J-F Bercher
email: jf.bercher@esiee.fr

A Diet
email: a.diet@esiee.fr

C Berland
email: c.berland@esiee.fr

G Baudoin
email: g.baudoin@esiee.fr

M Villegas
email: m.villegas@esiee.fr

Monte Carlo Estimation of Time Mismatch Effect in an OFDM EER Architecture

New 5 GHz Wireless Local Area Networks standards uses OFDM modulation in order to increase data rate transfer. OFDM transmitter needs linearization technique due to non-linearities of the power amplification operation. EER architecture can be used to solve this problem while keeping a high efficiency. However several sources of imperfections can lower the quality of the signal. Time mismatch has especially a great impact. This paper presents a Monte Carlo study of envelope/phase delay influence on the OFDM signal. The Autocorrelation is estimated considering the OFDM signal as complex Gaussian.

I. Introduction

New 3 rd Generation standards such as Hiperlan2 or IEEE 802.11a uses OFDM (Orthogonal Frequency Division Multiplex) at 5 GHz. The advantages are a high data rate transfer and robustness in multi-paths environment. Each of the sub-carriers (64 for Hiperlan2) uses a QAM modulation scheme. The high disadvantage of OFDM is that the envelope of the emitted signal exhibits a large amplitude range. Consequently the power is un-constant which results in distortions caused by the non-linearities in the radiofrequency transmitter. Linearization methods are necessary. EER (Envelope Elimination and Restoration) introduced by Kahn in 1957 [START_REF] Kahn | Single sideband transmission by envelope elimination and restoration[END_REF] is a solution to linearize the transmitter while keeping high efficiency. EER is based on the decomposition of the emitted signal in a magnitude signal (envelope varying) and a phase signal (constant power). Each signal is amplified separately. Recombination of the information is done by supply modulation of the high efficiency RF power amplifier (PA) as shown on Fig. 1.

The design of such architecture needs to precisely quantify the spectral effects of the previous defaults. A possible approach would be to study statistically the power spectrum obtained at the output of a simulated system, while varying values of the potential defaults. In fact, the impact of these defaults can be directly analysed using simulated signal generated with the same statistical properties as those of the potential outputs of the system. Hence, the results are obtained without resorting on a complete simulation of the whole system, that is interesting as far as computational load and simulation duration are concerned. It is also interesting in the fact that it focuses on the actual defaults under concern and not on other 'hidden' defaults on the simulation process.

II. Simulations

What we are ultimately interested in is the emitted spectrum. By Wiener-Kintchine theorem, it is the Fourier transform of the autocorrelation function ( ) τ Rxx . Because of time mismatch between envelope and phase components, the actual emitted signals is

( ) ∆ = = - signal distorted : ρ(t).e (t) x signal OFDM : ρ(t).e x(t) ∆) (t j ∆ (t) j ϕ ϕ
Hence, its autocorrelation writes:

( ) [ ] ( ) [ ](1) τ).e ρ(t ρ(t).e E τ x Rx τ) (t * (t)x x E τ x Rx τ) ∆ (t j ∆) (t j ∆ ∆ ∆ ∆ ∆ ∆ - - - - - = - = ϕ ϕ
where E[.] is the statistical average operator with the underlying probability distribution

{ } τ) - ∆ (t τ), ρ(t ∆), (t ρ(t), P - - - ϕ ϕ
The Autocorrelation may be estimated as the time average:

2 2 I Q + PWM Class D PA Class E PA 2 2 I I Q + 2 2 Q I Q + Frequency synthesizer +/- 90°2 2 I Q + PWM Class D PA Class E PA 2 2 I I Q + 2 2 Q I Q + Frequency synthesizer +/- 90°90° ( ) ( ) ( ) ∑ = - - - - = N 1 i ∆) (i j τ) ∆ (i j ∆ ∆ i .ρ .e τ - i ρ N 1 τ x x R ˆϕ ϕ from the output of a simulated system. (t) x ∆
The idea here is to estimate the autocorrelation (1) by ensemble average. This means that Rx ∆ x ∆ (τ) is computed as the mean of function of K independent realisations of 4 random variables { } ρ , , , ρ

) ( ) ( ) ∑ = - = K 1 k (k) j 4 (k) j 1 ∆ ∆ 3 2 k .ρ .e k ρ K 1 τ x x R ˆϕ ϕ (2)
In the OFDM context, the emitted signal can be considered as complex Gaussian if the number of subcarrier (Nsub) is high enough. This hypothesis (Central Limit Theorem) is valid for Nsub greater than 30. For Hiperlan2 (Nsub = 64), we model the OFDM signal as a Gaussian signal with known autocorrelation Rxx(τ).

Hence, the random variables { in [START_REF] Diet | Sensibilité d'une architecture EER à une modulation OFDM dans le système Hiperlan2[END_REF] must have the same joint distribution as the variables in [START_REF] Kahn | Single sideband transmission by envelope elimination and restoration[END_REF].

} ρ , , , ρ 4 3 2 1 ϕ ϕ { } τ) - ρ(t ∆), τ (t ∆), - (t ρ(t), - - ϕ ϕ
Equivalently, we shall generate a vector W,

{ } w , w , w , w W 4 3 2 1
= with same joint distribution as a vector of samples of our OFDM signal X:

{ } { } τ)e - ρ(t , τ)e - ∆ - ρ(t , ∆)e - ρ(t , ρ(t)e X τ) - x(t τ), - ∆ - x(t ∆), - x(t x(t), X τ) - (t j τ) - ∆ - (t j ∆) - (t j (t) j ϕ ϕ ϕ ϕ = =
As X, the samples of our OFDM signal, is a Gaussian vector, it suffice to impose that W and X have the same correlation matrix. This correlation matrix is analytically known (since it depends on the known autocorrelation Rxx(τ)), and it is easy to generate W as a simple transform of a random gaussian vector G with uncorrelated components. The correlation matrix is given by:

[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               - + - + = = 0 Rxx ∆ Rxx τ - ∆ Rxx τ - Rxx ∆) Rxx( 0 Rxx τ - Rxx τ - ∆ - Rxx τ) ∆ Rxx( τ Rxx 0 Rxx ∆ - Rxx τ Rxx τ ∆ Rxx ∆ Rxx 0 Rxx X X. E R * τ ∆,
(where * denotes the hermitian transpose)

The key of the simulation process is to remark that given any square root C of R (i.e. a cholesky factor or a matrix of eigenvectors) such that R = CC*, we have:

[ ] [ ] [ ] τ ∆, R * C C E * C * G G C E * W W E = = =
with W= CG and G a random zero mean Gaussian vector with uncorrelated components.

So, for a given ∆, the simulation consist in:

-generate a Gaussian random vector G with uncorrelated components. -compute W = CG for some values of τ.

-update R ( )

τ x x ˆ∆ ∆
Finally compute the spectrum by Fourier transform. Results of autocorrelation simulation are reported on Fig. 2 where the delay is varied from 0 to 10 nsec considering a 20 MHz OFDM signal using a root raised cosine shape filter (roll-off = 0.5). Results show that delay causes small variations on the autocorrelation response. Resulting spectrums (1 to 10 nsec delay, 10 curves) show spectral re-growths. When compared to Hiperlan2 spectral limit, a delay of 5 nsec is too high to fulfil standard requirements. Simulations are compared with HP-ADS Hiperlan2 ones (Fig. 4). Results showed in [START_REF] Diet | Sensibilité d'une architecture EER à une modulation OFDM dans le système Hiperlan2[END_REF] a limit of 3 nsec. The difference is explained by the difference in spectrum calculation and non-ideal Gaussian behaviour of Hiperlan2 simulated signal. The accuracy can be improved by increasing K (10000 here). Confidence intervals can be computed using Student tests.

Autocorrelation response for 0 and 10 nsec delay

Spectrum for 0 to 10 nsec delay

III. Conclusion

It was presented an estimation of envelope/phase time mismatch influence on an OFDM signal. This is particularly important in a sensibility analysis of EER architecture. Results showed a good agreement between simulated Monte Carlo study results and HP-ADS Hiperlan2 simulation ones. Characterizing the autocorrelation of the envelope delayed OFDM signal is possible. With simulated spectrums values, the delay imperfection can be analysed, and the impact of delay imperfection can be investigated.

Fig. 1 :

 1 Fig. 1: EER architecture principleIn this architecture, several sources of imperfections lower the quality of the transmission[START_REF] Diet | Sensibilité d'une architecture EER à une modulation OFDM dans le système Hiperlan2[END_REF]. The major impact is due to time mismatch ∆ between envelope and phase signals at the recombination. This is caused by different operations on each path Consequences are noise and phase rotation[START_REF] Baudoin | Influence of time and processing mismatches between phase and envelope in EER[END_REF]. A statistical Monte Carlo analysis enables a characterization of delay influence on the emitted signal.

(

Fig. 2 :

 2 Fig. 2: Autocorrelation response from simulations

FrequencyFig. 3 :

 3 Fig. 3: Spectrum from simulated autocorrelations

Fig. 4 :

 4 Fig. 4: Simulated spectrum with HP-ADS


Hernando Maldonado Colmán
email: hernando.maldonado-colman@centralesupelec.fr

Alberto Cuoci

Nasser Darabiha

Benoît Fiorina

A virtual chemistry model for soot prediction in flames including radiative heat transfer

Keywords: Kinetic modeling, Soot, Radiative heat transfer, Laminar flames

Modeling soot formation is a very difficult task which has been assessed by the numerical combustion community. Numerical simulations of turbulent sooting flames remain very challenging due to the complexity of soot formation phenomena, which implies development of elaborate soot models often too computationally demanding, or simplified soot models that are limited to a small range of operating conditions of interest. Here an innovative optimized global approach called virtual chemistry is proposed. It consists of a mathematical formalism including virtual species and virtual reactions, whose thermochemical parameters are optimized in order to describe a combustion system. Thermochemical properties are trained through a genetic algorithm, employing a learning database made of reference flame elements. This makes it possible to reproduce multiple combustion regimes and operating conditions. The virtual chemistry approach reproduces then

Introduction

Hydrocarbon/air combustion occurring in rich and incomplete conditions in practical industrial applications causes the formation of undesirable soot particles. Soot formation results from both highly complex gas phase chemistry and particles dynamics including nucleation, condensation, coalescence, and surface reactions such as surface growth and oxidation. Because of the complexity of these physico-chemical phenomena, numerical computations of gas chemistry together with particles dynamics request high amounts of CPU time, which quickly become an issue for the simulation of practical combustion chambers.

