LES investigation of soot formation in a turbulent non-premixed jet flame with sectional method and FGM chemistry

This study proposes a detailed soot modeling framework for large-eddy simulation (LES) to accurately predict soot formation and particle size distributions (PSD) in turbulent reacting ﬂows. The framework incorporates Flamelet Generated Manifold (FGM) chemistry and a soot model based on the discrete sectional method (DSM) to predict both qualitative and quantitative sooting behavior while keeping the computational cost affordable. Two elementary modeling strategies are considered in the LES formalism for describing soot formation rates. These strategies rely on an a-priori tabulation of soot formation rates and their run-time computation. The LES formalism is applied to the simulations of a well-characterized, non-premixed, turbulent jet ﬂame. A comparative analysis of strategies employed for ﬁltered soot source term treatment is conducted to investigate their impact on the prediction of soot quantities and the evolution of soot PSDs. The LES results for the gas phase and soot phase are compared against the available experimental data. A good prediction of soot evolution is achieved with the two methodologies. The tabulation of soot formation rates leads to a signiﬁcant reduction in computational cost compared to the model based on their explicit runtime computation. The LES results reveal that the modeling of ﬁltered soot source terms has a signiﬁcant impact on the quantitative prediction of soot formation. The possible reasons for the observed differences in the soot prediction are discussed. The run-time computation-based model provides a more consistent treatment of the non-linear interactions between the gas and soot phases in soot source terms compared to the tabulated soot chemistry approach. On the other hand, the tabulated soot chemistry model is an interesting and eﬃcient modeling approach for predicting soot formation in turbulent conditions. Overall, both approaches have their strengths and limitations, and the choice of approach may depend on the speciﬁc needs of the application. © 2023 The Author(s). Published by Elsevier Inc. on behalf of The Combustion Institute. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )


Introduction
Due to the negative effects of soot on the environment and human health, the legislative regulations on soot emissions from combustion devices have become stricter.The new regulations [1,2] will require monitoring the number of particles, particularly smaller ones, in addition to the total soot volume.Hence, predicting the evolution of soot particle size distribution is essential to develop appropriate mitigation strategies for their emissions.Predicting particle size distribution with soot models requires solving the population balance equation (PBE), which describes the dynamics of a polydispersed particulate system coupled with reactive flow equations [3] .As a result, sooting flame computations become CPU intensive [4] .Several approaches facilitate the description of the PBE, including stochastic methods [5] , moment-based methods [6,7] , kinetic models [8] , and discrete sectional methods [9][10][11] .However, due to limitations in terms of computational overhead, primarily moment-based [7,12,13] and, to a lesser extent, sectional methods [11,14] are more commonly used in large-scale flame simulations.Moment-based approaches allow an adequate description of the spatial and temporal evolution of soot particles, however, they provide only limited access to particle size distribution and often rely on approximations and closure models to estimate the unknown soot particle number density function (NDF).In contrast, discrete sectional method-based approaches offer direct information on the local size distribution of soot particles but are less commonly used as they are more computationally demanding.
The state of the art in soot modeling research suggests that detailed soot models coupled with detailed kinetics, transport, and radiation models provide a fairly good prediction of soot formation in laminar flames [15] after careful calibration of the models.The progression toward soot prediction in turbulent flames, however, remains a great challenge owing to its complex multi-physical nature, characterized by an intricate coupling between flow parameters, gas-phase chemistry, and soot properties.Numerical prediction of soot in turbulent conditions necessitates adequate characterization of the flame behavior as well as different physicochemical phenomena involved in soot formation.
Turbulent flows are characterized by a wide range of flow scales from energy-containing large scales to the smallest Kolmogorov scales [16] .In comparison, the time scales of soot formation are typically even larger than the integral flow time scales.This disparity makes the prediction of soot formation in turbulent flames a challenging task.Hence, although significant efforts have been made to describe turbulent-chemistry interaction, understanding the effects of turbulence on soot formation and their modeling are still open subjects to be addressed.
The prediction of soot formation in turbulent flames using the aforementioned soot modeling approaches is preferable to be conducted in the context of Direct Numerical Simulations (DNS), where a full description of all the temporal and spatial scales associated with turbulence and chemistry are resolved.However, the high computational cost required to resolve the coupled phenomena associated with the scales of turbulence, chemistry, and particle dynamics limits the DNS investigation to selected cases [17][18][19] .On the other end, Reynolds Averaged Navier-Stokes (RANS)-based methods, are more affordable and mainly used at industriallyrelevant conditions [20] .However, RANS approaches hardly provide information on the transient phenomena of the flow, flame dynamics, and soot formation.Therefore, methods like Large Eddy Simulation (LES), which facilitates information on transient features of the flow are more relevant despite their higher computational cost.In fact, LES has been applied very successfully for a wide range of problems including jet, swirl, or spray flames [4,11,[21][22][23][24][25][26] and it generally shows a good trade-off between accuracy and computational cost.Therefore, the development of reliable soot models for LES is essential for the study of soot formation in both fundamental and industrial configurations.In recent years, LES has been adopted to investigate soot formation in turbulent flames, departing from semi-empirical methods [27][28][29] and evolving towards more detailed soot modeling approaches [4,7,11,13,30,31] .
LES of turbulent sooting flames relies on three aspects: an accurate description of gas-phase chemistry, modeling of the dynamics of soot particle population, and integration of models for capturing interactions between soot-turbulence-chemistry. For reliable prediction of soot formation in turbulent combustion, a detailed understanding of gas-phase kinetics is necessary to properly capture the flame structure and formation of gas-phase species participating in the soot processes (such as polycyclic aromatic hydrocarbons (PAHs) or unsaturated hydrocarbons such as acetylene).However, using detailed kinetic schemes involving hundreds of species is impractical in LES employing finite-rate chemistry-based methods.As a result, tabulated chemistry methods [7,11] , reduced kinetic schemes [32] , and globally optimized chemistry [33] are often used instead.Concerning the soot-phase description, although quantitative modeling of soot formation in turbulent combustion is still beyond the frontier of current modeling advancements [15] , the state-of-the-art soot models can provide partial solutions to practical problems of interest when used judiciously along with appropriate predictions of the turbulent flow structures.
Another challenge in LES of turbulent sooting flames is modeling the subgrid-scale interactions between soot and turbulence, as soot chemistry involves non-linear interactions between the gas and solid phases.In the context of LES, subgrid turbulencesoot interactions refer to the correlations that arise from the cou-pling of the soot model with the turbulent flow.These include: a) the correlation between reactive scalars and the soot scalar (number density) concerning growth and oxidation terms, and b) the correlation between number densities in relation to coagulation terms.Different strategies have been proposed to model sootturbulence interactions in mono-dispersed and moment-based models [7,30] and more recently in sectional models [4,11] .Mueller and Pitsch [34] introduced a subfilter PDF model with double-delta PDF, considering both sooting and non-sooting modes for subgridscale turbulence-soot interaction.This approach was employed in several other studies [11,35] and subsequently reformulated by Berger et al. [36] , by defining the sooting mode sub-structure as a log-normal distribution.Furthermore, Yang et al. [30] proposed a presumed subfilter PDF that explicitly considers soot distribution in mixture fraction space.Recently, Colmán et al. [37] extended the presumed subfilter PDF model to account for finite-rate soot oxidation by analyzing the local relative motion of diffusionless soot particles relative to mixture fraction iso-contours.Although closure models for soot-turbulence interactions are important in the detailed description of the soot formation process, their impact on the overall prediction of global soot quantities appears less prominent compared to the sensitivity of soot production to model parameters involved in soot subprocesses [15] .As a result, in some cases [13,20] , interactions between turbulence and soot are neglected.
Earlier studies on the LES of turbulent sooting flames have been primarily focused on moment-based soot models.El-Asrag and Menon [38] employed the Linear Eddy Model (LEM) in combination with the moment method for soot prediction with LES.Later, Mueller and Pitsch [7,34] combined the moment-based soot model with an extended Flamelet/Progress Variable (FPV) [39] model by including lumped-PAH inception kinetics.In this LES approach, the presumed-PDF method was used to account for subgrid-scale interactions between turbulence and soot.Xuan and Blanquart [12] proposed an alternative approach to the lumped-PAH inception kinetics, where the filtered transport equations for aromatic species were solved with closure based on a PAH relaxation model.This aromatic chemistry-turbulence interaction was found to be important in accurately reproducing soot yield in turbulent flames.To obtain the information on particle size distribution in LES, the FPV presumed-PDF model of Mueller and Pitsch [34] was extended to a sectional method by Rodrigues et al. [11] .The qualitative trends in soot formation were reproduced in their work while providing detailed information on the evolution of local soot PSD.The decoupling of soot and gas-phase species in the LES presumed-PDF approach leads to narrow profiles of the mean soot volume fraction.Hence, the transported joint PDF (JPDF) method, which allows for a more accurate representation of the turbulence-chemistry interactions by directly coupling the full scalar space, is applied to the sectional model by Tian et al. [31] .Encouraging agreement was obtained with experimental measurements for soot quantities in turbulent non-premixed flames.
Besides flamelet-based methods, several other combustion models have been investigated in the LES framework to model soot formation in turbulent flames.The Conditional Moment Closure (CMC) approach was applied to a sectional soot model by Gkantonas et al. [24] to study soot formation and predict their size distribution in a lab-scale swirl Rich-Quench-Lean (RQL) combustor.In LES of lifted non-premixed turbulent flames with the sectional method, Grader et al. [14] used a finite-rate chemistry model, that requires no assumptions concerning the combustion regime.A good prediction of soot evolution was achieved in their study enabling detailed investigations of soot formation and oxidation processes.Sewerin and Rigopoulos [40] adapted the stochastic fields method to the LES of sooting flames, which recently extended to the sectional method by Sun and Rigopoulos [4] to predict the soot particle size distribution in turbulent non-premixed flames.Concerning sectional method-based approaches, simplifying the gasphase kinetics by the use of various combustion models in the LES can reduce CPU time, but the calculation of the soot source term remains computationally expensive.Recent research has explored different approaches to improve computational efficiency, including a three-equation model of Franzelli et al. [27] based on monodisperse closure of source terms.Machine learning models and virtual chemistry have also been proposed [33] for predicting soot formation, but they do not provide information on size distributions.Overall, developing detailed, and at the same time computationally efficient methods for predicting soot formation is crucial for their application in industrially relevant conditions.
Given the aforementioned challenges in accurately predicting soot formation in turbulent conditions, we propose an LES framework that combines a discrete sectional soot model with FGM tabulated chemistry [41,42] to efficiently predict soot formation and provide information on soot particle size distribution in turbulent flames.The complete tabulation of soot chemistry, similar to the treatment of gas-phase combustion chemistry in FGM, presents an interesting and efficient framework for its application to the simulations of turbulent sooting flames.While tabulated soot chemistry has been investigated with semi-empirical soot models [43] , detailed soot models [44] in laminar flames, accurately modeling soot formation rates in turbulent conditions remains a challenge due to the nonlinear interaction between gas-phase thermochemistry, soot particle dynamics, and turbulence.In this study, we present a detailed formalism of the tabulated soot chemistry model for LES based on the discrete sectional method.This new methodology is compared to a more detailed approach based on the local evaluation of the formation rates of the soot quantities.While the first strategy relies on an a-priori tabulation of soot formation rates for computationally efficient application of the sectional model, the second involves run-time computation of soot formation rates based on primitive tabulated quantities.The comparative assessment of these two strategies is conducted for soot formation in a laboratory-scale turbulent non-premixed sooting jet flame.The focus is given to the assessment of the tabulated approach to predict the complex interactions between the gas phase and the soot particles in turbulent conditions.
The paper is organized as follows.In Section 2 , the LES formalism is introduced, including a summary of the combustion and soot models.In Section 3 , the details of the target flame and numerical setup are presented.Subsequently, in Section 4 , the proposed LES formalism is applied to a turbulent non-premixed jet flame.Numerical results are first compared with experimental data, followed by the discussion and assessment of modeling approaches.Conclusions are presented at the end in Section 5 .

