Ph.D Bruno Renou
email: renou@coria.fr

Marie-Eve Clavel

Marie-Eve Clavel

Alexis Vandel

PhD Vincent Modica

PhD Zheng Chen

PhD Emilien Varea

PhD Vincent Moureau

M-E Clavel

Determination of spatially averaged consumption speed from spherical expanding flame: a new experimental methodology

Keywords: Laminar burning velocity, spherically expanding flames, Stretch effects, consumption speed, Displacement speed Laminar burning velocity, Spherically expanding flames, Stretch effects, Consumption speed, Displacement speed

Facilities using spherically expanding flame in closed vessels are generally used to report laminar burning velocities from extrapolated stretched flame speed. Considering the 1D planar reference case, flame displacement speed and consumption speed are rigorously identical. This is not true considering spherically expanding flames. While the flame displacement speeds are defined from a kinematic point of view, the consumption speed is linked to the integral of the reaction rate across the flame front. The latter is defined from a kinetic reference frame. Even though spherically expanding flames have been studied for decades, it is still challenging to experimentally determine the consumption speed. Recent developments define analytical expressions for the consumption speed and suggest experimental determination. One of the major issues consists in measuring the temporal evolution of the fresh gas density while the flame expands. In this work, an optical PIV-based method is developed to directly determine the fresh gas density ahead of the flame front and then the spatially averaged consumption speed. For the first time, experimental measurements of flame displacement speeds relative to fresh gases and burned gases, and spatially averaged flame consumption speeds are then reported for methane/air flames at different equivalent ratios. Experimental measurements are systematically compared with DNS which simulates the experimental geometry. A very good agreement is observed between experimental and numerical data irrespective to the equivalence ratio, underlying the robustness of the experimental methodologies. Finally, the sensitivity of the different flame speeds to flow confinement and extrapolation models is discussed.

spherically expanding flames. While the flame displacement speeds are defined from a kinematic point of view, the consumption speed is linked to the integral of the reaction rate across the flame front. The latter is defined from a kinetic reference frame. Even though spherically expanding flames have been studied for decades, it is still challenging to experimentally determine the consumption speed. Recent developments define analytical expressions for the consumption speed and suggest experimental determination. One of the major issues consists in measuring the temporal evolution of the fresh gas density while the flame expands. In this work, an optical PIV-based method is developed to directly determine the fresh gas density ahead of the flame front and then the spatially averaged consumption speed. For the first time, experimental measurements of flame displacement speeds relative to fresh gases and burned gases, and spatially averaged flame consumption speeds are then reported for Preprint submitted to Journal of L A T E X Templates August 25, 2021

Introduction

The development of new kinetic schemes requires efficient validation on ignition delays time, major and/or minor species profiles and unstretched laminar burning velocity S 0 L data. For 1D unstretched planar premixed flame, S 0 L represents the rate at which the fresh gases are consumed through the flame 5 front. By integrating the transport equation of the fuel mass fraction through the flame domain, one can derive the expression of the laminar burning velocity for 1D planar flames [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF],

S 0 L = 1 ρ u (Y F,b -Y F,u ) +∞ -∞ ωF dx (1)
where ρ u is the fresh gas density, ωF is the fuel reaction rate and Y F,b and Y F,u are the fuel mass fractions in burned and fresh gases, respectively. The 10 upper script 0 refers to the 1D and unstretched conditions. From Eq. ( 1), the laminar burning velocity S 0 L is equivalent to a consumption speed and corresponds to the fuel mass rate which enters the flame front. This expression of the laminar burning velocity is valid for any major species k. For the 1D planar configuration, the laminar burning velocity can also be defined as a kinematic 15 quantity which is called the density weighted flame displacement speed [START_REF] Giannakopoulos | Consistent definitions of "flame displacement speed" and 720 "markstein length" for premixed flame propagation[END_REF],

S 0 d = ρ ρ u (S f -u) • n = S 0 L (2) 2
The density-weighted displacement speed is defined to a specified temperature iso-surface. The temperature gradient defines the unit normal n to the flame surface, with n pointing towards the fresh reactants. S f = S f • n is the absolute flame speed or propagation speed, and u = u • n is the local 20 flow velocity normal to the temperature iso-surface. ρ is the local flow density obtained at the flame iso-surface where u is defined.

Laminar burning velocity is one of the few fuel parameters which can be experimentally measured at relevant conditions for engines or combustion chamber applications [START_REF] Egolfopoulos | Advances and challenges in laminar flame experiments and implications for combustion chemistry[END_REF]. Among the different techniques, experimental facility 25 using spherically expanding flame is generally chosen as it offers flexibility in terms of initial conditions.

Several methods have been developed to measure flame speeds from expanding flames in confined vessels. They should be considered as complementary.

30

The constant volume method is quite old and makes it possible to obtain flame speed values for high pressure and temperature conditions from the time analysis of the pressure signal in the combustion chamber and from a thermodynamic model for the mass fraction of burnt gases. Recently, experimental and numerical works have provided original solutions to overcome [START_REF] Sikes | An experimental study: laminar flame speed sensitivity from spherical flames in stoichiometric ch4-air 825 mixtures[END_REF] these models and demonstrated the potential of this method [START_REF] Hinton | Laminar burning velocity measurements of methane and carbon dioxide mixtures (biogas) over wide ranging temperatures and pressures[END_REF][START_REF] Xiouris | Laminar flame 730 speeds under engine-relevant conditions: Uncertainty quantification and minimization in spherically expanding flame experiments[END_REF][START_REF] Halter | Development of an optically accessible apparatus to 735 characterize the evolution of spherically expanding flames under constant volume conditions[END_REF].

The constant pressure method is widely used for a broad range of operating conditions (including low pressure) and fuel compositions. For this approach, spherical expanding flames are stretched due to the interaction between flame propagation and flow, and S 0 L is reported using extrapolation procedures to zero 40 stretch. Discussions about linear and non-linear models can be found in [START_REF] Kelley | Premixed flame propagation in a confining vessel with weak pressure rise[END_REF] for instance, and effects of finite flame thickness in extrapolation models in [START_REF] Liang | Extrapolation of laminar flame speeds from 740 stretched flames: Role of finite flame thickness[END_REF][START_REF] Shu | Extrapolations of laminar flame speeds from expanding spherical flames based on the finite-structure stretched flames[END_REF].

In the following, all the developments concern the constant pressure method.

In constant pressure method, unstretched laminar burning velocity is commonly

S d,b = ρ b ρ u (S f -u b ) • n (3)
This equation is systematically simplified into,

S d,b ρ eq b ρ u dR dt (4)
where ρ eq b is the burned gas density at adiabatic thermodynamic and chemical equilibrium conditions. dR/dt is the absolute flame speed, S f , deduced from the time derivative of the flame radius, R. R is experimentally obtained from high-speed schlieren, shadowgraphy or tomographic recordings. Note that 50 the burned gases are assumed to be at rest (u b = 0). So, in these conditions, Eq. ( 4) becomes the easiest way to experimentally determine the laminar burning velocity. However, the hypothesis required to reach this formulation are sometimes not fully verified, involving many biases on the S 0 L determination. Indeed, for small combustion chamber [START_REF] Burke | Effect of cylindrical confinement on the determination of laminar flame speeds using outwardly propagating flames[END_REF], for low pressure [START_REF] Faghih | On the determination of laminar flame speed from low-pressure and super-adiabatic propagating 750 spherical flames[END_REF] and when 55 radiation is non-negligible [START_REF] Chen | Effects of radiation and compression on propagating spherical flames of methane/air mixtures near the lean flammability limit[END_REF], these approximations fail and other experimental approaches are necessary.