The chemical kinetics describing the gaseous phase evolution including soot precursors needs a large number of species and reactions [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF] so that detailed chemical mechanisms generally cannot be directly employed in CFD of sooting flames. Three main routes for kinetics reduction techniques are in principle possible [START_REF] Fiorina | Challenging modeling strategies for les of non-adiabatic turbulent stratified combustion[END_REF][START_REF] Fiorina | Accounting for complex chemistry in the simulations of future turbulent combustion systems[END_REF]: analytically-reduced chemistry, tabulated chemistry and optimized global chemistry. Thanks to recent progresses in computational resources and in the automation of chemistry reduction techniques [START_REF] Pepiot | Automatic strategies to model transportation fuel surrogates[END_REF],

analytically-reduced schemes have been recently used for LES of turbulent sooting flames [START_REF] Felden | Impact of direct integration of analytically reduced chemistry in LES of a sooting swirled non-premixed combustor[END_REF][START_REF] Gallen | Lagrangian tracking of soot particles in LES of gas turbines[END_REF]. Despite impressive results, the remaining high number of species and the numerical stiffness of the system make difficult the systematic use of analytic schemes in a practical combustion chamber design process. Application of tabulated chemistry to soot formation is therefore more popular. It consists in storing complex chemistry up to polycyclic aromatic hydrocarbons (PAHs) in look-up tables, parametrized by a reduced set of thermochemical variables [START_REF] Mueller | LES model for sooting turbulent nonpremixed flames[END_REF][START_REF] Rodrigues | Coupling an LES approach and a soot sectional model for the study of sooting turbulent non-premixed flames[END_REF]. Despite its very low CPU cost, the tabulated chemistry approach is not able to well predict multi-regime combustion [START_REF] Fiorina | Modeling combustion chemistry in large eddy simulation of turbulent flames[END_REF]. Finally, literature presents a few attempts to optimize global mechanisms to predict soot precursor formation [START_REF] Said | Soot formation modeling for turbulent flames[END_REF][START_REF] Kollmann | Application of a soot model to a turbulent ethylene diffusion flame[END_REF][START_REF] Franzelli | Numerical modeling of soot production in aero-engine combustors using large eddy simulations[END_REF]. In these studies, a global mechanism is adopted to describe the gas chemistry and soot formation is described using a simplified chemistry.

The complex interaction between reactive gas phase and solid particles dynamics has motivated the development of a wide variety of mathematical approaches such as (sorted from high to low computational demand):

Lagrangian models [START_REF] Gallen | Lagrangian tracking of soot particles in LES of gas turbines[END_REF][START_REF] Celnik | Coupling a stochastic soot population balance to gas-phase chemistry using operator splitting[END_REF], Chemical Discrete Sectional Model (CDSM) [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF][START_REF] Pope | Simultaneous particle and molecule modeling (spamm): An approach for combining sectional aerosol equations and elementary gas-phase reactions[END_REF][START_REF] Pejpichestakul | Examination of a soot model in premixed laminar flames at fuel-rich conditions[END_REF], Aerosol Discrete Sectional model (ADSM) [START_REF] Rodrigues | Coupling an LES approach and a soot sectional model for the study of sooting turbulent non-premixed flames[END_REF][START_REF] Smooke | Computational and experimental study of soot formation in a coflow, laminar diffusion flame[END_REF][START_REF] Zhang | A numerical study of soot aggregate formation in a laminar coflow diffusion flame[END_REF][START_REF] Blacha | Soot predictions in premixed and non-premixed laminar flames using a sectional approach for pahs and soot[END_REF][START_REF] Eberle | A sectional PAH model with reversible PAH chemistry for cfd soot simulations[END_REF], or Method of moments (MOMs) [START_REF] Frenklach | Aerosol dynamics modeling using the method of moments[END_REF][START_REF] Blanquart | A joint volume-surface-hydrogen multi-variate model for soot formation[END_REF][START_REF] Mueller | A joint volume-surface model of soot aggregation with the method of moments[END_REF][START_REF] Salenbauch | A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth and oxidation[END_REF]. In addition, empirical and semi-empirical meth-ods [START_REF] Said | Soot formation modeling for turbulent flames[END_REF][START_REF] Tesner | Kinetics of dispersed carbon formation[END_REF][START_REF] Moss | Flowfield modelling of soot formation at elevated pressure[END_REF][START_REF] Leung | A simplified reaction mechanism for soot formation in nonpremixed flames[END_REF] have also been developed to track the soot volume fraction and particle number density, and eventually the particle size distribution [START_REF] Franzelli | A three-equation model for the prediction of soot emissions in les of gas turbines[END_REF].

Such pragmatic approaches are attractive because of their very low CPU cost, but in contrast, they are limited to a small range of operating conditions. In most cases, the parameters are a function of the fuel. New techniques using machine learning algorithms, such as artificial neural networks (ANNs), have been recently developed to tackle the soot prediction problem by using a post-processing tool and were tested in steady and unsteady laminar coflow flames [START_REF] Jadidi | An artificial neural network for the low-cost prediction of soot emissions[END_REF][START_REF] Jadidi | A long short-term memory neural network for the low-cost prediction of soot concentration in a time-dependent flame[END_REF].

The objective of the present work is to develop a highly-reduced soot model for the prediction of the soot volume fraction, encompassing both the detailed soot precursor chemistry and the physics of particles dynamics interactions. The impact of radiation on both gas and solid phase temperature, which is of primary importance in sooting flames, will be included. For that purpose, we follow a methodology based on a recent approach called virtual chemistry, introduced by Cailler et al. [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF] to predict pollutants formation such as CO [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF][START_REF] Maio | Virtual chemistry for temperature and CO prediction in LES of non-adiabatic turbulent flames[END_REF] or NOx [START_REF] Maio | A virtual chemical mechanism for prediction of NO emissions from flames[END_REF]. The virtual chemistry method consists in building-up an optimized set of reduced virtual chemical reactions including virtual species. A machine learning algorithm is applied to retrieve a set of user-defined constraints such as the temperature profiles, the heat release or targeted pollutant mass fraction. Here we focus on the prediction of the soot volume fraction. Predictions of the particle number density and the particle size distribution are not be considered in this article.

The virtual chemistry principle is presented in Section 2. In Section 3 an extension of the virtual approach is developed to handle soot formation in laminar flames. Also, a new procedure is developed in Section 4 to include radiative heat transfer in the computations. The optimization set-up of the virtual models and the computation cases are presented in Section 5.

The virtual approach is then validated in 1-D premixed and non-premixed ethylene-air configurations in Section 6. Finally, the virtual chemistry formalism is challenged in a 2-D premixed ethylene-air slot burner and a 2-D non-premixed ethylene-air coflow burner from Santoro et al. [START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF] in Section 7.

Virtual chemistry basics

The originality of virtual chemistry approach resides in virtual species and reactions whose thermodynamic and chemical properties are optimized using machine learning algorithms in order to retrieve properties of reference flames gathered in a learning database [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF]. The virtual reactions between the virtual species do not represent real chemical processes but they form a mathematical architecture designed to reproduce user-specified targets such as temperature and mass fractions of combustion products and pollutant species. A virtual scheme consists of a main block that models the heat release from the flame and sub-mechanisms dedicated to predict pollutants. Figure 1 depicts a schematic presentation of the approach. Virtual kinetic submechanisms have so far been developed to predict pollutants such as carbon monoxide (CO) [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF] and nitrogen oxides (NOx) [START_REF] Maio | A virtual chemical mechanism for prediction of NO emissions from flames[END_REF].

The virtual main mechanism structure presented in [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF][START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF] has shown that the following reaction scheme, composed of 8 virtual species and 2 virtual reactions, accurately captures the heat release and the flame temperature for a wide range of fuel compositions:

α M F,1 F + α M Ox,1 Ox → I (R M 1 ) I → N M P =4 k=1 α M P k,2 P k (R M 2 )
In this mechanism, F represents the fuel, Ox the oxidizer, I a virtual intermediate species and P k the virtual products. α M k,r is the stoichiometric coefficient per unit mass of the k th virtual species in the r th virtual reaction, and N M P is the number of virtual products. The superscript M stands for virtual main chemistry. The vector of reaction parameters χ M (see Supplementary Material) of the main mechanism is optimized through the machine learning algorithm proposed by Cailler et al. [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF].

Virtual species enthalpy and heat capacity are optimized to recover mixture properties at equilibrium. Unity Lewis number is assumed for the virtual species in the main mechanism.

Virtual soot sub-mechanism

Scheme architecture

As shown in Fig. 2(top), a classical soot model requires first a fine description of the detailed chemistry leading to the soot precursor formation (C 2 H 2 , PAHs, etc.) [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF]. Then, the formation of the particles is described through a one-precursor (C 2 H 2 or PAH) nucleation model [START_REF] Leung | A simplified reaction mechanism for soot formation in nonpremixed flames[END_REF][START_REF] Schuetz | Nucleation of soot: Molecular dynamics simulations of pyrene dimerization[END_REF] or multiple-PAHs approach [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF][START_REF] Selvaraj | A computational study of ethyleneair sooting flames: Effects of large polycyclic aromatic hydrocarbons[END_REF]. Some approaches consider that PAHs grow until they reach an intermediate state, called dimer, between the gas and solid phases [START_REF] Blanquart | A joint volume-surface-hydrogen multi-variate model for soot formation[END_REF][START_REF] Mueller | A joint volume-surface model of soot aggregation with the method of moments[END_REF].

