I Hannachi

R Stamm

J Rosato

Y Marandet

Effect of nonlinear wave collapse on line shapes in a plasma

Keywords: 32.70.Jz -Line shapes, widths, and shifts 52.35.Mw -Nonlinear phenomena : waves, wave propagation, and other interactions

The nonlinear interaction of waves can change the structural and radiative properties of plasmas. We describe the main features of a fully ionized unmagnetized plasma affected by strong Langmuir turbulence characterized by nonlinear wave collapse, and propose a simple model for evaluating the changes expected on a hydrogen line shape affected by such conditions. Our model is based on a stochastic renewal model using an exponential waiting time distribution and a half-normal probability density function for the electric field magnitude of the turbulent wave packet. The first results obtained with a simulation calculation of the hydrogen L  line show that strong Langmuir turbulence can provide an additional broadening to a Stark profile.

Introduction. -The influence of collective electric fields on a line shape emitted in a plasma has been studied for several decades in different types of plasmas [1,2,3]. Most of the collective fields considered were described by linear waves, e.g. the electronic Langmuir plane wave oscillating at the plasma frequency. The observed spectroscopic effects were the formation of satellites or dips, and changes in the overall shape of the line. In this work, we consider in an unmagnetized plasma the nonlinear coupling of Langmuir waves with ion sound and electromagnetic waves. In such conditions, the density fluctuations associated with ion sound waves affect the Langmuir waves, refracting them in regions of low density. Coherent plasma wave packets localize in such regions, and evolve to shorter scales and higher intensities [4]. Computer simulations reveal the existence of a wave packet cycle with a collapse arrested by dissipation, and of a nucleation mechanism allowing the creation of a new wave packet. If energy is supplied from an external source, localized wave packets may be created at a high rate, and the plasma will contain many of those coexisting wave packets. We consider in the following the effect of the presence of such collapsing wave packets on the line radiation emitted by an atom in the plasma. Interesting applications for this study concern plasmas in interaction with an energetic beam of particles. For example in astrophysical contexts, active galactic nuclei [START_REF] Miller | Active Galactic Nuclei[END_REF], or pulsar radio sources [START_REF] Asseo | [END_REF]. Relevant laboratory plasmas are laser plasmas [START_REF] Kruer | The Physics of Laser-Plasma Interactions[END_REF], radio experiments [START_REF] Bauer | [END_REF], or possibly also magnetic fusion plasmas, since such plasmas can be affected by energetic beams of runaway electrons [9]. The paper is organized as follows. In the next section, we make a short review of the main characteristics of strong Langmuir turbulence and present the wave packet cycle. The third section presents a stochastic model adapted for line broadening calculations in such conditions, and the last section is a discussion of the first calculations on the hydrogen L  line (transition from n=2 to n=1, where n is the principal quantum number).

Strong Langmuir turbulence and the wave packet cycle. -

We consider electron, ion and electromagnetic waves in a fully ionized unmagnetized plasma submitted to an external source of energy, such as a beam of energetic charged particles. This source of external energy may be coupled to the plasma waves, thus increasing their intensity, and allowing the start of nonlinear processes such as wave-wave interactions [START_REF] Bellan | Fundamental of Plasma Physics[END_REF]. Langmuir waves then couple with ion sound and electromagnetic waves and we enter in the domain of strong Langmuir turbulence and nonlinear wave collapse. A nonlinear plasma has a refractive index that depends on the intensity of waves present. The reason for this is found if we consider the nonlinear ponderomotive force per unit volume:

𝐹 ⃗ 𝑁𝐿 = - 𝜔 𝑝 2 𝜔 2 ∇ ⃗ ⃗⃗ 〈ε 0 E 2 〉 2 , ( 1
)
where E is the magnitude of the electric field of the wave oscillating at frequency , and brackets denote a time average. The electronic plasma frequency is defined by 𝜔 𝑝 = √𝑁 𝑒 𝑒 2 /𝑚𝜀 0 , with N e the electronic density, m and e the electron mass and charge, and 𝜀 0 the permittivity of free space. This force acts in a direction opposed to the gradient of the field. Due to this action, a coherent and intense Langmuir wave packet moves a part of the plasma out of the region of maximum field value. At the location of the wave packet, one observes a plasma density depression, and an increase of the refractive index. A localized wave packet refracts in these regions of low density, further increasing its intensity, and thus expelling even more plasma. The nonlinear dynamics for this process is well described by the Zhakarov equations [START_REF] Zakharov | [END_REF].

The first Zhakarov equation describes how density fluctuations modify and amplify Langmuir wave packets, and the second one takes account of the effect of intense waves on the density fluctuations. The study of these equations, together with numerical simulations, has revealed the complex dynamics of wave packets in cases where the ratio W of the wave energy density to the plasma energy density is not small. This ratio W is given by:

𝑊 = 𝜀 0 𝐸 𝐿 2 4𝑁 𝑒 𝑘 𝐵 𝑇 𝑒 , ( 2
)
where E L is the magnitude of the turbulent Langmuir field, k B the Boltzmann constant, and T e the electron temperature. Strong turbulence conditions have been found above a threshold for W depending on the plasma conditions. Zhakarov equations in one-dimension reduce to the nonlinear Schrödinger equation in the adiabatic limit, for which density fluctuations are changing slowly. This equation has been useful for understanding the qualitative features of turbulent wave packets. The same equation has been used in hydrodynamics and optics for analysing the behavior of rogue waves [12]. Stable soliton solutions have been found in plasmas, with a localized and oscillating electric field modulated by a Gaussian or Lorentzian envelope [13]. We will use in the following such a modulated wave packet as a basic piece for building our model for a line shape emitted in a plasma submitted to strong Langmuir turbulence.

Using more realistic 2D and 3D computer simulations, solitons lose their stability, and the medium exhibits numerous localized wave packets, each one undergoing a whole cycle with an average duration which scales as the inverse of the average of W [4]. Best fits to 3D numerical simulations predict a characteristic time T between collapses such as [14,15]:

𝑇 ≈ 40 〈𝑊〉 -1 𝜔 𝑝 -1 , (3)
where the brakets denote an average value of W. For the following calculations, we consider, as observed in experiments [4], that W can reach values of the order of one, and we will use a characteristic time between collapses in the range 30-50 times the inverse electronic plasma frequency. The cycle starts most frequently by a nucleation mechanism due to the presence of wave packets trapped as standing waves in density depressions with a high refractive index. If the wave packet accumulates enough energy from the background turbulence or from the external source of energy, the intensity of the wave increases and spatially narrows. This phase is the wave collapse, during which the amplitude of the wave electric field can reach values several hundred times larger than the background plasma microfield, according to eq. ( 2) written for values of W of the order of one. The arrest of wave collapse follows, and is mainly due to transit-time damping [14]. Finally, the wave energy is rapidly dissipated, and a renucleation takes place, allowing the start a new wave packet cycle. In strong turbulence conditions, many localized wave packets coexist, together with propagating waves. It is possible by coupling these two components to estimate the number density of collapse sites. The mean interpacket separation has been found to be about 2 to 3 times of the packet length scale at the start of collapse (close packing assumption) [4].

A model for the effects of wave collapse on a radiating emitter.

-We propose a simple statistical model to investigate the effect of multiple collapsing wave packets on a line shape emitted by an atom in a strongly turbulent plasma. We assume that the measurement time of the spectrometer is much larger than the characteristic times of Langmuir wave oscillations and collapse. We will also consider that the line formation depth is large compared to the spatial scale of the collapsing wave packets, a condition which is similar to what is assumed in the micro-turbulent case in astrophysics [START_REF] Marandet | High Energy Density Phys[END_REF]. Observed along a line of sight in optically thin conditions, many atoms contribute, and the spectral line appears as an average over the statistical properties of the turbulent Langmuir field. We propose to use a stochastic renewal process for modeling the effects of such a turbulent field. The Langmuir electric field applied on a single emitter is a sequence of oscillating fields with a phase  j and duration  j , each field 𝐸 ⃗⃗ 𝑗 having a different direction and magnitude:

𝐸 ⃗⃗ (𝑡) = { 𝐸 ⃗⃗ 1 𝑐𝑜𝑠(𝜔 𝑝 𝑡 + 𝜑 1 ) 𝑆 1 (𝑡), 𝐸 ⃗⃗ 2 𝑐𝑜𝑠(𝜔 𝑝 𝑡 + 𝜑 2 ) 𝑆 2 (𝑡), ⋮ 𝐸 ⃗⃗ j 𝑐𝑜𝑠(𝜔 𝑝 𝑡 + 𝜑 j ) 𝑆 𝑗 (𝑡), ⋮ 0 ≤ 𝑡 ≤ 𝑡 1 𝑡 1 ≤ 𝑡 ≤ 𝑡 2 ⋮ 𝑡 𝑗-1 ≤ 𝑡 ≤ 𝑡 𝑗 ⋮ (4)
The function S j modulates the amplitude of the field during the time interval  j , with an adjustable time width of the envelope function. In the following, we modulate each plane wave with a Lorentzian shaped envelope S j centered in the time interval  j , and with a peak value equal to 1. The time width at half maximum T L of the envelope can be changed in the time interval  j . If the ratio T L / j is large, we recover the plane wave model used in the linear regime. We study in the next section the effect of the shape of the envelope. Numerical simulations of wave packet collapse are a guide for choosing this shape in different plasma conditions [4]. With this model of modulated wave packet, we try to reproduce at the emitter's location the effect of a wave packet cycle, from the time of nucleation at the beginning of the time interval, to the final dissipation time, going through the maximum field magnitude value supposed to be attained at the middle of the time interval. The sequence of wave packets represented by eq. ( 4) is a modeling of the emitter's perturbation by successive wave packet cycles. We assume that each emitter is always perturbed by a nearby wave packet which is renewed at the end of each cycle, using the fact that the wave packets are close packed [4]. The renewal process requires a waiting time distribution (WTD) for sampling the wave packet durations  j .