LES formalism with FGM-DSM methodology
This section outlines the LES-FGM-DSM formalism for modeling the formation and evolution of soot in turbulent non-premixed combustion with FGM chemistry.In LES, the filtered governing equations for reacting flows are solved along with the equation of state.Details of the modeling approach and the numerical methods are given in the next subsections.

Reactive flow equations
Under the low-Mach number approximation, the Favre-filtered conservation equations for mass, momentum, and total enthalpy are obtained as follows [16,22] : where ρ, u , h , p , ν, ν t , S , I , D and P r t represent the density, velocity vector, total enthalpy, pressure, kinematic viscosity, subgridscale (SGS) turbulent viscosity, resolved strain tensor ( S = 1 2 [ ∇ u + (∇ u ) T ] ), identity tensor, thermal diffusion coefficient, and turbulent Prandtl number, respectively.In the conservation equations, the overline operator denotes the LES filtering, and the tilde represents the Favre-filtering operations.The subgrid stresses in the momentum equation are closed using the Boussinesq approximation [16] .The Vreman [45] model with modeling constant c k = 0 . 1 is adopted for the SGS turbulent viscosity.The viscous heating effects are considered to be relatively small and neglected in the enthalpy equation, while the unresolved heat flux is modeled using a gradient diffusion approach.Guided by the previous studies of the authors using the same numerical methods [22,46] , the turbulent Prandtl number P r t set constant with a value of 0.7.

Combustion modeling
Considering the complexity of the combustion process and the need for a regime definition to find a suitable combustion model, the combustion process is assumed to occur in the flamelet regime [47] .Therefore, turbulent combustion is modeled with the FGM [41,42,48] approach.In FGM, the thermochemical states of the flame can be described using a pre-computed flamelet database parameterized by a reduced set of control variables.In the context of non-premixed combustion, mixture fraction Z and reaction progress variable Y are selected as the control variables to represent mixing and combustion progress, respectively.
In the current work, the FGM database is constructed using a combination of non-premixed counterflow steady flamelets, which cover a range of different strain rates from lower values up to the extinction strain rate, along with unsteady quenching flamelets at the extinction strain rate.These flamelet solutions are subsequently mapped onto a cartesian grid with high resolution in both mixture fraction Z and progress variable Y spaces.After these manifolds are constructed for all the tabulated quantities, a nonuniform lookup table is created by interpolating the solutions from this database to the number of entries specified for the flamelet database.
The mixture fraction is determined from element mass fractions following the definition of Bilger [49] , while the progress variable is chosen as a linear combination of the mass fractions of selected species such that it preserves the unique mapping of flamelet in composition space.The progress variable is expressed as: where α j and Y j are the weight factor and mass fraction for the species j respectively.The specific definition applied in this work is detailed in Section 3.2 .To describe the combustion process during LES calculations, transport equations are solved for the filtered mixture fraction Z and filtered progress variable Y , which take the form: where D denotes the diffusion coefficient.The unresolved turbulent flux terms appearing in control variable transport equations after the LES filtering are closed using a gradient diffusion approach with a turbulent Schmidt number Sc t set equal to 0.7.
The term ˙ ω Y in Eq. ( 6) refers to the filtered progress variable source term, which is also obtained from the flamelet database.In Eqs. ( 5) and ( 6) , the diffusion coefficient D is evaluated under the assumption of a unity Lewis number and included as a lookup quantity in the manifold.It is worth highlighting that previous studies [50,51] have indicated that non-unity Lewis numbers can indeed have a significant impact on the transport of large species like pyrene (A4).Therefore, neglecting this effect by assuming unity Lewis numbers may result in quantitative underestimations of PAH, and thus soot formation in LES.In this context, more consistent models that account for the transport of control variables with non-unity Lewis numbers, as demonstrated in studies such as [21] , can be valuable for capturing preferential diffusion effects in sooting flame simulations.However, we recognize this aspect as an area for potential improvement and a direction for future investigations.
The normalized progress variable C is introduced for simplifying the construction and look-up of quantities from the FGM database.It is defined as: For the normalization of the progress variable Y during the construction of the flamelet database, Y max (Z) and Y min (Z) are used at each mixture fraction Z level of the flamelets spanning different strain rates.In Eq. ( 7) , the maximum value denoted as Y max (Z) at each mixture fraction Z level corresponds to the progress variable in the lowest strain rate flamelet, as the applied progress variable definition increases with the advancement of reactions and the stable flamelet solution at the lowest strain rate is the one closest to chemical equilibrium.Meanwhile Y min (Z) represents the value of the progress variable in the unburnt (mixing) state where the intermediate and product species used in the progress variable definition are all absent, hence Y min (Z) = 0 for all Z.It is important to note that in the CFD calculation, Y is an independent control variable that is being transported and does not depend on the transported mixture fraction.

Turbulence-chemistry interactions
In turbulent conditions, the scalar variables are exposed to spatio-temporal fluctuations that can not be resolved with the LES filter length, given in this case by the mesh size.Therefore, the flamelet database incorporating laminar flamelet solutions must be extended to account for the effect of turbulence on subgrid levels.Different methods for the modeling of subgrid-scale turbulencechemistry interactions exist in the literature.Here, the stochastic approach using presumed-shape Probability Density Functions (PDF) is used to model these interactions.Only fluctuations in Z and C are considered in the LES to describe the statistical impact of turbulence on the flame.The filtered thermochemical variable ψ , convoluted over the joint PDF of Z and C, can be expressed as: Commonly, statistical independence between Z and C is assumed.
Thus, the joint PDF P (Z, C) can be represented by individual presumed PDF distributions of Z and C. The Favre-filtered variable ψ and the filtered variable ψ can then be obtained by integrating the marginal PDFs of Z and C: where ψ refers to any quantity defined in the flamelet solutions.Several LES studies [52][53][54][55][56] have demonstrated β-function marginal PDFs as a suitable choice for the convolution of flamelet solutions.Hence, a β-function PDF is used to characterize the distribution of both mixture fraction and progress variable.For the application in LES, the thermochemical parameters of the mixture are mapped on four non-dimensional coordinates in the FGM database as: where S φ , called the segregation factor, represents the normalized variance of control variable φ, required in the β-PDF model.For S φ = 0 the variance φ 2 is zero, which leads to ψ ( φ) = ψ (φ) .The segregation factors for variables Z and C, are given as: and The subgrid variance of scaled progress variable C 2 is related to its unscaled variance Y 2 through [57] : Note that when the flamelet manifold is extended to account for subgrid-scale effects by the PDF integration, the scaled progress variable C also depends on the subgrid variance of mixture fraction: In this work, transport equations are solved for the closure of subgrid-scale variances of mixture fraction and progress variable.The transport equations for the sub-grid variances Z 2 and Y 2 are given by: where χ Z and χ Y respectively denote the subgrid-scale scalar dissipation rates for mixture fraction and progress variable, modeled as: where k is the unresolved turbulent kinetic energy and ε is its dissipation rate.To close χ Z and χ Y , the ratio of ε and k is defined following the expression of Both et al. [58] : where C ε = 1 .8 is a model parameter, and is the LES filter size (taken as the cube root of the volume of the grid element).

Soot modeling 2.4.1. Sectional soot model
The chemistry and dynamics of soot particles are described using the discrete sectional method [59] .In the present model, soot particle volume ranges are divided into a finite number of sections.In each section, i , the governing equation for the soot mass fraction Y s,i is solved by considering flow convection, diffusion, thermophoresis, and chemical rates.The sectional soot transport equation can be formulated as: where ρ, ρ s , u , v T , D s,i , ˙ ω s,i denote gas density, soot density (assumed to be equivalent to the density of solid carbon), velocity, thermophoretic velocity (calculated with Frienlander's [60] expression), soot diffusion coefficient (assumed to be constant for all particle sizes for numerical stability), and sectional source term, respectively.The soot source terms account for the chemical and physical processes involved in soot formation, including nucleation, condensation of Polycyclic Aromatic Hydrocarbons (PAHs), surface growth, oxidation, and coagulation.Nucleation is assumed to occur through the dimerization of two pyrene (A4) molecules, and PAH condensation onto soot particles is modeled using the method described by Roy [61] .Surface reactions, including growth and oxidation, are modeled using the standard hydrogen-abstraction-C 2 H 2addition (HACA) mechanism by Appel et al. [62] .Coagulation of soot particles is described using the model proposed by Kumar and Ramkrishna [63] , and the morphological properties (e.g.fractal dimension) of soot are not considered for simplicity.As a consequence, the present soot model does have limitations related to its omission of aggregation and the potential influence this may have on both the growth and size distribution of soot.Further details about the soot model and its validation can be found in previous work [59,64,65] .

Filtered sectional soot equations
For modeling soot formation and evolution in LES, the filtered equations for sectional soot mass fractions are obtained as: The thermophoretic velocity v T is modeled following [11] , as: Since the soot source term ˙ ω s,i includes contributions from the various sub-processes of soot formation, it depends on both gas-phase thermo-chemical variables ( φ g ) and soot variables ( φ s ).Therefore, the filtered source term ˙ ω s,i poses closure problems requiring models to account for subgrid-scale soot-chemistry-turbulence interactions.Here we consider two different strategies to treat the filtered soot source terms.
• In the first approach, the chemical source term for the soot section is evaluated during runtime using tabulated filtered thermochemical parameters and local concentrations of gas-phase species relevant to the soot model.The filtered soot source term is approximated by neglecting the subgrid-scale interaction between soot and turbulence, such that: ˙ In the current model, the transient behavior of soot precursors (e.g., A4) is partially considered, primarily during the 1-D flamelet calculation phase (using unsteady flamelets).The gas-phase consumption of precursor species due to soot formation is accounted for during the flamelet calculations step, and no additional transport equation is solved for their description in the LES.Nevertheless, there is also the possibility of extending the current model to explicitly solve transport equations for the precursor species, a technique adopted in several other studies [11,19] , allowing for the evolution of slower precursor species in FGM chemistry.For brevity, this approach, in which the soot source terms are computed on-the-fly is referred to as FGM-C here.
Additionally, an alternative variant of the FGM-C approach with a slightly different formulation for filtered soot source term is evaluated.This variant involves tabulating the gas-phase rate of the filtered chemical soot source term instead of calculating it from local gas-phase species concentrations.This particular variant, referred to as FGM-CR, is aimed at understanding the implications of using filtered concentrations of species to compute rates.Given that this method is a variant of the FGM-C model, it will be addressed separately to enhance clarity.A comprehensive analysis of this approach is provided in Section 4.2.5 .• The second model concerns the closure of the filtered soot source term through the presumed PDF approach.In a general form, the filtered soot source term convoluted with presumed PDF is written as: ˙ Using Bayes' theorem, the joint PDF can be split into two marginal PDFs for the thermochemical variable and for the soot such that P (φ g , φ s ) = P (φ g ) P (φ s | φ g ) .In this model, the soot source terms are parameterized through control variables and tabulated in the manifold.The marginal PDF P (φ g ) is assumed to have β-PDF function for Z and C, while the conditional PDF of soot variable P (φ s | φ g ) is treated as a δ-function.This approximation facilitates the partial inclusion of turbulence-soot interactions by accounting for the effect of subgrid-scale fluctuations in the mixture-fraction and progress variables on soot source terms.Besides, the assumption of δ-PDF function (neglecting subgrid-scale fluctuations) for P (φ g ) is also examined to understand the impact of PDF integration on the performance of the tabulated soot chemistry, which will be explained later.The approach employed in other studies [7,66] for the transport of slowly evolving species such as PAH and NO is applied for soot.Accordingly, the soot source term for the soot section is split into production ( ˙ The consumption part is linearized by soot mass fraction to avoid the un-physical consumption of soot when Y s,i = 0 .The filtered soot source term is modeled as: where the production and linearized consumption terms are parameterized through gas-phase thermochemical variables and tabulated (denoted by the superscript tab) in the manifold.This approach does not consider the separation of gas and soot parts for the soot production term.However, with the linearization of consumption terms, the partial separation of gas and soot time scales can be achieved.In general, soot concentrations are negligible in lean mixture fraction ( Z < Z st ) regions, as soot particles are oxidized through reactions with OH, whose concentration is high near the stoichiometric mixture fraction.In this model, since soot source terms are obtained from the flamelet solutions, the filtered consumption term ˙ ω − s,i ≈ 0 in lean regions.This can lead to the Modeling approaches for filtered soot source terms.

Approach Model Description
Computation FGM-C Runtime evaluation using tabulated primitive variables (gas-phase species) Computation FGM-CR Runtime evaluation using tabulated gas-phase rates for soot subprocesses Tabulation FGM-T Direct a-priori tabulation spurious existence of soot (during transport) in lean mixture factions if soot oxidation is not properly accounted for during the soot chemistry tabulation.To avoid this issue, the soot oxidation contribution, in the linearized consumption term in Eq. ( 24) is further approximated as: where ˙ ω −, ox s,i denotes the sectional soot consumption rate by oxidation subprocesses.For computational efficiency, the coagulation process of soot particles is not explicitly solved at runtime, but the inter-sectional mass transfer due to the coagulation process (or other subprocesses) is included in the flamelet computation.This approach, in which the soot source term is tabulated , is referred to as FGM-T from hereon.
The key differences between models are summarized in Table 1 .In brief, the FGM-T approach is based on the direct tabulation of soot source terms, and the application of these source terms in soot transport equations.Meanwhile, the FGM-C approach relies on the tabulation of primitive variables (e.g.concentrations of gas-phase species involved in surface growth and oxidation) and combines them with the local CFD solution of temperature, density, and fields representing soot to evaluate the soot source terms.The FGM-CR model, a variant of FGM-C, incorporates the tabulation of gas-phase rates related to various soot processes.The tabulated gas-phase rates are subsequently integrated with the local solutions of soot fields and gas-phase properties, akin to the FGM-C approach.
The schematic illustration of the LES-FGM-DSM implementation is shown in Fig. 1 .The LES formalism with the aforementioned soot source term models is applied to the simulation of soot production in a turbulent piloted jet flame.The piloted jet flame offers an ideal combustion environment to analyze the evolution of soot formation sub-processes, such as nucleation (near to burner) followed by soot growth (within flame mid-height), and soot oxidation (near the flame tip).The experimental details and numerical setup for the LES of this turbulent jet flame are presented in the following section.

Turbulent sooting flame configuration
The turbulent non-premixed ethylene-air piloted jet flame, experimentally investigated at Sandia National Laboratories [67] is considered here.This flame is also one of the target flames from the International Sooting Flame (ISF) workshop [68] .The selected flame presents well-characterized exit conditions for various nozzles.It provides an extensive experimental database of soot measurements, which justifies the appropriateness of this flame for soot model assessment and validation.Unfortunately, measurements for the velocity field and/or mixing are not available for the selected flame, which presents an additional challenge in the validation of the turbulent flow field in simulations.The details of the experimental configuration and simulation setup are presented below.

Experimental configuration
The selected non-premixed jet flame features a burner with a central jet of pure C 2 H 4 and low-speed pilot injection of fully reacted C 2 H 4 -air pre-mixture at an equivalence ratio of 0.9.The pilot tube (with an inner diameter of 15.2mm, and an outer diameter of 19.1mm) comprises an insert, which provides 64 small pilot flames for the stabilization of the main flame.The spacing and number of pilot flames have been designed to produce a uniform flow rate of burned products near the burner exit plane.The global mass flow rate of the pilot corresponds to 1.77 X 10 −4 kg/s.Fuel ( C 2 H 4 ) is injected at 294K with a bulk velocity of 54.7m/s, resulting in a Reynolds number of 20,0 0 0 (based on the diameter of the main jet d j = 3 .2 mm ) at the fuel nozzle exit.The pilot is surrounded by a co-flow of air at a bulk velocity of 0.6m/s.Additional details on the experimental conditions can be found in Ref. [67] .

Numerical setup
Based on the experimental observation of the visible flame height, the computational domain is designed to be 1m ×0.3m ×2 π in axial ( z), radial ( r) and circumferential directions, respectively.The computational domain is discretized via 384 ×192 ×64 cells in the axial, radial, and circumferential directions, respectively.A schematic illustration of the computational domain and grid is given in Fig. 2 a.The computational grid is selected such that the typical cell size near the fuel exit is ≈ 0.1mm.The grid is stretched in the axial and radial directions while the circumferential direction is uniformly spaced.In the radial direction, the grid is nonuniformly concentrated in the shear layers between the different injection streams.The structured grid is made of approximately 5 million hexahedral elements.The mesh has been also verified for Pope's criteria [69] in Fig. 2 b.More than 90% of the grid cells show a ratio of resolved to total kinetic energy ( M ) larger than 0.8, indicating the kinetic energy is sufficiently resolved with the selected mesh.
Synthetic turbulence derived from a Laplacian filter following the method proposed by Kempf et al. [70] is employed to define the inflow condition for the fuel.The mean axial velocity of the turbulent database is set to a power-law profile with an exponent of 1/7.Considering the Reynolds number of a pipe flow at the conditions of the fuel inlet, the turbulent intensity is around 5%.It is worth noting that there is no consensus in the literature regarding the selection of inlet boundary conditions in this case.Other numerical investigations [4,31] of similar flames have used turbulence intensity values ranging from 6% to 10%.Unfortunately, due to the unavailability of velocity measurements, the velocity profile and Reynolds stresses can not be compared and it was estimated with 5% in this study.The velocity inlet profiles for the pilot and coflow air streams are treated as uniform.The velocity of the air stream is specified as 0.6m/s while the bulk velocity of the pilot is adjusted to impose the mass flow rate satisfying the experimental condition.No-slip adiabatic wall conditions are imposed at the injector boundaries.The pilot inlet is assumed to have a composition close to the equilibrium state of an ethylene-air mixture at an equivalence ratio of 0.9 ( Z = 0 .0577 , C = 1 ) and temperature of 2256K.The fuel and coflow inlet temperatures are maintained at 294K.
For the creation of the FGM database, a series of 1-D nonpremixed counterflow flamelets are calculated with detailed chem-istry including soot kinetics using the code CHEM1D [71] .In LES simulations, 30 soot sections are transported to describe particle size distribution, hence the flamelets are also computed with 30 sections.To cover the composition space from chemical equilibrium to mixing in the flamelet database, first, a series of strained steady counterflow flamelets are computed by varying the applied stain rate from lower values (close to chemical equilibrium) until the extinction limit.Subsequently, the composition space between the extinction limit and the mixing solution is covered by simulating unsteady quenching flamelets at the extinguishing strain rate.The detailed kinetic scheme KM2 of Wang et al. [72] , involving 202 species and 1351 reactions, is used for the gas phase chemistry during the computation of flamelets.This mechanism has been extensively validated for soot formation prediction in ethylenefueled laminar [65,73,74] and turbulent flames [11] .Considering the large Reynolds number of the turbulent jet, turbulent diffusivities are expected to be higher than molecular diffusivities.Hence a unity-Lewis diffusion model is considered for the species transport in the flamelet computation.For the mapping of thermochemical variables, the progress variable ( Eq. ( 4) ) is defined based on H 2 O , CO 2 , CO, O 2 , H 2 , and A4 species mass fractions with their corresponding weight factors α H 2 O = 0 .0555 , α CO 2 = 0 .0228 , α CO = 0 .0357 , α O 2 = −3 .13 × 10 −4 , α H 2 = 0 .173 , α A4 = 0 .0988 .The progress variable definition is determined using a guess-and-check approach, and shown to preserve the unique mapping of Y in composition space [75] .Including pyrene (A4) in the progress variable definition ensures a unique mapping of the tabulated quantities under conditions where the soot models are especially sensitive to the local gas phase composition.However, since the A4 species is not transported in either of the approaches used for soot source terms in the present study, the accuracy of the current FGM-DSM framework is not significantly impacted by the inclusion of A4 in the progress variable Y.The thermochemical variables are stored in the FGM database with a non-uniform (refined near stoichiometric and equilibrium regions) resolution of 101 ×11 ×101 ×11 grid points in Z , S Z , C , and S C space respectively.
Previous studies [11,76] conducted on similar flames have demonstrated the importance of considering thermal radiation effects due to the temperature sensitivity of soot formation.The current LES-FGM-DSM framework can be expanded to include radiative heat transfer by further augmentation of the flamelet database for non-adiabatic conditions and additional parameterization in enthalpy space.However, the primary focus of this study is the comparative evaluation of two different methods for modeling soot formation chemistry.Hence, the effects of thermal radiation from gas and soot are neglected for simplicity.
The simulations are carried out in a Cartesian coordinate system using the multi-physics code Alya [77] , developed at the Barcelona Supercomputing Center (BSC).In the Alya code, a second-order conservative finite element scheme is used for spatial discretization, while an explicit third-order Runge-Kutta scheme is employed for the time integration.A low-dissipation scheme based on the fractional step algorithm proposed by Both et al. [46] is used for continuity and momentum equations under a low-Mach number approximation of reacting flows.The simulations are performed using the Hawk cluster equipped with AMD EPYC 7742 processors at the High-Performance Computing Center Stuttgart.The temporal statistics for the quantities are performed for a period of approximately 250ms after the simulations have reached a statistically steady state.

Gas-phase characteristics
To describe the main combustion features of the piloted turbulent jet flame, the instantaneous fields of temperature, and key species involved in soot chemistry are presented in Fig. 3 .An isocontour of stoichiometric mixture fraction ( Z = Z st ) characterizing the flame front is also shown.The flame is found to be stabilized by the pilot and attached to the burner.Under turbulent condi-tions, the flame reaction zones (indicated by OH contours) show local extinction events within the shear layers.The formation of PAH (A4) and C 2 H 2 is predominant in the fuel-rich region.However, a systematic lag can be noticed in the spatial locations of A4 formation compared to C 2 H 2 as the relatively large time scales governing A4 formation reflect in its downstream spatial evolution.The production of A4 is observed to initiate mainly after z/d j 50 , while C 2 H 2 , a precursor species responsible for surface growth is found to occur closer to the burner exit ( z/d j 20 ).Furthermore, the regions of higher OH concentration, in which soot oxidation is dominant, are shown to be prominent in the region beyond z/d j 170 .

Gas-phase validation
For the preliminary validation of gas-phase, computed timeaveraged radial profiles of mean OH mass fractions at different downstream heights are shown in Fig. 4 along with the experimentally measured OH signal.In addition, a comparison of the PAH signal against computed profiles of mean C 2 H 2 and A4 mass fractions at several axial locations is presented in Fig. 4 .Overall, the computed profiles show fair qualitative agreement with the experiments regarding downstream evolution, suggesting that the turbulent combustion models applied can favorably capture the main features of the flame structure in the gas phase.A slight overprediction of the jet spreading rate is observed in the radial direction at downstream positions leading to a wider spread in the species profile.This however is found to have a minor effect on the quantitative prediction of soot formation.

Characterization of soot formation 4.2.1. Instantaneous fields
To investigate the effects of tabulating the source terms and include the PDF integration on the evolution and distribution of soot, the instantaneous fields of soot volume fraction and soot formation rates (split into production ˙ ω + s and consumption ˙ ω − s parts) obtained with FGM-C and FGM-T approaches are compared in Fig. 5 .From the soot volume fraction fields, it is evident that soot is mainly formed in the fuel-rich zones beyond z/d j > 50 , characterized by high A4 mass fractions (see Fig. 3 ).Soot inception is predominant near the flame base, leading to the formation of smallsized particles and, consequently, low values of soot volume frac-  tion.From Fig. 5 , it can be observed that the soot production rate (dominated by surface growth primarily) is concentrated in the middle of the flame.Therefore, a high amount of soot volume fraction is noticed in the middle region of the flame, where incepted particles grow.On the other hand, consumption rates (oxidation) are prominent at the tip region of the flame, and near the stoichiometric iso-contour where particles are oxidized due to a high concentration of OH.Similar trends are noticed in the previous numerical studies [11,78] .In the presence of turbulent fluctuations, the soot formation is highly intermittent, forming sporadic pockets that detach from the fuel-rich zone and are convected downstream where soot oxidation takes place.
The FGM-T method results in a higher soot volume fraction in the middle region of the flame compared to FGM-C.Additionally, the FGM-T model predicted soot formation further upstream than the FGM-C model.This is because the soot production rate in FGM-T is higher near the flame base ( z/d j 50 ) and in the middle region ( z/d j 100 ) than compared to FGM-C, causing higher soot concentration.For a more quantitative illustration of soot formation characteristics between FGM-C and FGM-T, the scatter plots of the soot volume fraction and soot formation rates fields are compared in Fig. 6 (for the results from Fig. 5 ).The scatter plots are colored with the temperature.It is clear that soot formation is predominant in the fuel-rich regions spanning 0 . 1 < Z < 0 .4 .Within this sooting region of composition space, higher values of soot production rates are evident for FGM-T as compared to FGM-C, which explains the higher f v values obtained by FGM-T.Below the stoichiometric mixture fraction ( Z st ), almost all the soot is consumed for both methods since soot oxidation is predominant in the composition space close to the stoichiometric value.However, the distri-bution of soot consumption rates is found to be qualitatively and quantitatively somewhat similar in both the FGM-C and FGM-T approaches.Therefore, the noticed discrepancies in the quantitative prediction of soot with FGM-C and FGM-T methods are primarily the consequence of differences in the soot production rates.

Soot volume fraction fluctuations
Scatter plots of soot volume fraction (colored by temperature) against mixture fraction at different positions along the axial direction are compared in Fig. 7 .The scatter represents instantaneous values collected at several time instants.The conditional means of soot volume fraction are included for reference.It can be observed that a large extent of soot is present in fuel-rich mixture fractions while below the stoichiometric values, no soot is observed, as it is rapidly oxidized on the lean side.At lower heights, soot volume fraction samples span a large range of mixture fractions and temperatures.With an increase in axial location, higher values of soot volume fraction are detected, however, the spread in mixture fraction space tends to gradually decrease.The decreasing branch of soot volume fraction in mixture fractions beyond 0.2, vanishes at the downstream position, due to enhanced mixing leading to lower mixture fraction values.The qualitative trends of soot volume fraction scatter are similar in FGM-C and FGM-T simulations.However, more soot production is evident for FGM-T, especially at higher mixture fractions than their FGM-C counterparts.Far downstream of the flame ( z/d j 180 ), a lower soot volume fraction is found in FGM-T compared to FGM-C.
The correlation between mixture fraction and soot volume fraction can be further analyzed through conditional probability density functions ( P ( f v | Z ) ) of soot volume fractions in Fig. 8 .The PDFs are examined for three fuel-rich intervals.The P ( f v | Z ) at different axial positions indicate that fluctuations in soot volume fraction are mainly concentrated within rich regions, identified by 2 Z st < Z < 3 Z st .Moreover, a substantial amount of soot also exists in highly rich regions Z ≥ 3 Z st .The mean values of f v are lower for FGM-C as compared to FGM-T.For mixture fractions below 2 Z st , the peak value of P ( f v | Z ) approaches zero, confirming a minimal amount of soot in these regions, as soot oxidation is prominent within lean regions.In the FGM-C method, fluctuations in soot volume fraction primarily arise from resolved fluctuations in the flamelet independent variables ( Z, C) and turbulent transport of soot.In FGM-T, besides fluctuations in flamelet independent variables, and turbulent transport of soot, fluctuations in the chemical source term of soot sections are taken into account through the presumed-PDF integration during tabulation.
The fluctuations in soot volume fraction are often characterized by soot intermittency ( I s ).The soot intermittency is defined experimentally as the probability of observing an instantaneous value of soot volume fraction lower than 0.03ppm [11] .In Fig. 9 the experimental probe-resolved and numerical intermittency profiles are shown for sampled data at several time instants.The intermittency profiles are favorably captured beyond z/d j > 120 confirming the good prediction of soot particle oxidation and turbulent fluctuations (resolved).The simulation results tend to underestimate the intermittency close to the burner.The discrepancies in the simula-tion results compared to the experiment are more pronounced at lower heights for the FGM-T approach where soot volume fraction values are higher as evident in Fig. 5 .Nevertheless, the upstream translation with an underestimation is also noticed in several other numerical works [4,79] , hence the overall performance of the current LES-FGM-DSM approaches is quite reasonable in the context of the current state of the art in modeling turbulent sooting flames.Nevertheless, it is important to note that while subprocesses like soot growth and condensation contribute to the lower soot intermittency in downstream regions, it is also crucial to emphasize the impact of the LES resolution and the flapping of the jet on the intermittency.

Mean soot profiles
The time-averaged fields of soot volume fraction obtained from LES with FGM-C and FGM-T models are compared in Fig. 10 a.As can be observed, soot is predominantly restricted to fuel-rich regions within the stoichiometric mean mixture fraction iso-contour, while the peak soot volume fraction locations are found at approximately z/d j 125 .Because of the soot oxidation (predominantly through OH species near stoichiometric conditions) and flow fluctuations, soot particles do not exist over the complete mixture fraction space.For the FGM-T model, substantial soot concentration is observed at locations close to the burner exit ( z/d j 30 ),   while the soot formation is somewhat delayed in FGM-C simulations.For a more quantitative illustration, the radial profiles of computed mean and RMS soot volume fraction at several heights are compared against the measurements in Fig. 10 b.The RMS for computed soot volume fraction is calculated as: The qualitative trends in the experimental data are reasonably reproduced in the simulations.The normalized RMS profiles of soot volume fraction show very good agreement with measurements for both approaches, however, the magnitude of the soot volume fraction is underpredicted.The peaks observed in the soot volume fraction in the lean region away from the jet ( r/d j 5 ) are not well captured in the simulations.Contrary to measurements, soot almost ceases to exist beyond z/d j 180 in the computed results.The underprediction of soot volume fraction in far downstream regions could be attributed to either the overprediction of OH oxidation rates or the underprediction of the overall flame length in simulations.
The axial profiles of mean and RMS soot volume fraction along the centerline axis are compared in Fig. 11 .In the present turbulent  jet flame, the overall prediction of soot formation along the centerline is mainly controlled by surface growth (in the middle region) and oxidation (due to OH) as shown in [4] .A reasonable agreement between simulated and measured soot volume fraction profiles is obtained, but the peak value of f v is under-predicted by a factor of two in the simulations.The prediction of RMS fluctuations of soot volume fraction is similar in magnitude to the mean with approximately a factor 2 lower.This underprediction is aligned with the state-of-the-art results from this flame, where a comparable under/over prediction is found with other approaches [4,11,12,31] .From the present LES results, it can be inferred that the FGM-DSM formalism can fairly capture the soot volume fraction distribution under turbulent conditions.Besides, for both approaches, the position of the peak soot volume fraction is slightly shifted upstream compared to experiments, indicating that the soot consumption is predicted early in simulations.Nevertheless, an early formation of soot is evident for the FGM-T approach compared to FGM-C.
In the current LES, soot mass fractions slowly increase while going downstream from the inlet toward a statistically steady state from the initial no-soot condition.Therefore, capturing the chemical trajectories of soot evolution (from the gas phase to the steady state) becomes crucial in the accurate prediction of soot formation.In the FGM-C approach, soot subprocesses are explicitly computed using the local soot mass fractions.This yields a better qualitative description of unsteady soot evolution for the LES.On the contrary, in FGM-T soot source terms are calculated and stored for soot mass fractions in a steady-state flamelet, therefore, the chemical trajectories concerning the formation of soot from the gas phase to the steady state are not explicitly retained in the FGM tabulation strategy.Naturally, an extension of FGM to incorporate the chemical trajectories of soot formation would require augmentation of the database with unsteady flamelets (at every level of scalar dissipation rate), and an additional controlling variable to entirely parameterize the reaction progress of soot, making manifold generation more complex.Therefore, this aspect is left out of the scope of the present work.Moreover, the non-linear dependency of soot production rates on soot variables is not included in the present FGM-T formulation.Hence, the direct look-up of soot production rates may lead to their overestimation.As a result, higher soot volume fractions are noticed at lower axial positions for FGM-T compared to FGM-C.Nevertheless, the results for both approaches are overall in good agreement with the current state-of-the-art of this flame and the FGM-T approach shows high potential for LES due to the high computational efficiency and low computational cost.This aspect will be addressed in Section 4.3 .
As previously mentioned, the FGM-T model accounts for the subgrid-scale chemistry-soot-turbulence interactions, as the tabulated source terms are integrated with the presumed β-PDF function.To investigate the impact of the presumed-PDF model on soot prediction with FGM-T, additional simulation is carried out by neglecting the influence of subgrid-scale fluctuations on tabulated soot source terms by using a δ-PDF.In Fig. 12 , the axial profiles of mean and RMS soot volume fraction along the centerline are compared for the two PDF functions.The soot volume fraction profiles are only marginally influenced by the exclusion of the presumed-PDF treatment accounting for subgrid-scale interactions on soot source terms.Such a response of soot formation can be elucidated by analyzing the spatial distribution of soot volume fraction and mixture fraction variance (which characterize the influence of turbulence fluctuations on scalar mixing).The time-averaged fields of soot volume fraction (with δ-PDF) and mixture fraction variance are shown in Fig. 13 .As expected, the high variance of mixture fraction is primarily found in the shear layers generated by strong velocity gradients between the main jet and coflow.In the base region of the flame, close to the burner, the mixture fraction variance is significant.However, soot formation is not predominant in this region.On the contrary, the soot formation zone is mainly spanned in the mid-flame region, where mixture fraction gradients are low (leading to low variance).As a result, the subgrid-scale fluctuations only slightly affect the soot source terms.Consequently, overall soot formation is found to be only marginally affected by subgridscale turbulent fluctuations in the present flame.

Time-averaged particle size distributions
The coupling of LES with the sectional method provides information on the spatio-temporal evolution of the soot PSD.Hence, the calculated time-averaged PSDs at different axial locations along the flame centerline are plotted in Fig. 14 for FGM-T and FGM-C.The evolution of the PSD along its trajectory in the flame is strongly correlated to the particle history characterized by a succession of chemical and collisional processes associated with soot formation.The time-averaged soot PSDs feature mainly unimodal shapes for FGM-C and FGM-T approaches.The number density of larger-sized particles increases during this process as nucleated soot particles grow primarily through surface reactions.For FGM-T, the shift of PSD towards large diameters is observed as compared to FGM-C, which translates into higher soot volume fractions pre-  dicted by the FGM-T method.Beyond z/d j ≈ 150 , the number density starts to drop due to the combined effect of soot oxidation and fuel-lean conditions.As noticed earlier in Fig. 11 for both FGM-T and FGM-C, the soot concentration is almost negligible at around z/d j ≈ 200 since large soot particles are oxidized.This leads to a reduction in the number density of larger-sized soot particles downstream of the flame (as can be observed for z/d j ≈ 195 in Fig. 14 ).In general, both FGM-T and FGM-C demonstrate similar qualitative features of soot PSD evolution in this turbulent jet flame.The quantitative discrepancies in PSD profiles can be attributed to dif-ferences in the formation and unsteady evolution of soot in both approaches.For instance, in FGM-C, the coagulation of soot particles is explicitly computed, contrary to the FGM-T model.

Extension of the FGM-C model
Besides the two methods mentioned in Section 2 for soot source term closure, an additional model is considered as an extension of the FGM-C approach.Similar to FGM-C, this model relies on the run-time computation of soot source terms while tabulating the pre-computed gas-phase rate contribution in the FGM database.However, the filtered soot source term formulation differs slightly from the FGM-C approach, as discussed below.In a general form, the sectional soot source term can be expressed as: where j refers to the soot sub-process (e.g.nucleation), k g is the gas-phase factor independent of i , the ζ i term depends on the section (e.g.collision frequency factor), while i is a function of soot scalars (e.g.number density).The closure of filtered soot source term is achieved using the presumed-PDF approach: ˙ Since the time scales associated with the evolution of the thermochemical state are typically smaller than the time scales of soot production [7] , the independence between gas and soot phases is assumed, as discussed in Refs.[7,11] , which gives: where k j g represents the filtered rates for the soot subprocesses, and depend only on the gas phase.The k j g terms are tabulated in the manifold as a function of control variables Z and C .The second term j i in Eq. ( 30) depends on the soot variable ( φ s = Y s,i here).This term is computed at runtime.For model simplicity, the ζ i term, which depends on the sectional soot properties (such as soot volume) and gas-phase variables (such as temperature) is computed during simulation.To account for subgrid-scale turbulencechemistry interactions, the gas-phase contribution, k j g , is modeled with the presumed-PDF approach.The marginal PDF P (φ g ) is assumed to have a β function.The conditional PDF of solid phase contribution P (φ s | φ g ) is treated as a δ function.For brevity, this method is referred to as FGM-CR hereon.
It is important to note that FGM-C and FGM-CR methods primarily differ in the treatment of the gas-phase contribution in the closure of soot source terms.In the FGM-C case, k j g is computed on-the-fly using tabulated filtered gas-phase quantities φ g .
On the other hand in the FGM-CR, the filtered k j g is a-computed a-priori from flamelets and directly tabulated in the manifold.Since k j g (φ g ) = k j g ( φ g ) , the FGM-CR model essentially facilitates a more physically consistent treatment of subgrid-scale fluctuations of gas-phase variables, as compared to the FGM-C approach.At the same time, similar to FGM-C, subgrid-scale turbulence-soot interactions are not accounted for in the FGM-CR approach.This strategy, therefore, can be regarded as an extension of the FGM-C model.
The fields of mean soot volume fraction are compared in Fig. 15 a for the FGM-CR and FGM-C models.In general, the soot formation zones in FGM-CR and FGM-C show similarities in the spatial distribution as well as the magnitude of the soot volume fraction.A slight reduction in soot volume fraction is noticed near the flame tip.For more quantitative illustration, the radial profiles of mean and RMS (normalized with peak) soot volume fraction at several heights are compared against the measurements and FGM-C model in Fig. 15 b.The results obtained with the FGM-T method are also presented for reference.It can be observed the radial distribution of soot volume profiles for FGM-CR case exhibit qualitative similarities with FGM-C results.In the far downstream region, (close to the flame tip) where soot oxidation is prevalent, the slightly lower soot volume fraction is predicted by the FGM-CR strategy, as compared to FGM-C.The overall shape and peak location of the soot volume fraction obtained with FGM-CR agree closely with FGM-C.However, slight deviations between FGM-C and FGM-CR are evident in the downstream regions.The strong intermittency is evident ( Fig. 5 ) in downstream regions where soot pockets detach near the flame tip.This intermittent behavior is captured differently during soot source term filtering in the FGM-C and FGM-CR models.As a result, deviations are also seen in the soot volume fraction.

Remark on computational performance
Besides the gas-phase chemistry, in sooting flame simulations with the sectional method, the greatest CPU overhead is due to the computation of soot source terms.Especially the calculation of particle dynamics (coagulation) takes almost half of the total CPU time.Employing FGM chemistry leads to a significant reduction (up to factor 2) in CPU cost for gas-phase reactions.However, computation of soot source terms at runtime remains a CPU-intensive task, which is alleviated here with tabulation of the sectional source term in the FGM-T approach.In the current LES with 30 soot sections, the FGM-T approach yields a factor 3 reduction in CPU time per time-step as compared to FGM-C.The computational speedup with FGM-T for a higher number of sections can be even more significant since the computational time scales non-linearly with n sec .For instance, the FGM-T approach provides a computational speed-up of about a factor 7 compared to FGM-C when 60 soot sections are considered.
In summary, it is evident from the LES results that the modeling of filtered soot source terms strongly influences the soot formation prediction in turbulent flames.The complete tabulation of soot chemistry (FGM-T) tends to effectively capture the experimentally observed features of soot distribution in turbulent conditions.Compared to the FGM-C method, in which soot chemistry is calculated during runtime, the FGM-T method tends to overpredict the soot concentration.However, considering the state-of-the-art in the numerical prediction of soot formation in turbulent flames and the experimental uncertainty in the measured data, this discrepancy can be considered acceptable for practical applications.It is also important to emphasize the good computational efficiency of FGM-T within the context of the sectional method which makes it an attractive alternative to the FGM-C.However, as already highlighted, the complete tabulation of the soot chemistry has certain limitations, for example, soot-independent treatment of the soot production term, and a lack of information on the gas-tosoot history effects in flamelets.Therefore, in simulation applications where the unsteady evolution of soot quantities is not of interest, the FGM-T approach is more suitable for predicting the soot formation.Especially, in simulations of practical combustion systems, the FGM-T is an excellent strategy to gain an understanding of soot formation and information on size distribution, at affordable computational cost.On the other hand, for more fundamental and parametric studies, the FGM-C method is recommended as it accounts for the non-linear interactions between the soot and gas phase, without any modeling assumptions, and can be more reliable for highly transient cases.

Concluding remarks
This study presents two strategies based on the discrete sectional method coupled with FGM tabulated chemistry in the context of LES for the prediction of soot formation in turbulent non-premixed flames.The performance of the two strategies for modeling soot source terms is assessed on a turbulent non-premixed jet flame with a focus on the prediction of soot formation and particle size distributions.The LES results for the gas phase and soot phase are compared against the available experimental data.Despite some discrepancies, the results for the soot phase show reasonable agreement with the experimental results.
The LES study suggests that the soot source term closure in FGM-DSM coupling substantially influences the quantitative and qualitative prediction of soot formation in turbulent conditions.The observed differences in soot prediction for the investigated soot source-term models can be attributed to their capabilities in capturing unsteady chemical trajectories of soot formation, which evolve at a slower time scale than flame propagation.The computationally efficient model based on tabulated soot formation rates with a presumed PDF (probability density function) approach tends to yield good quantitative soot prediction, making it a promising tool to study soot production in light of industrial applications with LES.However, it was found that tabulating soot chemistry has limitations in capturing the transient evolution of soot.Therefore, the tabulated soot source term approach requires further improvements to meet limitations regarding capturing soot history effects.On the other end, the model that solves for the complete set of soot formation subprocesses provides a better qualitative description of soot formation and evolution.It is further observed that the separation of soot and gas phase terms only marginally impacts soot formation prediction for the run-time soot source term computation model in turbulent conditions.
In summary, the present study demonstrates that the sectional soot model coupled with FGM chemistry is a feasible and efficient approach to predict soot production in turbulent flames.The LES framework developed in this work for the FGM-DSM coupling does not consider subgrid-scale interactions between soot and turbulence.Including subgrid-scale soot-turbulence interaction models (e.g.Ref. [7,11] ) in the LES would be an important extension to investigate their impact on overall soot prediction.In addition, future work could consider incorporating developing new models or extending the linearization model used in the tabulated soot chemistry approach to account for the non-linear effects of the soot production rate.Moreover, the authors acknowledge that the modeling aspects related to thermal radiation, preferential diffusion, and soot agglomeration hold significant potential for enhancing the accuracy of the proposed soot modeling framework and are thus considered essential components for future research work.

Novelty and significance
Applications of detailed soot models, such as the discrete sectional method, have become increasingly important with emission regulations on soot particle size.However, several studies that employ sectional methods for soot prediction in turbulent conditions rely on the explicit computation of soot reaction rates, which suffer from high computational costs with an increase in the number of soot sections.To address this, the present work offers a computationally efficient tabulated soot chemistry-based sectional soot modeling framework for the LES approach.The comparative analysis presented in this study demonstrates the good predictive capabilities of the proposed method despite several modeling approximations.Overall, the proposed LES framework offers an attractive modeling choice for reducing the computational cost of sooting flame calculations in industry-relevant conditions while maintaining good predictive capabilities.The proposed tabulated soot chemistry model, therefore, has the potential to significantly advance the design and development of clean combustion systems.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.An overview of the LES-FGM-DSM implementation.Symbols are explained in the main text.

Fig. 2 .
Fig. 2. Illustration of the computational domain and grid (a), and the probability density function of the M parameter in Pope's criteria for the LES grid (b).

Fig. 3 .
Fig. 3. Instantaneous 2-D fields of temperature, OH mass fraction, C 2 H 2 mass fraction, A4 mass fraction.The stoichiometric mixture fraction is shown with dashed iso-lines.The instantaneous results correspond to the FGM-C case.

Fig. 4 .
Fig. 4. Comparison of radial profiles for computed (normalized with maximum) mean mass fractions against measured signals for OH (left panel), PAH (right panel) at different axial locations.

Fig. 5 .
Fig. 5. Instantaneous fields of soot volume fraction f v , soot production rate ( ˙ ω prod s ), soot consumption rate ( ˙ ω cons s ) for the FGM-C (top panel) and FGM-T (bottom panel) method.The dashed iso-lines denote Z st .

Fig. 8 .
Fig. 8.Comparison of soot volume fraction PDF conditioned on the mixture fraction at different axial locations along the flame for FGM-C (a), and FGM-T (b).

Fig. 9 .
Fig. 9. Comparison of soot intermittency I s profiles along the centerline for FGM-C and FGM-T approaches (lines) with measurements (symbols).

Fig. 10 .
Fig. 10.Time-averaged fields of soot volume fraction for FGM-T, and FGM-C closure models (a), and a comparison between experimental (symbols) and numerical (line) data for mean and normalized RMS of soot volume fraction profiles at several axial heights (b).

Fig. 11 .
Fig. 11.Comparison between experimental (symbols) and numerical (line) profiles of mean and RMS of soot volume fraction at the centerline for FGM-T and FGM-C approaches.

Fig. 12 .
Fig. 12.Comparison between experimental (symbols) and numerical (line) profiles of mean and RMS of soot volume fraction at the centerline for FGM-T case with β and δPDF integration applied to tabulated soot rates.

Fig. 13 .
Fig. 13.Time averaged fields of mean soot volume fraction and mixture fraction variance (range adjusted for better visualization) for FGM-T case with δ-PDF integration applied to tabulated soot rates.

Fig. 14 .
Fig. 14.Time-averaged soot PSDs at different axial locations along the centerline for FGM-T and FGM-C approaches.

Fig. 15 .
Fig. 15.Time-averaged fields of soot volume fraction for FGM-C, and FGM-CR closure models (a), and a comparison between experimental (symbols) and numerical (line) data for mean and normalized RMS of soot volume fraction profiles at several axial heights (b).