M De Zordo-Banliat
email: maximilien.de-zordo-banliat@safrangroup.com

X Merle

G Dergham

P Cinnella
email: paola.cinnella@ensam.eu

Bayesian model-scenario averaged predictions of compressor cascade flows under uncertain turbulence models

Keywords: Compressor Flow, RANS Models, Bayesian Model Averaging, Uncertainty Quantification

The Reynolds-Averaged Navier-Stokes (RANS) equations represent the computational workhorse for engineering design, despite their numerous flaws. Improving and quantifying the uncertainties associated with RANS models is particularly critical in view of the analysis and optimization of complex turbomachinery flows. In this work, we use Bayesian inference for assimilating data into RANS models for the following purposes: (i) updating the model closure coefficients for a class of turbomachinery flows, namely a compressor cascade; (ii) quantifying the parametric uncertainty associated with closure coefficients of RANS models and (iii) quantifying the uncertainty associated with the model structure and the choice of the calibration dataset based on an ensemble of concurrent models and calibration scenarios. Inference of the coefficients of three widely employed RANS models is carried out from high-fidelity LES data for the NACA65 V103 compressor cascade [1, 2]. Posterior probability distributions of the model coefficients are collected for various calibration scenarios, corresponding to different values of the flow angle at inlet. The Maximum A Posteriori estimates of the coefficients differ from the nominal values and depend on the scenario. A recently

 4], is used to build a prediction model than takes into account uncertainties associated with alternative model forms and with sensitivity to the calibration scenario. Stochastic predictions are presented for the turbulent flow around the NACA65 V103 cascade at mildly and severe off-design conditions. The results show that BMSA generally yields more accurate solutions than the baseline RANS models and succeeds well in providing an estimate for the predictive uncertainty intervals, provided that a sufficient diversity of scenarios and models is included in the mixture.

Introduction 1

The design of modern, highly-loaded axial compres- including aerodynamics [START_REF] Svetlana V Poroseva | Improving the predictive ca-944 pability of turbulence models using evidence the-945 ory[END_REF][START_REF] Meyer | Combined estimation of 948 hydrogeologic conceptual model, parameter, and 949 scenario uncertainty with application to uranium 950 transport at the hanford site 300 area[END_REF][START_REF] Duan | 954 Multi-model ensemble hydrologic prediction us-955 ing Bayesian model averaging[END_REF][START_REF] Tebaldi | The use of the multi-958 model ensemble in probabilistic climate projec-959 tions[END_REF][START_REF] Diomede | Discharge prediction based on 966 24 multi-model precipitation forecasts[END_REF][START_REF] Rojas | Application of a mul-971 timodel approach to account for conceptual model 972 and scenario uncertainties in groundwater mod-973 elling[END_REF]. Here 126 we focus on the Bayesian Model Averaging (BMA) 127 framework, initially proposed by Draper [START_REF] Draper | Assessment and propagation of model 976 uncertainty[END_REF] (see also 128 [START_REF] Hoeting | Bayesian model averaging: a tutorial[END_REF]). A significant extension to BMA is represented fidelity numerical and experimental data are available in the literature. For our study, we select three widely used RANS models, namely, the Spalart-Allmaras [START_REF] Spalart | One-equation tur-983 bulence model for aerodynamic flows[END_REF], Wilcox' k -ω [START_REF] David | Turbulence modeling for CFD. 986[END_REF], and Launder-Sharma k -ε [START_REF] Launder | Application of the 988 energy-dissipation model of turbulence to the cal-989 culation of flow near a spinning disc[END_REF] turbulence models. The purpose of the study is manifold:

1) we investigate if BMSA calibrated on elementary external flow configurations like those of [START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF]4] may still provide valuable information for the internal flow configuration of interest; 2) we set up a computationally efficient strategy for specifically calibrating BMSA for costly compressor flows; 3) finally, we apply BMSA to the NACA65 V103 compressor flow at operating conditions outside the calibration set, and we assess its capability to provide accurate predictions and the associated uncertainty intervals for new flows. The results are compared to those of BMSA based on the on-the-shelf sets of coefficients [4].

The paper is organized as follows. In Section 2, we recall the Bayesian framework, with special focus on BMSA. In Section 3, we describe the compressor flow configuration and the RANS models. In Section 4 we report BMSA results for the NACA65 V103 cascade at two off-design conditions. Finally, Section 5 summarizes the main findings and draws perspectives for future work.

Bayesian framework

In this section, we recall the theoretical framework for Bayesian model calibration and BMSA, following [START_REF] Merle | Robust prediction of 992 dense gas flows under uncertain thermodynamic 993 models[END_REF].

Bayesian calibration

Let us consider a physical model of the form:

∆ = M(θ) (1)
with ∆ = (∆ 1 , ..., ∆ N ) a vector of Quantities of Interest 185

The scope of Bayesian inference is to gain new knowledge about θ by constructing an improved representation of its pdf, based on prior knowledge and assimilating the observed data. For that purpose, let us note D the random vector of observed high-fidelity data.

