Karl Töpperwien
email: karl.topperwien@centralesupelec.fr

Stefano Puggelli

Ronan Vicquelin

Analysis of flame propagation mechanisms during light-round in an annular spray flame combustor: the impact of wall heat transfer and two-phase flow

Keywords: light-round, annular combustor, ignition, spray flames, dynamic modeling

Ignition in annular multi-burner combustors is marked by a succession of four phases, ending with a characteristic flame expansion from burner to burner, often referred to as light-round. During this last phase, flame propagation is prone to substantial change depending on the boundary and operating conditions. With realistic aero-engine conditions in mind, wall heat transfers can be enhanced during ignition in cold wall conditions, which aid an understanding of the main governing mechanisms of flame propagation. From a modeling perspective, several works have outlined the need for detailed descriptions of the liquid phase, turbulent combustion and wall heat transfer, which are all included in the present work for the first time. Large-Eddy Simulations

of light-round are performed in the annular MICCA-Spray combustor with cold walls, Lagrangian particle tracking, a dynamic closure for the sub-grid scale flame surface wrinkling as well as a custom tabulated wall model. The

Introduction

Light-round constitutes the final phase in a complex four-step process of forced ignition in annular combustion chambers [START_REF] Lefebvre | Gas Turbine Combustion: Alternative Fuels and Emissions[END_REF][START_REF] Mastorakos | Forced ignition of turbulent spray flames[END_REF]. After a spark is generated in phase I, and the initial flame kernel has grown (phase II) to establish a full-scale ignition of the first burner (phase III), a burner-toburner flame propagation can eventually be observed (phase IV) referred to as light-round [START_REF] Mastorakos | Forced ignition of turbulent spray flames[END_REF].

A multi-burner configuration is thus required to characterize this final phase which strongly depends on combustor design and operating conditions.

Extensive studies can be found in the literature for linear burner arrays observing distinct patterns for premixed (spanwise, axial and hybrid) [START_REF] Cordier | Experimental and Numerical Analysis of an Ignition Sequence in a Multiple-Injectors Burner[END_REF][START_REF] Barré | Flame propagation in aeronautical swirled multi-burners: Experimental and numerical investigation[END_REF] and non-premixed (propagation along flammable bridges) [START_REF] Machover | Experimental and numerical investigation on spark ignition of linearly arranged non-premixed swirling burners[END_REF] setups. These are different from patterns with liquid fuel injection systems which exhibit a branch propagation across neighboring spray branches or an arch-like propagation [START_REF] Marrero-Santiago | Experimental Study of Aeronautical Ignition in a Swirled Confined Jet-Spray Burner[END_REF][START_REF] Marrero-Santiago | Effect of injector spacing in the light-around ignition efficiency and mechanisms in a linear swirled spray burner[END_REF]. Marrero-Santiago et al. [START_REF] Marrero-Santiago | Effect of injector spacing in the light-around ignition efficiency and mechanisms in a linear swirled spray burner[END_REF] also note that the arch propagation mode substantially increases the ignition delay time. All works cite injector spacing as an influential parameter controlling the resulting pattern or causing the pattern to switch.

While linear burner arrays provide detailed insights into the flame spreading process, they lack key features of industrial annular combustors (e.g. flame propagation as arch, flame front merging). Lab-scale representations of annular designs-the subject of the present work-have emerged with first experimental studies by Bourgouin et al. [START_REF] Bourgouin | Ignition dynamics of an annular combustor equipped with multiple swirling injectors[END_REF], revealing the flame's distinct arch shape. Interestingly, Machover et al. [START_REF] Machover | Experimental investigation on spark ignition of annular premixed combustors[END_REF] report a sawtooth movement from burner to burner in their 12-burner premixed bluff-body configuration.

More recent works also outline an effect of common ignition modes "Spark First, Fuel Later" (SFFL) and "Fuel First, Spark Later" (FFSL) [START_REF] Xia | Experimental Investigation of the Flame Front Propagation Characteristic During Light-round Ignition in an Annular Combustor[END_REF], bulk flow velocity and equivalence ratio [START_REF] Gao | Experimental investigation on the flame propagation pattern of a staged partially premixed annular combustor[END_REF] on the observed propagation pattern.

Beyond propagation patterns, the propagation speed and the resulting light-round duration are subject to extensive research as well and are found to be controlled by (i) gas expansion, (ii) the laminar flame speed and (iii) wall heat transfer (among others). The volumetric expansion of burnt gases in the chamber, or thrust effect, constitutes a key mechanism [START_REF] Bourgouin | Ignition dynamics of an annular combustor equipped with multiple swirling injectors[END_REF][START_REF] Machover | Experimental investigation on spark ignition of annular premixed combustors[END_REF] governing the flow acceleration in azimuthal direction, which is proportional to the density ratio of fresh and burnt gases (ρ u /ρ b ).

The (mixing) bulk flow velocity is yet another driving mechanism inversely proportional to the measured light-round duration [START_REF] Bourgouin | Ignition dynamics of an annular combustor equipped with multiple swirling injectors[END_REF][START_REF] Bach | Spark ignition of single bluff-body premixed flames and annular combustors[END_REF][START_REF] Kao | Experimental Spray Structure and Combustion of a Linearly-Arranged 5-Swirler Array[END_REF][START_REF] Philip | Simulation of the ignition process in an annular multiple-injector combustor and comparison with experiments[END_REF][START_REF] Ciardiello | Effect of spark location and laminar flame speed on the ignition transient of a premixed annular combustor[END_REF]. This relation is investigated by Machover et al. [START_REF] Machover | Experimental investigation on spark ignition of annular premixed combustors[END_REF] in a premixed annular combustor featuring 12 or 18 bluff-body burners. The authors note that adding swirl, or increasing the equivalence ratio, increases the flame propagation speed, leading to shorter light-round durations. Interestingly though, the impact of flow velocity on the light-round duration is less pronounced for non-premixed operating conditions [START_REF] Machover | Spark ignition of annular non-premixed combustors[END_REF]. In turn, bulk flow velocity (precisely its circumferential component) becomes the main governing effect of flame propagation in the particular case of oblique injectors [START_REF] Ye | Ignition dynamics in an annular combustor with gyratory flow motion[END_REF]. As for linear setups, the same behavior in terms of flame propagation speed is reproduced in annular chambers when the inter-burner spacing is varied [START_REF] Machover | Experimental investigation on spark ignition of annular premixed combustors[END_REF][START_REF] Machover | Spark ignition of annular non-premixed combustors[END_REF].

The laminar flame speed should also be mentioned as an influential parameter for flame propagation. In the context of liquid fuels, detailed insights were initially gained from academic cases. Laminar flame speeds in droplet mists for overall lean conditions are generally lower than the gaseous laminar flame speed at the same equivalence ratio, suggesting that spray flames propagate at a leaner equivalence ratio than the overall equivalence ratio [START_REF] Ballal | Flame propagation in heterogeneous mixtures of fuel droplets, fuel vapor and air[END_REF][START_REF] Neophytou | Simulations of laminar flame propagation in droplet mists[END_REF][START_REF] De Oliveira | Mechanisms of flame propagation in jet fuel sprays as revealed by OH/fuel planar laser-induced fluorescence and OH* chemiluminescence[END_REF].

Such trends are confirmed by Prieur et al. [START_REF] Prieur | Ignition dynamics in an annular combustor for liquid spray and premixed gaseous injection[END_REF] who have carried out ignition experiments in the swirled annular combustor MICCA-Spray, with different liquid fuels and compared the data with premixed gaseous fuels. While the flame shapes remain comparable for all cases, liquid fuels generally tend to increase the light-round duration, depending on the fuel volatility. This implies that flame propagation speeds are lower than in the equivalent gaseous case.

More recently, similar studies were performed by Ciardiello et al. [START_REF] Ciardiello | Effect of spark location and laminar flame speed on the ignition transient of a premixed annular combustor[END_REF] for different gaseous fuels. Given a matched laminar flame speed, fixed bulk velocity and spark location, but otherwise variable parameters (fuel type, equivalence ratio, thermal power), the light-round duration of two setups is essentially constant. The authors conclude that the laminar flame speed has a first order impact on the resulting flame propagation speed during light-round.

Furthermore, the rate of burnt gas dilatation is characterized on a fundamental level for flame propagation in droplet mists and appears to be influenced by turbulence, initial droplet diameter and droplet equivalence ratio [START_REF] Wacks | Statistical Analysis of Turbulent Flame-Droplet Interaction: A Direct Numerical Simulation Study[END_REF][START_REF] Wacks | Flame Structure and Propagation in Turbulent Flame-Droplet Interaction: A Direct Numerical Simulation Analysis[END_REF].

Data on light-round durations for pre-heated and cold combustor walls have been reported in [START_REF] Philip | Simulation of the ignition process in an annular multiple-injector combustor and comparison with experiments[END_REF][START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF]. At constant bulk flow velocity, pre-heating the combustor walls drastically reduces the light-round duration by 50-70%, suggesting that combustor wall temperatures are crucial for the understanding of flame propagation.

Large-Eddy Simulations have proven to be a reliable tool in order to study light-rounds numerically after first pioneering works by Boileau et al. [START_REF] Boileau | LES of an ignition sequence in a gas turbine engine[END_REF] giving rise to further research with gaseous [START_REF] Philip | Simulation of the ignition process in an annular multiple-injector combustor and comparison with experiments[END_REF][START_REF] Philip | Large Eddy Simulations of the ignition sequence of an annular multiple-injector combustor[END_REF][START_REF] Zhao | Large Eddy Simulation of Multiple-Stage Ignition Process of n-Heptane Spray Flame[END_REF] and liquid fuels [START_REF] Lancien | Large Eddy Simulation of Light-Round in an Annular Combustor With Liquid Spray Injection and Comparison With Experiments[END_REF][START_REF] Collin-Bastiani | Modeling and Large Eddy Simulation of Two-Phase Ignition in Gas Turbines[END_REF][START_REF] Lancien | Leading point behavior during the ignition of an annular combustor with liquid n-heptane injectors[END_REF]. Most of these numerical studies rely on a simplified combustion chemistry to limit the computational costs of LES, and the thickened flame model (TFLES [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF][START_REF] Legier | Dynamically thickened flame LES model for premixed and non-premixed turbulent combustion[END_REF]). A tabulated chemistry approach based on filtered flamelets (F-TACLES [START_REF] Fiorina | A filtered tabulated chemistry model for LES of premixed combustion[END_REF]) was shown to perform similarly to TFLES [START_REF] Philip | Large Eddy Simulations of the ignition sequence of an annular multiple-injector combustor[END_REF].

Detailed kinetics have been included directly in [START_REF] Zhao | Large Eddy Simulation of Multiple-Stage Ignition Process of n-Heptane Spray Flame[END_REF] considering each cell as a perfectly stirred reactor. In terms of the liquid phase description, both monodisperse Euler-Euler (EE) [START_REF] Lancien | Leading point behavior during the ignition of an annular combustor with liquid n-heptane injectors[END_REF] as well as polydisperse Euler-Lagrange (EL) [START_REF] Collin-Bastiani | Modeling and Large Eddy Simulation of Two-Phase Ignition in Gas Turbines[END_REF] simulations of light-round were performed.

While LES is generally able to describe and qualitatively retrieve the light-round driving mechanisms observed experimentally, the results are quantitatively sensitive to the modeling choices. Detailed numerical models can improve the accuracy of the predicted light-round duration in LES, as separately highlighted and quantified by the following studies:

• The effect of dynamic combustion modeling has been investigated by Puggelli et al. [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF]. The authors proved that a constant ("static") flame surface wrinkling parameter β cannot be justified from a physical standpoint given the heterogeneity of the β field revealed by a dynamic evaluation of said parameter. Despite worse results in terms of the observed light-round duration (increase by 20% over a constant β approach), the argument for a dynamic combustion model is still upheld. A cancellation of errors originating from other modeling choices is cited as principal reason for such discrepancy. This error cancellation was more unfavorable (and more pronounced) in the case of the dynamic approach, while hidden (by coincidence) in the constant β case.

• An appropriate description of the liquid phase appears to be essential, in particular the inherent polydispersion of the fuel droplets, resulting from the fuel injection characteristics, droplet evaporation and the flow field. A polydisperse Euler-Lagrange approach similar to [START_REF] Collin-Bastiani | Modeling and Large Eddy Simulation of Two-Phase Ignition in Gas Turbines[END_REF] should then be favored over previous monodisperse Euler-Euler computations of light-round with liquid fuel [START_REF] Lancien | Large Eddy Simulation of Light-Round in an Annular Combustor With Liquid Spray Injection and Comparison With Experiments[END_REF][START_REF] Lancien | Leading point behavior during the ignition of an annular combustor with liquid n-heptane injectors[END_REF][START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF].

• Puggelli et al. [START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF] have demonstrated in an a priori study the intricate relation between variable thermodynamic properties of the boundary layer and the resulting wall heat fluxes. Their work postulated that detailed modeling of wall heat transfers would prove to be crucial in those cases in which hot burnt gases are in contact with cold combustor walls.

All these numerical works have anticipated or even proven an impact on light-round duration and thus the flame propagation speed, intentionally The study is organized as follows. A brief overview of the combustor geometry is given in Sec. 2, followed by details on the numerical setup in Sec. 3, in particular the custom approach to compute wall heat transfers, the liquid phase description and the dynamic combustion model. Results are presented in Sec. 4 focusing on the driving mechanisms of flame propagation.

The aspect regarding the low-order model, its relationship with the driving mechanisms and the impact of modeling choices is finally addressed in Sec. 5.

Experimental configuration

The MICCA-Spray combustor (Fig. 1(a)) is a test rig at the EM2C laboratory [START_REF] Prieur | Ignition dynamics in an annular combustor for liquid spray and premixed gaseous injection[END_REF][START_REF] Prieur | Flame and spray dynamics during the light-round process in an annular system equipped with multiple swirl spray injectors[END_REF] relative to which the chamber is divided into a positive (H+) and negative (H-) half (Fig. 1(c)).

Since the experimental setup is already covered in previous publications, the interested reader is directly referred to [START_REF] Prieur | Ignition dynamics in an annular combustor for liquid spray and premixed gaseous injection[END_REF][START_REF] Prieur | Flame and spray dynamics during the light-round process in an annular system equipped with multiple swirl spray injectors[END_REF] for more details on the experimental diagnostics and the available dataset.

Numerical setup

In light of recent numerical works on light-round ignition, the present work aims at including the previous individual findings in one single numerical setup. Its key elements are briefly summarized here and more details are provided for the tabulated wall model. Large-Eddy Simulations are performed with the AVBP solver [START_REF] Schönfeld | Steady and Unsteady Flow Simulations Using the Hybrid Flow Solver AVBP[END_REF] developed by CERFACS. The separate treatment for the gas and liquid phase is described in the following subsections.

Gas phase

The three-dimensional, filtered, compressible, reactive Navier-Stokes equations are solved on an unstructured tetrahedral grid using the Two-step Taylor-Galerkin Centered scheme (TTGC) [START_REF] Colin | Development of High-Order Taylor-Galerkin Schemes for LES[END_REF] with third-order accuracy in space and time (for arbitrary meshes). Subgrid-scale contributions are computed following the classical eddy viscosity assumption. The turbulent eddy viscosity is evaluated according to the SIGMA model [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF]. The turbulent species diffusivity and the turbulent heat conduction coefficient are determined from the turbulent Schmidt (Sc t ) and Prandtl (Pr t ) numbers (both equal to 0.6).

Dispersed phase

Assuming spherical fuel droplets and a dilute spray, the fundamental equations for droplet motion in a Lagrangian framework employed in this work are given as:

dx p,i dt = u p,i (1)
du p,i dt = f p,i m p,i (2)
dm p,i dt = ṁp,i (3)
dC p,l T p,i dt = 1 m p,i -φ c g,i + ṁp,i L v (T p,i ) (4)
with x p,i , u p,i , m p,i and T p,i denoting the droplet position, velocity, mass and temperature of the ith droplet. For the remainder of this work, indexing of droplets is dropped. C p,l is the liquid heat capacity which may vary with temperature. The droplet density is set constant (ρ l = 688 kg/m 3 ). External forces acting on the droplets are included in f p and are limited to drag force within this study, following the Schiller-Naumann drag force model [START_REF] Schiller | A drag coefficient correlation[END_REF]. Additional terms appear due to heat and mass transfer between the phases with the evaporation rate ṁp and gaseous conductive heat fluxes φ c g at the droplet surface given as:

ṁp = -πd p Shρ g D g ln (1 + B M ) (5)
φ c g = -πd p Nuλ (T ∞ -T p ) ln (1 + B T ) B T (6)
where d p , Sh, ρ g , D g , Nu and λ denote droplet diameter, the Sherwood number, the density of the gaseous mixture, the diffusion coefficient of the gas phase, the Nusselt number, and the thermal conductivity in the gas phase.

These equations represent the Spalding model [START_REF] Spalding | The combustion of liquid fuels[END_REF] with corrections proposed by Abramzon and Sirignano [START_REF] Abramzon | Droplet vaporisation model for spray combustion calculations[END_REF] to account for the presence of vapor films around evaporating droplets modifying the mass and thermal transport.

This affects in particular the evaluation of the Spalding numbers of mass and thermal transport, B M and B T . Assuming thermodynamic equilibrium, B M can be evaluated when the saturation pressure at the droplet surface is known, which is computed from the pretabulated Clausius-Clapeyron formula. Sherwood and Nusselt numbers, which appear in the correction of Abramzon and Sirignano [START_REF] Abramzon | Droplet vaporisation model for spray combustion calculations[END_REF], follow a correlation proposed by Frössling [START_REF] Frössling | Über die Verdunstung fallender Tropfen (On the evaporation of falling drops)[END_REF] and are functions of the droplet Reynolds number Re p , the Prandtl number Pr film and the Schmidt number Sc film in the vapor film. The thermal conductivity of the film is given by λ = µC p,g /Pr film , where µ and C p,g denote the gaseous viscosity and heat capacity in the film. These film properties are evaluated at a reference state, which can be interpolated between far-field and droplet surface conditions using the "third-rule" [START_REF] Hubbard | Droplet Evaporation: Effects of Transients and Variable Properties[END_REF]. Pr film and Sc film are functions of the film temperature and film mixture fraction. Those dimensionless quantities are evaluated from a polynomial fit validated in [START_REF] Töpperwien | Large-Eddy Simulation of Flame Dynamics During the Ignition of a Swirling Injector Unit and Comparison With Experiments[END_REF].

Ultimately, the latent heat of vaporization at the droplet temperature L v is read from lookup tables.

The governing equations are solved using a two-step Runge-Kutta scheme and coupled at every iteration to the gaseous solver.

Fuel injection model

A fully atomized, hollow-cone fuel spray is generated by the phenomenological FIM-UR model (fuel injection method by upstream reconstruction) [START_REF] Sanjosé | Fuel injection model for Euler-Euler and Euler-Lagrange large-eddy simulations of an evaporating spray inside an aeronautical combustor[END_REF][START_REF] Vignat | A Joint Experimental and LES Characterization of the Liquid Fuel Spray in a Swirl Injector[END_REF] and injected at the tip of each of the sixteen atomizers. The injected droplet distribution is a result of a calibration based on PDA measurements in non-reacting conditions performed at a height of x = 5 mm from the combustor backplane and a radius of r = 5 mm from the axis of rotation (cf. filled histogram in Fig. 2). This experimentally measured distribution is evaporation-corrected to account for the fact that droplets may evaporate and thus decrease in size during their path from the atomizer to the measurement position. For this correction, a constant value of the d 2 -law coefficient Vignat based on the methodology described in [START_REF] Vignat | A Joint Experimental and LES Characterization of the Liquid Fuel Spray in a Swirl Injector[END_REF] and applied to the same burner.

λ d2 = 1.65 × 10 -8 m 2 s -1 is derived from Ref.
an injection velocity of u d = 10 m/s is applied for all droplet sizes at the moment of their generation.

With these parameters, the histogram labeled "LES" in Fig. 2 is obtained from the calibration LES at the same position as in the PDA measurements.

It should be noted that each numerical droplet represents one physical droplet over the entire simulation. Secondary atomization is not taken into account.

Reaction kinetics and dynamic combustion model

Reaction kinetics are based on a global two-step scheme for n-heptane/air mixtures containing 6 species (C 7 H 16 , CO 2 , CO, H 2 O, O 2 , N 2 ) [START_REF] Paulhiac | Analysis of the spray flame structure in a lab-scale burner using Large Eddy Simulation and Discrete Particle Simulation[END_REF]. Reaction rates are computed from the Arrhenius law with adjusted pre-exponential factors depending on the local equivalence ratio [START_REF] Franzelli | A two-step chemical scheme for kerosene-air premixed flames[END_REF]. A validation against detailed schemes [START_REF] Smallbone | Experimental and modeling study of laminar flame speed and non-premixed counterflow ignition of n-heptane[END_REF] performed in [START_REF] Paulhiac | Analysis of the spray flame structure in a lab-scale burner using Large Eddy Simulation and Discrete Particle Simulation[END_REF] shows a reasonably good prediction of the laminar burning velocity at ambient pressure and over a wide range of equivalence ratios.

Since cell sizes are not sufficiently small to fully resolve the flame on the numerical grid, artificial thickening (TFLES) is applied in the flame region [START_REF] Colin | A thickened flame model for large eddy simulations of turbulent premixed combustion[END_REF][START_REF] Legier | Dynamically thickened flame LES model for premixed and non-premixed turbulent combustion[END_REF], ensuring correct laminar flame burning velocity retrieval. Thickening by a factor F is applied such that it is limited to the flame region only [START_REF] Legier | Dynamically thickened flame LES model for premixed and non-premixed turbulent combustion[END_REF] and ensures a reasonable resolution of flame profiles (7 grid points used here). As a consequence of the two-phase flow configuration, thickening is also applied to drag and evaporation [START_REF] Paulhiac | Analysis of the spray flame structure in a lab-scale burner using Large Eddy Simulation and Discrete Particle Simulation[END_REF], but restricted to zones dominated by premixed combustion detected by the Takeno sensor [START_REF] Yamashita | A numerical study on flame stability at the transition point of jet diffusion flames[END_REF]. Furthermore, this approach also implies that the flame surface must be corrected for unresolved sub-grid scale wrinkling effects, which may be described with a flame surface wrinkling model. Following the work of Charlette et al. [START_REF] Charlette | A power-law flame wrinkling model for LES of premixed turbulent combustion Part II: Dynamic formulation[END_REF], the flame surface wrinkling factor is expressed as:

Ξ ∆ = ∆ δ l β (7)
where ∆ corresponds to the local mesh size, δ 0 l to the laminar flame thickness and β to a free parameter. As highlighted in [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF], a dynamic evaluation of the β-parameter is recommended to account for different amounts of wrinkling of the propagating flame fronts on one hand, and stabilized flames downstream of these flame fronts on the other hand. Essentially, the wrinkling parameter β is determined following the work of Mouriaux et al. [START_REF] Mouriaux | Adaptation of a dynamic wrinkling model to an engine configuration[END_REF], using the same parameter set as in [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF] with the exception of a higher update frequency (every 250 iterations).

It should be reiterated that the primary aim is to study the light-round phase characterized by two separate flame fronts propagating through the combustion chamber. Hence, correctly predicting a flame speed is arguably the key requirement for any combustion model to fulfill under these circumstances -a goal the TFLES approach meets by design.

Wall modeling

When wall-resolved LES is not an option, the near wall flow may be described with wall models. Wall boundary profiles can then be computed from algebraic models for example, which in turn require a certain set of assumptions to hold true.

As stated in [START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF], an algebraic wall law inevitably falls short of correctly predicting wall heat fluxes, even if density variations are included as in [START_REF] Cabrit | Direct simulations for wall modeling of multicomponent reacting compressible turbulent flows[END_REF], since other properties (c p , µ, λ) can vary substantially in the boundary layer.

Puggelli et al. [START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF] have estimated an up to 70% lower heat flux under lightround conditions when standard algebraic wall models are used. Hence, in an attempt to improve the modeling of the near wall flow in the burnt gas region, a tabulated wall model approach is proposed here.

Similar to the work of Maheu et al. [START_REF] Maheu | Large-Eddy Simulations and of flow and heat transfer and around a low-Mach number turbine blade[END_REF], an interpolation database is generated to compute local wall heat fluxes and wall shear stresses as a function of known quantities at the first off-wall node of the mesh, also referred to as matching point. This approach appears to be a promising alternative to algebraic wall models.

Essentially, the one-dimensional Thin Boundary Layer Equations (TBLE) [START_REF] Schlichting | Boundary-Layer Theory[END_REF] are solved for a range of boundary conditions. The eddy viscosity µ t is y z approximated according to the expression proposed by Cabot and Moin [START_REF] Cabot | Approximate wall boundary conditions in the largeeddy simulation of high Reynolds number flow[END_REF]:

T MP , u MP Matching point T (y) δ MP
µ t = ρκyu τ 1 -exp - y A + 2 ( 8
)
where κ is the von Kármán constant, y the wall distance, u τ the friction velocity, y the normalized wall distance and A + a parameter. Superscripted stars denote semi-local scaling [START_REF] Huang | Compressible turbulent channel flows: DNS results and modelling[END_REF][START_REF] Patel | Semi-local scaling and turbulence modulation in variable property turbulent channel flows[END_REF] of the friction velocity u τ = (τ w /ρ) 0.5

and the normalized wall distance y = ρyu τ /µ which is generally employed for variable property flows. The turbulent Prandtl number Pr t appearing in the Thin Boundary Layer Equations is computed according to the formula of Kays [START_REF] Kays | Turbulent Prandtl Number-Where Are We?[END_REF].

The resulting wall heat fluxes and wall shear stresses obtained from TBLE are tabulated as a function of three parameters: the temperature T MP , the velocity u MP and the distance δ MP of the matching point (subscript "MP", see sketch in Fig. 3). These parameters constitute the inputs for the interpolation routine during LES, with a value range predetermined from precursor simulations. To safely cover the entire range of possible combinations which may be encountered during light-round, T MP is swept between [350, 2300] K in steps of 50 K, u MP between [0.5, 50] ms -1 in steps of 0.5 ms -1 and δ MP between [0.2, 1] mm in steps of 0.01 mm. At y = 0 (i.e. at the wall), the boundary condition is set to T = 300 K to solve the TBLE. For table generation, a theoretical burnt gas composition is imposed being equal to the global equivalence ratio, at which the combustor is operated (φ glob = 0.89) since variable compositions have not been considered in the generated database.

The entire interpolation database comprises 324 000 data points for each target variable (wall heat flux and wall shear stress). This database is created only once and is then accessed by the LES code at each iteration to compute the corresponding wall heat fluxes and wall shear stresses by simple table interpolation at each matching point based on its current input parameters.

In order to ensure that the tabulated wall model is only applied in the burnt gas region, the off-wall progress variable c MP is introduced to toggle between the tabulated wall model (for c MP ≥ 0.5) and the standard algebraic wall model (otherwise), as the TBLE have not been solved for a fresh gas composition where variable-property effects with temperature are absent.

The three-dimensional parameter space is cut at an arbitrary matching point distance δ MP = 0.4 mm for visualization purposes and plotted in Fig. 4 as a two-dimensional cut-plane showing the evolution of q wall and τ wall . At constant δ MP , the wall heat flux mainly increases with increasing matching point temperature T MP , implying an increasing temperature difference between the matching point and the wall at T wall = 300 K. τ wall is instead dominated by the matching point velocity u MP .

For an assessment of the interpolation accuracy, random tuples of T MP , 1000 2000 u MP and δ MP are generated within the database limits and used to solve the TBLE. Comparing the resulting wall heat flux and wall shear stress with the interpolated values from the database using the same tuple yields a relative error below 1% for either target variable. Liquid phase boundary conditions are set to allow elastic rebound in the combustion chamber. In the injectors, droplet-wall interactions are considered to be predominant, thus requiring a film-type boundary condition as developed in [START_REF] Chaussonnet | A new phenomenological model to predict drop size distribution in Large-Eddy Simulations of airblast atomizers[END_REF].

T MP [K]

Numerical domain and boundary conditions

Initial conditions and ignition procedure

Initial conditions are deliberately chosen to ensure a successful ignition and the initial flame kernel to survive the first stages of its development.

It is emphasized once more that these early stages are beyond the scope of this work. Instead, as mentioned in the introduction, we focus specifically on the light-round phase [START_REF] Lefebvre | Gas Turbine Combustion: Alternative Fuels and Emissions[END_REF][START_REF] Mastorakos | Forced ignition of turbulent spray flames[END_REF]. The combustion chamber is prefilled with a fuel/air mixture in the experiment for a few seconds before sparking. Since n-heptane is a volatile fuel, evaporation already occurs under ambient conditions. Stationary equivalence ratios are attained after around one second, but vary with chamber height due to a complex heterogeneous flow structure.

In an attempt to match these initial conditions of the experiment as best as possible, the simulation first starts with fuel injection and no combustion.

Taking advantage of the rotational symmetry of the combustor, pre-fueling is performed on one eighth of the full geometry including the atmosphere, greatly reducing computational costs. During this specific phase, periodic boundary conditions are applied to the tangential faces of the domain. The equivalence ratio is monitored over time until steady average values can be observed on several cut-planes at different heights. Second, the converged solution is cloned eight times (including the geometry) to yield the full annular combustor assembly. Third, the fuel injection phase is resumed in the actual full annular geometry for 24 ms to dissolve coherent flow structures resulting from the cloning procedure. The required simulation time of this step is derived from the autocorrelation rate of velocity fluctuations which is estimated as τ corr = 6 ms. Lastly, a 10 × 24 mm ellipse of burnt gases is introduced at the experimental spark plug position to act as an initial flame kernel and ensure a successful and robust ignition procedure.

Results

Before we derive a simplified model for flame propagation during lightround with cold combustor walls, we first draw the attention to the propagation mechanism under these conditions. The effects of liquid fuels and heat losses are studied both from a global perspective as well as locally on particular points on each propagating flame front. The role of the dynamic combustion model is also briefly outlined.

Flame propagation during light-round

To approach the simulation results from a global perspective, it is useful to examine instantaneous snapshots of different stages during light-round shown in Figure 6. Unless explicitly stated otherwise, the instant the simulation starts is used as time reference t 0 = 0 ms (flame kernel delivered in the outer recirculation zone of the first burner). The images are obtained by transforming the full cylindrical chamber into a rectangular system and integrating the heat release rate in line-of-sight direction. In these transformed images, the flames propagate from the center (position of the initial flame kernel shown in Fig. 6(a)) to the sides. The nozzle outlets of the sixteen injectors are included at the bottom of each image for reference.

All characteristic phases of light-round and their corresponding flame shapes are well retrieved in the simulation and are consistent with the experimental high-speed imaging performed by Prieur et al. [START_REF] Prieur | Ignition dynamics in an annular combustor for liquid spray and premixed gaseous injection[END_REF]. Identical flame shapes are also observed in Refs. [START_REF] Lancien | Large Eddy Simulation of Light-Round in an Annular Combustor With Liquid Spray Injection and Comparison With Experiments[END_REF][START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF] which provide detailed analyses for the interested reader.

Light-round duration

To compare the simulation with experimental data, the light-round duration is commonly used as a global metric, defined here as the elapsed time between two specific events during ignition. Synchronization between LES and experiment is usually established once the initial flame (starting from the sector S 0 marked with a bolt in Fig. 1(c)) has ignited the adjacent burner in sector S -1 (marked with a star in Fig. 1(c)). This common starting point eliminates at least those uncertainties associated with the stochastic behavior of the initial spark and the subsequent growth of the initial flame kernel.

Hence the focus on phase IV of light-round [START_REF] Lefebvre | Gas Turbine Combustion: Alternative Fuels and Emissions[END_REF][START_REF] Mastorakos | Forced ignition of turbulent spray flames[END_REF].

Conversely, the end of light-round can be determined more easily by taking the first recorded frame obtained from high-speed imaging (instantaneous solution in the case of LES) in which the two flame fronts start to overlap. The time difference between this instant and the synchronization point defined above yields the light-round duration τ LR .

Following this definition, light-round durations for four experimental runs (EXP1 -EXP4) [START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF] and one simulation with cold combustor walls (referred to as baseline case, abbreviated as BASE) are compiled in Tab. [START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF] with combustor walls at ambient temperature. The simulated light-round duration is given for the baseline case (BASE) for the same conditions as in the experiment. An additional simulation with adiabatic combustor walls (ADIAB) is also carried out for comparison. Numerical lightround durations are computed as visualized in Fig. 7.

release rate or the instant the propagating flame front in the negative chamber half H-has crossed a certain sector. In the former case, the heat release rate in sector S -1 is volume integrated and the instant of its peak value is used as synchronization point. The solid lines in Fig. 7 show the per-sector heat release rate for sectors S -1 to S -8 when performing the volume integral up to a height of 60 mm. Thus, the light-round duration (i) τ LR = 54.6 ms corresponds to the time difference between the two vertical dash-dotted lines in Fig. 7,i dashed line in Fig. 7, yielding a value of (ii) τ LR = 52.6 ms (cf. dashed box in Fig. 6(c)). The peak shift between the dashed line and the thick solid line (in Fig. 7) may be interpreted as uncertainty range (gray shaded area) and is less than 4% of the overall light-round duration τ LR .

A third possible definition of the light-round duration can be derived from the instant at which the circumferentially outermost flame edge has entirely crossed sector S -1: (iii) τ LR = 49 ms, equivalent to an uncertainty range of 10% at worst, when using definition (i) as reference.

In summary, the light-round duration is deemed to be only moderately impacted by the choice of the definition for the synchronization point, yielding very acceptable results in either case, compared to the duration reported in experiments. The modeling setup then allows to describe the flame prop- agation quantitatively in the considered configuration that is characterized by cold walls.

Thrust effect

The most important driving mechanism of flame propagation consists in a flow acceleration in azimuthal direction during the expansion of burnt gases inside the chamber. This acceleration is commonly referred to as thrust effect and is illustrated in Fig. 8 ahead of each flame front, although not uniformly over the chamber height.

Two instantaneous azimuthal velocity profiles are sampled along the dashed sampling line in Fig. 8 and plotted in Fig. 9. Initially (t = 1 ms), no mean azimuthal flow is observed except for the flow perturbations induced by the swirlers between the chamber bottom and up to 50 mm above (solid line in Fig. 9). The velocity profile exhibits at a later time t = 22 ms positive values across the entire chamber height (dash-dotted line in Fig. 9), which is an indication of the thrust effect. It is most pronounced at 30 mm above the chamber bottom, where it reaches u θ = 20 m/s, around four times of its initial value at the same height. This strong velocity peak raises the question how fuel droplets react to this effect and interact with the flame on a local scale. Droplet accumulations in the wake of the swirling jets are highlighted by red squares.

Sector layout as in Fig. 8.

Flame interaction with droplets

In Fig. 10, the flame/droplet interaction induced by the thrust effect is visualized on the same unwrapped cylindrical cut-plane at R = 0.175 m as before, with orange contour lines representing the progress variable (c = 0.76).

A representative time frame at t = 34.5 ms (propagation as two separate flame fronts) is chosen for an optimal visibility of the phenomenon on that plane. Dark shaded regions indicate an increased liquid volume fraction α l and can be interpreted as droplet accumulations. Crucially, these accumulations prevail mostly in the wake of the swirling jets close to the chamber bottom (see squares in Fig. 10). Droplets entering the combustion chamber ahead of the the propagating flame fronts deviate from their initial upward trajectory due to the thrust effect and are trapped in the recirculating flow between two injectors, similar to jet in crossflow configurations [START_REF] Lancien | Leading point behavior during the ignition of an annular combustor with liquid n-heptane injectors[END_REF]. As a consequence, the flame may locally burn in richer conditions than the overall global equivalence ratio, once it encounters such a droplet accumulation.

This effect can be revealed through the total equivalence ratio φ tot , the sum burner spacing enforces a sawtooth propagation pattern along connecting "bridges" of neighboring spray branches, since the inter-injector region close to the combustor backplane is too lean to be ignitable. Insufficient fuel/air mixing due to high equivalence ratios and low bulk-flow velocities constitute another reason for sawtooth propagation [START_REF] Gao | Experimental investigation on the flame propagation pattern of a staged partially premixed annular combustor[END_REF]. By contrast, the flame itself causes its characteristic sawtooth trajectory in the present work by generating upstream droplet accumulations through the thrust effect, which are less favorable to cross.

Impact of wall heat losses

Combustor wall temperatures not only affect the propagation pattern though. With the a priori study by Puggelli et al. [START_REF] Puggelli | Impact of Wall Temperature in Large Eddy Simulation of Light-Round in an Annular Liquid Fueled Combustor and Assessment of Wall Models[END_REF] in mind, a comprehensive a posteriori analysis of the effect of wall heat losses on flame propagation can now be provided in the present work. In particular, the relation between heat losses and the thrust effect is brought to the reader's attention. Differences are pointed out with regard to the complementary ignition simulation (ADIAB) which is based on the exact same setup except for the wall boundary condition set to adiabatic instead. This second ignition case also starts from the very same initial solution as its isothermal counterpart.

Heat release rate and wall heat fluxes

Similar to previous works, the heat release rates (HRR) follow a characteristic evolution during each stage of light-round (Fig. 13). Each curve is obtained by integrating the local heat release rate over the combustor volume without the surrounding atmosphere. While the slope is initially more moderate, a steep increase of the heat release rate is observed up to almost twice the nominal thermal power (P th ) of the combustor. Once the flamearch reaches the outlet of the chamber, heat release rates slightly decrease before increasing again, and reach roughly 2.5 times of P th at the instant both flame fronts start to merge (diamond marker in Fig. 13). Residual parts of the flame fronts leave the chamber during transition into steadystate operation, causing a steep decrease of the heat release rates until P th will eventually be reached. Both cases were stopped prior to steady-state operation due to constraints in available computational hours. Nevertheless, it can be inferred from Fig. 7 that each burner operates at approximately 5 kW after full ignition (already visible for burners in sectors S-1 to S-7), which totals in P th for all 16 burners.

For an overall quantification of the wall heat losses Q, all heat fluxes (both lateral walls and chamber bottom) are summed up and related to the nominal thermal power output P th of the combustor (when fully ignited) in Fig. 13. Note that these fluxes from the fluid onto the combustor walls are deliberately chosen to have a positive sign.

The sum of all wall heat fluxes across the chamber walls exhibits a steady increase up to 50% of P th (grey line in Fig. 13) towards the end of lightround. Viewed from the perspective of the simulation case ADIAB where all walls are adiabatic, heat losses account for a 27% longer light-round duration (τ LR ADIAB = 43 ms versus τ LR BASE = 54.6 ms). To understand how heat losses act on the driving mechanism of flame propagation, local effects have to be taken into account; these effects are investigated in the following subsection.

Index of heat loss

The heat loss index I HL may be considered as a useful metric to visually inspect the cooling of burnt gases (for case BASE), defined as:

I HL = h -h low h adiab -h low =      1 adiabatic 0 max. local heat loss (9)
where h denotes the local enthalpy, h low the enthalpy of the local mixture, if it had the temperature of the combustor walls (i.e. ambient temperature), and h adiab the enthalpy of the local mixture in perfect adiabatic conditions.

Since the values of h adiab (adiabatic mixture) and h low (mixture at ambient temperature) are chosen deliberately, I HL reaches unity, if the local mixture is equivalent to an adiabatic mixture, and lower values, if it incurs heat losses.

Upon examination of the unwrapped cutplane at R = 0.175 m, different regions with different levels of heat losses can be identified: burnt gases far downstream from each flame front (labeled as (1) in Fig. 14) take values of I HL < 0.7 as they have been exposed longer to the combustor walls than burnt gases immediately downstream of the flame fronts (labeled as ( 2)).

In fact, large parts of the flame fronts propagate in almost adiabatic conditions (I HL ≈ 1) across the full chamber height (on the presented cut-plane).

Therefore, the longer the burnt gases are in contact with the combustor walls, the more pronounced the heat loss effects, particularly towards the end of light-round. In turn, the fresh gas region labeled as (3) in Fig. 14 remains adiabatic, since fluid and walls are both at ambient temperature. The effect of evaporative cooling (particularly in the fresh gases) is comparably small and barely alters the fresh gas enthalpy.

As already discussed in [START_REF] Barré | Flame propagation in aeronautical swirled multi-burners: Experimental and numerical investigation[END_REF], the generated thrust, and thus the flame propagation speed, is proportional to the density ratio between burnt and fresh gases. Crucially, this density ratio may be altered due to heat transfer between hot burnt gases and combustor walls at ambient temperature.

Volume-averaging the density in the burnt gas volume inside the chamber (denoted by angular brackets with subscripted "V ") reveals a clear trend:

since density and temperature are inversely proportional, the burnt gas density steadily increases with decreasing burnt gas temperature. In Fig. 15, this relation is conveniently plotted as density ratio ρ u V / ρ b V to highlight the decreasing strength of the thrust effect (solid line). In turn, the thrust generated in the adiabatic case (dashed line in Fig. 15) remains almost constant during the propagation of two separate flame fronts. A moderate decrease is observed at the end of this case when fresh air from the environment is entrained at the combustor outlet.

Therefore, an appropriate wall model in the baseline case is deemed essential to correctly predict wall heat transfers and thus the light-round duration under such conditions. The intricate relation between heat losses, thrust effect and turbulent flame propagation speed is further discussed in Sec. 5

based on a mathematical model.

Flame/turbulence interactions

In light of the recent study published in Ref. [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF], the interaction of the flame with turbulence deserves further attention as well. This aspect is considered to be particularly relevant for the present case, since the flame surface wrinkling parameter β cannot be assumed constant over the entire flame. For a highly transient and inhomogeneous case such as light-round ignition, little physical argument can be made for a "universal" (i.e. constant) β. Therefore, it is more appropriate to employ a dynamic combustion model for light-round simulations, independently from the thermal boundary conditions (see also [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF]).

On a representative iso-surface of the progress variable c = 0.76 for a given instant t = 44 ms (Fig. 16), both propagating flame fronts exhibit a comparably low wrinkling parameter (β ≈ 0.18), while the stabilized flames downstream show higher values instead (β 0.4). Note that the color map is deliberately clipped to help distinguish β values on the flame fronts from ignited flames downstream.

The corresponding distribution of β on the flame surface is given for the same instant in Fig. 17 (solid line). The bin colors match the shading of the iso-surface in Fig. 16. Two major conclusions can be drawn from the PDF:

first, the distribution of values of the wrinkling parameter is non-uniform.

While the peak in the PDF at β ≈ 0.18 originates from the propagating flame fronts, the ignited and stabilized flames take a wider range of values of

β 0.4.
Second, the PDF considerably changes its shape over time. During the early stages of light-round, i.e. when the flame develops an arch-like form, only few burners are already ignited so that the peak in the PDF at β ≈ 0.18 (cf. Fig. 17, dotted line) is initially more pronounced. The plateau at β 0.4 in turn is much less developed. However, the more burners are ignited during flame propagation, the more the plateau develops while the peak associated with the propagating flame fronts gradually diminishes. Once the entire combustor is fully ignited and reaches a steady state, the PDF transitions into a single-peak PDF at β = 0.47, since there is no contribution from the propagating flame fronts any more (dashed line in Fig. 17). This evolution of the wrinkling parameter β is consistent with the analysis in [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF] and occurs independently of the wall boundary conditions.

Analysis

The previous section provides insights into the governing mechanisms of flame propagation during light-round without quantifying their individual impact on the propagation speed. Therefore, a macroscopic flame speed model is derived here for the present configuration to put forward the argument that all phenomena previously discussed have to be taken into account in the simulation. First, the model is benchmarked in terms of its prediction capabilities using the data obtained from the baseline case. Second, the flame speed model is decomposed to identify the terms and variables which are affected by the main physical phenomena studied in this work. Third, a priori estimations for the flame propagation speed under different conditions are computed using the macroscopic model to underscore the fact that neither phenomenon can be neglected.

Model for absolute flame propagation speed during light-round

Deriving an expression for the turbulent absolute flame speed S T a (also referred to as flame propagation speed) requires several macroscopic balance equations for the burnt gas mass. The equations are presented here in a condensed form, but the interested reader may find a complete step-by-step guide in Refs. [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF][START_REF] Philip | Dynamics of light-round in multi-injector annular combustors[END_REF] for adiabatic conditions. At this point, it should be clarified that the final expression for the flame speed is only valid when the flame propagation mode can be described as two separate, semi-confined flame fronts, limited by the chamber backplane, the lateral combustor walls and the combustor outlet. During this stage in the ignition process and with the approximation of the separate flame fronts as vertical planes, the rate change of the burnt gas volume V b inside the combustion chamber is written as:

dV b dt = S T a A 0 (10)
where the sectional surface area of the planes is A 0 = 2h∆R (chamber height h and width ∆R computed as difference between outer and inner chamber radii). Next, the macroscopic balance equation of the progress variable within the combustor is given as:

dm b dt = ṁin b -ṁout b + V ωc dV (11)
introducing the mass flow rates of burnt gases at the inlet (in case of any recirculation there) and outlet of the combustion chamber ( ṁin b and ṁout b ), and the source term of the progress variable ωc . The volume integral in Eq. 11 allows for introducing the LES flame consumption speed S ∆ c through the following expression:

ρ u A res S ∆ c = V ωc dV (12)
where ρ u denotes the averaged fresh gas density and A res the resolved flame surface. In the context of LES, the consumption speed can be computed from the laminar flame speed S l and the sub-grid scale flame surface wrinkling Ξ ∆ res averaged over the resolved flame surface

S ∆ c = Ξ ∆ res S l . (13)
Eventually, the mass of burnt gases m b can also be written as:

m b = ρ b V V b (14)
where ρ b V denotes the volume averaged burnt gas density. Differentiation of Eq. 14 with respect to time and substituting the corresponding terms with Eqs. 10 -13 yields the final expression for the absolute turbulent flame speed:

S T a = ρ u V ρ b V Ξ res Ξ ∆ res S l Term I - ṁout b -ṁin b ρ b V A 0 Term II - V b ρ b V A 0 dρ b dt Term III . (15)
The resolved flame surface wrinkling, Ξ res = A res /A 0 , is defined as ratio between the resolved flame surface and the sectional surface area A 0 of the combustor.

Figure 18 shows the temporal evolution of S T a computed according to Eq. 15 for the baseline case (top) and case ADIAB (bottom) as thick solid lines. The masked time span at the beginning (in gray) indicates that Eq. 15 cannot be applied during the first phase of light-round by construction, as the initial flame has not yet developed two separately propagating flame fronts. Averaging S T a over the valid time span up to flame front merging yields 8.81 m/s for the baseline case and 10.03 m/s for the adiabatic case.

A meaningful reference velocity may be computed from one sector arc length (i.e. s = 2πR/16) and the averaged time between two consecutive sector heat release peaks (obtained from Fig. 7) in order to assess the accuracy of the modeled flame propagation speed S T a . These propagation speeds are subscripted with "HR" in the following. For the baseline case, a velocity of S T a, HR = 7.6 m/s is obtained. This reference velocity is plotted as horizontal gray dashed line in Fig. 18 (for the flame branch in H-, see Fig. gible. The evolution of Term II is directly linked to ṁout b : it is zero until the initial flame arch has reached the chamber outlet. At that point, the outflux of burnt gases starts to increase (as does Term II) with the propagation of the flame fronts. Furthermore, comparing the baseline case to ADIAB shows that in the absence of wall heat losses Terms I and II are shifted to higher velocities. This suggests that the main physical phenomena described in Sec. 4 are interdependent, i.e. react and adapt to deliberately introduced changes in the operating or boundary conditions, for example deactivating wall heat transfers in Large-Eddy Simulations. These phenomena are therefore studied in detail in the following section.

Model decomposition

Apart from splitting Eq. 15 into its main terms, more insights into the main mechanisms of flame propagation can be gained when examining the governing physical phenomena individually and establishing links to the relevant variables in the model equation.

Laminar burning velocity and two-phase flow

The effects of the liquid phase on the absolute turbulent flame speed S T a enter through the laminar flame speed S l appearing in Term I of Eq. 15 and are twofold. First, the equivalence ratio field φ is strongly heterogeneous due to complex interactions of droplets with the flow and different amounts of liquid and pre-vaporized fuel along the propagating flame fronts. For example, φ ranges from roughly φ = 0.7 in the fresh gases (labeled as A in Fig. 19) up to φ = 0.95 in the burnt gases in the upper half of the combustion chamber where all droplets are fully evaporated (labeled as B). In rich recirculation zones between adjacent burners, φ is larger than unity as a consequence of droplet accumulations. The mixture inhomogeneity then presumably prevents the identification of S l with the theoretical laminar flame speed S 0 l (φ glob ) determined trivially from the global equivalence ratio, at which the combustor is operated.

Second, the gaseous equivalence ratio φ can also be seen to increase across the flame fronts in Fig. 19, underscoring the intrinsic two-phase flow structure of the flame front within which droplets are evaporating. This two-phase Therefore, it is more appropriate from a global viewpoint to consider an averaged laminar flame speed over the total flame surface A tot = Ξ ∆ res A res , computed as:

S l = 1 ρ u A tot Y eq c V ωYc dV (16)
where the integral is performed within the combustor volume V and normalized by the equilibrium value of the progress variable species Y eq c . Plotting S l according to Eq. 16 (or even according to Eq. 13 for S ∆ c ) yields S l ≈ 24 cm/s (or S ∆ c ≈ 30 cm/s), while the theoretical flame speed at global equivalence ratio is S 0 l (φ glob ) = 36 cm/s (see Fig. 20). Interestingly, the laminar flame speeds of case BASE and ADIAB are very similar.

Figure 20 shows that the actual laminar flame speed is, on average, much lower than S 0 l (φ glob ). This is attributed to the leaner conditions met by the flame during its propagation and to its two-phase flow nature. Indeed, it has already been established in the literature [START_REF] Ballal | Flame propagation in heterogeneous mixtures of fuel droplets, fuel vapor and air[END_REF][START_REF] Neophytou | Simulations of laminar flame propagation in droplet mists[END_REF][START_REF] Rochette | Effect of non-zero relative velocity on the flame speed of two-phase laminar flames[END_REF] mixtures. This implies that the effective (or averaged) equivalence ratio which corresponds to the averaged laminar flame speed plotted above must be smaller than the global equivalence ratio as well.

To provide further evidence for this argument, an average equivalence ratio of φ S l =24 cm/s = 0.70 can be obtained from the flame speed diagram in Fig. 21 assuming an average flame speed of S l = 24 cm/s. This average equivalence ratio can then be compared to the probability density function of φ, PDF(φ), on a characteristic iso-surface of the progress variable c = 0.76, shown in Fig. 22. The distribution of the gaseous equivalence ratio on this iso-surface ranges from φ = 0.65 to around φ = 1.2 with a broader peak at φ = 0.7 and matches the average equivalence ratio predicted by the flame speed diagram fairly well. As far as averaged quantities are concerned, the average (or effective) equivalence ratio for lean two-phase flames is proven equivalence ratios for T u = 300 K in fresh gases. Dashed line denotes reference data published in [START_REF] Paulhiac | Analysis of the spray flame structure in a lab-scale burner using Large Eddy Simulation and Discrete Particle Simulation[END_REF]. Crosses refer to the one-dimensional laminar consumption speed S ∆ c according to Eq. 17, which is computed for arbitrary locations along the flame front indicated in Fig. 19. of such flame is equivalent to its laminar flame speed allows to apply Eq. 17 in direction of the three-dimensional flame front normal for arbitrary positions along the propagating flame fronts.

S 1D c = 1 ρ u Y eq c ωYc dc (17)
These positions correspond to the labels 1 -4 in Fig. 19 and are plotted in Fig. 21 accordingly.

Given the assumptions made for the computation of such local flame speeds, all data points align remarkably well with reference S 0 l values, although the local total equivalence ratio (gaseous and liquid) can be quite different from the local gaseous condition. Furthermore, the laminar flame speed tends to increase with increasing height above the chamber backplane, since droplets are more likely to fully evaporate the farther they penetrate the combustion chamber (data points 1 to 3). Although closest to the chamber outlet, data point 4 exhibits the lowest equivalence ratio due to dilution effects resulting from entrained air of the surrounding atmosphere. Regardless of the position on the flame front, all laminar flame speeds are found to be inferior to the theoretical laminar flame speed at global equivalence ratio [19]. Compared to a light-round case with a fully premixed gaseous mixture, liquid fuel affects the laminar flame speed as well as the resulting absolute turbulent flame speed through Eq. 15.

Burnt gas density and heat losses

As foreshadowed in Sec. 4.5, wall heat losses act on the the burnt gas density ρ b V , which appears in all terms of Eq. 15. In particular, the generated thrust is affected as it scales with the density ratio ρ u V / ρ b V (see also Fig. 15). Including wall heat losses translates into a lower generated thrust, which in turn generates lower azimuthal velocities. This can be examined in Fig. 23 showing the azimuthal velocity u θ averaged across the chamber cross section at the position marked in the inset. The sampling plane is deliberately chosen farther away from sector S0 (sector of flame kernel deposition, filled burner in the inset plot) to ensure a clear visibility of heat loss effects on burnt gases, which are most pronounced towards the end of light-round (cf. Sec. 4.5.2). Both curves (BASE and ADIAB) exhibit a characteristic growth/decline with a positive peak before the flame front crosses the sampling plane, and a negative peak shortly after. The area-averaged azimuthal velocity u θ of case ADIAB however increases more steadily than in the baseline case (highlighted region in Fig. 23), eventually reaching the sampling plane faster (positive peak of both curves). Recalling that the flame propagates approximately at the speed of the accelerated flow [START_REF] Lancien | Large Eddy Simulation of Light-Round in an Annular Combustor With Liquid Spray Injection and Comparison With Experiments[END_REF], it can be inferred from the plot that flame propagation is generally faster when burnt gases are not subjected to wall heat losses.

Remarkably, S T a does not increase by the same amount as does the density ratio in Fig. 15 for example when passing form the baseline setup to an adiabatic setup. This behavior is attributed to the fact that the flame propagation speed is obviously not decoupled from Term II containing not only the burnt gas density ρ b V , but also the burnt gas mass flow rate at the combustor outlet ṁout b . When the generated thrust is stronger (i.e. when all walls are adiabatic), the outflux of burnt gases increases, which counterbalances the increase of Term I through an increased Term II as already seen in Fig. 18 (comparing dash-dotted lines for both cases). Therefore, the change of the density ratio does not control the resulting flame propagation speed alone.

Beyond the generated thrust and outflux of burnt gases, other phenomena have to be considered as well. While the laminar flame speed S l is essentially the same for both cases (see Fig. 20), as it is controlled primarily by the liquid phase, turbulence levels may be enhanced by a higher azimuthal velocity and increase the flame surface wrinkling. Consequently, all effects on the flame propagation speed have to be taken into account at the same time, since changes in one phenomenon are prone to affecting another. To corroborate this argument, flame/turbulence interactions are briefly covered in the next section, since a dedicated study on that subject is already published in [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF].

Flame/Turbulence interactions

The resolved flame surface wrinkling Ξ res = A res /A 0 can be examined in Fig. 24 (solid and dashed lines) to prove the interaction between thrust generated by the flame and turbulence. The grayed area masks the first phase of light-round, where Eq. 15 cannot be applied.

For both cases, the evolution of the resolved flame surface wrinkling Ξ res is similar, but case ADIAB exhibits higher values almost throughout the entire light-round. Up to the first peak, Ξ res is on average 25% larger in the adiabatic case at a given instant, and still 15% larger between t = 30 ms and t = 45 ms compared to the baseline case. This contributes entirely to a larger Term I in the model expression for S T a and thus to the higher absolute turbulent flame speed observed in the absence of wall heat losses.

Unlike the resolved flame surface wrinkling, the sub-grid scale flame surface wrinkling Ξ ∆ res averaged over the resolved flame surface is virtually the same for both cases (dash-dotted and dotted lines), and similar to the one examined in [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF]. This result is not surprising given the fact the the computational mesh is exactly identical in both simulations.

Model sensitivities

With the insights of the previous model decomposition, the last goal of this work can finally be addressed. Changes are introduced separately to the model expression for S T a in Eq. 15 in the following way for otherwise unchanged variables:

• instead of a dynamic evaluation of the wrinkling parameter β, a constant value of β ≡ 0.5 is assumed and S T a is recomputed accordingly from the model expression showcasing a different modeling approach of flame/turbulence interactions;

• a constant laminar flame speed at the global equivalence ratio S l = S 0 l (φ glob ) is imposed instead of a flame surface averaged value addressing flame/droplet interactions and the role of the liquid phase in general;

• S T a is computed by setting Term III (temporal density variations) to zero to emulate a quasi-steady state;

• instead of variable values for the density ratio ρ u V / ρ b V , the corresponding values under nominal conditions are used to recompute the flame propagation speed targeting changes in the modeling of wall heat transfers.

While the analysis has outlined the relation between the different mechanisms, each of these changes is deliberately chosen to aim at one physical phenomenon at a time in order to estimate a priori a new value for S T a . Compared to the (unaltered) baseline simulation case, these new values for the flame propagation speed can then give an indication about the relative importance of each phenomenon. Such an a priori sensitivity study allows to establish a common methodology to compare the underlying mechanisms. Complementary studies with a posteriori computations are further referenced, where available, to assess the bias of the a priori estimations.

The resulting flame propagation speed for all four cases listed above are plotted in Fig. 25 as grey lines. The baseline case is added for reference (solid black line).

Starting with a constant β formulation, it shows the most substantial impact on S T a . With an approximately 50% higher flame surface averaged sub-grid scale wrinkling factor Ξ ∆ res , the resulting flame propagation speed is around two times higher than in the baseline case. Compared to a 20% increase of the flame propagation speed as published in Ref. [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF] (a posteriori ), this modification of Eq. 15 certainly overestimates the flame propagation speed, since the higher thrust generated by a constant flame surface wrinkling parameter is not counterbalanced by an increased outflux of burnt gases in Term II. As expected with deactivating Term III, S T a increases marginally (6% on average), since temporal density variations are shown to be comparably small (see Sec. 5.1). It should be emphasized that neglecting the last term of Eq. 15 is not equivalent to case ADIAB, because Terms I and II are intentionally left unchanged for the sake of this a priori estimation.

Finally, recomputing the absolute turbulent flame propagation speed based on the nominal density ratio yields a 19% higher flame propagation speed.

Interestingly, the predicted a priori value (S T a, a priori = 10.44 m/s) matches the averaged a posteriori value (given in Sec. 5.1) obtained from case ADIAB quite well.

The main conclusion that can be drawn from these a priori estimations is the fact that all major governing mechanisms of flame propagation tend to substantially impact the resulting absolute turbulent flame speed. It also supports the argument that each of the underlying physical phenomena requires careful consideration in terms of the employed model approach in LES, since any unsubstantiated simplifications may yield incorrect values for S T a or cause hidden error cancellation at worst. Moreover, it justifies the modeling choices made for the baseline case in this work, which addresses all relevant phenomena at once, following the recommendations of previous studies.

However, the a priori study has revealed a certain bias in terms of the predicted values, which has to be taken into consideration when estimating the impact of any phenomenon solely from an a priori perspective.

Conclusions

The present work explores the driving mechanisms of flame propagation during light-round and the role of physical modeling. Large-Eddy Simulations are performed in MICCA-Spray, an annular swirled spray-flame combustor with 16 fuel injectors fed by liquid n-heptane. Simulations are carried out with cold combustor walls at T w = 300 K, allowing to assess the effect of wall heat transfer on flame propagation. Such conditions are highly relevant for flame propagation in confined multi-burner configurations, since a strong sensitivity to wall temperatures is observed in the available experimental data. Yet this relationship has not been profoundly investigated. In a broader context, the role of physical models required to perform LES of light-rounds is analyzed. Specifically, the need for detailed models for the main governing mechanisms is highlighted.

The numerical setup follows previous works in terms of a dynamic closure for subgrid-scale flame surface wrinkling in the TFLES framework, polydisperse Lagrangian particle tracking, fuel injection and droplet evaporation. A tabulation approach is proposed for the wall model in order to overcome the limitations of standard logarithmic wall models and is capable of accounting for all variable properties in the boundary layer. Unlike previous related works that have concentrated on a single aspect at a time to isolate its effect on flame propagation (heat losses, two-phase flow and dynamic combustion modeling) without addressing the other aspects, the current work attempts to incorporate the prior findings in a single numerical setup in order to draw general conclusions with respect to the impact of each model. Moreover, error cancellation can be avoided through this procedure as outlined by Puggelli et al. [START_REF] Puggelli | Impact of dynamic modelling of the flame subgrid scale wrinkling in large-Eddy simulation of light-round in an annular combustor[END_REF].

The predicted light-round duration is found to be in fairly good agreement with available experimental data for cold-wall conditions, in particular when considering the uncertainties associated with the synchronization procedure. The expansion of burnt gases induces a thrust effect as primary driving mechanism of flame propagation, causing an acceleration of the flow in azimuthal direction. It also acts on the liquid phase by creating a heterogeneous droplet distribution upstream of the flame fronts: droplets accumulate in the wake of the swirled jets, which in turn affects the trajectory of the flame. A characteristic sawtooth trajectory is observed for the leading point.

Another consequence of the presence of liquid fuel is a diminished laminar flame speed S l . Furthermore, cold combustor walls also enhance the effect of heat losses on burnt gases, which are predominantly found further downstream in the burnt gas region, while the propagating flame fronts encounter almost adiabatic conditions. As a consequence, the burnt gas density increases and leads to a The study, which assembles various models for dynamic flame wrinkling, polydisperse spray description and wall heat losses, shows that neither corresponding physical mechanism outweighs the other ones and underscores the argument that each physical phenomenon must be accurately modeled.

Variations in other modeling components have not been considered such as chemistry or subgrid-scale fluxes and could be investigated in future works.

In particular, the very first instants of the initial stage of ignition that is only briefly covered on purpose, is considered to be sensitive to detailed chemistry effects. Moreover, pre-heating combustor walls can presumably alleviate the characteristic sawtooth trajectory of the leading point by enhanced droplet evaporation, which may explain shorter light-round durations observed experimentally. This aspect is the subject of ongoing research.

  limiting their modeling changes to one aspect per case. Despite this, no study has to date included all these findings in light-round simulations for a comprehensive a posteriori analysis of the governing mechanisms of flame propagation during light-round. The goal of this work is therefore threefold: our objective is first to characterize the main governing mechanisms during light-round with liquid fuels and cold combustor walls by means of LES. For the first time detailed numerical models recently identified in the literature are combined: dynamic combustion modeling, polydisperse Euler-Lagrange formalism and advanced wall-modeling. The cold-wall case enhances the effect of wall heat transfer. It is therefore the subject of the present study. Second, a mathematical loworder model for turbulent flame propagation during light-round is derived. Links are established between the governing mechanisms and the expressions in the model on which they act. Ultimately, flame speeds are estimated for different modeling assumptions, which are plugged into the model to assess their respective effect on the flame propagation speed.

Figure 1

 1 Figure 1: a) Setup of the MICCA-Spray test rig at the EM2C laboratory [21, 35]. A simplified sketch of the injector assembly marked by the dashed rectangle is given in Fig. b). Figure c) shows the sector numbering, the flame propagation directions in each half of the chamber (thick arrows) and the swirl orientation (thin arrows). The spark plug is mounted in sector S0. The ignition of sector S-1 is used as reference to compare experimental and numerical light-round durations.

Figure 2 :

 2 Figure 2: Droplet diameter distribution measured experimentally by PDA (before evaporation correction) and sampled in precursor LES at a radius of r = 5 mm from the axis of rotation and x = 5 mm above the chamber backplane. Experimental data kindly provided by G. Vignat based on the methodology described in[START_REF] Vignat | A Joint Experimental and LES Characterization of the Liquid Fuel Spray in a Swirl Injector[END_REF] and applied to the same burner.

Figure 3 :

 3 Figure 3: Illustration of matching points (red off-wall nodes) providing the input variables for the tabulated wall model.

τ

  wall [kg/(ms -2 )]

Figure 4 :

 4 Figure 4: Tabulated wall heat flux q wall and wall shear stress τ wall at an arbitrarily chosen matching point distance δ MP = 0.4 mm.

Figure 5 :

 5 Figure 5: Azimuthal mesh cut of two adjacent sectors transformed into a rectangular system.

Figure 5

 5 Figure5shows a mesh cut at R = 0.175 m of the lower part of the combustion chamber clipped to two injectors. The full three-dimensional numerical domain (see Fig.1) comprises the air plenum, all 16 swirlers and the entire combustion chamber. For numerical reasons, the upper part of the combustion chamber is immersed in a larger cylinder representing the atmosphere around the combustor (not shown in Fig.5) and extends 60 cm beyond the chamber outlet. Typical mesh sizes in the flame region are of the order of 0.2 mm (at the combustor backplane) to 0.5 mm (at a height of 35 mm above the combustor backplane). The entire mesh contains 320 M thetrahedral cells.

Figure 6 :

 6 Figure 6: Line of sight integration of heat release rate in LES for different instants after transforming the chamber from a cylindrical into a rectangular system. Sectors are labeled according to Fig. 1(c). Image c) (here) illustrates the sector volume considered for heat release rate integration as discussed in Sec. 4.2: solid line: chamber clipped at a height of 60 mm; dashed line: full chamber height.

Figure 6 :

 6 Figure 6: Line of sight integration of heat release rate in LES for different instants after transforming the chamber from a cylindrical into a rectangular system (cont.).

  .e. the heat release peak in sector S -1 and the instant the propagating flame fronts start to overlap. Limiting the volume integration up to 60 mm aims at focussing on the attached flame at the burner while excluding heat release contributions from above and neighboring sectors (cf. solid line box in Fig.6(c)). Performing the volume integral over the full chamber height instead results in a delayed peak in sector S -1 as visualized by the

Figure 7 :

 7 Figure 7: Volume integrated heat release rate (gray shaded area) per sector in the negative half without clipping (dashed line, first sector only) and clipped to a height of 60 mm (solid lines). The light-round duration τ LR corresponds to the difference between the peak heat release in sector S -1 and flame front merging (vertical dash-dotted lines). The uncertainty range associated with the definition of the synchronization point corresponds to the peak shift between the thick solid and the dashed line.

Figure 8 :

 8 Figure 8: Azimuthal velocity u θ on an unwrapped cylindrical cut-plane at R = 0.175 m. Dark shades correspond to high azimuthal velocities (from left to right and vice versa). Solid contour lines of iso-values of the progress variable (c = 0.76) indicate the instantaneous flame position. Axial velocity profiles are plotted along the dashed vertical line in Fig. 9.

Figure 9 :

 9 Fig.1(c) and applies henceforth to all following unwrapped cutplanes.It is worth noting that the thrust effect can be observed several sectors

Figure 10 :

 10 Figure 10: Liquid volume fraction α l on an unwrapped cylindrical cut-plane at R = 0.175 m. Orange contour lines correspond to iso-values of the progress variable of c = 0.76.

Figure 11 :Figure 12 :

 1112 Figure 11: Total equivalence ratio φ tot on an unwrapped cylindrical cut-plane at R = 0.175 m. Contour lines correspond to iso-values of the progress variable of c = 0.76.

Figure 13 :

 13 Figure 13: Evolution of the volume-integrated heat release rate (HRR) for the baseline case (solid line) and adiabatic case (dashed line). In the baseline case, total wall heat fluxes Q are plotted as gray line in absolute values (left axis) and in percent of the nominal thermal power P th of combustor (right axis). Diamonds mark the instant at which the propagating flame fronts start to merge in each case.

Figure 14 :

 14 Figure 14: Heat loss index I HL on an unwrapped cylindrical cut-plane at R = 0.175 m. Unity values mark fully adiabatic zones, zero marks the highest local heat loss possible. Solid contour lines correspond to iso-values of the progress variable of c = 0.76. Dashed contours limit zones where I HL ≤ 0.7.

Figure 15 :

 15 Figure15: Density ratio (proportional to generated thrust) for the baseline case (solid line) and ADIAB (dash-dotted line). For reference, the nominal density ratio of adiabatic burnt gases at the global equivalence ratio φ glob = 0.89 and fresh gases is added as dashed line.

Figure 16 :

 16 Figure 16: Flame surface wrinkling parameter β.

Figure 17 :

 17 Figure 17: Distribution of flame surface wrinkling parameter β.

Figure 18 :

 18 Figure 18: Decomposition of Eq. 15 into three main terms (thin lines) plotted as absolute values for both cases. The absolute turbulent flame speed S T a (thick line) is added for reference. The horizontal dashed grey lines correspond to the average velocity obtained from the sector arc length and the averaged elapsed time between two adjacent sector heat release peaks of the respective case (see Fig. 7 for the baseline case).

  1(c)) and is fairly well retrieved by the modeled velocity suggesting that the major physical phenomena involved in flame propagation during light-round are accurately captured. The same holds true for case ADIAB, which yields S T a, HR = 9.98 m/s. Equation 15 may be split into its three main parts (indicated by curly braces) to identify the leading terms. It is recalled that Term I resembles the classical expression for a turbulent spherically propagating flame or a developing flame in a closed duct, Term II arises due to the fact that the control volume (i.e. the combustion chamber) is semi-confined and Term III appears as a consequence of temporal density variations. Each of these terms is plotted in absolute values in Fig. 18 (black dashed lines) and reveals an interesting result. Essentially, Eq. 15 is governed by Terms I and II, while the effect of temporal density variations (Term III) appears to be almost negli-

Figure 19 :

 19 Figure 19: Gaseous equivalence ratio φ on an unwrapped cylindrical cut-plane at R = 0.175 m. The one-dimensional consumption speed S 1D c is computed for Fig. 21 at the four labeled positions at the flame front. Contour lines correspond to iso-values of the progress variable of c = 0.76.

Figure 20 :

 20 Figure 20: Flame consumption speeds averaged over the total flame front surface (S l ) for case BASE (solid line) and ADIAB (dash-dotted line). The dotted grey line corresponds to the theoretical laminar flame speed at global equivalence ratio S 0 l (φ glob ). The consumption speed averaged over the resolved flame surface is given as dashed line for case BASE.

Figure 21 :

 21 Figure 21: Laminar flame speed diagram a gaseous n-heptane/air flame at varying

Figure 22 :

 22 Figure 22: Histogram of gaseous equivalence ratio φ on a characteristic iso-surface of the progress variable c = 0.76.

Figure 23 :

 23 Figure 23: Area averaged azimuthal velocity u θ sampled across the chamber cross section at the position indicated by a thick line in the inset plot. Filled burner is ignited first.

Figure 24 :

 24 Figure 24: Resolved flame surface wrinkling Ξ res for case BASE (solid line) and ADIAB (dashed line). The sub-grid scale flame surface wrinkling Ξ ∆ res averaged over the resolved surface is virtually the same for both cases (dash-dotted and dotted lines).

Figure 25 :

 25 Figure 25: Impact of modeling choices: ( ): S T a of baseline case; ( ): S T a for β ≡ 0.5; ( ): S T a for S l = S 0 l (φ glob ); ( ): S T a with Term III set to zero; ( ): S T a for nominal density ratio.

  lower generated thrust compared to light-round ignition with adiabatic walls. The flame surface wrinkling parameter β is briefly examined and shows a heterogeneous distribution consistent with previous research (low values across the propagating flame fronts versus higher values at the stabilized flames). The use of a dynamic evolution of the wrinkling parameter must therefore be preferred as no universal value exists which would result in the same flame surface wrinkling and thus the same flame propagation speed. The second part of this study is concerned with a mathematical expression to model the absolute turbulent flame propagation speed during light-round and accurately predicts averaged values for both cases under consideration (baseline and adiabatic). The equation is decomposed to study the governing mechanisms in detail and relate them to the identified terms in the model equation. Lastly, the expression for turbulent flame propagation is modified to estimate from an a priori perspective to what extent the absolute turbulent flame speed may change with different modeling choices.

Table 1 :

 1 Light-round durations of four experimental runs

	The inherent ambiguity for all definitions of a synchronization point be-
	tween experiment and simulation clearly deserves further discussion. An
	uncertainty range for the numerical light-round duration is worth defining,
	rather than considering it as an exact measure. Reasonable definitions (num-
	bered (i) to (iii)) can be based for example on a specific value of the heat

[START_REF] Lefebvre | Gas Turbine Combustion: Alternative Fuels and Emissions[END_REF]

. A complementary simulation (ADIAB) carried out using adiabatic wall boundary conditions only, is already listed here, but discussed later. Given the uncertainty associated with the definition of τ LR , the predicted duration (baseline case) agrees fairly well with the experimental data, if case EXP4 is considered as an outlier.

Acknowledgments

This project has received funding from the European Union's Horizon 2020 research and innovation program [grant number 765998] (ANNULIGHT). HPC resources of the GENCI allocations [grant numbers A0082B10118, A0082B10159] and the Mésocentre computing cluster of CentraleSupélec and École Normale Supérieure Paris-Saclay supported by CNRS and Région Îlede-France are gratefully acknowledged. The authors would like to show their gratitude to Dr. G. Vignat, Dr. D. Durox and Prof. S. Candel for the fruitful discussions and performing detailed experimental measurements of the droplet spray used for calibration in this work. We would also like to thank Prof. D. Veynante for sharing his unparalleled knowledge and his invaluable advice on the subject of dynamic combustion modeling. The assistance provided by Dr. E. Riber and Dr. B. Cuenot from CERFACS with the initial Euler-Lagrange setup in AVBP is greatly appreciated.