Uncertainties by Non-Intrusive Polynomial Chaos Method, CMES 22:(1) 41–64, 2007. 3 L Parussini, V Pediroda, Investigation of Multi Geometric Uncertainties by Different Polynomial Chaos

Methodologies Using a Fictitious Domain Solver, CMES 23:(1) 29–52, 2008. 4ES Oran, J.P. Boris. Numerical simulation of reactive flow, 2nd edn Cambridge Univ. Press, 2000. 5P Glaister. An approximate linearised Riemann solver for the Euler equations for real gases. J Comput. Phys.,

74:382-408,1988. 6J Hoffren, T Talonpoika, J Larjola, and T Siikonen. Numerical Simulation of Real-Gas Flow in a Supersonic

Turbine Nozzle Ring. J. Eng. Gas Turbine Power, 124:395–403, 2002. 7RS Miller, KG Harstad, J Bellan. Direct numerical simulations of supercritical fluid mixing layers applied to

heptane–nitrogen. J. Fluid Mech., 436:1-39, 2001. 8A Guardone, L Vigevano. Roe linearization for the van der Waals gas. J. Comput. Phys., 175:50-78, 2002. 9P Colonna, S Rebay. Numerical simulation of dense gas flows on unstructured grids with an implicit high

resolution upwind Euler solver. Int. J. Num. Meth. Fluids, 46:735–765, 2004. 10P Boncinelli, F Rubechini, A Arnone, M Cecconi, C Cortese. Real Gas Effects in Turbomachinery Flows: A

Computational Fluid Dynamics Model for Fast Computations. ASME J. Turbomachinery, 126: 268-276, 2004. 11P Cinnella, PM Congedo. Numerical solver for dense gas flows. AIAA J., 43:2457-2461, 2005.


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RKS, mean solution RKS, standard deviation RKS, deterministic solution

PRSV, mean solution PRSV, standard deviation PRSV, deterministic solution

MAH 3, mean solution MAH 3, standard deviation

MAH 6, mean solution MAH 6, standard deviation

MAH, deterministic solution

Fig.2 Results of stochastic computations for normally distributed input parameters. Top: stochastic pressure coefficient field: mean value (left) and standard deviation (right). Bottom: deterministic pressure coefficient field associated to the most probable (mean) values of the input parameters.

12F Rubechini, M Marconcini, A Arnone, M Maritano, S Cecchi. The Impact of Gas Modeling in the Numerical Analysis of a Multistage Gas Turbine. ASME Turbo Expo 2006, Barcelona, Spain, 8-11 May, ASME paper GT2006-90129, 2006.

13AJ White. A comparison of modelling methods for polydispersed wet-steam flows. Int. J. Numerical Meth. Eng., 57:819–834, 2003.

14RC Reid, JM Prausnitz, BE Poling. The Properties of Gases and Liquids, 4th edn, McGraw-Hill, 1987. 15PA Thompson. A fundamental derivative in Gas Dynamics. Physics of Fluids 14: 1843-1849, 1971. 16 A Jameson, W Schmidt, E Turkel. Solutions of the Euler Equations by Finite Volume Methods Using Runge-

Kutta Time-Stepping Schemes. AIAA Paper 8l-l259, June 1981. 17P Cinnella, PM Congedo. Inviscid and viscous aerodynamics of dense gas flows. J. Fluid Mech., 580: 179-217,

2007.


12

RKS PRSV

MAH, 3 uncertainties MAH, 6 uncertainties

Fig.3 Results of stochastic computations for normally distributed input parameters. Pressure coefficient along the airfoil surface: the mean value µµµµ and the uncertainty bars µµµµ±σσσσ are shown. 18G. Soave. Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering

Science, 27:1197-1203, 1972. 19 R. Stryjek, JH Vera. An improved Peng-Robinson equation of state for pure compounds and mixtures. The

Canadian journals of chemical engineering, 64:323-333, 1986. 20JJ Martin, YC Hou. Development of an equation of state for gases. AIChE Journal 1: 142-151, 1955. 21O Redlich, JNS Kwong. On the thermodynamics of solutions. Chem. Rev. 44:233-, 1949. 22DY Peng, DB Robinson. A new two-constant equation of state. Ind. Eng. Chem. Fundamentals, 15:59-64, 1976. 23G Emanuel. Assessment of the Martin-Hou equation for modeling a nonclassical fluid. ASME J Fluids Eng.,

116:883-884, 1994. 24A Guardone, B.M. Argrow. Non classical gasdynamic region of selected fluorocarbons. Phys. Fluids, 17:116102-

1,17, 2005. 25 N Wiener, The homogeneous chaos, Am. J. Math. 60:897–936, 1938. 26 G Loeven, J Witteveen, H Bijl. Probabilistic Collocation: An Efficient Non-Intrusive Approach For Arbitrarily

Distributed Parametric Uncertainties. 45th AIAA Aerospace Science Meeting and Exhibit, 8-11 January 2007, Reno, Nevada, 2007.

27MS Cramer, A Kluwick. On the propagation of waves exhibiting both positive and negative nonlinearity. Journal of Fluid Mechanics, 142:9-37, 1984. 28A Kluwick A. Handbook of shockwaves, vol. 1, chap. 3.4, pp. 339-411 (Academic Press, 2000). 29MS Cramer. Negative nonlinearity in selected fluorocarbons. Phys. Fluids A 1, 1894 (1989) 30P Cinnella, PM Congedo. GA-Hardness of Aerodynamic Optimization Problems: Analysis and Proposed Cures.

AIAA Paper 2007-3828 (2007).


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Fig.4 Probability Density Functions for the drag coefficient cd.

Fig. 5 : MAH equation with three uniformly distribu ted uncertain parameters. a) Mean pressure coefficient field, b) standard variation deviation of the pressure coefficient, c) wall distribution of the mean pressure coefficient with ±σσσσ error bars.

(a) (b)

Fig. 6 : Effect of mesh density on the computed results. a) Mean solution, b) standard variation.

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