Bayes rules states that :

f (θ|D = D) = f (D = D|θ) f (D = D) f (θ) (2)
Here, f (θ) is the prior pdf and represents the initial be-

223

In the present calculations, the observation error is modelled as an additive noise and the model inadequacy as a multiplicative term, as also done in [START_REF] Cheung | Bayesian un-938 certainty analysis with applications to turbulence 939 modeling[END_REF]. Specifically, the data D at a given location x i are related to the observation error by:

D(x i ) = D(x i ) + e i (x i ) (3)
with e i the observation noise at position x i and D(x i ) the The model-inadequacy η i is given by:

D(x i ) = η i ∆(x i , θ) (4)
with ∆(x i , θ) the model output at a point x i . We choose

229

the model errors to be independent and Gaussian, i.e. constraints. For more detailed discussion, see [START_REF] Draper | Assessment and propagation of model 976 uncertainty[END_REF].

239

The preceding choices for η i and e i lead to a likelihood function of the form:

f (D|∆, θ) = 1 (2π) N |K| exp - 1 2 (D -∆(θ)) T K -1 (D -∆(θ)) (5)
with

K = K e + K M
θ i,k ∼ θ|M = M i , S = S k , D = D k (6)
Aftewards, let us consider a new scenario S with no available data and a QoI ∆. Similarly to [START_REF] Draper | Assessment and propagation of model 976 uncertainty[END_REF], we use the law of total probabilities to state that :

f (∆|S , D, M, S) = I i=1 K k=1 f (∆|S , D k , M i , S k )p(M i |D k , S k )p(S k ) (7)
Here, f (∆|S

, D = D k , M = M i , S = S k ) represents
the distribution of ∆ obtained by propagating the posterior distribution θ i,k for the new scenario S , p is the probability mass function of a discrete random variable and we assumed that D and S are independent, as in [START_REF] Rojas | Application of a mul-971 timodel approach to account for conceptual model 972 and scenario uncertainties in groundwater mod-973 elling[END_REF]. For the sake of conciseness, we drop the clearer but redundant formulation M = M i or S = S k to simply write M i or S k in the rest of the paper. We also deliberately omit D, M and S in f (∆|S ) for the same reason.

Equation (7) leads to the following expression for the two leading moments of f (∆|S ) :

E ∆|S = I i=1 K k=1 E ∆|S , D k , M i , S k p(Mi|D k , S k )p(S k ) (8)
Var ∆|S =

I i=1 K k=1 Var ∆|S , D k , M i , S k p(Mi|D k , S k )p(S k )
within-model, within-scenario variance

+ I i=1 K k=1 E ∆|S , D k , M i , S k -E ∆|S , D k , M, S k 2 p(Mi|D k , S k )p(S k )
between-model, within-scenario variance

+ K k=1 E ∆|S , D k , M, S k -E ∆|S , Z, M, S 2 p(S k ) between-scenario variance (9)
p(M i |D k , S k ) = p(D k |M i , S k )p(M i |S k ) J j=1 p(D k |M j , S k )p(M j |S k ) ( 11
)
where p(M i |S k ) is a user-defined prior and p(D k |M i , S k )

is the model evidence :

p(D k |M i , S k ) = Θ f (D k |θ, M i , S k ) f (θ|M i , S k )dθ (12) p(M i |S k ) is generally chosen equiproportional, i.e. 296 p(M i |S k ) = 1/I.

297

The BMSA formulation is completed by selecting a prior probability mass function for the scenarios, i.e. an expression for p(S k ). It was shown in [START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF] that choosing a uniform prior for the scenario mass function brings unnecessary large variance. Following [4] and [START_REF] Merle | Robust prediction of 992 dense gas flows under uncertain thermodynamic 993 models[END_REF], we choose prior scenario based on model agreement:

             p(S = S k ) = ε -p k K k=1 ε -p k ε k = I i=1 E ∆|S , D k , M i , S k -E ∆|S , D k , M, S k 2 ( 13
)
with p = 2. In this formulation, scenarios for which 298 models give closer predictions are assigned higher prob-299 abilities. The NACA 65 V103 cascade therefore represents a challenging configuration for assessing the BMSA methodology.

For the purpose of BMSA calibration, we extracted from LES data selected quantities of interest, namely, the tangential velocity and turbulent kinetic energy profiles in the wall-normal direction and total pressure profiles in the wake. The LES data are in good agreement with the experiments of [START_REF] Leipold | The influ-1008 ence of technical surface roughness caused by pre-1009 cision forging on the flow around a highly loaded 1010 compressor cascade[END_REF]. Tangential velocity profiles at 4 streamwise positions (at x/l = 0.56, 0.64, 0.76 and 0.99 on the suction side, l being the chord and

x/l = 0 the leading edge), and total pressure (Pt inlet - are not all independent since they have to satisfy the fol-lowing relationships, derived for simple canonical flows [START_REF] Stephen B Pope | Turbulent flows[END_REF] (see also [START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF]):

Pt
κ 2 = σ ε C 1/2 µ (C ε2 -C ε1 ) (14)
P ε = C ε2 -1 C ε1 -1 (15)
Following [START_REF] Tavoularis | Further experiments 1023 on the evolution of turbulent stresses and scales 1024 in uniformly sheared turbulence[END_REF]

