Hernando Maldonado Colmán

Nasser Darabiha

Denis Veynante

Benoît Fiorina

A turbulent combustion model for soot formation at the LES subgrid-scale using virtual chemistry approach

Keywords: Soot, Turbulent flames, Soot subgrid-scale model, Virtual chemistry

Designing efficient and clean combustion devices to reduce soot formation in practical applications is a major concern nowadays. This could be achieved by performing predictive simulations in turbulent sooting combustors. Large Eddy Simulation (LES) is an attractive approach to predict flame/turbulence

interactions at small scales including soot phenomena. However, because of the peculiarly complex nature of soot production phenomena, turbulent sooting flames remain very difficult to simulate and are computationally expensive. Thus, reduced gas-phase kinetics and soot models should be included in numerical simulations. The reduced virtual chemistry approach is a great alternative to address this problem since it introduces an optimized and flexible strategy to tackle multi-mode turbulent combustion and pollutants prediction. Also, the virtual chemistry approach has recently demonstrated its capability in reproducing soot phenomena in laminar flames including radiative heat transfer. In the present work, the virtual chemistry methodology

Introduction

Soot modeling in turbulent flames is a great challenge because of the complexity of multi-physical phenomena involved. The numerical modeling strategy must tackle the complexity of the chemical process involved in the soot formation along with the interactions with the turbulence.

The gas-phase kinetics must reproduce not only the flame structure but also recover small hydrocarbon radicals as well as soot precursors and polycyclic aromatic hydrocarbons (PAHs) [START_REF] Frenklach | Detailed mechanism and modeling of soot particle formation[END_REF]. The prediction of these species needs the knowledge of the complex chemical pathways described using de-tailed mechanisms involving hundreds of species and thousands of reactions.

These mechanisms require prohibitive CPU times to be used in LES of practical applications. Consequently, kinetic reduction techniques such as tabulated methods [START_REF] Gicquel | Laminar premixed hydrogen / air counterflow flame simulations using flame prolongation of ILDM with differential diffusion[END_REF][START_REF] Ihme | Modeling of radiation and nitric oxide formation in turbulent nonpremixed flames using a flamelet/progress variable formulation[END_REF], analytically reduced chemistry approaches [START_REF] Pepiot | Automatic strategies to model transportation fuel surrogates[END_REF] and global optimized chemistry [START_REF] Franzelli | A two-step chemical scheme for kerosene-air premixed flames[END_REF] have been developed. In particular, a reduced chemistry method based on the virtual chemistry concept has recently been proposed by Maldonado Colmán et al. [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] to predict the soot volume fraction at very low CPU cost. This new approach accounts for a "soot chemistry" representation that envelops both soot particle dynamics and soot surface reactivity [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF].

Coupling detailed soot chemistry with the turbulence is a key modeling challenge [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF]. In LES, a number of developments have been proposed on subgrid-scale closures of the filtered soot source terms [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF][START_REF] El-Asrag | Simulation of soot formation in turbulent premixed flames[END_REF][START_REF] Mueller | Large eddy simulation subfilter modeling of soot-turbulence interactions[END_REF][START_REF] Berger | Modeling subfilter soot-turbulence interactions in Large Eddy Simulation: An a priori study[END_REF] and typically assume the same turbulence-flame interaction regime for all chemical species. However, the wide range of chemical time and length scales covered by detailed soot chemistry complicates the subgrid-scale (SGS) modeling of soot formation. Indeed, turbulent combustion models are generally valid in a specific flame regime identified in the Borghi diagram [START_REF] Peters | Laminar flamelet concepts in turbulent combustion[END_REF]. For instance, models based on geometrical approach [START_REF] Fiorina | Modeling combustion chemistry in large eddy simulation of turbulent flames[END_REF] are well adapted to the flamelet regime where the chemistry is much faster than the flow. At the opposite, when the chemistry is slow, the reactive layer is distributed, and PSR-like models are more appropriate to LES modeling [START_REF] Li | Large eddy simulation of mild combustion using finite rate chemistry: Effect of combustion sub-grid closure[END_REF][START_REF] Fooladgar | Characterization of flameless combustion in a model gas turbine combustor using a novel post-processing tool[END_REF]. The issue is therefore that interactions between turbulence and soot chemistry are expected to cover both flamelet and distributed regimes in the Borghi diagram. In the LES context, statistical PDF models which are independent of the turbulent combustion regime can be used to model turbulence-chemistry interactions. Examples of PDF models are found in recent works of Huo et al. [START_REF] Huo | A coupled mmc-les and sectional kinetic scheme for soot formation in a turbulent flame[END_REF], introducing a Multiple Mapping Conditioning for LES (MMC-LES) coupled to a detailed sectional kinetic scheme, and that of Sun and Rigopoulos [START_REF] Sun | Modelling of soot formation and aggregation in turbulent flows with the les-pbe-pdf approach and a conservative sectional method[END_REF], proposing a new methodology that includes the Population Balance Equation (PBE) of soot particles into the LES-PDF of the gas phase. Although both groups have successfully reproduced laboratory-scale turbulent sooting flames, these approaches are rather computationally expensive to simulate large-scale industrial applications [START_REF] Fiorina | Modeling combustion chemistry in large eddy simulation of turbulent flames[END_REF]. Therefore, a better alternative would be to keep a geometric approach, which reduces computational costs, but extends its validity to the PSR regime.