We have used a simple exponential law exp(-t), with  the jumping frequency chosen as the inverse of the characteristic time T between wave packet collapses. For each change of the field, the phase and direction are randomly distributed, and we need to prescribe a probability density function (PDF) for the magnitude value |𝐸 𝑗 ⃗⃗⃗⃗ |of the turbulent Langmuir field. PDFs of electric field strengths have been studied and measured in the case of electron-beam experiments. For ratios W of the order of one, a Gaussian distribution of the electric field strengths has been measured [START_REF] Levron | [END_REF], and is also predicted theoretically for the highest fields [15]. In the following, we have used a halfnormal distribution for the reduced electric field magnitude F=E/E 0 , with the Holtsmark microfield E 0 defined by E 0 =e/(4 0 r 0 2 ), where the average interparticle distance r 0 is given by (4/3)r 0

3 N e =1. The PDF P(F) can be written in function of the standard deviation of F:

𝑃(𝐹) = √2 𝜎√𝜋 exp (- 𝐹 2 2𝜎 2 ) . ( 5
)
For this PDF, the standard deviation  is related to the average magnitude 𝐸 𝐿 of the turbulent Langmuir field by the relation 𝜎 = √𝜋/2 𝐸 𝐿 /𝐸 0 . We have also made some calculations using a log-normal PDF with the same average field, without changing significantly the results described in the results section. Log-normal PDF have been measured e.g. for the fluctuation of density in tokamak plasmas [18,19]. Figure 1 presents, in units of the inverse plasma frequency, a piece of the history of one electric field component obtained by our model. The electric field with magnitude E j has been generated along a half-normal PDF with an average value E L equal to hundred times the Holtsmark field, and the component plotted is modulated by a Lorentzian envelope with a time width taken as 20 % of each time interval  j . For a temperature of 2.8x10 4 K, and a density of 10 18 m -3 , the jumping frequency has been taken equal to  p /50. In accordance with eq. ( 4) and the WTD, the electric field vector and phase are renewed for each new time interval. The fast oscillation seen on fig. 1 is at the plasma frequency, and the number of oscillations in each wave packet is sampled by the WTD.

Our model uses such electric field histories to obtain the dipole autocorrelation function (DAF) and the line shape of an emitting atom. Using a dipolar approximation, the time dependent interaction potential between the atom and an electric field 𝐸 ⃗⃗ is written as 𝑉(𝑡) =-𝐷 ⃗ ⃗⃗ . 𝐸 ⃗⃗ (𝑡), where 𝐷 ⃗ ⃗⃗ is the atomic dipole operator. The atomic DAF C(t) is given by [START_REF] Griem | Spectral Line broadening by Plasmas[END_REF]:

𝐶(𝑡) = 𝑇𝑟〈𝐷 ⃗ ⃗⃗ 𝑈 + (𝑡)𝐷 ⃗ ⃗⃗ 𝑈(𝑡)𝜌〉, ( 6
)
where the trace is over the atomic states, U(t) is the atomic evolution operator,  the density matrix, and the angle brackets imply an average over all the realizations of the turbulent Langmuir field. The evolution operator is obtained by solving the Schrödinger equation for a Hamiltonian H 0 +V(t), with H 0 the unperturbed atomic Hamiltonian. It is actually more convenient to use a time evolution operator 𝑈 ̅ (𝑡) in the interaction representation, using the relation 𝑈 ̅ (𝑡) = exp(𝑖𝐻 0 ℏ ⁄ ) 𝑈(𝑡). This operator obeys the Schrödinger equation:

𝑖ℏ 𝑑𝑈 ̅ (𝑡) 𝑑𝑡 = 𝑉 ′ (𝑡)𝑈 ̅ (𝑡), (7)
where V'(t) is the interaction representation of V(t). Using the electric field history given by eq. ( 4) we have solved numerically eq. ( 7) using an implicit scheme algorithm with a good compromise between accuracy and computing time [START_REF] Koonin | Computational Physics[END_REF]:

𝑈 ̅ (𝑡 + 𝑑𝑡) = [1 -𝑑𝑡 𝑉′(𝑡 + 𝑑𝑡) 2𝑖ℏ ] -1 [1 + 𝑑𝑡 𝑉′(𝑡) 2𝑖ℏ ] 𝑈 ̅ (𝑡). ( 8
)
The DAF is obtained after an average over a large number (10 4 ) of independent electric field histories. A discussion of this time dependent function provides specific information on the dynamics of the perturbation [START_REF] Alexiou | [END_REF]. The sampling of the stochastic variables is done on the computer with pseudorandom number algorithms, associated to numerical techniques such as transformation or rejection methods [START_REF] Vesely | Computational Physics, an Introduction[END_REF]. This simulation of the stochastic process is flexible, and well suited to the study of strong Langmuir turbulence. Such computer simulations have been used for more than 3 decades for obtaining accurate line shape calculations, but up to now mostly in equilibrium plasmas conditions [START_REF] Stamm R | [END_REF]25,[START_REF] Stambulchik | High Energy Density Phys[END_REF].

The calculations presented in the following concern only the hydrogen L  line, neglecting fine structure in order to obtain a fast numerical evaluation. Using the spherical quantum numbers n,l,m, the subspace for principal quantum number n contains 4 states, and the matrix representation of the quantum operators are 4x4 matrices. Results. -Most of the calculations of hydrogen L  DAF and line shape have been performed for a plasma temperature T e = 2.8x10 4 K, and a density N e = 10 18 m -3 . For such conditions the plasma exhibits most properties of a fully ionized plasma, and still contains numerous neutral atoms emitting hydrogen lines. Although being mainly studied for high density plasmas, detailed Stark line shapes are also needed for low density plasmas, since they enter in the modeling of radiative transfer together with Doppler broadening [START_REF] Rosato | [END_REF]. The numerical integration of the Schrödinger equation has been performed using a time step equal to 10 -3 times the inverse of the average Stark shift S. This frequency is defined by S=DE 0 /ħ, where D=3ea 0 for the substates n=2, and a 0 is the Bohr radius. We present in figs. 2-5 the DAF for L  subject to the single effect of the field defined by eq. ( 4), and calculated for a jumping frequency = p /50, and an average field E L =100 E 0 . These values have been chosen using eqs. (2)(3), and correspond to an average value of W equal to 0.8. We have plotted in figs. 2-3 four different DAF corresponding to different envelope shapes, which are governed by the ratio, expressed in percentages, of the Lorentzian time width  L to the time interval  between two successive steps. In fig. 2 the DAF with the strongest damping, corresponding to the broadest line shape, is obtained for plane waves (solid line), which is the case of infinite value for the ratio  L /. For values of this ratio larger than or of the order of 200 %, we observe that the decay of the DAF is similar to what is observed for plane waves. For ratios varying from 100 % (dashed line), to 70% (dotted line), and to 40 % (dash-dotted line), the decay is regularly reduced by a factor 2.5, if we examine the DAF at the exponential decay time. For values of the ratio between 50 % and 40 %, the decay is almost unchanged, announcing a change of behavior. For the same conditions, but different shapes of the wave packet, we have plotted in fig. 3 the DAF for four decreasing values of the ratio  L /. As this ratio is decreased, from 40 % (dash-dotted line) to 20 % (dotted line), and 10 % (dashed line), the decay is now increased, contrary to what is observed in fig. 2. For values of the ratio smaller than 5 %, corresponding to an envelope of the electric field appearing as a well isolated wave packet in each step of the renewal process, the DAF converges to a limit close to the DAF calculated for a ratio of 2 % (solid line). In this limit each wave packet contributes in an additive manner to the decay of DAF (impact limit for the field with the chosen PDF and WTD). The response of the DAF to the plasma frequency oscillation is visible in fig. 4, which shows its short time behavior. The oscillations seen are in phase with the plasma frequency, and have a maximum amplitude and duration for the plane wave case (solid line). For a ratio  L / equal to 100 % (dotted line) and 40 % (dashed line), one observes a regular decrease of the amplitude and duration of the oscillation as the width of the Lorentzian envelope is reduced. In fig. 5, it can be seen that the oscillations are reduced for a ratio of 20 % (dashed line), and that for smaller ratios the DAF converge to its limit plotted here for a ratio of 2 % (solid line). For the wave packet shapes presented in Figs. 3 and5, the DAF obtained with the narrowest Lorentzian envelopes are the least damped for early times, and the most damped for late times. This is consistent with a broadening by separated wave packets as for impact broadening. Several studies on the shape of a collapsing wave have been performed using Zhakarov equations or numerical simulations [4,28,29], suggesting that the wave packets are well separated in time during the wave packet cycle. Using the results of three dimensional collapsing wave packets, calculated numerically using the nonlinear Schrödinger equation [29], we have chosen a ratio  L / equal to 20 % in our line shape calculations. In accordance with our study presented in figs. 2-5, this value minimizes the damping effect on the DAF, and thus also the broadening effect of Langmuir turbulence on the line shape.