Recent developments in PIV post-processing [START_REF] Balusamy | Direct measurement of local 755 instantaneous laminar burning velocity by a new piv algorithm[END_REF][START_REF] Varea | Measurement of laminar burning velocity and markstein length relative to fresh gases using a new postprocessing procedure: Application to laminar spherical flames for 760 methane, ethanol and isooctane/air mixtures[END_REF][START_REF] Bradley | Flame speed and particle image velocimetry measurements of laminar burning velocities and markstein numbers of some hydrocarbons[END_REF] allow to directly measure the local fresh gas velocity, u u , ahead of the flame front. This region is characterized by high local velocity gradient, which makes the measurements 60 very difficult to perform. New algorithms were developed and the density weighted displacement speed relative to the fresh gases, S d,u , can be now experimentally determined,

S d,u = dR dt -u u (5)
The fresh gas velocity u u is approximated experimentally by the maximum of fresh gas velocity which occurs at the entrance of the preheat zone. S d,u allows 65 a direct measurement of laminar flame speed, without assumptions on the state of the burned gases, and on the velocity field. This approach must be considered to report laminar flame speed when conditions required to use Eq. 4 fail.

Consequently, the two kinematic approaches (Eqs. Marvin [START_REF] Fiock | The measurements of flame speed[END_REF], Bradley and Mitcheson [START_REF] Bradley | Mathematical solutions for explosions in spherical vessels[END_REF] or more recently Bonhomme et al. [START_REF] Bonhomme | Curvature and confinement effects for flame speed measurements in laminar spherical and cylindrical flames[END_REF] developed theoretical approaches for the spatially averaged consumption speed of the deficient species. Lefebvre et al. [START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF] recently developed a generalized expression for any major species k. However, the authors concluded that only 85 the equation of deficient species can be experimentally used since it does not require the determination of ambiguous burned gas parameters such as ρ b or the mass fraction of species k in the burned gases. These parameters are not necessarily spatially homogeneous and are almost difficult to measure. The expression of the spatially averaged consumption speed for the deficient species 90 k is [START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF],

S c = S c,k = S f - (R 3 0 -R 3 f,k ) 3R 2 f,k 1 ρ u dρ u dt = - 1 ρ u Y k,u R 2 f,k R0 0 ωk r 2 dr (6)
where R f,k is an equivalent flame radius based on the integral of a progress variable based on the partial density of species k and the radius. R f,k is given by

R f,k = 3 R0 0 1 - ρY k ρ u Y k,u 3r 2 dr (7)
In this equation, R 0 is the combustion chamber radius. Consequently, tomography [START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF]. This spatially averaged consumption speed is therefore not exactly equal to the local consumption speed (Eq.1) [START_REF] Giannakopoulos | Consumption and displacement speeds of stretched premixed flames -theory and simulations[END_REF], but it represents a convenient way to experimentally report the global fuel consumption rate, from scalar measurements. A promising method to measure the spatially averaged 105 consumption speed in a closed vessel was presented in the numerical work of Bonhomme et al. [START_REF] Bonhomme | Curvature and confinement effects for flame speed measurements in laminar spherical and cylindrical flames[END_REF]. The authors proposed to combine both the temporal evolution of the flame radius R f (t) and the internal chamber pressure P (t) assuming an isentropic compression for the fresh gases, as

1 ρ u dρ u dt = 1 γ u P dP dt (8)
The spatially averaged consumption speed yields 110

S c = dR f dt - (R 3 0 -R 3 f ) 3R 2 f 1 γ u P dP dt (9)
Prima facie, Eq. 9 seems to be a good candidate for the evaluation of the consumption speed as it only requires the temporal evolution of the flame radius and the pressure signal P (t). However, from an experimental point of view, the values of the consumption speed could lack accuracy since this formula is based on the difference of two large numbers. This was originally mentioned by Andrews and Bradley [START_REF] Andrews | Determination of burning velocities: A critical 780 review[END_REF]. Therefore, the experimental accuracy of the pressure measurement must be excellent to avoid any noise during the time derivative computation of the pressure signal. It implies that the pressure sensor has to be accurate enough during the early stages of flame development (pre-pressure period), where the pressure rise is very small (in the range of 120 hundreds Pascal) and at the same time, the pressure sensor must also be capable of withstanding high pressure loads due to the combustion process in a constant volume chamber. From these observations, the consumption speed can therefore be estimated from Eq. ( 6) where the fresh gas density needs to be evaluated, or from Eq. ( 9) where the pressure trace needs to be accurately 125 recorded.

The aim of this study is therefore three-fold:

• First, a new experimental approach to estimate the instantaneous density field ahead of the flame front is developed and used to report the 130 spatially averaged consumption speed for spherical expanding flames.

It is based from the mass conservation written for a material control volume, and requires the measurement of the local fresh gas velocity.

This methodology is validated by comparing DNS simulations of laminar spherically expanding flame and experimental data for methane/air 135 mixtures (lean, rich and stoichiometric).

• The experimental methodology initially developed by [START_REF] Varea | Measurement of laminar burning velocity and markstein length relative to fresh gases using a new postprocessing procedure: Application to laminar spherical flames for 760 methane, ethanol and isooctane/air mixtures[END_REF] to report fresh gas velocity profile ahead of the flame front is reconsidered and improved, to provide more accurate data. An extensive work of optimization of post-processing routines, camera properties and laser frequencies are 140 conducted on classical methane/air mixtures at atmospheric conditions.

A direct comparison with DNS data is also performed to assess the quality of the post-processing steps.

• Finally, the suitability of the spatially averaged consumption speed formulation ( S c ) and kinematic approaches ( S d,u and S d,b ) for lean, rich 145 and stoichiometric methane/air flames is discussed from both experimental and numerical data.

Experimental and numerical apparatus

In this section, the experimental setup for flame speed measurements is briefly described. Details on the numerical simulations performed with the DNS 150 code (A-SURF) are also presented.

Experimental setup

The apparatus and post-processing methods are described in detail in [START_REF] Varea | Measurement of laminar burning velocity and markstein length relative to fresh gases using a new postprocessing procedure: Application to laminar spherical flames for 760 methane, ethanol and isooctane/air mixtures[END_REF][START_REF] Varea | Pressure effects on laminar burning velocities and markstein lengths for isooctane-ethanol-air mixtures[END_REF] and are briefly reminded. Experiments are conducted in a spherical stainless steel combustion chamber with an equivalent chamber radius R 0 = 8.243 cm.