RANS solver and computational setup 390

The simulations presented in this study are con-391 ducted by using the CFD solver elsA, developed by 392 ONERA [START_REF] Cambier | The onera elsa 1027 cfd software: input from research and feedback 1028 from industry[END_REF]. We solve the 2D steady compressible

Surrogate modelling

To n is defined as :

Q 2 n = 1 - 1 200 200 i=1 (∆ n ) true i -(∆ n ) pred i 2 Var((∆ n ) true ) (18)
For each model and scenario, we present in Figure 2a shows typical calibration results for the κ coefficient of the Spalart-Allmaras model. As also observed in [START_REF] Wn Edeling | Bayesian estimates of parameter variability in the k-ε turbulence model[END_REF], the coefficient is well informed by the data but is highly sensitive to the calibration scenario.

It can be noticed that calibration may assign high probabilities to values of κ that are very different from the standard value 0.41, especially for off-design scenarios farthest from the nominal conditions. Similar results are obtained for other coefficients and models, not reported for brevity. In the next Figure 2b, we present calibration results for the hyper-parameter σ η , which is also well informed by the data. As described in Section 2.1, σ η can be interpreted as a measure of model accuracy in the calibration scenario. We notice that for all calibrations the mean of the hyper-parameter σ η is smaller than 10%.

BMSA prediction for S 2 511

For the rest of this paper, we present BMSA pre- 2.

518

In this section we first report results of BMSA of the 519 NACA65 V103 configuration at mildly off-design conditions, namely, scenario S 2 . The results are discussed for selected velocity and total pressure profiles, representative of typical BMSA predictions. However, similar considerations hold for other locations in the flow.

We present in Figure 3 the tangential velocity profile at x/l = 0.99. The x-axis represents the normalized distance to the wall y n /l, y n being the distance to the blade.

The BMSA results are based on the three sets of scenarios described in the above. Predictions of the baseline RANS models are also reported for comparison. These exhibit significant differences, even for the present attached 2D flow. The k -ω and Spalart-Allmaras models provide rather close predictions, in better agreement with the LES data than the kmodel, which performs noticeably worse than the two other for this case. BMSA predictions of a normalized total pressure profile in the compressor wake are presented in Figure 5.

Results are reported again for BMSA1 and for BMSA2 based on full posterior distributions and MAP estimates of the coefficients. The quantity on the xaxis (namely y/l) represents the normalized crossflow position, with the origin aligned with the trailing edge.

For this QoI, the BMSA models exhibit a behavior similar to the velocity profiles. Specifically, the kbaseline model predicts a wake profile farther from the LES reference compared to the two other baselines.

Second, the prediction using the BMSA1 model predicts a wake profile relatively close to the best performing nominal RANS model, with LES reference data falling within two standard deviations from the mean prediction. As for the velocity profile, the BMSA2 model provides results in very good agreement with the reference data (Figure 5b), especially for the peak and the left-hand side of the profile. For the right-hand side, corresponding to flow coming from the suction side (characterized by a more challenging physics), BMSA still improves over the nominal models but with higher standard deviations than for the rest of the profile. The results do not change much when using MAP estimates instead of full posteriors. In fact, Figure 6 shows that, once again, the contribution of parametric uncertainty to the total variance is very small, which justifies the use of MAPs. For BMSA1, the calibration scenarios are labelled as in [START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF]. The scenario weighting criterion automatically assigns higher probabilities to scenarios corresponding to mixed pressure gradients (airfoil-like cases like S 2100 )

or to zero-gradient (S 1400 ) and mildly favorable cases (S 6300 ), which is a bit counter-intuitive. This is probably due to the fact that model agreement for the prediction scenarios is better for regions of favorable pressure gradient (the left part of the blade), leading to lower errors and the higher weighting of such scenarios. For BMSA2, scenario weighting is little affected by the MAP approximation. In both cases, the scenarios are assigned similar probabilities, with scenarios S 1 and S 3 being preferentially weighted with respect to S 4 . This can be explained by the proximity of the inlet flow angle

E ∆|S ± √ Var [∆|S ] ( ), E ∆|S ±2 √ Var [∆|S ] ( ), Base- line k -ω ( ), Baseline Spalart-Allmaras ( ) and Baseline k -ε ( ).
The scenario probabilities are reported in Figure 10 722 for the three BMSA. Once again we focus only on sce-723 narios with a probability of 5% or higher. For BMSA1 Further reduction of the computational cost can be achieved by using alternative criteria to assign weights to the BMSA scenarios. In this work, we used a criterion based on model agreement in the prediction scenario derived in [START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF]4]. This criterion has proved to be effective in assigning high weights to scenarios in the mixture that are more similar to the prediction case.