The objective of the present paper is to develop a turbulent combustion model for soot formation. Chemical kinetics will be based on the virtual chemistry methodology recently extended to soot [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] and validated on laminar flame configurations. An original hybrid turbulent combustion model formulation is proposed, which degenerates to the flamelet regime in the flame front region and to a PSR formalism in the post-flame zone, characterized by slow soot chemistry. Then, the model is challenged in the turbulent non-premixed sooting flame from Sandia [START_REF] Zhang | Design of "model-friendly" turbulent non-premixed jet burners for C2+ hydrocarbon fuels[END_REF], in which the SGS model has a significant impact on soot formation [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF].

In the present work, the LES formalism for mass fraction filtered transport equations is first presented in Section 2. Section 3 briefly introduces the fundamentals of the virtual chemistry methodology. Section 4 expands the virtual chemistry approach to an LES framework for turbulent combustion and presents the derivation of a new hybrid SGS model for turbulence-chemistry-soot interactions. The new hybrid SGS model for virtual chemistry is then applied to a laboratory-scale turbulent sooting flame. The experimental and LES numerical setups are described in Section 5. Finally, in Section 6, results using the new hybrid model is challenged to experimental measurements, as well as other numerical results found in the literature.

LES formalism: Mass fraction filtered transport equations

By employing the Thickened Flame model for LES (TFLES), the Favrefiltered transport equation of the k th species mass fraction Y k is typically expressed as:

∂ ρ Y k ∂t + ∂ ∂x i ρu i Y k = ∂ ∂x i F T Ξ ∆ µ Sc k + (1 -S F ) µ t Sc t,k ∂ Y k ∂x i + ωk , (1)
where ρ is the mixture density and u i is the i th component of the velocity.

µ and µ t are the laminar and turbulent viscosity, respectively. Sc k and Sc t,k are respectively the k th species laminar and turbulent Schmidt numbers. Ξ ∆ is the subgrid flame wrinkling. Dynamic thickening [START_REF] Legier | Dynamically thickened flame LES model for premixed and non-premixed turbulent combustion[END_REF] is applied by using the flame sensor:

S F = max(S F,k , 0),
where S F,k is defined equal to 1 within the k th species reactive layer and 0 outside [START_REF] Cailler | Prediction of CO emissions in LES of turbulent stratified combustion using virtual chemistry[END_REF]. The thickening factor is F T = max(F T,k , 1), with:

F T,k = max 1 + (F max T,k -1)S F,k , 1 . (2)
Naturally, the TFLES model considers the flame front region (S F = 1) to be in the flamelet regime [START_REF] Poinsot | Theoretical and numerical combustion[END_REF] and therefore the diffusive term outside the flame, hereafter referred as the post-flame region, includes a turbulent diffusive contribution. Further analyses in the following sections require a more explicit notation in terms of these two regions, and therefore, Eq. 1 is rewritten as follows:

∂ ρ Y k ∂t + ∂ ∂x i ρu i Y k = ∂ ∂x i S F F T Ξ ∆ µ Sc k ∂ Y k ∂x i + ∂ ∂x i (1 -S F ) F T Ξ ∆ µ Sc k + µ t Sc t,k ∂ Y k ∂x i + ωk , (3)
where the first term in the RHS corresponds exclusively to the flame front region (S F = 1) and the second one to the post-flame region (S F = 1). Now, note that when S F = 0, following Eq. 2, F T Ξ ∆ = 1. One then can write:

∂ ρ Y k ∂t + ∂ ∂x i ρu i Y k = ∂ ∂x i S F F T Ξ ∆ µ Sc k ∂ Y k ∂x i + ∂ ∂x i (1 -S F ) µ Sc k + µ t Sc t,k ∂ Y k ∂x i + ωk , (4)

Soot virtual chemistry modeling

A virtual chemical scheme consists of virtual species and reactions whose thermo-chemical properties are optimized by machine learning algorithms [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF]. The range of application of reduced virtual schemes is given by the operating conditions and flame configurations covered by the learning database [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF]. The virtual chemistry model is composed of a main scheme, dedicated to capture heat release and temperature, and one or more satellite sub-mechanisms devoted to pollutant prediction such as CO [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF], NO x [START_REF] Maio | A virtual chemical mechanism for prediction of NO emissions from flames[END_REF] or soot [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF]. In the present study, the main mechanism for ethylene-air combustion from [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] is retained:

α M F,1 F + α M Ox,1 Ox → α M I,1 I (R M 1 ) I → N M P k=1 α M P k,2 P k (R M 2 )
Also, the following virtual soot sub-mechanism proposed in [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] is employed:

α s F,1 F + α s 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 ) 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 )
In these reactions, F and Ox correspond to the fuel and oxidizer respectively.

I, P k , V 1 , V 2
and V 3 are virtual species. Analysis conducted in [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF] showed that an accurate prediction of equilibrium flame temperature over the whole range of equivalence ratios is obtained with four products, N M P = 4. The virtual species S corresponds to soot. α M k,r and α s k,r are stoichiometric coefficients per unit mass of the k th virtual species in the r th virtual reaction, where the superscript M and s respectively correspond to the main and soot sub-mechanism. The soot volume fraction is obtained from f s v = ρY s S /ρ s where the soot mass fraction Y s S is tracked by the virtual species S. The soot density ρ s is set to 1800 kg m -3 . The rate of progress of the r th virtual reaction in a mechanism v (M or s) is modeled by the Arrhenius law:

q v r = A v r • T β v r exp E v a,r RT N v s k=1 [ψ v k ] n v k,r , (5)
where

N v s is the number of virtual species in mechanism v. A v r , β v r , E v a,r
and n v k,r are kinetics constants optimized by a machine learning algorithm. [ψ v k ] denotes the molar concentration of the virtual species ψ v k . To ensure consistency between the main and soot mechanisms,

α s F,1 = α M F,1 , α s Ox,1 = α M
Ox,1 , and kinetics rate parameters of reactions R M 1 and R s 1 are identical. The chemical source term of the k th virtual species in mechanism v is then given by:

ωv k = N v r r=1 α v k,r q v r (6)
where N v r is the number of reactions in mechanism v. Virtual main and soot mechanisms have been optimized in [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] to target an ensemble of reference flame solutions computed using the CRECK detailed mechanism [START_REF] Saggese | Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame, Combust[END_REF], including 297 species and 16797 reactions.

To optimize the main mechanism, a learning database made of 26 premixed flames with equivalence ratios ϕ in the range [0.5; 3.0] is used. As explained in [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF], the soot sub-mechanism is optimized on a different database made of 11 premixed flames with ϕ ∈ [1.8; 2.8] and 6 non-premixed flames of strain rates a ∈ {6, 10, 25, 45, 60, 90} s -1 . The virtual soot sub-mechanism reproduces faithfully the soot volume fraction given by detailed chemistry simulations in both premixed and non-premixed radiative laminar flames [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF].

Soot turbulent modeling

Chemical time and length scales

The modeling of filtered chemical reaction rates ωv k should consider the regime of interactions between chemistry and turbulence. In most cases, chemical phenomena responsible of heat release exhibit time scales much shorter than that of turbulence. Filtered chemical reaction rates of thickened species involved in the main mechanism are then usually modeled under the flamelet regime assumption [START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. However this hypothesis is questionable for soot chemistry which exhibits a broader range of characteristic time scales.

To evidence this issue, a 1-D freely-propagating premixed ethylene-air flame calculated using the virtual chemistry model including radiative effects [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] for a rich mixture of equivalence ratio of ϕ = 2.4 is analyzed here.

Figure 1(top) presents a sample of thermochemical quantities corresponding

to the main mechanism: the temperature (T ), an intermediate virtual species (I) and one virtual product (P 1 ). Quantities of P 2 , P 3 and P 4 are similar to those of P 1 and therefore are not shown. In the left column, all quantities evolve within the thermal flame thickness, δ 0 L = (T max -T min )/ max(dT /dx), here of the order of δ 0 L ≈ 1 mm. All quantities in Fig. 1, except for the temperature, are normalized by their maximum value and are indicated by the operator ⟨ • ⟩.

Chemical species involved in the soot sub-mechanism are presented in Fig. 1, with a focus on the thermal layer (middle) and the post-flame region (bottom). Quantities V 3 are similar to those of V 1 and consequently is not plotted. As the species I in the main mechanism, the intermediate species V 1 , directly linked to the consumption of F and Ox through reaction R s 1 is produced within the flame front. However, this is not the case of virtual species involved in R s 2 to R s 4 , such as S or V 2 , which are mainly produced and consumed in the post-flame region (see bottom subfigures). The thickness of the soot reactive layer δ 0 soot is defined by the zone where | ωs S | > 0.001× max(| ωs S |). For ϕ = 2.4, δ 0 soot ≈ 100mm which is much higher than δ 0 L . Thus, the soot species cannot be localized by the flame sensor.

Chemical time scales τ M c and τ s c , representative of the main and soot virtual mechanisms, respectively, are now introduced as:

τ M c = δ 0 L S 0 f g , τ s c = δ 0 soot S 0 bg , (7)
where S 0 f g and S 0 bg are the flame displacement speeds in the fresh and burnt gases, respectively [START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. These chemical time scales associated with the flame front and the soot formation are plotted as a function of the equivalence ratio in Fig. 2. As the chemical characteristic time of soot formation is about three orders of magnitude slower than that of the main chemical species, interactions with the turbulence will therefore be different. Nevertheless, it should be noted that other chemical species of the soot sub-mechanism, such as V 1 and V 3 , are also interacting in the flame front region as mentioned before (see Fig. 1), and consequently turbulence-chemistry interactions of such species belong to multiple regimes and further measures must be taken in the post-flame region.

Soot subgrid-scale model

Turbulence interactions with the flame front and the post-flame reactive layers are characterized here by introducing the following subgrid-scale Damkhöler numbers:

Da M = τ t τ M c , Da s = τ t τ s c . (8)
The turbulence time scale, τ t = ∆ x /u ′ sgs , is evaluated from the LES filter characteristic length ∆ x and the subgrid-scale (SGS) velocity fluctuation

u ′ sgs .
As shown in Fig. 2, chemical time scale involved in the flame front (main mechanism) is of the order of 10 -4 s, which is expected to be smaller than the turbulence time scales encountered in practical combustion chambers.

Consequently Da M > 1, and the flamelet regime is retrieved. As in the classical TFLES approach [START_REF] Fiorina | Modeling combustion chemistry in large eddy simulation of turbulent flames[END_REF], the filtered species reaction rates in Eq. 4 involved in the main mechanism are modeled with the subgrid-scale flame wrinkling Ξ ∆ :

ωM k = Ξ ∆ F T ωM k . ( 9
)
This approach is quite enough to resolve the turbulent-chemistry interactions in the flame front region and has been used in multiple occasions in practical turbulent combustion applications (to resolve for instance heat release and CO) [START_REF] Cailler | Prediction of CO emissions in LES of turbulent stratified combustion using virtual chemistry[END_REF][START_REF] Maio | Virtual chemistry for temperature and CO prediction in LES of non-adiabatic turbulent flames[END_REF][START_REF] Maio | Capturing multi-regime combustion in turbulent flames with a virtual chemistry approach[END_REF]. However, the same closure cannot be retained for species involved in the current virtual soot sub-mechanism which involves both short and long time scales. The following hybrid formulation is therefore proposed to model ωs k :

ωs k = S F • Ξ ∆ F T ωs k + (1 -S F ) • ωs k PostFlame . (10)
In the flame front region, S F takes the value of unity, and Eq. 10 is then simplified to ωs k = Ξ ∆ ωs k /F T and flamelet regime assumption is retrieved. Recalling Eq. 4, the source term is activated together with the laminar diffusive term as adopted for the flamelet regime. At the opposite, in the post-flame region, S F = 0, and the filtered reaction rate is given by ωs k PostFlame , and in Eq. 4, both laminar and turbulent diffusive terms are also triggered.

The model is written so that Eq. 4 degenerates towards a classical TFLES formalism at the flame front (S F = 1) while no thickening is applied in the post-flame region (S F = 0). Now considering the soot sub-mechanism, the reaction R s 1 , which produces soot precursor V 1 , is mainly effective at the flame front while reactions R s 2 through R s 4 act in the post-flame region. Therefore, in order to account for the turbulent mixing, an efficiency factor η r is introduced into the filtered soot reaction rate:

ωs k PostFlame = N s r =4 r=1 η r α s k,r q s r . (11)
At the post-flame region, the following features must be ensured:

• S F = 0 should be guaranteed with the adequate main mechanism indicators [START_REF] Cailler | Prediction of CO emissions in LES of turbulent stratified combustion using virtual chemistry[END_REF],

• Rates of consumption of fuel (F ) and oxidizer (Ox) should be consistent between main and soot mechanism (reactions R s 1 and R M 1 ),

• The SGS turbulence-soot-chemistry interactions should be in concordance with the previous item.

Consequently for the first reaction in Eq. 11, one should have:

η 1 = Ξ ∆ F T . ( 12
)
Soot characteristics time scales involved in reactions R s 2 through R s 4 are (primarily) expected to be slower than subgrid turbulent time scale. Therefore Da s << 1 and the combustion regime at the SGS may be assimilated to perfectly-stirred reactor (PSR) regime [START_REF] Poinsot | Theoretical and numerical combustion[END_REF]. The efficiency factor η r is then modeled by the Damkhöler number as:

η r = max 1 Da s , 1 for r = 2, to 4. ( 13
)
Turbulence will then enhance the combustion by a factor 1/Da s . When the turbulence vanishes at the subgrid-scale, i.e., when the system behaves as a laminar reacting flow, then u ′ sgs → 0, and according to Eq. 8 the soot Damköhler number Da s → +∞. Then, Eq. 13 ensures that η r → 1 and that the laminar flame regime is well recovered.

Prior to the 3-D calculations of the following Section, an analysis of the phenomenological behavior of 1-D pseudo-turbulent flames is performed, using the hybrid turbulent combustion model, which is included in Appendix A.

Radiation model

Sooting flames may present significant radiative heat losses. Therefore, the new virtual radiative model proposed by Maldonado Colmán et al. [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] is employed here. This model is able to produce the radiative effects of both gas and soot in hydrocarbon-air sooting flames. The radiative source term is modeled by optically-thin approximation:

Qr = 4κ v Planck σ T 4 -T 4 b , (14)
where T b is the background cold temperature, σ is the Stephan-Boltzmann constant and κ v Planck is the virtual Planck absorption coefficient. κ v Planck is composed of the contributions of gas and solid phases as κ v Planck = κ M gas +κ v soot . The gas absorption coefficient κ M gas is modeled as a fifth-order polynomial function of the inverse of the temperature, whose set of polynomial coefficients is optimized according to the methodology described in Ref. [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF]. The learning database consists of 26 non-adiabatic premixed flames including gas radiation only based on Ref. [START_REF] Barlow | Scalar profiles and no formation in laminar opposed-flow partially premixed methane/air flames[END_REF] (ϕ ranging in [0.5; 3.0]). The set of gaseous Planck mean absorption parameters is optimized using the machine learning algorithm of Ref. [START_REF] Cailler | Building-up virtual optimized mechanism for flame modeling[END_REF], by comparing the results of temperature and laminar flame speed against the learning database [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF]. The set conducing to the best fitness compared to detailed chemistry results is finally selected [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF].

The soot absorption coefficient κ v soot is proportional to the temperature T and the soot volume fraction f s v using the main and soot sub-mechanisms, respectively. The modeling constant is based on Ref. [START_REF] Modest | Radiative Heat Transfer[END_REF]. For further information about the virtual radiative model, we refer the reader to Ref. [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF].

The filtered radiative source term is calculated by neglecting SGS temperature and composition fluctuations, using Eq. 14, as follows:

Qr ≃ Qr ( T , Y v k ). ( 15
)

LES of a turbulent sooting flame

The experimental configuration retained for validation is the turbulent non-premixed ethylene-air jet flame, the Sandia burner [START_REF] Zhang | Design of "model-friendly" turbulent non-premixed jet burners for C2+ hydrocarbon fuels[END_REF], studied by the International Sooting Flame (ISF) workshop [START_REF]ISF5, International sooting flame workshop[END_REF]. The burner consists of a central jet of pure ethylene with an inner diameter of D = 3.2 mm. The central jet is surrounded by a coflow of air and a pilot flame. The central jet bulk velocity is u bulk f uel = 54.7 cm • s -1 . The pilot flame is fed with a premixed ethylene-air mixture of ϕ = 0.9 at a mass flowrate of 1.772 × 10 -4 kg • s -1 , corresponding to 2% of the heat release rate of the central jet. The air coflow mean velocity is 0.6 m • s -1 . All gas flows are injected at atmospheric pressure and T 0 = 294 K. For further details on the experimental configuration, the reader may refer to [START_REF] Zhang | Design of "model-friendly" turbulent non-premixed jet burners for C2+ hydrocarbon fuels[END_REF].

Numerical setup

The computational domain has a cylindrical shape with a diameter of 188D and a length of 500D. The unstructured grid is made of 43.4 millions of tetrahedral cells. The smallest cells of length ∆ x ∼ 0.1 mm are located at the main jet exit.

No-slip condition is imposed at the injector walls. Fully-developed velocity profiles (including velocity fluctuations) issued from a pipe flow simulation is prescribed at the main inlet as in [START_REF] Rodrigues | Coupling an LES approach and a soot sectional model for the study of soot-ing turbulent non-premixed flames[END_REF]. Uniform velocity profiles are assumed at the coflow and pilot inlets boundary conditions. Burnt gases at thermodynamic equilibrium, corresponding to the mixture previously mentioned, are injected at the pilot inlet. The pilot walls are non-adiabatic and a uniform temperature distribution of 1000 K is imposed. Other lateral walls are adiabatic.

The hybrid turbulent combustion model for soot from Section 3 and the models for turbulence-chemistry interactions and radiation developed in Section 4 are implemented in the low-Mach, unstructured finite-volume YALES2 code [START_REF] Moureau | Design of a massively parallel cfd code for complex geometries[END_REF]. In this code, the convective terms are solved with a 4 th order scheme in time and space. The subgrid Reynolds stress tensor is closed using the sigma model [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF]. The filtered radiative source term in the energy equation is approximated by using Eq. 15 and neglecting subgrid-scale temperature and mixture fraction fluctuations.

The SGS flame wrinkling Ξ ∆ is closed with the Charlette model [START_REF] Charlette | A power-law flame wrinkling model for LES of premixed turbulent combustion part I: non-dynamic formulation and initial tests[END_REF][START_REF] Charlette | A power-law flame wrinkling model for LES of premixed turbulent combustion part II: dynamic formulation[END_REF],

with the constant parameter β = 0.5. A constant Prandtl number Pr = 0.7 is considered. The Lewis number for virtual species is set to unity except for the virtual soot species S, then Schmidt numbers are Sc k = 0.7, for k̸ = S. Indeed, according to [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF], the Lewis number for the virtual soot species S is set to Le S = 25, which corresponds to a laminar Schmidt number of Sc S = 17.5. The turbulent Schmidt number of all virtual species, including S, is dynamically evaluated as in [START_REF] Moin | A dynamic subgrid-scale model for compressible turbulence and scalar transport[END_REF].

The time and Favre averaging for the statistics are accumulated over 250 ms of physical time (about 3.5 times the flow-through time on the flame length).

Results

Figure 3 gives a typical 2-D representation of instantaneous temperature (a) and soot volume fraction (b) using the virtual non-adiabatic model as mentioned above. The radiative heat losses decrease the peak temperature along the stoichiometric iso-line by 250 K.

As expected, the high concentration of soot is located in the fuel-rich region. The maximum soot volume fraction is about 1.6 ppm, which is similar to what is observed in the experiments [START_REF] Shaddix | Understanding and predicting soot generation in turbulent non-premixed jet flames[END_REF].

Temperature and major species radial profiles

To highlight the importance of radiative effects in turbulent sooting flame simulation, both adiabatic and non-adiabatic simulations are performed. x/D =175. Adiabatic simulations significantly overestimate the temperature whereas non-adiabatic solutions are in good agreement with experimental data [START_REF] Kearney | Temperature, Oxygen, and Soot-Volume-Fraction Measurements in a Turbulent C2H4-Fueled Jet Flame[END_REF]. The non-adiabatic model gives less than 10% of discrepancy in the mean profiles.

Numerical RMS curves show differences up to 50% compared to experiments. However, RMS in simulation (i.e., of the resolved field) are not directly comparable to the experimental RMS because subgrid fluctuations are not taken into account [START_REF] Veynante | Comparison between les results and experimental data in reacting flows[END_REF]. The computation accuracy is comparable to results shown in [START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a monte carlo method coupled to large eddy simulation[END_REF], obtained by combining the RFPV turbulent combustion model [START_REF] Ihme | Modeling of radiation and nitric oxide formation in turbulent nonpremixed flames using a flamelet/progress variable formulation[END_REF] and a Monte Carlo radiative solver.

Further validation of the reacting mixture is now done by comparing X O 2 /X N 2 profiles of the LES results against experimental measurements [START_REF] Kearney | Temperature, Oxygen, and Soot-Volume-Fraction Measurements in a Turbulent C2H4-Fueled Jet Flame[END_REF].

Figure 5 presents the mean (top) and RMS (bottom) profiles of X O 2 /X N 2 for two different distances from the injector. At both position, mean profiles using the non-adiabatic model are in good agreement with experimental measurements, with an underestimation of about 25 %. Without radiation, the mixture profile is under predicted a bit more. Numerical results show that LES can qualitatively capture the RMS fluctuations but underestimates them by a factor of about 2 to 3 at x/D = 134 and lower than 30 % at x/D = 175. [START_REF] Zhang | Design of "model-friendly" turbulent non-premixed jet burners for C2+ hydrocarbon fuels[END_REF] and to three recent LES results from other groups involved in the ISF workshop. The various numerical strategies employed to model soot formation and corresponding references are indicated in Table 1. In Results #1, a tabulated chemistry was considered for the gas phase based on [START_REF] Wang | A PAH growth mechanism and synergistic effect on pah formation in counterflow diffusion flames[END_REF], a sectional method for soot formation with a lumped PAH for the nucleation model [START_REF] Rodrigues | Coupling an LES approach and a soot sectional model for the study of soot-ing turbulent non-premixed flames[END_REF] and Monte Carlo method for radiation [START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a monte carlo method coupled to large eddy simulation[END_REF]. In Results #2 and #3, the tabulated chemistry was based on [START_REF] Narayanaswamy | A consistent chemical mechanism for oxidation of substituted aromatic species[END_REF], the soot formation was modeled using a method of moments [START_REF] Blanquart | Analyzing the effects of temperature on soot formation with a joint volume-surface-hydrogen model[END_REF][START_REF] Mueller | A joint volume-surface model of soot aggregation with the method of moments[END_REF] and optically thin approximation for radiation [START_REF] Barlow | Scalar profiles and no formation in laminar opposed-flow partially premixed methane/air flames[END_REF]. Both works [START_REF] Blanquart | Analyzing the effects of temperature on soot formation with a joint volume-surface-hydrogen model[END_REF][START_REF] Mueller | A joint volume-surface model of soot aggregation with the method of moments[END_REF] consider PAH dimerization to couple the gas-solid transition, however in Results #2 a multiple species nucleation strategy is used, while a lumped PAH transport equation used in Results #3.

Soot volume fraction profiles

Regarding the turbulence-soot interactions, SGS models based on the work of Mueller and Pitsch [START_REF] Mueller | Large eddy simulation subfilter modeling of soot-turbulence interactions[END_REF][START_REF] Mueller | LES model for sooting turbulent nonpremixed flames[END_REF] are used for results #1, #2 and #3. The model from Results #3, however, took a further measure by adapting the soot SGS model to capture soot phenomena only in mixture compositions where soot growth rates are higher than the oxidation rate and, in this way, then the spurious soot oxidation is reduced [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF]. Further information about the chemistry and soot models, the reader is referred to the references shown in Table 1.

The non-adiabatic hybrid turbulent combustion model predicts correctly maximum soot volume fraction, while neglecting the radiation effects causes a significant underprediction. Soot is however produced too fast by the hybrid turbulence-soot model while the consumption rate is fairly captured after the peak. Indeed, it should be noted that the building-up process of the hybrid

Effect of the soot SGS model

To highlight the influence of the hybrid combustion regime model, a simulation has been performed by neglecting the impact of the subgrid-turbulence on the soot chemistry in the post-flame region, i.e., η r = 1 for r = 1 to 4 in Eq. 11. Figure 8 compares the solution of this simulation against the hybrid model solution and experiments. A significant underestimation of f v is revealed when the hybrid model is turned off (factor 3). Same effect on soot production in the Sandia burner was also observed in [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF], where a difference of an order of magnitude is observed when changing the SGS model. An estimation of the resolved soot reaction layer is given by the following relationship:

A posteriori validation

δ 1 S = 1 max{grad ⟨Y s S ⟩} , (16)
where ⟨Y s S ⟩ stands for the mean soot mass fraction. For instance, at x/D = 101.6: δ 1 S ≈ 9m. As expected, soot evolves at a scale much larger than the mesh size (here max ∆ x ≈ 2mm) and there is no flame resolution issues in the post flame region.

These observations validates a posteriori the PSR model selected to close the filtered reaction rates related to soot virtual chemistry.

Conclusions and discussion

The virtual chemistry and radiative models developed for sooting laminar flames [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF] have been implemented into an LES framework to simulate turbulent sooting flames. As soot chemistry exhibits a time scale much slower than the flame front chemistry, an original subgrid-scale hybrid reaction rate closure is proposed to cover both the flamelet regime chararacteristic of the thermal flame layer and a PSR-like regime observed in the post-flame region, where slow soot chemistry is dominant.

Three-dimensional LES of the Sandia burner is performed using the virtual chemistry, radiative and hybrid SGS models, giving good agreement with experimental measurements. The new virtual soot sub-mechanism captures well the soot physical phenomena and results are comparable to those of other groups obtained by using detailed models. The SGS model has a big impact on soot results, as observed by [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF]. An analysis of the Damköhler numbers is performed a posteriori validating the closure for the filtered reaction rates related to soot virtual chemistry. In overall, soot-turbulent-chemistry interactions in the Sandia burner are fairly retrieved by the non-adiabatic hybrid approach.

Further modeling developments should however be considered to improve the results accuracy. This can be done at different levels by:

• enlarging the learning database target during the virtual chemical scheme optimization [START_REF] Cailler | Development of a virtual optimized chemistry method. Application to hydrocarbon/air combustion[END_REF] in order to better capture the soot oxidation process;

• retaining a dynamic closure for the subgrid-scale flame wrinkling in the flame front region [START_REF] Wang | Implementation of a dynamic thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF];

• introducing a more detailed closure in the distributed combustion regime encountered in the post-flame region [START_REF] Li | Large eddy simulation of mild combustion using finite rate chemistry: Effect of combustion sub-grid closure[END_REF][START_REF] Fooladgar | Characterization of flameless combustion in a model gas turbine combustor using a novel post-processing tool[END_REF];

• a priori testing of turbulent source term closure against DNS data of a turbulent sooting reactive flow representative of post-flame conditions. Such comparison aims to challenge the modeling of the efficiency parameter η r used in Eq. 13. The configuration will be similar to the one computed by Caudal et al. [START_REF] Caudal | Modeling interactions between chemistry and turbulence for simulations of partial oxidation processes[END_REF] to study turbulent chemistry interaction in partial oxidation processes.

Additional efforts are needed to assess the CPU cost required by each methodology shown in this paper. Estimating the costs by combining different approaches to solve the gas phase kinetics (tabulated, analytically reduced, global, and virtual mechanisms) and soot (MoM, sectional, virtual models) using the same computational resources will help the users in the decision-making process based on their applications. Finally, it is worth noting that the proposed hybrid model can be used for the prediction of other pollutants whose chemical formation exhibits also both short and long characteristic time scales such as NOx.

1. An artificially thickened flame with thickening factor equal to T F = 5, referred to as the "Flamelet" case. The impact of the subgridturbulence on the soot chemistry in the post-flame region is neglected:

i.e., η r = 1 for r = 1 to 4 in Eq. 11.

Figure 1 :

 1 Figure 1: Laminar ethylene-air flame structure, ϕ = 2.4. Top: Temperature T (solid red line), normalized source terms ⟨ ωk ⟩ (symbols with dotted lines in left column), and normalized species mass fractions ⟨Y k ⟩ (lines in right column) profiles of the main mechanism; Middle and Bottom: values for the soot sub-mechanism. Normalization is defined as: ⟨φ⟩ = φ/ max(|φ|).

Figure 2 :

 2 Figure 2: Chemical time scales associated with the flame front τ M c and the soot formation τ s c in 1-D laminar premixed ethylene-air flames for different equivalence ratios ϕ.

Figure 4

 4 Figure 4 compares experimental and numerical radial profiles of the mean (top) and RMS (bottom) temperature at two axial locations x/D =134 and

Figure 6

 6 Figure 6 compares mean (top) and RMS (bottom) radial profiles of soot volume fraction profiles at three axial positions. Numerical results using the adiabatic and non-adiabatic hybrid turbulent combustion model are compared to experimental data [17]. The non-adiabatic virtual chemistry model fairly reproduces the soot formation. Although the radial profiles meet correct orders of magnitude of soot volume fraction and the sooting region (|r/D| ≲ 15) compared to the experimental measurements, other qualitative features such as the double-peak are not well captured. Nevertheless, itshould be noted that results are in good rapport with other state-of-the-art numerical results[START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a monte carlo method coupled to large eddy simulation[END_REF][START_REF] Jain | Effects of large aromatic precursors on soot formation in turbulent non-premixed sooting jet flames[END_REF].

Figure 7

 7 Figure 7 presents mean soot volume fraction along the burner axis. Adiabatic and non-adiabatic profiles obtained by the hybrid turbulent combustion

Figure 9

 9 Figure 9 shows the scatter plot of Da M and Da s in the radial direction, at three horizontal planes located at x/D = 9.375, 31.25, 93.75 and 187.5 for the non-adiabatic simulation using the hybrid combustion regime model. The flame front chemistry is dominant at x/D = 9.375 while the soot chemistry starts further downstream at about 31.25 and is well established at x/D = 93.75. Soot consumption continues far downstream in the post-flame region, at x/D = 187.5, where the flame chemistry is completed. As anticipated in Section 2, Da M is larger than unity, which is in accordance with the flamelet assumption retained to close the filtered reaction rates involved in the main mechanism. Conversely, Da s is smaller than one for all axial positions.

2 .

 2 An artificially thickened flame with thickening factor equal to T F = 5 including the new hybrid turbulent combustion model of Section 4, simply referred to as the "PSR" case. Three simulations are conducted by imposing constant values for the efficiency: η = 1.1, 1.5, and 2 Results plotted in Fig. A.1, show the temperature (left), soot volume fraction (center), and soot source terms (right) profiles of the 1-D pseudoturbulent flames. As expected, the temperature profile is not impacted by the soot SGS closure. However significant differences are observed on the soot production in the post-flame region, for x > 10 mm: the soot volume fraction is enhanced when the efficiency η increases. This is also evidenced in Fig. A.1 (right), which shows soot source term normalized as:⟨ ωs S ⟩ Flamelet = ωs S | PSR max ( ωs S | Flamelet ) . (A.1)The contribution of the soot source term in the flame front is negligible compared to that of the post-flame (see solid lines in zoomed subfigure), with a difference of about two orders of magnitude. Since the soot source term is significant at about 10mm away from the flame front, there is no considerable impact from the thickening and wrinkling operations. This confirms the hypothesis made to establish the hybrid flamelet-PSR model presented in Section 4.

Figure 3 :

 3 Figure 3: Instantaneous fields of (a) temperature and (b) soot volume fraction. The black continuous line on each figure corresponds to the stoichiometric mixture fraction Z st indicating the flame front.

Figure 4 :

 4 Figure 4: Radial profiles of mean (top) and RMS (bottom) temperatures at x/D = 134 (left) and x/D = 175 (right). Numerical results of adiabatic (blue) and non-adiabatic(red) cases are compared to experimental data (symbols)[START_REF] Kearney | Temperature, Oxygen, and Soot-Volume-Fraction Measurements in a Turbulent C2H4-Fueled Jet Flame[END_REF].

Figure 5 :

 5 Figure 5: Radial profiles of mean (top) and RMS (bottom) X O2 /X N2 at x/D = 134 (left) and x/D = 175. Numerical results of adiabatic (blue) and non-adiabatic (red) cases are compared to experimental data (symbols) [37].

Figure 6 :

 6 Figure 6: Radial profiles of mean (top) and RMS (bottom) soot volume fraction at three different axial positions x/D. Numerical results of the adiabatic (blue) and non-adiabatic (red) hybrid model are compared to experimental data (symbols) [17].

Figure 7 :

 7 Figure7: Mean soot volume fraction profile along the burner axis. See Table1for inforabout Results #1-3.

Figure 8 :

 8 Figure 8: Mean soot volume fraction profile along the burner axis. Effect of the SGS model on soot formation.

Figure 9 :

 9 Figure 9: Scatter plot of flame (red) and soot (black) Damköhler numbers in the radial direction at a horizontal section y-z for four axial positions x/D = 9.375, 31.25, 93.75 and 187.5. Da M is conditioned into the flame front region (S F = 1), and Da s outside the reaction zone (S F = 0).

Figure A. 1 :

 1 Figure A.1: Temperature (left), soot volume fraction (center), and normalized soot source terms (right) profiles of 1-D seudo-turbulent premixed ethylene-air flames with equivalence ratio ϕ = 2.5: comparison of numerical results using the Flamelet and PSR models. The soot source terms are normalized following Eq. A.1, and separated by their contributions in the flame front (solid lines) and the post-flame (dashed lines with symbols) regions according to Eq. 10.

Acknowledgments

Vincent Moureau and Ghislain Lartigue from CORIA laboratory and the SUCCESS scientific group are acknowledged for providing the YALES2 code. This work was granted access to the HPC resources of IDRIS and TGCC under the allocations A0112B10253 made available by GENCI (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.

Results

Gas-phase

Soot Radiative Reference chemistry model Model #1 Tabulated chemistry Method of moment Optically Thin [START_REF] Jain | Effects of large aromatic precursors on soot formation in turbulent non-premixed sooting jet flames[END_REF] #2 Tabulated chemistry Sectional model Monte Carlo [START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a monte carlo method coupled to large eddy simulation[END_REF] #3 Tabulated chemistry Method of moment Optically Thin [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF] Table 1: Modeling strategies of the simulation results shown in Fig. 7.

turbulent combustion model did not cover effects such as pyrolysis, gradual PAH formation, or another chemical path that could help capture this delay.

In addition, the learning database does not cover the situation where soot convects towards the flame front, and soot growth in fuel-lean regions might be underpredicted as observed in a laminar non-smoking coflow flame [START_REF] Maldonado Colmán | A virtual chemistry model for soot prediction in flames including radiative heat transfer[END_REF].

Therefore, the learning database needs to be enlarged. However, this misprediction of soot region position was observed in other LES of turbulent jet flames in the literature using detailed combustion and soot models [START_REF] Mueller | LES model for sooting turbulent nonpremixed flames[END_REF][START_REF] Sewerin | An les-pbe-pdf approach for predicting the soot particle size distribution in turbulent flames[END_REF]. Compared to the other numerical models, the virtual chemistry model is up to par with the state-of-the-art of this configuration. The maximum soot volume fraction is underpredicted by a half in Result #1 [START_REF] Jain | Effects of large aromatic precursors on soot formation in turbulent non-premixed sooting jet flames[END_REF], and overpredicted by a factor greater than 2.5 in Results #2 [START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a monte carlo method coupled to large eddy simulation[END_REF] and #3 [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF]. However, the position of the peak of soot volume fraction is only well predicted by Result #1 [START_REF] Jain | Effects of large aromatic precursors on soot formation in turbulent non-premixed sooting jet flames[END_REF]. Result #3 [START_REF] Yang | Large eddy simulation of soot evolution in turbulent reacting flows: Presumed subfilter PDF model for soot-turbulence-chemistry interactions[END_REF] overestimates the peak position by 23D, while Result #2 [START_REF] Rodrigues | Analysis of radiative transfer in a turbulent sooting jet flame using a monte carlo method coupled to large eddy simulation[END_REF] and the hybrid turbulent combustion solution underestimate it by 16D and 58D, respectively. In terms of computational costs, the LES using the virtual model required a total of about 2.