155

The mixture is seeded with micronical silicon oil droplets with a theoretical boiling temperature of T = 580K. High-speed laser tomography technique is used to capture the temporal evolution of the flame front and particle images by using a circle least-square fitting method. A double cavity Nd:YLF laser atm and a temperature of 293K.

Numerical simulation with A-SURF code

175 Spherical flame propagation is simulated using the DNS code A-SURF.

The simulation domain reproduces the experimental apparatus described above in terms of chamber geometry and thermodynamical properties. A-SURF solves the one-dimensional, unsteady, compressible, conservation equations for a multi-component reactive flow in spherical coordinates using the finite volume 180 method. The details on governing equations, numerical schemes, and code validation can be found in [START_REF] Chen | Effects of lewis number and ignition energy 785 on the determination of laminar flame speed using propagating spherical flames[END_REF]. Simulations are performed for CH 4 /air flames with the GRI-mech 3.0 kinetic scheme [START_REF] Smith | Gri-mech homepage[END_REF]. It is noted there are several recent kinetic models for methane oxidation. According to Wang et al. [START_REF] Wang | Laminar flame speeds of methane/air mixtures at engine conditions: Performance of different kinetic models and power-law correlations[END_REF], there is good agreement among laminar flame speeds predicted by different mechanisms 185 for ambient conditions and large discrepancy appears only for engine-relevant conditions with pressure above 20 atm. Since here we consider the ambient conditions for which GRI-mech 3.0 was optimized, GRI-mech 3.0 is used in our simulations. Initial conditions are atmospheric thermodynamic conditions, i.e. a temperature of 293 K and a pressure of 1 atm. In all the simulations, 190 the mixture is initially static. In order to maintain adequate resolution of the propagating flame, dynamically adaptive mesh is utilized in A-SURF with a finest grid size of 8 µm. It is noted that radiation heat loss is not considered here. Since the equivalence ratios are not close to the flammability limits of methane/air mixtures and the corresponding laminar flame speeds are above 15 195 cm/s, radiation heat loss has negligible effect on propagating spherical flames whose radii are below 2 cm [START_REF] Yu | Radiation-induced uncertainty in laminar flame speed measured from propagating spherical flames[END_REF]. This DNS database will be used for a direct and blind comparison with the different laminar flame speeds reported from the experiments, and the validation of the post-processing routines as well.

Unstretched laminar burning velocities are computed by Cantera [START_REF] Goodwin | Cantera: An 800 object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes[END_REF] from

200
the one-dimensional freely-propagating configuration, by using the same kinetic scheme [START_REF] Smith | Gri-mech homepage[END_REF] and the same operating conditions (T = 293 K, P = 1 atm).

In these simulations, the mixture-averaged diffusion model is considered. The computational domain is from 0 to 5.0 cm. The minimum values of the gradient and curvature for the last iteration are set to 0.02. ) obtained from a sub-pixel fit (7 points Gaussian) gives the fresh gas velocity u ahead of the flame front at the location of the first IW, so that

u(t * , ∆r 2 ) = R T (t * + ∆t) -R T (t * ) ∆t + ∆r 2 -∆r opti 1 (t * ) ∆t ( 10
)
where R(t * ) and R(t * + ∆t) are the flame radii at time t * and t * + ∆t, ∆t is the time between two successive images and ∆r opti Original details of the procedure can be found in [START_REF] Varea | Measurement of laminar burning velocity and markstein length relative to fresh gases using a new postprocessing procedure: Application to laminar spherical flames for 760 methane, ethanol and isooctane/air mixtures[END_REF]. There is a compromise to find on the thick of the interrogation window (IW) in the ROI. The thicker the IW, the better the correlation since much more droplets will be included in the correlation. But the spatial filtering of the velocity profiles will be more 235 important and the maximum of the velocity profile would not be captured. We did several tests, and the conclusions give that the compromise is at 2 pixels.

It is worth noting that the first term in the right hand side of Eq. 10 is equal to the flame propagation speed S f • n approximated by finite difference.

Improvement and validation of the experimental methodology 240

The availability of the DNS database is a good opportunity to revisit and optimize the post-processing routines which were originally developed in [START_REF] Varea | Measurement of laminar burning velocity and markstein length relative to fresh gases using a new postprocessing procedure: Application to laminar spherical flames for 760 methane, ethanol and isooctane/air mixtures[END_REF]. First, we address the question of the tomographic image post-processing, since the image quality impacts strongly the contour detection and correlation

processes. An automatic contrast adjustment procedure of tomographic images 245 is performed by using a linear stretching of intensity range. At the same time, laser sheet in-homogeneity is corrected by using temporally averaged seeded tomographic recordings (without flames). Consequently, black and white are saturated to make flame front detection easier and particles more identifiable.

In a second step, the interval time between two-successive images must be From the last improvements which have been discussed previously, some numerical and experimental profiles of the fresh gas velocity are reported in Figure 3 for two different equivalence ratios and two different flame radii. The experimental velocity data, u(r), in the decreasing zone (green zone) is 280 fitted with the theoretical model suggested by Poinsot et al. [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF], following a (R 3 0 -r 3 )/r 2 shape. Simultaneously, the experimental and numerical radii at the maximum fresh gas velocity are reported for a quantitative comparison between experiments and simulations. This figure highlights the capability of the experimental methodology to report the gas velocity profiles both for the 285 fresh gas side and in the preheat zone up to the evaporation of oil droplets accurately. The spatial resolution of the measurements is high enough to capture the maximum of the gas velocity which occurs at the entrance of the preheat zone. The interval times in two successive images have been adapted such as the particle displacement at the maximum velocity profiles remains close to 3 290 pixels, as suggested above. One can notice also the perfect agreement of the experimental profiles with the numerical data.

Regarding Fig. 3, it is clear that the maximum of fresh gas velocity doesn't peak exactly at T u but at a slightly higher temperature. Numerical data analysis shows that this temperature is around 310 K and remains almost constant as 295 the flame propagates. This is very similar to the results presented in [START_REF] Jayachandran | Determination of laminar flame speeds using stagnation and spherically expanding flames: Molecular 805 transport and radiation effects[END_REF].

Consequently, the experimental displacement speed corresponds to

S Exp d,u = dR dt -u g,max (11)
where u g,max is the velocity at which the fresh gas velocity is maximum. We assume implicitly equality between the definition of S d,u from Eq. 5 and S Exp d,u

from Eq. 11. This observation underlines also the difficulty to select the fresh 300 gases iso-surface when considering the displacement speed relative to the fresh gases as it was originally discussed by Gianakopoulos et al. [START_REF] Giannakopoulos | Consistent definitions of "flame displacement speed" and 720 "markstein length" for premixed flame propagation[END_REF] and Jayachandran et al. [START_REF] Jayachandran | Determination of laminar flame speeds using stagnation and spherically expanding flames: Molecular 805 transport and radiation effects[END_REF]. The choice of the fresh gas temperature as a reference location for the determination of S ( d, u) is poorly conditioned and should be used with care. However, even with this limitation, this approach must be considered as Jayachandran et al. [START_REF] Jayachandran | A study of propagation of spherically expanding and counterflow laminar flames using direct measurements and numerical simulations[END_REF] and Faghih et al. [START_REF] Faghih | On the determination of laminar flame speed from low-pressure and super-adiabatic propagating 750 spherical flames[END_REF]. It is worth noting that this method is also limited by the flame thickness. The thinner the flame front, the 310 more difficult it becomes to capture and spatially resolve the maximum velocity.

Consequently, this approach is not suited for high pressure conditions but it is rather well adapted for low pressure conditions (large flame thickness), when the assumptions used to apply Eq. 4 fail.

d dt Vm(t) ρdV = 0 (12)
According to the transport theorem, an arbitrary control volume V a (t) which coincides with V m (t) at time t but doesn't necessarily move at the fluid velocity can be introduced and Eq. ( 12) is

335 d dt Vm(t) ρ dV = d dt Va(t) ρ dV + Aa(t) ρ u -Ẋ .ndA ( 13
)
where Ẋ is the local velocity of the area A a (t) bordering the volume V a (t) and u is the fluid velocity at the location X. Using an arbitrary volume is more convenient in experiments since the boundaries of the surface can be well selected. The sketch in Fig. 1 shows a spherical volume (only a quarter is represented) which uniquely contains fresh gases from a radius R = X to the 340 vessel wall at R = R 0 . Note that X is adjusted so that the distance d to the flame front is kept constant at each time step. Integrating Eq. ( 13) between time t n and t n+1 yields

ρ dV n+1 - ρ dV n t n+1 -t n - Aa(t n+1/2 ) ρ u -Ẋ dA = 0 ( 14
)
Note that only the radial component is considered as the geometry is spherical. Assuming that the density at time step n + 1/2 is the average density 345 between time t = n and t = n + 1, one can estimate the temporal fresh gas density evolution at each time step as

ρ n+1 u = ρ n u R 3 0 -(X n ) 3 3∆t + 1 2 X n+ 1 2 2 u n+ 1 2 -Ẋn+ 1 2 R 3 0 -(X n+1 ) 3 3∆t -1 2 X n+ 1 2 2 u n+ 1 2 -Ẋn+ 1 2 (15) with X n+ 1 2 = X n+1 +X n 2 and Ẋn+ 1 2 = X n+1 -X n ∆t
. This equation requires the initial fresh gas density ρ 0 u , the position X of the area A a (t) at each time step and the fresh gas velocity u through the area A a (t) between two consecutive 350 time steps. This later is available unambiguously thanks to the developments presented above. The experimental method for determining these parameters as well as its validation are presented in the next section.

Validation from DNS data

Direct numerical simulations of CH 4 /air flames at atmospheric pressure and 355 temperature are used to access the time evolution of the fresh gas density. It is worth noting that the exact same methodologies, i.e. post processing techniques, are used for both experimental and numerical data. This induces consistency for comparisons. From the numerical simulations, the temporal fresh gas density evolution is evaluated using Eq. ( 15). This equation written in a discrete sense is 360 computed with the position X which corresponds to the edge of a control volume and the velocity u of the fresh gases at this specific position X. Moreover, from the simulations, the fresh gas density is also directly available. Therefore, one can evaluate the suitability of the Eq. ( 15) to report the fresh gas density, directly from the DNS. The results are shown in Figure 4 (Top). Note that the 365 variables X and u have been computed from DNS data at an acquisition rate equal to the experimental one, i.e. 5 kHz. A perfect agreement between the fresh gas density from the simulations and the one calculated using Eq. ( 15) from the simulations are reported. This result confirms the reliability of Eq. ( 15) to accurately determine the fresh gas density ahead of the flame front. Moreover, 370 an acquisition rate of 5 kHz seems to be sufficient enough to describe well the temporal evolution of the fresh gas density. In Eq. 15, the parameter X is defined according to the flame front position so that X = R T + d, where d is a constant. The distance d should be set to avoid the influence of the preheat zone, so that the velocity profile displays a decreasing behavior, following a For the experimental determination of ρ u , the distance d will be set to 70 pixels (3.3 mm). This value of d = 70 pixels is extracted from the velocity profile fit in Figure 3 and corresponds to the green fill diamond.

Application to experimental data and validation

Experimental data are processed to estimate the temporal evolution of the 385 fresh gas density from Eq. ( 15). The main difficulty remains in the ability to accurately measure the parameter X, the position of the area A a (t) at each time step, and u, the fresh gas velocity through the area A a (t) between two consecutive time steps. Both experimental and numerical data of the compression term 1/ρ u • dρ u /dt plotted against the flame radius are reported in 390 Fig. 5 for different equivalence ratio. Note that the experimental repeatability -shot-to-shot variation-is shown by plotting the 10 experimental trials. A perfect agreement is reported as both the experimental and numerical data do overlap. From the previous statements, both the repeatability of experimental data as well as the accuracy of the post-processing techniques to access the 395 compression term of Eq. ( 15) are demonstrated. From the thermodynamic perspective, the rate of change of the fresh gas density is linked to the rate of change of the pressure assuming isentropic compression, see Eq. [START_REF] Liang | Extrapolation of laminar flame speeds from 740 stretched flames: Role of finite flame thickness[END_REF]. Figure 5 reports also experimental data for the compression term. Data corresponding to the time derivative of the pressure trace -isentropic compression -are widely 400 dispersed. Therefore no trend of the rate of change of the pressure trace can be distinguished neither computed. The data lead to a high level of inaccuracy. It is worth noting that the accuracy of the pressure signal is responsible for this.

Since the increase of the pressure signal is a low as 3 percents for a typical flame (radius of 2 cm for a stoichiometric CH 4 /air flame), no pressure transducers 405 were found to deal with small signal detection, high frequency sampling, huge scaling range -up to several bar -and still affordable setup. Therefore, it clearly appears that the measure of the pressure signal for the evaluation of the compression term and the evaluation of Eq. ( 9) is not relevant enough. From an experimental point of view, there is no possibility to achieve a reliable evaluation 410 of the consumption speed assuming isentropic compression hypothesis when considering the constant pressure method in spherical expanding flames.

Experimental uncertainties on stretched flame speeds

Experimental uncertainties on the different stretched flame speeds are computed as follows. First, we need to consider the uncertainties relative 415 to the operating conditions, mainly the initial mixture temperature, pressure, equivalence ratio and spark energy. In addition to the intrinsic accuracy of each sensor, the initial conditions can be slightly different from one experiment to another. For this study, we consider a maximal variation of ±1.5 K, ±0.001 bar and ±1% for initial temperature, pressure and equivalence ratio, respectively. the circle fit based on the least-square method. The mean relative uncertainty over all the images is on the order of 0.03%. The uncertainty on the maximum fresh gas velocity u u includes the contribution of the sub-pixel fit of correlation peak, the uncertainty in the flame front position after dewarping, and the contribution of the six degree polynomial fit of the fresh gas velocity profile. The 435 mean relative uncertainty of u u over all the images is on the order of 0.5%. Once these uncertainties are evaluated, they are propagated when flame stretch and the different flame speeds are computed, according to methodology suggested by [START_REF] Gum | Guide to the expression of uncertainty in measurement[END_REF][START_REF] Hughes | Measurements and their Uncertainties. A practical guide to modern error analysis[END_REF]. Consequently, for each test, a one σ uncertainty is computed for each instantaneous stretched flame speed S, hereafter e(S), where S is either

S = k S k e 2 (S k ) k 1 e 2 (S k ) (16)
The corresponding variance is :

σ 2 (S) = 1 k 1 e 2 (S k ) (17)
Finally, an expanded uncertainty U (S) of the averaged speed S is calculated from this variance σ 2 (S), and a coverage factor k, calculated from a Student's law based on the number of value used on the average process (k > 10), and on 450 a level of confidence of 95%.

