Ariane Emmanuelli
email: ariane.emmanuelli@onera.fr

Maxime Huet
email: maxime.huet@onera.fr

Thomas Le Garrec
email: thomas.le_garrec@onera.fr

Sébastien Ducruix

Sebastien Ducruix
email: sebastien.ducruix@centralesupelec.fr.

Study of Entropy Noise through a 2D Stator using CAA

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

T contribution of combustion noise to the overall noise generated by turbofan engines is becoming increasingly significant as fan and jet noise are reduced. In addition, combustion noise can impact thermo-acoustic instabilities in the combustion chamber. As well as being destructive, their understanding can help reduce NO x emissions. Present in all gas turbines, combustion noise is comprised of two distinct contributions: direct and indirect noise. The former originates from the fluctuations in the heat released by the flame in the combustion chamber. The latter is generated by the acceleration of heterogeneities resulting from combustion through the rest of the engine, namely the turbine stages and the nozzle. These heterogeneities may be compositional, vortical or entropic, leading to entropy noise which is the subject of this paper.

The basis on the theory of indirect combustion noise was established in the seventies when Marble and Candel developed a model for compact nozzles [START_REF] Marble | Acoustic Disturbance from Gas Non-uniformities Convected Through a Nozzle[END_REF]. It was followed by the work of Cumpsty and Marble which led to a model for compact turbine stages [START_REF] Cumpsty | The Interaction of Entropy Fluctuations with Turbine Blade Rows; A Mechanism of Turbojet Engine Noise[END_REF]. Interest in indirect combustion noise has renewed in the last couple of decades. Numerical [START_REF] Richter | On the Application of CAA -Methods for the Simulation of Indirect Combustion Noise[END_REF] [4] [START_REF] Leyko | Numerical and Analitycal Modelling of Entropy Noise in a Supersonic Nozzle with a Shock[END_REF] [START_REF] Becerril | Numerical Investigation of Combustion Noise: The Entropy Wave Generator[END_REF] and experimental [START_REF] Bake | The Entropy Wave Generator (EWG): a Reference Case on Entropy Noise[END_REF] [8] [START_REF] Kings | Experimental and Numerical Investigation of Direct and Indirect Combustion Noise in a Lean Premixed Laboratory Swirled Combustor[END_REF] studies first focused on the simpler nozzle configuration, but numerical studies of a turbine stage [START_REF] Duran | Numerical simulation of acoustic and entropy waves propagating through turbine blades[END_REF] [11] also validated Cumpsty and Marble's compact model. They highlighted the presence of shear dispersion attenuation mechanisms in the turbine [START_REF] Leyko | Simulation and Modelling of the Waves Transmission and Generation in a Stator Blade Row in a Combustion-Noise Framework[END_REF] and suggested entropy noise remains nonetheless significant [START_REF] Duran | Numerical simulation of acoustic and entropy waves propagating through turbine blades[END_REF]. This was confirmed by recent experimental results [START_REF] Knobloch | Experimental Assessment of Noise Generation and Transmission in a High-Pressure Transonic Turbine Stage[END_REF]. Another experimental study which focused on the effect of hot streak migration, confirmed the temperature attenuation across the blades and showed the effect of hot streak injection position as well as the impact on secondary flow [START_REF] Gaetani | Hot Streak Evolution in an Axial HP Turbine Stage[END_REF]. Although advances are made numerically and experimentally, such studies remain complex and costly. Analytical investigations are a solution. Their low computational cost may also enable entropy noise to be taken into account in an industrial context. Modelling has mainly focused on the nozzle configuration. Marble and Candel's compact model gave way to many extensions in 1D, in particular to nozzles of arbitrary shapes and the non-linear regime [START_REF] Huet | A Nonlinear Model for Indirect Combustion Noise Through a Compact Nozzle[END_REF] [15] [START_REF] Moase | The Forced Response of Chocked Nozzles and Supersonic Diffusers[END_REF] [17] [START_REF] Duran | Solution of the Quasi-one-dimensional Linearized Euler Equations Using Flow Invariants and the Magnus Expansion[END_REF]. The model CHEOPS-Nozzle was developed to take 2D radial variations into account [START_REF] Zheng | A 2D-axisymmetric Analytical Model for the Estimation of Indirect Combustion Noise in Nozzle Flows[END_REF] [START_REF] Zheng | Analytical and Numerical Study of the Indirect Combustion Noise Generated by Entropy Disturbances in Nozzle Flows[END_REF], and showed that 1D models are only valid at low frequency because entropy wave distortion greatly affects results in the rest of the frequency range [START_REF] Emmanuelli | CAA Study of Entropy Noise in Nozzle Flow for the Validation of a 2D Semi-analytical Model[END_REF].

