Uncertainty Quantification-Driven Robust Design

INCA manuscript No. (will be inserted by the editor)

Uncertainty Quantification-Driven Robust Design Assessment of a Swirler’s Geometry Pamphile T. Roy · Guillaume Daviller · Jean-Christophe Jouhaud · B´ en´ edicte Cuenot

Received: date / Accepted: date

Abstract This paper presents an Uncertainty Quantification (UQ) on a swirl injector. Based on a mesh refinement strategy and Large Eddy Simulations (LES), the present study goes addressing a problem with high dimensional parameter space consisting in geometrical variables. Using real measurements, a Highest Density Region (HDR) method was used to reduce the parameter space by uncovering correlations due to the Additive Manufacturing (AM) process. Results confirm the sensitivity of the flow to the swirler geometry. Moreover, it shows the robustness of the design to manufacturing dispersions. This work highlights the potential of UQ for robust design by being able to take into account the manufacturing process. Keywords Uncertainty Quantification, Dimension reduction, Highest Density Regions, LES

P.T. Roy · G. Daviller · J-C. Jouhaud · C. Benedicte CFD Team, CERFACS, 42 Avenue Gaspard Coriolis, 31057 Toulouse cedex 1, France E-mail: [email protected]

1 Introduction At the heart of a turbomachine is the combustion chamber which fuel is provided by injectors. Depending on the velocity of the laminar flame speed and on the power required, different devices are to be considered to stabilize a flame. Swirled injectors have been designed to overcome this issue [14]. By impinging a movement of rotation on the flow, a recirculation zone is created directly at the outlet of the device. This ensures enough residence time for the fuel to be consumed. The design of such geometry is of prime importance when looking at reducing combustion consumption and pollutant emissions. Indeed, these objectives are met by lean combustion which may lead to: combustion instabilities and reduced extinction margins. When considering environmental aspects, efficiency, reparability or even durability, the problem becomes a multiobjective optimization which is hard to tackle. As a matter of fact, numerical simulations—and here computational fluid dynamics (CFD) simulations—, have undoubtedly become reliable design aids [18, 8]. Recently, the use of a mesh adaptation strategy and Large Eddy Simulations (LES) has been demonstrated in order to identify the relevant physics of swirling flows [6]. From expert knowledge, the design of a new swirler might proceed in three steps before the production of the device: (i ) an exploratory phase is performed using Reynolds-averaged Navier-Stokes (RANS) computations. ((ii ) the resulting geometry installed in the combustion chamber is simulated using a reactive LES. (iii ) Experiments are performed. The first step allows to try all possibilities in terms of geometrical modifications (angles, number of channels, etc.) to meet certain requirements such as the swirl number, the effective surface and the permeability. The second step is

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2 Swirler 2.1 Swirler and its Uncertainties Figure 1 presents a pattern of the swirler used in this work (also described in [6]). It consists in two counter rotating sets of 8 tangential vanes that feed the swirler with air. This design results in a rotation of the flow that creates recirculation bubbles which are paramount to combustion stability. In order to simplify the geometry, no fuel will be injected in this study and a plug replaces the fuel injection system in the primary flow. As the geometry is built with metal AM, some discrepancy in the size of the channels have been experimentally observed. These changes are also observed numerically. Figure 2, where all channels’ dimensions have been changed according to Section 3, shows the total