Finally, the solid phase is modeled employing dedicated particles interaction and distribution formalisms such as for instance sectional [START_REF] Rodrigues | Coupling an LES approach and a soot sectional model for the study of sooting turbulent non-premixed flames[END_REF] or MOMs [START_REF] Mueller | A joint volume-surface model of soot aggregation with the method of moments[END_REF] approaches. These successive modeling stages lead to the prediction of the soot characteristics such as volume fraction, particle number density and particle size distribution.

Most of chemical reduction methods, such as tabulated [START_REF] Fiorina | Premixed turbulent combustion modeling using tabulated detailed chemistry and pdf[END_REF] or analyticallyreduced [START_REF] Pepiot | Automatic strategies to model transportation fuel surrogates[END_REF] approaches, are limited to the gas-phase kinetics. The description of the solid phase, based on classical approaches, remains very time consuming. The virtual soot sub-mechanism proposed in the present work (Fig. 2 bottom) has the advantage of reducing the overall process by targeting the final quantities of interest only.

The soot volume fraction f s v is here targeted. The following chemical structure is proposed to predict the formation of the soot particles species, represented by the virtual chemical species S:

α M F,1 F + α M Ox,1 Ox → α s V 1 ,1 V 1 + α s V 3 ,1 V 3 (R s 1 ) V 1 → α s S,2 S + α s V 2 ,2 V 2 (R s 2 )
• Detailed gas-phase kinetics including soot precursors: C2H2, PAHs, etc. Soot model

• Sectional • MOMs • Semi-Empirical Virtual main mechanism • T • F • OX CLASSICAL VIRTUAL GAS PHASE SOLID PHASE Transition Nucleation model
Virtual soot sub-mechanism

• Virtual soot pseudo-species • Soot mass fraction, number density …..

• • Np

• NDF

Learning Database

✦ Equilibrium compositions ✦ 1-D Premixed Flames ✦ 1-D Non-premixed Flames ✦ etc.

Detailed kinetic mechanism

Optimization Optimization

Reduced gas-phase kinetics

• Tabulated • ARC Soot precursors C2H2, PAHs, etc. f s v f s v
S + V 1 + V 2 → α s S,3 S + α s V 3 ,3 V 3 (R s 3 ) S + Ox + V 2 → Ox + 2V 3 (R s 4 )
where V 1 , V 2 and V 3 are three virtual species and α s k,r is the stoichiometric coefficient per unit mass of the k th virtual species in the r th virtual reaction.

The superscript s stands for virtual soot sub-mechanism.

The soot production is initiated by reaction R s 1 . The rate of progress of this reaction must be equal to that of reaction R M 1 in order to be consistent with the fuel oxidation through the main virtual mechanism. The three virtual species V 1 , V 2 and V 3 together with virtual reactions R s 2 , R s 3 and R s 4 represent the nucleation, surface growth and coagulation, and oxidation of soot particles. This approach encompasses all the mechanisms of soot formation in both gas and solid phases, without modeling individually each physical step.

Kinetic parameters optimization

The virtual soot sub-mechanism reaction rates of reactions R s 1 through R s 4 are expressed as follows:

q s 1 = [F ] n M F,1 [Ox] n M Ox,1 A M 1 exp - E M a,1 RT , (1)
q s 2 = [V 1 ] n s V 1 ,2 A s 2 exp - E s a,2 RT , (2)
q s 3 = [S] n s S,3 [V 1 ] n s V 1 ,3 [V 2 ] n s V 2 ,3 A s 3 exp - E s a,3 RT , (3)
q s 4 = [S] n s S,4 [Ox] n s Ox,4 [V 2 ] n s V 2 ,4 A s 4 T β s 4 exp - E s a,4 RT , (4)
where E s a,r is the the activation energy, A s r is the pre-exponential constant, n s k,r is the order of species k in reaction r, and β s r is the temperature dependence exponent. As previously mentioned, kinetic rate parameters of Eq. 1 are given by the first reaction R M 1 of the main mechanism.

The ensemble of virtual chemical parameters of the virtual soot submechanism constitutes a vector χ s = {A s r , E s a,r , α s k,r , n s k,r , β s 4 } to be optimized. It has been observed that the vector χ s depends on the nature of the fuel and of the mixture fraction Z [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF][START_REF] Maio | A virtual chemical mechanism for prediction of NO emissions from flames[END_REF]. Also, the mass conservation imposes:

α s V 1 ,1 + α s V 3 ,1 = 1, α s S,2 + α s V 2 ,2 = 1 and α s S,3 + α s V 3 ,3 = 3.
The kinetic parameters in vector χ s are optimized by minimizing the following cost function:

C s (χ s ) = Nc i=1 w s f s v,i -f d v,i L 2 f d v,i L 2 + (1 -w s ) gradf s v,i -gradf d v,i L 2 gradf d v,i L 2 , (5)
where f s v,i and f d v,i are the soot volume fractions of the i th flame solution, respectively computed with virtual and detailed reference chemistries and gradf v,i indicates the gradient of f v,i . The superscript d stands for reference detailed chemistry solutions. N c is the number of 1-D reference flame solutions which constitute the learning database. The weight w s is set to 0.9, as in the virtual main mechanism [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF].

The soot volume fraction f s v is calculated from the virtual soot mass fraction Y s S :

f s v = ρ ρ s Y s S ( 6
)
where ρ is the gaseous mixture density, and ρ s is the soot density considered constant, ρ s =1800 kg m -3 .

Radiative heat transfer modeling

Radiative heat transfer in the gas phase may significantly impact the heat release and the flame temperature. To account for this phenomenon, radiative gas properties of the reactive flow must also be included in the optimization process of the virtual main mechanism. These properties depend on gas temperature and mixture composition [START_REF] Barlow | Scalar profiles and no formation in laminar opposed-flow partially premixed methane/air flames[END_REF]. For sooting flames, the Planck mean absorption coefficient κ Planck can be decomposed as κ Planck = κ gas +κ soot [START_REF] Modest | Radiative Heat Transfer[END_REF], where κ gas and κ soot correspond to gas and solid phases, respectively.

Gas radiation model: virtual approach

The gaseous Planck mean absorption coefficient κ M gas is expressed with the polynomial development presented in [START_REF] Barlow | Scalar profiles and no formation in laminar opposed-flow partially premixed methane/air flames[END_REF]:

κ M gas (Z,T ) = γ M 0 (Z) + γ M 1 (Z)(1000/T ) + γ M 2 (Z)(1000/T ) 2 + γ M 3 (Z)(1000/T ) 3 + γ M 4 (Z)(1000/T ) 4 + γ M 5 (Z)(1000/T ) 5 , (7)
where the polynomial coefficients γ M l (Z) are chosen to retrieve solutions of canonical combustion configurations including radiative heat transfer. The

vector of polynomial coefficients Γ M = {γ M 0 , γ M 1 , . . . γ M 5
} is in practice optimized through the evolutionary algorithm proposed by Cailler et al. [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF].

The optimization procedure consists in minimizing the following cost function:

C M (Γ M ) = Nc i=1 w M T M i -T d i L 2 T d i L 2 + (1 -w M ) S M L,i -S d L,i S d L,i , (8)
where N c is the number of flame elements included in the learning database, and S d L,i and T d i are the targeted reference laminar flame speed and temperature retained for the optimization, respectively. The weight w M is set to 0.9, as in Ref. [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF].

Soot particles radiation model

The Planck soot absorption coefficient κ v soot is expressed from the soot volume fraction f s v predicted by the virtual soot sub-mechanism and the gas temperature T given by the main mechanism:

κ v soot = αf s v T, (9)
where α is a modeling constant set to 1302 m -1 K -1 as in Ref. [START_REF] Modest | Radiative Heat Transfer[END_REF].

Virtual schemes optimization set-up

Learning database and virtual scheme optimization

The reference flame database is generated using the detailed soot kinetics model proposed by the CRECK Modeling Group [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF] which consists of 16797 reactions and 297 species. Among these species, 20 lumped pseudo-species called "BIN" are considered. BIN 1 through BIN 4 correspond to heavy PAHs with 20, 40, 80 and 160 carbon atoms respectively. The nascent soot particles are classified as BIN 5 and they grow from spherical shape (BIN 5 to BIN 12 ) until they take an aggregate form (BIN 13 to BIN 20 ) [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF]. This kinetic mechanism has been extensively validated on several experimental data in many different systems [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF][START_REF] Pejpichestakul | Examination of a soot model in premixed laminar flames at fuel-rich conditions[END_REF][START_REF] Pejpichestakul | Buoyancy effect in sooting laminar premixed ethylene flame[END_REF]. This mechanism is used here for all detailed chemistry ethylene-air combustion simulations. Radiative heat losses are calculated using the optically-thin narrow-band radiation model based on Ref. [START_REF] Soufiani | Application of statistical narrow-band model to coupled radiation and convection at high temperature[END_REF], which is included in REGATH package [START_REF] Darabiha | The influence of the temperature on extinction and ignition limits of strained hydrogen-air diffusion flames[END_REF]. The reference soot volume fraction is defined from the detailed chemistry flame solutions as [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF]. The virtual main mechanism is trained by employing the genetic algorithm detailed in [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF]. The reference flame solutions, used to evaluate the cost function given by Eq. 8, comprise 1-D freely-propagating laminar premixed flames including radiative heat transfer for a range of equivalence ratios covering the flammability limits (φ = 0.5 and φ = 3.0). It has been shown in [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF][START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF] that a learning database with similar limits was sufficient to recover heat release and temperature in both premixed and non-premixed flame regimes.

f d v = (ρ/ρ s )Y d soot , where Y d soot = i Y BIN i for i = 5, 6, . . . , 20
However, to account for multi-mode combustion in the virtual soot submechanism, the learning database is made of both 1-D freely-propagating premixed flames and 1-D counterflow non-premixed flames as in [START_REF] Maio | A virtual chemical mechanism for prediction of NO emissions from flames[END_REF]. The database includes 11 1-D premixed flames at atmospheric pressure and initial temperature of 298 K with equivalence ratios from φ = 1.8 through φ = 2.8, by a step of ∆φ = 0.1. It also includes 6 non-premixed counterflow flames, where pure ethylene at 298 K is injected against air at 298 K, covering strain rates from a = 6s -1 to a = 95s -1 , see Table 1 detailed and virtual chemistry, are performed using the REGATH package [START_REF] Darabiha | The influence of the temperature on extinction and ignition limits of strained hydrogen-air diffusion flames[END_REF].