The objective of this study is to build a reference case using Computational AeroAcoustics (CAA) to investigate the entropy noise generated in a 2D stator cascade. The resulting noise levels will be compared to those obtained with a model which is currently under development, based on CHEOPS-Nozzle. After describing the chosen case, numerical and post-processing methods will be presented, and the entropy noise results will be discussed in a third section. The geometry chosen for this application case is based on the stator used for the European-FP7 project RECORD [START_REF] Knobloch | Experimental Essessment of Noise Generation and Transmission in a High-pressure Transonic Turbine Stage[END_REF] [23] [START_REF] Bake | Indirect Noise Generation in a High Pressure Turbine Stage[END_REF], during which entropy noise was studied for the full turbine stage using the high-pressure turbine test-rig at Politecnico di Milano for both subsonic and supersonic operating points. The subsonic conditions are used in the present study, and the domain consists of a 2D profile extracted at 50% of the blade height, with lateral periodic boundaries and regions of length of about 20 axial-chords upstream and downstream for post-processing reasons. The mean flow field is simulated using the ONERA software C [START_REF] Refloch | CEDRE Software[END_REF], developed for energetics and propulsion applications. The fluid is considered inviscid and calorically perfect (constant heat capacity c p ). The Euler equations are solved on an unstructured mesh and simulations are run for air with a heat capacity ratio γ = 1.4. Spatial discretisation is of second order, and a pseudo-transient first order implicit scheme is used in time. The inlet temperature is set to 322 K, the inlet velocity to 42 m/s in the axial direction and the outlet pressure is 109216 Pa in accordance with the subsonic operating point of Politecnico di Milano's experimental study. The Mach number reaches 0.66 at the throat and its evolution is shown in Fig. 1. Note that this geometry generates strong acceleration and turning of the flow.

II. Case presentation

The objective is to investigate the entropy noise generated by the acceleration of a normalised plane entropy wave

σ = s /c p (1)
though this mean flow field. Figure 2 shows the different waves involved. The axial direction is noted x and the azimuthal direction y. Indices 1 and 2 describe the inlet and outlet positions, while exponentsand + relate to acoustic waves travelling in the upstream and downstream directions respectively. The stator is forced with an incoming entropy wave σ 1 , which generates the normalised transmitted and reflected acoustic waves P 2 + and P 1 -. In the case where the pressure and velocity fluctuations p and u are purely acoustic, they may be expressed with the Riemann invariants:

P + = p + γp 0 = 1 2 p γp 0 + u c 0 (2)
P -= p - γp 0 = 1 2 p γp 0 - u c 0 (3)
In addition, P 1 + and P 2 -are waves potentially originating from the reflection of P 1 -and P 2 + on the inlet and outlet boundaries of the domain respectively. The noise generated by the acceleration of the entropy wave σ 1 is studied using the thermo-acoustic transfer functions (TATF) [P 1 -/σ 1 ] and [P 2 + /σ 1 ], which caracterise the downstream and upstream propagating waves, respectively. The scattering of acoustic waves by the stator is also investigated. To do so, acoustic waves are injected through the domain boundaries in additional simulations. Two cases are of interest: acoustic forcing from upstream with a wave P + 1 , and downstream forcing for which the wave P - 2 is injected. In the same way as for the TATF, the results can be analysed using the transfer functions [P 1 -/P ]. All six of these transfer functions will be discussed in section IV.