Primary vanes LES domain 81 mm

performed to assess the combustion stability and the performances. Finally experiments are conduct to validate the design of the swirl injector. This work focuses on swirler produced by metal Additive Manufacturing (AM) [9, 19]. These technologies have gained a lot of attentions as they allow a total liberty from a design point of view without sacrificing mechanical properties [13]. However, as pointed out by these studies, some uncertainties remain regarding the structural composition of these metallic systems, making this an active topic of research. Our concerns are not about these microstructure characteristics, but rather about the geometrical uncertainties due to the manufacturing itself. Depending on the deposition method, the scanning method or even the powder size, the dimensions seen by the flow are impacted. Indeed, due to the complexity of some designs, it might not be possible to polish correctly the surface using traditional mills. This may leave some defects which sizes would depend on the quality of the whole process. Consequently, this paper seeks to measure the effect of the AM on the flow. This is achieved through an uncertainty quantification (UQ) study [11] performed with LES by varying the geometry. To reduce the number of parameters, a principal component analysis (PCA) is performed. This paper is organized as follows; Section 2 presents the configuration which is considered as well as the numerical setup used to simulate it. Section 3 details the methodology used to reduce the parameter space. Results are presented and analysed in Section 4 and Section 5, respectively.

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Inlet

Plug 300 mm

Secondary vanes 90 mm

Fig. 1: Schematic view of the LES computational domain. dissipation field Φ:  Φ = (µ + µt )

∂ui ∂uj + ∂xj ∂xi

2 (1)

where µ is the molecular viscosity and µt the turbulent viscosity. This quantity controls the irreversible losses (see [6]) and explain the difference between the two fields, especially on the plug tip and on the measured pressure drop.

2.2 Numerical Setup 2.2.1 LES configuration All simulations are performed using the compressible Navier-Stokes solver AVBP [20]. The finite-element scheme TTGC [3] is used on a tetrahedral mesh and LES equations are closed using SIGMA model [15]. At the inlet and outlet boundaries Navier-Stokes Characteristic Boundary Conditions (NSCBC) are used [16], imposing the mass flow rates and the pressure, respectively. All wall boundary conditions are specified as wall no-slip conditions. 2.2.2 Mesh Adaptation Strategy The Adaptive Mesh Refinement (AMR) strategy used in this study is based on the methodology proposed by [6]: from the result obtain on an initial arbitrary mesh, a metric based on the time-averaged viscous dissipation field is computed. Then the mesh of the computational domain is adapted using the MMG3D library [5]. This procedure is repeated until the experimental pressure loss is obtained (see Fig. 4). The computational details of one simulation are summarized on table 1.

UQ-Driven Robust Design Assessment of a Swirler’s Geometry

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˜ ≤ 3 · 106 W/m3 . Left Fig. 2: Impact of geometrical modifications on the axial velocity. Visualization with 0 ≤ Φ case is at ∆P ∼ 4 065 Pa and right case is at ∆P ∼ 4 143 Pa.

Fig. 3: From left to right: coarse mesh, first adaptation and second adaptation. 2.2.3 Simulations toward UQ In the end, the following study was conducted using the second mesh adaptation. In order to quantify the uncertainty on the flow physics, a sample of 30 LES

was considered. The procedure used to generate this sample is detailed in Section 3. For each simulation, a new geometry has been constructed. Then, the mesh was automatically adapted using the aforementioned methodology. This procedure ensures that the meshes

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Table 1: Summary of mesh adaptation for one simulation

α  number of cells (×106 ) time step (×10−7 s) number of CPU hours number of cores ∆P error

Coarse

AD1

AD2

1.6 0.59 5h30 560 57,8 %

50 0.3 4.1 0.47 2h20 2800 6,5 %

30 0.4 12.8 0.20 9h30 2800 4,8 %

AVBP Exp

6000 5000

P (Pa)

4000 3000 2000 Coarse

1000

AD1

AD2

0 0.00

0.02

0.04

0.06 Time (s)

0.08

0.10

0.12

Fig. 4: Temporal evolution of the pressure loss computed with LES on the three different meshes. The dash line represents the experimental measure of ∆P . are optimum for each configuration. It is only based on these meshes that the pressure loss was captured on a statistically significant period of 0,4 ms. The total computational cost for this UQ study using LES is about 1 000 000 CPUhours. Thanks to the high performance computing ressources provided by the CINES on OCCIGEN, the return time was only 2 days. The BATMAN (Bayesian Analysis Tool for Modeling and uncertAinty quaNtification) tool [17] was used to handle all simulations and perform the analysis. This python framework is able to treat any black-box code seamlessly.

of parameters. Indeed, AM is a layer-by-layer process which precision relies on the ability of the machine to build up and polish these layers. Using this correlation, it is possible to drastically reduce the parameter space dimensions. From expert knowledge, we have been granted access to a set of swirlers’ geometries. From this data, the goal has been to infer a link between the discrepancy in the channels dimensions. Due to the dimension differences and the layering process, two groups have been considered: (i ) the inner set of channels noted s = in, and (ii ) the outer set of channels noted s = ou. Each of these two groups makes a spatial curve: ds (i) with d the channel dimension, i a given channel and the subscript s standing for the inner or the outer set. Using highest density region boxplot method [10], the median curve can be recovere along with some confidence intervals. Furthermore, this allows to infer new realizations that are statistically relevant. The dataset is represented as a matrix, each line corresponding to a curve. This matrix is then decomposed using a principal components analysis (PCA) [1], which allows to represent the data using a finite number of modes. This compression process turns the functional representation into a scalar representation on a reduced space. The retained variance is ensured to be maximum in this projection. Within this space, the median curve corresponds to the Higher Density Region (HDR) in terms of probability. Moreover, the farther a point is from this HDR, the more the curve is unlikely to be similar to the other curves. A multivariate kernel density estimation [21] technique is used to find the probability density function (PDF) fˆ(xr ) of the multivariate space. This density estimator is given by Ns 1 X fˆ(xr ) = Khi (xr − xri ), Ns i=1

(2)

with hi the bandwidth for the i th component and Khi (.) = K(./hi )/hi the kernel which is chosen as a modal probability density function that is symmetric about zero. K is the Gaussian kernel and hi are optimized on the data. From this KDE, the HDR reads

3 Dimensionality Reduction

Rα = xr : fˆ(xr ) ≥ fα

Performing an UQ study requires a large number of numerical simulations in order to converge statistical moments. A large number of parameters requires an even larger number of samples. In this case—Section 2—, the uncertain parameters are the dimensions of all channels, and these 16 quantities are not totally independent. The manufacturing process can lead to a correlated set

R with fα such that Rα fˆ(xr )dxr = 1 − α: the region has a probability of containing 1 − α points that verify a highest density estimate. Thus the median curve is defined as the inverse transform—from the reduced space into the original space—of arg sup fˆ(xr ). Also 50% and 90% highest density regions are computed—with respectively α = 0.5 and α = 0.1.

(3)

UQ-Driven Robust Design Assessment of a Swirler’s Geometry

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(a) Inner channels

(b) Outer channels

Fig. 5: Bivariates principal component scores plots of the 30 samples in the reduced space of parameters. Each point shows a curve sample. Point’s distributions for each components are plotted along the diagonal. This method allowed to reduce the uncertain parameters from 16 independent parameters to 6 independent parameters while still being able to account for 90% of the total variance inherent to the original dataset. The default assumption stating that each of these parameters were independent was clearly wrong and would have led to an enormous number of simulations.

4 Results From the PCA, a set of 30 points was sampled from a low discrepancy sequence [2, 4] of Sobol’ (scrambled). Figure 5 is the result of the PCA using 3 modes per set. As said previously, 3 modes ensure that the compression step retains more than 90% of the original variance. Figure 6 shows the inverse transform leading to a sample of 30 curves that are statistically relevant to the original dataset. It can be noted that the HDRs— computed on the original dataset—are well covered by the sampling. The justification behind the number of high fidelity simulations is twofold. (i ) Needless to say that such analysis requires an enormous computational time Section 2. Hence, a budget of 30 HF simulations encompassing this work was set. (ii ) From [7], one should construct a model with at least Ns > 10ndim , with ndim the number of parameters. But the parameters being chosen as modes of a PCA, the importance of the parameters are uneven so that it can be said with confidence that 30 simulations are enough to capture the intrinsic physics of the underlying parameter space.

5 Discussion Figure 7 shows the evolution of the pressure loss by varying the channels’ geometries. As the DoE consists in six dimensions, the response is presented on a bivariate plot. The first modes of both inner and outer channels was used, as they contribute most to the variance of the quantity of interest. It can be noted that the variance of the pressure loss among the sample is high. Furthermore, it seems that a model cannot be drawn from the cloud as no clear shape appears. We verified this assumption by trying to fit the data using several surrogate model strategies available in BATMAN—Section 2—such as Gaussian process, polynomial chaos and radial basis functions. None of these methods were able to fit the data and infer a surrogate model. The reported quality—assessed by Leave-oneout cross validation [12]—was extremely low. Although the construction of a surrogate model is not feasible in this case, the study highlights the intrinsic and chaotic variability present in the data. This phenomenon can be explained by: (i ) the geometrical modifications that compensate themselves, on one hand some channels are widened whereas on the other end some other are tightened ; and (ii ) the amplitude of these modifications. In this work, only small manufacturing defects were considered (approximatively 2% of cross section area of a channel). The ability of the solver to distinguish the physics from one case to another is paramount. Finally, the relative numerical error coming from the solver is estimated less than 1% which is sufficient to capture with accuracy the pressure drops

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(a) Inner channels: din (i)

(b) Outer channels: dou (i)

Fig. 6: Channels’ dimensions. The median curve is represented in bold solid line and shaded area respectively represent the 50% and the 90% HDR. Along these are plotted 30 samples. induced by geometrical modifications (in the magnitude of ' 5%).

4250 4200 ∆P (Pa)

4150 4100 −0.04 −0.02 0.00 0.02 0.04

1i

nn

r

C

0.00 P C 1 0.02 oute

0.04

P

−0.02

er

4050

0.06

Fig. 7: Resulting pressure loss plotted on the first components of the reduced space.

6 Conclusions The purpose of this work was to perform an Uncertainty Quantification (UQ) on a swirler’s geometry. More precisely, this study focused on modifications due to the manufacturing process and their effects to the pressure loss. As the number of parameters required to characterize the configuration was high, a Highest Density Region (HDR) technique was used. This HDR methodology allowing to cut down the number of parameters

from 16 to 6 independent variables. It highlighted the non independence of these geometrical parameters and thus the impact of the Additive Manufacturing (AM). Thanks to a mesh adaptation strategy, a set of 30 different cases was accurately simulated with Large Eddy Simulations (LES). This leads to a visualization of the variation of the pressure drop according to changes in the geometry. This study shows nothing out of the expected as slight modifications lead to slight differences in the pressure loss. Nevertheless, the intrinsic variability of the system did not exhibit any noticeable pattern. As a consequence, no surrogate model could be built. This is explained by the nature of the modifications and most importantly by their amplitudes. By augmenting the ranges of variability in the system, in order not to model the AM but rather to explore other designs, the impact on the pressure loss is expected to be higher. This could allow to build a surrogate model. For the same reason increasing the mass flow rate could lead to the same outcome. Here the mass flow rate was taken as in the experiment [6], where it was restricted to low values. Finally, the method used in this work paves the way to more general uncertainty studies which would take into account manufacturing defects. Acknowledgements The authors acknowledge Patrick Duchaine from Safran Helicopter Engines for his insights on their technologies as well as Bertrand Iooss and Michael Baudin from MRI (EDF R&D) for helpful discussions on uncertainty quantification and support on OpenTURNS. The financial support provided by all the CERFACS shareholders (AIRBUS Group, Cnes, EDF,

UQ-Driven Robust Design Assessment of a Swirler’s Geometry M´ et´ eo-France, ONERA, SAFRAN and TOTAL) is greatly appreciated and we thank them to enable the achievement of such research activities. This work was granted access to the HPC resources of [CCRT/CINES/IDRIS] under the allocation [x20162a6074] made by GENCI.

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