Antoni Alomar

Samir Beneddine

Denis Sipp

Aurélie Nicole

Aurelie Nicole

Wave-packets in a reacting, imperfectly-expanded supersonic jet

Keywords: Imperfectly-expanded jet, reacting jet, coherent structures, PSE

The present work focuses on the large coherent structures of a turbulent, reacting and imperfectly-expanded supersonic jet. These have been extracted from a Large Eddy Simulation (LES) including chemical reactions by Spectral Proper Orthogonal Decomposition (SPOD). While SPOD is now a standard tool for non-reactive flows, very few studies have dealt with reactive jets. The article shows that the pressure and axial velocity fields ex-hibit coherent wave-packets with a well-defined wavelength and amplitude envelope. The shock/ recompression cells have a strong effect on the amplitude of the wave-packet, but a weak effect on the wavelength. The temperature field exhibits a lower self-coherence and exhibits small scale structures upstream, generated within the mixing layer, and an irregular pattern of larger structures downstream in agreement with Prasad and Morris (J Sound Vib 476 115331 (2020)). A high-energy coherent structure has been observed at a Strouhal num-ber of St = 0.4 (normalized by the jet diameter and the jet exit velocity), which appears to result from a mutual reinforcement between the main shear layer instability and the vortex shedding from the normal shock. An attempt is made to model the wave-packets using the parabolized stability equations (PSE) about the mean flow, ignoring the chemical reactions. The PSE solution fails to model the temperature fluctuations, but shows rather good agree-ment with the leading SPOD modes of pressure and axial velocity downstream. This reveals that the flame impacts the pressure and velocity wave-packets in a weak manner: the coher-ent structures in the pressure and velocity field are generated by hydrodynamic convective instability, without being significantly altered by the reactive nature of the flow.

Introduction

High temperature, reacting supersonic jets are commonly used to propel space launchers, rockets and missiles. In order to optimize their performance and to minimize their environmental and acoustic impact, an understanding and modeling of these flows are crucial. It is therefore a major subject of research, as proved by the numerous articles dedicated to reactive jets published in the past few years (for instance Barzegar Gerdroodbary et al. [START_REF] Barzegar Gerdroodbary | Computational investigation of multi hydrogen jets at inclined supersonic flow[END_REF], Gerdroodbary et al. [START_REF] Gerdroodbary | The influence of upstream wavy surface on the mixing zone of the transverse hydrogen jet at supersonic free stream[END_REF], Li et al. [START_REF] Li | The influence of the sinusoidal shock generator on the mixing rate of multi hydrogen jets at supersonic flow[END_REF][START_REF] Li | Effect of inclined block on fuel mixing of multi hydrogen jets in scramjet engine[END_REF]).

Turbulent jets are noise-amplifier flows, in which the (linear) convective instabilities amplify upstream perturbations, creating a coherent unsteady field (Huerre and Monkewitz [START_REF] Huerre | Local and global instabilities in spatially developing flows[END_REF],

Sipp et al. [START_REF] Sipp | Dynamics and control of global instabilities in open-flows: A linearized approach[END_REF]). Non-reacting, subsonic and perfectly-expanded supersonic jets have been shown to contain energetically-dominant coherent structures (Lepicovsky et al. [START_REF] Lepicovsky | Coherent large-scale structures in high reynolds number supersonic jets[END_REF], Schmidt et al. [START_REF] Schmidt | Spectral analysis of jet turbulence[END_REF]). The temporal homogeneity and the approximate axial homogeneity lead to wavepackets, that is, spatio-temporal coherent structures in the form of waves propagating axially.

In unforced jets the coherent structures are generated continuously from random upstream perturbations, and all frequencies/wavelengths manifest simultaneously and randomly. Their extraction is more difficult than in forced jets, where phase-averaging can be used (Gaster et al. [START_REF] Gaster | Large-scale structures in a forced turbulent mixing layer[END_REF], Sinha et al. [START_REF] Sinha | Incorporating actuation effects in reduced-order models for feedback control of axisymmetric jets[END_REF]). Several methods have been successfully used in the past for this purpose: multi-point pressure and velocity correlations (Fuchs [11], Lau et al. [START_REF] Lau | The intrinsic structure of turbulent jets[END_REF]), triggerevent-based conditional averaging (Lau and Fisher [START_REF] Lau | The vortex-street structure of turbulent jets. part 1[END_REF]), linear stochastic estimation (Adrian [START_REF] Adrian | Turbulence in Liquids[END_REF][START_REF] Adrian | Eddy Structure Identification[END_REF]), or Proper-Orthogonal-Decomposition (POD) (Berkooz et al. [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF], Citriniti and George [START_REF] Citriniti | Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition[END_REF], Delville et al. [START_REF] Delville | Examination of large scale structures in a turbulent plane mixing layer. part 1: Proper orthogonal decomposition[END_REF], Davoust et al. [START_REF] Davoust | Dynamics of m=0 and m=1 modes of streamwise vortices in a turbulent axisymmetric mixing layer[END_REF]).

In both subsonic and supersonic jets the coherent structures were early associated to linear, local instabilities of the mixing layer (Gill [20], McLaughlin et al. [START_REF] Mclaughlin | Experiments on the instability waves in a supersonic jet and their acoustic radiation[END_REF], Tam and Hu [START_REF] Tam | On the three families of instability waves of high-speed jets[END_REF], Luo and Sandham [START_REF] Luo | Instability of vortical and acoustic modes in supersonic round jets[END_REF]). The parallel-flow stability equations around the mean flow applied to turbulent jets well recover the early stages of amplification of the instability waves (Suzuki and Colonius [START_REF] Suzuki | Instability waves in a subsonic round jet detected using a near-field phased microphone array[END_REF], Kerherve ´ et al. [START_REF] Kerhervé | Educing the source mechanism associated with downstream radiation in subsonic jets[END_REF]), but fail as soon as the streamwise profile becomes locally stable. If non-parallel effects are weak, Parabolized-Stability-Equations (PSE, Herbert [START_REF] Herbert | Parabolized stability equations[END_REF]) reproduce well this decaying part of the wavepacket. The PSE are a good approximation of the more expensive global resolvent approaches (Sipp et al. [START_REF] Sipp | Dynamics and control of global instabilities in open-flows: A linearized approach[END_REF], McKeon and Sharma [START_REF] Mckeon | A critical-layer framework for turbulent pipe flow[END_REF],

Semeraro et al. [START_REF] Semeraro | Modeling of coherent structures in a turbulent jet as global linear instability wavepackets: Theory and experiment[END_REF], Gomez et al. [START_REF] Gomez | A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator[END_REF], Beneddine et al. [START_REF] Beneddine | Conditions for validity of mean flow stability analysis[END_REF]), which studies the input-output properties of the flow. The PSE well describe the evolution of the wave-packets of subsonic and ideally-expanded supersonic jets, especially at low jet temperatures (Gudmundsson and Colonius [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF], Cavalieri et al. [START_REF] Cavalieri | Wavepackets in the velocity field of turbulent jets[END_REF]). For ideally-expanded supersonic jets the agreement has been more limited, especially for the decay rates, but the axial wavelength and radial structures of the wave-packets are certainly captured by the PSE (Rodriguez et al. [START_REF] Rodriguez | Inlet conditions for wave packet models in turbulent jets based on eigenmode decomposition of large-eddy simulation data[END_REF], Sinha et al. [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF]).

Recent theoretical studies have shed some new light on the link between resolvent analysis around the mean-flow and spectral POD (SPOD). McKeon and Sharma [START_REF] Mckeon | A critical-layer framework for turbulent pipe flow[END_REF], Beneddine et al. [START_REF] Beneddine | Conditions for validity of mean flow stability analysis[END_REF] have established a sufficient condition for a linear stability analysis around the mean flow to recover a dominant SPOD mode: the singular value of the leading resolvent mode must be much greater than the others. On the other hand, it has been established that SPOD modes are equivalent to resolvent modes when the expansion coefficients of the latter are uncorrelated (Towne et al. [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]). Rigas et al. [START_REF] Rigas | One way navier-stokes and resolvent analysis for modeling coherent structures in a supersonic turbulent jet[END_REF] have successfully used resolvent analysis to model the coherent structures of perfectly expanded supersonic jets.

Fewer studies have been published concerning the coherent structures in supersonic imperfectly expanded jets. Edgington-Mitchell et al. [START_REF] Edgington-Mitchell | Coherent structure and sound production in the helical mode of a screeching axisymmetric jet[END_REF] have recently studied experimentally screech tone generation in such a jet. They observed that while the non-coherent velocity component is only weakly affected by the shock/recompression cells, the coherent structures suffer a strong modulation by them. The modeling of the coherent structures in such jets is more challenging due to the higher non-parallelism, which may render necessary to consider the global stability equations Beneddine et al. [START_REF] Beneddine | Global stability analysis of under-expanded screeching jets[END_REF]. However, the non-parallelism diminishes downstream due to the weakening of the recompression cells, and the PSE may still be a good approximation in the downstream region.

Another family of jets which are still relatively unexplored concerning the coherent structures are reacting jets. Reacting jets are characterized by the existence of a number of chemical reactions within the flowfield, typically the combustion of a fuel by an oxidant, which generates a high temperature flame. While a significant number of works exist in the literature studying the impact of the coherent flow structures on the flame (Mungal et al. [START_REF] Mungal | The visible structure of turbulent jet diffusion flames: Large-scale organization and flame tip oscillation[END_REF], Yu and Schadow [START_REF] Yu | Role of large coherent structures in turbulent compressible mixing[END_REF], Schlimpert et al. [START_REF] Schlimpert | Hydrodynamic instability and shear layer effects in turbulent premixed combustion[END_REF]), the effect of the flame on the jet has been much less studied. Zhou and Hitt [START_REF] Zhou | Proper orthogonal decomposition analysis of coherent structures in simulated reacting buoyant jets[END_REF] used POD to extract the coherent structures of a re-acting, moderate Reynolds number subsonic jet. Obviously, the flame modifies strongly the mean and unsteady temperature field, but its effect on the velocity and pressure is usually thought to be limited. In fact, a common modeling approximation has been to neglect the impact of the flame on the flow (see Delhaye et al. [START_REF] Delhaye | Simulation and modeling of reactive shear layers[END_REF] for example). The linear PSE around the mean flow correctly describes a compressible reacting mixing layer in the case of a simple chemical model with infinitely-fast reactions (Day et al. [START_REF] Day | Nonlinear stability and structure of compressible reacting mixing layers[END_REF]).

