Florian Berton
email: florian.berton@inria.fr

Ludovic Hoyet
email: ludovic.hoyet@inria.fr

Anne-Helene Olivier
email: anne-helene.olivier@univ-rennes2.fr

Julien Pettre
email: julien.pettre@inria.fr

Gaze Behaviour During Collision Avoidance Between Walkers: A Preliminary Study to Design an Experimental Platform

Keywords:

When walking, vision is the main source of information that allows us to navigate safely by detecting potential collisions with other walkers. In order to gain a better understanding of the relationship between gaze activity and kinematics of motion during pedestrian interactions, we present in this paper a preliminary study towards designing a more comprehensive experimental platform. In this study, participants are asked to avoid collisions with an upcoming virtual character using a joystick, while we measure their gaze behaviours using an eye-tracker. As we are interested in the effects of potential collisions on gaze activity, i.e., where and when participants look to avoid potential future collisions, we display in our experiment a virtual character for which we vary the initial Time To Closest Approach (ttca) and Distance of Closest Approach (dca) values, to change its risk of collision with our participant. We then measure participant trajectory adjustments and gaze activity during the interaction. Our preliminary results show which type of data this platform produces, and demonstrate the interest of designing more comprehensive experiences and tools to analyze both gaze activity and kinematics.

INTRODUCTION

In recent years, a lot of progress has been made on improving the visual realism of virtual crowds. However, these crowds are far from looking like real ones as they still lack some of the subtle behaviours displayed by humans. Introducing such subtleties in crowd simulators therefore requires to further understand how humans behave in crowds in order to reproduce these behaviours.

This paper follows up this trend by attempting to further our understanding of human behaviours during collision avoidance between two pedestrians. In particular, vision being our main source of information to detect and avoid potential collisions with other walkers, we are interested in this paper in the relationship between gaze activity and locomotion in such situations. Previous work suggests that optic flow has a strong impact on pedestrian locomotion (e.g., [START_REF] Patla | Understanding the roles of vision in the control of human locomotion[END_REF][START_REF] Warren | Visually controlled locomotion: 40 years later[END_REF]). While multiple studies have involved users wearing eye-trackers in real environments to analyze their gaze activity (e.g., [START_REF] Jovancevic-Misic | Adaptive gaze control in natural environments[END_REF][START_REF] Kitazawa | Pedestrian vision and collision avoidance behavior: Investigation of the information process space of pedestrians using an eye tracker[END_REF][START_REF] Liao | Identifying user tasks in map based-pedestrian navigation from eye tracking data[END_REF]), such studies can be difficult to organize in real crowds because of technical, human, and experimental organization. A solution is therefore to design these experiments in Virtual Reality, which was demonstrated to show common properties with behaviours in real environments [START_REF] Olivier | Walking with virtual people: Evaluation of locomotion interfaces in dynamic environments[END_REF]. However, gaze activity during collision avoidance situations has not been explored yet in virtual environments. Therefore, the goal of this paper is to present such a preliminary study in order to prepare the ground for the design of a more comprehensive VR experimental platform.

In order to create these foundations, we designed an experiment in a simple virtual environment where participants are asked to avoid collisions with an upcoming virtual character using a joystick, while we measure their gaze behaviours using an eye-tracker. As we are interested in the effects of potential collisions on gaze activity, i.e., where and when participants look at to avoid potential future collisions, we display in our experiment a virtual character for which we vary the initial Time To Closest Approach (ttca) and Distance of Closest Approach (dca) values, to change its risk of collision with our participant. We then measure participant's trajectory adjustments and gaze activity during the interaction.

In this paper, we therefore present the design of the experimental platform and the framework of the preliminary experiment. While we have not yet produced any results, our contribution is also to present and discuss the type of data expected from such experiments, in particular through data obtained in preliminary tests. This data has been obtained by running a pilot version of the platform on a desktop screen for testing purposes, but our goal is to perform such experiments using an immersive setup in future, such as a CAVE or a Head Mounted Display.

This paper is organized as follows. Section 2 reviews the literature in connection with this work. Then, Section 3 introduces our objectives, and Section 4 describes our experimental framework. Section 5 presents some preliminary results. Then, Section 6 concludes on the experiment and presents our directions for future work.

RELATED WORK

Collision avoidance

Collision avoidance has been extensively studied in previous works. Especially, numerous authors focused on the kinematics aspects by describing motion adaptations performed to avoid a collision, such as speed and orientaion adaptations [START_REF] Olivier | Collision avoidance between two walkers: Role-dependent strategies[END_REF]. Moreover, avoidance strategies depend on the characteristics of the situation (e.g., walking speed, angle of crossing) rather than the characteristics of the walkers (e.g., gender, personality) [START_REF] Huber | Adjustments of Speed and Path when Avoiding Collisions with Another Pedestrian[END_REF] [START_REF] Knorr | Influence of Person-and Situation-Specific Characteristics on Collision Avoidance Behavior in Human Locomotion[END_REF]. They also showed that walkers are able to accurately predict the future risk of collision during the interaction [START_REF] Olivier | Minimal predicted distance: A common metric for collision avoidance during pairwise interactions between walkers[END_REF]. This notion of future risk of collision is based on the definition of minimum predicted distance (mpd), and depends on the current speed and orientation of the walkers. This is the closest crossing distance walkers would meet if they do not adapt their trajectory (constant velocity). Therefore, walkers adapt their motion if and only if the value of mpd at the beginning of the interaction is below a certain threshold (1m), i.e., when there is a future risk of collision if none of the walkers adapt their motion. When considering the evolution of mpd over time, Olivier et al. [START_REF] Olivier | Minimal predicted distance: A common metric for collision avoidance during pairwise interactions between walkers[END_REF] described 3 stages in the interaction: 1) an observation stage where mpd is constant, meaning that no motion adaptation is performed, 2) a reaction stage, where motion adaptations are performed so as to increase the future crossing distance and avoid a collision, 3) a regulation stage, where mpd is maintained to a constant level. This latter stage shows that the collision avoidance task is solved before the end of the interaction, meaning that the task is performed by anticipation. Also, it is important to mention that mpd is similar to the future distance of closest approach (dca), used in this article.

Because of the complexity of performing such studies in real situations, recent studies aimed at validating Virtual Reality (VR) as a powerful tool to study interactions between walkers. For example, virtual trajectories performed in a CAVE using a joystick to avoid a virtual walker showed very similar qualitative patterns as compared to real conditions [START_REF] Olivier | Walking with virtual people: Evaluation of locomotion interfaces in dynamic environments[END_REF]. In that context, Varma et al. [START_REF] Varma | Assessing the Relevance of Eye Gaze Patterns During Collision Avoidance in Virtual Reality[END_REF] used an immersive VR platform to study the influence of the virtual agents' eye gaze direction on the behaviour of participants in a collision avoidance task. While participants' preferential tendency was to avoid the virtual agent turning right, participants changed their strategy turning left when the virtual agent shows non verbal cues through head and eye orientation indicating the intent to go to the right. Other studies use VR to investigate how participants avoid a group of virtual walkers, i.e., if they go through or around such groups [START_REF] Bruneau | Going through, going around: A study on individual avoidance of groups[END_REF], or to investigate the motion cues used by participants to successively interpret the motion of the virtual walker they are interacting with [START_REF] Lynch | Collision avoidance behavior between walkers: global and local motion cues[END_REF]. VR therefore offers new opportunities to study motion adaptations during interactions between walkers as well as more perceptual mechanisms allowing a perfect control of the information displayed to the participants.

Gaze and interactions between walkers

While all the previously mentioned studies focused on collision avoidance and virtual environments to study interactions between walkers, almost none of them used or explored information about participants' gaze activity in relation with the trajectory adaptations. However, vision is a fundamental perceptual system, that allows us among others to control locomotion (e.g., [START_REF] Patla | Understanding the roles of vision in the control of human locomotion[END_REF][START_REF] Warren | Visually controlled locomotion: 40 years later[END_REF]), giving us information about our own position and motion in the environment but also information about the static and dynamic characteristics of the environment. For instance, humans are able to perceive collisions, even in crowded environments, as was demonstrated by Andersen et al. [START_REF] Andersen | Perceptual information and attentional constraints in visual search of collision events[END_REF] who designed an experiment where participants had to detect potential collisions with moving spheres on a computer screen. They showed that participants often detected collisions, but that accuracy decreased with the number of spheres, suggesting that collision detection is based on a visual search and is influenced by the number of objects.

In more ecological contexts, several works investigated gaze behaviour of walkers using an eye-tracking device in real environments. Kitazawa et Fujiyama [START_REF] Kitazawa | Pedestrian vision and collision avoidance behavior: Investigation of the information process space of pedestrians using an eye tracker[END_REF] studied the relation between gaze and the Information Process Space (IPS) during a collision avoidance event with participants walking on a platform. They noticed that participants do not gaze more at objects and other pedestrians located in the scene than at the ground. Furthermore, they deduced that the IPS shape is not a homogeneous circle around the walker, but presents a more important anterior area. Similarly, Jovancevic-Misic and Hayhoe [START_REF] Jovancevic-Misic | Adaptive gaze control in natural environments[END_REF] demonstrated that participants adapt their gaze strategies depending on the behaviour of surrounding persons. By asking participants to walk around an oval track while other people acted in specific ways with predefined risks of collision, they showed that pedestrians with risky behaviours were more gazed at by participants. However, gaze behaviours were also demonstrated to be affected by other factors, e.g., related to the environment. For instance, Fotios et al. [START_REF] Fotios | Using eye-tracking to identify pedestrians critical visual tasks, part 1. dual task approach[END_REF] demonstrated that participants gaze more at the ground when the lighting is diminished in the virtual environment, and that the probability of looking at other pedestrians was increased when they were in the range of 8 to 12m [START_REF] Fotios | Using eye-tracking to identify pedestrians critical visual tasks. part 2. fixation on pedestrians[END_REF].

OBJECTIVES

A lot of work has been done to describe motion adaptations performed by a walker to avoid collisions with another walker. However, since vision is fundamental to control locomotion, other studies investigated the gaze behaviour performed by walkers when navigating in their environment. However, little is known about the coupling between gaze behaviour and locomotor adaptation during collision avoidance. For example, while three stages in the interactions have been detailed (i.e., observation, reaction and regulation) based on the kinematics of locomotion [START_REF] Olivier | Minimal predicted distance: A common metric for collision avoidance during pairwise interactions between walkers[END_REF], no information about gaze behaviour has been provided. Our objective is then to study the relation between gaze and trajectory adaptations in a collision avoidance task between two walkers. We aim at analyzing the duration of gaze fixation as well as the timing of gaze with respect to trajectory adaptations and the risk of collision.

This paper is a work-in-progress article. We start answering this objective by developing an experimental framework involving a collision avoidance task between a participant and a virtual walker. The experimental platform design is detailed in the next Section.

EXPERIMENTAL FRAMEWORK

Apparatus

Apparatus is illustrated in Figure 1. We used Unity 5.5.4 to design the 3D environment display and character animation. Participants navigate in the virtual environment using a joystick. The longitudinal axis of the joystick controls speed linearly from 0.8m.s -1 to 2m.s -1 . The lateral axis controls the angular rotation speed linearly from -25deg.s -1 to 25deg.s -1 . If participants do not touch the joystick (rest position), the speed is set at 1.33m.s -1 , which represents a comfort human walking speed [START_REF] Bohannon | Comfortable and maximum walking speed of adults aged 2079 years: reference values and determinants[END_REF], and the angular rotation speed is set to 0deg.s -1 (i.e., straight-ahead walk). This steering navigation interface combined with this transfer function was previously validated for the study of human walkers interactions in a virtual environment [START_REF] Olivier | Walking with virtual people: Evaluation of locomotion interfaces in dynamic environments[END_REF]. To ensure that ground displacements are perceived in the optic flow as participants are moving forward in the virtual environment, we chose a texture with a random noise for the virtual ground as suggested by Geri et al. [START_REF] Geri | Simulating time-to-contact when both target and observer are in motion[END_REF]. While participants are seated in front of a 24-inch screen for the experiment, gaze activity is recorded using a desktop eye-tracker ("TheEyeTribe", accuracy: 0.5-1 • , sampling frequency: 60Hz).

Task

Participants are instructed to navigate in the virtual environment towards a target located at 800m from the starting point (see Figure 2), while avoiding any collision with the virtual human walking within the same area. The virtual human displays a neutral appearance, to avoid any influence on gaze behaviour of factors such as gender, clothes, color, eye contact, and is driven by a RVO crowd simulator [START_REF] Van Den | Interactive navigation of multiple agents in crowded environments[END_REF] with a preferred speed of 1.33m.s -1 . It therefore walks along a straight path at a constant speed as long as there is no immediate collision with the participant, but also presents late reactions in cases when participants do not avoid it.

Factors

As detailed in Section 3, we hypothesized that a strong link exists between the user's gaze behaviour and the risk of collision with the virtual walker. Therefore, to assess this hypothesis, we propose to manipulate two factors related to the risk of collision: the future distance of closest approach (dca) and the time to closest approach (ttca). These factors can be computed at each frame of the interaction as illustrated in Figure 3. For a frame n, assuming that the user and the virtual human would keep a constant speed and orientation, we can linearly extrapolate their future trajectories and compute the minimum crossing distance (dca(n)) as well as the time remaining before the dca(n) will occur (ttca(n)). In this study, we will therefore manipulate the initial values of dca and ttca, i.e., when the user can first see the virtual human, in order to introduce variations in the initial risk of collision:

• dca will vary between -3 to 3m (following a uniform distribution), to consider situations were the future crossing distance goes from high values without collision if no motion adaptation is performed to full contact. The sign of dca relates to whether the user is expected to be in front of the virtual human at the time of crossing (dca>0), or behind at the time of crossing (dca<0).

• ttca will be set to 4s and 8s, to consider short term and longer term potential interactions.

• the initial position of the virtual human in the user's field of view will be randomly chosen (following a uniform distribution) to avoid any confounding effects of the initial position.

PRELIMINARY RESULTS

During a preliminary experiment, one participant (male, 26 years old) completed a number of 20 trials. In addition to gaze data, we recorded the trajectory of both the participant and the virtual human, which provides us with their positions and velocities over time. This data and the gaze position were re-sampled and interpolated for all the scenarios with an interval of 0.05s (20 frames per second). We defined a fixation when the participant gazed at the same point for more than 0.1s. Furthermore, we defined a collision threshold at 0.5m, i.e, the approximated distance between two pedestrians from which a physical contact almost always occurs. Figure 4 (Left) describes the trajectory of the participant and the virtual agent over time for 2 trials performed by the participant. Figure 4 (Right) shows the corresponding evolution of dca until the time of closest approach (ttca). A change in dca means that motion adaptations were performed by the participant. With respect to these Figures, we superimposed the gaze behaviour highlighting with several colours the elements of the environment the participant was looking at.

Figure 4, upper part, illustrates a trial with an initial value of dca equal to -0.44m and a ttca of 4s. This means that given the initial conditions of the interaction, if the participant does not adapt his trajectory, a collision would occur because dca is below the collision threshold. During this trial, we can observe that the participant gazed at the different elements in the scene: the target, the ground, the virtual human and the sky. In particular, we observe that the participant fixated the virtual human (between 2.1s and 2.6s) just before adapting his trajectory, i.e., when the dca value starts changing.

Figure 4, lower part, illustrates a second trial with an initial value of dca equal to 1.66m and a ttca of 8s. Contrary to the first trial, the initial value of dca is above the collision threshold. Theoretically, there is no major risk of collision in this interaction even if the participant does not modify his trajectory. As in the previous example, the participant gazed at the different elements of the virtual scene. Furthermore, we also noticed a fixation during the phase of observation as defined previously by Olivier et al. [START_REF] Olivier | Minimal predicted distance: A common metric for collision avoidance during pairwise interactions between walkers[END_REF] (between 2.4s and 4.2s). While this fixation is longer than the one of the previous example, the trajectory adaptation is however less important, as expected from the difference in dca.

In this paper, we only observe few trials. They illustrate the nature of data we can extract from our experimental framework. More specifically, they show the relevance of the framework to perform analysis of collision avoidance in space and time, coupling gaze and trajectory data. For example, in our examples, we can notice that participants gaze at the character when they initiate adaptations (dca start variating). Also, gaze patterns seem different before and after participants start adapting their trajectory, with shorter fixations during avoidance than before. We now need to perform a complete the virtual environment are color-coded on the curves. Up-Right: participant and virtual human trajectories during one trial where initial dca is -0.44m and ttca is 4s. Up-Left: evolution of dca over time during this scenario, until the time of closest distance. Bottom-Right: trajectories for one trial where initial dca is 1.66m and a ttca of 8s. Bottom-Left: evolution of dca over time during this scenario, until the time of closest distance.

experimental campaign in order to confirm our conclusions.

CONCLUSION & FUTURE WORK

In this paper we present an experimental framework to study the gaze activity during a collision avoidance task between a participant and a virtual walker. Using an eye-tracker to record eye movements, a joystick to navigate in a virtual environment and a desktop setup, we are able to know what participants are looking at and to make the link between their gaze behaviour and the adaptation of their trajectory to avoid potential collision.

Using this platform, we performed a preliminary experiment, whose very first results show trends in line with our initial hypotheses. For instance, in the two examples described in Section 5, the participant looked at the virtual human just before performing adaptations. However, more work is now required to explore these factors, including more participants on a complete experimental plan. We also plan to conduct this experiment in an immersive setup such as a CAVE, rather than on a desktop PC. Technical issues will therefore need to be tackled to use the eye tracker in such a setup.

Concerning the virtual environment, we are currently considering an improvement of its visual realism. In particular, we plan to add character shadows because of their importance on perceiving 3D trajectories, which is a main parameter for navigation. We are also considering using more realistic characters and environments (e.g., street or city square), but such changes will require to carefully account for potential confounding factors, e.g., character gender or the color of its clothes.

In terms of analysis, we plan to compute the duration of fixations, as well as their timing during the interaction. They will be studied with respect to the dca values as well as motion adaptations, which will be made possible by our experimental control of the dca factor. One main objective will then be to identify standard gaze behaviour patterns during a collision avoidance task. If some invariant patterns are found, they could be used to design new animation techniques based on real-user gaze behavior. This can result in more realistic interactions between a user and a virtual walker. Finally, this platform will allow us to subsequently study the relationship between gaze activity and locomotion under the influence of other variables such as the number of people in the scene or their facial expressions.

Figure 1 :

 1 Figure 1: Example of a participant during our experiment. Participants are seated in front of a 24-inch screen. They use a joystick to navigate in the virtual environment and to avoid a virtual walker. An eye-tracker is located below the screen and records their eye movements.

Figure 2 :

 2 Figure2: Illustration of the virtual environment, with the virtual walker to avoid, the goal to reach and the location of the participant's gaze in the virtual environment (not displayed during the experiment).

Figure 3 :

 3 Figure3: dca (distance of closest approach) and ttca (time to closest approach) computations. For each frame n of the interaction, these variables are computed by a linear extrapolation of trajectories given the current state of the user and walkers at frame n.

Figure 4 :

 4 Figure 4: All Figures: Gaze locationsin the virtual environment are color-coded on the curves. Up-Right: participant and virtual human trajectories during one trial where initial dca is -0.44m and ttca is 4s. Up-Left: evolution of dca over time during this scenario, until the time of closest distance. Bottom-Right: trajectories for one trial where initial dca is 1.66m and a ttca of 8s. Bottom-Left: evolution of dca over time during this scenario, until the time of closest distance.

  Figure 4: All Figures: Gaze locationsin the virtual environment are color-coded on the curves. Up-Right: participant and virtual human trajectories during one trial where initial dca is -0.44m and ttca is 4s. Up-Left: evolution of dca over time during this scenario, until the time of closest distance. Bottom-Right: trajectories for one trial where initial dca is 1.66m and a ttca of 8s. Bottom-Left: evolution of dca over time during this scenario, until the time of closest distance.


Colette Nicoli

Pierre Haldenwang

Bruno Denet

Premixed flame dynamics in presence of mist

Keywords: Water-sprayed flame, misted flame, flame mitigation, premixed flame speed-up, Darrieus-Landau instability

come

can be observed when particular water mists are injected within the premixture.

To contribute to shed light upon these less than intuitive observations, we propose to study the propagation of a nearly stoichiometric premixed flame within a 2D-lattice of water droplets. Main parameters of investigation are droplet size and droplet inter-distance (or equivalently, lattice spacing). When the droplet inter-distance is small, the results confirm that a sufficient amount of water quenches combustion. For larger droplet inter-distance, we observe a flame speed enhancement for suitable droplet size. Concomitantly, the flame front folds subjected to Darrieus-Landau instability. The final discussion, which invokes a Sivashinsky-type model equation for DL instability, interprets such a speed promotion in presence of mist as a secondary non-linear enhancement of the flame surface.

Introduction

Deflagration of fuel-air mixture is an omnipresent threat in the enclosures that contain industrial processes or energy production systems. To prevent or mitigate the risk of explosion, systems of water sprinkling are expected to be efficient devices for safety. As a matter of fact, numerous experiments have studied the propagation of a flame within a premixture composed of air, gaseous fuel and water mist. The situation corresponds to the so-called "misted" flame.

It has clearly been established that the interaction between these components can lead to quench deflagration and/or detonation [START_REF] Thomas | Studies of detonation quenching by water sprays[END_REF][START_REF] Thomas | Influence of water sprays on explosion development in fuel-air mixtures[END_REF], to mitigate gas explosion [START_REF] Kees Van Wingerden | The influence of water sprays on gas explosions. part 2: Mitigation[END_REF], or to extinguish diffusion flame [START_REF] Shimizu | Experiments and numerical simulation on methane flame quenching by water mist[END_REF]. In other situations, premixed flame interaction with water mist can delay the transition to detonation [START_REF] Boeck | Influence of water mist on flame acceleration, DDT and detonation in H2-air mixtures[END_REF]. The interaction can be restricted to only modify the structure of the deflagration [START_REF] Thimothée | Experimental investigation of the mechanisms of cellular instabilities developing on spherical two-phase flames[END_REF] or the detonation [START_REF] Jarsalé | Ethylene-air detonation in water spray[END_REF]. Microgravity facilities offer an ideal environment for controlling the water spray characteristics and studying the effects of a quiescent water mist on the propagation of a premixed flame [START_REF] Abbud-Madrid | The influence of water mists on premixed flame propagation in microgravity[END_REF][START_REF] Thimothée | Experimental investigation of the mechanisms of cellular instabilities developing on spherical two-phase flames[END_REF], the objective being the deduction of quantitative data about the effects of mean droplet inter-distance and droplet radius on extinction.

On the other side, and surprisingly, experimentalists often report on the opposite effect: injection of water mist can promote flame propagation [START_REF] Gieras | Flame acceleration due to water droplets action[END_REF][START_REF] Zhang | Enhancement effects of methane/air explosion caused by water spraying in a sealed vessel[END_REF]. In these articles, the aerodynamic effects produced by the injection of water droplets are invoked for creating large scale turbulence inside the premixture, and resulting in an acceleration of the flame propagation. By contrast with this interpretation, which involves turbulence added by the mist injection, the purpose of the present contribution is to show that premixed acceleration is also possible with quiescent water mist, and to conclude that water droplets can intrinsically accelerate a flame already subjected to the Darrieus-Landau instability. More generally, since fuel storage in liquid phase is known to achieve -for the time being-the best ratio of energy to weight, spray combustion is at the root of a wide field of research. This has long motivated numerous investi-gations dealing with liquid-injected combustion and spray-flame dynamics. The present numerical investigation follows a series of studies dedicated to fuel/air spray-flames. To schematize the initial state of the premixture, we have proposed to consider the fuel/air spray as a 2D-lattice, at the nodes of which fuel droplets are located. A set of contributions [START_REF] Nicoli | Lean flame dynamics through a 2D lattice of alkane droplets in air[END_REF], 2015, 2016[START_REF] Thimothée | Microgravity experiments and numerical studies on ethanol/air spray flames[END_REF] has investigated the various roles played by the main spray parameters: s, the lattice spacing, ϕ, the equivalence ratio of the fuel-air premixture that initially surrounds the droplets, and R d , the typical droplet radius. By contrast with those contributions, we presently consider a singlephase premixed flame (i.e. with no fuel in liquid phase) which interacts with "passive" droplets of water. Actually, the present contribution can be seen as directly inspired by our most recent paper [START_REF] Nicoli | Darrieus-landau instability of premixed flames enhanced by fuel droplets[END_REF]] that shows that "passive fuel" droplets are able to first trigger the Darrieus-Landau instability, then induce secondary small-scale non-linear perturbations which provoke additional flame speed-up. Fuel droplets are considered as "passive" when their vaporization is too slow for participating to propagation. This can be met in rich sprays where the so-called "spray Péclet number" is large [START_REF] Nicoli | Darrieus-landau instability of premixed flames enhanced by fuel droplets[END_REF].

The present topic consists in substituting water droplets for "passive fuel droplets". We can here select a near stoichiometric flame ϕ = 1.1 and droplets that can vaporize in such way that they are able to interact with the flame propagation. In other words, the numerical approach considers the mist as a thermal sink, and -when vaporizing-the droplet surroundings become diluted and colder loci where combustion slows down, or even is quenched. Since the presently selected chemical scheme is elementary, we are aware that the chemical role played by water on flame slowdown will not be taken into account. It is however admitted that this effect is generally limited, since it is known that injecting a steam rather than a spray is less efficient in flame quenching (see the review in Ref. [START_REF] Philip | Suppression of methane-air explosions with water in the form of 'fine' mists[END_REF] ).

Model for premixed flame propagation through a water droplet lattice

The article is based on 2D numerical simulations, which start from the basic equations characterizing any reactive medium. The model of spray-flame propagation resorts to the general approach we have used in the 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 water droplets are posi- tioned. The lattice spacing is denoted by s, in such a way that the droplet

inter-distance 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 single-phase medium of heterogeneous composition, where the equivalence ratio varies from ϕ far from the droplets to an unknown 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], vaporization, Stefan flow and heat-mass 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, the current droplets are here fully resolved. This explains why 3-D DNS is hardly affordable for a parametric exhaustive study. The droplets belong to the same continuum as the gas; therefore, they can move, be carried by the flow, heat, follow the properties of a real gas, and continuously switch from (liquid) dense fluid to (vapour) light fluid; they also can distort themselves into non-circular patterns, in absence of surface tension. The initial droplet radius varies in a range from 0 to 2 in terms of flame thickness units δ * f : at standard pressure, the droplets are hence submillimetric. The present simulation takes advantage of the fact that surface tension is neglected: the liquid fuel containment is simply achieved by the fact that heat and mass transfers are frozen at the spray initial temperature (the spray being at rest far upstream from the flame). The latter point is carried out by using the non-linear dependence of the diffusion coefficients with respect to temperature. Lastly, the latent heat of water is accounted for by a strongly non-linear dependence of heat capacity with temperature. Note additionally that the experiments in cloud Wilson chambers, carried out in microgravity [START_REF] Thimothée | Experimental investigation of the mechanisms of cellular instabilities developing on spherical two-phase flames[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 and the implemented numerical methods have largely been described in previous publications [START_REF] Nicoli | Rich spray-flame propagating through a 2D-lattice of alkane droplets in air[END_REF][START_REF] Nicoli | Darrieus-landau instability of premixed flames enhanced by fuel droplets[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. More precisely, the units for time, length, mass and temperature are derived from the theoretical properties of this ideal flame, in terms of flame thick-

ness δ * f = D * th,b /U * L , transit time τ * f = D * th,b /U * 2
L , flame temperature T * b and burnt gas density ρ * b . Hereinafter, the the subscript "u" is hence associated with the fresh mixture, at the initial temperature T u . 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.

The numerical experiments are conducted as follows. Water 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 appears as a dense water puff belonging to the same continuum as its surrounding premixture. Heat-mass transfers, vaporization and the related gas expansion start when the medium is heated by the proximity of the flame. Among the main quantities computed, the flame spreading is considered. Since 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, t). We then define x F (t), the "mean flame" position at time t, as the locus where T y (x F , t) = 0.5. U , the flame spreading rate, is hence ẋF (t), the derivative of x F (t).

Results

From the classical theory of DL-instability, it is known that the Darrieus-Landau instability (DLI)presents a threshold in wavelength. This indicates that the development of DL-instability within the framework of periodic conditions transverse to the propagation 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. Thus, before working with droplets, singlephase 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 DL-instability in single-phase premixture, and the threshold L y = L DL is the so-called cut-off scale of DL instability.

Next, we shall only consider transversally large box (i.e. L y > L DL ). More precisely, L y will take the values L y = 24, 36 and 48. As we shall always set L y = 2s = 2 √ 2L d , the boxes will correspond to three cases of droplet inter-distance, which are hereafter called small, moderate and large droplet inter-distance L d , respectively. In the course of its propagation, the premixed flame regularly meets the water droplets of the lattice. As can be observed in Fig. 1, droplets of large size do not vaporize entirely when crossing the flame. As a result, the initial amount of liquid water is not the sole relevant parameter of the study, since it often does not fully interact with flame propagation. It directly acts for quenching only when the water droplet entirely vaporizes in the vicinity of the front, i.e. for small droplet radii or, more precisely, when the spray Péclet number is small enough. Hence, the main parameters are the droplet inter-distance -or the lattice spacing-and the droplet radius (or equivalently the water loading). In what follows, we give different fixed values to s, the lattice spacing, and vary the droplet radius. In the range of the studied parameters, we have observed flame extinction, flame mitigation, and flame acceleration.

Small droplet inter-distance

The figure 2 presents x F (t), the position of the "mean flame" against time, for L d = 12/ √ 2. The mean slope of these curves will be called flame speed. As can be remarked in the figure, when the flame meets a droplet, this can modify the instantaneous "mean flame" velocity, all the more so that the actual flame remains flat (normally to its propagation towards the x-direction). This is clearly the case of the curve for R d = 1.09, the propagation of which is affected by the droplets. For the intermediate water loading R d = 2.18, flame quenching occurs very early (actually, shortly after the lattice entrance). As for the smaller water loading obtained for R d = 0.545, the flame still propagates, being affected by the DL-instability. Its velocity appears slightly smaller than that without water loading. The figure 2 illustrates what is commonly accepted: water mist can quench or mitigate combustion spreading.

Moderate droplet inter-distance

Let us now consider the same variation of water droplet radii for the larger lattice spacing s = 18. The figure 3 again presents x F (t), the position of the "mean flame" against time. Only, the water loading of R d = 2.18 is sufficient to quench the premixed flame (actually, the oscillations in the figure indicates that the "mean flame" position hesitates between two distinct positions just before extinction). On the other hand, a striking phenomenon of premixed flame acceleration occurs for the other water loadings. As a matter of fact, the mean slope for R d = 0.545 and R d = 1.09 appears faster that the single-phase flame non-loaded with water droplets (i.e. R d = 0). In other words, only big droplets can affect a zone large enough for quenching the flame. Contrarily, when this zone is limited, it ends up with the slight enhancement of the premixed velocity. Here, the increase remains modest, being estimated to equal about 10%.

Large droplet inter-distance

Here, we consider the largest lattice spacing (s = 24). We have once more studied the "mean flame" position versus time. The result is presented in Fig. 4 where x F (t) is drawn against time. We did not find any water loading that is sufficient to quench the premixed flame. More precisely, all the mean slopes for R d = 0.545, R d = 1.09 and R d = 2.18 indicate that premixed flame propagates faster in the presence of water droplets, provided that the droplet inter-distance is suitable. In other words, the domain of influence of each droplet -essentially due to Stefan flow-is too small -in comparison with the flame surface-to damp or muffle the flame. Contrarily, the presence of water droplets is responsible of flame speed-up, which is here of 15%. In the forthcoming section, we discuss the mechanisms involved either in quenching or in speed promotion. Moreover, it will be shown that a flame speed-up larger than 50% can be expected in very large boxes.

Discussion on mist effects

Evidently, the mechanism leading to extinguish a premixed flame resorts to the classical arguments for flame quenching, as the presence of efficient heat sinks. By contrast, the premixed flame acceleration by quiescent water mist requires a novel argumentation that necessitates new developments.

Mechanism for premixed flame extinguishing

In the course of its propagation through the lattice of water droplets, the flame front pattern results from the reconnection (or "wound healing") of two types of local behaviours: a) zones in the vicinity of the droplets, where a thermal sink is active and where the flame does not propagate, b) zones where the premixture seemingly can burn more or less adiabatically. If zones b) does exist, they burn even though they are submitted to lateral heat loss due to zones a). This picture relates to a classical problem: a premixed flame propagating in a channel which enforces heat losses at its walls. An approximate approach conducted with a simple diffusive-thermal model [START_REF] Alliche | Extinction conditions of a premixed flame in a channel[END_REF] showed that -in a narrow channel of a given width-the premixed flame bends, all the more so that the lateral heat losses are large. There is nevertheless a critical value of the flame curvature radius, below which quenching occurs.

In the previous section, flame extinction has been observed for L d = 12/ √ 2 ≈ 8.5. The model mentionned above [START_REF] Alliche | Extinction conditions of a premixed flame in a channel[END_REF] predicted that propagation is allowed in a channel larger that seven times the flame thickness, whatever the strength of the thermal losses at the wall. Therefore, increasing moderately the droplet inter-distance will allow the propagation even for quite large water droplets. This is what appends when

L d is enlarged from L d = 12/ √ 2 to L d = 18/ √ 2 [resp. L d = 24/ √ 2] in paragraph 3.2 [resp. 3.3].
The latter situation is also considered in the next paragraph.

Flame "wound healing" and flame acceleration

Figure 5 illustrates the propagation (from the right to the left) of a premixed flame within the water droplet lattice: set initially flat, the flame front rapidly exhibits the classical cusped form of the flames subjected to Darrieus-Landau instability. Concomitantly, secondary wrinkles occur when the flame front meets a water droplet. Considering Fig. 5, the mechanism now investigated starts from the following considerations: the droplet interdistance is large, in such a way that the premixed flame possesses large zones of free propagation (i.e. more or less adiabatic). As observed for the rich spray-flames of the reference [START_REF] Nicoli | Darrieus-landau instability of premixed flames enhanced by fuel droplets[END_REF] in large boxes, "passive" droplets trigger the Darrieus-Landau instability in the same manner as in Fig. 5. This result was found in agreement with the classical results on DL instability, since the 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 presence of droplets can additionally provoke a more striking behaviour: the role of the droplets on the flame speed in the context of the fully-developed Darrieus-Landau instability. The DL instability in the non-linearly saturated regime appears as a folded front with one cusp [sometimes with two cusps]. One cusp corresponds to the DL non-linear pattern with the fastest propagation. As described by Fig. 5, such a front perturbed by "holes of no combustion" exhibits additional wrinkles of shorter wavelengths, which result from the wound healing of the DL-affected pattern drilled by the extinction holes that accompany the vaporizing droplets. 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 spreading enhancement is observed. This 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.

The last point is illustrated by Fig. 6, where three different quantities are gathered and compared. More precisely, ẋF , the flame mean front velocity as previously defined, is plotted against time in Fig. 6.b, while the instantaneous overall heat release is drawn in Fig. 6.a and whereas instantaneous flame surface area is depicted in Fig. 6.c. In each sub-figure, we have plotted a moving average of the corresponding quantity performed on the floating time interval [-10, 10], which more or less relates to the fluctuation period.

The quantity ẋF , as plotted in Fig. 6.b, calls for the following comments. It presents large fluctuations which do not sound quite physical, especially the velocity fluctuations that become smaller than 1. This is certainly due to the artificial definition of x F . Only the main values obtained by moving average has a clear meaning; it exactly corresponds to the mean slope measured in Fig. 4,i.e. what we have called "misted flame speed". To interpret the large fluctuations, we need the other two sub-pictures (6.a and 6.c), where the fluctuations appear rather in phase with those of ẋF (Fig. 6.b). In Fig. 6.c, the minimum value of the flame area is constant and more or less corresponds to that of the DL-pattern. This indicates that every increase in flame area is due the encounter of a droplet, the influence of which depends on the impact position along the DL-pattern. In others words, the local extinction due to a passive droplet increases the front area. After a "wound healing" process which depends on the droplet location along the DL-pattern, the front retrieves its more classical DL-pattern, and its corresponding velocity. As for the overall heat release plotted in Fig. 6.a, its fluctuations confirm those of the surface, and show particularly complex peaks, the interpretation of which being not easy for the time being.

Note that the moving average of front velocity and heat release gives a rather constant value, while it is not exactly the case for the flame surface, for which a slight increase is observed. We interpret the drift as owing to the droplet diffusion -as time goes-due to the absence of confinement by surface tension in our droplet model. The same argument might prevail for interpreting the shift in peak intensity in Fig. 6.c. Nevertheless, the converging informations brought by Fig. 6 confirm that the mist effects on the DL-affected premixed flame increase the flame front area, the combustion rate, and the flame velocity. As the flame velocity enhancement is clearly established, its strength remains rather modest: about 15%. We observed that the strength of the speed promotion increases with L y . When L y become very large, the numerical cost dramatically increases (all droplets are resolved!), and we decide to explore the tendency with a Sivashinsky-like model equation. This is the purpose of the forthcoming paragraph.

Further considerations for very large L y

To make the study in large box affordable, we select the Sivashinsky-like model equation proposed by [START_REF] Joulin | On a tentative approximate evolution equation for markedly wrinkled flames[END_REF], which allows to handle DL-affected premixed flames with large amplitude wrinkles. The idea consists [START_REF] Nicoli | Darrieus-landau instability of premixed flames enhanced by fuel droplets[END_REF] in considering a propagation domain where some local zones of no propagation are incorporated. After several attempts for choosing the size of the droplet influence zone, it turns out that varying the size of the hole does not greatly change the results, provided that the size remains reasonably small in comparison with the lattice-spacing. The (costless) numerical experiments achieved with the model equation offer an evident theoretical sustainment to the numerical simulations: water droplets induce secondary wrinkles in the Darrieus-Landau pattern, which increase the flame surface and promote the spreading velocity. The model equation parameters are then adjusted to retrieve the flame speed promotion observed in the simulations.

The (lateral) size of the box is now increased from L y = 48 to L y = 144, while the time of integration is prolonged. The results are reported in Fig. 7. The results of propagation without droplet exhibit the same asymptotic value whatever the box size. The value correspond to the DL-affect flame speed, i.e. U DL = 1.1 (in adiabatic, stoichiometric, premixed flame speed units). When the flame is misted, the mean front speed oscillates with various frequencies. For instance for L y = 48, as the flame meets droplets, its velocity starts oscillating with a low frequency and exhibits no speed promotion, while asymptotically the frequency doubles and the speed increases to 1.2 as mean value. Such a non-linear behaviour appears complex and its study stands beyond the scope of the article. If the box size is doubled (L y = 96), or tripled (L y = 144), the mean value of the velocity oscillations increases to 1.3 and 1.4 respectively. The latter value corresponds to a 30% enhancement. This gives cause for hope that much higher promotion can occur in combustion chamber of realistic size.

To sum-up, the present article shows a premixed flame dynamics in presence of mist, which is particularly rich: from extinction to acceleration. The decisive parameter is the droplet inter-distance. When this quantity is small, a critical value of the water amount can quench the premixed flame. When the inter-distance is large, the droplets impose additional flame wrinkles to the Darrieus-Landau pattern. It results in flame speed-up.

Figure 1 :

 1 Figure 1: Water droplet 2D-lattice: H2O mass fraction. On the left side of the figure, the initial field results from the positioning of the droplets at the nodes of a face-centred 2D-lattice of spacing s. On the right side, a nearly planar single-phase premixed flame propagates to the left (ϕ = 1.1, R d = 1.09, s = 12). The computational domain represented is L y = 2 × s high and L x = 7.1 × s long.

Figure 2 :

 2 Figure 2: Small droplet inter-distance: position of the "mean flame" vs. time for several normalized radii of water droplets (ϕ = 1.1, L y = 24, s = 12).

Figure 3 :

 3 Figure 3: Moderate droplet inter-distance: position of the "mean flame" vs. time for several normalized radii of water droplets (ϕ = 1.1, L y = 36, s = 18).

Figure 4 :

 4 Figure 4: Large droplet inter-distance: position of the "mean flame" vs. time for several normalized radii of water droplets (ϕ = 1.1, L y = 48, s = 24).

Figure 5 :

 5 Figure 5: Superimposed successive snapshots of the reaction rate for a premixed flame, where DL-instability has been triggered by droplets (ϕ = 1.1, R d = 1.09, s = 24 and L y = 48).

Figure 6 :

 6 Figure 6: Comparison of three different instantaneous characterizations of flame spreading: (a) overall heat release, (b) ẋF , the flame mean front velocity, and (c) the overall flame surface area. Each quantity is additionally drawn (doted line) smoothed by a moving average on a time interval of 20 normalized time units (ϕ = 1.1, R d = 1.09, s = 24 and L y = 48).

Figure 7 :

 7 Figure 7: Premixed flame velocity vs. time as results of a Sivashinsky-like model equation in large boxes L y = 48, 96 and 144: the DL-velocities are compared with and without droplets (the droplets are modelled by holes of no combustion).

Acknowledgements

The present work has received the support of the Research Program "Micropesanteur Fondamentale et Appliquée", GDR CNRS n2799, under contract CNES/170048.


Elias Al Sarraf

Christophe Almarcha

Joël Quinard

Basile Radisson

Bruno Denet
email: bruno.denet@irphe.univ-mrs.fr

Quantitative analysis of flame instabilities in a Hele-Shaw burner

Keywords: Premixed flames, Darrieus-Landau instability, Hele-Shaw burner

We show in this paper that a Hele-Shaw burner can be used for studying the development of premixed flame instabilities in a quasi-two dimensional configuration.

It is possible to ignite a plane flame at the top of the cell, and to measure quantitatively the growth rates of the instability by image analysis. Experiments are performed with propane and methane-air mixtures. It is found that the most unstable wavelength, and the maximum linear growth rate of perturbations, directly measured in the present experiments, have the same order of magnitude as those previously measured on flames propagating freely downwards in wide tubes.

Introduction

Hydrodynamic flame instabilities (see [START_REF] Clavin | Combustion waves and fronts in flows[END_REF] for a review) can be important for turbulent burning, particularly at high pressure [START_REF] Kobayashi | Flame instability effects on the smallest wrinkling scale and burning velocity of high-pressure turbulent premixed flames[END_REF][START_REF] Savarianandam | Burning velocity of premixed turbulent flames in the weakly wrinkled regime[END_REF][START_REF] Fogla | The turbulent flame speed for low-to-moderate turbulence intensities: Hydrodynamic theory vs. experiments[END_REF] . Many studies of turbulent combustion consider the weakly wrinkled regime where intrinsic instabilities are of primary importance to describe the evolution of the burning velocity [START_REF] Savarianandam | Burning velocity of premixed turbulent flames in the weakly wrinkled regime[END_REF]. These instabilities result in an enhanced flame front area that leads to self-turbulization of combustion, so the flame does not follow simply the incoming velocity fluctuations.

Despite the ever increasing capacity of computer simulations, it is still interesting to validate simplified models that can predict the flame front evolution starting from a reduced set of parameters. More specifically, the Sivashinsky equation [START_REF] Sivashinsky | Nonlinear Analysis of Hydrodynamic Instability in Laminar flames I. Derivation of Basic Equations[END_REF] is known to generate a front dynamics that is qualitatively in agreement with the propagation of unstable premixed flame fronts. This equation in its non dimensional version (see [START_REF] Almarcha | Premixed flames propagating freely in tubes[END_REF] for instance), can be written as: φ t + 1/2(φ x ) 2 = I(φ ) + νφ xx [START_REF] Clavin | Combustion waves and fronts in flows[END_REF] where φ is the flame front position, x the transverse coordinate, I(φ ) is the Landau operator corresponding to multiplication by |k| in Fourier space. The right hand size corresponds to the linear dispersion relation, of the form σ = akbk 2 , where σ is the growth rate of perturbations, k the wavenumber. This non dimensional version of the equation contains only one parameter ν: the number of unstable modes in the domain (ratio domain width/cut-off length scale), which can be obtained from the cutoff wavenumber k c such that every perturbation with a larger wavenumber is damped. This cut-off wavelength is also supposed to be the scaling length that controls the velocity of weakly turbulent flames [START_REF] Savarianandam | Burning velocity of premixed turbulent flames in the weakly wrinkled regime[END_REF] or of a self-turbulent flame with a fractal structure [START_REF] Gostintsev | Self-Similar Propagation Law and Fractal Structure of the Surface of a Free Expanding Turbulent Spherical Flame[END_REF]. This length can be evaluated by direct measurements [START_REF] Almarcha | Premixed flames propagating freely in tubes[END_REF] or from the knowledge of the stability limits of planar premixed flames [START_REF] Quinard | Limites de stabilité des structures cellulaires dans les flammes de prémélange, étude expérimentale[END_REF][START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF], but with a lot of uncertainties.

Precise experimental studies of the hydrodynamic instability are rare, and even the propagation of a laminar flame in a tube can lead to a very complex non-linear evolution if the tube is large enough [START_REF] Almarcha | Premixed flames propagating freely in tubes[END_REF]. It has been suggested by Joulin and Sivashinsky in [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF] and first obtained experimentally by Ronney in [START_REF] Sharif | Premixed-Gas Flame Propagation in Hele-Shaw Cells, Spring Technical Meeting[END_REF] (see also [START_REF] Wongwiwat | Flame Propagation in Narrow Channels at Varying Lewis Number, 25th ICDERS[END_REF]), to study flames in a quasi two dimensional configuration in a Hele-Shaw cell. We study here, following the paper [START_REF] Almarcha | Experimental Two Dimensional Cellular Flames[END_REF], where qualitative results were presented, this configuration, where the flame dynamics is more easily recorded with a high-speed camera than in a cylindrical burner. We will show in the present paper that this Hele-Shaw configuration allows quantitative measurements of the growth rates of the instability at different equivalence ratios for propane and methane-air flames. We measure important parameters of the non-linear modelling : the cut-off wavelength λ c and a characteristic growth rate σ max of the most unstable perturbation with wavelength λ max ∼ 2λ c .

Let us recall that in the literature, the growth rates of the Darrieus-Landau instability have only been measured twice, first by Clanet and Searby for very lean propane air flames [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF] by using an acoustic field to create initially a plane flame, then by Truffaut and Searby [START_REF] Truffaut | Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF] for rich propane air flames by studying the development of the instability on an oblique flame. On the other hand, several direct numerical simulations measuring the instability growth rate exist, but often with a one step global chemistry (see for instance [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF]) and even in the present Hele-Shaw configuration in an effective 2D formulation [START_REF] Kang | A Computational Study of Saffman-Taylor Instability in Premixed Flames[END_REF][START_REF] Kang | Effects of heat and momentum losses on the stability of premixed flames in a narrow channel[END_REF].

It turned out possible to control the initial flame shape in order to study the development of corrugations starting from a planar flame, thus measuring the growth rate of perturbations with various wavelengths. We use a plate with periodic indentations to force a specific wavelength on the front. The protocol of the experiment will be presented first before discussing the procedure that can be used to determine the cut-off wavelength.

Experimental methods

The Hele-Shaw burner consists of two glass plates 50cm wide and 150cm high, separated by an adjustable gap width that will however in this article be kept constant at 4.7mm (Figure 1). The 50cm width has been chosen to accommodate several unstable wavelengths, typically 100 close to stoichiometry (we actually measure the most unstable wavelength, which is approximately λ max ∼ 2λ c for a Darrieus-Landau dispersion relation σ ∝ kk 2 ). The 4.7 mm thickness is a compromise, sufficiently small not to have cells in the thickness, but not very small in order to be far from extinction.

The burner is mounted vertically with the inlet gas mixture at the bottom, open at the upper end and it is closed on both vertical sides. The equivalence ratio (φ ) and dilution (δ = O2/(O2 + N2)) of the premixed gas are controlled via a PC interface connected to mass-flow regulators Bronkhorst EL-Flow with an accuracy of 1%.

The operating procedure is as follows : after each run, the air flow is opened and maintained until the burner walls have cooled to ambient temperature. The flow of combustible mixture is then adjusted to the desired equivalence ratio and dilution, and a 2D inverted V-flame (a 2D Bunsen flame) is ignited at the top of the burner where it remains anchored thanks to a flow rate in excess compared to the laminar flame velocity.

Closing the valve at the bottom of the burner stops the flow and the downwards flame propagation is recorded using a high-speed video camera (Photron FASTCAM SA3).

The decrease of the flow is sufficiently smooth so that the deceleration effect is much smaller than that of gravity and doesn't perturb the growth rate measurement. In addition, the acoustic modes of the cell are not observed during the early stages of destabilization where we measure the growth rate.

Mass flow controller

Mass flow controller

Mass flow controller

Free propagation

The successive positions and shapes of the flame front drawn on Figure 2 show the natural development of corrugations induced by the intrinsic flame instability. Starting from a quasi planar shape when the flame enters the cell, the flame evolves classically first in a linear stage followed by a crest coalescence period that would end with a stationary regime controlled by coalescence and creation of new cells [START_REF] Almarcha | Experimental Two Dimensional Cellular Flames[END_REF]. Let us note that the Sivashinsky equation has exactly the same type of behavior for a large number of unstable modes in the presence of noise. Discrete Fourier transforms of these profiles confirm this tendency with a linear development of corrugations up to the time t = 0.1s. After that time perturbations with wavenumbers around and slightly smaller than k max reach important amplitudes (see Figure 3).

Figure 4 shows the amplitude of different Fourier modes in logarithmic scale versus time, that can be used to measure a linear growth rate with an accuracy of about ±10%.

In Figure 5 (what is actually plotted is the amplitude divided by the wavelength), we illustrate the different regimes in the development of perturbations: a noise dominated regime at low amplitude, a linear regime followed by a non linear saturation. The Such an analysis gives access to the linear growth rate for each mode with noticeable amplitude (see Figure 6), but perturbations with wavenumbers slightly higher than k max were not easily measurable, as their initial amplitude was too small.

Forced propagation

To bypass the difficulties with the free propagation method, we used a forcing of the inverted V flame at the top of the cell. A plate with periodic indentations was placed a few centimeters above the burner exit, thus forcing the flame response at the desired wavelength.

This plate is placed at the top of the burner as shown in figure 7. Its position relative to the glass plate is a compromise: the perturbation amplitude must not be too large to allow a sufficient growth in the linear regime, but not too small either to overcome the noise and imprecisions of the measures. with a forcing at a wavelength 20mm. At the beginning of propagation, the flame is almost plane, then a very regular perturbation appears and a linear evolution is seen up to a time of approximately 0.1s. After this time, the non linear effects come into play, leading to a complex evolution controlled by the coalescence of the crests present on the front. Fourier analysis of the flame front position confirms that the amplitude of the forced perturbation is larger than the one of spontaneous corrugations (Figure 9).

Figure 10 shows the perturbation amplitude corresponding to the forced wavelength in logarithmic scale, compared to the most unstable wavelength λ max which is present anyway even without specific forcing. The indentations on the forcing plate are shown in Figure 7, the forcing at a particular wavelength is only approximate, there are also other wavelengths being created. These wavelengths have a much lower initial amplitude, but the amplification is exponential, after some time the most unstable wavelength is always observed. As seen in Figure 10, it is relatively easy to obtain the growth rate corresponding to the forcing wavelength. Let us note however that it The linear growth rates of the perturbations can be measured in a range of wavenumbers where σ is typically larger than half its maximal value. Outside this range the measurement of the perturbation is biased by the spontaneous evolution of the most unstable wavelength. Depending on the experiment, the accuracy of each measurement of the linear growth rate can be estimated between 5 to 10% of the measured value, depending on the equivalence ratio and amplitude of initial forcing.

As shown in Figure 11 where growth rates obtained by the forcing method (empty green diamonds) are superimposed to the previous results obtained by free propagation (crosses), the use of forcing represents a marked improvement. Values of growth rates in the descending part of the dispersion relation can now be obtained, the precision is generally better, the most unstable wavelength, an important physical parameter, can be obtained in a much easier way.

We show in Figure 12 the error bars corresponding to the two wavelengths of Figure 10. It can be seen that the typical error for a forced wavelength is much smaller than in an unforced case. In this last case the initial noise present for small times makes it difficult to choose accurately the slope of the line fitting the experimental points, leading to a large error bar on the growth rate.

3 Propane-air and methane-air mixtures

Growth rates

The results for propane-air flames are plotted for different equivalence ratios in Figure 13. The lines in this figure are best parabolic fits (of the form σ = akbk 2 ) through the data [START_REF] Clavin | Combustion waves and fronts in flows[END_REF], the symbols are measurements using the forced propagation method. The Fit by a second order polynomial: blue curve by Pelcé and Clavin [START_REF] Pelcé | Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames[END_REF] (see also [START_REF] Clavin | The influence of the temperature dependence of diffusivities on the dynamics[END_REF] for temperature dependent diffusivities), Matalon and Matkowsky [START_REF] Matalon | Flames as gasdynamic discontinuities[END_REF], Frankel and Sivashinsky [START_REF] Frankel | The effect of viscosity on hydrodynamic stability of a plane flame front[END_REF]. These articles do not include heat and viscous losses. The two last papers obtain, for zero gravity, the growth rates in the form σ = akbk 2 , where a depends on gas expansion and laminar flame velocity and b depends on Lewis number : this is apparently valid for small wavenumbers and for every Lewis number except very low Lewis numbers where b would become negative. Figure 14 shows the same type of results for methane-air mixtures (symbols, experimental measurements by the forcing method, lines, best parabolic fit). Note that this is the first time that growth rates of methane-air flames are obtained experimentally, the results of the literature being limited to propane-air flames. We have less difficulties for methane than for propane air flames, the results for methane can be obtained even for rich flames.

The maximum growth rate first begins to increase with the equivalence ratio up to φ = 1.1, then sharply decreases for rich flames. On the lean side, the growth rate decreases less than in the propane case (the limiting component is methane, with a Lewis number close to unity). A parabolic fit through such measurements gives access to the maximum growth rate σ max and to the most amplified wavenumber k max , We will report these quantities in the next subsection.

The growth rates of the flames shown in Figures 13 and14 correspond to downward propagating flames. In the propane case, we compare in Figure 15 the growth rates of upward and downward propagating flames for an equivalence ratio of 0.9 and a cell thickness of 3.5mm (for this thickness the growth rates are slightly smaller). As expected (see [START_REF] Pelcé | Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames[END_REF][START_REF] Clavin | The influence of the temperature dependence of diffusivities on the dynamics[END_REF]), upward propagating flames are more unstable, but the difference is not extremely large. According to Joulin and Sivashinsky [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF] the main effect of viscous losses (Saffman Taylor effect) is to act like upward gravity, and thus increase the growth rate. We will see that our measurements in the Hele-Shaw burner are lower than previous measurements, suggesting that the effect of viscous losses is small and much lower than the effect of heat losses, as in the numerical simulations of Kang Baek and Im [START_REF] Kang | Effects of heat and momentum losses on the stability of premixed flames in a narrow channel[END_REF]). In accordance with the theoretical analysis of Joulin and Sivashinsky, we find that the primary instability in the Hele-Shaw burner is due to Darrieus-Landau effects. However these effects are reduced by heat losses. .Figure 15 shows a small effect of buoyancy. With the cell thickness we use (in order to have a quasi two dimensional propagation), flames with a very small burning velocity are quenched, which limits the role of buoyancy. Slow flames can be obtained for very lean or rich flames, but another possibility is to vary the dilution by adding nitrogen. We show in Figure 16 the growth rates of downward propagating methane flames (φ = 1.1) for different dilutions (we recall that δ = O2/(O2 + N2) is a dilution in oxygen). Reducing δ (adding nitrogen) reduces the flame speed and increases the role of gravity. Let us note however that we were not able to measure growth rates for flames close to the stability threshold far for the most unstable wavelength.

Maximum growth rate

In Figure 17, we show for propane and methane-air flames our values for the maximum growth rate σ max in the Hele-Shaw burner versus equivalence ratio compared to the previous results of the literature for the propane case. Let us first consider the propane case, our values are plotted in red, the values of Clanet and Searby [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF] in magenta, the values of Truffaut and Searby [START_REF] Truffaut | Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF] in black. These different authors all report the maximum growth rate in a narrow range of equivalence ratio. We have already explained in a previous section the limitations we have in the present Hele-Shaw experiment, let us discuss these two previous articles. In the Clanet Searby case, there is a limitation to very lean propane flames because the method they use to obtain a plane flame (parametric restabilization of the front by an acoustic field) works only for very slow flames. Let us also note that Clanet and Searby do not report actually the maximum growth rate but measure only the growth rate for one wavenumber (expected to be close to the maximum). Truffaut and Searby on the other hand, measure the growth of perturbations on oblique flames ; there is apparently a limitation to the rich case in order to anchor the flame.

The values that we measure for the growth rate are of the same order of magnitude as these two previous experiments but seem to have a lower value, particularly compared to Truffaut and Searby. We interpret these lower values in the Hele-Shaw cell as being caused by the heat losses to the walls. We include in appendix A measurements of velocity profiles, which seem to indicate that the effective gas expansion in the Hele-Shaw burner is smaller than for flames propagating in tubes.

Unfortunately, we could not obtain growth rates for the same equivalence ratio as the previous measurements. As in the case of propane growth rates decrease quickly for lean flames, the difference between our values in the Hele Shaw cell and measurements of Clanet and Searby [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF] and Truffaut and Searby [START_REF] Truffaut | Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF] is probably around 30%, except close to extinction.

For the methane-air case our experimental measurements are the first ones in the literature. For lean or stoichiometric flames, σ max is close to the value obtained for propane, then the growth rate decreases on the rich side.

Figure 18 reports the values of k max , the wavenumber corresponding to the maximum growth rate. Clanet and Searby do not measure this value since they force their front at one wavelength, but we have added the results of Quinard [START_REF] Quinard | Limites de stabilité des structures cellulaires dans les flammes de prémélange, étude expérimentale[END_REF][START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF] obtained at the threshold (for σ max = 0) for flames highly diluted with nitrogen. This high dilution leads to smaller values of k max [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF], so it is not surprising that the Quinard values are smaller, both in the propane and the methane case. For the propane case, the values we obtain (in red) seem smaller than the Truffaut Searby values, but naturally larger than the Quinard values (light blue). For the methane case (green points), we The values of k max can be used to obtain the cut-off wavenumber k c ≈ 2k max , leading to our estimates of the number of unstable modes (typically 100 close to stoichiometry)

in our 50cm burner. The Hele-Shaw burner thickness we use (4.7mm) is close to the cut-off wavelength in this case, since it has been chosen to prevent the appearance of cells in the thickness while being sufficiently large to reduce the heat losses effects. presented here, the forcing method is more precise than the free propagation method.

We would like however to compare the forcing method to the methods previously used in the literature to measure growth rates.

Let us start with a general comment. In direct numerical simulations, measurements of growth rates have become relatively common, at least with a one-step chemistry (see for instance [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF][START_REF] Kang | Effects of heat and momentum losses on the stability of premixed flames in a narrow channel[END_REF]). In the experimental literature, on the other hand, we have only two papers, Clanet and Searby [START_REF] Clanet | First experimental study of the Darrieus-Landau instability[END_REF] and Truffaut and Searby [START_REF] Truffaut | Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF], both for propane-air flames.

Clanet-Searby method

Clanet and Searby use an acoustic field of sufficient amplitude to stabilize a plane flame.

They stop the acoustic forcing, perturb the flow with a grid (array of wires), and look at the evolution to measure the growth rate. This is the first experiment where growth rates of the hydrodynamic instability were measured. However, this experiment has a number of drawbacks.

1. First of all, the amplitude of the acoustic field must be chosen in order to stabilize the plane flame. For slow flames (< 20 cm/s), e.g. very lean propane-air flames, this is indeed possible, but for relatively fast flames, for instance close to stoichiometry, there is simply no acoustic amplitude able to stabilize the plane flame. On the other hand, the Clanet Searby method would probably also work for flames diluted with nitrogen.

2. The evolution after the acoustic forcing is stopped starts with an important acceleration. Normally, a Darrieus Landau instability for any wavenumber generates two modes, a growing mode (the one where we measure growth rates in the present paper) and a decaying mode. With a large initial acceleration the amplitude of the decaying mode is large. As a result the precision of the measurement is low, Clanet and Searby use some fit with two exponentials to measure the growth rate.

Acknowledgements

We thank Guy Joulin for many stimulating discussions on the Hele-Shaw configuration. which can be compared to the value obtained by Cantera [START_REF] Goodwin | Cantera: An object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes[END_REF] with a San Diego mechanism [26] E = 7.57 for propane at φ = 0.9. As expected the value obtained experimentally is slightly lower, probably because of heat losses. The typical thickness of the velocity profile is also close to the one given by complex chemistry, shown as a dashed line on the figure. Note that these profiles are obtained by PIV, it is possible that out-of-plane effects exist in this configuration. It seems that as suggested by Joulin and Sivashinsky [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF] heat losses have an important role in the Hele-Shaw cell, which

FuelFig. 1

 1 Fig. 1 Experimental set-up

Fig. 2

 2 Fig. 2 Flame front at successive moments, close to the top of the Hele Shaw cell, showing the linear development of the hydrodynamic instability, free propagation, propane-air mixture φ = 0.81.

Fig. 3 Fig. 4 Fig. 5

 345 Fig. 3 Temporal evolution of the Fourier transform of the fronts of Figure 2.

Fig. 6

 6 Fig.6Growth rate vs wavenumber for a propane-air mixture at φ = 0.81, free propagation

Fig. 7 Figure 8 ,

 78 Fig. 7 Position of the metallic forcing plate at the top of the burner

Fig. 8

 8 Fig. 8 Flame front at successive moments, close to the top of the Hele Shaw cell, showing the linear development of the hydrodynamic instability, forced propagation. propane-air mixture φ = 0.81, forcing wavelength 20mm.

Fig. 9

 9 Fig. 9 Temporal evolution of the Fourier transform of the fronts of Figure 8.

Fig. 11

 11 Fig.11Growth rate vs wavenumber for a propane-air mixture at φ = 0.81, forced propagation (green empty diamonds) superimposed to the free propagation results of figure6(black crosses),

Fig. 12 1 (Fig. 13

 12113 Fig. 12 Error bars for the two wavelengths of Figure 10. Propane-air mixture at φ = 0.81

Fig. 14

 14 Fig.14Growth rate vs wavenumber for methane-air mixtures.

Fig. 15

 15 Fig.[START_REF] Truffaut | Study of the Darrieus-Landau instability on an inverted-'V' flame, and measurement of the Markstein number[END_REF] Growth rate of downward and upward propagating flames for propane-air flames at φ = 0.9

Fig. 16

 16 Fig. 16 Growth rate of downward propagating methane flames for different dilutions

Fig. 17

 17 Fig. 17 Maximum growth rate versus equivalence ratio for methane-air and propane-air mixtures

4 A

 4 comparison of the different experimental methods to measure growth rates We have presented in this article experimental techniques to measure growth rates of the flame hydrodynamic instability in a Hele-Shaw cell. Among the two techniques

Fig. 18

 18 Fig. 18 wavenumber corresponding to the maximum growth rate versus equivalence ratio for methane-air and propane-air mixtures .

3 .

 3 The initial wavelength is created by a grid in the tube. ApparentlyClanet and Searby have tried grids with several wavelengths, but were only able to report results for one wavelength. It is difficult to know exactly what were the problems at the time (they claim in the article that the method should work for several wavelengths), but let us contrast that with our method with the forcing plate at the top of the cell. If the forcing is too large (so that we have no linear regime) or too small (then we have noise) we simply move the forcing plate and perform a new experiment. It seems difficult to control the initial amplitude in such a way in the Clanet Searby setup. than the two previous experiments, suggesting that heat losses is the dominant effect.Nevertheless except close to extinction, our values of growth rates are not completely different from the previous experiments. The general form of the dispersion relation is also the one typical of a Darrieus-Landau instability.5 ConclusionWe have shown in this article that a Hele-Shaw burner, a configuration proposed twenty years ago by Joulin and Sivashinsky, can be used for a quantitative analysis of flame instabilities. The instability in a Hele Shaw cell, although apparently smaller because of heat losses, is close to the usual Darrieus Landau instability observed in tubes. Because of the quasi two dimensional geometry, the experiments are relatively easy to perform. Very unstable flames (hundred of unstable modes, a situation typical of high pressure premixed flames) can be obtained here at atmospheric pressure. It was possible to obtain a plane flame at the top of the burner and study the development of perturbations by a simple image analysis. In this article, we have insisted on the experimental technique needed to obtain quantitative results. We plan to study in future papers if our results are comparable to existing experiments in terms of Markstein length and if the dispersion relation measured here can help in the non linear modelling of the front propagation.

Fig. 19 Figure 20

 1920 Fig. 19 Velocity field and normals at different positions on the front in a small cell 60 cm×1.5 cm×5 mm

7 Compliance with Ethical Statements Funding : This work was supported by the French National Research Agency (project PDF) under agreement ANR-14-CE05-0006. The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University -A*MIDEX, a French "Investissements d'Avenir" programme. It has been carried out in the framework of the Labex MEC.

Truffaut-Searby method

Truffaut Searby use an electrostatic forcing on one side of an inverted V-flame and study the development of the perturbation to measure the growth rates. It seems limited to rich flames (to anchor the flame probably) and to relatively large wavenumbers : the authors consider that the two sides of the flame perturb each other for low wavenumbers and do not report the growth rates in this case. It is not obvious anyway that the perturbations on an oblique flame behave exactly in the same way as for plane flames for small wavenumbers. The Truffaut Searby method is interesting compared to Clanet and Searby method since they can control the initial amplitude of the perturbation with the electrostatic forcing in order to stay in the linear regime. This experiment is probably more precise than that of Clanet and Searby, and works for fast flames. There is however the possibility of having vortices created behind the forcing rod which were apparently encountered in some cases [START_REF] Truffaut | Etude expérimentale de l'origine du bruit émis par les flammes de chalumeaux[END_REF] and could perturb the measurements.

Present experiment

In the present experiment, the initial objective was to study flame instability in Hele-Shaw cells quantitatively. It was found possible thanks to the forcing method, which allows us to control the initial amplitude of the perturbation. Compared to the experiment of Truffaut and Searby, the method works for lower equivalence ratio in the propane-air case and lower wavenumbers, but until now we were unable to perform measurements for rich propane-air flames. Of course there are two main effects, viscous losses and heat losses, which exist in a Hele-Shaw cell and are not present in a tube.

Viscous losses (Saffman-Taylor effects) are supposed to increase the growth rate, heat losses to lower it. We found definitively that the growth rates we measured were lower Conflict of Interest: The authors declare that they have no conflict of interest.

Appendix A Gas expansion

We have shown in this article that the growth rates of the hydrodynamic instability in the Hele-Shaw cell, are generally smaller than those in a tube, which suggests that heat losses play a role. It would be interesting to measure the temperature profile in the burner, but this is a difficult task. We show in this section that by measuring the velocity field on both sides of the flame , burnt and fresh gases, by a PIV technique using cesium oxide particles as seeding material, experimental values of the expansion ratio E can be obtained. It is known [START_REF] Frankel | On effects due to thermal expansion and Lewis number in spherical flame propagation[END_REF] that the normal mass flux is not exactly conserved across the flame, so measuring the velocity jump between fresh and burnt gases does not give exactly the expansion ratio, however for our purpose we only need approximate values of the expansion ratio (these values will be used to discuss in more detail the Darrieus Landau instability in the Hele-Shaw burner in future papers).

Figure 19 (propane φ = 0.9) shows the velocity field in the fresh and burnt gases and the normals at different positions on the front in a small cell (60 cm×1.5 cm× 5 mm).

We use a smaller cell here in order to reduce the presence of cusps on the front, the heat losses however are slightly increased. could be described by a lower gas expansion.