The line shape is obtained by a numerical Fourier transform, and is first calculated for the single effect of strong Langmuir turbulence. This pure Langmuir line shape may also be convolved with an equilibrium Stark broadening line shape, leading to a full profile which would be emitted in a plasma containing a large number of collapsing wave packets [14]. Our model assumes that the two broadening mechanisms are independent, and that it is possible to neglect the effect of density fluctuations on the Stark profile. Such turbulent fluctuations are known to affect Stark profile in inhomogeneous plasmas submitted to unstable drift waves [START_REF] Marandet | High Energy Density Phys[END_REF]30].

For the same plasma conditions already studied, and the same jumping frequency = p /50, and field E L =100 E 0 , we present in fig. 6 three different L  line profiles. We have plotted the profiles using alpha units defined by the reduced wavelength =/E 0 , expressed in angström per cgs field strength units as in standard line shape tables [START_REF] Griem | Spectral Line broadening by Plasmas[END_REF]31] The effect of strong turbulence is thus much reduced for this density, but still significant since the full profile (solid line) is 34 % broader than the Stark profile. We have performed other line shape calculations for temperatures between 10 4 and 10 5 K, and densities varying from 10 17 to 10 21 m -3 . The effect of strong Langmuir turbulence is always significant as compared to equilibrium Stark broadening on this line.

Conclusion. -

We have reviewed the main characteristics of the physical aspects of wave collapse and strong Langmuir turbulence which may affect the spectra of atomic emission in a plasma. In such conditions, many localized wave packets coexist if the plasma is coupled to an external source, and their cyclic dynamics includes a wave collapse during which the electric field reaches values well above the equilibrium plasma microfield. We have proposed a stochastic renewal process for modeling the history of an electric field experienced by an atom in conditions of strong Langmuir turbulence. The model requires the PDF for the turbulent electric field and a WTD for sampling the times of jumping from one field to the following. An exponential law is used for the WTD, with a jumping frequency taken as the inverse of the mean duration for the wave collapse cycle, and the wave packet field magnitudes are sampled along a half-normal PDF. The DAF of the hydrogen line L  has been calculated with a numerical simulation for different shapes of the wave packet, revealing different behaviors as the width of the envelope is changed. We have performed DAF and line profile calculations for a temporal width of the wave packet equal to 20 % of the step duration. This shape suggested by simulations of the wave packet collapse [4], is also among those minimizing the broadening effect on the line shape. The main conclusion of the paper is that for plasmas at about 2 eV, the Stark width of the L  line is increased in presence of strong turbulence, and this for densities between 10 18 and 10 21 m -3 . The width of the line is dependent on the parameters of strong Langmuir turbulence like the value of W, and the plasma conditions. A future direction for this work is the investigation of other plasma conditions, and the calculation of lines which are better suited for a plasma diagnostic, such as the hydrogen Balmer lines.

Fig. 1 :

 1 Fig. 1: Piece of history of one electric field component obtained by our model for a Lorentzian envelope function with a time width taken to 20 % of each time interval between two successive jumps (time in units of the inverse plasma frequency).

Fig. 2 :

 2 Fig. 2: (Color on-line) Dipole autocorrelation functions (DAF) for L  submitted to strong Langmuir turbulence calculated for a jumping frequency = p /50, and an average field E L =100 E 0 , in a plasma with a density N e = 10 18 m -3 and temperature T e = 2.8x10 4 K. We have plotted the DAF for four different shapes of the envelope (Time in S -1 units)

Fig. 4 :

 4 Fig. 4: (Color on-line) Same as fig. 2 for short times and three shapes of the Lorentzian envelope.

Fig. 5 :

 5 Fig. 5: (Color on-line) Same as fig. 3 for short times and three shapes of the Lorentzian envelope.

Fig. 6 :

 6 Fig. 6: (Color on-line) Profile of the L  line for a density of 10 18 m -3 , and a temperature of 2.8x10 4 K. The pure Stark profile (dashed line) is compared to the pure Langmuir profile (dotted line) with E L =100 E 0 and a jumping frequency = p /50, and to a full profile obtained by convolution of the two previous (solid line).

Fig. 7 :

 7 Fig. 7: (Color on-line) Same as fig. 6 for a density of 10 20 m -3 , E L =60 E 0 and = p /30.

Acknowledgments

This work is supported by the Agence Nationale de la Recherche, project SEDIBA, ANR-11-BS09-023.


Olivier Herbinet

Paul-Marie Marquaire

Frédérique Battin-Leclerc

René Fournet

Thermal decomposition of n-dodecane: experiments and kinetic modeling

Keywords: Pyrolysis, n-dodecane, Thermal decomposition, Kinetic modeling, Aromatic compounds, Jet fuels

The thermal decomposition of n-dodecane, a component of some jet fuels, has been studied in a jet-stirred reactor at temperatures from 793 to 1093 K, for residence times between 1 and 5 s and at atmospheric pressure. Thermal decomposition of hydrocarbon fuel prior the entrance in the combustion chamber is an envisaged way to cool the wall of hypersonic vehicles. The products of the reaction are mainly hydrogen, methane, ethane, 1,3butadiene and 1-alkenes from ethylene to 1-undecene. For higher temperatures and residence times acetylene, allene, propyne, cyclopentene, 1,3-cyclopentadiene and aromatic compounds from benzene to pyrene through naphthalene have also been observed. A previous detailed kinetic model of the thermal decomposition of n-dodecane generated using EXGAS software has been improved and completed by a sub-mechanism explaining the formation and the consumption of aromatic compounds.

Introduction

One of the main issue of hypersonic flight is the thermal management of the vehicle and more especially of the engine [1,2]. Studies are currently conducted in order to develop an active cooling system, called regenerative cooling, that consists in using the fuel (typically a blend of heavy hydrocarbons) heat sink capability due to the endothermicity of its thermal decomposition [2][3][4][5]. Because of the large heat load found in a hypersonic vehicle, engine temperature is expected to increase well above 800 K leading in the thermal decomposition of the fuel circulating at their walls before the injection in the combustion chamber. A consequence of the decomposition of the heavy hydrocarbon fuel is the production of lighter species and then of aromatic and poly-aromatic compounds. In order to quantify the heat transfer through the walls of the engine and the composition of the fuel entering the combustion chamber, a detailed kinetic model of the thermal decomposition of the fuel is required.

Jet fuels are composed of hundreds of aliphatic and aromatic hydrocarbon compounds of which major components are normal alkanes, branched alkanes, cycloalkanes, aromatics and alkenes. Surrogate mixtures have been developed in order to reduce the size of pyrolysis and oxidation models of jet fuels. N-dodecane, a major component of some jet fuels (kerosene, NORPAR12, JP7, JP8 [4]) is one of the surrogates usually chosen for normal alkanes [START_REF] Humer | Proceedings of the Combustion Institute[END_REF]. A very limited work was published about n-dodecane pyrolysis. Zhou et al. studied the thermal decomposition of n-dodecane in an unpacked flow reactor at atmospheric pressure and at temperatures from 623 to 893 K [START_REF] Zhou | [END_REF]. Yoon et al. performed the pyrolysis of pure n-dodecane in a batch reactor at a pressure of 10 bars and at temperatures from 673 to 523K [8,9]. More recently a new study of the thermal decomposition of n-dodecane allowed us to propose a detailed kinetic model made of 1175 reactions involving 175 molecular and radical species [10]. This model was mainly validated from experimental results obtained in a plug flow reactor at higher temperatures (950-1050 K) than the previous papers [START_REF] Zhou | [END_REF][8][9] and residence times ranging from 0.05 to 0.2 s. In order to get experimental data in a different type of reactor and for a broader range of experimental conditions the new study presented here has been performed using a jet-stirred reactor, which is a well suited apparatus for gas phase kinetic study [11][12][13][14].

Experimental method

The experimental apparatus (Figure 1) which was developed for the study of the thermal decomposition of liquid hydrocarbons involves a new device used for the liquid hydrocarbon feed and evaporation control and a quartz jet-stirred reactor (JSR).

Figure 1

Liquid n-dodecane (reactant) is contained in a glass vessel pressurized with nitrogen.

Before performing a series of experiments, oxygen traces are removed from the liquid hydrocarbon through nitrogen bubbling and vacuum pumping. Liquid hydrocarbon flow rate between 1 and 50 g.hr -1 is controlled, mixed to the carrier gas (helium) and evaporated in a Controlled Mixer and Evaporator (CEM) provided by "Bronkhorst". This apparatus is composed of a liquid mass flow controller followed by a mixing chamber and a single pass heat exchanger. Carrier gas flow rate is controlled by a "3 nL.min -1 Tylan -RDM280" gas mass flow controller set upstream the CEM. Molar composition of the gas at the outlet of the CEM is 2% hydrocarbon and 98% helium. Working with high dilution presents two advantages: the variation of molar flow rate due to the reaction is very weak and can be neglected and strong temperature gradients inside the reactor due to endothermic reactions are avoided.

The thermal decomposition of the fuel was performed in a quartz jet-stirred reactor that was developed by Matras and Villermaux [15]. This type of reactor (Figure 2) which was already used for numerous gas phase kinetic studies [11][12][13][14] is a spherical reactor in which diluted reactant enters through an injection cross located in its center and that can be considered as well stirred for a residence time (τ) between 0.5 and 5 s [15,16]. The diameter of the reactor is about 50 mm and its volume is about 90 cm 3 . The stirring of the whole reactor volume is achieved by the mean of four turbulent gas jets directed in different directions and produced by the four nozzles of the injection cross in the center of the reactor.

Inside diameter of the nozzles is about 0.3 mm. The reactor is preceded by a quartz annular preheating zone in which the temperature of the vapor mixture is increased up to the reactor temperature before entering inside. The annular preheater is made of two concentric tubes, the inter-annular distance of which is about 0.5 mm. Gas mixture residence time inside the annular preheater is very short compared to its residence time inside the reactor.

Both spherical reactor and annular preheating zone are heated by the mean of "Thermocoax" heating resistances rolled up around their wall. Temperature control of each part is made by "Eurotherm 3216" controllers and type K thermocouples. Reaction temperature measurement is made by means of a thermocouple which is located inside the intra-annular space of the preheating zone and the extremity of which is on the level of the injection cross.

Figure 2

Pressure inside the JSR is close to atmospheric pressure (about 106 kPa). It is manually controlled thanks to a control valve placed after the liquid hydrocarbon trap. 32 products of n-dodecane thermal decomposition were identified and quantified by gas chromatography and can be classified in two groups. First, light species (on-line analysis) which are hydrogen and C 1 -C 5 hydrocarbons: methane, acetylene, ethylene, ethane, allene, propyne, propene, 1,3-butadiene, 1-butene and 1-pentene. Second, C 5 -C 16 species:

1,3-cyclopentadiene, cyclopentene, benzene, 1-hexene, toluene, 1-heptene, phenyl-ethylene, 1-octene, indene, 1-nonene, naphthalene, 1-decene, 1-undecene, 1-and 2-methyl-naphtalene, biphenyl, acenaphthalene, phenanthrene, anthracene, fluoranthene and pyrene.

Light species samples are taken at the outlet of the reactor and are analyzed on-line by two gas chromatographs set in parallel. The first gas chromatograph is a "Hewlett Packard 5890" equipped with a thermal conductivity detector (TCD) for hydrogen detection, a flame ionization detector (FID) for hydrocarbons detection and a carbosphere packed column. A first analysis is made at an oven temperature of 473 K for methane and C 2 -hydrocarbons detection. A second analysis at an oven temperature of 303 K is necessary to separate hydrogen peak from helium peak (experiment carrier gas). Argon was chosen as column carrier gas and TCD reference gas in order to detect hydrogen with a better sensibility. The second gas chromatograph used for C 3 and C 4 -hydrocarbons is a "Schimadzu 14A" fitted with a FID and a Haysep-D packed column with nitrogen as carrier gas. Oven temperature profile is: 313 K held 30 min, rate 1 K.min -1 , 473 K held 30 min. Heavy species (Figure 3) are accumulated in a glass trap quenched in liquid nitrogen (about 77 K) during time typically between 5 and 15 min according to reactant feed. Then the glass trap is disconnected from the reactor and products are dissolved in a solvent (acetone).

A known amount of internal standard (n-octane) is added to the mixture which is injected by an auto-sampler in an "Agilent 6850" gas chromatograph equipped with a FID and a HP-1 capillary column with helium as carrier gas. Oven temperature is: 313 K held 30 min, rate 5 K.min -1 , 453 K held 62 min. Heavy species were identified by a gas chromatograph-mass spectroscopy (GC-MS) system working in the same conditions as the "Agilent 6850" gas chromatograph.

Figure 3

Experimental results

Experimental results have been obtained at temperatures from 793 K to 1073 K, residence times ranging from 1 to 5 s, and with a reactant mole fraction at the inlet of the reactor of 2%.

On the following graphs (Figure 4 to , Figure 6), symbols correspond to experiments and curves to modeling.

Figure 4 represents the evolution of the conversion of n-dodecane vs. residence time (Figure 4a) and vs. temperature for a residence time of 1 s (Figure 4b). It shows that the conversion strongly increases with temperature between 873 K and 1023 K.

Figure 4

Figure 5 and Figure 6 present the evolution of the mole fractions of the products of the reaction vs. residence time (Figure 5) and vs. temperature (Figure 6). The major products are hydrogen and ethylene. For most of the pyrolysis products, an increase of the residence time or of the temperature leads to an increase of the mole fraction (e.g. hydrogen, ethylene and aromatic compounds), while the mole fractions of 1-alkenes from 1-butene to 1-undecene show an increase with residence time and temperature followed by a decrease. These 1-alkenes are probably primary products which decompose when temperature or residence time increase. Primary products of n-dodecane thermal decomposition were determined from a selectivity analysis performed at 873K (selectivity is the ratio between the mole fraction of the considered product and the mole fractions of all products). In these conditions, the conversion of n-dodecane is less than 20% (maximum conversion corresponding to the residence time of 5s).

If the extrapolation to origin of the selectivity of a species vs. residence time is different from zero then this product is probably a primary product (Figure 7a). This is the case for hydrogen, methane, ethylene, ethane, propene and heavier 1-alkenes from 1-butene to 1-undecene from which extrapolated initial selectivities are summarized in Table 1. The sum of the initial selectivities of primary species is equal to 0.991 (theoretical value: 1). Species with the highest initial selectivities are ethylene, methane, propene and hydrogen. At 873 K, residence time has little effect on selectivities. Selectivities of hydrogen and 1-alkenes from 1-hexene to 1-undecene slightly decrease in a monotonous way with residence time whereas selectivity of ethane increases. Selectivities of 1-butene, methane and propene decrease first with residence time and then increase whereas selectivities of 1-pentene and ethylene present the opposite trend.

If the extrapolation of the selectivity vs. residence time gives a value close to zero, then the product may be a secondary product. This is the case for benzene (Figure 7b).

Modeling of n-dodecane thermal decomposition

A first detailed kinetic model of the thermal decomposition of n-dodecane, made of 1175 reactions involving 154 species, was built using EXGAS software [14,[17][18][19] according to rules of construction previously described [10]. Agreement between available experimental data [10] and computed results was reasonable taking into account that no kinetic parameters fitting was attempted and that only few species were quantified. Some changes were brought to this previous model: the kinetic parameters of two types of very important reactions have been modified according to data taken from literature and a sub-mechanism involving poly-aromatic compounds was added.

Comparison with available experimental data [20,21] shows that in the case of the recombination of two large radicals (containing more than two carbon atoms) the pre-exponential factors previously deduced from thermochemical data and from the A factors of the related reverse combination given by the modified collision theory [14,[START_REF] Benson | Methods for the Estimation of Thermochemical Data and Rate Parameters[END_REF] were under-estimated. Pre-exponential factor of unimolecular initiations by bond fission have been multiplied by a factor 4 (except for the unimolecular initiation involving methyl radical) in order to be in better agreement with data from the literature.

Retroene reaction is a molecular reaction which consists in a 1,5 hydrogen shift reaction followed by a dissociation [START_REF] Richard | [END_REF]. Products of the reaction are a lower 1-alkene and propene (Equation 1).

H + ‡ (1)

In the previous model [10], pre-exponential factor of retroene reactions, involving 1-alkenes (from 1-pentene) were taken equal to 8.0×10 12 cm 3 .mol -1 .s -1 (A factor used for the retroene reaction of decyclbenzene) [24]. Comparison with data from the literature shows that this value is slightly over-estimated in the case of 1-alkenes [25,26]. The value proposed by Tsang [25] and King [26] (4.0×10 12 cm 3 .mol -1 .s -1 ) has been used in the model presented here.

A set of 273 elementary reactions involving 117 new species (mainly aromatics, poly-aromatics and related radical species) has been added to the modified existing model in order to explain the formation and the consumption of species such as cyclopentene, 1,3-cyclopentadiene and aromatic compounds from benzene to pyrene. These reactions have been described in two papers about propane pyrolysis [27,28]. It appears that cyclopentadiene, benzene and their related radicals (cyclopentadienyl, phenyl and benzyl radicals) play an important role in the formation of heavier aromatic and poly-aromatic species. The model explains the formation of cyclopentadienyl radical and cyclopentadiene from the addition of acetylene on propargyl radical (C 3 H 3 ) [29]. Benzene is formed through three pathways (Figure 8): the combination of two propargyl radicals or of propargyl and allyl radicals (C 3 pathway) [29,30], the combination of cyclopentadienyl and methyl radicals through the formation of methyl-cyclopentadiene (C 5 pathway) [31][32][33][START_REF] Lifshitz | Proceedings of the Combustion Institute[END_REF], and the addition of acetylene to C 4 radicals followed by a ring closure (C4 pathway) [START_REF] Westmoreland | [END_REF].

Figure 8

Validation of the model of the pyrolysis of n-dodecane

Simulations were run with CHEMKIN II software [START_REF] Kee | Sandia Laboratories Report[END_REF] using the detailed kinetic mechanism presented here. Mole fractions of n-dodecane pyrolysis products have been computed as a function of the residence time for a temperature of 973 K (Figure 5) and as a function of the temperature for a residence time of 1 s (, Figure 6).

The agreement between computed and experimental mole fractions vs. residence time is globally satisfactory in the studied conditions both for conversion of n-dodecane (Figure 4a) and for the formation of hydrogen, ethylene, methane, 1,3-butadiene, indene, naphthalene and other 1-alkenes (Figure 5). Mole fraction of benzene is over-predicted for smaller residence times. Mole fractions of cyclopentene, toluene, styrene, propyne and allene are slightly under-predicted. Mole fractions of cyclopentadiene and acetylene are slightly over_predicted for short residence times and slightly under-predicted for longer residence times. Simulated conversion vs. temperature for a residence time of 1 s (Figure 4b) is in a good agreement with experimental results over a wide range of temperature (848-1073 K).

Agreement between computed molar fractions and experimental data is satisfactory (, Figure 6) except for hydrogen, ethane, benzene cyclopentadiene which are slightly over-predicted and for anthracene which is under-predicted by a factor 5. The mechanism reproduces with a good accuracy the increase and the decrease of the mole fraction of 1-alkenes (from 1-pentene to 1-undecene) with increasing T.

Figure 9 displays a comparison between simulations using the new and the previous models and the experimental results obtained in a plug flow reactor between 950 and 1050 K that are previously used for validations. The model presented here leads to a higher conversion than the previous one and than experimental values, whereas it represents much better the evolution of mole fractions of main products (methane, ethylene, ethane, propene and 1-alkenes such 1-hexene) vs. residence time.

Figure 9 6. Analysis of the mechanism

Flow rate analysis has been performed for a temperature of 973 K and a residence time of 1 s corresponding to a n-dodecane conversion around 55%. As the flow rate analyses concerning the conversion of n-dodecane and the formation of the main products are similar to those previously published [10], the flow rate analysis presented in Figure 10 only shows the main pathways leading to the formation of aromatic and poly-aromatic species involved in the present mechanism. The arrows in the diagram represent the flow of consumption of the reacting species through this channel (as a percentage). This diagram underlines the importance of cyclopentadienyl, phenyl and benzyl radicals in the formation of heavier aromatic and poly-aromatic compounds. Cyclopentadienyl radical (which is mainly obtained by addition of acetylene on the propargyl radical followed by a cyclisation) is mainly consumed to form benzene through the C 5 -pathway (Figure 8). Over-prediction of the mole fractions of benzene and cyclopentadiene vs. temperature by the model are related.

Cyclopentadienyl radical is also consumed to form benzyl radical by addition to acetylene [START_REF] Colket | Reaction mechanisms for toluene pyrolysis. 25 th International Symposium on Combustion[END_REF] and to produce naphthalene by combination with itself followed by several rearrangements [START_REF] Melius | [END_REF]. Phenyl radical mainly reacts by metatheses to give back benzene. It is also consumed through minor pathways to form styrene by addition to ethylene, phenyl-acetylene by addition to acetylene and biphenyl by addition to benzene. Benzyl radical is consumed to form toluene by metathesis and combination with hydrogen radical, ethyl-benzene by combination with methyl radical and indene by addition of acetylene. Indene reacts by metatheses to lead to indenyl radical which is then consumed to form phenanthrene by combination with cyclopentadienyl radical and methyl-indene by combination with methyl radical. Methyl-indene reacts by metatheses to form methyl-indenyl radical, which leads to naphthalene by an intramolecular rearrangement. Naphthalene is consumed through metatheses to give radicals leading to acenaphthalene (addition of acetylene) and methyl-naphtalene (combination with methyl radical). Phenanthrene leads to pyrene (through metatheses and addition on acetylene) and to anthracene.

Figure 10

that the conversion of n-dodecane is mainly controlled by unimolecular initiations and by H-abstractions from reactant with methyl radicals at low temperature and by H-abstractions from reactant with hydrogen atoms at high temperature (Figure 11a). Combination of two methyl radicals is important at low temperature because the reaction is competitive with metatheses involving methyl radicals.

The formation of long 1-alkenes is mainly ruled by the reaction of decomposition by β-scission of linear dodecyl radicals obtained from metatheses of radicals with n-dodecane.

Two types of reactions have a great influence on their consumption as shown in Figure 11b for 1-hexene: the decomposition by retroene reaction and the decomposition by unimolecular initiation reaction. Unimolecular initiation (energy of activation of 70.7 kcal.mol -1 ) becomes more important compared to retroene (energy of activation of 57.5 kcal.mol -1 ) when temperature increases.

As far as benzene is concerned sensitivity analysis shows that it is very influenced by the reaction leading from the methyl-cyclopentadiene to the methyl-cyclopentadienyl radical by H-abstraction with hydrogen and methyl radicals at low and high temperatures. At low temperature (873 K) unimolecular initiations of n-dodecane, the reaction of combination of two methyl radicals (this reaction is competitive with the reaction of combination of methyl and cyclopentadienyl radical) and the reactions of metathesis involving ethyl radical and n-dodecane (this reaction forms ethane which indirectly leads to intermediate species in the formation of benzene) have a great influence on benzene formation (Figure 11c).

Figure 11

Conclusion

The thermal decomposition of n-dodecane has been studied in a jet-stirred reactor over a wide range of temperatures from 793 K to 1073 K and for residence times between 1 and 5 s. 32 products of the reaction have been analyzed. Major products are hydrogen and ethylene.

Other products are mainly 1-alkenes from propene to 1-undecene and aromatic compounds.

The formation of aromatic compounds intervenes rather quickly; it takes importance for conversions higher than 25 %. The formation of large amount of aromatic and polyaromatic compounds may me troublesome: these species are precursors of the formation of soot and even coke at high temperature [39Mendiara et al., 2005] which can generate a solid deposit in the cooling channel (before the injection in the combustion chamber) and a fouling in the combustion chamber.

A detailed kinetic model generated by the EXGAS software has been completed by a set of elementary reactions involving aromatic and poly-aromatic compounds. The new model obtained is made of 1449 reactions and contains 271 species. It is expected that this detailed kinetic model (composed of elementary steps) will be accurate enough for temperatures from 500 to about 1500 K.

Agreement between new experimental results presented here and computed results is satisfactory especially for primary products such 1-alkenes. As far aromatic compounds are concerned a light variation between experiments and modeling is observed especially for long residence times. This may be related to uncertainties on kinetic parameters of reactions involved in the formation of benzene (C5-pathway) and poly-aromatics compounds. The new model reproduces well experimental data presented in a previous paper [10] and obtained with a different type of reactor working in different conditions. Simulations of the cooling a engine will be performed from the model of the thermal decomposition of n-dodecane which will be coupled with computational fluid dynamics codes. The size of the detailed kinetic model (hundreds of species and thousands of reactions) involves very long computational times. Models can be reduced and adapted to have lower calculation times. The lumping is one of the methods which can be used for the reduction of detailed kinetic models. It consists in gathering molecules which have the same formula, present identical functional groups and include rings of the same size into one species. This method was used in the secondary mechanism of the present model and applied to linear alkanes and dienes which are formed in small quantities [10]. A second way of reducing models is to eliminate redundant species and reactions from a kinetic analysis (rate of production and rate sensitivity analyses) of the mechanism. In some cases this method allows to eliminate one third of the species and two thirds of the reactions [START_REF]Mathematical tools for the construction, investigation and reduction of combustion mechanisms[END_REF]. Reduction based on the investigation of time-scales consists in identifying QSSA species and then to eliminate them by solving the QSSA expressions or by writing global reactions [START_REF]Mathematical tools for the construction, investigation and reduction of combustion mechanisms[END_REF]. At least another method for the reduction of the kinetic models is the mathematical fitting (by a system of ordinary differential equations for example) of simulated results obtained with the detailed model [START_REF]Mathematical tools for the construction, investigation and reduction of combustion mechanisms[END_REF].