Conclusions

However, this criterion requires computing the new flow with the K models using the coefficient from all of the I scenarios, even when many of them are in the end assigned a very low probability. Alternative criteria have been proposed in the literature (e.g. [START_REF] Merle | Robust prediction of 992 dense gas flows under uncertain thermodynamic 993 models[END_REF]) that allow selecting the more suitable scenarios a priori, thus excluding from the beginning scenarios that are affected a probability below a given threshold and finally reduc-

2 sors requires accurate predictions of stagnation pressure 3 losses

 23 at the early stages of the design process. Com-4 pressor flows are characterized by high relative speeds, 5 leading to the formation of shock waves interacting 6 coefficients are now represented as probability distri-111 butions, the model output is also a random quantity, 112 characterized by a probability distribution. In other 113 terms, the solution is naturally equipped with uncer-114 tainty intervals. Parametric approaches can be easily 115 applied to small, noisy datasets and can be successively 116 updated as soon as new or better data become available.

117

  Refs[START_REF] Cheung | Bayesian un-938 certainty analysis with applications to turbulence 939 modeling[END_REF][START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF] used Bayesian inference for calibrating the 118 Spalart-Allmaras and k -ε models, respectively, by us-119 ing experimental data for turbulent flat plate boundary 120 layers. Although parametric approaches only infer on 121 model coefficients, they can also be used for estimating, 122 to some extent, model-form uncertainties. One way to 123 do that is to adopt multi-model ensemble techniques, 124 which have been used in a variety of applications, 125

129by

  Bayesian Model-Scenario Averaging (BMSA) 130 [21, 3, 4]. Like BMA, BMSA combines the predictions 131 from multiple models, thereby providing a measure 132 for model uncertainty, using posterior distributions 133 of the coefficients inferred from different calibration 134 scenarios. In [3], a BMSA model was constructed 135 by averaging five RANS models calibrated on 14 136 scenarios, corresponding to turbulent flat plate flows 137 subject to various external pressure gradients. BMSA, 138 calibrated on the scenarios of [3], was successfully 139 applied to a transonic wing configuration in [4]. 140 141 In the present work we investigate the potential of 142 BMSA for robust predictions of turbomachinery flows 143 under uncertain RANS models, with focus on a com-144 pressor cascade. We focus more particularly on the 145 NACA65 V103 compressor cascade, for which high-146

177(

  QoI) computed by a model M given a set of parameters 178 θ of dimension P. 179 In the deterministic framework, the components of 180 θ are perfectly known and have a fixed value. In the 181 Bayesian framework, the unknown parameters vector θ 182 is treated as a random vector, characterised by a joint 183 probability density function (pdf), noted f . Due to the 184 uncertainty on θ, ∆ is also a random vector.

186

  lief about θ, f (D = D|θ) is the likelihood and corre-187 sponds to the probability to observe D, a realisation of 188 the random variable D, if θ is known exactly. The pos-189 terior pdf f (θ|D = D) represents the updated knowledge 190 of θ given the observed data vector D, of size N. In prac-191 tice, calibration compares the model prediction and the 192 observations to extract the pdf of the parameters vector 193 θ that is the most likely to capture the data. In our case, 194 θ is the set of closure parameters associated with a given 195 RANS model. 196 From Eq. (2), it appears that the posterior distribution 197 is entirely determined by the prior and likelihood func-198 tion. Following Arnst [28] we use uninformative priors, 199 i.e. uniform priors, for each component of θ (supposed 200 independent). As RANS models have been carefully de-201 signed, we are confident in assuming that the standard 202 values should be included in the range of the prior. We therefore choose uniform priors that include standard 204 values (reported in

215

  The likelihood function f (D = D|θ) is a statistical 216 model for observation errors (discrepancies between the 217 data and their true, unobserved, values) and model in-218 adequacies. The latter accounts for the fact that part of 219 the physics is missed by the model due to any approx-220 imation introduced in its construction, so that the true 221 phenomenon can never be exactly captured, even with 222 the best possible model coefficients.

224(

  unobserved) true value of the QoI vector. We choose 225 the components of the observation noise to be indepen-226 dent and normally distributed, with zero mean and a 227 standard deviation equal to 1% of the observed value.

230η

  i ∼ N(1, σ 2 η ) where σ η is an additional uncertain 231 hyper-parameter that needs to be calibrated, and there-232 fore is concatenated to the vector of parameters θ. The 233 hyperparameter σ η is a measure of the magnitude of the 234 model inadequacy and thus can be taken as an indica-235 tor of the accuracy of a given model, calibrated for a 236 given scenario. Considering a model error mitigates the 237 influence of overfitting on the calibration, as it relaxes 238

EE

  ∆|S , D k , M, S k in Eqs (8) and (9) represents the mean of ∆ averaged over all the models being calibrated on the same scenario. It is computed through : E ∆|S , D k , M, S k = I i=1 ∆|S , D k , M i , S k p(M i |D k , S k )p(S k ) (10) The posterior model probability p(M i , S k |D k ) reflects how well the model M i fits the data D k for the scenario S k . It can be computed through the Bayes rule:

300 3 . 3 . 1 .Figure 1 :

 3311 Figure 1: Sketch of the compressor Cascade V 103 adapted from [32]. Ma 1 is the inlet Mach and β 1 is the angle of attack.