III. Numerical methods

A. CAA

The ONERA Computational AeroAcoustics (CAA) code s AbrinA_v0 [START_REF] Redonnet | Numerical Simulations of Propagation of Small Perturbations Interacting with Flows and Solid Bodies[END_REF] [27] is used for entropy wave convection and acoustic generation and propagation. The linearised Euler equations are solved in the time domain with a perturbation form of the conservative variables built using the mean flow and a disturbance field. A sixth order finite difference scheme is used combined with tenth order explicit filtering in space to avoid numerical dispersion and dissipation effects, and a third order explicit compact Runge-Kutta scheme is applied in time. The mesh is 2D structured, made of eight domains and 260000 points. The mean flow field described in section II is interpolated onto it. The mesh is dimensioned to have at least 20 points per acoustic or entropic wavelength, which is sufficient to propagate waves without significant numerical error in amplitude or in phase. A time step of 1.67 × 10 -7 second is chosen so that the CFL reaches a maximum at 0.78.

The boundary conditions derived by Tam et al. [START_REF] Tam | Radiation and Outflow Boundary Conditions for Direct Computation of Acoustic and Flow Disturbances in a Nonuniform Mean Flow[END_REF] [29] from the asymptotic solutions of the linearised Euler equations are used both to ensure the almost perfectly non-reflective behaviour of the boundaries and to inject plane waves into the domain. As the linearised equations are solved, the computation of the noise levels at all frequencies considered is achieved with limited numerical cost by running a single simulation for each type of injected wave with multi-harmonic forcing varying from 100 Hz to 1000 Hz with a step of 100 Hz. Two harmonic simulations with entropy forcing at 100 Hz and 1000 Hz are also run to validate the numerical methodology. The resulting harmonic and multi-harmonic TATF are in good accordance as shown in Fig. 3.

B. Post-processing

Wave separation. The time signals obtained with s AbrinA_v0 must be post-processed to construct the acoustic waves. The Riemann invariants can give these directly under the condition pressure and velocity fluctuations are of acoustic origin only. However, in this particular case velocity fluctuations u are not purely acoustic but also include a contribution from vorticity perturbations [START_REF] Duran | On the Reflection and Transmission of Circumferential Waves Through Nozzles[END_REF]. This is visible in Fig. 5 which also shows pressure fluctuations p remain one-dimensional and purely acoustic. A direct mode matching wave separation method based solely on the pressure fluctuations can therefore be used [START_REF] Rienstra | An Introduction to Acoustics[END_REF]. In the harmonic regime, it writes:

p = p + + p - (4) ∂p ∂ x = -ik + x p + -ik - x p - (5)
where p -and p + are the pressure fluctuations associated with the upstream and downstream propagating acoustic waves respectively, and the wavenumbers are expressed as:

k + x = ω u 0 + c 0 (6) k - x = ω u 0 -c 0 ( 7
)
where ω is the angular frequency, u 0 the mean axial velocity and c 0 is the velocity of sound. The derivatives are evaluated using a finite difference scheme. Eqs. ( 4) and ( 5) form a linear system that can easily be solved to provide p + and p -. To reduce numerical errors, the wave separation process is achieved for several planes in the upstream and downstream regions. The waves are then phase-shifted to a reference plane in each zone (the domain inlet for waves noted 1, and the outlet for position 2) before averaging and construction of the TATF. Non-reflective post-processing. As the boundary conditions used may not be perfectly non-reflective, the postprocessing described in this paragraph is used to cancel out any wave reflection arising from numerical errors. This is achieved by using results given by entropic forcing, upstream acoustic forcing and downstream acoustic forcing simulations, and by considering the acoustic transfer functions [P 1 -/P

+ /σ 1 ].
The following system of six equations and six unknowns may be solved to obtain the non-reflective transfer functions:

P 1 -(σ 1 ) = P 1 - σ 1 nr σ 1 (σ 1 ) + P 1 - P 1
+ nr P 1 + (σ 1 ) + P 1 - P 2 - nr P 2 -(σ 1 ) (8)
P 2 + (σ 1 ) = P 2 + σ 1 nr σ 1 (σ 1 ) + P 2 + P 1
+ nr P 1 + (σ 1 ) + P 2 + P 2 - nr P 2 -(σ 1 ) (9)
P 1 -(P 1 + ) = P 1 - σ 1 nr σ 1 (P 1 + ) + P 1 - P 1
+ nr P 1 + (P 1 + ) + P 1 - P 2 - nr P 2 -(P 1 + ) (10)
P 2 + (P 1 + ) = P 2 + σ 1 nr σ 1 (P 1 + ) + P 2 + P 1 + nr P 1 + (P 1 + ) + P 2 + P 2 - nr P 2 -(P 1 + ) (11)
P 1 -(P 2 -) = P 1 - σ 1 nr σ 1 (P 2 -) + P 1 - P 1 + nr P 1 + (P 2 -) + P 1 - P 2 - nr P 2 -(P 2 -) (12)
P 2 + (P 2 -) = P 2 + σ 1 nr σ 1 (P 2 -) + P 2 + P 1
+ nr P 1 + (P 2 -) + P 2 + P 2 - nr P 2 -(P 2 -) (13)
The different quantities result from the forcing given in parenthesis, and the subscript nr indicates that the transfer functions are non-reflective.

IV. Results

The noise levels computed using the CAA approach described above are presented in this section. As the plane entropic wave injected into the domain is accelerated through the stator (recall the mean flow Fig 1) it is heavily deformed as illustrated in Fig 4 . It is both chopped by the blade row and turned with the flow. The acceleration and deformation of this wave is responsible for the generated entropy noise in the form of the pressure and velocity fluctuations presented in Fig 5 . As noted in section IIIB, the pressure fluctuations are one-dimensional throughout the domain, but the velocity fluctuations are not downstream of the blade because of a coupling with the vortical mode as the entropy wave is accelerated [START_REF] Duran | On the Reflection and Transmission of Circumferential Waves Through Nozzles[END_REF]. One may also note the colour contour is saturated in this region where the velocity fluctuations are larger than acoustic amplitudes. The resulting TATF [P 1 -/σ 1 ] nr and [P 2 + /σ 1 ] nr are presented in Fig. 6. The amplitude decreases sharply with frequency and there is a break in the curves around 500Hz after which the amplitude decreases very slowly. This evolution deserves further investigation. It is also worth noting the results for the upstream and downstream TATF are very close. The phase is presented in the range [-π; π] as it is too poorly discretised to be unwrapped. This will however be possible in the prospect of comparison with analytical results, as their low computational cost allows greater sampling. The upstream acoustic transfer functions [P 1 -/P 1 + ] nr and [P 2 + /P 1 + ] nr , and [P 2 + /P 2 -] nr and [P 2 + /P 2 -] nr resulting from downstream forcing are presented in Fig. 7 and Fig. 8 respectively. In addition, the compact solution, for which the dimension of the geometry is negligeable compared to the wavelength, is plotted for all three forcing cases. It was computed using the model developed by Cumpsty and Marble [START_REF] Cumpsty | The Interaction of Entropy Fluctuations with Turbine Blade Rows; A Mechanism of Turbojet Engine Noise[END_REF] [START_REF] Leyko | Simulation and Modelling of the Waves Transmission and Generation in a Stator Blade Row in a Combustion-Noise Framework[END_REF]. Although the CAA results could not be computed up to zero Hertz, they are in good accordance with the transfer functions resulting from the compact model for entropic forcing and both cases of acoustic forcing. Contrary to the entropic case, the amplitude of the acoustic transfer functions remains of the same order of magnitude as the compact solution over the range of frequencies considered, which can be explained by looking at the wavelengths involved. The smallest wavelengths (at 1000 Hz) upstream of the blade are between 0.3 and 0.4 m for the acoustic waves, whereas it is of only 0.04 m in the entropic case and the axial chord is 0.0306 m. The stator is therefore compact for both cases of acoustic forcing, whereas non-compactness affects the amplitude of the transfer functions significantly from about 200Hz.