Figure captions:

Figure 5 ,

 5 Figure 5, Figure 6

Figure 1 :

 1 Figure 1 : Experimental apparatus flow sheet.

Figure 2 :

 2 Figure 2 : Scheme of the jet stirred reactor and the annular preheating.

Figure 3 :

 3 Figure 3 : Heavy products separation (HP-1; FID).

Figure 4 :

 4 Figure 4 : Conversion of n-dodecane versus (a) residence time and (b) temperature (τ=1s).

Figure 5 :,Figure 6 :

 56 Figure 5 : Evolution of products mole fraction with residence time (T=973 K).

Figure 8 :

 8 Figure 8 : The three pathways to benzene formation.

Figure 9 :

 9 Figure 9 : Comparison between previous experimental data (plots) and modeling using the

Figure 10 :Figure 11 :

 1011 Figure 10 : Aromatics and poly-aromatics consumption flux analysis (T=973 K; τ=1s)

  Figure 2

  Figure 3

  Figure 8

Table 1 , Figure 7

 1

Table 1

 1 Initial selectivities of primary products obtained by extrapolation (T=873 K).

	Primary product	Initial selectivity
	hydrogen	0.101
	methane	0.135
	ethylene	0.322
	ethane	0.030
	propene	0.124
	1-butene	0.065
	1-pentene	0.040
	1-hexene	0.050
	1-heptene	0.035
	1-octene	0.030
	1-nonene	0.026
	1-decene	0.024
	1-undecene	0.009
	total	0.991

Acknowledgments

This work was supported by MBDA-France and the CNRS. The authors are very grateful to E. Daniau, M. Bouchez and F. Falempin for helpful discussions.


Olivier Léon
email: olivier.leon@onera.fr

Estelle Piot

Delphine Sebbane

Frank Simon

Frank Simon Mesure

Mesure de vitesse acoustique dans un écoulement de conduite turbulent par LDV et TR-PIV

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

Introduction

La réduction des nuisances sonores issues de turboréacteurs est un enjeu important pour l'industrie aéronautique, requérant le développement de techniques de mesure et de localisation du bruit. Si l'utilisation de microphones pour de la métrologie acoustique dans un environnement au repos est une solution technique adéquate, plusieurs difficultés surviennent lorsqu'il s'agit de réaliser des mesures acoustiques en présence d'un écoulement. En effet, en premier lieu, la turbulence aérodynamique peut significativement bruiter le signal acoustique, requérant alors l'utilisation de méthodes de filtrage et de traitement spécifiques. De plus, l'utilisation de sondes microphoniques placées en écoulement est par nature intrusive, pouvant altérer les caractéristiques de ce dernier. Enfin, l'utilisation de sondes placées en paroi, sur une maquette ou dans une veine d'essai, permet généralement d'obtenir des informations sur le champ acoustique et les sources associées, mais n'est parfois pas réalisable expérimentalement et ne permet pas une mesure volumique directe. Afin de répondre à ces problématiques, divers développements techniques ont été proposés au cours des dernières décennies, dont plusieurs reposant sur des méthodes optiques [START_REF] Valière | Acoustic Particle Velocity Measurements Using Laser: Principles, Signal Processing and Applications[END_REF]. Particulièrement, Minotti et al [START_REF] Minotti | Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow[END_REF] à l'ONERA ont développé une approche basée sur la vélocimétrie laser à frange (LDV) dans le but de mesurer simultanément de manière ponctuelle les vitesses acoustiques de particules (APV pour Acoustic Particule Velocity) et les caractéristiques turbulentes de l'écoulement pour des études de traitements acoustiques de nacelles, appelés liners acoustiques. Leur méthode de mesure de composantes de vitesse acoustique en présence d'écoulement repose sur l'estimation de l'intercorrélation dans le domaine spectral du signal de vitesse mesuré par LDV avec un signal de référence fortement corrélé au signal acoustique recherché (fluctuations de pression en paroi ou signal d'excitation acoustique). Par la suite, ces auteurs ont appliqué avec succès leur approche à la mesure de composantes de vitesse acoustique au-dessus de liners pour diverses conditions d'écoulements rasants et niveaux sonores [START_REF] Lavieille | Measurement of acoustic quantity fields in duct flow by Laser Doppler Velocimetry[END_REF][START_REF] Betgen | Implementation and non-intrusive characterization of a hybrid activepassive liner with grazing flow[END_REF]. Plus récemment, Haufe et al [START_REF] Haufe | Multi-scale measurement of acoustic particle velocity and flow velocity for liner investigations[END_REF] ont quant à eux mis en oeuvre une méthode de mesure par vélocimétrie Doppler globale avec modulation de fréquence (FM-DGV) pour des applications similaires. La méthode d'estimation mise en oeuvre par ces auteurs repose quant à elle sur l'estimation directe de paramètres d'amplitude et de phase d'une onde au moyen de théories de l'information. Ces auteurs ont ainsi mis en évidence l'applicabilité de leur technique pour des mesures d'APV engendrées par des excitations acoustiques à des niveaux sonores supérieurs à 120dB dans des écoulements de conduite jusqu'à M=0.3. Même si cette approche autorise une mesure simultanée en plusieurs points de manière a priori moins complexe qu'avec une technique LDV, la résolution spatiale de cette approche reste cependant relativement faibleles résultats présentés étant contenus dans une grille de 23x7 points de mesures. Afin d'obtenir une résolution spatiale plus importante à moindre coût, il peut être intéressant d'utiliser une technique de mesure de type PIV (Particle Image Velocimetry). Entre autres, Fischer et al [START_REF] Fischer | Acoustic PIV: Measurement of the acoustic particle velocity using synchronized PIV-technique[END_REF] ont ainsi explorés l'utilisation de la PIV afin d'étudier le champ de vitesse acoustique induit au-dessus de liners. Comparée aux approches précédentes, la technique mise en oeuvre par ces auteurs ne permet pas une résolution temporelle suffisante permettant d'utiliser les méthodes d'estimations évoquées, imposant alors un travail en moyenne de phase. Si une telle approche s'avère efficace pour des excitations mono-fréquentielles, son extension à des excitations multi-fréquentielles semble non triviale de par la complexité potentielle du signal temporel de synchronisation.

Les travaux ici présentés visent à reprendre la méthode développée par Minotti et al [START_REF] Minotti | Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow[END_REF] afin d'étudier son extension à la mesure par TR-PIV, qui permettrait alors de réaliser une estimation des composantes de vitesse acoustique directement dans un plan, avec une résolution spatiale élevée, contrairement à l'approche développée par Haufe et al [START_REF] Haufe | Multi-scale measurement of acoustic particle velocity and flow velocity for liner investigations[END_REF]. L'intérêt de cette approche face à celle développée par Fischer et al [START_REF] Fischer | Acoustic PIV: Measurement of the acoustic particle velocity using synchronized PIV-technique[END_REF] réside dans son applicabilité directe au cas communément étudié d'un écoulement turbulent portant une excitation acoustique multi-fréquentielle. En effet, la technique ici étudiée s'appuie sur l'estimation d'interspectres entre la mesure de vitesse résolue en temps et un signal de fluctuation de pression acoustique en paroi : cette approche permet d'isoler dans l'espace spectral les composantes de vitesse associées aux ondes acoustiques introduites dans la conduite tout en opérant une réjection de la turbulence stochastique. Typiquement, comme montré par la suite, une mesure de vitesse acoustique dans un écoulement présentant un taux de turbulence de l'ordre de 5 % et pour des niveaux sonores de 130dB, soit un rapport signal sur bruit d'environ -20dB, est réalisable en supposant l'acquisition d'un nombre d'échantillons suffisant.

Cet article est organisé de la manière suivante. La section 2 détaille la méthode d'estimation utilisée pour l'évaluation des composantes de vitesse acoustique et apporte une première estimation de l'incertitude de la mesure en fonction du rapport signal sur bruit et de du nombre d'échantillons considérés. La section 3 développe ensuite la mise en oeuvre expérimentale de cette approche sur un cas simple d'écoulement de conduite pleinement turbulent portant une onde acoustique plane. Les statistiques de l'écoulement sont alors données dans la section 4. Enfin, la section 5 regroupe des résultats d'estimation de vitesse acoustique pour les mesures par LDV et TR-PIV.

Méthode d'estimation de la composante de vitesse acoustique

Nous cherchons ici à donner un estimateur robuste des amplitudes RMS et des différences de phase (par rapport à un signal de référence) de composantes de vitesse acoustique lorsqu'une excitation multi-tonale est introduite dans un écoulement de conduite pleinement turbulent.

Théorie

En notant le champ de vitesse 𝒖 = {𝑢 𝑖 } 𝑖∈⟦1,3⟧ 𝑇 , nous pouvons décomposer celui-ci de telle sorte que 𝒖(𝑥, 𝑡) = 𝒖 0 (𝑥) + 𝒖 ̃(𝑥, 𝑡) + 𝒖 ′ (𝑥, 𝑡) + 𝒃(𝑥, 𝑡) avec 𝒖 0 le champ moyen, 𝒖 ̃ le champ de vitesse acoustique engendré par l'excitation, 𝒖 ′ le champ de fluctuations turbulentes et 𝒃 le bruit inhérent à la mesure. Le champ de vitesse acoustique instantané associé à une excitation multi-tonale peut s'écrire de la façon suivante :

𝑢 ̃𝑖(𝒙, 𝑡) = ∑ √2 𝐾 𝑘=1 𝐴 𝑢 ̃𝑖 (𝒙, 𝜔 𝑘 ) cos[𝜔 𝑘 𝑡 + 𝜑 𝑢 ̃𝑖𝑠 (𝒙, 𝜔 𝑘 ) + 𝜑 𝑠 (𝜔 𝑘 )]
avec 𝐴 𝑢 ̃𝑖 (𝒙, 𝜔 𝑘 ) l'amplitude RMS de vitesse acoustique à la pulsation 𝜔 𝑘 , 𝜑 𝑢 ̃𝑖𝑠 (𝒙, 𝜔 𝑘 ) la différence de phase par rapport à un signal de référence 𝑠(𝑡) à la pulsation 𝜔 𝑘 , 𝜑 𝑠 la phase du signal de référence et 𝐾 le nombre total d'ondes acoustiques planes considérées.

Dans le cadre de ce formalisme, nous supposerons que les fluctuations turbulentes 𝒖 ′ sont totalement décorrélées des composantes de vitesse acoustique 𝒖 ̃, ce qui est le cas lorsque les perturbations acoustiques n'altèrent pas la dynamique turbulente de l'écoulement et lorsque la turbulence elle-même n'engendre pas d'acoustique.

Avec ces hypothèses, l'amplitude RMS et la différence de phase des composantes de vitesse acoustique, respectivement 𝐴 𝑢 ̃𝑖 et 𝜑 𝑢 ̃𝑖𝑠 , peuvent être obtenue par évaluation de la fonction d'interspectre 𝐺 𝑢 𝑖 𝑠 (𝒙, 𝜔 𝑘 ) entre le signal de vitesse mesuré 𝑢 𝑖 (𝒙, 𝑡) et le signal de

Estimation de l'incertitude en présence de bruit blanc Gaussien

En pratique, puisque le nombre d'échantillons de mesure 𝑁 est limité, l'estimation de l'interspectre de 𝑢 𝑖 avec 𝑠 sur un intervalle de temps donné, soit 𝐺 ̂𝑢𝑖 𝑠 , présentera une dépendance à la turbulence de l'écoulement et au bruit de mesure. Les relations théoriques précédentes ne seront ainsi pas exactement vérifiées. Par souci de simplicité, considérons que les deux variables aléatoires 𝑢 𝑖 ′ et 𝑏 peuvent être modélisées par un bruit blanc Gaussien 𝑤 = 𝑁(0, 𝜎 𝑤 2 ), et considérons une seule onde acoustique plane, ces résultats étant indépendant du nombre d'ondes acoustiques 𝐾. Dans un tel cas [START_REF] Kay | Fundamentals of statistical signal processing[END_REF], la déviation standard minimum que l'on peut espérer en présence d'un bruit blanc Gaussien pour l'estimation de 𝐴 𝑢

Mise en oeuvre expérimentale

Afin de mettre en oeuvre les développements théoriques précédents et d'en valider les résultats pour les différentes moyens métrologiques ici considérés (TR-PIV et LDV), nous présentons le cas simple de l'estimation des composantes d'une onde acoustique mono-fréquentielle dans un écoulement pleinement turbulent dans une conduite entièrement rigide. Des travaux démontrant l'extension directe de cette approche à l'estimation de composantes de vitesse acoustique pour des excitations multi-tonales et en présence de liners acoustique par métrologie LDV sont rapportés par Minotti et al [START_REF] Minotti | Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow[END_REF] et Betgen et al [START_REF] Betgen | Implementation and non-intrusive characterization of a hybrid activepassive liner with grazing flow[END_REF].

Conditions d'essai

Les mesures ici présentées ont été réalisées sur le banc aéro-thermo-acoustique de l'ONERA Toulouse. Une représentation schématique de ce dernier est donnée par la figure 2. La section de la veine d'essai est carrée, de dimensions 50mm x 50mm (2ℎ × 2ℎ

L [dB] |𝑝 ′ | 𝑟𝑚𝑠 [Pa] |𝑢 ′ | 𝑟𝑚𝑠 [m/

Mesures de pression

La section de sortie de la veine n'étant pas parfaitement anéchoïque, des ondes acoustiques de retour sont engendrées par les ondes aller issues de l'excitation imposée. Ainsi, avant de réaliser des mesures par vélocimétrie laser, des mesures d'amplitude des ondes acoustiques aller ont été réalisées au moyen de sondes microphoniques placées en parois. Ces amplitudes sonores sont celles données dans le tableau 1. Deux microphones B&K 4182 espacés de 28mm sont ainsi utilisés afin de réaliser ce tri d'onde [START_REF] Bodén | Influence of errors on the twomicrophone method for measuring acoustic properties in ducts[END_REF]. Ces mesures microphoniques permettent également d'estimer l'évolution théorique de l'amplitude et de la phase du système d'ondes de pression acoustique aller-retour dans la veine d'essai. En supposant un écoulement uniforme, nous pouvons alors évaluer les composantes théoriques de vitesse acoustique associées et comparer ces résultats aux mesures par LDV et TR-PIV.

Système LDV

Un système LDV 2 composantes a été monté sur un banc de déplacement

Statistiques de l'écoulement

Les grandeurs statistiques de l'écoulement de conduite considéré ont été mesurées au moyen du système PIV, en l'absence d'excitations acoustiques et à fréquence d'acquisition réduite. Le champ moyen de vitesse axiale rendu sans dimension par la vitesse débitante est donné par la figure 3a, montrant une vitesse sur l'axe de l'ordre de 1.2 𝑈 𝑏 . Dans la zone de mesure considérée, nous pouvons observer qu'une approximation de l'écoulement par un champ de vitesse uniforme reste raisonnable. Les amplitudes RMS de fluctuation de vitesse axiale sont données par la figure 3b, montrant ainsi l'état pleinement turbulent de l'écoulement et des taux de turbulence en centre veine de l'ordre de 5%, s'accroissant vers les parois. Les estimations de composantes de vitesse acoustique issues des mesures TR-PIV sont présentées sous forme de champs et des coupes à trois positions transverses 𝑦 sont tracées. Les intervalles de confiance à 90% associés à ces dernières estimations sont obtenus par une méthode de type bootstrap et sont représentés par des courbes pleines semi-transparentes grises. Les résultats issus des mesures LDV sont globalement en très bon accord avec les résultats de mesure TR-PIV, les écarts observés pouvant être attribués à de légères différences de niveaux sonores d'excitation, les mesures ayant été réalisées à plusieurs semaines d'écart. Les résultats théoriques basés sur des mesures microphoniques en amont de la zone mesure sont également en bon accord et montrent que les disparités de vitesse observées sont globalement inférieures à des amplitudes sonores équivalente de ±0.5dB, classiquement observées lors de mesures microphoniques. Des résultats équivalents sont observés sur l'ensemble des cas listés dans le tableau 1.

Conclusions

Ces travaux montrent d'une part l'applicabilité à la mesure par TR-PIV de la méthode d'estimation de composantes de vitesse acoustique dans un écoulement de conduite pleinement turbulent, telle que développée par Minotti et al [START_REF] Minotti | Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow[END_REF]. D'autre part, il est montré que cette approche est asymptotiquement efficace. Par comparaison à des résultats issus de mesure LDV et microphoniques, nous concluons sur la faisabilité de l'approche pour des niveaux sonores jusqu'à 130dB et un taux de turbulence d'environ 5%, soit un rapport signal sur bruit d'environ -20dB. La suite de ces travaux cherchera à appliquer cette méthode en proche paroi pour l'étude de liners acoustiques avec écoulement rasant en présence d'excitations multi-fréquentielles.

Figure 1 :

 1 Figure 1 : Déviation standard des estimateurs d'amplitude RMS 𝐴 ̂ et de différence de phase 𝜑 ̂ pour différents rapports signal-sur-bruit (en dB) ; les lignes noires continues représentent les limites de Cramér-Rao obtenues pour 3 nombres d'échantillons N ; 200 simulations indépendantes sont utilisées pour l'estimation des déviations standards représentées par des symboles (+).

Figure 2 :

 2 Figure 2 : Représentations schématiques (a) du banc aéro-thermo-acoustique à l'ONERA Toulouse et (b) du champ de mesure dans la cellule d'essai Tableau 1 : Amplitudes sonores d'excitation acoustique explorées pour la mise en oeuvre de l'estimation de composantes de vitesse acoustique par vélocimétrie laser (LDV et TR-PIV), et paramètres des conditions d'essais.

  (a) Vitesse axiale moyenne (b) Fluctuations de vitesse axiale

Figure 3 :

 3 Figure 3 : Statistiques de l'écoulement au centre de la cellule d'essai

Figure 4 :

 4 Figure 4 : Estimations des composantes de vitesse acoustique (amplitude et phase) pour deux niveaux d'excitation sonore ; Résultats issus des mesures TR-PIV (champs et lignes de couleur) comparés aux résultats obtenus par LDV (croix) et aux solutions théoriques (tiret-pointillé). 5 Résultats d'extraction de vitesse acoustique La méthode développée dans la section 2 est appliquée aux mesures réalisées par TR-PIV et LDV pour l'ensemble des cas donnés par le tableau 1. Par souci de concision, seuls les résultats d'estimation d'amplitude et de phase des vitesses acoustiques pour les plus faibles excitations

  référence 𝑠(𝑡). En effet, par linéarité des opérateurs, la fonction d'interspectre s'écrit 𝐺 𝑢 𝑖 𝑠 (𝒙, 𝜔 𝑘 ) = 𝐺 𝑢 ̃𝑖𝑠 (𝒙, 𝜔 𝑘 ) + 𝐺 𝑢 𝑖 ′ 𝑠 (𝒙, 𝜔 𝑘 ) + 𝐺 𝑏 𝑖 𝑠 (𝒙, 𝜔 𝑘 ) avec théoriquement 𝐺 𝑢 𝑖 ′ 𝑠 = 0 et 𝐺 𝑏 𝑖 𝑠 = 0 en suivant les hypothèses précédentes. Les grandeurs recherchées s'expriment alors de la manière suivante,

	𝐴 𝑢 ̃𝑖 (𝒙, 𝜔 𝑘 ) =	|𝐺 𝑢 𝑖 𝑠 (𝒙, 𝜔 𝑘 )| √|𝐺 𝑠𝑠 (𝜔 𝑘 )|
	𝜑 𝑢 ̃𝑖𝑠 (𝒙, 𝜔

𝑘 ) = 〈𝐺 𝑢 𝑖 𝑠 (𝒙, 𝜔 𝑘 )〉 avec |. | désignant l'opérateur de norme et 〈. 〉 de phase d'un nombre complexe.

Dans ces travaux, l'estimateur 𝐺 ̂𝑢𝑖 𝑠 de la fonction d'interspectre 𝐺 𝑢 𝑖 𝑠 est obtenu par une méthode de type WOSA (Welch's Overlapped Segmented Average). Comme l'estimation de l'amplitude et de la phase de la composante acoustique est réalisée dans le domaine fréquentiel, la richesse du contenu spectral n'est pas un facteur limitant tant que le théorème d'échantillonnage de Nyquist-Shannon est respecté et que fréquences des composantes acoustiques sont bien séparées dans l'estimation des fonctions d'interspectre.

  ̃𝑖 et et 𝜑 𝑢 ̃𝑖𝑠 est donnée par la limite inférieure de Cramér-Rao (CRLB pour Cramér-Rao lower bound). Celle-ci s'exprime pour ces deux estimateurs Afin d'illustrer ces résultats et d'estimer l'efficacité de la méthode d'extraction de la composante de vitesse acoustique présentée, nous considérons ici le cas synthétique d'un signal échantillonné à la fréquence 𝑓 𝑠 et défini par 𝑢(𝑡 𝑛 ) = √2𝐴 𝑟𝑚𝑠 cos(2𝜋𝑓𝑡 𝑛 + 𝜑) + 𝑤(𝑡 𝑛 ) où 𝑓 est la fréquence du signal synthétique, 𝑤 est du bruit blanc gaussien et 𝑡 𝑛 = 𝑛/𝑓 𝑠 avec 𝑛 ≤ 𝑁, 𝑁 étant le nombre total d'échantillons considérés. Pourvu que le théorème d'échantillonnage est vérifié, les valeurs de 𝑓 𝑠 et de 𝑓 n'importent pas. Le signal de référence utilisé est défini par 𝑠(𝑡 𝑛

		𝜎 𝐴 ̂𝑢 ̃𝑖 ≥	𝜎 𝑤 √𝑁
	𝜎 𝜑 ̂𝑢 ̃𝑖𝑠 ≥	1 √𝑁	𝜎 𝑤 𝐴 𝑢 ̃𝑖
	où 𝑁 est le nombre total d'échantillons. En écrivant le rapport signal sur bruit 𝑆𝑁𝑅 = 𝐴 𝑢 ̃𝑖 2 /𝜎 𝑤 2 , ces
	bornes s'écrivent alors
	𝜎 𝐴 ̂𝑢 ̃𝑖 𝐴 𝑢 ̃𝑖 ≥	1 √𝑁	10 -𝑆𝑁𝑅[𝑑𝐵]/20
	𝜎 𝜑 ̂𝑢 ̃𝑖𝑠 ≥	1 √𝑁	10 -𝑆𝑁𝑅[𝑑𝐵]/20

  ). La cellule d'essai est connectée à un pavillon exponentiel et une chambre assourdie assurant une sortie quasianéchoïque. Le repère orthonormé direct de mesure est défini tel que 𝑥 soit aligné avec l'axe veine, 𝑦 dans la direction transverse et 𝑂 au centre de la cellule d'essai.Les mesures sont réalisées sous atmosphère normale, avec une régulation en température de l'écoulement à la température ambiante 𝑇 0 = 20 °C, donnant une vitesse du son 𝑐 0 = 343 m/s. Le débit massique est régulé à 𝑄 = 100 g/s, donnant une vitesse débitante de 𝑈 𝑏 = 33.2 m/s et un nombre de Mach débitant 𝑀 𝑏 = 0.1. Le nombre de Reynolds basé sur la vitesse débitante et la demi-hauteur de la section d'essai est alors 𝑅𝑒 𝑏 = 8.27 × 10 5 .Un ensemencement par particules de silice amorphe est utilisé pour les deux techniques de mesure LDV et TR-PIV. Cet ensemencement est injecté en amont de la chambre de tranquillisation et d'un redresseur d'écoulement afin d'assurer son homogénéité.Une excitation acoustique à une fréquence 𝑓 𝑎 = 800 Hz est imposée par les deux haut-parleurs à plusieurs niveaux sonores 𝐿 définis dans le tableau 1 et tels que

  3 axes dans une configuration de diffusion avant afin de maximiser le rapport signal-sur-bruit et d'assurer une cadence d'acquisition élevée, supérieure à 15kHz. Ce système LDV est composé d'un laser argon Spectra-Physics d'une puissance de 1W. Les signaux sont traités par un analyseur spectral IFA755 et les statistiques de vitesse sont obtenues au moyen du logiciel ASSA[START_REF] Micheli | Assa, un outil de référence pour le traitement du signal en vélocimétrie laser[END_REF] développé à l'ONERA. Les dimensions du volume de mesure sont estimées à 0.05 mm x 0.7 mm. Un minimum de 200,000 échantillons est acquis pour chaque point de mesure.Les mesures TR-PIV ont été réalisées au moyen d'un système composé d'un laser Litron LDY-304, utilisé à une fréquence de répétition 𝑓 𝑙 = 3 kHz et avec un temps inter-pulse ∆𝑡 = 15 𝜇s, et d'une caméra Phantom v711 utilisée en pleine résolution (1280 px x 800 px). Un objectif macro Sigma de 180mm de focale monté sur cette dernière permet d'obtenir un champ de mesure de dimensions 1.6h x 1.0 h. Ce système est arrangé en configuration 2D2C afin de mesurer les composantes de vitesse dans le plan (𝑥𝑦) à 𝑧 = 0, comme représenté sur la figure2b. Le posttraitement des acquisitions est réalisé au moyen du logiciel FOLKI-SPIV[START_REF] Champagnat | Fast and accurate piv computation using highly parallel iterative correlation maximization[END_REF], développé à l'ONERA et permettant un traitement précis et rapide sur GPU. En effet, pour chaque cas considéré dans le tableau 1, un minimum de 22,000 paires d'images est enregistré afin d'obtenir des statistiques acceptables pour l'évaluation des composantes de vitesse acoustique, comme illustré dans la section 2.2.

	3.4 Système TR-PIV