  )/(Pt inlet -P inlet ) profiles at 2 positions downstream of the trailing edge (x/l = 1.02 and 1.1) were used for the calibration/assessment of all models considered in the study. For RANS models involving a transport equation for the turbulent kinetic energy (TKE), such as the k -ω and k -ε models, we also considered TKE profiles at the same positions on the suction side as the velocity profiles. For the calibrations reported in the following we used data for a small number of observation points along each profile, clustered in the near wall region and toward the wake center. The data are then concatenated to form the vector D. In total, we used 82 probes for k -ω and k -ε models, and 50 for Spalart-Allmaras model. As a general rule, the number of data used in the calibration is a tradeoff between the necessity of inform-353 ing the model coefficients and computational cost asso-354 ciated with the construction and inversion of the corre-355 lation matrices involved in the likelihood function.

356 3 . 2 .

 32 figuration based on pure expert judgement, we adopt a 363

393

  RANS equations for perfect Newtonian gases by using394 a cell-centered finite volume approximation on struc-395 tured multi-block grids. The upwind scheme of Roe 396 with second-order MUSCL extrapolation is used for ap-397 proximating the spatial fluxes. For time stepping, we 398 use the first-order backward Euler scheme. 399 The computational domain contains a single blade 400 profile and periodic boundary conditions are applied at 401 the upper and lower boundaries to simulate an infinite cascade. The domain extends from 0.4 chords upstream of the leading edge to 0.5 chords downstream the trailing edge. The top and bottom boundaries are separated by a distance equal to the gap between neighboring blades, t/l = 0.59, with l the axial chord. In addition to the periodicity conditions at the upper and lower boundaries, non-slip adiabatic boundary condition is applied at the blade wall, and characteristic conditions are imposed at the inlet and outlet boundaries. At the inlet, the total pressure, enthalpy and angle of attack are prescribed; a constant static pressure is enforced at the outlet.The computational grid is composed by 200,000 cells distributed on 12 blocks. The near-wall grid resolution leads to an average height of the first cell closest to the wall (in wall coordinates) such that y + < 1.0 on both the suction and the pressure side of the blade. For all computations, we assume that the solution has converged to the steady state when the L 2 norm of the residuals is reduced by five order of magnitude with respect to the initial value. The simulations are run in parallel on 12 cores and the typical CPU time for obtaining a converged solution is of the order of 20 minutes. Since a very large number of numerical simulations is required for the calibration of model coefficients using MCMC, the solver output for the observed data is approximated by means of a surrogate model, described in the next section.

  For a given model and a given scenario, the 200 samples are used to build a surrogate for each one of the observed QoIs involved in the likelihood function (namely, velocity, TKE and total pressure values at selected points in the flow field, as discussed in section (3.1)). We verified the accuracy of the surrogate models by Leave-One-Out cross-validation. For each model and scenario, we compute the Q 2 n criterion for every element ∆ n of ∆ = (∆ 1 , ..., ∆ N ). By definition, Q 2

  Hyper-parameter σ η .

Figure 2 :

 2 Figure 2: Posterior probabilities in case of Spalart-Allmaras model: Scenario 1 ( ), Scenario 2 ( ), Scenario 3 ( ) and Scenario 4 ( ). Priors has been chosen uniform on [0.36 , 0.56] and [0 , 1]respectively.

512

  diction in blue color, with first and second standard-513 deviations in degrading shades of blue. Red color is 514 reserved for the LES reference data from [1]. Black, 515 green and orange colors are used respectively for the 516 baseline k -ω, kand Spalart-Allmaras models, with 517 the nominal closure coefficients of Table

Figure 3a displays results{S 1 , S 3 ,

 13 Figure 3a displays results for the BMSA1 model, i.e. using on-the-shelf MAP estimates of model coefficients calibrated on flat plates from [4]. The prediction expectancy for this model does not yield better results than the best baseline model but performs much better than the worst one. Moreover, the prediction error bars, corresponding to ±2 standard deviations, encompass rather well the reference data, except in the region closest to the wall. In Figure 3b we report the results for BMSA2, calibrated on compressor scenarios {S 1 , S 3 , S 4 } and applied to S 2 . In this figure, the complete posterior distributions are propagated through the models based on kriging surrogates of the output QoI. Propagation of the full posterior distributions is based on surrogate models for each RANS model in the mixture and each QoI, as discussed in the above. The predictive accuracy of BMSA improves significantly when we consider closeby scenarios for model calibrations. In particular, the mean

Figure 3 :

 3 Figure 3: Prediction of the normalized tangential velocity profile at x/l = 0.99 on the suction side for scenario 2. LES data from Leggett et al. [1] ( ), E ∆|S ± √ Var [∆|S ] ( ), E ∆|S ±2 √ Var [∆|S ] ( ), Baseline k -ω ( ), Baseline Spalart-Allmaras ( ) and Baseline k -ε ( ).

Figure 4

 4 Figure4shows the variance decomposition accord-

Figure 4 :

 4 Figure 4: Variance decomposition of prediction for the normalized tangential velocity profile on the suction side at x/l = 0.99 for scenario 2.within-model, within scenario variance, between models, within scenario variance and between scenario variance

  Complete distributions obtained on S = {S 1 , S 3 , S 4 }. (c) MAP coefficients obtained on S = {S 1 , S 3 , S 4 }.