V. Conclusions and prospects

A CAA reference case has been set up for the study of entropy noise in a 2D stator cascade. Noise levels were investigated for three forcing types: entropic, upstream acoustic and downstream acoustic. The resulting non-reflective transfer functions compare well with the compact solution [START_REF] Cumpsty | The Interaction of Entropy Fluctuations with Turbine Blade Rows; A Mechanism of Turbojet Engine Noise[END_REF]. The noise levels computed using a semi-analytical model currently under development will be compared with those presented in this paper. It follows the same main hypothesis as the model CHEOPS-Nozzle, in particular the assumption acoustic waves are one-dimensional, which has been verified in this paper. It is also based on the Euler equations, making a comparison with the present reference case all the more relevant. The 2D stator is studied in the context of a project aiming at developing a model for entropy noise through the turbine stages. In future studies, the model will be extended to the 3D stator and to the rotor. This will finally lead to a model for a full turbine stage, whose results could be compared to existing experimental data.

Fig. 1

 1 Fig. 1 Contour of the Mach number.

Fig. 2

 2 Fig. 2 Diagram of the waves propagating in the stator.

Fig. 3

 3 Fig. 3 Amplitude (left) and phase (right) of upstream (blue) and downstream (red) TATF obtained using harmonic and multi-harmonic simulations (data at frequencies not simulated with harmonic forcing is shown in grey).

Fig. 4

 4 Fig. 4 Contour of normalised entropy fluctuations σ for harmonic forcing at 1000 Hz computed using CAA.

Fig. 5

 5 Fig. 5 Contour of pressure fluctuations p' (left) and axial velocity fluctuations u' (right) for harmonic forcing at 1000 Hz computed using CAA.

Fig. 6

 6 Fig. 6 Amplitude (left) and phase (right) of upstream and downstream non-reflective TATF computed using CAA and in the compact case[2].

Fig. 7

 7 Fig. 7 Amplitude (left) and phase (right) of upstream and downstream non-reflective transfer functions resulting from the injection of a planar acoustic wave P + 1 upstream using CAA, and in the compact case[2].

Fig. 8

 8 Fig. 8 Amplitude (left) and phase (right) of upstream and downstream non-reflective transfer functions resulting from the injection of a planar acoustic wave P - 2 downstream using CAA, and in the compact case[2].


LES GRANDES TRANSFORMATIONS PAR LES OBSERVATIONS ET L'EXPÉRIMENTATION

Pauline Lecomte-Grosbras a , Jean-François Witz a , Annie Morch a a Univ. Lille, CNRS, Centrale Lille, UMR 9013 -LaMcube -Laboratoire de Mécanique, Multiphysique, Multi-échelle, F-59000 Lille, France ;

Mots-clés : Grandes déformations, mécanique des milieux continus, mesures de déformation

Introduction

Les grandes transformations ou transformations finies concernent les transformations entraînant des déplacements, déformations et/ou rotations qui ne permettent plus de se placer dans le cadre de l'hypothèse des petites perturbations ou des déformations infinitésimales. Dans ce cas, les configurations non-déformée et déformée du milieu continu sont significativement différentes. C'est généralement le cas dans les matériaux grandement déformables comme les élastomères ou tissus biologiques mous ou encore dans le cas de déformations plastiques dans les matériaux. Une transformation décrit l'essemble des déplacements, rotations et changements de formes (ou déformations) appliqués sur le matériau. Les grandes transformations peuvent donc être la conséquence de grands déplacements, grandes rotations et/ou changements de forme importants. Différentes combinaisons de ces changements sont possibles tels que petits déplacements et grandes déformations comme par exemple dans le cas de localisations de déformations ; grands déplacements et petites déformations, par exemple dans le cas de transformations rigides importantes avec de faibles déformations ; ou encore grands déplacements et grandes déformations. Ce cours s'articulera en deux parties. La première partie présentera les outils permettant de décrire les grandes transformations et la seconde partie se concentrera sur les aspects expérimentaux et en particulier sur les méthodes de mesures.