The optimized reaction rate parameters to account for premixed and nonpremixed flames in the virtual soot sub-mechanism are given in the Supplementary Material. The convergence of the cost function C s (χ s ) defined in Eq. 5 is illustrated in Fig. 3. Here, several realizations of the optimization procedure have been conducted to ensure that the final solution corresponds to a global minimum cost function value as suggested in [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF].

Computed cases

The virtual mechanisms and radiation models are now challenged on 1-D and 2-D ethylene-air flames. The following series of computations are performed: • D-A: Adiabatic flames using the detailed CRECK mechanism [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF];

• D-R: Non-adiabatic flames using the detailed CRECK mechanism [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF].

Only gas radiation is considered using the narrow-band radiation model based on Ref. [START_REF] Soufiani | Application of statistical narrow-band model to coupled radiation and convection at high temperature[END_REF] for 1-D configurations, and on that of [START_REF] Barlow | Scalar profiles and no formation in laminar opposed-flow partially premixed methane/air flames[END_REF] for 2-D configurations;

• D-Rs: Non-adiabatic flames using the detailed CRECK mechanism [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF]. Gas radiation is modeled as in the case D-R, whereas solid phase radiation is modeled using Eq. 9;

• V-A: Adiabatic flames using the developed virtual mechanisms;

• V-R: Non-adiabatic flames using the the virtual mechanism. Only gas radiation is considered;

• V-Rs: Non-adiabatic flames using the the virtual mechanism. Both gas and particle radiation are considered.

Table 2 summarizes the computed cases, indicating the kinetic model and radiative effects involved. anisms reproduce very well the reference temperature profiles. The impact of radiation on the temperature, especially visible for long distances (long residence times), is well captured by the virtual chemistry and radiation models.

Computed case D-A D-R D-Rs V-A V-R V-Rs Kinetic model Detailed mechanism Virtual mechanism Adiabatic √ - - √ - - Gas radiation - √ √ - √ √ Soot radiation - - √ - - √

1-D non-premixed counterflow flames

The gas phase virtual radiative model is validated in this section for non-premixed flames. Four series of 1-D ethylene-air counterflow flames are computed at different strain rates a again for the four conditions D-A, D-R, V-A, and V-R. Figure 6 illustrates the evolution of the maximum flame temperature with the strain rate. In both adiabatic and radiative cases, the virtual main chemistry and virtual radiative models predict very well this evolution for a very wide range of strain rates, from equilibrium up to quenching conditions. As expected, radiative heat losses affect the maximum flame temperature at very low strain rates. Temperature and soot volume fraction profiles are shown in Fig. 7 for equivalence ratios equal to 2.1 and 2.4, respectively. Solutions are plotted for the six simulation cases D-A, D-R, D-Rs, V-A, V-R and V-Rs.

Validation of soot virtual sub-mechanism and virtual radiative model 6.2.1. 1-D freely-propagating premixed flames

Again, for both φ = 2.1 and φ = 2.4, comparisons between D-A/V-A and D-R/V-R results in Fig. 7 (top) show that the virtual main mechanism accurately captures the temperature profiles with and without considering gas radiation. The soot volume fraction profiles, plotted in Fig. 7 (bottom) are also very well predicted by the virtual soot sub-mechanism for these operating conditions. The soot production, enhanced by radiative heat transfer, also significantly increases by more than 2 orders of magnitude between φ = 2.1 The influence of solid phase radiation on temperature is only significant for φ = 2.4, where the soot volume fraction is high enough (cases D-Rs and V-Rs). The virtual model which includes both gas and solid particles radiation predicts very well the temperature and soot reference solutions.

1-D non-premixed counterflow flames

Figure 8 presents temperature and soot volume fraction profiles for an adiabatic 1-D non-premixed counterflow flame (strain rate a = 10 s -1 ) computed with both detailed (D-A) and virtual (V-A) chemistry approaches.

Pure ethylene is injected from right while air is injected from left. Soot formation takes place in a rich region located between the maximal flame temperature position (x = -4mm) and the stagnation plane (x = 0). The soot volume fraction is well predicted by the virtual soot sub-model trained to capture both premixed and non premixed flame archetypes as indicated on Table 1.

It is interesting to focus on the dashed-dotted profile that shows the soot volume fraction profile obtained with a virtual scheme sub-mechanism (V-A-Prem.) trained to target only 1-D premixed flames. The target premixed flames are accessible only within the flammability limits, highlighted by the grey area in Fig 8 . The peak of the soot volume fraction is located at the rich side of the grey zone. It can be seen that such target flame is unable to capture the soot produced here above the rich extinction limit.

As indicated in Sec. 5.1, the virtual scheme used in case V-A has been optimized with a Lewis number of species S set to Le S = 25 whereas Le=1 has been assumed for all other virtual species (see the Supplementary Material for the choice of Le S = 25). As soot particles diffusion is very small compared to gaseous species, assuming unity Lewis also for the soot would have been unrealistic. This is illustrated in Fig. 9 where the solution obtained with Le=1 for all species considerably underestimates the soot production.

Finally, the evolution of maximum temperature T max and maximum soot volume fraction f max v as a function of strain rate a are presented in Fig. 10 for adiabatic simulation cases D-A and V-A as well as non-adiabatic cases D-Rs and V-Rs including both gaseous and solid phases radiative heat losses. The virtual soot sub-mechanism predicts well the soot volume fraction maximum value f max v 1-D counterflow ethylene-air flames for a large range of strain rates a.

Application of virtual models to burner-stabilized flames

The reduced chemistry approach is now applied to a burner-stabilized ethylene-air flame studied within the International Sooting Flame (ISF) Workshop [START_REF]ISF5, International sooting flame workshop[END_REF]. Two equivalence ratio of φ = 2.34 and φ = 2.64 are investigated at atmospheric pressure and temperature of T = 298 K. Simulations are conducted by imposing at the inlet the fresh gases injection velocity measured in experiments [START_REF] Menon | Addition of no2 to a laminar premixed ethylene-air flame: effect on soot formation[END_REF][START_REF] Xu | Soot formation in laminar premixed ethylene/air flames at atmospheric pressure[END_REF] (6.73 cm•s -1 ). Figure 11 (top) show that for both detailed and virtual radiative simulations, numerical prediction of temperature agrees very well with experimental measurements [START_REF] Menon | Addition of no2 to a laminar premixed ethylene-air flame: effect on soot formation[END_REF]. Soot volume fraction profiles plotted in Fig. 11 (bottom), exhibit also a fair agreement between simulations and experimental data. Therefore, even if burnerstabilized flames where not included in the learning database, the virtual chemistry is able to capture their chemical structure.

Application of virtual models to 2-D laminar flames

In this section, the three simulation cases V-A, V-R and V-Rs (see Table 2) are challenged in two two-dimensional laminar flame configurations. First, a 2-D premixed ethylene-air slot burner is simulated. Next, the non-premixed Santoro coflow burner [START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF] from ISF workshop is studied.

2-D laminar ethylene-air premixed slot burner

The geometry of the premixed slot burner is shown in Fig. x(mm)

0 -2 -4 2 4
x(mm)

Detailed Virtual 0 -2 -4 2 4
x(mm)

0 -2 -4 2 4
x(mm) mm, the mismatch is even limited to 25 % for the three cases.

Table 3 presents a comparison of the computational cost required by virtual and detailed chemistries to simulate the premixed slot burner. A speed-up of more than 2000 is observed when using the virtual scheme.

2-D laminar Santoro Coflow Burner

The non-premixed flame experimentally studied by Santoro et al. [START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF] is now computed with the virtual soot model. The burner consists of a central pipe fed with pure ethylene surrounded by a concentric tube for air injection.

The fuel and air flow rates are 3.85 cm 3 •s -1 and 713.3 cm 3 •s -1 , respectively. Further details on the experimental setup are given in [START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF]. The virtual chemistry simulation includes both gas and solid phases radiation phenomena (V-Rs), which are both significant in this configuration [START_REF] Santoro | The transport and growth of soot particles in laminar diffusion flames[END_REF].

2-D iso-contour of temperature and soot volume fraction fields are shown in Fig. 17.

Radial profiles of temperature are plotted in Fig. 18 (a) for two heights above the burner at z = 10 mm and 50 mm. Numerical results using the shows the axial evolution of the maximum soot volume fraction in the radial direction. The virtual approach (V-Rs) reproduces the maximum value of f v as in the detailed simulations (D-Rs), but with a peak position mm closer to the exit. Soot is well oxidized at the flame tip when using the virtual model (V-Rs), where the difference of maximum soot volume fraction compared to the detailed model results (D-Rs) is about 50 % at z = 60 mm.

Axial profiles of radially-integrated soot volume fraction, defined as F v (z) = results of Blacha et al. [START_REF] Blacha | Soot predictions in premixed and non-premixed laminar flames using a sectional approach for pahs and soot[END_REF] have been obtained using a detailed chemistry mechanism including 71 species and 803 reactions. Soot particles distribution has been modeled with a sectional method whereas radiative heat transfer from gas and soot particles are captured with the assumption of an optically thin medium according to Di Domenico et al. [START_REF] Di Domenico | Development and validation of a new soot formation model for gas turbine combustor simulations[END_REF]. Simulation results of Akridis and Rigopoulos [START_REF] Akridis | Modelling of soot formation in laminar diffusion flames using a comprehensive cfd-pbe model with detailed gas-phase chemistry[END_REF] have been obtained using a detailed mechanism including 75 species and 529 reactions [START_REF] Wang | A comprehensive reaction model of ethylene and acetylene combustion[END_REF]. A discretized population balance equation for soot the particles is solved, where its source terms closures are based on different models from the literature [START_REF] Liu | Numerical modelling of soot formation and oxidation in laminar coflow non-smoking and smoking ethylene diffusion flames[END_REF][START_REF] Smooke | Soot formation in laminar diffusion flames[END_REF][START_REF] Hall | Predictions of soot dynamics in opposed jet diffusion flames[END_REF]. An optically thin approximation for the radiation of gas species (H 2 O, CO 2 and CO) and soot particles from Kronenburg et al. [START_REF] Kronenburg | Modeling soot formation in turbulent methane-air jet diffusion flames[END_REF] here considered. Finally, experimental measurements from ISF database [START_REF]ISF5, International sooting flame workshop[END_REF] are also added in this figure (blue circles).

These results highlight qualitative differences between the three detailed soot formation mechanisms solutions. Soot volume fraction tendency predicted by the virtual soot sub-mechanism is consistent with the three complex chemistry solutions and with the experimental data: soot particles are produced in the first half of the domain before being consumed downstream.

Deepening the analysis of the differences between the virtual chemistry solutions and experimental data not relevant here, as the accuracy of the reduced model is naturally limited to that of the reference detailed mechanism retained (CRECK) to generate the learning database.

In comparison with the CRECK detailed mechanism (D-Rs), the rate of soot oxidation process downstream the burner is underestimated by the virtual scheme (V-Rs). We attribute this drawback to the flame archetypes retained in the learning database. Indeed, close to the fuel injection, the chemical flame structure follows a classical non-premixed flame where pure fuel diffuses into pure air. Such flame archetype has been well targeted during the optimization process. However, further downstream, burnt gases mix with surrounding air coflow before entering the soot chemistry reactive region. In this case, the flame archetype is therefore closer to a diluted air/burnt gases stream, a configuration which has not been considered in the learning database during the virtual soot sub-mechanism optimization process.

To further pursue these analyses, the learning database used to train the virtual scheme, which includes premixed and non-premixed flames is projected in subspaces (T, Z) and (f v , Z) in Figs. 21 (a) and (b), respectively. However, the projection of the Santoro flame solution in the (f v , Z) subspace reveals some incompleteness of the training database. Indeed, for Z between 0.063 and 0.12 (equivalence ratios between 1 and 2 respectively), several trajectories exceed by an order of magnitude the upper envelopes given by premixed and non-premixed flame solutions. Complementary flame archetypes should then be targeted during the optimization step to entirely capture the soot formation process.

Conclusions and discussion

A new reduced virtual chemistry model is developed to predict the temperature profiles, heat release rate and soot volume fraction in laminar ethyleneair flames. Also, a virtual approach is proposed in order to include gas phase radiative heat transfer in the simulations. This new virtual approach integrates for the first time both gaseous and solid phases in a virtual mechanism, by bypassing the gas-solid complex formalism without explicit mathematical distinction between both phases.

Numerical results using the new virtual model are validated in 1-D ethylene air flames in canonical configurations, such as premixed and non-premixed flames, against reference detailed chemistry calculations. Additionally, the virtual model is challenged in a 1-D burner-stabilized flame configuration, giving very good results when compared to reference detailed chemistry calculations and experimental measurements. This shows that the virtual model is capable to predict soot formation in configurations other than those used to build the reference database.

Furthermore, simulations in two 2-D configurations are carried out using the virtual models: a premixed slot burner and a non-premixed coflow burner from the ISF workshop. In both configurations, very good results of the flame structure and soot characteristics are obtained. Also, the new virtual radiative model well provides the heat losses. In overall, the new virtual soot sub-mechanism captures soot physical phenomena with good level of accuracy and is comparable to other numerical works in the literature with more detailed models. By adding representative flames in the reference database used for the generation of the virtual model, one would be able to cover both smoking and non-smoking conditions in Santoro's flame [START_REF] Santoro | Soot particle measurements in diffusion flames[END_REF][START_REF] Santoro | The transport and growth of soot particles in laminar diffusion flames[END_REF][START_REF] Bonnety | Probing the local radiative quenching during the transition from a non-smoking to a smoking laminar coflow ethylene/air nonpremixed flame[END_REF].

In general, a massive reduction of the soot kinetics model is obtained, passing from 297 species and 16797 reactions in the reference detailed mechanism to 12 virtual species and 6 virtual reactions in the virtual mechanism, without need of additional models for solid particles. This chemical reduction has a big impact on the computational time of the simulations. When using the virtual model in 2-D laminar flames simulations, the CPU cost is reduced by a factor of about 2100 compared detailed models.

Perspectives to this work are to apply the soot virtual scheme to turbulent flames encountered in practical applications. The main difficulty would be to handle the different chemical paths followed by the various flame elements encountered in turbulent reactive flows. By training the virtual scheme on both premixed and non-premixed strained flamelets, it is expected to accurately capture most local combustion elements encountered in practical turbulent flames. If this is not sufficient, additional flame configurations may be added to the training database.

Figure 1 :

 1 Figure 1: Schematic presentation of virtual chemistry methodology.

Figure 2 :

 2 Figure 2: Diagram of soot formation models; top: current soot models, bottom: virtual soot chemistry approach.

  . The total number of operating conditions N c in the database is N c = N fpf c + N cf c = 17, with N fpf c premixed φ = {0.5, • • • , 3.0}; ∆φ = 0.1 flames Soot sub-mechanism Adiabatic and Radiative premixed φ = {1.8, • • • , 2.8}; ∆φ = 0.1 and non-premixed flames a = {6, 10, 25, 45, 60, 95} s -1

Figure 3 :

 3 Figure 3: Evolution of the cost function C s (χ s ) during the optimization process. Each line represents a single realization and the best kinetic parameters set χ s corresponds to the lowest value of the cost function C s (χ s ) [31].

6. 1 -

 1 D laminar ethylene-air flame computations 6.1. Validation of virtual main mechanism 6.1.1. 1-D freely-propagating premixed flames Simulations of 1-D freely-propagating premixed ethylene-air flames are first conducted. The laminar consumption speed S L is plotted in Fig.4as a function of the equivalence ratio φ. The virtual approach predicts very well the flame speed in all conditions. As the radiative heat losses take place essentially in hot gases, the flame front and the flame speed S L are not affected by radiation.

Figure 4 :

 4 Figure 4: Evolution of the laminar flame speed of 1-D premixed ethylene-air flames (P = 1 atm, T 0 = 298 K): comparison between the four cases D-A, D-R, V-A, V-R.

Figure 5 :

 5 Figure 5: Impact of radiative heat losses on temperature profile in a 1-D freely-propagating premixed ethylene-air flame (φ = 2.3): comparison between the four cases D-A, D-R, V-A, V-R.

Figure 6 :

 6 Figure 6: Evolution of the maximum temperature T max with the strain rate a in 1-D nonpremixed ethylene-air counterflow flames: comparison between the four cases D-A, D-R, V-A, V-R.

Figure 5

 5 Figure 5 presents temperature profiles obtained at φ = 2.3 for four cases D-A, D-R, V-A, and V-R. Both the adiabatic and non-adiabatic virtual mech-

Figure 7 :

 7 Figure 7: Temperature (top) and soot volume fraction (bottom) profiles of premixed ethylene-air flames, with and without considering solid particle radiation for different equivalence ratios φ = 2.1 (left) and φ = 2.4 (right): comparison between the six cases: D-A, D-R, D-Rs, V-A, V-R, V-Rs.

Figure 8 :

 8 Figure 8: Temperature and soot volume fraction profiles of adiabatic and non-adiabatic 1-D non-premixed counterflow flame using detailed and virtual models, cases D-A and V-A. Soot volume fraction profile obtained by a virtual soot sub-mechanism optimized on a learning database including only 1-D premixed flames (V-A-Prem.) is also plotted. The region covered by the learning database of only 1-D premixed flames within the extinction limits is shown by the grey area.

Figure 9 :

 9 Figure 9: Influence of the virtual soot species S Lewis number, Le S , on soot volume fraction in a counterflow ethylene-air flame.

Figure 10 :

 10 Figure 10: Evolution of the maximum temperature T max (left), and maximum soot volume fraction f max v

Figure 11 :

 11 Figure 11: Burner-stabilized premixed ethylene-air flames for φ = 2.34 and φ = 2.64. Comparison of numerical results (D-Rs and V-Rs) of temperature (top) and soot volume fraction (bottom) profiles against experiments; (M-L) Menon et al. [43] and (X-S, X-L) Xu et al. [44].

Figure 12 :

 12 Figure 12: Premixed slot burner: geometry of the numerical setup used in simulations.

12

 12

  . The domain is long enough to allow soot formation. The cell size in the flame front are ∆ x ∼ 0.03 mm, leading to 15-20 cells across the thermal layer.Boundary conditions are indicated in Fig.12. The burner walls are adiabatic. The computation is carried out on half of the domain because of the symmetric geometry of the burner. Symmetry is also assumed for outer lateral boundary conditions. Ethylene-air mixture with equivalence ratio of φ = 2.5 at T = 298 K and p = 1 atm is injected at the inlet. A parabolic axial velocity distribution, with mean axial velocity u m = 30 cm • s -1 , is imposed to mimic a fully-developed channel flow. The six numerical cases presented in Sec. 5.2 are computed. Simulations are carried out with the laminarSMOKE package[START_REF] Cuoci | Numerical modeling of laminar flames with detailed kinetics based on the operator-splitting method[END_REF][START_REF] Cuoci | A computational tool for the detailed kinetic modeling of laminar flames: Application to c2h4/ch4 coflow flames[END_REF] based on the OpenFOAM framework, designed to handle arbitrarily complex kinetic mechanisms.

Figure 13 :

 13 Figure 13: Premixed Slot Burner: comparison of a) temperature and b) soot volume fraction fields using the virtual and detailed models, case V-R (left) and case D-R (right).

Figure 13

 13 Figure 13 compares the virtual soot sub-mechanism (case V-R) and the detailed model (case D-R) results. The analysis of predicted temperature fields shows the ability of the virtual chemistry to capture accurately the flame position. Flame heights of 6 mm and 6.3 mm are found respectively for D-R and V-R cases (5 % error). A good agreement is also observed on the soot volume fraction fields. Temperature and soot volume fraction profiles along the centerline are plotted in Fig. 14 for the six simulation cases D-A,

Figure 14 :

 14 Figure 14: Premixed Slot Burner: Temperature (black) and soot volume fraction (green) profiles along the centerline. Comparison of simulation (a) between cases D-A and V-A, (b) between cases D-R and V-R, and (c) between cases D-Rs and V-Rs.

Figure 15 displays

 15 Figure 15 displays the radial temperature profiles for the six simulation cases at two different heights above the burner z = 3 mm and z = 20 mm.At z = 3 mm, the difference of the flame thickness is about 0.1 mm between detailed (D-A, D-R, D-Rs) and virtual (V-A, V-R, V-Rs) results. Further downstream at z = 20 mm, the temperature profile is almost uniform and exhibits a difference of less than 2% between detailed and reduced simulations.

Figure 15 :

 15 Figure 15: Premixed Slot Burner: radial temperature profiles at two different heights above the burner, z = 3 mm (blue) and z = 20 mm (red). Comparisons (a) between cases D-A and V-A, (b) between cases D-R and V-R, and (c) between cases D-Rs and V-Rs.

Figure 16 displays

 16 Figure 16 displays the radial soot volume fraction profiles for the six different cases at z = 3 mm and z = 20 mm. At z = 3 mm, the virtual adiabatic model (a) underestimates the peak of soot volume fraction given by the reference solution by two orders of magnitude. The discrepancies are reduced to a factor of two when radiative effects are included. At z = 20

Figure 16 :

 16 Figure 16: Premixed Slot Burner: radial soot volume fraction profiles at two different heights above the burner, z = 3 mm (blue) and z = 20 mm (red). Comparisons (a) between cases D-A and V-A, (b) between cases D-R and V-R, and (c) between cases D-Rs and V-Rs.

The 2 -Figure 17 :

 217 Figure 17: Santoro Coflow Flame: Temperature (left) and soot volume fraction (right) fields using the virtual chemistry and radiative models.

Figure 18 :Figure 19 :

 1819 Figure 18: Santoro Coflow Flame: Temperature profiles; (a) radial, and (b) axial. Comparison between detailed (D-Rs), solid lines, and virtual chemistry (V-Rs), dashed lines.

Figure 20 :

 20 Figure 20: Santoro Coflow Flame: Integrated soot volume fraction profiles along the centerline. Comparison of detailed and virtual chemistry results against experimental measurements. Numerical results from other groups are also included.

Figure 21 :

 21 Figure 21: Scatter plot in the (a) (T, Z) and (b) (f v , Z) phase spaces using the reference detailed mechanism results: points of 1-D premixed (blue) and non-premixed counterflow (red) flames as well as results from the 2-D coflow flame (black) simulation from Section 7.2 are presented.

Table 1 :

 1 List of operating conditions of 1-D flames included in the learning database.

All detailed chemistry simulations include thermophoretic effects for all species BIN 5 to BIN 20 . Unity Lewis number is assumed for virtual species present in the main and virtual mechanisms excepted for the virtual soot species S for which Le S = 25 is considered. This value is estimated by averaging the Lewis number of BINs found in the detailed mechanism. See the Supplementary Material for more details. All flame calculations, with both

Table 2 :

 2 Summary of computed cases. Kinetic model and radiative effects involved are indicated.

Table 3 :

 3 Comparison of the CPU time (per core) with the virtual scheme(12 species, 6

		Computational time using Computational time using
	Physical time			Speed-up
		Virtual Chemistry	Detailed chemistry
	1 ms	30 s	17.5 h	2100
	50 ms	25 min	850 h	2040
	reactions) and the detailed CRECK scheme [1] (297 species and 16797 reactions) . The
	numerical cost has been estimated by computing the 2-D premixed slot burner on 120
	cores (Intel Xeon CPU E5-2670), without accounting for radiation.

r 2πf v (r, z)rdr, are shown in Fig 20. In addition to D-Rs solutions, detailed chemistry solutions obtained by other groups are also indicated.

Acknowledgments

This work was granted access to the HPC resources of TGCC under the allocations A0092B10253 made available by GENCI (French Grand Equipement National de Calcul Intensif), and HPC resources from the Mésocentre computing center of CentraleSupélec and Ecole Normale Supérieure Paris-Saclay supported by CNRS and Région Ile-de-France (http://mesocentre.centralesupelec. fr/).


Yacine Bechane
email: yacine.bechane@centralesupelec.fr

Benoît Fiorina

Numerical investigations of turbulent premixed flame ignition by a series of Nanosecond Repetitively Pulsed discharges

Keywords: Plasma-assisted combustion, Nanosecond Repetitively Pulsed discharges, Turbulent premixed flame, Large Eddy Simulation Colloquium: New concepts

Nanosecond Repetitively Pulsed (NRP) discharges are an efficient way to promote turbulent flame ignition in lean regimes. The energy released by NRP discharges leads first to an ultra-fast species dissociation and heating phenomena, followed by a slow heating process. A phenomenological plasma model is presented to capture the influence of NRP discharges on the combustion process at low CPU cost. The model is here implemented in a LES

flow solver to simulate the ignition sequence of a bluff-body turbulent premixed flame by a series of NRP discharges. Two numerical computations are performed. First, only the thermal effects of the discharge (ultra-fast heating and slow heating due to vibrational energy relaxation) are taken into account. Then both the thermal and chemical effects (mainly O 2 dissociation into O) are considered. The results show that in the first simulation the ignition never occur, whereas in the second simulation flame ignition occurs after only 5 pulses. The ignition success or failure results from a competition

Introduction

To reduce pollutant emissions such as nitrogen oxides, lean combustion chambers are under development in many energy-related sectors. However, a recurring problem to industrial applications is that lowering the flame temperature slows the chemical reactions, causing difficulties to ensure successful ignitions over the wide range of operating conditions covered by practical applications.

An emerging solution to promote ignition is to generate high-voltage electric discharges between two electrodes located inside the combustion chamber. It generates locally a plasma, which interacts with the combustion. Beneficial effects of plasma-assisted combustion have been observed in laboratory-scale burners under the action of microwave discharges, gliding arc discharges, plasma torches, nanosecond dielectric barrier discharges or Nanosecond Repetitively Pulsed (NRP) discharges [START_REF] Starikovskiy | Plasma-assisted ignition and combustion[END_REF][START_REF] Ju | Plasma assisted combustion: dynamics and chemistry[END_REF]. Among these various types of discharges, Nanosecond Repetitively Pulsed (NRP) discharges [START_REF] Galley | Plasma enhanced combustion of a lean premixed air[END_REF][START_REF] Pilla | Stabilization of a turbulent premixed flame using a nanosecond repetitively pulsed plasma[END_REF] have shown to be a particularly energy-efficient way to initiate and control combustion processes especially when conventional ignition systems such as spark plugs are rather ineffective or too energy costly [START_REF] Ju | Plasma assisted combustion: dynamics and chemistry[END_REF].

Efficiency of NRP discharges is high because the energy of the electric discharge is not only spent to increase the gas temperature but also to generate active species [START_REF] Rusterholtz | Ultrafast heating and oxygen dissociation in atmospheric pressure air by nanosecond repetitively pulsed discharges[END_REF]. In typical conditions, around 35% of molecular oxygen is dissociated in the inter-electrode region [START_REF] Rusterholtz | Ultrafast heating and oxygen dissociation in atmospheric pressure air by nanosecond repetitively pulsed discharges[END_REF][START_REF] Lo | Space and time analysis of the nanosecond scale discharges in atmospheric pressure air: I. gas temperature and vibrational distribution function of n2 and o2[END_REF]. This high concentration of radicals has a positive effect on ignition phenomena [START_REF] Macheret | Energy efficiency of plasma assisted combustion in ram/scramjet engines, 36th AIAA Plasma dynamics and Lasers Conference[END_REF]. Experiments conducted in closed vessel showed that NRP discharges greatly improve ignition probability over different flow conditions [START_REF] Lovascio | Effects of pulsation frequency and energy deposition on ignition using nanosecond repetitively pulsed discharges[END_REF] but also revealed the complexity of the ignition mechanism. For instance, a larger amount of energy deposition or higher pulse frequency do not necessarily lead to a higher ignition probability [START_REF] Lefkowitz | An exploration of inter-pulse coupling in nanosecond pulsed high frequency discharge ignition[END_REF]. To fully control the ignition by NRP discharge a complete understanding of the governing physics and the impact of the different discharge parameters is necessary. Numerical simulations are a good candidate to investigate the fundamental mechanisms of plasma-assisted ignition. For example, the phenomenological plasma-assisted combustion model developed by Castela et al [START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF] has highlighted the key role played by the radicals created by the NRP discharges in the turbulent flame ignition. These simulations also evidenced the flow recirculation induced by the discharge which strongly impact the probability of ignition success [START_REF] Castela | A 3-d dns and experimental study of the effect of the recirculating flow pattern inside a reactive kernel produced by nanosecond plasma discharges in a methane-air mixture[END_REF]. While these simulations gave a first understanding of plasma-assisted ignition, the mechanisms of turbulent flame ignition by a series of NRP discharge remains unknown.

The objectives of the present work are to understand, through numerical investigations, the mechanisms of turbulent premixed flame ignition by a series of NRP discharges. Simulations of a bluff-body premixed turbulent burner, designed and experimented at EM2C laboratory, are performed for that purpose. The numerical strategy relies on the implementation of the recently developed plasma combustion model [START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF] in an Large Eddy Simulations (LES) flow solver. The modeling approach is described in Sec. 2, whereas the configuration is detailed in Sec. 3 and results are presented in Sec. 4. Finally, a scenario of plasma assisted ignition mechanism supported by a low-order model is proposed in Sec. 5.

Modeling plasma-assisted turbulent combustion

Nanoseconds pulsed discharges model

NRP discharges are characterized by high voltage pulses (5 to 20 kV) that last few nanoseconds and are repeated at frequencies of the order of tens of kHz. Each NRP pulse causes an increase of the electrons kinetic energy, which is transferred to the surrounding heavy particles through collisional processes. 0D modeling studies [START_REF] Popov | The effect of nonequilibrium excitation on the ignition of hydrogen-oxygen mixtures[END_REF][START_REF] Williams | Model and igniter development for plasma assisted combustion[END_REF] on the influence of the atomic particles (O, H, N) and excited species (O 2 (a 1 ∆ g ), H 2 (v), N 2 (A,B,C,a ) , ...), produced by nanosecond discharges, identified new kinetic pathways in the combustion process. 1-D and 2-D simulations of lean laminar flame ignition by NRP discharges [START_REF] Takana | Numerical simulation of nanosecond pulsed dbd in lean methaneair mixture for typical conditions in internal engines[END_REF][START_REF] Tholin | Influence of fast-heating processes and o atom production by a nanosecond spark discharge on the ignition of a lean h2/air premixed flame[END_REF][START_REF] Casey | Simulations of planar non-thermal plasma assisted ignition at atmospheric pressure[END_REF] showed a discharge chemical effects on the combustion with an important production of radicals such as O, OH, H, CH 3 that causes a kinetic enhancement and a thermal effect that leads to a thermal enhancement. These numerical methods aims to solve the governing equations for the electric field and the electron energy, as well as the continuity equations for neutral and excited species, and the energy, mass and momentum balance equations for the gas mixture. Despite their accuracy, these strategies require high computational resources which are too expensive for 3-D turbulent flame simulations.

A semi-empirical model has been proposed by Castela et al. [START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF] to capture the effects of the NRP discharge on combustion without solving the electric field and the detailed plasma chemistry. The model considers that the NRP discharges induces mainly an electronic and vibrational excitation of nitrogen molecules as observed in [START_REF] Popov | Investigation of the mechanism for rapid heating of nitrogen and air in gas discharges[END_REF]. The nano-second scale relaxation of nitrogen electronic states leads to an ultra-fast increase of gas temperature and dioxygen dissociation into atomic oxygen [START_REF] Rusterholtz | Ultrafast heating and oxygen dissociation in atmospheric pressure air by nanosecond repetitively pulsed discharges[END_REF], whereas the relaxation of the vibrational states causes a a slow gas heating at the milliseconds scale.

Ultra-fast and slow processes are modeled by spliting the rate of energy Ėp deposited at each pulse in the reactive mixture into three contributions:

Ėp = Ėp chem + Ėp heat + Ėp vib (1)
The chemical Ėp chem and thermal Ėp heat terms refer to the amount of energy transferred into the ultra-fast dissociation of O 2 molecules and gas heating, respectively. The rate of energy Ėp vib contributes to the vibrational excitation of N 2 . According to [START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF], these energy contributions are modeled as follow:

Ėp chem = η Y O 2 Y f O 2 (1 - e O 2 e O ) Ėp (2)
Ėp heat = [α -η Y O 2 Y f O 2 (1 - e O 2 e O )] Ėp (3) Ėvib = (1 -α) Ėp ( 4
)
where Y f O 2 represents the mass fraction of O 2 in the fresh mixture prior to each pulse. e k is the internal energy of species in mass units. α is the ratio of energy deposited at the ultra-fast time scale, whereas η is the amount of O 2 dissociated into O atoms. These parameters are here set to α = 55% and η = 35%, according to experimental [START_REF] Rusterholtz | Ultrafast heating and oxygen dissociation in atmospheric pressure air by nanosecond repetitively pulsed discharges[END_REF] and numerical [START_REF] Popov | Fast gas heating initiated by pulsed nanosecond discharge in atmospheric pressure air[END_REF] studies of the NRP discharge properties. The ability of the NRPD model to reproduce the discharge effects (O 2 dissociation, ultra-fast and slow heating) has been validated against measurements in [START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF].

LES balance equations

The plasma-combustion is coupled with the LES reactive flow governing equations combined with the Thickened Flame model for TFLES [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF].

It consists in artificially thickening the reaction zone to enable the proper resolution of the flame front on a practical LES grid. In addition to the mass and momentum equations, the following balance equations are solved for total energy e, vibrational energy e vib and species mass fractions Y k :

∂ρ e ∂t + ∂(ρ u i e) ∂x i = ∂ ∂x i C p F Ξ µ P r + (1 -S) µ t P t r ∂ T ∂x i + ∂ ∂x i N k=1 F Ξ µ S c,k + (1 -S) µ t S t c,k ∂ Y k ∂x i h s,k + ∂(σ ij u i ) ∂x i + Ėp chem + Ėp heat + Ṙp V T (5
)

∂ρ e vib ∂t + ∂(ρ u i e vib ) ∂x i = ∂ ∂x i µ S c,evib + µ t S t c,evib ∂ e vib ∂x i + Ėp vib - Ṙp V T (6) ∂ρ Y k ∂t + ∂(ρ u i Y k ) ∂x i = ∂ ∂x i F Ξ µ S c,k + (1 -S) µ t S t c,k ∂ Y k ∂x i + Ξ F W k ωc k + W k ωp k (7)
where ϕ and ϕ are Reynolds and Favre filtering of variable ϕ. µ, S c and P r are the viscosity, the Schmidt and Prandtl numbers, respectively, where t superscript denotes for turbulent quantities. The dynamical formulation of the thickening factor reads F = 1 + (F max -1)S, where the flame sensor S equals 0 and 1 outside an inside the reactive layer, respectively. Ξ is the subgrid scale flame wrinkling. ωc k , the chemical reaction rate due to combustion, is modeled by a detailed mechanism.

The plasma chemical reaction rate ωp k is closed to account for the O 2 dissociation induced by the electrical discharge:

ωp O = η Y O 2 Y f O 2 Ėp e O W O (8) ωp O2 = - W O W O2 ωp O (9) ωp k = 0 if k = O 2 , O (10)
As the plasma chemistry is much faster than flow mixing, sub-grid scale interactions between turbulence and plasma chemistry have been neglected.

The term Ṙp V T in both Eq. ( 5) and Eq. ( 6) refers to the relaxation rate of the vibrational energy into gas heating. It is modelled considering the Landau-Teller harmonic oscillator approach [START_REF] Landau | Zur theorie der sohdldispersion[END_REF]:

Ṙp V T = ρ e vib -e eq vib ( T ) τ V T ( 11
)
where the equilibrium value of the vibrational energy, e eq vib (T ) , at a given gas temperature T, is given by e eq vib (T ) = r Θ 1 /(exp (Θ 1 /T ) -1), with Θ 1 = 3396K and r = R/W N 2 , R being the gas constant and W N 2 the molar mass of nitrogen. τ V T is computed according to [START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF]

MiniPAC configuration

Experimental set up

The MiniPAC configuration, sketched in Fig. 1

Numerical set up

LES of the MiniPAC burner ignition sequence are performed using the YALES2 unstructured finite-volume low-Mach number solver [21]. The time integration is performed using a fourth-order temporal scheme with a centred fourth order scheme for spatial discretization. The sub-grid Reynolds stresses tensor is closed using the Dynamic Smagorinsky model [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF]. The COFFEE kinetic scheme involving 14 species and 38 reactions is used for the methaneair mixture chemistry [START_REF] Coffee | Kinetic mechanisms for premixed, laminar, steady state methane/air flames[END_REF], species equations are coupled with turbulence using the Thickened Flame model for LES (TFLES) [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF] and the Charlette sub-grid scale flame wrinkling model [START_REF] Charlette | A power-law wrinkling model for les of premixed turbulent combustion: Part i non dynamic formulation and initial tests[END_REF].

The discharges properties have been set to be consistent with experimental measurements for the temperature and the atomic oxygen spatial Tab. 1, are performed to understand the role of species dissociation in the turbulent flame ignition mechanism. In both simulations, 55% of the energy is deposited at the ultra-fast scale (α = 0.55) whereas 45% is transferred into the vibrational energy, which relax slowly into gas heating. In LES-T case, only thermal effects are considered in the plasma discharge model (η = 0). In LES-TC case, in accordance with experimental and numerical observations [START_REF] Rusterholtz | Ultrafast heating and oxygen dissociation in atmospheric pressure air by nanosecond repetitively pulsed discharges[END_REF][START_REF] Popov | Fast gas heating initiated by pulsed nanosecond discharge in atmospheric pressure air[END_REF] 35 % of O 2 is dissociated into O atoms during the pulse. For both cases, the same amount of energy of 3.5 mJ per pulse is transferred to the flow.

Two different simulations, indicated as cases LES-T and LES-TC in

Results

For both cases, the NRP discharges are applied at time t = 0 with a pulse frequency of 20 kHz. Figure 3 effects of the plasma are considered. The temperature in the discharge zone increases after each pulse until it reaches a steady state after approximately 15 pulses. The heat brought by the discharge diffuses in the recirculation zone located downstream the bluff body but the mixture does not ignite.

Although the pulse generator is always turned on, no flame kernel is formed.

Figure 4 shows instantaneous fields of temperature for LES-TC solution,

where both thermal and chemical properties of the plasma are considered.

A reactive kernel is now clearly identified after only 5 pulses at t = 200 µs, leading to local ignition. Even if the pulse generator is turned off after the 10 th discharges, the kernel still growths and transits to a turbulent V shaped flame, which reach a steady-state regime at t = 7.5 ms.

These observations are confirmed by the temporal evolution of computational domain maximum temperature, plotted in Fig. 5. When only thermal effects are considered, the temperature never reaches the burnt gases temperature level, despite the unceasing plasma pulsations. Indeed, despite two peaks at 2000 K observed at t = 0.75 ms and t = 2.75 ms, the maximal temperature fluctuates around 1200 K. At the opposite, when both chemical and thermal plasma effects are considered, ignition is clearly identified after the 5 th pulse. A peak above 4000 K is then reached at the center of the discharge. The combustion is even sustained when the NRP discharges are turned off after the 10 th pulse. .

Scenario of ignition mechanism by NRP discharge

The NRP energy deposition causes an ultra-fast increase of the gas temperature, which generates a gas expansion. It creates two recirculation zones, promoting fresh gas entrainment into the discharge channel [START_REF] Castela | A 3-d dns and experimental study of the effect of the recirculating flow pattern inside a reactive kernel produced by nanosecond plasma discharges in a methane-air mixture[END_REF]. This phenomena is evidenced in Fig. 7 Residence time of the mixture in the PSR is therefore given by τ = d/V . The PSR balance equations reads:

dh dt = 1 τ K k=1 Y k0 h k0 - K k=1 Y k h k - Q h ρV + Ėp chem + Ėp heat + Ṙp V T ( 12
)
dY k dt = 1 τ (Y k0 -Y k ) + W k ωc k ρ + W k ωp k ρ ( 13
)
de vib dt = 1 τ (e vib0 -e vib ) + Ėp vib -Ṙp V T (14)
where 0 subscript denotes gases properties flowing in the PSR. The rate of deposited thermal and chemical energies Ėp chem , Ėp heat and Ėp vib are closed according to Eq. ( 2), Eq. ( 3) and Eq. ( 4). The relaxation rate of vibrational energy Ṙp

V T is modeled by Eq. ( 11) and the plasma chemistry sources terms ωp k is closed by Eq. ( 8), Eq. ( 9) and Eq. ( 10). where ignition occurs between 5 and 10 pulses. The blue area is however located outside the ignition conditions whatever the number of discharges, which is consistent with the ignition failure observation in LES-T.

Conclusion

This article presents the first LES of turbulent flame ignition by a series of NRP discharges. The numerical investigations have been conducted with the massively parallel YALES2 solver in which a phenomenological plasmaassisted combustion model has been implemented. The retained configuration is a premixed bluff-body turbulent flame where successful ignition by NRP discharges has been experimentally observed at EM2C. Ignition is also observed in the simulation but only if the ultra-fast dis- Results presented in this article confirms that the major interest of NRP discharges is the production of radical species which directly interact with the combustion chemistry mechanism. The simulations has here been per-formed by assuming that the production of O species by the discharge is the key chemical effect of plasma on combustion chemistry. This assumption is not restrictive with respect to the modeling strategy, which will be extended in the future to account for other radicals production and possible fuel dissociation induced by the plasma discharge.

  (a), is a premixed methaneair bluff-body stabilized turbulent flame. The injection tube and the bluffbody diameters are D i = 1.6 cm and D b = 1 cm, respectively. The burner is fed with a methane-air mixture (φ = 0.95), injected at a constant mass flow rate ṁ = 15 m 3 h -1 which corresponds to a bulk velocity U b = 37.5 m.s -1 . 10 kV discharges, corresponding to a reduced electric field (E/N ) of 100 Td to 300 Td (1Td = 10 -21 V.m 2 ) are applied using a 2 mm diameter electrode positioned 5 mm above the bluff body and connected to a high frequency pulse generator. The images of the discharge during the pulses show that the discharge has a quasi-cylindrical shape of 5 mm length corresponding to the inter-electrode distance, and a radius equal to 1.75 mm. Electrical measurements of the voltage and the current showed that each discharge pulse transfers 3.5 mJ to the flow. The deposit lasts 50 ns and is repeated at a frequency of 20 kHz.

Figure 1 :

 1 Figure 1: (a) MiniPAC burner scheme. (b) mesh in the near-electrode region

  distribution and the deposited energy. The discharge has a cylindrical shape filtered with the spatial function F (r) = erf c[(r/a) b ] where r is the radial distance from the discharge axis. a and b are geometric parameters, fitted to model the measured discharge radius and to ensure a sufficient numerical resolution of the discharge interface.A fraction of the energy deposited by the discharge leaves the domain in the form of acoustic waves causing locally a temperature decay[START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF]. Since the low-Mach number formulation can not capture acoustic waves, the amount of acoustic discharge energy leaving the domain has been evaluated from the DNS computations[START_REF] Castela | Modelling the impact of non-equilibrium discharges on reactive mixtures for simulations of plasma-assisted ignition in turbulent flows[END_REF] and then deducted from the amount of energy deposited in the low-Mach LES solver[21].The computational domain includes half of the injection tube and covers 35 D b and 30 D b in downstream and radial directions, respectively. To identify the optimal spatial resolution of the the plasma, a series of five electrical pulses has been computed on three different grids characterized by cells sizes of 25 µm, 50 µm, and 100 µm in the discharge zone. Figure.2 shows that the evolution of the maximum temperature and O concentration in the discharge zone is similar on each grid, even on the coarser mesh 100 µm which is therefore retained to conduct the LES. As the discharge interface is well resolved by this grid, no influence of TFLES model is expected in this region.The final mesh therefore consists of 40 millions of tetrahedral elements. In the plasma discharge zone the cells size equals 0.1 mm while in the flame zone it ranges from 0.1 mm to 0.5 mm. A 2-D view of the mesh in (x,z) plane is illustrated in Fig.1(b) in the near-electrode region.

Figure 2 :

 2 Figure 2: Sensitivity study of the grid resolution in the discharge zone. Temporal evolution of maximal values of temperature and atomic oxygen O concentration reached in the discharge are plotted during 5 pulses. Three different mesh size are tested, corresponding to cell size of 25 µm (black), 50µm (blue) and 100 µm (red, retained for the LES) in the discharge zone.

Figure 3 :

 3 Figure 3: Simulation of the ignition attempt of the MiniPAC configuration by a series of NRP discharges, case LES-T, Only the thermal effects are taken in account (η = 0). Instantaneous 2-D temperature fields are shown at the: 1 st (a), 10 th (b), 30 th (c), 60 th (d), 90 th (e) plasma pulses. Ignition never occur even after 90 pulses.

Figure 4 :

 4 Figure 4: Simulation of the ignition attempt of the MiniPAC configuration by a series of NRP discharges, case LES-TC, both the thermal and the chemical effects are taken in account (η = 0.35). Instantaneous 2-D temperature fields are shown at the: 1 st (a), 5 th (b), 10 th (c) plasma pulse and at t = 5.8 ms (d) and t = 7.5 ms when the pulses generator is off. The flame ignites at the 5 th pulse (b) and transits to form a V flame in (d) and (e).

Figure. 6 Figure 5 :Figure 6 :

 656 Figure 5: Temporal evolution of the maximum value of gas temperature in the MiniPAC simulations. The mixture ignites after 5 pulses for case LES-TC (red), whereas ignition never occurs for case LES-T (blue).

  (a), which shows the velocity stream-lines superimposed on the temperature field in the inter-electrode region, at t = 25 µs after a pulse discharge. Ignition success or failure results from competition between the residence time of the reacting gases in the discharge zone and the combustion chemistry time scale. This phenomena is modeled by a PSR schematized in Fig. 7(b), whose volume matches the discharge channel of diameter d. The flow of gas passing through the PSR reproduces the recirculating gases at velocity V .

Figure 7 :Figure 8 :

 78 Figure 7: (a) flow recirculation in the discharge channel after a pulse. (b) PSR modeling of plasma-assisted ignition mechanism.

Figure 9 :

 9 Figure 9: plasma-assisted ignition diagram. The number of pulses required to form a flame kernel in terms of the residence time. Blue symbols correspond to the PSR-T case. Red symbols correspond to the PSR-TC case. Residence times τ T for the LES-T case and τ T Cfor the LES-TC case are indicated by the red and the blue hatched area, respectively.

  . The filtered rate of energy,

	Ėp chem ,	Ėp heat and	Ėp vib are closed from Eq. (2), Eq. (3) and Eq. (4) by neglecting
	sub-grid scale contribution, i.e. by assuming that Ėp

y = Ėp y ( Y O 2 , Ėp ).

Table 1 :

 1 Numerical case properties. The same amount of energy per pulse E p =3.5 mJ is deposited for each case.

	shows a series of 2-D instantaneous snapshots
	in the plane (y,z) of the temperature field for case LES-T, where only thermal

Acknowledgments

This work was supported by the ANR programs PASTEC (ANR-16-CE22-0005-01) and granted access to the HPC resources from IDRIS, TGCC and CINES under the allocation A0032B10253 and A0052B10253 made by GENCI (Grand Equipement National de Calcul Intensif) and HPC resources from the Mesocentre computing center of CentraleSupelec and Ecole Normale Superieure Paris-Saclay supported by CNRS and Rgion Ile-de-France ( http:// mesocentre.centralesupelec.fr/ ). Vincent Moureau and Ghislain Lartigue from CORIA lab, and the SUCCESS scientific group are acknowledged for providing the YALES2 code.