Figure 5 :Figure 6 :

 56 Figure 5: Prediction of the normalized pressure wake profile at x/l = 1.10 for scenario 2. LES data from Leggett et al. [1] ( ), E ∆|S ± √ Var [∆|S ] ( ), E ∆|S ± 2 √ Var [∆|S ] ( ),Baseline k -ω ( ), Baseline Spalart-Allmaras ( ) and Baseline k -ε ( ).

662 of S 2 ,

 2 S 3 and S 1 . For the first two scenarios, the flow 663 is qualitatively similar to S 2 , which is not the case for 664 S 4 , as discussed in the next section. In all BMSA, the 665 Spalart-Allmaras model is generally assigned the high-666 est probability, and kthe lowest. Using the MAP ap-667 proximation changes slightly the model evidences, and 668 subsequently model weighting within each scenario, but 669 the results are overall very close to the BMSA2 using 670 the full posterior distributions.(a) MAP coefficients calibrated on flat-plate [4] Obtained with complete distributions on S = {S 1 , S 3 , S 4 }. Obtained with MAP estimates of the distributions on S = {S 1 , S 3 , S 4 }.

Figure 7 :

 7 Figure 7: Distribution of p(S k ) and p(M i |D k , S k ) in case of scenario 2.Only scenarios with probability superior to 5% are shown on Figure7a. Each bar sums to the probability of the scenario. Each probability of scenario is then decomposed into probabilities of models, given this scenario. k -ε ( ), k -ω ( ) and Spalart-Allmaras ( ).

  Figure 8: 2-D contour of first two moments of the BMSA prediction for normalized total pressure for scenario 2. In this case, we considered MAP estimates on scenarios S = {S 1 , S 3 , S 4 }

6824. 3 .

 3 BMSA prediction for S 4The BMSA mixture models are then applied to the prediction of a more challenging off-design conditions, i.e. the separated flow scenario S 4 . We show in Figure 9 the predictions for the tangential velocity profile at x/l = 0.99 on the suction side in this case. The BMSA1 solution is reported in Figure 9a. The solution clearly under-estimates the size of the backward flow region. Nevertheless, the predicted velocity profile exhibits incipient separation and the ±2σ error bars encompass reasonably well the reference LES solution.

Figure

  Figure9breports results for BMSA2 calibrated on scenarios S 1 , S 2 and S 3 . In this case, the mean solution compares poorly with the reference LES. Since BMSA2 has been calibrated on attached scenarios, the posterior coefficients tend to provide even fuller velocity profiles than the baseline models, which already fail to predict flow separation, except for the baseline k -ω that underestimates the size of the reversed flow. We also observe that, in this case, the error bars are small and do not encompass the reference data. This is due to the fact that the models in the mixture strongly agree on the wrong solution. This result shows the importance of including sufficientlu diverse scenarios in BMSA models. In the present BMSA, predictions are based on models with similar characteristics (linear eddy viscosity), furthermore calibrated on similar attached flow scenarios. As a consequence, the resulting BMSA model is very good at predicting flow scenarios similar to the calibration ones but generalizes badly to a different flow, leading to less accurate results than BMSA1. In Figure9cwe present results for BMSA3, which aggregates together the flat plate scenarios and the NACA65 scenarios. Increasing the diversity of scenarios in the model mixture has a beneficial effect on the solution, which is not worst than

Figure 9 :

 9 Figure 9: Prediction of the tangential velocity profile at x/l = 0.99 on the suction side for scenario 4. LES data from Leggett et al. [1] (),

  MAP estimates obtained on S = {S 1 , S 2 , S 3 }. {S 1400 , .....S 2134 , S 1 , S 2 , S 3 }.

Figure 10 :

 10 Figure 10: Distribution of p(S k ) and p(M i |D k , S k ) in case of scenario 4. Only scenarios with probability superior to 5% are shown on Figures 10a and 10c. Each bar sums to the probability of the scenario. Each probability of scenario is then decomposed into probabilities of models, given this scenario. k-ε ( ), k-ω ( ) and Spalart-Allmaras ( ).

significantly improves the predictions compared to the 792 baseline

 792 RANS models when it is used to predict sce-793 nario characterized by an intermediary inlet angle with 794 respect to those included in the BMSA. Additionally, 795 the predicted error bars encompass the reference data. 796 However, this strategy may leads to overfitting prob-797 lems. When applied to a scenario with operating con-798 ditions leading to radically different flow features com-799 pared to the training scenarios, BMSA provides less ac-800 curate predictions than the baseline models. In addition, 801 the error bars are strongly underestimated due to the in-802 sufficient diversity of models and scenarios included in 803 the mixture. 804 Since it is difficult to select a priori the most suit-805 able scenarios to be included in the BMSA based on 806 pure expert judgement (for instance, one could argued 807 that flat plate scenarios are a priori less suitable than 808 the NACA65 scenarios to predict another NACA65 flow 809 condition), it is very important to include in the mixture 810 sufficiently diverse flow scenarios and RANS model to 811 mitigate overfitting and avoid underestimation of vari-812 ance. For instance, predictions of the strongly off-813 design scenario based on a mixture of the flat plat and 814 NACA65 scenarios preserved or improved the average 815 prediction with respect to the baseline RANS models 816 and delivered sufficiently large error bars to encompass 817 the reference data.

818A

  serious limitation to the number of models and 819 scenarios in a BMSA is the computational cost of the 820 stochastic prediction. In fact BMSA combines stochas-821 tic predictions of a new flow scenarios from K several 822 models using posterior pdf of the closure coefficients 823 calibrated for I flow scenarios. Each stochastic predic-824 tion involves an UQ calculation, corresponding to a high 825 number (O(100) or more, according to the UQ method 826 in use and to the number of uncertain coefficients), lead-827 ing to an unacceptably high number of costly determin-828 istic RANS simulations, O(100 K×I). A first method for 829 drastically reducing the required number of determinis-830 tic simulations, first proposed in [4] and further assessed in this work, is to approximate the posterior pdf by Dirac pdf based on Maximum A Posteriori (MAP) estimates of the closure coefficients. This approximation is shown to affect weakly the quality of the BMSA predictions, both in terms of mean and variance, while reducing the number of deterministic simulations to only K × I, i.e. nine deterministic simulations in the present application. All the required simulations are independent and can be run in parallel and the BMSA intervenes as a post-processing step. Since MAP-based BMSA does not rely on any surrogate model for the UQ propagation step, it can be use to extract potentially any QoI at any point in the flow, provided that such QoI is computable with the baseline models in the mixture (for instance, a BMSA prediction of a QoI like the turbulent kinetic energy k or the turbulent dissipation ε can be obtained only if all the models in the mixture are at least two-transport equation models).

  ing the number of deterministic calculations required for the prediction of a new flow. The development 866 and assessment of smarter and computationally efficient 867 scenario-selection criteria will make the object of fur-868 ther research. 869 of high-fidelity simulations for main gas path tur-898 bomachinery components and their industrial im-899 pact. Flow, Turbulence and Combustion, 102(2), K. Duraisamy, G. Iaccarino, and H. Xiao. Tur-902 bulence modeling in the age of data. Annual

Table 2 )

 2 , as done in[START_REF] Cheung | Bayesian un-938 certainty analysis with applications to turbulence 939 modeling[END_REF] [START_REF] Wn Edeling | Predic-888 tive rans simulations via bayesian model-scenario 889 averaging[END_REF]. Further-

	205
	206	more, there is no evidence that model predictions would
	207	be improved by choosing closure coefficients with sig-
	208	nificant deviations from the standard values. The prior
	209	intervals are therefore chosen to be large enough to al-
	210	low a good exploration of the parameter space, while
	211	avoiding values preventing the CFD solver to converge.
	212	Also note that excessively large prior distributions may
		lead to overfitting problems, resulting in posterior coef-

213

ficients that fit very well the calibration data, but deteri-214 orate predictions of unobserved quantities of interest.

Table 1 :

 1 Flow conditions for various compressor cascade scenarios.

	325

32, 33, 1], and the 313 high-fidelity data available in the literature are suitable 314 for BMSA calibration and assessment. Hereafter we 315 consider in particular the LES data from Leggett [1, 2], 316 who investigated the cascade at four off-design condi-317 tions, corresponding to calibration/prediction scenarios 318 in the present Bayesian framework. The scenarios have 319 different values of inlet angle but similar inlet Reynolds 320 and Mach numbers and inlet turbulence intensities. 321 Flow conditions characterizing each scenario are 322 reported in Table 1. Previous study [1] pointed out 323 that RANS models provide rather accurate results for 324 near design conditions, but behave poorly at off-design.

  C w1 , C w2 , C w3 , C v1 and κ . Since the coefficient C w1 is related to the other coefficients by the relation:

							Model	Closure Coefficient Standard value
							C ε2	1.92
							k -ε	C µ σ k	0.09 1.0
							κ	0.41
	378	we set	P ε	= 2.09 in equation (15). By	κ	0.41
	379	enforcing the preceding relationships, we are finally left	σ do	0.125
		with 4 uncertain closure coefficients, namely C ε2 , C µ ,	k -ω	σ * σ	0.6 0.5
							β *	0.09
							β	0.0708
							κ	0.41
							C w2	0.3
							Spalart-	C w3	2.0
							Allmaras	C v1	7.1
							C b1	0.1355
		The coefficients must satisfy the relation [25]	C b2 σ	0.622 2/3
		α =	β β * -	2	κ 2 √ β *	(16)
		so that only six independent coefficients are left. In the
		C w1 =	C b1 κ 2 +	1 + C b2 σ	(17)
		only 7 independent closure coefficients are left, whose
	389

380

σ k and κ. The standard values of these coefficients for 381 the Launder-Sharma model are given in Table

2

. 382 3.2.2. Wilcox k -ω (2006) 383 The second model is Wilcox' k -ω model [25], based on transport equations for the turbulent kinetic energy k and the turbulent dissipation rate ω = ε/k. This model has seven closure coefficients denoted α, β, β * , σ, σ * , σ do and κ, whose standard values are given in Table 2. 384 following, α is computed a posteriori once the other co-385 efficients have been calibrated from data.

386 3.2.3. Spalart-Allmaras model 387 The Spalart-Allmaras model [24] is a singletransport-equation model for a viscosity-like quantity ν, which merges with turbulent viscosity ν t far from the walls. It involves 8 closure coefficients: C b1 , C b2 , σ, 388 standard values are given in Table

2

.

Table 2 :

 2

Standard values of the closure coefficient for the k -ε, k -ω and Spalart-Allmaras models, according to

[START_REF] Launder | Application of the 988 energy-dissipation model of turbulence to the cal-989 culation of flow near a spinning disc[END_REF]

[25]

[START_REF] Spalart | One-equation tur-983 bulence model for aerodynamic flows[END_REF]

, respectively.

  reduce the number of expensive RANS simulations involved in model calibrations, we approximate the QoIs required in the argument of the likelihood func-

	whose hyperparameters are determined by optimizing
	the likelihood. For that purpose, we use the L-BFGS-
	B [38] optimizer available in the scipy library [39]. The
	initial RANS calculations required as an input to the sur-
	rogate model are distributed in the parameter space by
	Latin Hypercube Sampling (LHS) [40] optimized un-
	der the Maximum Projection Design criterion. More
	precisely, this criterion ensures optimal space filling by
	maximizing the minimal distance between points of the
	LHS, for every projection in parameter sub-spaces. We
	construct a separate surrogate based on 200 RANS sam-

tion , (i.e. ∆(θ)) by means of surrogate models based on Gaussian process regression. For that purpose, we use the Gaussian Process Regression module available in scikitlearn

[START_REF] Pedregosa | Scikit-learn: Ma-1035 chine learning in Python[END_REF]

. We select a Matern -3/2 kernel, ples for each concurrent turbulence model and each calibration scenario in Table

1

, for a total of 2400 CFD calculations, run in parallel on a multi-processor computer. This is a considerable computational effort, but it is done one for all prior to the calibration phase.

Table 3 the

 3

	S 1	0.976 0.991	0.975
	S 2	0.967 0.968	0.965
	S 3	0.995 0.997	0.970
	S 4	0.996 0.985	0.994

431

mean value of the Q 2 n criterion, averaged on the N sur-

432 rogate models. 433 Average Q 2 k -ε k -ω Spalart-Allmaras

Table 3 :

 3 Average values of Q 2 n for models and scenarios. S 3 and S 4 to predict scenario S 2 .

	4. Results

434

In this section, BMSA is used to predict two of the 435 scenarios presented in section 3.1 (namely, S 2 and S 4 ) 436 . In scenario S 2 , the flow remains attached all over the 437 suction side of the blade, whereas flow separation is ob-438 served in scenario S 4 , which is very different from the 439 other scenarios in the database and represents a chal-440 lenging configuration for assessing BMSA predictions 441 far outside the training set. The BMSA results reported 442 in the following are based on different ensembles of cal-443 ibration scenarios. First, a baseline BMSA model, noted 444 BMSA1, is constructed by propagating the maximum-445 a-posteriori (MAP) estimates of model coefficients and 446 the model posterior probabilities of [4]. Although such 447 coefficients were obtained for flat plate flows, we may 448 expect that the thin NACA65 V103 blades can be ap-449 proximately modelled as flat plates subject to a variable 450 (mostly adverse) pressure gradient. It is then interest-457 Afterwards, another BMSA model, noted BMSA2, 458 is developed by calibrating the RANS models against 459 compressor configurations. More precisely, we cali-470

In all cases the smart scenario weighting of Eq. (

13

) 471 is used to assign a priori probabilities to the scenarios 472 involved in the BMSA models.

473

Specifically, the error term k in Eq. (

13

) is de-474 termined by computing the ||.|| 2 of local errors on the 475 normalized velocity and total pressure profiles at the 476 streamwise stations of section 3.

477 4.1. Calibration results 478 In order to construct a BMSA model specifically 479 trained for compressor configurations, we apply the sta-480 tistical calibration framework described in section 2.1 481 to infer on the parameters of the RANS models for the 482 scenarios of section 3.1. 483 For each model and scenario, we assign to the clo-484 sure coefficients non-informative uniform marginal dis-490 The hyper-parameter σ η is assigned a uniform prior 491 in the range [0, 1]. For illustrative purpose, we present

Table 4: Lower and upper bounds for the prior of the Spalart-Allmaras closure coefficient for the scenario 3.

Table 5 :

 5 Root-Mean Square values for the baseline models (averaged

		Scenario 2	Scenario 4
		U t,RMS P t,RMS U t,RMS P t,RMS
	Baseline	0.728	0.492	0.797	0.345
	Flat plate	0.710	0.448	0.567	0.232
	NACA65	0.275	0.183	1.199	0.554
	configuration
	NACA65	0.561	0.356	0.843	0.372
	and flat plate
	value for the 3 considered models), BMSA with models calibrated on
	flat plates [4], BMSA with models calibrated on NACA65 configura-
	tions and BMSA with models calibrated on NACA65 and flat plates
	together.

https://github.com/pymc-devs/pymc

[12] S.B. Pope. A more general effective-viscosity hy-

Acknowledgements

Agency for Research and Technology (ANRT) for pro-872 viding support through the CIFRE PhD grant number

A recently developed Bayesian framework is as-