Description de la transformation

Cette section décrira les outils mathématiques permettant de décrire la transformation combinant déplacements, rotations et déformations dans les configurations initiale (Lagrangienne) et courante (Eulérienne) (Figure 1) [START_REF] Forest | Mécanique des milieux continus[END_REF][START_REF] Sidoroff | Les Grandes Déformations[END_REF][START_REF] Ogden | Non-linear elastic deformations[END_REF]. Cette section posera les bases pour la section suivante qui se focalisera sur les grandeurs accessibles expérimentalement.

FIGURE 1 -

 1 FIGURE 1 -Configuration initiale et courante


Alice Michel

Bénédicte Langlois

Simon Le Hello

François Gravey

Prevalence of mutations within RamAR operon associated with cross resistance phenotype among ESBL Enterobacteriaceae isolated in intensive care units

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

References:

1. Gravey F, Cattoir V, Ethuin Enterobacteriaceae producing ESBL (ESBL-E) are a major public health problem leading to an increase of morbi-mortality.

Enterobacteriaceae can adapt to their hosts and/or hospital environment through diverse mechanisms, in particular modification of efflux systems expression conferring resistance to a large panel of biocides.

AcrAB-TolC is a resistance-nodulation-division (RND) pump, which is in part regulated by a two-component system: RamAR operon.

Mutations in this operon drive to a pump over-expression leading to a cross-resistance (CR) phenotype regarding quinolones, tigecycline, chloramphenicol, temocillin as well as a higher tolerance to biliary salts and resistance to phagocytosis.

The aim of the study was to determine the prevalence of the mutations inside the RamAR operon leading to a CR phenotype among ESBL-E isolated and sequenced in a French Teaching hospital.

BACKGROUND:

In 2019 in our ICUs wards, approximatively 10% of the ESBL-expressing E.cloacae and K.pneumoniae, showed mutations in their ramR regulator, disabling this gene. This suggests that RamR could be an important vector of antimicrobial resistance.

CONCLUSION:

Throughout 2019, we studied the proportion of CR phenotype associated with a RamAR mutation among ESBL-E isolated from ICUs patients in a French University hospital (1,2) . ü All the first ESBL-E isolated among patients hospitalized in medical, surgical or pediatric ICUs during 2019 have been collected and their genome sequenced by Illumina technology. ü Cross-resistances to ciprofloxacin, tigecycline, temocillin and chloramphenicol were searched by broth microdilution using EUCAST breakpoints. ü For each strain which presented a CR phenotype to the four molecules, MICs have been determined in presence of 20mg/L PAbN to highlight involvement of a RND pump in the CR phenotype observed. ü At the same time RamAR operon mutations have been looked for all the sequenced strains. ü Finally phenotypical and genomical data have been put together in order to find all the relevant RamAR mutation regarding the CR phenotype.

MATERIAL & METHODS:

In total 165 BLSE-E have been included. Thirteen strains presented a CR phenotype of which (7/49) Enterobacter cloacae complex, (6/53) Klebsiella pneumoniae, whereas none of the 63 E.coli tested presented the expected phenotype. MICs slightly decreased with addition of PAbN suggesting an RND pumps deregulation (TABLE 1). For all strains, no resistance gene could explain the CMIs increased of CR phenotype, but nine of them presented mutation in ramR gene. Mutations such as nucleotides substitutions or deletions were specific to resistant strains and affected the helix-turn-helix site of the protein, which drive to a disability of RamR to bind and inhibit the ramA expression.

RESULTS:

  within RamAR operon associated with cross resistance phenotype among ESBL Enterobacteriaceae isolated in intensive care units A. Michel a , B. Langlois a , S. Le Hello a,b , F. Gravey a,b a DYNAMICURE Inserm 1311, UNICAEN-université de Rouen Normandie, b Department of Clinical Microbiology, Caen University Hospital, Caen, France 3. Bacterial susceptibility & resistance Poster Session 3b. Resistance surveillance & epidemiology: Gram-negatives