U (S) = kσ(S) (18)
Consequently, S will be plotted according to stretch with error bars equal to ±U (S), that encompass large fraction (95%) of the distribution of values that could reasonably be attributed to S. Obviously, the higher the number of flame recordings per condition, the lower the σ(S).

455

In the following sections, stretch flame speeds, S, will be fitted with linear or non-linear models using the least squares method to find the extrapolated flame speed and the corresponding Markstein length. The uncertainty on these two parameters is estimated from the covariance matrix of the fit and the dispersion of the residuals. All these data are reported in different tables below. R 0 [START_REF] Burke | Effect of cylindrical confinement on the determination of laminar flame speeds using outwardly propagating flames[END_REF][START_REF] Bradley | Burning velocities, markstein lengths, and flame quenching for spherical methane-air flames: A computational study[END_REF][START_REF] Chen | On the accuracy of laminar flame speeds measured from outwardly propagating spherical flames: Methane/air at normal temperature and pressure[END_REF][START_REF] Sikes | An experimental study: laminar flame speed sensitivity from spherical flames in stoichiometric ch4-air 825 mixtures[END_REF]. This range is represented by filled circles in Figure 6. A direct comparison between experimental stretched flame speed and DNS data is performed and a very good agreement is then observed for all the equivalence ratios. For rich and stoichiometric conditions, the difference is marginal, especially for flame kernels free from ignition effects. For lean condition, the 485 difference is larger and on the order of 1 cm/s. Ignition of very lean mixtures requires high amount of energy to get spherical kernels which may affect the displacement speed S d,b a little beyond the minimum flame radius of 1 cm.

For flame radii larger than 2 cm, the slope decreases progressively to become strongly negative. This confinement effect is visible both from the experiments 490 and the simulations. This blind comparison confirms the capability of the 1D spherical DNS with GRI3.0 kinetic scheme to reproduce the dynamic of the flame front. Linear extrapolation (Eq. 19) at zero stretch for lean and stoichiometric conditions, and a classical non-linear extrapolation (Eq. 20) for rich conditions are used as recommended by [START_REF] Kelley | Nonlinear effects in the extraction of laminar flame speeds from expanding spherical flames[END_REF]. A deviation of S 0 L from experiments with those from predicted by Cantera 500 simulation is observed for all the cases and it is about 2-3 cm/s (Table 1). This difference in S 0 L between experiments (based on outwardly propagating spherical flame) and simulation (based on unstretched, adiabatic, freely-propagating planar flame) is commonly observed in the literature ( [START_REF] Chen | On the accuracy of laminar flame speeds measured from outwardly propagating spherical flames: Methane/air at normal temperature and pressure[END_REF], Figure 3) and has been explained by many possible experimental biases: mixture preparation, 505 ignition effect, radiation effect, extrapolation model, confinement effect. More surprisingly, S 0 L values extrapolated from DNS data with the same range of flame stretch free from ignition and confinement effects are also lower than Cantera values, whereas the same kinetic scheme is used. Similar behaviors were observed previously for confined spherical expanding flames but never discussed.

495 S d,b = S 0 L -L b K (19)
S d,b S 0 L 2 ln S d,b S 0 L 2 = -2 L b K S 0 L (20)

510

For instance, stoichiometric methane/air flames were simulated by [START_REF] Faghih | On the determination of laminar flame speed from low-pressure and super-adiabatic propagating 750 spherical flames[END_REF] with GRI3.0 mechanism with larger radius combustion chamber. S 0 d,b was estimated at 268 cm/s, leading to S 0 L = 35.45 cm/s, which is below the target given by 1D planar flame calculation. Similar observations can be found in [37] and [START_REF] Beeckmann | 835 Flame propagation speed and markstein length of spherically expanding flames: Assessment of extrapolation and measurement techniques[END_REF] still with GRI3.0. In order to be consistent to the extrapolation methodology used for S d,b , the following linear extrapolation (Eq. 21) at zero stretch is used [START_REF] Giannakopoulos | Consistent definitions of "flame displacement speed" and 720 "markstein length" for premixed flame propagation[END_REF],

515 S 0 L [cm/s] σ S 0 L [cm/s] L b [mm] σ L b [mm] fit Cantera 18.49 - - - - φ = 0.
S d,u = S 0 L -L u K (21)
It is important to note that using traditional extrapolation formula for 535 S d,u (affected by thermal dilation) is not strictly correct, as pointed out by [START_REF] Tien | On the burning velocity of stretched flames[END_REF]. Consequently, such extrapolation performed (using a thermal dilatation affected value) may result in errors as pointed out by [START_REF] Jayachandran | Determination of laminar flame speeds using stagnation and spherically expanding flames: Molecular 805 transport and radiation effects[END_REF]. However, the [START_REF] Varea | Measurement of laminar burning velocity and markstein length relative to fresh gases using a new postprocessing procedure: Application to laminar spherical flames for 760 methane, ethanol and isooctane/air mixtures[END_REF], as well as more recent works from Bradley et al. [START_REF] Bradley | Flame speed and particle image velocimetry measurements of laminar burning velocities and markstein numbers of some hydrocarbons[END_REF]. From these studies, L u for stoichiometric methane/air flame at ambient temperature and 550 atmospheric pressure are very close to those obtained in this work, -0.12 mm and -0.11 mm, respectively. For their numerical work, Giannakopoulos et al.

[2] demonstrated that flame sensitivities to stretch were more dependent to the isotherm chosen when the fresh gas side was used compared to the burned gas side. In this study, we show that experimental measurements can report 555 quantitative and accurate data on S d,u and its associated extrapolated value when care is taken on the experimental protocol and post processing. Moreover, as mentioned by Jayachandran at al. [START_REF] Jayachandran | Determination of laminar flame speeds using stagnation and spherically expanding flames: Molecular 805 transport and radiation effects[END_REF][START_REF] Jayachandran | A study of propagation of spherically expanding and counterflow laminar flames using direct measurements and numerical simulations[END_REF] developments in [START_REF] Bonhomme | Curvature and confinement effects for flame speed measurements in laminar spherical and cylindrical flames[END_REF] and [START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF]:

S c,DN S = - 1 ρ u Y k,u R 2 f,k R0 0 ωk r 2 dr (23)
Equality between Eq. 22 and Eq. 23 was already numerically demonstrated 575 in [START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF]. We present now the experimental measurements of S c by Eq. 22

and the comparison with the DNS data from Eq.23. For the three different equivalence ratios, the spatially averaged consumption speeds are reported according to flame stretch (Figures 8, 9 Similarly to the density weighted displacement speeds, extrapolation at zero 605 stretch of the spatially averaged consumption speed allows to introduce the Markstein length L c , according to the following linear equation,

S c = S 0 L -L c K (24)
Values of extrapolated flame speed and Markstein length are reported in In this work, it was decided to introduce a flame surface which does not suffer from any ambiguity and which is defined by an equivalent radius based on the 625 total mass of fuel (if lean flame) or of oxidant (rich flame) [START_REF] Bonhomme | Curvature and confinement effects for flame speed measurements in laminar spherical and cylindrical flames[END_REF][START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF], R * = R f,k .

For this particular value of R * , Eq. 25 becomes equal to Eq. 24. We remind that this equivalent radius corresponds to an isotherm close to 600 K, and it is obtained experimentally from tomographic recordings. By considering Tables 1 and3, one can see the very small differences between the Markstein lengths L b 630 and L c .

The evolution of numerical data of L * c and L b according to the isotherm of the flame radius is plotted in Figure 11. x = 0 refers to the location of the maximum heat release, and x = 0.55 corresponds to the fresh gas side. The difference between these two Markstein lengths decreases when one approaches 635 towards the gases fresh, and cancels out when R * is close to R f,k . Thus, with this reference surface, the spatially averaged consumption speed is expected to be very similar to S d,b . Note that even if the lengths L b and L c are similar, the spatial averaged consumption speed is not sensitive to confinement effects, and seems to exhibit a linear behavior at high equivalence ratio, unlike S d,b . Note 640 that this Markstein length L c is different from that introduced by asymptotic studies [START_REF] Giannakopoulos | Consumption and displacement speeds of stretched premixed flames -theory and simulations[END_REF], where the local consumption speed is considered (Eq. 1). This Markstein length noted L c,Local is systematically greater than the length L b and very weakly dependent on the fuel, according to the following relation [START_REF] Giannakopoulos | Consumption and displacement speeds of stretched premixed flames -theory and simulations[END_REF],

L b -L c,Local = 1 σ -1 σ 1 λ(x) x dx • l f ( 26
)
where σ is the thermal expansion ratio, λ is the non-dimensional thermal 645 conductivity and l f is the flame reaction zone thickness estimated from the ratio of the thermal diffusivity of the fresh gases with the laminar flame speed.

Conclusion

This paper highlights the experimental tools which have been developed for measuring laminar flame speeds in the expanding flame velocity configuration L . Extension of this linear extrapolation to non equi-diffusive fuel/air mixtures requires additional tests and discussion. It is also worth noting that this method presents a quite important scatter in spite of the improvements made. It is then necessary to carry out a larger number of tests (here about ten) to converge.

The flame speed S d,b is very easy to measure with the assumptions usually 700 made (burnt gases at equilibrium and at rest), but remains very sensitive to the effects of flame confinement. In this geometry, extrapolation at zero stretch leads to a systematic underestimation of the laminar burning velocity S 0 L of the order of 1 to 3 cm/s, regardless of the equivalence ratio of the mixture. For small combustion chambers, this effect is noticeable even when respecting the classic 705 radius range criteria for extrapolation. A careful look of the literature also shows that this phenomenon is observed even for larger combustion chambers. This quantity also requires an accurate knowledge of the fuel composition in order to determine the density of the burnt gases by a chemical equilibrium calculation.

These remarks partly explain the disparities observed in the values reported in 710 recent years.

  3 and 5) are complementary and must be used together to address the difficult task to measure laminar 70 burning velocity with the constant-pressure method. The two previous flame speed definitions are related to kinematic information. It is however possible to introduce the consumption speed based on the integral of the reaction rate. Similar to 1D condition (Eq. 1), a local consumption speed can be introduced and calculated in spherical expanding 75 flames [16]. This definition is appropriate for unidirectional flows, e.g., planar or spherical flames, where the normal to the flame is properly defined, but it can not be measured experimentally. To overcome this difficulty, the spatially averaged consumption speed was introduced a long time ago by normalizing the global consumption rate, obtained by integrating the fuel reaction rate 80 throughout the entire combustion volume by the flame surface area. Fiock and

95

  the spatially averaged consumption speed S c corresponds to the global consumption rate normalized by the surface defined by the radius R f,k based on the total mass of species k contained into the sphere of radius R f,k . For lean and stoichiometric mixtures, k = Fuel, and for rich conditions, k = O 2 . It was recently shown that the isotherm corresponding to the flame 100 radius R f,k is very closed to that of experimental radius obtained by laser

205 3 .

 3 Measurements of fresh gas velocity profiles ahead of the flame front3.1. PrinciplesAn in-house PIV post-processing technique is used to compute the fresh gas velocity field ahead of the flame front, to report S c and S d,u . For two successive tomographic images where the flame radius R T can be extracted, the 210 algorithm defines a region of interest (ROI) around the spherical flame. The ROI is an angular sector of the fresh gases with an angle θ which is adjusted to 320 • which fits the flame contour of a radius R T . The width of the ROI is restricted to 110 pixels (5.22 mm), where the velocity profiles are measured (Figure1). This ROI is then dewarped into a Cartesian coordinate system using 215 a bilinear interpolation scheme. An interrogation window (IW) with a width varying from 1 to 5 pixels is located within this ROI. A matching method based on a two-dimensional normalized cross-correlation is used between IWs of the two successive ROIs. First, the best correlation between both IWs is obtained by shifting only the first IW in increments of one pixel in the direction normal to 220 the flame front. The location of the maximum of correlation (∆r opti 1

1 is the optimum displacement 225 of

 1225 the first IW obtained for the displacement of the second IW, ∆r 2 . Fixing the second IW and searching the best correlation on the first one avoids problems of evaporated droplets. To characterize the velocity profile ahead of the flame, the second stage of post-processing consists in moving the second IW by step of one pixel (∆r 2 ) in the direction normal to the flame front. For each position 230 of this window, the first step of the template matching procedure is restarted.

Figure ( 1 )

 1 Figure (1) Sketch of flame kernel propagation in the confined combustion chamber of radius R 0 . (a) A quarter of the flame with a flame thickness δ L is represented. The red curve is associated to the flame contour obtained by laser tomography and corresponds to the oil droplet evaporation isotherm. (b) A 1D view of a dewarped tomographic image with the corresponding tomographic flame radius R T . The control volume used for the determination of fresh gas density is represented with yellow dotted lines. (c) Similar dewarped tomographic images at two successive times used for the determination of the fresh gas velocity profiles ahead of the flame front. The ROI is represented by a yellow dotted line rectangle and the IW is the gray zone with an adjustable width of 1-5 pixels.

250opti 1 (

 1 considered since it may impact the accuracy of the distance ∆r between the value of the optimum displacement of the first IW ∆r opti 1 (t * ) with the value of the displacement of the second IW ∆r 2 (Second term of the RHS in Eq. 10).The classical problem of correlation peak fitting procedure is clearly evidenced in Figure2, where different correlograms have been obtained for different values 255 of ∆t. The laser frequency is kept constant at f = 5000Hz, and the images are processed with a ∆t = n/f , n varying from 1 to 4. For each, a basic 7-points Gaussian fit is used to report the values of ∆r opti 1 (t * ). The lower the interval time ∆t, the smaller the values ∆r opti 1 (t * ), and the greater the impact of the accuracy of the sub-pixel position of the maximum correlation peak on 260 ∆r t * ). However, a high interval time may act as a low-pass spatial filtering when computing the fresh gas velocity which may fail to capture the maximum flow velocity. From our operating conditions, the best compromise seems to get a minimum value of ∆r around 3 pixels which ensure a acceptable uncertainty and a spatial resolution of the velocity profile. Consequently, image frequency 265 (nominal frequency or a fraction of the nominal frequency) is selected according to the flame propagation speed values for a specific operating conditions.

270

  These selected plots are representative of the experimental measurements. At the same time, numerical temperature profiles are plotted. Due to inherent time shift from the numerical initialization of the flame kernel, computations and experimental data are time-scaled for a similar radius. The shift in time is as low as 200 µs which is more or less one experimental time step. The 275 experimental velocity profiles obtained from the methodology presented above are plotted with black open circles. The fresh gas velocity at the entrance of the flame front, u u , is obtained from the maximum (filled red diamond) of a

Figure ( 2 )

 2 Figure (2) Correlation peaks (open circles) and corresponding 7-points Gaussian fit (continuous lines) for different interval times between two successive images. Value of ∆r 2 is set to 7 pixels, corresponding to a location in the fresh gas velocity profiles close to the maximum values. Results for φ = 1, and t * = 7.0 ms

305a

  complementary to the classical methodology that might introduce issues due to unknown fuel chemistry and burned gas composition, differential diffusion or radiation effects, and low pressure as mentioned in the work of Varea et al.[START_REF] Varea | Determination of burning velocities from spherically expanding h2/air flames[END_REF],

Figure ( 3 )

 3 Figure (3) Fresh gas velocity and temperature profiles from experimental and numerical data, for two different operating conditions. (Top) Methane/air flame (φ = 0.7 and R T = 15.2 mm) and (Bottom) Methane/air flame (φ = 1.0 and R T = 19.8 mm)

375 1 /

 1 R 2 shape. This condition is satisfied when d ≥ 1 -2 mm. Effects of larger values d on the calculation of the compression term 1/ρ u • dρ u /dt are reported in Fig. 4 from the simulations. Relative differences of the rate of change of the fresh gas density calculated at d = 1.2 mm and d = 3.5 mm are marginal, and demonstrate very negligible effects of the distance d on the compression term.

  380

Figure ( 4 )

 4 Figure (4) (Top) Relative difference of the numerical fresh gas density ρu calculated by Eq. 15 using DNS data and directly extracted from DNS database (φ = 1). (Bottom) Influence of the position d on the evaluation of the compression term 1/ρu • dρu/dt of fresh gases. The relative difference of the value of this compression term computed with distance from tomographic radius d = 3.5 mm and d = 1.2 mm is reported as a function of the flame radius.

420

  Spark energy, shape and volume may change also slightly from one ignition to another one. These four sources of uncertainties may impact the flame dynamic, and their contribution to the different flame speeds will be estimated by performing a statistical analysis with a number of flame recordings per condition, N ≥ 10. The second source of uncertainty comes from the image 425 processing steps for each flame recording. Both flame radius and fresh gas velocity determination from tomographic recordings are associated to individual uncertainties coming from the capability of the routines to extract the quantities of interest from noisy signals. The uncertainty of flame radii is estimated for each flame contour by considering the relative distance between the flame contour and 430

Figure ( 5 )

 5 Figure (5) Comparison of the fresh gas density compression term from simulations and experimental data (based from more than 10 different flames) with respect to the experimental flame radius, for lean (Top), stoichiometric (Middle) and rich (Bottom) conditions. Comparison between the terms 1/ρu • dρu/dt and 1/γuP • dP/dt for experiments when φ = 1.0.

440S

  d,b , S d,u or S c . In the next section, the contribution of these two sources of uncertainties on the different flame speeds is reported as follows. All the instantaneous stretched flame speeds S k are averaged per class of stretch of 8s -1 over the N tests, weighted by their respective instantaneous uncertainty e 2 (S k ), according to [31, 445 32]:

460 5 . 5 . 1 .Figure 6

 5516 Figure 6 presents a direct comparison of density weighted displacement speed relative to burned gases for lean, rich and stoichiometric conditions. Flame 475

  It is worth noting that the Markstein Lengths L b in Eqs 19 and 20 are related to the density weighted displacement speed relative to burned gases, and differ from most of the data available in the literature by a factor equal to the density ratio, since S d,b is considered instead of S d,b = dR/dt.

Figure ( 6 )

 6 Figure (6) Evolution of S d,b with flame stretch at different equivalence ratios.

Figure 7

 7 Figure 7 reports the evolution of experimental and numerical S d,u over flame stretch rate at different equivalence ratios. As for S d,b , the flame speeds from the 10 measurements are averaged per class of stretch of 8s -1 . The statistical approach used to report data allows a good and relevant comparison with the 520

  zero stretch extrapolation of the experimental data converges with accuracy to the Cantera 1D values. The target values are reached for both DNS and 540 experimental data for the three equivalence ratios. S d,u measurement technique is shown to be not affected by the small size of the combustion chamber and therefore to confinement and ignition effects, allowing accurate measurements of extrapolated values for low or intermediate chamber radii.

Figure ( 7 )

 7 Figure (7) Direct comparison of density weighted displacement speed relative to fresh gases for experimental and DNS data.

5. 3 .

 3 Spatially averaged consumption speed, S c

  and 10). At the same time, the first and the second term of the RHS of Eq.22 are also plotted both from the experiments 580 and the DNS to validate the experimental determination of these two terms. S c is calculated as the difference of two large numbers, and therefore its accuracy will depend directly on the accuracy of each contribution. For the first term, error bars are similar to those indicated for S d,b which were mainly due to the shot-to-shot repeatability, multiplied by the density ratio. The second term 585 has similar error bars. Both terms are well captured by the experiment, with stretched values that are very close to the DNS (slightly underestimated for φ = 1.0 and 1.2, and quite clearly overestimated for φ = 0.7). For high stretch values, this difference slightly increases, before diverging when the influence of ignition becomes predominant. The difference of these two contributions 590 gives the spatially averaged consumption speed S c , which is in very good agreement with the consumption speed computed from the integral of the reaction rate. For stretched flames free from confinement and ignition effects, experiments and DNS values are superimposed. For φ = 0.7, the agreement is slightly worse, as already observed for S d,b . For this condition, ignition 595 effect seems to persist beyond the usual minimum value of flame radius. The error bars on experimental S c are slightly lower compared to those of the two contributions, which underlines the compensating effect when considering the shot to shot fluctuation. Stretched consumption speeds seem much less sensitive to confinement effects than S d,b , with a globally linear behavior at large radii, 600 observed both numerically and experimentally, with the range of flame radius accessible through the chamber. The extrapolated values are similar to each other, slightly lower than the values given by the 1D calculation by Cantera. The slopes are also well captured by the experiments.

Figure ( 8 )

 8 Figure[START_REF] Liang | Extrapolation of laminar flame speeds from 740 stretched flames: Role of finite flame thickness[END_REF] Direct comparison of spatially averaged consumption speed for experimental and DNS data, φ = 1.0. DNS data are represented by continuous lines. Experimental data are reported in this figure with circle symbols. Red color is dR T /dt, blue color is the density ratio term in Eq. 22 and the spatially averaged consumption speed is represented in green by using Eq. 22 for experiments and Eq. 23 for the DNS. Linear extrapolations over the bold line and filled symbols are reported in dotted and dashed lines.

Figure ( 9 )

 9 Figure (9) Direct comparison of spatially averaged consumption speed for experimental and DNS data, φ = 1.2. DNS data are represented by continuous lines. Experimental data are reported in this figure with circle symbols. Red color is dR T /dt, blue color is the density ratio term in Eq. 22 and the spatially averaged consumption speed is represented in green by using Eq. 22 for experiments and Eq. 23 for the DNS. Linear extrapolations over the bold line and filled symbols are reported in dotted and dashed lines.

Figure ( 10 )

 10 Figure (10) Direct comparison of spatially averaged consumption speed for experimental and DNS data, φ = 0.7. DNS data are represented by continuous lines. Experimental data are reported in this figure with circle symbols. Red color is dR T /dt, blue color is the density ratio term in Eq. 22 and the spatially averaged consumption speed is represented in green by using Eq. 22 for experiments and Eq. 23 for the DNS. Linear extrapolations over the bold line and filled symbols are reported in dotted and dashed lines.

Figure ( 11 )

 11 Figure (11) Effect of reference surface, expressed in flame radius and corresponding isotherm, on the Markstein lengths L * c and L b . φ = 1

650(

  kinematic and kinetic velocities) whose definitions and expressions have been rigorously derived previously. These laminar flame speeds were measured for methane/air mixtures at different equivalence ratios and compared directly with DNS performed by A-SURF code simulating the exact geometry of the combustion chamber, in spherical 1D coordinates. This numerical approach 655 allowed validation of all the post-processing steps, giving access to the local fresh gas velocity at the entrance of the flame front, as well as to the density of the fresh gases, and the values of the stretched laminar flame speeds. With this numerical database, the experimental measurements enable the following conclusions to be drawn about the relevance and the accuracy of the different 660 flame speeds. This work reports a new experimental approach to evaluate the spatially averaged consumption speed S c for spherically expanding flames from tomographic flame radius and fresh gases density measurements, Eq. (22). This methodology was derived recently from the global reaction rate normalized 665 by a reference flame surface defined by an equivalent flame radius based on the total mass of fuel (if lean flame) or of oxidant (rich flame) [20]. The novelty of this work is associated to the experimental measurement of fresh gas density from the integration of the continuity equation through a material control volume. Considering the initial fresh gas density, the location of the 670 boundary of the control volume and the fresh gas velocity, it is possible to report the temporal evolution of the fresh gas density with an excellent level of accuracy, as demonstrated by the direct comparison with the DNS. The spatially averaged consumption speed is very close to the flame speed S d,b with similar positive Markstein lengths. S c is however less sensitive to flame 675 confinement and behaves more linearly with flame stretch for rich conditions.Even interesting to report an experimental measurement of total fuel reaction rate, it seems therefore difficult to extend this technique to other fuel/air mixtures without any other preliminary validations, such as the equivalence between tomographic flame radius and the equivalent flame radius, R f,k . It is 680 also important to note that this spatially averaged consumption speed differs from the local consumption speed introduced in asymptotic theories and used in 1D configurations, and must be considered as complementary.The displacement speed S d,u requires an accurate measurement of the maximum velocity of the fresh gases at the entrance to the flame front. In 685 this paper, an important improvement of post-processing tools was carried out and validated by a direct comparison with DNS simulations. To our knowledge, this is the first time that the fresh gas velocity profiles across the flame front in spherical expanding flames have been compared directly with DNS. The experimental fresh gas velocity profiles overlap perfectly with the 690 DNS, and the agreement with the stretched flame speed S d,u is excellent. For the operating conditions (quasi equi-diffusive fuel/air mixtures), this flame speed behaves linearly with stretch, and allows a direct estimation of the laminar burning velocity S 0 L by a linear extrapolation. This quantity is not influenced by confinement effects and does not underestimate the extrapolated flame speed 695 S 0

Table ( 1

 ( ) Extrapolated flame speed from S d,b and corresponding Markstein length, L b , obtained from Eq. 19 and 20 by fitting S d,b = f (K) for tomographic radius between 0.8cm and 1.9cm.5.2. Displacement speed relative to fresh gases, S d,u

	7 Experiments	16.82	0.036	0.030	0.0024	Linear
	DNS	17.28	-	0.085	-	Linear
	Cantera	36.30	-	-	-	-
	φ = 1.0 Experiments	33.38	0.078	0.099	0.0022	Linear
	DNS	33.71	-	0.094	-	Linear
	Cantera	32.00	-	-	-	-
	φ = 1.2 Experiments	29.56	0.047	0.105	0.0011	Non-Linear
	DNS	29.50	-	0.095	-	Non-Linear

Table 2

 2

	summarizes

Table 3 .

 3

		S 0 L [cm/s]	σ S 0 L	[cm/s]	Lc [mm]	σ Lc [mm]	R T range [cm]
	Cantera	18.49		-	-	-	-
	φ = 0.7 Experiments	15.81	1.21	0.015	0.100	[1.4; 2.2]
	DNS	17.90		-	0.135	-	[1.4; 2.2]
	Cantera	36.30		-	-	-	-
	φ = 1.0 Experiments	31.88	0.54	0.028	0.017	[1.0; 2.2]
	DNS	35.40		-	0.084	-	[1.0; 2.2]
	Cantera	32.00		-	-	-	-
	φ = 1.2 Experiments	27.95	0.50	0.010	0.018	[1.0; 2.2]
	DNS	31.82		-	0.047	-	[1.0; 2.2]

Table ( 3

 ( ) Extrapolated flame speed from Sc and corresponding Markstein length, Lc, obtained from Eq. 24 by fitting Sc = f (K) on a specific tomographic radius range.

Acknowledgements

Financial supports from the French National Research Agency, under the project 'RAFBIOALG (ANR-18-CE43-0009)' and National Natural Science Foundation of China (no. 51861135309) are gratefully acknowledged.

The spatially averaged consumption speed of the deficient species is now reported from experimental measurements of flame radius R T , obtained from tomographic recordings, and fresh gas density deduced from fresh gas velocity profile as presented above. This is reminded in Eq.22.

We implicitly assume that the experimental flame radius R T is equal to 570 R f,eq in Eq. 7. This was already validated in [START_REF] Lefebvre | Formalism for spatially averaged consumption speed considering spherically expanding flame configuration[END_REF]. Simultaneously, the spatially averaged consumption speed from the DNS can be exactly computed from the integral of the mass reaction rate of the deficient species, according to the

Declaration of interests

☒ 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.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: