Colette Nicoli

Pierre Haldenwang

Bruno Denet

Darrieus-Landau instability of premixed flames enhanced by fuel droplets

Recent experiments on spray-flames propagating in Wilson cloud chamber have established that spray-flames are much more sensitive to wrinkles or corrugations than single-phase flames. To propose certain elements of explanation, we numerically study the Darrieus-Landau (or hydrodynamic) instability developing in premixtures that contain an array of fuel droplets. Two approaches are compared: the numerical simulation starting from the general conservation laws in reactive media (DNS), and the numerical computation of Sivashinsky-type modelequations (SME) for DL-instability. Both approaches provide us with results in deep agreement.

It is first shown that the presence of droplets in fuel/air premixtures induces initial perturbations which are large enough to trigger the Darrieus-Landau (DL) instability. Second, the droplets are responsible of additional wrinkles when the DL instability is developed. The latter wrinkles are of length scales shorter than those of the DL-instability, in such a way that the DL-unstable spray-flames have a larger front surface and therefore propagate faster than the single-phase ones when subjected to the same instability. In combustion systems, fuel is often injected under liquid phase as in diesel, aerospace engines or furnaces. This has long motivated studies dealing with spray-flame dynamics. The experimental literature reports numerous situations where the spray-flame is more corrugated, and faster, than the equivalent single-phase flame. We have particularly in mind the recent experiments conducted in Wilson chamber [START_REF] Bradley | Laminar mass burning and entrainment velocities and flame instabilities of isooctane, ethanol and hydrous ethanol/air aerosols[END_REF] and [START_REF] Thimothee | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF], the latter experiments being performed in microgravity. Consequent wrinkles have also been reported in earlier works about spray-flames [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF][START_REF] Hayashi | Propagation velocity and structure of flames in droplet vapor air mixtures[END_REF][START_REF] Nomura | Effects of pressure on flame propagation in a premixture containing fine fuel droplets[END_REF], or particle cloud combustion [START_REF] Hanai | Pulsating flame propagation of pmma particle cloud in microgravity[END_REF].

There is hence an important number of observations which indicate that spray-flame fronts are particularly prone to fold, if the droplet size is large enough. The purpose of the present numerical work is to contribute to understand this phenomenon.

The experiments on combustion in sprays at moderate and high pressures have generally revealed behaviors in large departure from the equivalent single-phase premixed flames. For instance, two-phase flames have been found to propagate with a pulsating regime [START_REF] Atzler | Burning rates and flame oscillations in globally homogeneous two-phase mixtures (flame speed oscillations in droplet cloud flames)[END_REF][START_REF] Hanai | Pulsating flame propagation of pmma particle cloud in microgravity[END_REF] without the resort to differential diffusive effects [START_REF] Suard | Different spreading regimes of spray-flames[END_REF][START_REF] Nicoli | Analysis of pulsating sprayflames propagating in lean two-phase mixtures with unity Lewis number[END_REF][START_REF] Greenberg | Propagation of rich premixed polydisperse spray flames[END_REF]. In addition, certain results obtained in microgravity [START_REF] Nomura | Microgravity experiments of flame propagation in ethanol droplet vapor air mixture[END_REF] have shown that spray-flames in lean ethanol-air mixtures can propagate faster than the equivalent gaseous flames, and slower in globally rich mixtures. Recent numerical studies [START_REF] Nicoli | Lean flame dynamics through a 2D lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Numerical study of flame dynamics through a 2D-lattice of alkane droplets in air[END_REF] concerning globally lean mixtures with a fixed overall equivalence ratio ϕ T = 0.85 brings complementary results concerning the mixture composition effects and the droplet size influence on spray-flame dynamics. On the other hand, classical ground experiments [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF][START_REF] Hayashi | Propagation velocity and structure of flames in droplet vapor air mixtures[END_REF][START_REF] Bradley | Laminar mass burning and entrainment velocities and flame instabilities of isooctane, ethanol and hydrous ethanol/air aerosols[END_REF] have exhibited opposite observations to the ones made in [START_REF] Nomura | Microgravity experiments of flame propagation in ethanol droplet vapor air mixture[END_REF]: spray-flames in rich mixtures of ethanol (or iso-octane) and air are faster than the equivalent gaseous premixed flame.

To interpret these complex aspects of the flame propagation in dropletvapor-air mixture, Hayashi and Kumagai [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF][START_REF] Hayashi | Propagation velocity and structure of flames in droplet vapor air mixtures[END_REF] introduced the concept of effective fuel-air ratio. This corresponds to an attempt at accounting for the unvaporized fuel and incompletely mixed fuel, which depends in a complex manner on droplet size and fuel-air ratio. Our recent investigations on sprayflames dynamics [START_REF] Nicoli | Lean flame dynamics through a 2D lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Spray-flame dynamics in a rich droplet array[END_REF] have established the various roles played by the main spray parameters: ϕ G , the equivalence ratio of the fuel-saturated premixture that initially surrounds the droplets, ϕ L , the liquid loading or the equivalence ratio of the fuel under liquid phase to the overall quantity of air, and R d , the typical droplet radius. Note that we obviously have ϕ T = ϕ G + ϕ L and L d , the typical droplet interdistance, is a function of R d and ϕ L . In particular, in both recent contributions [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Spray-flame dynamics in a rich droplet array[END_REF] on rich sprayflames, we have proposed to consider the "spray Péclet number" defined as follows:

P e s ∝ ρ L ρ G 2R d * U L (ϕ G ) D * th,b 2R d L d (1)
The "spray Péclet number" corresponds to the ratio of the droplet vaporization time

τ vap ∝ (ρ L /ρ G )(2R d ) 2 /D * th,b to the propagation time τ prop ∝ L d /U L (ϕ G ), where U L (ϕ G ) and D *
th,b are the laminar single-phase flame speed related to the initial fuel-saturated mixture and the thermal diffusivity in the burnt gases of the stoichiometric single-phase laminar flame, respectively. Note that R d , the 2D droplet radius (or more properly the disk radius), is a function of both droplet interdistance L d and liquid loading ϕ L . This obviously takes account of the present 2D planar geometry, so that the actual 2D radius is about one third of the radius that would give the same liquid equivalence ratio in 3D.

When the "spray Péclet number" is large, the flame propagation between two droplets is so fast that droplet vaporization does not modify the initial premixture that surrounds the droplets. In other words, spray-flame speed is simply the one of the single-phase flame propagating in the initial premixture. This result has numerically been established clearly, as long as the spray-flame remained planar. Now, to simplify the forthcoming discussions on folded flames, we decide to assume that the "spray Péclet number" is large, in such a way that the role of the droplets is restricted to create small regions of rich combustion imposed to a single-phase flame front locally propagating with U L (ϕ G ). Accordingly with the concept of effective fuel-air ratio introduced by Hayashi and Kumagai [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF][START_REF] Hayashi | Propagation velocity and structure of flames in droplet vapor air mixtures[END_REF], the effective fuel-air ratio is here reduced to ϕ G for spray with P e s > O(1).

A situation where the "spray Péclet number" is large generally corresponds to a rich spray. Even though engines are desirable in lean configuration, rich spray-flames are known to locally exist close to the system of liquid injection.

Because in numerous experiments the spray-flame front was found cor-rugated with a large number of cells, it is a difficult task to determine the fundamental laminar velocity of the spray-flames from those experimental data. Moreover, the spray-flame speed enhancement observed in the experiments could possibly be explained by instabilities of the front, only. As a matter of fact, the complex interplay between instabilities and droplets seems to have a peculiar importance for the spherical flames, a case that has extensively been studied in the recent years [START_REF] Nomura | Microgravity experiments of flame propagation in ethanol droplet vapor air mixture[END_REF][START_REF] Nassouri | Flame structure of ethanol-air premixed mixtures at high pressures in microgravity[END_REF][START_REF] Bradley | Laminar mass burning and entrainment velocities and flame instabilities of isooctane, ethanol and hydrous ethanol/air aerosols[END_REF], and that follows the pioneering work of [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF] [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF]. The present work, which adopts the simplified configuration P e s > O(1), allows us to cancel the part of the interplay in relation with the role of the vaporization in the premixture establishment. On the theoretical side, the creation of many cells in spherical premixed flames (without droplets) has been modeled with the Sivashinsky equation, particularly in 2D by Karlin and Sivashinsky (2007) [START_REF] Karlin | Asymptotic modelling of self-acceleration of spherical flames[END_REF]. This DL-model equation contains the main two effects, creation of many cells and self-acceleration.

Two models for DL-unstable spray-flame propagation

The article is based on the predictions of two different approaches that are able to study the Darrieus-Landau instability (or the hydrodynamic instability of premixed flame). The first one uses the 2D numerical simulation (DNS) that starts from the basic equations characterizing any reactive medium. As for the second approach, we use a much lighter mathematical approach (SME) that corresponds to the DL-model equation due to Sivashinsky.

Solving the conservation 2-D laws for reactive media (DNS)

This model of spray-flame propagation resorts to the general approach that we have used in our recent contributions [START_REF] Nicoli | Lean flame dynamics through a 2D lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF]. The initial state of the spray is schematized by a 2D face-centered lattice, at the nodes of which the droplets are positioned. The lattice spacing is denoted by s, in such a way that the droplet interdistance is

L d = s/ √ 2.
The spray-flame propagation is governed by the usual set of conservation laws for mass, momenta, energy and species, with the simplest chemical scheme [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF] that allows us to handle a flame propagating through a medium of heterogeneous composition, where the equivalence ratio varies from ϕ G far from the droplets to a very large value close to the droplets. As described in [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF], the spray is considered as a non-homogeneous one-phase medium. Vaporization, Stefan flow and heatmass transfers at the liquid-gas interface are solved from the conservation laws and do not require particular sub-models.

By contrast with most literature on spray-flames (for a recent review, see [START_REF] Wacks | Flame structure and propagation in turbulent flame-droplet interaction: A direct numerical simulation analysis[END_REF]), the current droplets are here fully resolved. This explains why 3-D DNS is hardly affordable for a parametric exhaustive study. The droplet radius, being a function of both lattice spacing and liquid loading, has a typical value which varies in a range from 0.1 to 1 in flame thickness units. The droplets belong to the same continuum as the gas; therefore, they can move, heat, follow the properties of a real gas, and continuously switch from (liquid) dense fluid to (vapour) light fluid. The present simulation takes advantage of the fact that latent heat and surface tension are neglected for the following reasons: a) latent heat of alkanes corresponds to a small fraction of the reaction heat (e.g. see Ref. [START_REF] Lewis | Combustion, Flames and Explosions of Gases[END_REF]) and can be neglected in the thermal budget of the pre-heat zone; b) liquid fuel containment can simply be achieved by the fact that heat and mass transfers are frozen at the spray initial temperature (the spray being at rest far upstream of the flame). The latter point is carried out by using the non-linear dependence of the diffusion coefficients with respect to temperature. The droplets are initially set identical at the nodes of a face-centered lattice. The vaporization process is hence freed from the possible weakness related to the vaporization model selected. Note additionally that the experiments in cloud Wilson chambers, carried out in microgravity [START_REF] Thimothee | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF], lead to nearly monodisperse sprays, with a nearly regular density for the droplets. This is why the current initial conditions are seemingly relevant with regard to such experiments.

The chemical scheme, based on the classical one-step Arrhenius law, can be interpreted as resulting from a flamelet theory: the flame front is conceived as a series of small planar flames connected to each other, that individually propagate through a locally homogeneous medium. This consideration requires that the spray-flame thickness remains small in comparison with the typical droplet interdistance. This requirement is generally fulfilled for sprays at moderate and high pressures. The dynamics of these flames depends on the equivalence ratio of the local premixture burnt by each flamelet. More precisely, the local equivalence ratio, denoted by ϕ u , is assessed [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF] at each computational point in such a way that it represents the "upstream" equivalence ratio, as seen ahead by each small planar flame. Furthermore, it is known that the classical one-step Arrhenius law largely overestimates the adiabatic flame temperature of the rich mixture.

To overcome the difficulty to correctly assess the laminar flame speed, hereinafter denoted U L , we considered an easy modification [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF] of this simple scheme: heat release is a linear function of ϕ u , the fresh premixture equivalence ratio. This model has been proven to correctly mimic the premixed single-phase flame behaviors (adiabatic flame temperature and flame speed) [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF]. As a matter of fact, such a modification of heat release takes account of all the species existing at the actual flame temperature. From the numerical point of view, at each time step, the classical system of reactiondiffusion equations is solved first. Then, it is coupled with the Navier-Stokes equations. The overall scheme that computes the Navier-Stokes equations within the context of DL-instability has previously been described in Denet and Haldenwang [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF].

The set of equations, as well as the results we shall present, are handled under non-dimensional form. The scales retained for non-dimensioning are those of the adiabatic, stoichiometric, premixed flame of the considered fuel (presently a heavy alkane). More precisely, the units for time, length, mass and temperature are derived from the theoretical properties of this ideal flame, in terms of flame thickness The numerical experiments are conducted as follows. The droplets are initially positioned at the nodes of a face-centred lattice of spacing s (in δ * f units), as illustrated by Fig. 1. The lattice is embedded in a computational box of length L x and height L y . For periodicity reason, we have chosen either L y = s or L y = 2s. The surface tension of the droplets being neglected, a droplet appear as a dense fuel puff belonging to the same continuum as its surrounding premixture. Furthermore, the containment of the fuel within the droplet results from the freezing of the mass diffusion at the unburnt temperature. The droplets can move, carried by the flow resulting from gas expansion. Heat-mass transfers, vaporization and the related gas expansion start when the medium is heated by the proximity of the flame.

δ * f = D * th,b /U * L , transit time τ * f = D * th,b /U * 2 L ,
If s and ϕ L are given, R d , the droplet radius is hence determined. In addition, if ϕ G is given, we can estimate the "spray Péclet number". All the numerical experiments considered in what follows satisfy the condition P e s > O(1). Therefore, accordingly with our recent studies [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF], [START_REF] Nicoli | Spray-flame dynamics in a rich droplet array[END_REF], the presence of droplets has no impact on the general dynamics of the planar spray-flame. We shall show that this is no longer the case when the sprayflame is DL-unstable. Among the main results, the spray-flame speed is considered. For its evaluation, we have to define the rate of flame propagation. As the front is strongly corrugated and changing in space and time, we choose an averaged definition of the front position by performing the partial integration of the temperature field in the transverse (periodic) y-direction.

We hence obtain the one-dimensional quantity < T > y (x). We then define x F , the front position, as the locus where < T > y (x F ) = 0.5.

Solving the Sivashinsky DL-model equation (SME)

The results of our DNS that will be shown below suggest that if the premixture between droplets is not lean, the main effect of the droplets is to locally slow down the premixed flame. We suggest to model this phenomenon by using a variant of the Sivashinsky equation proposed by Joulin and Cambray for large gas expansion [22]. Note that, originally, the Sivashinsky DL-model was derived within the context of small gas expansion. The extension proposed by Joulin-Cambray is known to allow the description of larger gas expansion (the current gas expansion through the single-phase flames of our numerical experiments is about 5, a value reputed affordable by the Joulin-Cambray extension). The idea is to include a position-dependent flame velocity in the non-linear term of this model-equation. It reads

ψ t + U ef f (x, y)(1 + a(γ) 2 ψ 2 x ) + (1 -a(γ)) 2 < U ef f (x, y)ψ 2 x > = Ω(γ)U g (I(ψ) + ψ xx /k n ) ( 2
)
where ψ is the position of the front γ = (T b -T u )/T b = 0.8 is the expansion parameter, Ω(γ) = ((γ + 1/(1 -γ)) the laminar flame propagation speed, outside the droplets, and to U g /5 in a circle of influence around each droplet. Note that this lowered flame velocity accounts for different effects: a very rich and slow flame around the droplets and an hydrodynamic effect of the flow field caused by droplet evaporation.

The Sivashinsky equation will be solved in various cases corresponding to a different number of unstable modes of the Darrieus-Landau instability (the stationary solutions of the equation without droplets being reported in [START_REF] Denet | Stationary solutions and Neumann boundary conditions in the Sivashinsky equation[END_REF] for different numbers of unstable modes). The general idea sustaining the SME approach consists in checking whether a Sivashinsky DL-model, that incorporates some local weak propagation, is able to retrieve the overall properties of the spray-flame. After several attempts for choosing both the size of the droplet influence zone and the drop in flame velocity, it turned out that varying both depth and size of the hole did not greatly change the results, provided that the size remains reasonably small in comparison with the lattice-spacing.

Triggering DL-instability with droplets

The classical theory of DL-instability indicates that the development of DLinstability within the framework of periodic conditions transverse to the propagation generally requires a wide enough "channel", namely that L y , the imposed periodicity, should be of a few tens times as large as the flame thickness. However, before working with droplets, single-phase numerical experiments with strong initial noise have established that the minimal channel width allowing the DL-instability development, and denoted here by L DL , presently corresponds to about 14 times the flame thickness (i.e. L DL ≈ 14). In other words, L y > L DL is needed for observing the DLinstability in single-phase premixture, and the threshold L y = L DL is the so-called cut-off scale of DL instability. Next, we consider two situations of DNS: a transversally small box (i.e. L y < L DL ) and a transversally large box (i.e. L y > L DL ).

3.1 DNS results in transversally small computational domain When L y < L DL , and whatever the initial perturbations of the flame front, the single-phase flame front is found to remain flat, indicating that the transverse threshold of the DL instability is not reached. Our studies that concern sprays with rich overall equivalence ratios observe a propagation regime already described by Hayashi and Kumagai [START_REF] Hayashi | Flame propagation in fuel droplet-vaporair mixtures[END_REF][START_REF] Hayashi | Propagation velocity and structure of flames in droplet vapor air mixtures[END_REF] when the droplets are large enough [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Spray-flame dynamics in a rich droplet array[END_REF], i.e. when the fuel under liquid phase does not contribute to combustion spreading. For smaller radii R d , the droplets vaporization would enrich the surrounding gaseous phase that can become greater than ϕ G , leading to a spray-flame velocity than depends on droplet radius in a rather complex manner [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF]. As recalled above, to determine the critical radius for the occurrence of the "large droplet" regime, we have proposed a criterion based on the "spray Peclet number", namely P e s > O(1).

In the rest of the article, the latter assumption will hold, as well as the choice L y > L DL .

DNS results in transversally large computational domain

As shown later by Fig. 4a, where the DL instability will not develop within the framework of a periodicity of 24 times the flame thickness, the condition L y > L DL is however not sufficient for realistic development of the DL instability. Turbulence or noise are known to trigger the DL-instability. The issue treated now is whether the droplets can play the same role.

Let us first describe how flame and droplet interact. As mentioned above, the initial premixture is supposed to sustain spray combustion without the recourse to droplet vaporization. Hence, the planar spray-flame speed is U L (ϕ G ). If the initial surrounding mixture is lean (i.e. ϕ G < 1), a triple flame occurs when the combustion burns a droplet. The lean wing is nothing but the initially premixed flame, the velocity of which is U L (ϕ G ). The rich wing surrounds the droplet which is transported towards the burnt gases by the rapid gas expansion due to vaporization. When the triple flame shape disappears, the rich wing transforms into a closed flame, richer and richer, the velocity of which finally vanishes. This rich flame accompanies the droplet during its motion and takes the form of an ellipse, when carried downwards by the flow.

As for the diffusion flame, it delimits the remaining oxygen located behind the lean wing. Since the overall equivalence ratio is rich, the diffusion flame burns the rest of oxygen. The figure 2, which presents a sequence of four snapshot of the reaction rate, illustrates how the lean flame is locally deformed by the presence of a droplet.

On the other side, when the initial surrounding premixture is already rich, no triple flame exists and the presence of the droplet only modifies the equivalence ratio of the mixture in its surroundings. In other words, the flame that essentially burns the initial premixture meets rich puffs of mixture at the nodes of the lattice, which locally hinders the spray-flame propagation. In the forthcoming section, it will nevertheless be shown that those loci of hindrance induce folds that increase the overall propagation of the spray-flame. A sequence of four snapshots for the present rich case is provided in Fig. 3. -the first one considers a single-phase flame of equivalence ratio 1.1 (i.e. ϕ G = 1.1 and ϕ L = 0). A planar flame is initiated without initial perturbation on the right hand side of a 24δ * f high computational box. In absence of noise or perturbations, the flame remains flat, as shown in Fig. 4.a, where we have superimposed 10 successive snapshots of the reaction rate field.

-the second situation concerns the same single-phase flame, still initiated on the right hand side of a 24δ * f high box, but the front has been perturbed by a y-sinusoidal x-displacement of 0.3 times the flame thickness. As long known [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF], this perturbation is sufficient to trigger the DL instability on a relatively short distance as reported in Fig. 4.b, where are superimposed 13 successive snapshots of the reaction rate.

-the third case treats of a planar spray-flame with the same equivalence ratio of the initial surrounding mixture. But this initially unperturbed flat flame meets droplets, the perturbations of which initiate the DL instability of the spray-flame. The parameters of this numerical experiment are ϕ G = 1.1, ϕ L = 0.5, s = 24 and L y = 24. As indicated in Fig. 5, the presence of droplets triggers the folding of the spray-flame which then experiences rather complex corrugations. One the left-hand side of Fig. 5 (i.e. when leaving the lattice), a single-phase premixture is retrieved and we can recognize the typical non-linear DL-shape of the single-phase premixed flame.

Sivashinsky DL-model equation (SME) results

As described above, the SME approach mimics the presence of droplets thanks to holes of partial extinction distributed at the nodes of the lattice.

The box height is here fixed to L y = s. The simulations of the Sivashinsky equation are carried out with the following parameters; density ratio between burnt and unburnt gases equals to 5; number of unstable Darrieus Landau modes: 2.4; the effective flame velocity inside the droplet-affected zone is fixed to 0.2 times the laminar flame velocity; the droplet-affected zone is inside a radius of influence around the droplets, which equals 0.1 times the total width L y . These values have been chosen in agreement with the 2-D simulations of the flame propagating in the (rich) close vicinity of the droplets.

The front shape at different times is plotted in Fig. 6, showing the effect of the droplets on the hydrodynamic instability. When the front reaches a droplet, the associated partial quenching creates wrinkles which triggers the DL instability, and then sustains the particular pattern of a DL affected front.

Droplets enhance spreading of DL-unstable sprayflames

The purpose of the present section concerns the assessment of the velocity of the spray-flames that are subjected to Darrieus-Landau instability, or equivalently what occurs when a single-phase flame subjected to DL instability meets a lattice of droplets.

SME approach

We first present the results obtained with the Sivashinsky DL-model equation. The simulations are with and without droplets, for the following numbers of unstable Darrieus-Landau modes: 1.2, 2.4 and 4.8 (1.2 corresponds to a small width). We here use initial conditions allowing the flame to reach quickly a shape with one large cell in the domain, and we now show the effect of the number of unstable modes (or equivalently domain height L y ) on the observed propagation velocity. The mean position of the front versus time is plotted in Fig. 7. It can be noticed in the figure that for a small width (1.2 unstable modes) there is not much difference between the results without droplets (black, dashed line) and with droplets (black, solid line). With 2.4 unstable modes (magenta -or dark grey-, dashed/dotted without droplets, magenta -or dark grey-, solid line with droplets), the perturbations caused by the droplets lead to larger flame amplitude and flame speed that in the case without droplets. This effect is even more important in the case of 4.8 unstable modes, green -or light grey-, dashed/dotted line without droplets, green -or light grey-, solid line with droplets.

Evidently, in what follows, the DNS approach provides us with more details on the manner that allows a flame to propagate through a lattice of droplets. However, the SME approach, implies much lighter computational efforts and allowed us to investigate a larger field of parameters. It also serves to highlight the most important (and quite paradoxical) point of this paper: the flame speed of a DL-unstable flame is increased because the flame propagates more slowly around the droplets. We will see below that the same results are obtained in our DNS.

DNS approach

We carry out the following DNS numerical experiments. Before meeting droplets, a single-phase flame (characterized by the set of parameters ϕ G , ϕ L = 0, L y ) exhibits folds because it develops the DL instability from an initial perturbation, and propagates towards the lattice of droplets. The initial conditions are such that the DL instability reaches its non-linear regime before the single-phase flame meets the first droplet. On the right hand side of Fig. 8, the single-phase flame folds as it propagates towards the droplet. When the first droplet is on the point to be met, the flame adopts the classical cusped shape of the DL instability. Then, when the flame propagates within the droplet lattice (i.e. we have now ϕ L = 0.5), it is however noticeable that the front pattern is more corrugated than in the single-phase DL instability. On the left hand side of the figure, the flame finally leaves the droplet lattice and tends to recover the classically cusped pattern.

We have performed a series of such numerical experiments for various ϕ G and s, the liquid loading of the droplet lattice being set to ϕ L = 0.5 and L y = s or L y = 2s. In Fig. 9, where combustion front x-position is plotted against time, we observe that the slope of the curve is not constant. It increases as the flame propagates within the lattice. It is now clear that the presence of droplets enhances the propagation of the DL-affected flame. In the figure, it is also observed that the change in slope occurs whatever ϕ G , the most intense effect corresponding to the nearly stoichiometric conditions of the surrounding mixture (i.e. ϕ G ≈ 1.1).

The slope of the combustion x-positioning vs. time within the lattice is hereinafter called spray-flame speed. The magnitude of this quantity incorporates two cooperative effects: the velocity increase due to the Darrieus-Landau instability which cooperates with the additional wrinkling due to the vicinity of the droplets where combustion is reduced (since too rich).

Evidently, the spray-flame speed will depend on the height of the computational domain (L y ), since DL single-phase flame speed depends on L y . Furthermore, the spray-flame speed also depends on the droplet size. But, it turns out that this effect remains weak, the number of the droplets (i.e. the role of s) for given L y and ϕ L playing a more important role.

To illustrate the essential part of these results, we have gathered the spray-flame speeds as a function of ϕ T for various values of s and L y , and for fixed ϕ L = 0.5. This corresponds to Fig. 10, which requires several comments. First of all, the curves with label s = 0 corresponds to a situ- ation where vaporization is immediate, so that the flame is of single-phase type. Second, only the region where P e s > O( 1) is easy to interpret: for L y = 12, the spray-flame is DL-stable; therefore, the curve with s = 12 is nothing but the curve with s = 0, translated to the right. For higher L y (i.e. L y = 24, L y = 48), the single-phase flame becomes DL-unstable as indicated on the left-hand side of Fig. 10 by a velocity higher than 1. As for the corresponding spray-flame (i.e. located at ϕ T = ϕ G + 0.5), the presence of droplet increases its speed, especially when the droplets are numerous (i.e. when L y = 2s). This corresponds to what was already predicted by the SME approach. Since a reliable DNS is sufficient to predict all the spray-flame properties in terms of DL-instability, it could be argued that the SME approach is unnecessary. Let us nevertheless stress on the fact that SME provides us with the following important (supplementary) results:

-SME gives the same qualitative results for a cost 10 4 times cheaper. -SME confirms the validity of our interpretation about the DNS results in terms of holes of low propagation. In return, this confirmation recalls the interest in simplified models as tools for analysis, and fast investigation.

-SME gives the same qualitative results, and, by nature, cannot imply the diffusive-thermal instability. This confirms our interpretation of the DNS results that only the Darrieus-Landau instability is involved, and not some differential diffusive effects.

Discussion and conclusion

The numerical experiments of the paper actually concern the propagation of three successive different flames: (a) a single-phase premixed flame of equivalence ratio ϕ G that meets droplets and becomes unstable versus Darrieus-Landau instability; (b) a spray-flame -in the P e s > O(1) regime-fully subjected to DL instability, for which the droplets create additional wrinkles; (c) a single-phase premixed flame of equivalence ratio ϕ G that leaves the domain occupied by droplets and remains unstable versus Darrieus-Landau instability.

To sustain the discussion and illustrate the point, in Fig. 11 we have simultaneously drawn two pictures, namely, a picture that show the various successive front patterns, using numerous snapshots of the reaction rate field, and a graph that depicts the mean front position, the slope of the latter curve giving the front velocity. Hence, the mean velocity that corre-Figure 11: Superimposition of twenty one snapshots of the reaction rate field, positioned above the plot of the mean flame location, in such a way that the instant of each snapshot can be read on the time abscissa of the mean front location curve; four zones of particular flame speed are pointed out; parameters are L y = 3s = 72, ϕ G = 1.1 and ϕ L = 0.5 sponds to a particular flame pattern can be read on the position curve, as the slope of the curve vertically below the considered pattern. Fig. 11 invites us to identify four different periods when the front propagates: from the right to the left of the figure, we notice -a first time during which the flame speed remains constant at about U L (ϕ G ), the quasi-planar front pattern being perturbed by the droplets, acting as "holes of weak combustion".

-a second period where the previous perturbations trigger an instability of much larger amplitude and length scale, namely the Darrieus-Landau instability, which superimposes on the droplet-induced pattern. It corresponds to a period of flame speed-up. -a third time during which the flame propagation attains a quasi-steady state of high velocity, i.e. U s = 1.53. The front pattern corresponds to the addition of two kinds of wrinkles, namely, the DL corrugations which are now of maximum amplitude, complemented with droplet-induced foldings which appear as perturbations of length-scale smaller than the DL corrugations.

-the final period corresponds to the propagation of a single-phase flame subjected to DL-instability. The corresponding (last) part of the position curve indicates a constant slope at about U DL = 1.37, which retrieves the propagation velocity of a DL-unstable premixed flame in a periodic box of 72 times the flame thickness. The above sequence of events confirms that droplets trigger the DL-instability, then enhance propagation velocity of the DL-unstable front. In other words, DL-unstable spray-flame is faster than DL-unstable single-phase flame.

To sum-up, the Darrieus-Landau instability has been studied in the framework of spray combustion, where we have focused on the simplest situation of spray-flames that corresponds to P e s > O(1), i.e. at large "spray Péclet number" the planar spray-flame propagates at the same rate as the single-phase premixed flame that shares the same initial fuel-saturated mixture as the spray. Two numerical approaches -a DNS and a Sivashinsky model equation-have brought results in profound agreement. In particular, it has been shown that the droplet action on the premixed flame triggers the Darrieus-Landau instability on a reasonably short distance of propagation. This result is in agreement with the classical results on DL instability, since DL instability is known to require perturbations or noise to develop in a satisfactory manner [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF]. The same triggering effect has been obtained when including "holes of slow combustion" in SME. Furthermore, the DL-cells that appear in the experiments by Chauveau et al. [START_REF] Thimothee | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF] or in the Bradley-Lawes experiments [START_REF] Bradley | Laminar mass burning and entrainment velocities and flame instabilities of isooctane, ethanol and hydrous ethanol/air aerosols[END_REF] are of sub-centrimetric length scales. Now, one centimetre corresponds in the experiments to about 25 times the flame thickness. Therefore, our computational domain with L y = 48 can simulate several practical DL-cells, which is enough to numerical handle the overall DL-dynamics.

A more striking result established in the paper, either by DNS or by SME, concerns the role of the droplets on the flame speed in the context of the fully-developed Darrieus-Landau instability. The DL instability in the nonlinearly saturated regime appears as a folded front with only one cusp, which corresponds to the DL non-linear pattern with the fastest propagation. Such a front perturbed by "holes of slow combustion" exhibits additional wrinkles of shorter wavelengths. Interpreted accordingly with a linear point of view in DL-theory, those short wavelengths would lead the spray-flame to a slower propagation. By contrast, a spread enhancement is observed. This spray-flame speed promotion has to be interpreted in a non-linear manner, in the sense that secondary wrinkles are additionally imposed to the fastest DL-pattern, and still increase the effective flame surface. Such a phenomenon of secondary folds often results from a small scale turbulence, that transports the flame in both positive and negative manners with respect to propagation. Here, the droplets always act in a negative manner, the droplets locally slowing down the propagation. Nevertheless, local slowdown allows the spray-flame to accelerate. Furthermore, both approaches (DNS and SME) agree with each other, claiming that an important mechanism related to the rich spray-flame speed enhancement at high "spray Péclet number" is due to the simple fact that a droplet produces a (rich) zone of weak combustion, which induces additional folds; we are hence faced with the quite paradoxical situation that local quasi-extinction can accelerate overall propagation.

2 T

 2 keyword: spray-flame; two-phase combustion; Darrieus-Landau instability; hydrodynamic instability; droplet combustion Nomenclature δ * f characteristic thickness of the stoichiometric single-phase flame γ expansion parameter ψ front position in Sivashinsky DL-model equation ρ G density of the fuel vapour ρ L density of the fuel liquid phase τ * f characteristic transit time of the stoichiometric single-phase flame τ prop characteristic time of flame propagation between two droplets τ vap characteristic time of droplet vaporization according to D-2 law ϕ G equivalence ratio of the fuel-saturated premixture ϕ L equivalence ratio of fuel under liquid phase to overall quantity of air ϕ T overall equivalence ratio of the spray ϕ u local equivalence ratio of unburnt gases upstream of the non-homogeneous flame D * th,b thermal diffusivity of burnt gases for a stoichiometric mixture L d typical droplet interdistance L x length of the computational domain, parallel to flame propagation L y height of the computational domain, transverse to flame propagation L DL computational box height above which DL-instability develops P e s spray Péclet number R d droplet radius s lattice spacing s = L d √ * b adiabatic temperature of the stoichiometric single-phase flame U L single-phase laminar flame velocity U * L adiabatic single-phase laminar flame velocity at stoichiometry x F mean front position 1 Introduction

Figure 1 :

 1 Figure 1: Alkane/air droplet lattice: initial fuel mass fraction resulting from the positioning of the droplets at the nodes of a 2D-lattice of spacing s. On the very right side of the figure, a planar single-phase premixed flame propagates to the left (in the lattice: ϕ G = 1.1 , ϕ L = 0.5, s = 12). The computational domain represented is L y = 2 × s hight and L x = 9.7 × s long.

  flame temperature T * b and burnt gas density ρ * b . Hereinafter, the subscript "b" is associated with the burnt mixture, at the flame temperature T b . The superscript " * " is associated with the values assessed for the adiabatic, stoichiometric, single-phase flame.

Figure 2 :

 2 Figure 2: Four successive snapshots of the reaction rate, when a planar spray-flame interacts with a droplet in the lean case of initial surrounding mixture (ϕ G = 0.9, ϕ L = 0.5, s = 24, P e s > O(1)).

Figure 3 :Figure 4 :

 34 Figure 3: Four successive snapshots of the reaction rate, when a planar spray-flame interacts with a droplet in the rich case of initial surrounding mixture (ϕ G = 1.1, ϕ L = 0.5, s = 24, P e s > O(1))

Figure 5 :

 5 Figure 5: Superimposed successive snapshots of the reaction rate for the spray-flame subjected to DL-instability that has been triggered by droplets (ϕ G = 1.1, ϕ L = 0.5, s = 24 and L y = 24).

Figure 6 :

 6 Figure 6: Successive front positions provided by the Sivashinsky DL-model equation triggered by "combustion holes"

Figure 7 :

 7 Figure 7: Mean front position vs. time computed with the Sivashinsky equation: when the number of unstable DL modes increases, the slope enhancement due to droplets is more marked.

Figure 8 :

 8 Figure 8: Superimposed successive snapshots of the reaction rate for the spray-flame subjected to subjected to Darrieus-Landau instability (DL) that has been initially triggered outside the droplets (ϕ G = 0.9, ϕ L = 0.5, s = 24 and y-periodicity of L y = 48).

Figure 9 :

 9 Figure 9: x-position of the combustion front vs. time for three different initial surrounding mixtures ϕ G = 0.9, ϕ G = 1.1 and ϕ G = 1.3; DL-affected flame accelerates when entering the droplet lattice (ϕ L = 0.5 and L y = 2s = 48).

Figure 10 :

 10 Figure 10: Normalized spray-flame speed as a function of overall equivalence ratio for various s and L y , the liquid loading being fixed to ϕ L = 0.5 (curves labelled s = 0 correspond to the single-phase flames).

  1/2 -1)/(2 -γ) is the exact Darrieus Landau coefficient, a(γ) is a coefficient controlling the flame amplitude given in [22]. I(ψ) corresponds to the multiplication by |k| in Fourier space, k n corresponding to the number of unstable modes in a box of width 2π. U ef f is an effective normal propagation speed, position-dependent, i.e. equal to U g = 1,

Acknowledgements

The present work has received the support of the Research Program "Micropesanteur Fondamentale et Appliquée", GDR CNRS n2799, under contract CNES/150515.


Elias Al Sarraf

Christophe Almarcha

Basile Radisson

Joel Quinard

Bruno Denet

Premixed

Hele-Shaw Burner

E Al-Sarraf

Premixed flame instability in a Hele Shaw burner

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Combustion instabilities are of primary importance for turbulent burning in actual burners and there are some hopes so that they can be simulated using a model equation such as the Sivashinsky nonlinear equation first proposed in 1977 for the flame front propagation [1]. Large-scale flame front in a number of configurations were studied theoretically and numerically [2,[START_REF] Michelson | Thermal expansion induced cellular flames[END_REF] in good qualitative agreement with the experiments [START_REF] Joulin | Potential-flow models for channelled twodimensional premixed flames around near-circular obstacles[END_REF]. However a quantitative comparison is hardly expected unless a 2D-flame can be observed.

To this end, we study a flame propagating in a Hele-Shaw burner where the flame dynamics is more easily recorded than in a cylindrical burner. The only parameters of the model are a cut-off wavelength λ c and a characteristic growth rate σ of the most unstable perturbation with wavelength λ max ~2λ c . These parameters can be estimated knowing the unstable wavelength at the onset of instability for a planar flame propagating downwards [START_REF] Truffaut | Experimental Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF], and some direct measurements of the rate of growth σ were performed as a function of the wavenumber of the flame perturbation [START_REF] Joulin | Potential-flow models for channelled twodimensional premixed flames around near-circular obstacles[END_REF][START_REF] Almarcha | Premixed flames propagating freely in tubes[END_REF]. But these values are not directly to be carried forward on the present configuration, since a point is to know whether the mechanism of instability can be described in the same way in a 2D configuration and for a 3D planar flame [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF].

The burner consists of two glass plates 0.50 m wide and 1.5 m high, separated by a gap width of 5 or 10 mm. An acoustic damper, situated at the bottom, prevents the thermo-acoustic instabilities by damping the reflection of acoustic waves. After each run, the airflow is opened and maintained until the tube walls have cooled to ambient temperature. The flow of combustible, methane or propane, is then adjusted to the desired equivalence ratio and a 2D inverted V-flame is ignited at the top of the burner with a lighter. Closing the valve at the bottom of the burner then stops the flow and the downward flame propagation is recorded thanks to a high-speed camera (Fig. 12).

EXPERIMENTAL RESULTS

Experiments were performed with propane and methane-air mixtures whose dynamic properties are a priori known [START_REF] Truffaut | Experimental Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF]. Near stoichiometric mixtures were diluted with nitrogen in order to reduce the growth rate of the perturbations and to improve the accuracy of the measurements. The initial flame contour appears relatively flat (Fig. 2), so Fourier analysis of the successive records of the flame contour can give access to the linear growth rate of the spontaneously excited * Corresponding author. Email: quinard@irphe.univ-mrs.fr perturbations of the flame (Fig. 3). Using this procedure, we only access to the growth rate of large wavelengths of perturbations (Fig. 4, colored symbols) as the information on the domain of large wavenumbers is quickly hidden by the most unstable response of the flame. Introducing a plate with periodic bights a few centimeters above the burner exit, it is possible to force the flame response at the desired wavelength, so extending the measurements of the dispersion relation to larger wavenumbers k=2π/λ (Fig. 4, black symbols). A best parabolic fit through these data (blue line) compares favorably with previous measurements (large red symbol) and with the classical relation [START_REF] Truffaut | Experimental Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF][START_REF] Almarcha | Premixed flames propagating freely in tubes[END_REF][START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF]: the slope dσ/dk at small wavenumber is close to the one calculated by considering only the Darrieus-Landau instability, and the general trends of this curve resemble the hypotheses used in Sivashinsky's simulations. In particular, the large-scale cut-off wavelength is close to the one calculated with gravity effects, the small-scale cut-off wavelength is about half the most unstable wavelength, and its value varies with the equivalence ratio of the combustible mixture (not shown here) in relation with Lewis number effects [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF].

However, these wavelengths are slightly larger than those measured at the onset of instability of planar flames propagating downwards, probably because of 3D-effects that modify the 2D flame speed and the transverse flux, inducing changes in the expansion ratio that controls the rate of growth.

CONCLUDING REMARKS

These experiments confirm that the dynamics of 2D-flame that can be observed in a Hele-Shaw burner is analogous to the dynamics of planar flames previously described, with only small changes needed to correct the cut-off wavelength and the growth rate for 3D effects. It is thus probable that these flames could be relevantly simulated using a model equation such as the Sivashinsky's equation.

Figure 1 :

 1 Figure 1: Experimental set-up Figure 2: Flame front at successive moments separated from 16 ms (vertical scale expanded)

Figure 3 :Figure 4 :

 34 Figure 3: Spectrogram of successive flame contours with forcing at λ=15mm; propane-air, φ=0.7, δ= 0.21

Acknowledgements: This work was supported by the French National Research Agency under agreement ANR-14-CE05-0006 and it has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02).


Colette Nicoli

Pierre Haldenwang

Bruno Denet

Specificities of Rich Spray-Flame Spreading

come

Introduction

This works deals with the dynamics of rich two-phase flames. Combustion experiments in sprays at high pressure have revealed behaviours in large departure from the equivalent gaseous premixed flames. The part played by droplet size has clearly been evidenced. Recent microgravity experimental results by Nassouri et al. and Thimothee et al. [START_REF] Nassouri | Combustion of aerosols: structure and speed flame study in microgravity[END_REF][START_REF] Thimothee | Rich spray-flame propagating through a 2d-lattice of alkane droplets inair[END_REF][START_REF] Thimothee | Microgravity experiments and numerical studies on ethanol/air spray-flames[END_REF] indicated that spray-flame is much more unstable than the equivalent single-phase flame. In the classical ground experiments conducted by Hayashi et al. [START_REF] Hayashi | Propagation velocity and structure of flames in droplet-vapor-air mixtures[END_REF] in Wilson chamber, the spray-flame speed measurements had already shown a displacement of the rich flammability limit towards higher equivalence ratio. Recent similar experiments by Bradley et al. [START_REF] Bradley | Laminar Mass burning and entrainment velocities and flame instabilities of isooctane, ethanol and hydrous ethanol/air aerosols[END_REF] in Wilson chamber confirm these trends, and even indicate that increasing liquid loading for very large droplets does not strongly modify the spray-flame speed. Among the earliest experiments on spray-flames, on the rich side U S ϕ T ( ) , the spray-flame speed is found larger than U L ϕ T ( ) , the single-phase flame of the same overall equivalence ratio. An explanation for this velocity increase has been proposed by Hayashi and Kumagai [START_REF] Hayashi | Propagation velocity and structure of flames in droplet-vapor-air mixtures[END_REF][START_REF] Hayashi | Flame Propagation in Fuel Droplet-Vapor-Air Mixtures[END_REF] : the spray-flame speed is simply U L ϕ G ( ) , the velocity of the premixed 1-phase flame with the equivalence ratio of the gaseous surrounding mixture. In other words, droplets do not participate to propagation. Another possible cause for spray-flame speed enhancement, clearly observed in the experiments particularly at high pressure [START_REF] Bradley | Laminar Mass burning and entrainment velocities and flame instabilities of isooctane, ethanol and hydrous ethanol/air aerosols[END_REF][START_REF] Nomura | Effects of pressure on propagation in a premixture containing fine fuel droplets[END_REF], is that spray-flames are subjected to front instabilities, whereas 1-phase flames remain planar in the same conditions.

The present numerical study aims to interpret the increase of spray-flame speed observed on the rich side and to look at the specific role of the droplets on the flame front structuration. This work is devoted to large transversal length, L Y , of the computational box for which the flame front becomes unstable with respect to Darrieu-Landau (DL instability). Recent work [START_REF] Nicoli | Darrieus-Landau instability of premixed flames enhanced by fuel droplets[END_REF] has proved that spray-flame speed promotion on the rich side can be seen as an intrinsic effect, as soon as the droplet size is large enough.

The configuration of spray presently studied consists in schematizing the initial unburnt two-phase medium thanks to a centered 2D-lattice of heavy alkane droplets surrounded by vapour pressure and air (see Fig. 1). In the spirit of the effect sought after, the droplet inter-distance (or lattice spacing s ) is considered to be large in comparison with the characteristic reaction-diffusion scale. In other words, s is here supposed to be too large for modelling combustion through some homogenization process [START_REF] Suard | Different spreading regimes of spray-flames[END_REF][START_REF] Nicoli | Analysis of pulsating spray flames propagating in lean two-phase mixtures with unity Lewis number[END_REF]. In such a configuration, ϕ u , the local equivalence ratio of the fresh mixture, can vary from a very rich (large) value close the droplets to the equivalence ratio far from the droplets (i.e. that due to the vapour pressure in the mixture), hereafter denoted by ϕ G . In the same manner, ϕ L expresses the equivalence ratio of the fuel under liquid phase (or liquid loading), while ϕ T (with ϕ T = ϕ G + ϕ L ) is the overall equivalence ratio .

Because combustion spreads in a medium of variable composition, a chemical scheme taking account of the local equivalence ratio in the unburnt mixture has been developed [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF]. This (simplest) model retains a global irreversible one-step reaction, governed by an Arrhenius law, the heat reaction of which depends on ϕ u . In other words, heat release takes account of all the species existing at the actual flame temperature [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF].

The spray flame speed is determined as a function of the overall equivalence ratio or the liquid loading (i.e the droplet radius effect), when the rich gaseous mixture is fixed at is considered ϕ G = 1. 1 , in a large computational domain L Y = 48 (with the lattice spacing ( s = 24 ).

Modelling Spray-Flame

The present numerical modelling uses the usual set of conservation laws: mass, momenta, energy and species. Since the accurate chemical schemes for alkane are too complex for efficient simulations, a standard approach for a heterogeneous medium considers a simplified chemical kinetics. The simplest manner consists of choosing an irreversible 1-step reaction, the parameters of which are adjusted to mimic the actual flame temperature and dynamics.

It is known that the classical one-step Arrhenius law largely overestimates the adiabatic flame temperature in the rich side. To overcome the difficulty of assessing the main flame characteristics correctly, a modification [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF] of the standard chemical scheme is considered: heat release becomes a linear function F ϕ u ( ) of the fresh pre- mixture equivalence ratio ϕ u . F ϕ u ( ) is adjusted to mimic the premixed single-phase flame behaviours (adiabatic flame temperature and flame speed) [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF]. As for the estimate of ϕ u , two combustion invariants allow us to derive ϕ u from any (burnt or unburnt) point-wise mixture composition [START_REF] Nicoli | Rich Spray-Flame Propagating through a 2DLattice of Alkane Droplets in Air[END_REF].

Non-dimensioning

Non-dimensioning is performed with the use of the theoretical data related the stoichiometric (gaseous) premixed flame as derived in the theoretical papers by Joulin and Mitani [START_REF] Joulin | Linear stability analysis of two-reactant flames[END_REF] and Garcia-Ybarra et al. [START_REF] Garcia-Ybarra | Soret and Dilution Effects on Premixed Flames[END_REF]. The stoichiometric flame temperature, as * b T , is given by ( )

f f p u u b M C Q Y T T ν * f * + =
Temperature and species mass fractions are handled under the reduced forms

θ = ( T -T u ) T b * -T u ( ) ; ψ i = Y i Y i ,u *
(i=f for the alkane and i=o for oxygen) As for the time and length scales, we use D * th,b , the thermal diffusivity coefficient of the burnt gases, and the stoichiometric (gaseous) flame speed as given by

( ) [ ] ) / exp( 4 ) ( * * 2 * * 2 * * 3 2 * , b A b f f b p b b u b L T T B Y M C Ze U u o - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ν ρ ρ λ ρ ρ

Heterogeneous Medium and Combustion model

This allows us to establish the scalar conservation laws under the following dimensionless form

∂θ ∂ t + V .∇θ = 1 ρC p div( λ∇θ ) + F(ϕ u ) W ( ρ ,ψ i ,θ ) ∂ψ i ∂ t + V .∇ψ i = 1 ρ div( ρ D i ∇ψ i ) -ν i M i W ( ρ ,ψ i ,T )
where the reaction rate is now defined as

W(ρ,ψ i ,θ ) = Ze 3 4 ρ u * ρ b * ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 ψ f ψ o ρ ρ b * ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 3 exp Ze θ -1 1 + γ (θ -1 ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥
with the reduced activation energy (Zeldovich number)

Ze = T A ( T b * -T u ) T b * ( ) 2 and γ = ( T b * -T u ) T b * .
As in the conservation laws appears the velocity field V , the reaction-diffusion system is coupled with the Navier-Stokes equations. The overall scheme that computes the Navier-Stokes equations has previously been described in Denet and Haldenwang [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF]. Accordingly with the above non-dimensioning, the general form given to the adjustment function ) F( u ϕ can be approximated by the following quantity

F(ϕ u ) = 1-α(ϕ u -1) [ ] if ϕ u ∈ 0.5, 2 [ ] ,
whereas

W = 0 if ϕ u < 0. 5 or ϕ u > 2.
. α is a coefficient depending on the fuel. For octane-air pre-mixtures, α = 0.33 and Ze = 7 have been chosen from experimental data [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF].

In what follows, the x-coordinate corresponds to the direction of propagation of a folded flame front. To define the mean position of this front, we perform an averaging in the transverse (y) direction of the temperature field to get < θ > y (x) . By definition, x F , the front position, is given by < θ > y (x F ) = 0.5 .

Results: Spray-Flame spreading in transversally large computational domain

The numerical simulations have been carried out using the following parameters: the Lewis number of fuel, oxygen and nitrogen are respectively Le F = 1. 8 , Le O = 0.9 , Le N = 1 . The overall equivalence ratio ϕ T is varied in the rich range 1.6 , 3

⎡ ⎣ ⎤ ⎦ . The spray flame front position, and consequently the spray flame speed, is assessed by following a given isotherm, < θ > y (x F ) = 0.5 , as in [START_REF] Nicoli | Rich Spray-Flame Propagating through a 2DLattice of Alkane Droplets in Air[END_REF].

In a general manner, combustion spreading is found to be not uniform in time: it combines stages of vaporization and stages of flame propagation (see figure 4). The highlevel of non-homogeneity in a spray flame and the complexity of the spreading are illustrated in figures 1 and 2.

L X = 233 ; L Y = 48 ; s = 24 ;ϕ G = 1.1 ; ϕ T = 2 ; R d = 1.48 ( )
As mentioned above, the computational domain of Fig. 1 is transversally large enough to sustain DL instability. The retained value of the transversal box length is L Y = 48 . As ignition is carried out in the single- phase pre-mixture, DL instability classically develops as a cusped front. Then, the cusped pattern meets the successive rows of droplets. To illustrate the interaction between the DL affected premixed flame with the spray droplets, we have plotted in figures 2, three successive snapshots of the fuel mass fraction field and reaction rate field. As in our model the droplets can move, they are carried along, and stretched, by the flow (to the right) due to vaporization and gas expansion. ( )

The DL instability of the spray-flame -in the HK regimeprovokes a stronger speed-up. The great difference between rich spray flame speed and equivalent premixed gaseous flame speed is undoubtedly due to the role of the droplets that bring additional wrinkles to the DL-affected flame front and still accelerate the flame for large enough droplet. This acceleration has not a monotone dependence on the droplet radius. To sum up this numerical study, in

Conclusion

This numerical study on rich spray combustion presents the influence of global spray composition (liquid loading, or droplet radius) on spray-flame speed, with a quasi-stoichiometric gaseous pre-mixture in a large computational domain. The spray flame speed is larger than the speed of the purely gaseous flame with an equivalent equivalence ratio. Purely DL instabilities can develop and increase the flame front surface. In a second time, other wrinkles due to the droplet are added: a new acceleration , depending of the droplet size exists.

Figures 2

 2 represent two fields at three instants in the spray flame soon after the ignition up the consumption of the first droplets. When ( L Y > 14 ), the computational box is large enough to allow the front to develop wrinkles, if initial perturbations are introduced. For the sake of an easy interpretation of the numerical results, we keep the spray Peclet number large enough to maintain the spray-flame in the Hayashi-Kumagai (HK) regime [17]. The spray Peclet number Pe s U L (ϕ G ) ⎡ ⎣ ⎤ ⎦ is defined as the ratio of the droplet vaporization time on the propagation time (through the lattice spacing at the speed U L (ϕ G ) ) We decide to choose the initial spray parameters in such manner that 4.≤ Pe S ≤15. 5 . More precisely, the composition of the spray, has a fixed gaseous pre-mixture ϕ G = 1.1 and the global equivalence ratio is in the range 1.6 ≤ ϕ T ≤ 3. The parameter of the study is ϕ T (and concomitantly R d ) Typical spray example is given in Fig.1, where ignition is performed at right hand side of the computation domain and propagation occurs from the right to the left.

Figure 1 .

 1 Figure 1. Initial fuel mass fraction: dark blue zone (on the right hand edge) indicates gases already burnt; flame then spreads to the left,

Fig.2a indicates

  Fig.2aindicates that the droplets are transported by the flow back to the burnt gases much before their complete vaporization and mixing. This confirms that we are faced with the Hayashi-Kumagai (HK) regime of spray-flames. We note the classical cusped pattern of the flame front with a large length scale that corresponds to L Y = 48 . The main cusped pattern of the reaction rate is marked by additional wrinkles of smaller length scales due droplets.

Figures 2 :

 2 Figures 2: Alkane air mixture: ϕ T = 2. , ϕ G = 1. 1 , ϕ L = 0. 9

  Figures 2: Alkaneair mixture: ϕ T = 2. , ϕ G = 1. 1 , ϕ L = 0. 9 and L Y = 48 ; s = 24 ( )

Figures 3 :

 3 Figures 3: Spray-flame affected by Darrieus-Landau instability ϕ T = 2. , ϕ G = 1. 1 , ϕ L = 0. 9 L Y = 48 ; s = 24;R d = 1.47 ( ) a) Superimposition of successive snapshots of the reaction rate field (b) Mean x-position of the combustion front vs. (time) : sprayflame speed is given by the slope of the curve. Figures 3 illustrate a case of a spray flame spreading in the Hayashi-Kumagai (HK) regime, submitted to the DL instabilities. It appears that spray-flame speed is larger than U DL (ϕ G = 1. 1 ) : the promotion on the rich side can be seen as an intrinsic effect, due to the fact that the droplet size is large enough [17]. Droplet radius influence on the spray flame speed. On figure 4, the spray flame position of quite a few spray mixtures (global equivalence ratio ϕ T or droplet radius) versus the time are compared. ϕ T The mean slope of the curves gives is the spray flame speed. Until time (t=32), the flame front of each studied spray crosses a purely gaseous pre-mixture ϕ G = 1.1 in a large box: the flame position versus time is the same for each spray mixture.After t=32, when the flame front meets the droplets, it can be observed that small differences exist: spray flame is faster for the smallest droplets. For larger values of ϕ T , ie

Figure 4 :

 4 Figure 4 : Flame front position versus the time for various global equivalence ratio (or various droplet radius) L X = 300 ; L Y = 48 ; s = 24 ; ϕ G = 1.1 ( ) On figure 5, the spray flame speed versus droplet radius, shows that as soon as R d ≤ 1. 5 , spray flame speed is about U s ≈ 1. 45 , while if R d ≥ 1. 8 , U s ≈ 1. 33 .

Figure 5 :

 5 Figure 5: Spray flame speed versus droplet radius L X = 300 ; L Y = 48 ; s = 24 ; ϕ G = 1.1

Table 1

 1 defined as the ratio of the droplet vaporization time on the propagation time (through the lattice spacing at the speed U L (ϕ G ) )All the cases studied have a very large Pe s U L (ϕ G ) ⎡ ⎣ ⎤ ⎦ :but, when Pe s U L (ϕ G ) ⎡ ⎣ ⎤ ⎦ ≤ 10 droplets are more efficient to contribute at the acceleration of the spray flame. An optimal droplet size seems to exist to increase the flame front wrinkling.

	ϕ T		1.6	1.7	2	2.5	3
	R d		1.09	1.2	1.475	1.83	2.15
	Pe s U L (ϕ G ) ⎡ ⎣	⎤ ⎦	4	4.84	7.3	11.27	15.5
	U s		1.44 1.46	1.47	1.32	1.33

we have gathered the spray-flame speed when the DL instabilities are developed, for various rich global equivalence ratio. The first line corresponds to a droplet radius for each spray. The second line is the spray Peclet number Pe s U L (ϕ G ) ⎡ ⎣ ⎤ ⎦

Table 1 .

 1 Spray-flame speed U s for various values of R d ϕ G = 1. 1 , L Y = 48 and s =24 ; U L ϕ G = 1.1

	(	) = 0.99

Acknowledgements

This study was granted by the Research Program " Micropesanteur Fondamentale et Appliquée "-GDR 2799 -CNRS/CNES under the contract CNES/140569.


Giri Wahyu Alam

Etienne Pihan

Marie Benoit

Nathalie Mangelinck-Noël

Impact of the initial growth interface on the grain structure in HPMC-Si ingot

Keywords: EBSD, grain structure, HPMC-Si, photoluminescence, seeded growth

In High Performance Multi-Crystalline-Silicon (HPMC-Si) ingots, small seed grains generate grain boundaries that can terminate the propagation of dislocation clusters. Here, we focus on the seed template formation and its impact on the initial growth by directional solidification utilizing metallography, photoluminescence, and EBSD analysis. In the seed region, two randomly oriented grain morphologies are found: a genuine nonmelted seed from the poly-Si chunks, and a re-solidified infiltrated molten silicon region. All grains grow by epitaxy on the seed grains and grains grown from wider grains in the seed, reach a higher solidification height.

I. INTRODUCTION

In both research and industry, seeding is an explored way to control the final structure of Silicon photovoltaic (PV) ingots. In particular, HPMC-Si method makes use of fine grained seeds aiming at creating grain boundaries that can terminate the propagation of dislocation clusters [START_REF] Lan | Grain control in directional solidification of photovoltaic silicon[END_REF][START_REF] Lan | Engineering silicon crystals for photovoltaics[END_REF][START_REF] Ciftja | Growth of High Performance Multicrystalline Silicon: A Literature Review[END_REF]. Indeed, dislocation clusters coupled to impurities are very detrimental to PV properties [START_REF] Supawan Joonwichien | Effects of crystal defects and their interactions with impurities on electrical properties of multicrystalline Si[END_REF] so that it is necessary to limit their propagation. Numerous characterization studies concerning the grain morphology during ingot growth have led to the conclusion that RAGB presence in the growth area and small grain size are of primary importance to provide sufficient termination sites for dislocations clusters [START_REF] Stokkan | Study of evolution of dislocation clusters in high performance multicrystalline silicon[END_REF]. Consequently, the crystalline defect density in the growth ingot can be reduced. However, initial growth on seeds and seed morphology has only been studied in details by few authors. Ronit et al. [START_REF] Prakash | Grain growth of cast-multicrystalline silicon grown from small randomly oriented seed crystal[END_REF] describe the growth from poly-Si seeds as a three stage process. Those authors evidenced an initial growth with spherical grain morphology (stage I), from which grains grow in the solidification direction together with grain selection (stage II) before to reach a steady state in terms of columnar grain size (stage III). From the observation of the grain structure in both FBR (Fluidized Bed Reactor) and poly-Si seeds, Ekstrøm et al. [START_REF] Ekstrøm | Microstructure of multicrystalline silicon seeded by polysilicon chips and fluidized bed reactor granules[END_REF] observed a coarsening of the microstructure of the two seed materials. In the case of poly-Si chips, the authors also observed a larger grain structure initiated from silicon infiltrated in-between poly-Si chunks gaps below the main seed interface.

It is the aim of this contribution to deepen the knowledge of initial grain template formation in seeds and to discuss its consequences on initial grain structure at the start of HPMC-Si ingot growth. The ultimate objective is to tailor the seed layer morphology to control the final grain structure. As a consequence, we focused on metallographic observations, photoluminescence imaging, etch pits analysis and EBSD mapping of:  initial seed layer composed of poly-Si chunks grain structure and inter-chunks gaps,  seed layer formation from genuine non-melted seed from the poly-Si chunks and from re-solidified infiltrated molten silicon,  grain structure in the vicinity of the seed / initial growth interface.

II. EXPERIMENTAL TECHNIQUES

A G2 (60kg) silicon ingot was directionally solidified from high purity (> 250 Ω.cm) WACKER poly-Silicon (Size 4) in a silicon nitride coated VESUVIUS silica crucible. Boron doped silicon was added close to the top of the poly-Si chunks to achieve 1 -2 Ω.cm resistivity target. The melting and growth interface was concave and a dipping rod provided regularly the information on residual solid height in the middle of the ingot. The seed arrangement consisted of 20 mm poly-Si chunks Size 1 (length range 0 -15 mm chunk size). During the melting segment, the top and bottom zone temperatures were set at 1510°C and 1410°C, respectively while the growth rate was set at 1.2 cm/h. The position of the sample extracted from the ingot and studied in details in the following can be seen in red Fig. 1. This sample is situated at the seed layer-initial growth interface in a region of concavity of the solid/liquid interface. Besides, a reference sample (30 mm in diameter) was created by introducing Epoxy Araldite AY-103 resin in size 1 poly-Si chunks to simulate the initial seed morphology and to observe initial grain size and gaps between chunks. After polishing, the selected sample was characterized using photoluminescence (LIS-R2- [START_REF] Trupke | Photoluminescence Imaging for Photovoltaic Applications[END_REF]) to identify seed layer-initial growth interface from doping variation influence on carrier density under illumination. Etch pit studies were performed using Wright etchant [START_REF]CRC Handbook of Metal Etchants[END_REF] to reveal dislocations and grain boundaries. Lastly a SEM -Nova Nanosem 630 was used for EBSD mapping with a 9.9 µm step on a selected 10 x 8.85 mm 2 area across the seed-initial growth interface to study grain structure and crystallographic orientation.

III. ANALYSIS OF THE RESULTS

A. Initial seed layer

The reference sample was observed during resin pouring. There were many gas bubbles escaping from the gaps simultaneously to the resin filling in the gaps between poly-Si chunks. Gas pockets were trapped and concentrated on the isolated areas between the stacked poly-Si chunks or are due to its irregular shape. After resin hardened, it was horizontally polished and vertically cut in the middle revealing the cross section of the poly-Si chunks. Gap widths between poly-Si chunks are found highly variable in the range of 0.1 mm to 15 mm horizontally and 0.06 mm to 7.3 mm vertically. At the same time, the cross section of poly-Si chunk can be very thin around 0.09 µm in thickness.

Metallography of the initial poly-Si chunks reveals two main microstructure morphologies and are measured by Abrams Three-Circle Procedure (Fig. 2) [START_REF]Standard Test Methods for Determining Average Grain Size[END_REF]: isolated concentric columnar grains with grain size over 10 µm, some even reaching 150 µm (red arrows) and abundant small elongated grains, below 10 µm diameter (yellow arrows). Concentric columnar grains are also found in [START_REF] Fancher | Grain growth and mechanical properties in bulk polycrystalline silicon[END_REF] and are called "feather" like structure. This measurement is comparable to the measurement in [START_REF] Fancher | Grain growth and mechanical properties in bulk polycrystalline silicon[END_REF] that used Heyn Intercept Method with a circular line. Additionally, many circle shape defects (yellow rectangle) which, behave as cracks in the chunks are observed on the surface. Their origin is unknown to the authors. Dislocation etch pits are found in all grains of the initial poly-Si under 500x magnification and above. Moreover, Σ3 twins are revealed particularly in the large columnar grains.

B. Seed layer in the grown ingot

PL imaging (Fig. 3) allows identifying precisely the position of the concave initial growth interface separating seed layer (darker bottom domain) and growth (light grey top domain) at the beginning of the process. The unambiguous use of PL is probably related to the different doping concentrations in the seed layer region and in the molten silicon just before the crystallization started. The observation of very bright domains near the growth interface is an indication that dopants are imbedded in the re-solidified areas in the seed layer during the early stages of the melting segment. However, it must be reminded that the PL contrast reveals a complex combination of doping and crystalline quality in the sample. The growth interface is found almost continuous separating spherical and columnar grain structures. In Fig. 3, the genuine non-melted seed domains and the resolidified infiltrated melt domains are revealed by a dark PL intensity and a brighter PL intensity, respectively. Similarly to the reference sample, the gap width between non-melted chunks is around 10 mm at maximum.

Chemical etching of the same area (Fig. 4) allows to identify the associated morphology. One can distinguish two main grain structures and two sublayers in the seed layer. From the interface to a position about 4 mm below it, most of the grains have an aspect ratio close to unity. Below that level, we clearly distinguish genuine non-melted seed domain of poly-Si chunks (size < 200 µm) from re-solidified infiltrated melt domains (size 472 µm in average and Coefficient of Variation (CV) 0.37) [START_REF] Abrams | Grain Size Measurement by the Intercept Method[END_REF].

Moreover, we can see in Fig. 4 that genuine non-melted seed from poly-Si chunks undergoes a strong grain coarsening generating an increasing grain size toward the growth interface (from 18 -72 µm at the seed layer bottom to around 300 µm close to the initial growth interface), everywhere far above the initial grain size. C. Grain structure in the vicinity of the seed layer-growth interface The 10 x 8.85 mm2 area identified as a red box in Fig. 3 was analyzed in more details using EBSD (Fig. 5). The EBSD technique provides both crystallographic orientation (IPF: Inverse Pole Figure, Fig. 5a) and grain boundary types (Fig. 6a).

At first, EBSD results also confirm that all grains from the seed intersecting the initial interface grow as columnar grains due to the directional solidification process. Second, regarding grain size and morphology measured from Heyn Lineal Intercept Method with straight lines [START_REF]Standard Test Methods for Determining Average Grain Size[END_REF] and aspect ratio, it appears that:  the seed layer up to 4 mm below the initial interface position (Fig. 5a) has an average grain size of 465 µm (CV 0.78) and 453 µm (CV 0.94) in the horizontal and vertical directions respectively. The average aspect ratio is around 1 indicating spherical grains in longitudinal sections.  columnar grains above the initial interface position (Fig. 5a) grow on seeds with an average grain size of 352 µm (width) near the seed layer-initial growth interface line.  the average width almost doubles to reach 639 µm after 5 mm of growth on the top side of observed area indicating strong grain selection in the very first 5 millimeters of growth Both morphologies of seed grains (spherical and columnar) have a random orientation as shown in Fig. 5a. In the growth region just above the interface (Fig. 5c), the crystallographic orientations along the growth direction are mostly in direction near-<115> and nearly none near-<101> and <111>. The texture in the initial growth region (Fig. 5b) and seed layer (Fig. 5c) are not identical due to the fact that the orientations of the grains in the seed region are not all present at the level of the initial growth interface. To understand in deeper details the grain formation and competition from the seed, we measure the growth length of the different grain boundary types. In our process, the length percentage of RAGBs (Fig. 6b) is the highest then followed by Σ3, Σ9, Σ27a, and Σ27b length percentage. The growth lengths of Σ9, Σ27a, and Σ27b are the smallest. These observations confirm that grains start there growth by nucleation on the Fig. 6. The grain boundary map showing selected twin grain boundaries (Σ3, Σ9, Σ27a, Σ27b) and RAGB (a) and Length proportion of each grain boundary types in the seed (green) and growth (blue) regions just above the initial interface (b).

randomly oriented seed grains leading to a randomly oriented grain structure at the beginning of growth. This is consistent with the lower occurrence of Σ3 grain boundary type. Another study will be devoted to the evolution of the grain structure upper in the sample where Σ3 grain formation followed by the formation of Σ9, Σ27a, and Σ27b by competition could be expected [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of 〈110〉 seeded growth Si[END_REF] oppositely to random nucleation.

Measuring the grain width at the level of the initial growth interface shows that the grain width is linked to the columnar grain propagation length. In the EBSD observation area, about 50% of the grains have a width between 200 -400 µm, and the rest have a width between 400 -1200 µm.

The growth length of columnar grains issued from a grain width lower than 400 µm is below 1 mm for 70% of them and none reaches a growth length of 3 mm. In the case of grains with an initial width higher than 400 µm, there is no stop of growth before 1 mm, around 20% does not reach a higher growth length than 2 mm and 40% reaches a growth length above 4 mm. Fig. 7. Growth propagation length perpendicularly to the interface as a function of grain width measured at the initial interface from Fig. 5.

IV. DISCUSSION

A. Formation of the seed layer

During the resin pouring, the resin easily reached the bottom of the mold and filled in the gaps between the chunks. The viscosity of this resin (1800 -2400 mPa.s at 25°C [14]) is higher than the molten silicon viscosity at the melting point (0.575 mPa.s [START_REF] Sato | Viscosity of molten silicon and the factors affecting measurement[END_REF]). As a consequence, we can conclude that the molten silicon can flow down and fill in the seed layer gap without retardation and can finally reach the bottom surface of the crucible. In parallel, there is an intense exchange in the seed layer between the argon gas that previously fills the gap between the poly-Si chunks and the infiltrated molten silicon. However, this exchange cannot completely be carried out in some areas because the escaped gas can be trapped in between the stacked chunks, thus creating gas holes as also observed in metal casting. The cavities are not always caused by the trapped gas. Indeed, due to the heat distribution in the seed layer, a colder area in the corner crucible of a concave furnace type is created. When approaching this area, the temperature of flowing molten silicon reduces rapidly and might freeze before completely filling the seed layer gap.

Our results in Fig. 3 and Fig. 4 show that, on the one hand, some large spherical grains (499 µm (CV 0.68) in average) grow during the melting stage inside the seed layer gap and that on the other hand, poly-Si coarsening is significant (from size lower than 100 µm to grain size larger than 200 µm). In the following, the first and the second regions are denoted respectively re-solidified infiltrated molten silicon and genuine non-melted silicon. The first region is created after the infiltrated silicon is accumulated in the bottom crucible and resolidified between the poly-Si chunks. The low temperature difference and the higher solidification rate creates re-solidified grains with an aspect ratio close to 1. In average, these grains have a bigger size than the second region of genuine non-melted silicon at the bottom crucible. This phenomenon is also discussed in [START_REF] Trempa | Defect formation induced by seed-joints during directional solidification of quasi-mono-crystalline silicon ingots[END_REF][START_REF] Wang | Metal-like growth of silicon during rapid solidification by quenching undercooled droplets on a chill plate[END_REF].

The falling molten silicon corresponds to the first silicon melted and can be doped with various amounts of dopants depending on its initial location in the crucible. Indeed, Boron doped silicon was initially situated at the top part of the feedstock arrangement. As a consequence, it is believed that the varying PL contrasts in the seed layer in (Fig. 3) is due to successive melt infiltrations downwards in the gaps even though, the PL contrast can also indicate a defective area. Upon measuring the resistivity, the target value of 1 -2 Ω.cm in the whole ingot growth is attained only from the middle height of the ingot growth. It implies that some dopant still remains somewhere in the seed layer after the melting segment ends. As a consequence, the molten silicon has a lower than expected concentration of boron in the initial growth.

In the vertical direction, the seed layer structure can be divided in two main regions: re-solidified infiltrated molten silicon and genuine non-melted silicon.

In parallel, the applied thermal gradient also creates conditions for grain coarsening inside the seed. A strong grain coarsening on the genuine non-melted seed increases the seed grain size. The very dark PL intensity as found on the bottom crucible is to be related to a smaller grain sizes and a high etch pits density. When grains reach a specific size (> 300 µm), their emitted PL intensity is similar as the one from re-solidified infiltrated molten silicon. This condition is observed in the seed layer up to 4 mm below the growth interface. Consequently, it is difficult to distinguish between grains originating from solidified infiltrations in-between the seed chunks (~ 500 µm) and large coarsened grains of poly-Si chunks (> 200 µm). However, some genuine non-melted seed is still present close to the interface as indicated from the circular crack in poly-Si chunk also evidenced in the literature [START_REF] Ekstrøm | Microstructure of multicrystalline silicon seeded by polysilicon chips and fluidized bed reactor granules[END_REF] and from the dark PL intensity region for different experimental conditions.

Moreover, comparison to the grain structure in the seed layer of another run, with different thermal history during the melting segment, shows that genuine non-melted seed of poly-Si grain coarsening is dependent on thermal history during the melting segment. The grain coarsening might start from the beginning of the melting segment, i.e. earlier than in the case of the infiltrated molten silicon, as long as the poly-Si chunk temperature reaches 1000°C or 1150°C [START_REF]Standard Test Methods for Determining Average Grain Size[END_REF][START_REF] Ouwens | Recrystallization processes in polycrystalline silicon[END_REF]. At this temperature, the grain coarsening could be generated gradually from the poly-Si chunk surface towards the center depending on the heat on its surface. The thickness of the poly-Si chunk might also induce a temperature difference between the surface and the inner area and, finally, influence its grain size as seen in the cross section of genuine non-melted seed near to the crucible bottom.

One of the most important factors in thermal history, that control the grain size, is the temperature setting of heaters in the furnace. The effect of thermal gradient is only found in the vertical cut due to the vertical heat effect whereas the horizontal cut shows similar grain structure

B. Consequences on ingot grain structure

Combined PL and EBSD results prove that initial growth is governed by nucleation on seed so that the seed layer structure is reproduced at the initial growth interface. This simple but important point implies that the initial grain structure is controlled fully by the seed layer properties. As a consequence, the grain size on the seed layer will control the initial size of the columnar grains. Moreover, the wider the grains in the seed layer, the longer the columnar grains whereas a high density of grain boundaries associated to small grains might be preferred in HPMC-Si process. Another important point is that grain selection is strong in the first 5-millimeter of growth, larger grains having a higher survival probability. Further investigation needs to be conducted to find which kind of initial grain size could generate lower dislocated area in the whole ingot.

In the case of the utilization of poly-Si chunks as seed, the combination of the initial grain size variation and of the different thermal gradient might provide a higher variation of grain size in the seed layer. However, lower grain size variation and a bigger grain size might develop in the re-solidified seed. As the larger initial grains have a higher survival probability, the columnar grains grown from the re-solidified seed can develop longer and can be found at the top part of the ingot. We can expect that an appropriate adjustment between the gradient temperature and the poly-Si size can ensure homogenous grain size. Having homogenous grain size might also have an impact on the occurrence of dislocated areas as observed when using FBR as seed in comparison to the poly-Si chunk seed [START_REF] Ekstrøm | Microstructure of multicrystalline silicon seeded by polysilicon chips and fluidized bed reactor granules[END_REF]. From this, we can conclude that we should control the formation of specific grain properties from the seeds to be able to tailor the ingot properties.

Additionally, as the growth initiated from the remaining seed grains, the initial crystallographic orientation is mainly random as expected in the growth of HPMC-Si ingot. This orientation is not promoted only by the genuine non-melted seed, but also by the combination with the re-solidified infiltrated silicon melt seed providing a random orientation base for the HPMC-Si initial growth. The random orientation in the genuine non-melted seed is expected from the initial grain orientation in the poly-Si chunks. The random orientation in the re-solidified infiltrated silicon melt seed is a consequence of the small grains development after arrival of the infiltrated melt in the cold region. These small grains are solidified without preferred orientation and without an assistance from the poly-Si chunk. It leads to a random orientation as in the genuine non-melted seed.

At the start of growth, the occurrence of RAGB and Σ3 boundaries in the vertical section are similar to what is also observed in the literature. This initial growth, in the first 5milimeter, shows similar length of grain boundary type as found in the seed layer near the growth interface. The RAGB length has the highest proportion above 50% followed by Σ3 boundaries. There are several factors that influence the length of RAGB. As the RAGB grow in the same direction together with the columnar grains, they elongate perpendicular to the growth interface. Oppositely, Σ3 boundaries grow in geometrical directions imposed by the {111} facets with a short growth length [START_REF] Tsoutsouva | In situ investigation of the structural defect generation and evolution during the directional solidification of 〈110〉 seeded growth Si[END_REF]. Therefore, the growth length of RAGB is much longer in the initial stage.

However, the Σ3 twin grains are easily formed due to their lowest energy of formation and thus they can be found associated to almost all columnar grains. In the columnar grains, the presence of Σ3 boundaries, which is associated with twin growth, can also be initiated by the seed layer. The grains of the seed layer grow jointly with associated Σ3 twin grains. The Σ3 grain boundaries in the seed layer are not only formed by the re-solidified infiltrated melt seed or during the synthesis of poly-Si chunk in the Siemens process. They can also be formed as twin annealing growth during the coarsening of poly-Si chunk [START_REF] Smallman | Modern Physical Metallurgy and Materials Engineering: Science, Process, Application[END_REF]. This coarsening is clearly seen by comparing the microstructure of poly-Si chunk before and after the ingot growth.

V. SUMMARY OF THE WORK

The seed layer morphology from HPMC-Si ingots is found to vary from the bottom of the crucible to the initial growth interface. There are two domain morphologies at the bottom of the seeds: a genuine non-melted seed, and a re-solidified seed. The genuine non-melted seed undergoes a strong grain coarsening generating an increasing grain size toward the growth interface due to the thermal history.

The comparison of PL imaging and EBSD shows that at the seed layer-growth interface, growth takes place on all grains intersecting the growth interface. Consequently, a strong correlation between the seed layer grain sizes and crystallographic orientation exists in the first growth area. Grains are randomly oriented, contain twins and RAGB boundaries with equivalent radial grain size in this region. The larger seed grains regrown at the initial growth interface have a longer survival length. It suggests that the largest grains in the seeds are more prone to generate the final grain structure after further growth.

As a consequence, two major parameters control grain size and grain selection: seed layer formation during the melting stage and the first stage of columnar growth for the given processing parameters. In the future, we will also focus on the grain crystallographic orientation in the propagation length analysis and on the analysis of the dislocation density and distribution related to the seeding.

Fig. 1 .

 1 Fig. 1. Position of the sample inside the ingot (red parallelepiped).

Fig. 2 .

 2 Fig. 2. Reference typical initial poly-Si microstructure (polished and wright etched reference resin infiltrated sample).

Fig. 3 .

 3 Fig. 3. Photoluminescence imaging using high magnification lens at the bottom of the grown ingot (40x20 mm 2 ) showing both seed layer and growth areas delineated by the initial growth interface.

Fig. 4 .

 4 Fig. 4. Build-up of the same area as in Fig. 3 (optical images after chemical attack). The yellow dashed line delineates the initial growth interface determined from PL. Local grain size in genuine non-melted seed are also indicated.

Fig. 5 .

 5 Fig. 5. (a) EBSD IPF (Inverse Pole Figure) map along the growth direction X0. Texture of the seed layer area (b), and of the initial growth area (c).

ACKNOWLEDGMENTS

This work was partially funded by the CEA-INES, France and the Ministry of Research, Technology and Higher Education Republic of Indonesia, through its Scholarship Program for Research and Innovation in Science and Technology (RISET-Pro) World Bank Loan No. 8245-ID. The authors would like to acknowledge N. Plassat and D. Ponthenier for ingot growth.