One of the most important motivations for the study of wave-packets is their impact on noise radiation. Even at high Reynolds numbers, where the energy of the large coherent structures is a limited portion of the total fluctuating energy of the jet (Gudmundsson and Colonius [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF]), they are responsible for most of the radiated noise at low polar angles, in both subsonic (Jordan and Colonius [START_REF] Jordan | Wave packets and turbulent jet noise[END_REF], Brouzet et al. [START_REF] Brouzet | Role of coherent structures in turbulent premixed flame acoustics[END_REF]) and supersonic jets (Tam [47], Tam et al. [START_REF] Tam | The sources of jet noise: experimental evidence[END_REF], Sinha et al. [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF]). There is indirect evidence that large coherent structures exist also in imperfectly-expanded supersonic jets based on their radiated sound (Tam et al. [START_REF] Tam | On the two components of turbulent mixing noise from supersonic jets[END_REF][START_REF] Tam | The sources of jet noise: experimental evidence[END_REF]).

Recently, Shen et al. [START_REF] Shen | Extraction of large-scale coherent structures from large eddy simulation of supersonic jets for shock-associated noise prediction[END_REF] have demonstrated the impact of large coherent structures on the broadband shock associated noise generated by an imperfectly-expanded supersonic jet.

The present study has two main goals. First, showing that a complex jet, namely a turbulent, reacting and imperfectly-expanded supersonic jet, contains energetically-dominant wave-packets. The present work is based on a reactive LES simulation that was extensively described in Alomar et al. [START_REF] Alomar | Reduced order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition[END_REF]. In particular, the classical Smagorinsky subgrid model (Smagorinsky [START_REF] Smagorinsky | General circulation experiments with the primitive equations: I. the basic experiment[END_REF]) has been used with Arrhenius chemical equations. In Alomar et al. [START_REF] Alomar | Reduced order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition[END_REF], the spatio-temporal structures of the flowfield have been analysed with a classical space-only POD methodology, combined with a temporal reduction of the leading sets of chronos via an energybased selection of Fourier modes. In the present article, we will perform an SPOD analysis of this same data-set and extract for each frequency and azimuthal wavenumber the coherent structures of the fluctuation field. The second goal is the modeling of these wave-packets by a weakly-non-parallel approach of the dynamics, namely by a PSE model in which the perturbation equations for the chemical reactions are ignored. Such an analysis takes into account only partly the effect of the flame on the fluctuation dynamics: the increased temperature is taken into account but not the effect of the chemical fluctuations. The comparison between the SPOD and PSE analyses will therefore provide new information regarding the flame effect on the jet dynamics: if they display a good agreement, the reactive nature of the flow has little effect on the corresponding wavepackets since they are correctly recovered when neglecting the chemical reactions. This study can be seen as a first step prior to more complete studies, taking into account non-parallel effects and chemical reactions.

The outline of the article is the following. First ( §2) the flowfield obtained from the LES simulation is presented, including the main features of the mean flow and unsteady content.

The sampled dataset and the SPOD algorithm used to obtain the coherent structures are then described in §3. In §4 the coherent structures of the pressure, axial velocity and temperature are described and discussed. Finally, in §5 the PSE algorithm is described and its predictions are compared against the coherent structures obtained from SPOD, with attention to the impact of the mean flow non-parallelism on the accuracy of the PSE.

Flowfield

Simulation

The simulated jet mimics the existing experimental 'MASCOTTE' rig at ONERA (Grisch et al. [START_REF] Grisch | Cars measurements at high pressure in a ch4/o2 jet flame[END_REF]). In this particular case, the oxidant oxygen (gas) and the fuel methane (gas) react in the combustion chamber to generate a temperature of 2400 K, and a pressure of 8.5P ∞ . The pressure, temperature, and the mass fractions are approximately uniform at the outlet section of the combustion chamber and correspond to those resulting from a complete reaction. These conditions are taken as the inlet stagnation conditions for the LES simulation. All species are treated as ideal gases, with heat capacities varying with temperature.

An axial slice of the flowfield geometry is shown in Fig. 1. The flow domain is axisymmetric. A convergent-divergent nozzle links the combustion chamber to the ambient. The nozzle has an area expansion ratio of 2.37. The far field lateral boundary consists of a prescribed inlet velocity, equal to a low velocity parallel to the jet axis (U ∞ = 0.015c ∞ ). Adiabatic no slip conditions are applied to all solid-wall boundaries (thick solid line in Fig. 1). The ambient pressure is imposed at the downstream boundary.

The multi-physics in-house code CEDRE Refloch et al. [START_REF] Refloch | Cedre software[END_REF] has been used to simulate the flow. This code has been previously validated in the case of hot supersonic (Langenais et al. [START_REF] Langenais | Numerical investigation of the noise generated by a rocket engine at lift-off conditions using a two-way coupled cfd-caa method[END_REF]) and reacting supersonic jets (Gueyffier et al. [START_REF] Gueyffier | Numerical simulation of ionized rocket plumes[END_REF], Rialland et al. [START_REF] Rialland | Numerical simulation of ionized rocket plumes[END_REF], Guy et al. [START_REF] Guy | Ionized solid propellant rocket exhaust llume: Miles simulation and comparison to experiment[END_REF]).

It uses a second order finite volume scheme in space, and a second order implicit Runge-Kutta scheme in time. An unstructured, isotropic tetrahedral mesh containing 32 million cells has been generated using the meshing module of CEDRE. It is refined at the nozzle walls and progressively coarsens downstream and in the radial direction away from the jet lip line, in order to avoid boundary reflection. At the nozzle lip the cell size is 0.01D (D is the nozzle exit diameter). Along the jet lip line, the cell size increases from 0.01D at the nozzle exit to 0.03D at x/D = 5, and to 0.06D at x/D = 12. The LES mesh quality index Celik et al. [START_REF] Celik | Index of resolution quality for large eddy simulations[END_REF] is higher than 75% in the mixing layers and in the self-similar region, which indicates an adequate resolution. The classical Smagorinsky model for the sub-grid scales has been used, with model constants C = 0.1, C µ = 0.09. The use of a one-equation sub-grid scale model shows only marginal differences to the classical Smagorinsky model in turbulent under-expanded jets with multiple species Li et al. [START_REF] Li | Large-eddy simulation of time evolution and instability of highly underexpanded sonic jets[END_REF].

All chemical species are treated as ideal gases with heat capacities varying with temperature, and forming a perfect mixture in local thermal equilibrium. For each species k:

P k = ρ k R k T, P k = X k P, Y k = µ k X k /( µ l X l ).
For the mixture:

P k = P, X k = Y k = 1, P = ρ k R k T = ρRT (R k
and R are the species and the mixture specific gas constant, respectively, X k is the molar fraction, Y k is the mass fraction, and µ k is the molar mass).

The reaction rates are described through the Arrhenius model.

A re-combustion flame spontaneously appears and grows along the shear layer, generated by the interaction of the hydrogen and the carbon monoxide ejected by the jet with the oxygen in the ambient. This turbulent diffusion flame should strictly involve a chemistry-subgrid scale model coupling (Gao and OBrien [START_REF] Gao | A large-eddy simulation scheme for turbulent reacting flows[END_REF], Jimenez et al. [START_REF] Jimenez | A priori testing of subgrid models for chemically reacting non-premixed turbulent shear flows[END_REF], DesJardin and Frankel [START_REF] Desjardin | Large eddy simulation of a non-premixed reacting jet: application and assessment of subgrid-scale combustion models[END_REF]).

As a first approximation, it has been assumed that the chemical reactions and the turbulence subgrid-scale model are uncoupled. This implies that the reaction occurs as if the flow was locally laminar. This might not be accurate everywhere in the flow domain, but is deemed a reasonable approximation, especially since our focus is not the flame itself.

The simulation was initialized from a steady RANS solution. The simulation time step is 10 -7 s, which was set by the chemical processes. A total of 77 runs of 2500 time steps per run were performed, corresponding to a simulation time of 0.01925 s. After an initial transient of 0.008 s, the sampling and the calculation of the mean flow carried on during 0.01125 s, corresponding to 1750D/U J .

Based on the conditions at the section x/D = 0.2, where the mean flow is approximately uniform, the jet is weakly over-expanded, with an exit pressure of 0.74 times the ambient pressure, an axial velocity of 2290 m/s, a temperature of 1450 K and a Mach number of 2.15.

The exit Reynolds number based on ambient conditions is 3.6 • 10 6 .

The current study is focused on the axial velocity, temperature and pressure. Pressure and velocity are commonly used flow variables to study the coherent structures. In the present hot reacting jet case, there is also interest in the temperature field, as it is presumably most affected by the flame. In a previous study of the same jet, the chemical mass fractions have been observed to be strongly and globally correlated with the temperature field (Alomar et al. [START_REF] Alomar | Reduced order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition[END_REF]). Therefore, their coherent structures are expected to be similar to the temperature. The velocity, temperature and pressure have been normalized with the conditions immediately upstream of the normal shock, at x/D = 0.85: U J = 2750 m/s, T J = 953 K and P J = 6404 Pa.

There the velocity and the pressure attain their maximum and minimum values, respectively.

The exit velocity and the diameter are used to normalize the frequency as a Strouhal number,

Mean flow

A Reynolds decomposition of the flow variables into mean and perturbation components is applied:

g(x, r, θ, t) = g 0 (x, r) + g ′ (x, r, θ, t). (1)
The mean flowfields of the axial velocity, radial velocity, temperature and pressure are shown in Figs. 3a,b,c,d, respectively. Imperfectly expanded supersonic jets contain a series of shock/ recompression cells along the axis, whose strength decays downstream. This structure is clearly observed in Fig. 3. There is a normal shock within the first cell, located at x ≈ 0.9D.

The normal shock generates a re-circulation region, where the axial velocity reaches a negative value. In contrast to other flow variables, the temperature increases downstream. This is due to the re-combustion flame that appears first within the mixing This parameter may take order one values outside of the core of the jet since U 0 decreases to small values there. Comparing these profiles with the profiles of the axial velocity and temperature, a relationship can be inferred between V 0 /U 0 and the width of the mean temperature field. The values corresponding to the radial distance r * such that

T 0 (r * ) = 1.3T ∞ , i.
e. an indicator of the width of the mean temperature field, are highlighted (filled circles). This issue is further discussed in Section 1 of the Supplementary material.

Frequency and azimuthal content

The point frequency spectrum (in energy per unit area) of axial velocity measured by a sensor within the mixing layer, at x/D = 5, is shown in Fig. 5. Similar trends are found for the other flow variables. The most energetic portion of the spectrum is within The vertical green lines show the shock and recompression cells locations.

0.07 < St < 0.4, with a peak around St = 0.2. This value is close to the far field noise peak frequency measured at shallow angles of subsonic (Cavalieri et al. [START_REF] Cavalieri | Wavepackets in the velocity field of turbulent jets[END_REF]) and ideally-expanded supersonic jets (Sinha et al. [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF]).

A series of sensors located at a fixed distance from the jet axis and distributed uniformly in the azimuthal direction has been used to determine the azimuthal content of the fluctuations. The one-sided spectra are shown in Figs. 6a (pressure), 6b (axial velocity) and 6c (temperature), at various axial locations. The pressure shows a monotonous decay as a function of the azimuthal wavenumber m which becomes steeper for increasing x. The temperature exhibits a similar decaying profile, but less steep than the pressure. In the case of the velocity, on the other hand, m = 0 fluctuations dominate only within x < 4D. For 4D < x < 12D, modes m = 1, 2 and 3 exhibit higher fluctuation levels than m = 0 perturbations. The contribution of the leading azimuthal modes to the amplitude is presented in Figs. 7a (pressure), 7b (axial velocity) and 7c (temperature). For the pressure and the axial velocity, the impact of the shock/recompression cells is highest on m = 0. In this case a clear modulation along the jet by the shock/recompression cells is observed. In the case of the temperature, the levels of the three azimuthal modes are close to each other and strongly modulated within x < 8D. In the following we restrict the analysis to m = 0, 1, 2.

Dataset from the simulation

Snapshots

The flowfield database has been obtained from directly sampling the simulated field, leading to a set of three-dimensional snapshots. Since no temporal pre-filtering of the simulated field has been applied, the sampling frequency of the snapshots needs to be high enough to maintain aliasing low. At the same time the total number of extracted snapshots is limited by simulation time and data size constraints. In the present case, a sampling frequency of

St samp = f samp D/U J = 1.
12 has been deemed a good compromise. It has been checked that the energy associated to higher unresolved frequencies is less than 7% of the total fluctuating energy in the region of interest, and aliasing is thus weak. Note also that the spectrum shifts to lower frequencies downstream, diminishing the aliasing even further. The dataset contains S = 1792 uniformly distributed snapshots which span a total time of 1602D/U J .

The spatial region of the three-dimensional snapshots is cylindrical: 0.5D < x < 13D, r < 6D. Along the axis the snapshots contain 135 cells, whose size varies smoothly (logarithmic growth) from 0.0375D to 0.545D. Then the cell size increases linearly to 12D. Beyond 12D, the cell size remains constant. In the radial direction there are 50 cells, and the cell size decreases from 0.065D at the axis to 0.02D at the nozzle lip line, and then grows until 0.365D, at the outer cells. In azimuth, the grid contains N θ = 36 cells uniformly distributed. The snapshots contain N x × N r × N θ = 135 × 50 × 36 = 243000 cells.

SPOD of the snapshots

The azimuthal periodicity of the flowfield allows to decouple the problem into a numerable set of azimuthal modes :

ĝm (x, r, t) = N θ i=1 g ′ (x, r, θ i , t)exp(-jmθ i ), m = -N θ /2, ..., N θ /2. ( 2
)
The cross-spectral density Ŝm,f is defined as

Ŝm,f (x, x ′ , r, r ′ ) = E[ĝ m,f (x, r)ĝ m,f (x ′ , r ′ ) * ], (3)
where ĝm,f corresponds to the Fourier transform of ĝm (x, r, t) at frequency f , E[ ] is the ensemble average and * the complex conjugate. The SPOD modes φm,f are the eigenvectors of this tensor:

Nx p=1 Nr q=1 Ŝm,f (x, x ′ p , r, r ′ q )Q(x ′ p , r ′ q ) φm,f (x ′ p , r ′ q ) = λ m,f φm,f (x, r), (4)
where λ m,f are the SPOD energies and Q(x, r) is a given scalar product. In this study, a standard L 2 scalar product is considered. It accounts for both the cell size quadrature, the cylindrical coordinate system and the choice of the flow variable of interest, which are for the present study the pressure, the axial velocity or the temperature. For instance, for an azimuthaly Fourier transformed state vector containing these three variables g ˆ = (p ˆ, u ˆ, T ˆ)T and if the temperature is the variable of interest, Q(x, r) reads

Q(x, r) =      0 0 0 0 0 0 0 0 rΩ      , (5)
with Ω the local mesh cell volume, such that g ˆT Qg ˆ = T ˆ .

The eigenvalue problem ( 4) is solved in practice in the following steps. First, the scalar product Q is integrated into the cross-spectral tensor by introducing a scaled version of g ˆm, g ˜m:

gm (x, r, t) = ĝm (x, r, t)Q(x, r) 1/2 . ( 6
)
Then the matrices gm are divided into N b temporal bins of length l b = S/N b , and a windowed Fourier transform is performed on each bin:

gb m,f (x, r) = l b k=1 gb m (x, r, t k )w(t k )exp(-j2πf t k ), (7)
f = f samp /l b , ..., f samp /2; b = 1, ..., N b ,
where w(t k ) denotes the Hann window. The scaled cross-spectral tensor is then the average

E[g m,f (x, r)g m,f (x ′ , r ′ ) * ] = 1 N b N b b=1 gb m,f (x, r)g b m,f (x ′ , r ′ ) * . (8)
The following scaled eigenvalue problem is then solved using the PYTHON functions scipy.linalg.eigh and scipy.linalg.eigvalsh (note the use of optimized routines for Hermitian matrices):

Nx p=1 Nr q=1 Sm,f (x, x ′ p , r, r ′ q ) φm,f (x ′ p , r ′ q ) = λ m,f φm,f (x, r), (9)
whose solutions are the numerable set of eigenvalues and eigenmodes λ n m,f , φn m,f . The energy fraction of the nth SPOD mode is λ n m,f / n λ n m,f . The SPOD mode φm,f is then recovered by re-scaling back:

φm,f (x, r) ≈ λ 1 m,f φ1 m,f (x, r) Q(x, r) 1/2 . ( 10
)
The length of the temporal bins results from a trade-off between the frequency resolution and the convergence of the ensemble average. Here, the total number of snapshots used is 1792, corresponding to a total time of 1602D/U J . The set of snapshots has been divided into 24 bins of 64 snapshots. An overlap of 50% between the bins has been used, leading to 47 effective bins.

The chosen scalar product Q(x, r) accounts for both the numerical integration over cells of non-uniform volumes and the choice of flow variables. While the spatial grid is always the same, three different scalar products have been considered, corresponding to the pressure, the axial velocity and the temperature. Considering one variable at a time leads in general to a higher energy decay rate, and thus to a better focus on the coherent structures.

Finally, the authors would like to comment on the relationship between SPOD and classical (space-only) POD, which has been recently addressed in detail by Towne et al. [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]. The fundamental difference lies in the fact that in SPOD stationarity is imposed explicitly through a harmonic temporal dependence. In classical POD (or the equivalent snapshots POD) the chronos are only orthogonal functions of time, and cannot in general be associated to single frequencies. Since chronos in classical POD belong to a larger set of orthogonal functions, they diminish the error of the low-rank approximations for the ensemble of frequencies. For example, in the case of traveling wave-packets of similar energy, but different frequencies and wavelengths, it is possible that classical POD identifies them with modes, but flips eventually from one mode to the other if the resulting global POD mode minimizes the overall low-rank reconstruction error (this has been observed by Alomar et al. [START_REF] Alomar | Reduced order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition[END_REF]). In general, classical POD is appropriate for optimizing the low-rank reconstruction of a particular dataset, but it is less appropriate to identify wave-packets.

Wave-packets

In this section, the coherent structures of the jet obtained from the SPOD analysis are discussed, with focus on the overall spatial patterns, the amplitude and the wavelength. pressure and the axial velocity, which is close to the values obtained in previous studies for non-reacting perfectly-expanded jets (Rodriguez et al. [START_REF] Rodriguez | Acoustic field associated with parabolized stability equation models in turbulent jets[END_REF]). The axial velocity shows the highest overall self-coherence. This may seem unexpected since pressure is an integrated quantity depending on the surrounding velocity field. Therefore, this point may need further investigations in future work. Except at St = 0.4, the self-coherence is lower for the temperature, with the leading SPOD mode containing a maximum of 20% of the total energy. This Fig. 9), an overall decrease of the energy fraction of the leading SPOD mode is observed, especially for the axial velocity. This is evidence that the coherent structures are mostly dominant at low frequencies. A peak of the energy fraction of the leading SPOD mode is observed at St = 0.4 and m = 1 for all variables, and is especially strong for the temperature.

While for the pressure and the axial velocity, the energy fraction of this SPOD mode is comparable to the SPOD modes at lower frequencies, it is by far the SPOD mode with the largest energy fraction for the temperature.

Spatial structure

Figures 10 and 11 illustrate the spatial structure of the first and second SPOD modes for each of the three flow variables, all corresponding to m = 1. Note that the apparent pixelation in these plots is actually due to the spatial sampling and not due to plotting. In the case St = 0.2, a wave-like pattern is clearly seen, especially for the pressure and the axial velocity. These patterns exhibit a well-defined axial wavelength, a growth, saturation and decay in the downstream direction. The coherent structure is associated to the mean shear layer and peaks relatively downstream of the jet, where the flame is already thick. This coherent mode may therefore be presumably affected by the flame, although a clear impact of the flame is not observed on the pressure and axial velocity modes. This is in agreement with Prasad and Morris [START_REF] Prasad | A study of noise reduction mechanisms of jets with fluid inserts[END_REF]. This paper uses Doaks Momentum Potential Theory which separates the hydrodynamic, acoustic and thermal modes (Doak [START_REF] Doak | Analysis of internally generated sound in continuous materials: 3. the momentum potential field description of fluctuating fluid motion as a basis for a unified theory of internally generated sound[END_REF]) and shows that hydrodynamic modes are confined to the shear layer of the jet and the thermal modes have coherent structure restricted to the turbulent region and are not radiating. In this case, for the temperature, both SPOD modes reveal small scale structures appearing well upstream and developing along the mixing layer. As opposed to the wave-packets of the pressure and velocity, the characteristic length of these structures corresponds to the thickness of the mixing layer, implying that they are associated to the mixing of the jet gases with the ambient. They are also probably affected by the flame that grows in that same region. Actually, these small structures propagate downstream at almost vanishing speed. They start to grow immediately downstream of the crossing point between the oblique shock and the mixing layer (see Fig. 3), and they appear as bursts that propagate away from the jet in the radial direction. Larger structures emerge downstream, but a clear dominant wavelength cannot be discerned. This is in agreement with the low self-coherence of the temperature field, and indicates that the wave-packets are much weaker than for the pressure and the axial velocity.

In the case St = 0.4, for which a high coherence was observed in the dominant energy fraction, a highly coherent wave-packet is observed in all flow variables, including the temperature. Its peak is located in the upstream region. The pressure shows a strong coupling with the radiated noise. The dominant frequency of the wave-packet approximately coincides with that expected by vortex shedding in the recirculation region behind the normal shock:

St = f D ns U J = f D U J D ns D ≈ 0.4 × 0.5 = 0.2, ( 11
)
where D ns is the diameter of the normal shock. The normal shock acts as a hemispherical bluff body, with a detachment and formation of an axisymmetric shear layer (see Figs. At the same time, the axial wavelength of this mode can be accounted for by a simple shear layer instability model for which the phase speed is equal to the mean velocity, here U J /2: λ/D = λf /(U J St) ≈ 1/(2St) = 1.2, which is consistent with the observations in Figure 11. The features of this coherent mode are therefore compatible with tonal instability associated to bluff-body vortex shedding. This leads to a remarkably high energy and coherence.

The amplitudes of the leading SPOD modes for each of the flow variables are plotted in Figs.

12, 14 and 16. These graphs also include the PSE predictions, which will be introduced and discussed in section 5, and can be ignored for the moment. The amplification, saturation and decay of the coherent structures is observed for all variables, though less clear for the temperature. In the pressure and axial velocity, we can see oscillations of the amplitude along the jet, of wavelengths which are globally compatible with a shock/recompression cell modulation. Note however that the recompression waves become weaker as we go further downstream (see figure 3), which may explain a less obvious modulation downstream. The axial velocity shows more intense amplitude oscillations than the pressure, and thus a stronger interaction with the shock and recompression cells. The second SPOD modes differ from the leading SPOD mode in general. On the other hand, the amplitude envelopes of the temperature are more irregular and contain a larger noise component. The lower convergence of the temperature is explained by its low self-coherence. Much longer sampling times would be required in this case to achieve the same level of convergence of the SPOD modes than the other flow variables.

Figures 13, 15 and 17 show the phases of the leading SPOD modes along the jet lip line (again, the PSE prediction is included and can be ignored for the moment). Note that SPOD modes are defined up to a complex multiplicative constant. Therefore, to ease the analysis Table 1: Phase speeds normalized with U J along the line r = 0.65D. of the results, we have shifted all phases so that they match at about x/D = 4, since we are more interested in the downstream region, where the non-parallelism is weaker. The phase of both leading SPOD modes reveal an approximately constant growth rate all along the jet for the pressure and the axial velocity, highlighting a single well defined wavelength and single wave-speed. In general, the phases of the leading and second SPOD modes of pressure and axial velocity have similar growth rates, and therefore similar wavenumbers and wave-speeds. In the case of the temperature, the growth rate of the leading SPOD modes is more irregular than for the pressure and axial velocity. But the trends are still compatible with wave-packets. Overall, it appears that the wavenumber of the wave-packet as described by the leading SPOD mode remains remarkably constant as x increases, while its amplitude may be prone to more or less severe oscillations, linked to the interaction with the shock/recompression cells.

St = 0.1 St = 0.2 St = 0.3 St = 0.4 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2
The normalized phase speeds from the leading SPOD modes, defined as u ph /U J = ω/(kU J ) = 2πSt/(kD), are shown in Table 1. As a general trend, the phase speed decreases with m and slightly increases with frequency. For the pressure and the velocity, they are close to 0.5, especially for m = 0. The phase speeds of the temperature, on the other hand, only approach 0.5 for St = 0.4 and m = 1. This agrees with the previous observation that only in this case the SPOD modes of the temperature are compatible with well-defined wave-packets.

In Fig. 18 we have compared the phase evolution of the two leading m = 1 SPOD modes along r = 0.65D for the three variables, in the case (a) St = 0.2 and (b) St = 0.4. At St = 0.2, it clearly appears that the wave-packet is here composed of two distinct components: in the upstream part of the domain (x/D < 4), the temperature exhibits small wavelength and low phase speed oscillations linked to the flame, while further downstream (x/D > 4), it exhibits longer wavelength oscillations. These low phase speed structures associated to the flame have been detected experimentally by Vorontsov et al. [START_REF] Vorontsov | Investigation of the structure of a diffusion hydrogen plume in a supersonic high-enthalpy air jet[END_REF]. At St = 0.4 all variables exhibit the same wavelength associated to the highly coherent structure that dominates the flow at this frequency and azimuthal number.

The main conclusion of the previous analysis is that, as opposed to the pressure and axial velocity, which are dominated by well-defined wave-packets for all frequencies and azimuthal numbers considered, the temperature is only driven by wave-packets for St = 0.4 and m = 1, 2. In this case all flow variables reveal a strong wave-packet associated, presumably, to a reinforcement between the tonal vortex shedding from the normal shock and the main shear layer instability.

Wave-packet modeling using linear PSE

In this section an attempt is made at modeling the wave-packets using the Parabolized Stability Equations (PSE) around the mean flow [START_REF] Herbert | Parabolized stability equations[END_REF]. After a presentation of the PSE approach ( §5.1), we compare the PSE and SPOD modes to assess the validity of the simplified PSE approach to reconstruct the SPOD modes of the jet ( §5.2).

An assessment of the non-parallelism of the mean-flow with respect in particular to the presence of shock/recompression cells is included in the Supplementary material (Section 1).

The robustness of the PSE solution to the initial condition has been also addressed in the supplementary material (Section 2)

PSE

We consider here the compressible Navier-Stokes equations and neglect the chemical reaction terms. The state vector is defined as: q = (ρ, u, v, w, T ) (where u, v, w are the axial, radial and azimuthal velocity components). The mass, momentum and energy conservation equations are made non-dimensional using the reference values: D, U J , ρ J , T J , R g = R air , c p = γR g /(γ -1), µ 0 and κ 0 . The ideal gas law, p = ρR g T , is used to eliminate the pressure:

∂ρ ∂t + u • ∇ρ + ρ∇ • u = 0, ρ ∂u ∂t + ρu • ∇u + 1 γM 2 (ρ∇T + T ∇ρ) = 1 Re (µ∇ • s + ∇µ • s), ρ ∂T ∂t + ρu • ∇T + (γ -1)ρT ∇ • u = γ ReP r ∇ • (κ∇T ) + M 2 γ(γ -1) Re (µs : ∇u),
where the deformation tensor is:

s = ∇u + (∇u) T - 2 3 (∇ • u)I, (12)
and Re = ρ J U J D/µ J , M = U J / γR g T J , P r = c p /(µ 0 κ 0 ), γ = c p /c v . The notations '•' and ':' refer to the simple and double contract product operators; '∇' and '∇•' are the gradient and divergence operators. While the gas is assumed ideal and calorically perfect (constant R g , c p and c v ), the viscosity and thermal expansion coefficients, µ and κ, are modeled by Sutherland's law:

µ(T ) = T 3/2 1 + C s /T 0 T + C s /T 0 , (13)
where Sutherland's viscosity, temperature and constant are µ s = 1.711 × 10 5 Pa s, T s = 273.15 K, C s = 110.4 K, respectively. A Sutherland law is also used for the thermal conductivity, whose non-dimensional form is analogous to the viscosity.

The linearized PSE equations are obtained as follows:

1. Decompose all variables including the viscosity and thermal conductivity in mean and fluctuating quantities q(x, r, θ, t) = q 0 (x, r) + q ′ (x, r, θ, t), introduce this decomposition into the Navier-Stokes equations and keep only linear terms in the perturbations and terms of order 1/Re.

2. Consider a perturbation field of the form:

q(x, r, θ, t) = Âm,f (x, r)exp(j x α(ψ)dψ -jmθ -j2πf t), (14)
and assume the conditions of weak non-parallelism:

V 0 , ∂ ∂x ∼ 1 Re , (15)
3. Consider the following additional normalizing condition:

∂ ∂x ∞ 0 Â * m,f Âm,f rdr = 2 ∞ 0 Â * m,f ∂ Âm,f ∂x rdr = 0. ( 16
)
This condition removes the exponential dependence on x of the shape function Âm,f (x, r),

effectively separating the short wavelengths associated to the local instabilities (contained within the exponential in Eq. 14) from the slow axial variation of the mean flow (contained within the amplitude envelope). Here, all five flow variables have been included in the energy norm (normalized with their reference values), as this led to the best numerical convergence.

The differential system can be recast in the compact form:

A∂ x Âm,f = B Âm,f + C ∂ Âm,f ∂r + D ∂ Âm,f ∂r 2 . ( 17
)
which leads to:

∂ x Âm,f = (A -1 B + A -1 C ∂ ∂r + A -1 D ∂ ∂r 2 ) Âm,f = L m,f Âm,f . (18)
The resulting system, together with the condition

∞ 0 Â * m,f ∂ Âm,f ∂x rdr = 0 ( 19
)
is approximately parabolic in nature (Malik and Chang [START_REF] Malik | Nonparallel and nonlinear stability of supersonic jet flow[END_REF]), and can be solved advancing along x from an inlet condition upstream:

Âm,f (x + ∆x, r) -Âm,f (x, r) = L m,f Âm,f (x + ∆x, r). (20)
The final solution to be compared with the SPOD modes is: qm,f (x, r) = Âm,f (x, r)exp(j

x α(ψ)dψ). (21)
In order to prevent the singularity at r = 0, both sides of Eq. ( 17) are multiplied by U 0 r 2 .

The matrices A, B, C, D (multiplied by U 0 r 2 ) can be found in §Appendix A. A non-reflective boundary condition in the far field is applied through the LODI relations (Poinsot and Lele [START_REF] Poinsot | Boundary conditions for direct simulations of viscous compressible flows[END_REF]). On the jet axis, the following kinematic conditions have been imposed (Batchelor and

Gill [START_REF] Batchelor | Analysis of the stability of axisymmetric jets[END_REF]):

ûm,f (r = 0) = pm,f (r = 0) = 0, for m > 0, vm,f (r = 0) = ŵm,f (r = 0) = 0, for m = 1, vm,f (r = 0) = -ŵm,f (r = 0), for m = 1.

The chemical reactions are manifested through a heat source term in the temperature equation. Although this term does not appear in the final linearized equations, the effect of the flame is indirectly accounted for in the mean flow around which the equations are linearized, since we consider for that term the actual mean temperature field from the LES simulation (similarly to a linear stability analysis capturing successfully truly nonlinear features of the dynamics by considering solely linear interactions around the mean-flow, see

Hammond and Redekopp [START_REF] Hammond | Global dynamics of symmetric and asymmetric wakes[END_REF], Pier [START_REF] Pier | On the frequency selection of finite-amplitude vortex shedding in the cylinder wake[END_REF], Barkley [76], Sipp and Lebedev [START_REF] Sipp | Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows[END_REF]). One of the goals of the present study is to assess the validity of that approximation.

Comparison of leading SPOD modes with the PSE solution

The projection coefficient of the PSE solution on the two leading SPOD modes has been considered in order to quantify their overall match. The coefficient related to the ith SPOD mode is:

Λ i m,f = xf x0 ∞ 0 qm,f (φ i m,f ) * dxrdr xf x0 ∞ 0 qm,f (q m,f ) * dxrdr xf x0 ∞ 0 φ i m,f (φ i m,f ) * dxrdr . ( 22
)
Tables 2, 3 and 4 contain the projections for each of the flow variables, together with the energy fractions of the SPOD modes.

We first focus on the three lower frequency cases. For the pressure, the PSE solution matches well the leading SPOD mode for all m (Λ 1 larger than 0.6). For the axial velocity, the match is less accurate overall, but still, Λ 1 is larger than Λ 2 in all cases. In general, when Λ 1 and Λ 2 are close to each other, the energy fractions of the first and second SPOD modes are also close. It is thus possible that the linear model may represent a mix of the two dominant modes. In the case of temperature, the mismatch between the PSE and the leading SPOD mode is the rule in general. Furthermore, in a number of cases, the PSE solution is closer to the second SPOD mode than the leading SPOD mode. An alignment of the PSE solution with the second SPOD mode instead of the leading SPOD mode has been previously observed experimentally in subsonic moderately heated jets Gudmundsson and Colonius [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF]. These results indicate that the effect of the flame may not be neglected for the modeling of the temperature fluctuations.

At St = 0.4, for which a coherent structure resulting from the interaction between the vortex shedding behind the normal shock and the main shear instability is dominant, the situation is more complex and erratic, even in cases where the dominant SPOD mode is much more energetic than the second one (see for example pressure and axial velocity at m = 1 or temperature at m = 1 and m = 2). Possible reasons for this failure are analysed here. A consistent growth of the amplitude of this coherent structure is limited to the region x < 2D, followed by a series of strong amplitude oscillations presumably due to the shock/recompression cells, and a decay downstream. This is not a favorable scenario for the PSE approach, as it requires not only an initial stage of amplification, but also that the mean flow is approximately parallel there. This latter condition is clearly not satisfied here as the non-parallelism is high within x < 2D. Furthermore, the oscillations due to the shock/recompression cells suggest that this coherent structure is highly sensitive to these cells.

There are several other and more general reasons that may explain why a PSE approach may not recover a leading SPOD mode. First, we recall that we have neglected the effect of the fluctuations of the chemistry variables. This may have a large impact especially in the case of the temperature SPOD modes. Second, in Beneddine et al. [START_REF] Beneddine | Conditions for validity of mean flow stability analysis[END_REF], it was argued that the PSE-SPOD match should actually hold if the dominant singular value of the resolvent operator is much larger than the second one. Yet, the PSE approach only provides an approximation of the first resolvent mode and therefore does not provide any hint about the separation of the singular values. In general, the presence of a spatially unstable local mode promotes the existence of a resolvent mode with a large singular value reflecting the strength of the instability and the spatial extension of the unstable region. If the strength is weak and/or the extension is small, the separation may be weak. If there are multiple unstable local modes, there must actually be multiple resolvent modes with large singular values. This may account for the fact that the PSE solution may project on the first and / or second SPOD modes. Another possible explanation are deficiencies of the PSE approach to reconstruct the resolvent mode. In so far, enforcing parabolicity in the PSE approach or non-parallel effects through interaction with the shock/recompression cells for example (certain SPOD modes may be strengthened through interaction with the shock/recompression cell structure, while this cannot be captured by the present PSE approach) would provide possible explanations.

St = 0.1 St = 0.2 St = 0.3 St = 0.4 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 Λ 1 0
St = 0.1 St = 0.2 St = 0.3 St = 0.4 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 Λ 1 0
St = 0.1 St = 0.2 St = 0.3 St = 0.4 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 m = 0 m = 1 m = 2 Λ 1 0
C xf x0 ∞ 0 C qm,f -φ 1 m,f 2
dxrdr
where C is the multiplicative constant to be determined.

There is a remarkable qualitative agreement between the leading SPOD modes (in some cases the second one) and the PSE prediction for the pressure and the axial velocity. Overall, the radial patterns agree, with the exception of the features associated with the interaction with the shock/recompression cells within the jet core, and far away from the jet, where the Mach waves dominate. In accordance with previous observations, the impact of the shock/recompression cells is mostly observed on the m = 0 mode. In the case of the axial velocity, the leading SPOD modes reveal strong radial gradients, leading to complex spatial patterns in the downstream region. These features are remarkably well captured by the PSE.

The Mach waves away from the jet, which are clearly seen in the leading SPOD modes of the pressure, are poorly captured by the PSE. Earlier studies on ideally-expanded supersonic jets, both iso-thermal and moderately heated, have shown good agreement for the acoustic or Mach wave propagation angle (Rodriguez et al. [START_REF] Rodriguez | Inlet conditions for wave packet models in turbulent jets based on eigenmode decomposition of large-eddy simulation data[END_REF], Sinha et al. [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF]). The poor prediction in the present jet is attributed to the high temperature ratio (T J /T ∞ = 8) and the presence of the flame, which increase the non-parallelism parameter |V 0 /U 0 | (for further discussion on the nonparallelism of the jet see Section 1 in the Supplementary material).

The amplitude and phase along the lip line predicted by the PSE are included in Figs. 12 to 17 (dashed lines). Overall, the evolution of the amplitude of the wave-packet is predicted only qualitatively by the PSE for the pressure and the axial velocity, and even less accurately for the temperature. For the pressure and the axial velocity, the amplitude oscillations induced by the shock/recompression cells are mostly ignored by the PSE. The impact of the shock and recompression cells on the helical SPOD modes is weaker, which is also true for the ensemble of frequencies as shown in Fig. 7. This explains why the helical wave-packets are better predicted than the m = 0 wave-packets by the PSE. On the other hand, the phase and thus the axial wavelength of the leading SPOD modes are reasonably well predicted overall for all frequencies and azimuthal numbers considered. At the lowest frequencies (St = 0.05, 0.1), the PSE prediction of the pressure and axial velocity misses the wave-packet decay (not shown), leading to an over-prediction of the wave-packet amplitude downstream. This is attributed to a failure of the nearly-parallel assumption at low frequencies. A better prediction should be obtained from a model using global modes Gudmundsson and Colonius [START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF]. For the temperature, the PSE predictions are rather poor in amplitude, and less accurate in phase than the pressure and axial velocity. This is not surprising, as temperature is the variable which is most sensitive to the flame. Previous attempts at modeling ideally-expanded, supersonic moderately heated jets using linear PSE failed at predicting the amplitude envelope of the pressure along the jet for many of the pairs (St, m) (Rodriguez et al. [START_REF] Rodriguez | Inlet conditions for wave packet models in turbulent jets based on eigenmode decomposition of large-eddy simulation data[END_REF], Sinha et al. [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF]). It is not surprising that in the present extreme case of a non-parallel reacting jet, the prediction of the SPOD modes of temperature remains limited.

Conclusion

The question that motivated this study was two-fold. First, to determine and to characterize the large-scale coherent structures of a turbulent, reacting and imperfectly-expanded supersonic jet. And second, attempt at modeling them using the PSE about the mean-flow, neglecting the chemical reactions (in a similar way to Hammond and Redekopp [START_REF] Hammond | Global dynamics of symmetric and asymmetric wakes[END_REF], Pier [START_REF] Pier | On the frequency selection of finite-amplitude vortex shedding in the cylinder wake[END_REF],

Barkley [START_REF] Barkley | Linear analysis of the cylinder wake mean flow[END_REF], Sipp and Lebedev [START_REF] Sipp | Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows[END_REF]) to assess whether or not the observed wavepackets from the high-fidelity simulation could be recovered from a non-reactive model. The results αµ 0 U 0 r 2 ; C 25 = 1 Re ∂ T µ 0 ∂ r U 0 U 0 r 2 ;

C 31 = - 1 γM 2 T 0 U 0 r 2 ; C 32 = 1 3Re
αµ 0 U 0 r 2 ;

C 33 = -ρ 0 U 0 V 0 r 2 + 4 3Re µ 0 U 0 r - 4 3Re
∂ T µ 0 ∂ r T 0 U 0 r 2 ;

C 34 = - 1 3Re
jmµ 0 U 0 r; C 35 = -1 γM 2 ρ 0 U 0 r 2 ; C 43 = - where

∂ (•) q = ∂ q ∂(•) , µ 0 = µ(T 0 ), κ 0 = κ(T 0 ), ∂ T µ 0 = ∂µ ∂T | 0 , ∂ T κ 0 = ∂κ ∂T | 0 . (A.1)
The remaining entries to the matrices are zero.

The local stability equations can be derived from the PSE equations by removing the

x-dependence of the mean flow, and substituting the α function by a constant wavenumber α LS . The resulting system is then put as an eigenvalue problem for α LS and the state vector q, for given m, f . The details can be found, for example, in Beneddine et al. [START_REF] Beneddine | Conditions for validity of mean flow stability analysis[END_REF].

Figure 1 :

 1 Figure 1: Sketch of the geometry of the simulation and the boundary conditions.

Figure 2 :

 2 Figure 2: Colormaps of a snapshot of the pressure p t /P J (a,b), axial velocity u t /U J (c,d), and temperature T t /T J (e,f); transverse snapshots at x/D = 8.

Figure 3 :

 3 Figure 3: Colormaps of (a) mean pressure, b) mean axial velocity u 0 /U J , (c) mean temperature T 0 /T J , and (d) mean mass fraction of OH Y OH . Note that the flow contains a single shock, and then successive recompression cells with compression waves that become weaker as one goes downstream.

Figure 4 :Figure 5 :

 45 Figure 4: (a) Mean axial velocity profiles at x/D = 1.75 (black solid), 3.75 (dashed), 6.25 (dash-dotted), 8.75 (dotted), 10.25 (solid with circles) (b) mean temperature profiles at x/D = 1.75, 3.75, 6.25, 8.75, 10.25 (legend as in (a)), and (c) profiles of the non-parallelism parameter V 0 /U 0 at x/D = 3.75, 6.25, 8.75, 10.25 (legend as in (a)), with the large circles corresponding to the values at r * such that T 0 (r * ) = 1.3T ∞ .

Figure 6 :

 6 Figure 6: Azimuthal spectrum on the jet lip line for pressure (a), axial velocity (b) and temperature (c), at various distances from the nozzle exit (note that the over-bar in the definition of the spectrum is a time mean).

Figure 7 :

 7 Figure 7: Contribution of the three leading azimuthal modes along the jet lip line for the pressure (a), the axial velocity (b) and the temperature (c) (note that the over-bar in the definition of the spectrum is a time mean).

4. 1 .

 1 Energy fall-off: self-coherence In real turbulent flows, the coherent portion of the unsteady flowfield, characterized by the leading SPOD modes, accounts only for a small part of the total fluctuating energy. In reacting flows the energy fraction of the leading SPOD mode is potentially even lower, as the flame introduces new dynamics. The energy fraction of the five leading SPOD modes are shown in Figs. 8, considering respectively pressure, axial velocity and temperature fluctuations. The relative energies of the leading SPOD mode lie in the range 20-40% for the

Figure 8 :

 8 Figure 8: Energy fractions of the 5 leading SPOD modes corresponding to the pressure (circles), axial velocity (crosses) and temperature (gradients).

Figure 9 :

 9 Figure 9: Energy fractions of the two leading SPOD modes as a function of the frequency, for m = 0 (a), m = 1 (b) and m = 2 (c).

Figure 10 :

 10 Figure 10: Colormaps (real part) of the leading SPOD mode (a) and second SPOD mode (b), corresponding to St = 0.2, m = 1.

Figure 11 :

 11 Figure 11: Colormaps (real part) of the leading SPOD mode (a) and second SPOD mode (b), corresponding to St = 0.4, m = 1.

3 and 4 )

 4 . Vortex shedding at high Reynolds numbers around axisymmetric bluff bodies, such as disks and spheres, has been previously observed at St ≈ 0.2, m = 1 and was associated to the existence of an unstable global mode (Achenbach[START_REF] Achenbach | Vortex shedding from spheres[END_REF], Vilaplana et al.[START_REF] Vilaplana | Global mode of a sphere turbulent wake controlled by a small sphere[END_REF], Meliga et al.[START_REF] Meliga | Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion[END_REF]).

Figure 12 :

 12 Figure 12: Amplitude of the pressure field along the nozzle lip line from PSE (dashed), first SPOD mode (crosses) and second SPOD mode (points) corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 13 :Figure 14 :

 1314 Figure 13: Phase of the pressure field along the nozzle lip line from PSE (dashed), first SPOD mode (crosses) and second SPOD mode (points), corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 15 :

 15 Figure 15: Phase of the axial velocity field along the nozzle lip line from PSE (dashed), first SPOD mode (crosses) and second SPOD mode (points), corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 16 :

 16 Figure 16: Amplitude of the temperature field along the nozzle lip line from PSE (dashed), first SPOD mode (crosses) and second SPOD mode (points), corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 17 :

 17 Figure 17: Phase of the temperature field along the nozzle lip line from PSE (dashed), first SPOD mode (crosses) and second SPOD mode (points), corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 18 :

 18 Figure 18: Phase along the line x = 0.65D of the leading m = 1 SPOD mode of the temperature (solid), the 2nd SPOD mode of the temperature (dashed), the leading SPOD mode of the pressure (solid with circles), and the 2nd SPOD mode of the pressure (dashed with circles), corresponding to St = 0.2 (a) and St = 0.4 (b).

  The spatial structure of the leading SPOD and PSE modes of pressure and axial velocity, for m = 0 and m = 1, are shown in Figs. 19, 20, 21 and 22, corresponding to St = 0.2 (a) and St = 0.4 (b). It is reminded that the pixelation is due to the spatial sampling and not

Figure 19 :

 19 Figure 19: Pressure field (real component) from first POD mode and PSE for m = 0, corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 20 :

 20 Figure 20: Pressure field (real component) from first POD mode and PSE for m = 1, corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 21 :

 21 Figure 21: Axial velocity field (real component) from first POD mode and PSE for m = 0, corresponding to St = 0.2 (a) and St = 0.4 (b).

Figure 22 :

 22 Figure 22: Axial velocity field (real component) from first POD mode and PSE for m = 1, corresponding to St = 0.2 (a) and St = 0.4 (b).

B 54 =r α 2 κ 0 U 0 r 2 - γ ReP r m 2

 5422 jm(γ -1)ρ 0 T 0 U 0 r;B 55 = j2πf ρ 0 U 0 r 2 -αρ 0 U 2 0 r 2 -(γ -1)ρ 0 V 0 U 0 r -(γ -1)∂ r V 0 ρ 0 U 0 r 2 -(γ -1)∂ x U 0 ρ 0 U 0 r 2 + γ ReP κ 0 U 0 + γ ReP r ∂ T κ 0 ∂ r T 0 U 0 r + γ ReP r (∂ r T 0 ) 2 ∂ T T κ 0 U 0 r 2 + γ ReP r ∂ T κ 0 ∂ rr T 0 U 0 r 2 + γ(γ -1)M 2 Re ∂ T µ 0 (∂ r U 0 ) 2 U 0 r 2 ; C 11 = -V 0 r; C 13 = -ρ 0 r; C 22 = ρ 0 U 0 V 0 r 2 + 1 Re µ 0 U 0 r + 1 Re ∂ T µ 0 ∂ r T 0 U 0 r 2 ; C 23 = 1 3Re

C 44 =

 44 -ρ 0 U 0 V 0 r 2 + 1 Re µ 0 U 0 r + 1 Re ∂ T µ 0 ∂ r T 0 U 0 r 2 ; C 52 = -2γ(γ -1)M 2 Re µ 0 ∂ r U 0 U 0 r 2 ; C 53 = -(γ -1)ρ 0 T 0 U 0 r 2 ; C 55 = -ρ 0 U 0 V 0 r 2 + γ ReP r κ 0 U 0 r + γ ReP r ∂ T κ 0 ∂ r T 0 U 0 r 2 ; D 22 = 1 Re µ 0 U 0 r 2 ; D 33 = 4 3Re µ 0 U 0 r 2 ; D 44 = 1 Re µ 0 U 0 r 2 ; D 55 = γ ReP r κ 0 U 0 r 2 ,

Table 2 :

 2 Projection coefficients of the PSE pressure field with the first and second SPOD modes (Λ i ), and energy fractions of the first and second SPOD modes (λ

i ).

Table 3 :

 3 Projection coefficients of the PSE axial velocity field with the first and second SPOD modes (Λ i ), and energy fractions of the first and second SPOD modes (λ

			.71	0.77	0.71	0.65	0.80	0.68	0.68	0.83	0.76	0.60	0.71	0.69
	Λ 2 0.38	0.33	0.18	0.37	0.12	0.33	0.25	0.17	0.15	0.36	0.65	0.28
	λ	1	0.38	0.27	0.21	0.29	0.31	0.20	0.20	0.26	0.23	0.14	0.39	0.19
	λ	2	0.14	0.17	0.10	0.13	0.11	0.12	0.17	0.10	0.11	0.12	0.11	0.11

i ).

Table 4 :

 4 Projection coefficients of the PSE temperature field with the first and second SPOD modes (Λ i ), and energy fractions of the first and second SPOD modes (λ i ). The cases where Λ 2 > Λ 1 , i.e. where the projection onto the second SPOD mode is greater than the projection on the leading SPOD mode, are in bold.

			.49	0.68	0.58	0.49	0.41	0.36	0.46	0.30	0.32	0.36	0.18	0.15
	Λ 2 0.41	0.22	0.31	0.35	0.19	0.22	0.37	0.24	0.01	0.30	0.11	0.13
	λ	1	0.40	0.39	0.36	0.39	0.40	0.30	0.21	0.21	0.26	0.20	0.46	0.30
	λ	2	0.17	0.20	0.17	0.13	0.14	0.13	0.18	0.13	0.12	0.18	0.04	0.11

  due to the plotting. The PSE fields shown are limited to the region x > 4D, where the PSE solution is more accurate. As for any linearized model, predictions are up to a multiplicative constant, which here has been determined from a best-fit with the SPOD modes: min

			.46	0.16	0.06 0.35	0.14	0.12	0.37	0.30	0.09	0.02	0.14	0.12
	Λ 2 0.18	0.13	0.12 0.45	0.13	0.03	0.46	0.10	0.07	0.26	0.17	0.06
	λ	1	0.12	0.10	0.09	0.14	0.09	0.09	0.15	0.10	0.09	0.19	0.53	0.28
	λ	2	0.09	0.09	0.09	0.08	0.08	0.07	0.10	0.09	0.08	0.10	0.03	0.03

Acknowledgments

The authors would like to thank Franois Vuillot, Valrie Rialland, Aurlien Guy and Sidonie Lefebvre for a number of useful comments and suggestions.

Funding Sources

This work has been funded by the ONERA internal project PRF SIMBA.

coherent structures within the jet.

The leading SPOD modes for pressure and axial velocity were found to contain up to 40%of the fluctuation energy, and exhibit a clear wave-packet structure for all frequencies and azimuthal numbers tested. The spatial structures of the wave-packets for pressure and axial velocity are well predicted by PSE, except in the region away from the jet core and in the vicinity of the shock/recompression cell structures where non-parallel effects are strongest.

The present results therefore show that the large-scale structures of pressure and velocity are well captured by a model accounting partially for the presence of the flame (for the meantemperature increase in the mean-flow but not the chemical reaction fluctuations). Yet, the amplitude growth, saturation and decay along the jet is only approximately recovered, similarly to previous studies in hot subsonic and ideally-expanded supersonic jets. As for the temperature fields, we have shown that they exhibit a lower self-coherence overall. Their leading SPOD modes contain in general small scale features directly linked to the flame dynamics, and the lack of a well-defined wavelength of the large structures. The result is that the temperature field can hardly be identified to wave-packets, and the temperature evolution predicted by PSE only poorly recovers the SPOD modes.

For the particular case St = 0.4, m = 1, 2, a highly coherent structure exists in all variables including the temperature, associated to a mutual reinforcement between the main shear layer instability and the vortex shedding induced in the recirculation region behind the normal shock. The PSE are not able to capture this mode due to a rather upstream location of the amplitude peak location, where strong non-parallel effects exist due to the presence of a strong shock and strong recompression cells. A global resolvent analysis may be able to better capture this mode. Also, accounting for chemistry fluctuations in the linearized equations would be an interesting future research direction.

Appendix A. Details of the PSE 513

The sub-matrices forming A, B, C, D in Eq. 17 are:

A 33 = ρ 0 U 2 0 r 2 ; A 44 = A 33 ; A 52 = (γ -1)ρ 0 T 0 U 0 r 2 ; A 55 = A 33 ;

α∂ T µ 0 ∂ r T 0 U 0 r 2 ;

B 52 = -∂ x T 0 ρ 0 U 0 r 2 -(γ -1)αρ 0 T 0 U 0 r 2 ; B 53 = -∂ r T 0 ρ 0 U 0 r 2 -(γ -1)ρ 0 T 0 U 0 r + 2 Re γ(γ -1)M 2 αµ 0 ∂ r U 0 U 0 r 2 ;


Alexis Théron

MD,PhD Anissa Touil

MD Noémie Résseguier

MD,PhD; Gwenaelle Collod-Beroud

PhD Giulia Norscini

Gaelle Odelin

MD Anne-Sophie Simoni

Gilbert Habib

PhD Frédéric Collart

MD,PhD; Stéphane Zaffran
email: stephane.zaffran@univ-amu.fr

MD Gaëlle Odelin

PhD Gilbert Dvm

MD Habib

PhD Jean-François Avierinos
email: avierinosjfavierinos@ap-hm.fr

Clinical insights into a tertiary care center cohort of patients with bicuspid aortic valve

Keywords: Heart disease, Bicuspid aortic valve, Aortic stenosis, Aortic regurgitation

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

Introduction

Bicuspid aortic valve (BAV) is defined as a spectrum of altered aortic valves presenting with two functional cusps and less than three zones of parallel apposition without fusion. [START_REF] Sievers | A classification system for the bicuspid aortic valve from 304 surgical specimens[END_REF] It is the most frequent congenital heart disease (CHD) affecting 0.5 to 2% of the population with a strong male predominance. [START_REF] Garg | Mutations in NOTCH1 cause aortic valve disease[END_REF] BAV is associated with excess mortality in some cases [START_REF] Michelena | Sex Differences and Survival in Adults With Bicuspid Aortic Valves: Verification in 3 Contemporary Echocardiographic Cohorts[END_REF] and accounts for more complications than all other CHDs combined, 4,5 6 related to increased risk of ascending aorta aneurysm, dissection and valve degeneration, the latter occurring 15 to 20 years earlier as compared to normal tricuspid aortic valve. [START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF][START_REF] Michelena | Incidence of aortic complications in patients with bicuspid aortic valves[END_REF] Aortic valve and/or ascending aorta replacements are the only therapeutic option, [START_REF] Baumgartner | 2017 ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease[END_REF] with 25% incidence of this end-point at 20 years in patients without significant valvular dysfunction or aorta dilatation at first diagnosis. [START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF] Predictors of surgery include age>50 years and presence of valve degeneration at diagnosis, associated with a 4-fold increased risk of valve replacement at 20 years. [START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF] However, mechanisms underlying degeneration from normal to leaking or stenotic BAV remain poorly understood. 2 11 12 The Bicuspid Aortic valve Project (BAP) is designed to collect clinical and genetic data on consecutive patients prospectively admitted with a definite diagnosis of BAV. We sought to describe the clinical and echocardiographic spectrum of consecutive BAV patients as they present in a tertiary care center and to determine factors associated with each type of valve degeneration.

Materials and Methods

Study population

The BAP is a French single center prospective study including consecutive patients with nonsyndromic BAV for clinical and genetic purpose. Patients were recruited between 2014 and 2018 in La Timone Hospital, Marseille, France. Patients were not included if they had connective tissue disease, previous aortic valve surgery and less than 18 years old. This study was performed in accordance with institutional guidelines. Written consent was obtained from all patients.

BAV diagnosis

Definite

Baseline Clinical data

Clinical data were prospectively collected at the time of inclusion as well as family history of cardiac surgery, aortic dissection and CHD. Surgical correction of valve dysfunction and associated aortic dilatation (BAV aortopathy) was indicated according to the European and US guidelines. [START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease[END_REF][START_REF] Baumgartner | 2017 ESC/EACTS Guidelines for the management of valvular heart disease[END_REF] Baseline Echocardiographic data Aortic regurgitation (AR) was graded through an integrative approach. 9,10 Mechanism of AR was separated into prolapse of the fused leaflet, cusps restriction or both. Aortic stenosis (AS) was considered as present when mean transvalvular gradient>10mmHg with a restriction of the aortic valve opening. Severe AS was defined by a mean gradient>40mmHg and an indexed AVA<0,6cm 2 /m 2 . 9 Normal valve function was defined by the absence of AR even mild and of AS with a mean trans-valvular gradient<10mmHg. Aortic dilatation was defined by an ascending aorta diameter≥40mm and separated into "root phenotype", "tubular phenotype" or "combined phenotype" according to the site of main dilatation. Baseline clinical and echo data were compared according to aortic valve function (AS vs normal valve function, AR vs normal valve function, AS versus AR).

Follow-up assessments

Patients were monitored by their personal physicians during follow up. The primary endpoint was the indication for cardiac surgery at 6month follow-up.

Statistical analysis

Statistical analyses were performed using R-software 3.0.3. Quantitative variables are expressed as mean±SD and categorical variables are presented as numbers and percentages. Comparisons between groups were performed using the Student's t test or the Mann-Whitney U test for quantitative variables, and using the chi-2 test or the Fisher's test for categorical variables. Univariate logistic regression analyses were performed to assess the association between potential determinants of valve dysfunction. Firth's correction was applied by performing Firth's penalized-likelihood logistic regression to take into account the small number of patients without valve dysfunction. [START_REF] Firth | Bias Reduction of Maximum Likelihood[END_REF] Potential factor associated with valve dysfunction were considered as candidate for the multivariable analysis when their p-value was less than 0.20 according to the univariate analysis. A backward selection procedure was then applied to build a final regression model.

Results

Baseline patient characteristics are shown in Table 1. Two hundred twenty-three patients were prospectively included in the cohort Mean age was 53±17years, majority were asymptomatic and male (sex ratio = 3:1). Mean follow-up was 242.3±249.5 days. None had history of aortic dissection or family history of unexplained sudden death in the youth. Among 17 patients with infective endocarditis (IE) at presentation, 12 (5% of all BAV patients) had isolated aortic valve endocarditis and 4 (24% of all IE) peri annular complications.

Echocardiographic characteristics

BAV phenotype are shown in Table 2. According to Sievers classification, majority of patients (80%) had type 1 BAV, localized between left and right coronary cusps in 88% of them, 15% had type 0 and only 2% had type 2 BAV. According to embryologic classification, L-R BAV with or without raphe was observed in 83% of all BAV patients, NC-R BAV in 10% and NC-L BAV in 1% (Figure 1). Aortic dilatation was present at diagnosis in half BAV patients, evenly distributed between root and tubular phenotype.

Normal and pathological measurements are shown in Tables 1 and2. Twenty-four patients (11% of all patients) had normal valve function and 199 patients (89%) presented with valvular dysfunction at the time of inclusion. Patients with normal valve function were predominantly male, younger than patients with AS or combined AS and AR (p<0.01) but were of similar age than patients with isolated AR at diagnosis (p=0.4) (Figure 2). Only four of those patients with normal valve function were older than 60 years old. They presented less frequently with raphe than patients with valve dysfunction (54%vs.84%, p<0.01).

Majority had L-R BAV as patients with valve dysfunction (75%vs.84%, p=0.58). Prevalence of aortic dilatation (BAV aortopathy) was similar to that observed in patients with any type of valve dysfunction (50%vs.49%, p=0.41).

Sixty-eight patients (31% of all patients) had isolated AS, including 58 (26% of all BAV patients) with severe AS. Mean indexed aortic valve area was 0.49±0.24 cm 2 /m 2 and mean trans-valvular gradient was 56±21mmHg. Patients with AS were significantly older than patients with normal function or with isolated AR. They were predominantly males, but proportion of women (39%) was higher than among AR patients (14%). Majority had raphe and L-R BAV as patients with AR. Prevalence of aortopathy was lower than among patients with isolated AR and of different phenotype, mostly tubular.

Ninety-one patients (41% of all patients) had isolated AR, including 50 (22% of all BAV patients) with severe AR. Mean effective regurgitant orifice area 34±15mm 2 and mean regurgitant volume was 83±39ml. As expected, LV diameters and volumes were higher than in AS patients. Mechanism of AR was pure leaflet prolapse in 40 patients (44% of all AR patients), restriction in 5 (6%), combined prolapse and restriction in 46 (50%). Among AR patients, male predominance was more prominent than among AS patients, mean age was 22 years younger, comorbidity was lower but endocarditis higher at diagnosis. Majority had raphe and L-R BAV. Prevalence of aortopathy was twice higher than among patients with AS, mostly of root phenotype.

Forty patients (17%) had combined significant AS and AR. They were predominantly male and clinical and echocardiographic pattern was in-between isolated AS and AR patients.

Majority had raphe and L-R BAV. Aortopathy was mostly of tubular or combined phenotype.

Surgical event during follow-up

Cardiac surgery was performed within 6 months of diagnosis in 145 patients (65%), including 141 aortic valve replacements, mainly with bio-prothesis(n=85) and 4 isolated ascending aorta tubular graft replacement. Mean age at cardiac surgery was 56±15years and 72(49%) of operated patients were so before the age of sixty. Forty-two patients (42/145 = 29%) were NYHA 1 class and 80 patients (80/145=56%) were NYHA 2. Among NYHA 1 class patients, indication for surgery was driven by an ascending aorta dilatation (>55mm or 50mm in case of risk factors) in 12 patients (12/42=28.5%, aortic root dilatation (n=3), tubular aorta dilatation(n=9) ; 12 patients had severe asymptomatic AS with Vmax >5.5m/s (n=5) or positive exercise test (n=7) ; 8 patients had severe asymptomatic AR with severe LV dilatation (mean LVES volume index =57mL/m²).

Indication for surgery was AR in 26 patients (18% of all aortic valve replacements, mean age= 44±17 years), AS in 92 patients (65%, mean age=62±13years) and ascending aorta dilatation in 27 patients (including 23 combined surgery).

Clinical factors associated with BAV dysfunction

Discussion

The present study conducted on 223 consecutive patients diagnosed with BAV in routine clinical practice shows that in a tertiary care center 1) prevalence of severe AS and AR are similar, observed in a quarter of all patients at diagnosis, 2) AR is mainly due to leaflet prolapse and occurs at a much younger age than AS, 3) presence of raphe is associated with valve dysfunction mainly whereas embryologic phenotype is not.

BAV is the most common CHD and is associated with high rates of valve degeneration. [START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF][START_REF] Michelena | Incidence of aortic complications in patients with bicuspid aortic valves[END_REF][START_REF] Michelena | Sex Differences in Bicuspid Aortic Valve Adults: Who Deserves Our Attention, Men or Women?[END_REF] Development of ascending aorta aneurysm is observed in one fourth of patients 25 years after diagnosis of BAV with an excess risk of aortic dissection. [START_REF] Michelena | Incidence of aortic complications in patients with bicuspid aortic valves[END_REF] Subsequently BAV patients experience high rates of surgical aortic interventions at younger age than patients with tricuspid aortic valve. [START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF][START_REF] Michelena | Bicuspid aortic valve: identifying knowledge gaps and rising to the challenge from the International Bicuspid Aortic Valve Consortium (BAVCon)[END_REF][START_REF] Michelena | Incidence of aortic complications in patients with bicuspid aortic valves[END_REF][START_REF] Roberts | Frequency by Decades of Unicuspid, Bicuspid, and Tricuspid Aortic Valves in Adults Having Isolated Aortic Valve Replacement for Aortic Stenosis, With or Without Associated Aortic Regurgitation[END_REF] However, data from populations referred to tertiary care centers at evolving clinical and echocardiographic stages unveil excess mortality in some subsets as compared to expected survival in an age-and sex-matched population due to higher prevalence of valvular dysfunction. [START_REF] Michelena | Sex Differences and Survival in Adults With Bicuspid Aortic Valves: Verification in 3 Contemporary Echocardiographic Cohorts[END_REF] Our study is consistent with these data 3 reporting low prevalence of strictly normally functioning aortic valve at diagnosis of BAV, barely exceeding 10% of patients despite the young age of the study population. Almost none of them were older than 60, making valve degeneration an early process. Conversely, the vast majority of BAV patients presented with significant valvular dysfunction at diagnosis, associated with aortic dilatation in half of them, explaining the high incidence of surgery within 6 months of diagnosis reaching two thirds of patients at a mean age of 56 years-old, even younger when indication was driven by AR. In accordance with recent studies confirming that early surgery offers better outcome than conservative care in severe aortic valve disease, majority of patients were asymptomatic or with low symptoms at time of surgery.

Such high burden of morbidity and mortality imposes comprehensive assessment of determinants of clinical outcome and mechanisms of valve degeneration, critical for follow-up and management of patients with a first diagnosis of BAV. To date, identified risk factors of cardiac events, include presence of valve degeneration at first diagnosis of BAV, baseline ascending aorta diameter>40mm, [START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF] female gender particularly with significant AR 3 and older age. [START_REF] Michelena | Sex Differences and Survival in Adults With Bicuspid Aortic Valves: Verification in 3 Contemporary Echocardiographic Cohorts[END_REF][START_REF] Michelena | Natural history of asymptomatic patients with normally functioning or minimally dysfunctional bicuspid aortic valve in the community[END_REF] In turn, risk factors of outcome in patients with strictly normal BAV are poorly defined and BAV ability to maintain normal valve function or to become dysfunctional is not understood. BAV phenotype has been correlated to the type of dysfunction in one study, typical phenotype, defined by 11-17 or 10-16 o'clock positions of commissural axis, being associated with AR and atypical phenotype, defined by 13-19 or 12-18 o'clock positions of commissural axis, being associated with AS. [START_REF] Kang | Association Between Bicuspid Aortic Valve Phenotype and Patterns of Valvular Dysfunction and Bicuspid Aortopathy[END_REF] Our results do not confirm this association since embryologic phenotype was not associated with neither type of dysfunction, which if consistent with the overwhelming majority of L-R fusion in community and tertiary series 3 , making hazardous any attempt of risk stratification using this parameter. Proportions of typical and atypical phenotype were indeed almost identical in our series in both types of degeneration and close to those observed in normal BAV. Likewise, within each phenotype, normal valve function, leaking or stenotic degeneration were evenly distributed. Presence of raphe has been associated with higher prevalence of AS and AR and increased incidence of aortic surgery. [START_REF] Kong | Prognostic implications of raphe in bicuspid aortic valve anatomy[END_REF] BAV differs from the tricuspid valve by number of cusps and sinus, by the elliptical shaped of the aortic ring and the restrictive and asymmetrical opening of aortic valve during systole. [START_REF] Bäck | Biomechanical factors in the biology of aortic wall and aortic valve diseases[END_REF][START_REF] Robicsek | The congenitally bicuspid aortic valve: how does it function? Why does it fail?[END_REF] MRI revealed eccentric systolic jet and abnormal downstream helical flow patterns through normal BAV resulting in greater flow acceleration and higher shear stress. [START_REF] Sun | Ex vivo evidence for the contribution of hemodynamic shear stress abnormalities to the early pathogenesis of calcific bicuspid aortic valve disease[END_REF][START_REF] Marom | Fully coupled fluidstructure interaction model of congenital bicuspid aortic valves: effect of asymmetry on hemodynamics[END_REF] This phenomenon attributed to commissural fusion could lead to higher sensitivity to valve degeneration and calcification [START_REF] Sun | Ex vivo evidence for the contribution of hemodynamic shear stress abnormalities to the early pathogenesis of calcific bicuspid aortic valve disease[END_REF] and could be magnified by the increased stiffness of the fused leaflet due to the presence of raphe. Our data corroborated this hypothesis by showing an independent association between raphe and valve dysfunction mainly AS and a lower prevalence of raphe in normal BAV. However, raphe can be seen as the expression of an already ongoing degenerative process and could be acquired. [START_REF] Yang | The bicuspid aortic valve raphe: an evolving structure[END_REF] In addition, in our population more than half of normally functioning BAV presented with raphe and almost 20% of AR patients without, leaving unanswered the issue of valve dysfunction in many patients and imposing to identify determinants of degeneration independently of phenotype.

The most striking observation of our series was the major age difference at diagnosis of AR as compared to AS, patients with significant AR being more than 20 years younger than patients with significant AS. Interpretation of such age marked differences at initiation of leaking or stenotic process are unclear. Specifics interactions between genetic pathways and environmental factors linked to aging could be argued. For instance, overexpression of RUNX2 signal has been observed in mice models of calcified aortic valve. [START_REF] Steitz | Smooth muscle cell phenotypic transition associated with calcification: Upregulation of Cbfa1 and downregulation of smooth muscle lineage markers[END_REF] This transcription factor is connected to NOTCH1, a single-pass transmembrane receptor expressed in valvular endothelial cells during development, which mutations have been reported in patients with BAV associated with valve calcification. [START_REF] Garg | Mutations in NOTCH1 cause aortic valve disease[END_REF] Conversely, dysregulation of genetic signals implicated in valvular interstitial matrix homeostasis has not been reported to date but could shift BAV towards myxomatous degeneration, leaflet prolapse and AR.

Limitations

Any cohort of BAV patients is susceptible to diagnostic errors between tricuspid and bicuspid valve, with the risk of data contamination by non BAV patients. The prospective design of this study and the cautiousness taken in scanning with exclusion of all unclear diagnosis particularly among AS patients should reduce this bias. The fact that many patients with wellfunctioning BAV were not diagnosed and not captured in this tertiary cohort creates also a selection bias. A part of elective patients with BAV had been referred from another clinic and only those with valve dysfunction could have been referred to our tertiary care center. This could have falsely increase the prevalence of valve dysfunction in our population. Close follow-up of BAV patients from birth to occurrence of valve dysfunction would be the ideal way to disclose mechanisms of degeneration but is not conceivable. Therefore, cross sectional comparison of BAV patients according to valve function at first diagnosis of BAV is an imperfect but acceptable method.

Valvular degeneration is obviously a complex interplay of various factors including hemodynamic constraints due to the fused leaflets and genetic pathways dysregulation yet to be determined, leading to maladaptive response of valvular ECM to the aortic flow environment.

Conclusion

This study reported high prevalence of valve dysfunction at first diagnosis of BAV among patients referred to a tertiary care center. The leaking or stenotic degenerative process is poorly associated with clinical and phenotypic markers, dominated by age, gender and presents of raphe. BAV degeneration might be a complex phenomenon involving several contributors, among which genetic pathways dysregulation will deserve specific investigation.

  diagnosis of BAV was based on visualization by ultrasound imaging in short axis view of two functional cusps in mid-systole; partial fusion between cusps was considered as BAV. Transesophageal echocardiography (TEE) was performed when diagnosis was uncertain after transthoracic echocardiography (TTE). Raphe was considered as definitely present when clearly visible by TTE or TEE. BAV phenotypes were specifically classified regarding both Siever's classification and embryologic classification 1,6 and all echocardiograms were performed by two experienced cardiologists (JFA,AT) to ensure homogeneous collection of data. Schematic presentation of the developmental phenotypes of the aortic valve and typical characteristics according to Sievers et al. J Thorac Cardiovasc Surg 2007;133:1226-1233 BAV classification Sievers classification is based on presence or absence of raphe (fusion) between fused cusps. Embryologic classification refers to type of embryologic cusp fusion -or nonseparation-based on commissural axis orientation in short axis view. Typical BAV is defined by 11-17 or 10-16 o'clock positions of commissural axis, corresponding to left-right coronary cusps fusion (L-R BAV). Atypical BAV is defined by 13-19 or 12-18 o'clock positions of commissural axis, corresponding to non-coronary and right coronary cusps fusion (NC-R). The rare non-coronary-left coronary cusps fusion (NC-L) is also labeled atypical and defined by 14-20 o'clock position. Embryologic classification according to Michelena et al. Circulation. 2014;129:2691-2704

  We compared normal with AS valve function. By univariate analysis (OR[95%CI]), age>50 years (18.6[6.2-64], p<0.001), presence of raphe (5.7[1.8-19.5], p=0.003), diabetes (8.8[1.1-1147.3], p=0.04) and dyslipidemia (3.8[1.2-15.5], p=0.02) were significantly associated with BAV stenosis whereas male gender (1.4[0.5-3,6], p=0.5) and orientation of commissures (1.5[0.3-14.8], p=0.7) were not. By multivariable analysis, age>50 years (41.6[10.3-248.2], p<0.001) and presence of a raphe (12.8[2.4-87.4], p<0.001) were significantly associated with AS. We also compared normal with AR valve function. By univariate analysis, male gender (5.0[1.6-16.4],p=0.005), aortic annulus diameter(1.2[1.02-1.56],p=0.03) and presence of raphe (3.6[1.1-11.9],p=0.03) were significantly associated with BAV regurgitation whereas age (0.9[0.3-2.5],p=0.8) and orientation of commissures (1.9[0.4-19.0],p=0.5) were not. By multivariable analysis, only male gender (5[1.6-16.4],p=0.005) was significantly associated with AR. Finally, we compared AR with AS valve function. By univariate analysis, male gender (3.6[1.7-8.7], p=0.001), younger age (10[5.2-20.0], p<0.001) and higher aortic annulus diameter (1.27[1.14-1.43], p=0.001) were significantly associated with AR whereas dyslipidaemia (3.3[1.6-7.1, p=0.002) was associated with AS. Multivariable analysis identified younger age (1.9[1.85-1.94], p<0.001) as independently associated with AR and dyslipidemia (3.3[1.22-8.22], p<0.001) with AS, while male gender was of borderline significance for predicting AR (1.15 [0.70-2.54],p=0.07).

Figure 1 :

 1 Figure 1: Phenotypic patterns of bicuspid aortic valve according to embryologic and Sievers classification. (A) Typical BAV with 10-16 o'clock position of commissural

Figure 2 :

 2 Figure 2: Phenotypic patterns of bicuspid aortic valve according to valve dysfunction.

Acknowledgements We thank Marine Herbane for her pretty shema

Funding There was no official financial support from any company

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest