Antoine Bottenmuller
email: antoine.bottenmuller@gmail.comgavet@emse.fr

Léo Théodon

Johan Debayle

Daniel Tobón Vélez

Mallorie Tourbin

Christine Frances

Yann Gavet

Granulometric Analysis of Maltodextrin Particles Observed by Scanning Electron Microscopy

Keywords: granulometry, SEM, numerical simulation, overlapping particles

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction 1.Context

Nowadays, an increasing number of active compounds coming out of discovery (food, pharmaceutics, cosmetics, etc.) have limited applications mainly due to problems of poor stability, water insolubility or high toxicity for example. In order to increase the bioavailability and improve the stability of these compounds, different nanotechnology-based delivery systems have been developed. Among these, nanoencapsulation represents an effective approach to improve solubility, minimize degradation process, reduce toxicity and control the active absorption and biological response of molecules like polyphenols [START_REF] Conte | Polyphenols nanoencapsulation for therapeutic applications[END_REF]. Polyphenolic compounds among which the Chlorogenic acid (CGA) group, and particularly the 5-O-caffeoylquinic acid (5-CQA), exert important therapeutic roles such as antioxidant, antibacterial, anti-inflammatory or cardioprotective [START_REF] Naveed | Chlorogenic acid (cga): A pharmacological review and call for further research[END_REF]. Encapsulation could be performed by spray-drying technique, using the Nano Spray-Dryer B-90 from Büchi Labortechnik AG (Fig. 1), which allows to coat different active ingredients (AI), producing dry nanoparticles [START_REF] Arpagaus | Nano spray drying for encapsulation of pharmaceuticals[END_REF].

The wall materials in which the AI is confined provide protection from the surrounding environment, preserving their chemical, physical and biological properties, as well as enable to control its release [START_REF] Fang | Encapsulation of polyphenols -a review[END_REF]. In this study, maltodextrin has been used as a wall material to coat the 5-CQA. When encapsulating, the release of the active molecule depends on several properties of the particles, especially their size distribution, which itself depends on the process parameters. It is consequently very important to determine the size distribution of the produced particles to optimize the delivery properties of the active molecule.

Figure 1: Schematic Nano Spray-Dryer set-up for the encapsulation of CGA by maltodextrin. Adapted from [START_REF] Piñón-Balderrama | Encapsulation of active ingredients in food industry by spray-drying and nano spray-drying technologies[END_REF] and [START_REF] Cano-Sarabia | Nanoencapsulation[END_REF].

Images acquisition

A sample of the dry powder collected at the end of the spray-drying process has been analyzed by Scanning Electron Microscopy (SEM), in order to characterize their size and morphology. The equipment used is a Field-Emission Scanning Electron Microscope JEOL JSM 7100F. The dry particles are placed on a carbon film and metallized by coating with a thin layer of gold using a plasma coater, to avoid excessive electron absorption. The surface of the sample is scanned by an electron beam produced from a tungsten filament and the observation of the particle surface is done by detecting secondary electrons (with a voltage of 10 kV). Electron micrographs were taken at 20000X of magnification. The sample was scanned in order to take about twenty images representative of the entire population of particles constituting the powder (Fig. 6a).

Segmentation methods

To perform a granulometric analysis in such images, the use of a circle detection method is needed. Two of the most efficient and used methods from the literature for circle detection have been studied and tested for this problem: the Stochastic Watershed (SW) [START_REF] Angulo | Stochastic watershed segmentation[END_REF] and the Circular Hough Transform (CHT) [START_REF] Ioannou | Circle recognition through a 2d hough transform and radius histogramming[END_REF]. As these two methods have some major problems in this situation, a new method has been developed to deal with them: the Curvature Analysis Method (CAM).

Stochastic watershed

Although the watershed segmentation is a strong method for contours extraction in smooth images for which the contours of the objects are clear enough and well defined, it is not robust for objects with major light effects and heterogeneous 3D aspects, and too many false contours may be detected [START_REF] Zhang | A watershed segmentation algorithm based on an optimal marker for bubble size measurement[END_REF], [START_REF] Lou | Watershed segmentation of topographical features on freeform surfaces and its application to additively manufactured surfaces[END_REF]. The Stochastic Watershed (SW) enhances the results by accumulating multiple watershed realizations with random markers [START_REF] Malmberg | Exact evaluation of targeted stochastic watershed cuts[END_REF]. A distance transform is then applied on these extracted contours [START_REF] Fabbri | 2d euclidean distance transform algorithms: A comparative survey[END_REF], and the local maxima are calculated and defined as the centres of the different circles on the map, and the value of the distance gives their radius.

Circular Hough transform

The Circular Hough Transform (CHT) is one of the most widely used methods for circle detection [START_REF] Smereka | Circular object detection using a modified hough transform[END_REF], [START_REF] Yuan | Lattice strain mapping using circular hough transform for electron diffraction disk detection[END_REF], [START_REF] Adem | Impact of activation functions and number of layers on detection of exudates using circular hough transform and convolutional neural networks[END_REF]. After a low-pass filter on the gradient magnitude image and a binarization of the contours, a discrete threedimensional CHT space is built and the circles are extracted from it [START_REF] Mirzaei | An automatic algorithm for determination of the nanoparticles from tem images using circular hough transform[END_REF].

Proposed method: curvature analysis

The two previous methods produce however either under or over-segmentation, especially in the presence of overlapping grains, and provide a grain size distribution far from reality. The Curvature Analysis Method (CAM) has been thought and developed to give a solution to this problem. This is a procedural algorithm divided into three steps:

1. From the original image, the so-called "linear minimum MSE map" is built; it represents the probability for each pixel to be on the contour of a grain, by minimizing a weighted Mean Squared Error (MSE) function in a pre-defined window for each pixel of the image (Fig. 2).

2. Then, a clean binary image of the thin arcs (portion of the contours) of the grains is extracted from the linear minimum MSE map (Fig. 3).

3. The associated set of circles best fitting with these arcs is finally estimated (Fig. 4).

These steps will now be detailed.

Building of the linear minimum MSE map

This map is constructed from a circular window scanning the grayscale gradient magnitude image ∇. The line going by the centre p c of this window, and minimizing the weighted MSE regarding all pixels in the window, is obtained by regression. Its slope is denoted by α.

Thus, the error function E pc which must be minimized for each pixel p c from the gradient magnitude ∇, is defined as follows:

E pc (α) = 1 n i=1 ∇(p i ) n i=1 d(p i , L pc,α ) 2 ∇(p i ) (1) with:
(p i ) i∈ 1;n the family of the n pixels surrounding the centre p c in the pre-defined window;

∇(p i ) the gray value of the pixel p i in the gradient magnitude image;

L pc,α the line of slope α ∈ ] -∞; +∞] crossing the point p c in the image's coordinates ;

d(p i , L pc,α
) the Euclidean distance between the point p i and the line L pc,α in the 2D Euclidean vector space:

d(p i , L pc,α ) = (y i -y c ) -α(x i -x c ) √ 1 + α 2
By associating to each pixel p c the corresponding value of the minimum error E pc (α), the "linear minimum MSE map" (Fig. 2e) is finally built and linearly normalized (its values go from 0 for the highest MSE values to 1 for the lowest ones).

Extraction of the thin arcs of the grains' contours

The second step's is extracting the binary image of the thin arcs of the grains' contours from the linear minimum MSE map previously built. It is divided into five sub-steps:

1. First, the linear minimum MSE map is binarized through a K-means clustering on the gray values of the map's pixels, with K = 2 in our case. (Fig. 3a).

2. Then the topological skeletonization [START_REF] Lakshmi | A survey on skeletons in digital image processing[END_REF] is applied on this binary image of the grains' contours, to get the thin topology-preserving version of these contours (Fig. 3b).

3. This skeleton is then cleaned by removing the isolated binary objects for which the area and the maximum Feret diameter are smaller than defined thresholds (Fig. 3c).

4. The remaining arcs are then split by applying an intersection detector: each positive pixel surrounded by three positive pixels or more is considered as an intersection, and is then removed (green parts in Fig. 3d).

5. Again, the arcs are split, according to a curvature criterion (red parts in Fig. 3d). Only arcs with a quite constant curvature will be remaining (Fig. 3e).

Circles association and rearrangement

The last step aims to get the final detected circles from the binary thin arcs of the grains' contours. It is also divided into four sub-steps:

1. First, to each distinct arc is associated one unique circle, defined by the position of its centre p c = (x c , y c ) and its radius r, computed by considering the arc A as a 2D point cloud (A = (p i ) i∈ 1;n ) and minimizing the error function E circle (p c , r), which has a well-defined solution to its minimum, defined as follows:

E circle (p c , r) = 1 n n i=1 ((x i -x c ) 2 + (y i -y c ) 2 -r 2 ) 2 (2)
A threshold on E circle removes the less probable circles. The first remaining circles can be seen on Fig. 4a. 2. Close circles are then detected and merged with the same previous formula. The proximity of the circles is simply computed by the Euclidean distance in the space (x c , y c , r) (Fig. 4b).

3. Circles for which the arcs are fully on another circle are merged with the one circle (Fig. 4c).

4. Finally, knowing the arcs associated to each computed circle, the circles for which the average gray value on the original image near their arcs is higher outside the circle than inside are removed (Fig. 4d). The remaining circles are then the definitive detected circles (Fig. 4e).

Validation

To test the three different segmentation methods in images of grains for which the ground truth (i.e. the true centre position and the true radius of the equivalent disk for each visible grain) is known, a grains simulation model has been developed, based on several properties of the real images. This model will be used to validate the results on the particle size distribution obtained with the three segmentation methods from the ground truth, and compare the accuracy of the three methods between each other.

Grains simulation model

The grains simulation model is inspired by the dead leaves model [START_REF] Bordenave | The dead leaves model: a general tessellation modeling occlusion[END_REF], [START_REF] Jeulin | Dead leaves models: From space tessellations to random functions[END_REF]. It takes three input parameters: the average number of grains n avg on the image, the distribution law L radii of the grains' radii, and the vector of the direction d light of the light source in the half-sphere in the 3D space above the image's plan. It starts by defining a definitive number of grains by choosing it through a uniform law between n avg /2 and 3n avg /2.

The grains are added iteratively to a grayscale image. Each grain is built and added over all the previous grains as follows (see also Fig. 5):

1. The random parameters of the disk, its radius and position, are first generated. The centre is here randomly chosen on the maxima of the distance map computed on the empty space in the image.

2. A random deformation of the shape is then computed, and the properties of the disk (radius, position) are updated according to this new shape. This allows to move the disk to avoid collisions with previous ones. 3. A gray value is generated as a function of the iterative step, and lighting effects (for volume simulation) and a shadow (based on lighting direction) are finally added.

After all the grains are placed and drawn (Fig. 5d), the generated output image is blurred at different depth layers, and little blurred Gaussian noises are finally added to it (Fig. 5e).

Evaluation of the segmentation methods through the model

In this study, we consider two kinds of laws for the Particle Size Distribution (PSD): a bimodal law, and a log-normal law. For each law, 100 images have been generated with the grains simulation model. The ground truth (labels) for all the images is well known and concerns only the visible grains.

The three segmentation methods are then applied on all generated images, with the best inner parameters possible set for each method, resulting in a list of radii of detected circles.

The PSD densities, in number and in volume, of the three segmentation methods as well as the ground truth, are retrieved for the two laws. The four resulting diagrams are presented Fig. 8, and the values of the mean and the standard deviation (STD) of each distribution are given in the Tables 1 and2. The volume densities aim to highlight the largest radii, which are mostly false detections in our simulations when getting over 1µm, even if they aren't abundant in number. They show that, on the one hand, a noticeable proportion (in volume) of false large grains over 1µm of radius have been detected by the CHT, where the ground truth indicates almost no true circles. The SW and CAM, on the other hand, have almost no false detection over this threshold.

The three methods seem all to underestimate the size of the grains in average from the ground truth (see Tables). The graphs of densities in number show that CHT and CAM have both more false little grains than they should have, even if this is less noticeable for the CAM. But the global shape of the true distribution law can be identified on both methods' curves in the four graphs, where this one shape fades with the SW.

More generally, the SW's density curves seem to be quite concentrated around their mean value: their standard deviation is much smaller than the ground truth compared to the two other methods' ones, and their minimum and maximum radii never get far from the mean, preventing detection of much larger or much smaller true grains. On the contrary, the CHT seems to detect a noticeable amount of false grains in extreme radius values despite the thresholds put in its parameters, and its large standard deviations illustrate well this. The CAM, on the other hand, seems to be more stable for both number and volume densities, despite mean values still a little smaller than the truth's ones, and its density curves, as well as its mean and standard deviation values, seem to be the closest ones from the ground truth's in every cases.

Application on real images 4.1 Results

A total of 20 real images of particles of maltodextrin observed by SEM is provided for this study. They all have the same scale. The three segmentation methods are applied (see Fig. 6 for the main processing steps of the CAM, and Fig. 7 for an example of segmentation comparison between the three methods on a real image), and just like for the validation part, the diagrams of the PSD from the lists of the detected circles' radii are built, for densities in both number and volume.

Ground data have been collected by Laser Diffraction (LD) in a dry way with the Mastersizer MS 3000 particle size analyser [START_REF] Faé | Making soil particle size analysis by laser diffraction compatible with standard soil texture determination methods[END_REF]. But as this method of particle size measurement suffers from a few defaults [START_REF] Varga | On the reliability and comparability of laser diffraction grain size measurements of paleosols in loess records[END_REF], it cannot be considered as ground truth. The resulting PSD is illustrated Fig. 9, and the corresponding means and STDs are given in Table 3. Like in the validation process, the graphs and the standard deviations show that the SW's density remains quite concentrated around its mean, where the CHT's one appears very large, with a high proportion of small (Fig. 9a) and large (Fig. 9b) radius values at the same time, and has even once again in its volume density what seems to be a significant proportion of false grains radii over 1µm, limited by the threshold given in its parameters. The CAM seems once again to be more moderated, extended in both small and large radii zones, but in reasonable proportions.

Although the LD's curve seems to be quite close to the ones from the segmentation methods in the graph in number, it is way far from these ones in the graph in volume, showing extremely large values of grains radii up to 60µm. When looking at real images, no grain seems to have a radius over 1µm, and, in reality, no grain could be as large as the LD's curve indicates. It is likely due to grains clusters which are considered as only one single grain when physically analysed by this technique to get its corresponding radius. Based on the CAM results, it can then be concluded that the real PSD on the particles of maltodextrin must be close to a log-normal distribution law, for which the mean is around 0.207µm and the standard deviation around 0.093µm.

Discussion

When applying the SW, a major problem appears: if the number of markers is small, only the main contours of the largest grains are extracted, the tiniest ones having just a little chance to get targeted by a marker, and when increasing this number N , false contours begin to appear very quickly inside the largest grains. Because the size of the grains is far from being constant and because there are many overlapping cases in such images, it seems that there is no proper value for N to detect small and large grains at the same time. These observations could explain the fact that the SW's density curve seems to be quite concentrated around its mean.

Although the CHT seems to be robust on images with well-defined and separated objects, it seems to be quite defective when the grains are condensed and overlapping, and when heterogeneous 3D aspects and major light effects affect the grains appearance. The 3D CHT space building process doesn't consider contours' properties (such as the curvature) and does not make the difference between the pixels of the contours from two different objects. It can therefore easily mix them and detect false circles between two or more different grains if they are close enough or overlapping. This could explain the relatively high proportion of very small and very large false grains in the granulometric analysis.

As the CAM analyses only parts (the arcs) of the contours of the grains, and as the grains are far from being perfectly circular, the detected circle completely depends on the curvature of the one part of contour: if the arc has a too small radius of curvature compared to the radius of the true grain to which it belongs, the associated circle will be smaller than the true corresponding grain, and, on the contrary, if the arc is almost flat, the associated circle will be bigger. Moreover, when the arc is only a little piece of the grain's contours and when its curvature is not steady, the eq. 2 tends to give a smaller radius than the ground truth. This could explain the underestimates of the mean radius of the circles detected by the CAM.

This application finally shows that the LD technique provides a PSD way far from the ones obtained by any of the three segmentation methods and even from the observations by eye. It can not then be considered as trustworthy for such types of particles and configuration.

Conclusion

The particles of maltodextrin from this study are facing particular features: the irregularity of the contours of the grains, the rough aspect of their surface combined with strong light effects, and the way the particles are condensed and mostly overlapping, make the granulometric analysis on such images quite challenging.

A new segmentation approach -the Curvature Analysis Method (CAM) -has been developed to address the problems raised by the existing methods from the literature such as the Circular Hough Transform (CHT) and the Stochastic Watershed (SW). A 2D grains simulation model has also been created to assess and compare the accuracy of the three segmentation methods with a known ground truth.

The analysis of the PSD obtained from each of the three methods on simulated images shows that the CAM's density curves are more accurate than the ones obtained from the SW and the CHT, with less false detections and a better estimate of the mean radius and of the standard deviation. The same results can be observed on real images.

As this study focuses only on topology-based segmentation methods, and to go further, deep learning approaches, such as CNN or U-Nets for objects localisation, could be used as segmentation tools and could give efficient and accurate results to this PSD evaluation problem.

  (a) Original image. (b) Gradient magnitude with the circular window (blue) centred on a pixel pc. (c) View on the one window with the centre pc (green). (d) Line Lp c ,α (red) going by the center pc and minimizing the weighted MSE function. (e) Final computed linear minimum MSE map.

Figure 2 :

 2 Figure 2: Construction steps of the linear minimum MSE map.

  (a) Binarized image of the linear minimum MSE map from Fig. 2e. (b) Binary image skeleton. (c) Clean skeleton. (d) Intersection area (green) and curvature irregularities (red) on clean skeleton.(e) Final split arcs.

Figure 3 :

 3 Figure 3: Steps of the extraction of the thin arcs.

  (a) All circles from arcs with an error below a threshold. (b) Circles close enough in the space (xc, yc, r) to be merged, green and blue being two clusters of merging circles. (c) Circles sharing arcs close enough from the circles edges to be merged, green being one cluster of two merging circles. (d) Circles being brighter inside their arc than outside (green) are removed. (e) Final detected circles.

Figure 4 :

 4 Figure 4: Steps of circles association and rearrangement.

  (a) Binary shape of a grain with random contours. (b) Grayscale light effects added to the shape, where light comes from the left. (c) Shadow of the grain computed from the shape and light direction. (d) An image with several grains added one on top of the other. (e) Final image after being blurred and noised.

Figure 5 :

 5 Figure 5: Example of the generation of a simulated image.

  (a) Original image. (b) Grayscale normalized linear minimum MSE map. (c) Clean binary skeleton from MSE map with curvature splits (red) and intersections (green). (d) Circles computed for all arcs from the skeleton. (e) Final circles segmentation.

Figure 6 :

 6 Figure 6: Example of application of the CAM on a real image: view of the different steps.

  Segmentation with SW (N = 40). (c) Segmentation with CHT. (d) Segmentation with CAM.

Figure 7 :

 7 Figure 7: Example of results from the three segmentation methods on real image.

  Log-normal law: volume densities

Figure 8 :

 8 Figure 8: Diagrams of densities (PSD) retrieved from the simulated images with SW, CHT and CAM methods.

Figure 9 :

 9 Figure 9: Diagrams of densities (PSD) on real images.

Table 1 :

 1 Mean and STD of densities on bimodal law.

	Properties Ground truth SW	CHT	CAM
	Mean	0.199 µm	0.190 µm 0.280 µm 0.191 µm
	STD	0.082 µm	0.058 µm 0.101 µm 0.082 µm
	Properties Ground truth SW	CHT	CAM
	Mean	0.199 µm	0.287 µm 0.282 µm 0.190 µm
	STD	0.082 µm	0.054 µm 0.104 µm 0.081 µm

Table 2 :

 2 Mean and STD of densities on log-normal law.

Table 3 :

 3 Mean and STD of densities on real images.

	Properties LD	SW	CHT	CAM
	Mean	0.217 µm 0.285 µm 0.284 µm 0.207 µm
	STD	0.133 µm 0.052 µm 0.105 µm 0.093 µm


UNIMECA Dominique Morvan
email: dominique.morvan@univ-amu.fr

WIND EFFECTS, UNSTEADY BEHAVIORS, AND REGIMES OF PROPAGATION OF SURFACE FIRES IN OPEN FIELD

Keywords: Fire modeling, Unsteady phenomena, Wildfire behavior, Wind effects

Wind effects, unsteady behaviors, and regimes of propagation of surface fires in open field

Dominique Morvan

INTRODUCTION

As formalized by [START_REF] Countryman | Pacific Southwest Forest and Range Experiment Station[END_REF], the three factors affecting the most behavior of wildfires and forming the fire triangle are: weather, topography, and fuel (see also [START_REF] Pyne | Introduction to Wildland Fire[END_REF]. With ambient temperature and relative air humidity, the wind intensity is a part of the weather conditions that affect significantly the conditions of ignition and propagation of wildfires [START_REF] Benson | Climatic and weather factors affecting fire occurrence and behaviour[END_REF]. As underlined by [START_REF] Benson | Climatic and weather factors affecting fire occurrence and behaviour[END_REF], three kinds of "wind" can affect the behavior of wildfires: geostrophic wind resulting from the general meteorological conditions, local thermal wind resulting from a differential of temperature between sea and earth, and the wind induced by the fire itself under the action of the buoyancy. Many experimental studies carried out at small and large scale, through various vegetation layers (dead fuel beds, grasslands, shrublands) have highlighted a power law relationship between the rate of spread (ROS) of surface fires and the 2 m or 10 m open wind velocity (U W = U 2 or U 10 ) [START_REF] Catchpole | Rate of spread of free-burning fires in woody fuels in a wind tunnel[END_REF][START_REF] Fernandes | Fire spread prediction in shrub fuels in Portugal[END_REF][START_REF] Fernandes | Shrubland fire behaviour modelling with microplot data[END_REF][START_REF] Rothermel | Fire spread characteristics determined in the laboratory[END_REF][START_REF] Trabaud | Etude du comportement du feu dans la garrigue de chêne kermes à partir des températures et des vitesses de propagation[END_REF]:

ROS = A × U B W (1)
with an exponent B, which can vary from 0.4 [START_REF] Trabaud | Etude du comportement du feu dans la garrigue de chêne kermes à partir des températures et des vitesses de propagation[END_REF]in shrubland) to 2 [START_REF] Mcarthur | Weather and Grassland Fire Behaviour[END_REF]in grassland). In grassland, experimental fires and wildfires observations [START_REF] Fogarty | A field guide for predicting grassland fire potential[END_REF] have shown that the curve characterizing the relationship ROS versus U W can be decomposed in three zones: a linear zone (B ∼ 1) for wind speed ranged between 5.5 and 11 m/s, bounded by two nonlinear zones for weak (U W < 1m/s and B > 1) and strong (U W > 14 m/s and B < 1) wind conditions. The same kind of behavior has been also observed for shrub land covers, from numerical simulations performed using a multiphase formulation [START_REF] Morvan | Modeling the propagation of a wildfire through a Mediterranean shrub using a multiphase formulation[END_REF]. At small scale, for experimental fires through solid fuel beds, the variations of this exponent have been related to the surface area volume ratio [START_REF] Rothermel | Fire spread characteristics determined in the laboratory[END_REF]. Exceeding relatively strong wind conditions, experimental data highlighted that the rate of variation of the curve ROS versus U W changed and exhibited a decreasing part [START_REF] Mcarthur | The Tasmanian Bushfire of 7th February 1967, and Associated Fire Behaviour Characteristics[END_REF][START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF]. At large scale for surface fires in grasslands, [START_REF] Cheney | Prediction of fire spread in grasslands[END_REF] have showed that the linear relationship between ROS and U W can be only observed until the wind speed velocity was less than 1.4 m/s; above this value, a power law relation has been observed characterized by an exponent B < 1( B ∼ 0.84). In the same paper, the authors suggested that this exponent must vary with the wind conditions. [START_REF] Pitts | Wind effects on fires[END_REF], [START_REF] Cheney | Prediction of fire spread in grasslands[END_REF], and [START_REF] Morvan | Interaction between head fire and backfire in grasslands[END_REF] reported the role played by the orientation of the wind vector compared to the direction of fire propagation (backing fire/heading fire transition). [START_REF] Sullivan | Convective Froude number and Byram's energy criterion of Australian experimental grassland fires[END_REF], [START_REF] Morvan | Modeling the propagation of a wildfire through a Mediterranean shrub using a multiphase formulation[END_REF], and [START_REF] Morvan | Physical phenomena and length scales governing the behaviour of wildfires: A case for physical modelling[END_REF] have underlined the importance of two regimes of propagation (wind driven and plume dominated) upon the effect of wind conditions on the propagation of surface fires. The transition between these two regimes of propagation has been very often analyzed, in introducing two non-dimensional numbers: the Froude number (Fr) and the Byram's convective number (Nc). These two physical parameters represent the ratio between two forces (expressed in module or in power), namely, the inertia of the wind and the buoyancy due to the thermal plume, governing the trajectory of the flame and consequently the heat transfer between the flame and the vegetation:

Fr = U 2 W gL f Nc = 2gI B ρC P T 0 (U W -ROS) 3 (2)
where g is the acceleration of gravitation; I B and L f are the fireline intensity and a flame length scale; ρ, C P , and T 0 are the density, the specific heat, and the temperature of the ambient air; ROS is the fire rate of spread. Experimental results obtained in a fire wind-tunnel [START_REF] Beer | The interaction of wind and fire[END_REF] highlighted a sudden change of slope in the curve ROS versus U W , which can be attributed to a change in the mode of energy transfer (by convection and radiation) between the flame and the solid fuel. [START_REF] Beer | The interaction of wind and fire[END_REF] underlined the role played by vertical fluctuations of the wind flow upon the propagation of plume dominated fires.

Many phenomena affecting the coupling between wildfires and atmosphere are nonlinear; this characteristic is certainly at the origin of the difficulty in reducing the relationship between the fire rate of spread and the wind speed to a single power law function [START_REF] Clark | Analysis of small-scale convective dynamics in a crown fire using infrared video camera imagery[END_REF]. As underlined in a recent article [START_REF] Finney | On the need for a theory of wildland fire spread[END_REF], we are far from fully answering the following question: "How does fire spread in dry grass, and why does it spread faster when the wind blows?" In the same paper, Finney has also underlined that the 3D structure of the flame front, characterized by the formation of vertical peaks and breaks, allowed the wind flow to cross the fire front, which affected significantly the fire dynamics and the fire/atmosphere interaction.

From the point of view of safety of people in charge of fire fighting or prescribed burning operations, the sudden modifications of fire behavior is as important as its average behavior. Infrared images of experimental crown fires [START_REF] Coen | Infrared imagery applied into wildland fire dynamics[END_REF] allowed in highlighting the existence of bursts of flame, characterized by velocity vectors with a module one order of magnitude larger than the local wind speed, which can affect significantly the fire front dynamics especially for a fire propagating along a slope terrain. Two experimental campaigns carried in grassland in Texas (Pilot Study in 2005, FireFlux in 2006) [START_REF] Clements | First observations of turbulence generated by grass fires[END_REF] have shown that turbulence (TKE) measured during the propagation of a fire can be four times larger than atmospheric turbulence (TKE) measured near the ground far from fire.

The behavior of fires driven by radiation heat transfers (such as plume dominated fires) is certainly more difficult to predict than fires driven by convective heat transfer (wind driven fires). The main reason of this difference of predictability is that, when we analyze the two terms representing these two mechanisms of heat transfer in the energy balance equation in the solid fuel, the first one (because of the Stefan-Boltzman σ T 4 law) is much more nonlinear than the second one (which is more or less linear) [START_REF] Morvan | Physical phenomena and length scales governing the behaviour of wildfires: A case for physical modelling[END_REF]. For a long time the wind effects and consequently the wind driven fires have been considered to be the most critical conditions in terms of safety for structures and people potentially affected by this natural hazard. This is also the more convenient conditions (if the wind is not too strong) to conduct experimental fires. For all of these reasons, this regime of fire propagation is more known than the other one (plume dominated fires). Unfortunately, when the vegetation is very dry and characterized by a great accumulation of biomass, the power of the buoyant flow generated by the thermal plume can force the transition toward a plume dominated fire, which can also be catastrophic (such as the situation observed during the black Saturday in February 2009 in Australia) [START_REF] Cruz | Anatomy of a catastrophic wildfire: The black Saturday Kilmore east fire in Victoria, Australia[END_REF].

From a simplified wind and flame configuration, [START_REF] Baines | Physical mechanisms for the propagation of surface fires[END_REF] proposed an analysis on the relative importance of the two main mechanisms of heat transfer (by radiation and convection) governing the propagation of surface fires. The analysis was conducted in introducing a non-dimensional parameter P, representing the ratio between the heat flux received by the solid fuel by radiation and the convective cooling. Baines concluded rapidly that heat transfers between the flame and the vegetation were dominated at small scale by convection and at large scale by radiation. This conclusion is not fully confirmed by experimental observations carried out at large scale for crown fires [START_REF] Clark | Analysis of small-scale convective dynamics in a crown fire using infrared video camera imagery[END_REF][START_REF] Stocks | Overview of the International Crown Fire Modelling Experiment (ICFME)[END_REF]. The authors underline that at large scale the convective exchanges were enhanced by the turbulent flow and the radiation heat transfer increases because of the flame stretching.

The objective of this article was to study some unsteady behavior affecting the propagation of surface fires through a homogeneous vegetation layer (reproducing a tall grass) on a flat terrain. This problem has been studied using 2D numerical simulations. We know that for strong enough wind conditions, wildfire fronts exhibit a 3D behavior, and appear as a more or less regular succession of peaks and troughs [START_REF] Beer | The interaction of wind and fire[END_REF][START_REF] Finney | A study of flame spread in engineering cardboard fuelbeds. Part I: Correlations and observations[END_REF][START_REF] Linn | Using periodic line fires to gain a new perspective on multi-dimensional aspects of forward fire spread[END_REF], which cannot be taken into account in 2D simulations. Unfortunately, 3D simulations need very large computing resources and, for this reason, some compromises must be introduced in the space resolutions [START_REF] Linn | Using periodic line fires to gain a new perspective on multi-dimensional aspects of forward fire spread[END_REF], which can also limit the conclusions that can be extracted from these numerical simulations. For the same reasons, parametric studies are not always possible in 3D or remain always very expensive in terms of CPU time. A good compromise to study wildfires in 3D can be in using periodic boundary conditions along the vertical lateral boundaries, as it was proposed by [START_REF] Linn | Using periodic line fires to gain a new perspective on multi-dimensional aspects of forward fire spread[END_REF]. Despite the limitations introduced in using a 2D approximation, this study must be considered as a first step in studying unsteady phenomena associated with wildfires behavior. This rough approach can highlight some general trends, for example, the character more or less predictable of some wildfire regimes (wind driven and plume dominated), and more generally the physical mechanisms associated with fires behavior. Of course, this analysis and the conclusions that we can extract from this simplified approach, must be refined in the future in performing 3D simulations.

This article was voluntarily limited to surface fires propagating on a flat terrain through a homogeneous vegetation layer. Heterogeneity of the vegetation and local variations of slope, which constitute also sources of uncertainties during the propagation of fire, have not been considered.

In the first part of the article, unsteady signals of fireline intensity have been analyzed for various steady average wind conditions (vertical logarithmic profile), ranged between 1 and 25 m/s. This relative large range of variations has allowed in covering the two regimes of fire propagation, namely, plume dominated and wind driven. Then the effect of a sinusoidal time variation of the wind flow has been explored for relatively moderated average wind conditions (U 10 = 2m/s, U 10 =±1m/s), for frequencies ranging between 0.5 Hz and 3 Hz.

PHYSICAL AND MATHEMATICAL MODEL

The problem of the propagation of a surface fire through a vegetation layer has been simulated numerically using a multiphase formulation. This approach is similar to the model initially proposed by [START_REF] Grishin | Mathematical Modelling of Forest Fires and New Methods of Fighting Them[END_REF], which has been extended with success to simulate in 2D and 3D various situations, such as surface fires in grassland [START_REF] Morvan | Physical modelling of fire spread in grasslands[END_REF][START_REF] Morvan | Interaction between head fire and backfire in grasslands[END_REF] and shrubland [START_REF] Morvan | Modeling the propagation of a wildfire through a Mediterranean shrub using a multiphase formulation[END_REF], the efficiency of surface fuel reduction to prevent the vertical transition from surface to crown fire [START_REF] Dupuy | Physical modelling of forest fire behaviour in a fuel break[END_REF], head fire/backfire interaction [START_REF] Morvan | Numerical simulation of the interaction between two fire fronts in grassland and shrubland[END_REF][START_REF] Morvan | Interaction between head fire and backfire in grasslands[END_REF], and the burning of a Douglas tree [START_REF] Mell | Numerical simulation and experiments of burning Douglas fir trees[END_REF]. The approach has been partially validated in comparing numerical results with experimental data obtained at small and large scale in various configurations [START_REF] Mell | Numerical simulation and experiments of burning Douglas fir trees[END_REF][START_REF] Morvan | Modeling of fire spread through a forest fuel bed using a multiphase formulation[END_REF][START_REF] Morvan | Physical modelling of fire spread in grasslands[END_REF][START_REF] Morvan | Interaction between head fire and backfire in grasslands[END_REF].

In a few words, the model can be summarized as follows: The coupled system formed by the vegetation and the surrounding atmosphere has been represented as a sparse porous media (the vegetation) coupled with a gaseous phase (surrounding atmosphere, gaseous mixing coming from the pyrolysis and combustion processes). The various elements constituting the structure of the vegetation have been represented as a set of families of solid fuel particles (foliage, twigs, trunk, etc.). The internal composition of each solid fuel element has been represented as a mixing of four components (water, dry matter, char, and ash), and each one was described using its own mass fraction. Assuming that the solid fuel elements of the vegetation stayed at rest, the evolution of the state of the vegetation (mass, volume, composition, temperature) was governed by a set of ordinary differential equations (ODE) representing the balance equations for mass and energy, and reproducing the different steps of degradation of the vegetation (drying, pyrolysis, and char combustion) resulting from the intense heating coming from the fire front. The degradation rates appearing on the right-hand side of these equations have been evaluated experimentally from thermal analysis of Mediterranean fuel samples and represented using an Arrhenius law process (pyrolysis and heterogeneous combustion) and threshold type process (drying).

The evolution of the gaseous phase is governed by the various physical mechanisms occurring inside the atmosphere surrounding the fire front: atmospheric turbulence and canopy turbulence interaction, mixing and combustion between gaseous pyrolysis products and ambient air, and radiation from the gas + soot mixture. Taking into account the important role played by the radiation heat transfer in the propagation of wildfires and considering the level of temperature fluctuations in the flame (see [START_REF] Morvan | Physical phenomena and length scales governing the behaviour of wildfires: A case for physical modelling[END_REF] for the justifications), a particular effort has been done in this new version of the fire behavior model to simulate the turbulence/radiation interaction (TRI). Consequently, the radiation transfer equation (RTE) has been formulated using an optically thin fluctuation approximation (OTFA) [START_REF] Coelho | Numerical simulation of the interaction between turbulence and radiation in reactive flow[END_REF][START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF], as follows:

dα G I ds = α G σ σ a T 4 π -σ a I + σ S α S 4 σ T 4 S π -I σ a T 4 ≈ σ a T 4 1 + 6 T ′2 T 2 + 4 σ ′ a T ′ σ a T = σ a T 4 1 + 6 T ′ 2 T 2 + 4 T ′ 2 σ a T ∂σ a ∂T ∂σ a ∂T = 1862 × α soot (3)
The transport equation governing the temperature variance T ′ 2 = θ has been approximated as follows [START_REF] Kenjeres | Contribution to elliptic relaxation modeling of turbulent natural and mixed convection[END_REF]:

∂ ρθ ∂t + ∂ ρ ũj θ ∂x j = ∂ ∂x j µ eff Pr T ∂θ ∂x j + 2P θ -2ε θ P θ = µ T Pr T ∂T ∂x j 2 ε θ = ρ θ 2 × R × ε K (4)
where, in agreement with experimental observations, the ratio between scalar and velocity dissipation time (R) has been assumed to be equal to 0.5 [START_REF] Béguier | Ratio of scalar and velocity dissipation time scales in shear flow turbulence[END_REF]. We are conscious that the characteristics of flames for our problem, and especially the optical thickness defined as the product of the absorption coefficient by the turbulent integral length scale σ a × l t , do not verify strictly the condition necessary for the use of the OTFA (σ a × l t << 1) (Coehlo, 2007). A crude approximation of these two terms gives an evaluation of the order of magnitude of optical thickness ranging between 1 and 5 (in assuming a temperature ranged between 1000 and 1500 K, a soot volume fraction ranged between 10 -6 and 4 × 10 -6 , and a turbulent integral length scale is equal to 0.5 m). From these values, we can conclude that the flames studied in this article can be considered as intermediate between thin and thick. However, one can consider that, despite this limitation, the use of the OTFA to simulate TRI constituted a progress in comparison to do nothing.

The numerical simulations were performed in a 2D domain: 170 m long by 35 m high. These dimensions were considered as sufficient to guarantee that the simulations reproduced correctly the propagation of a surface fire through a semi-infinite, 0.7 m depth fuel layer, without artificially forcing the behavior of the fire with open boundary conditions (at the top and at the outlet on the right end side). To control the direction of propagation of the fire, the fuel was fully removed within the first 20 m of the domain. Consequently, homogeneous fuel layer (0.7 m depth) was distributed between the position X = 20 m and X = 170 m (all of the fuel properties are reported in Table 1). Following the adaptive mesh refinement method described and validated in a previous study in 1D [START_REF] Morvan | Modeling of one dimensional fire spread in pine needles with opposing air flow[END_REF] the mesh has been automatically refined in a region (20 m long) located on both sides of the fire front. The mesh size in this refined region has been chosen in following some physical considerations [START_REF] Morvan | Physical phenomena and length scales governing the behaviour of wildfires: A case for physical modelling[END_REF], i.e., the grid size must be smaller than the extinction length scale and the depth of the vegetation layer. From these two constraints, the grid size along the two directions of the problem were fixed as follows: X = 0.25 m, Z = 0.175 m (see Table 1).

The physical properties characterizing the solid fuel layer were similar to the values collected on the field for a tall grass (fuel model #3 as described by [START_REF] Anderson | Aids to determining fuel models for estimating fire behaviour[END_REF]). A 2-m-long gaseous burner (in injecting gaseous CO at 1600 K from the ground, with an injection velocity equal to 1 m/s) used to ignite the solid fuel layer, has been activated only 30 s after the beginning of the calculations (to be sure that the turbulent boundary layer flow was established before introducing the fire). In the absence of strong turbulence before that, the fire was fully developed; we cannot exclude that the combustion rate at the burner was partially piloted by the Arrhenius law and, consequently, the injected gaseous fuel must be hot enough to obtain a significant reaction rate. After having proceeded to some numerical tests, an arbitrary value of 1600 K has been chosen. Nevertheless, the properties characterizing the fire behavior (rate of spread, intensity, etc.) were evaluated after the fire had spread along a sufficient distance, to be sure to be not affected by these Source: [START_REF] Anderson | Aids to determining fuel models for estimating fire behaviour[END_REF]; [START_REF] Burgan | Revisions to the 1978 National Fire-Danger Rating System[END_REF]. ignition conditions. Preliminary numerical tests have also highlighted that an initialization time equal to 30 s was long enough to stabilize the turbulence boundary layer profile inside and above the vegetation layer. To facilitate the ignition of the fuel, the first 2 m were artificially dryed (FMC = 5%), once the ignition of the vegetation layer was confirmed, BS the burner was stopped. A logarithmic velocity profile has been imposed at the inlet of the computational domain (left-hand side) and as an initial condition for the velocity vectors field:

U X (z) = A × U 10 × Ln z + z 0 z 0 (5)
The constant A was adapted to fix the velocity magnitude 10 m above the ground level (10 m open wind velocity, i.e., U x (z = 10 m) = U 10 ), in assuming a nude ground at the entrance of the domain, a value equal to z 0 = 0.01 m has been used for the roughness length z 0 in Eq. ( 5).

To quantify the behavior of fires, some global parameters have been extracted from numerical results, such as the rate of spread (ROS) evaluated from the first derivative of the time trajectory of the pyrolysis front, in assuming that this last one can be located by the isotherm in the solid fuel, T S = 500 K temperature at which the rate of pyrolysis reaches its maximum value [START_REF] Morvan | Modeling the propagation of a wildfire through a Mediterranean shrub using a multiphase formulation[END_REF][START_REF] Pyne | Introduction to Wildland Fire[END_REF], the fireline intensity (in kW/m), and the Byram convective number, defined as follows:

I B =ṁ × HN C = 2gI B ρC P T 0 U 10 -ROS 3 ( 6
)
where ṁ is the mass loss rate (only the dry fuel, in kg/m.s), H = 18,000 kJ/kg designates the heat of combustion [START_REF] Burgan | Revisions to the 1978 National Fire-Danger Rating System[END_REF][START_REF] Pyne | Introduction to Wildland Fire[END_REF], g is the acceleration of gravity, ρ is the air density (standard conditions), T 0 and C P are the temperature and the specific heat of ambient air. Moreover, the propagation of the fire front reached rapidly a quasi steady state, and all the trajectories of the solid fuel temperature were parallel. Consequently, the fact of choosing a particular value to evaluate the ROS did not affect the results.

UNSTEADY BEHAVIOR OF A SURFACE FIRE SUBMITTED TO A STEADY WIND FLOW

A first set of numerical simulations, under steady wind conditions has been performed, using a wind speed ranged between 1 and 25 m/s. This large domain of variation for this input parameter, allowed in covering the two regimes of propagations identified for surface fires, namely, the plume dominated fires and the wind driven fires. A first illustration of the temperature field (gaseous phase) obtained for three values of the wind speed (U 10 = 1, 8, and 20 m/s) can be seen in Figure 1. This figure illustrates perfectly one of the consequences resulting from the balance between the two forces governing the behavior of the fire, namely, the inertia of the wind and the buoyancy from the difference of temperature (and consequently the density) between the plume and the ambient air. As the wind speed increased, the trajectory of the plume was more and more affected by this external forcing and was more and more tilted toward the unburned vegetation. From a more general point of view, one of the parameters characterizing the behavior of the fire, the ratio between the rate of spread and the 10 m open wind velocity (ROS/U 10 ), can be expressed as a function the inverse of the convective Byram number (1/N C ) (see Eq. ( 2) and Figure 2). With the values imposed for the wind speed and those obtained for the ROS and the fireline intensity, the resulting range of variation for 1/N C was included between 7 × 10 -4 to 10, which covered a large range of situations, from plume dominated fire (1/N C << 1) to wind driven fire (1/N C >> 1). The numerical results shown in Figure 2 highlight clearly that the maximum value of the ratio ROS/U 10 was obtained for the smallest value of the parameter 1/N C . Due to quite low fuel moisture content conditions, the ROS can represent 46% of the 10 m open wind speed. Experimental observations for grassland fires in Australia reported by [START_REF] Cheney | Prediction of fire spread in grasslands[END_REF] have shown values larger than 50%. The same curve has shown also that for larger values of 1/N C (>>1)1), the ratio ROS/U 10 converged more or less toward a constant value (around 8%), revealing a nearly linear relationship between ROS and U 10 . Variations of the fire behaviors can also be highlighted for the time variations of the fireline intensity, both the average value <I B >(Figure 3) and the standard deviation <I B '2 > 1/2 (Figure 4). The two curves in Figure 3 and representing the time evolution of the fireline intensity signal obtained for two values of the 10 m open wind speed (U 10 ) show clearly a great modification of the dynamic of the fire in terms of amplitude and frequency of the fire intensity signal. The general trend shown for the curves representing standard deviation, normalized using the average value, versus the inverse convective Byram number (Figure 4), can be summarized as follows: for plume dominated fires (1/N C << 1) the fire front exhibited (in relative values) larger fluctuations than for wind driven fires (1/N C >> 1). As indicated in the introduction, this feature (plus the impact upon the ratio ROS/U 10 ) increases the character more unpredictable of plume dominated fires. This particularity can certainly be attributed to the increase of nonlinear mechanisms (such as radiation) governing the heat transfers between the fire front and the vegetation.

As indicated in the introduction, two forces, namely, the inertia of the wind and the buoyancy due to the difference of temperature between the plume and the ambient atmosphere, can affect the trajectory of the flame and consequently the behavior of fires. Therefore, two mechanisms of instability (associated to these two forces) can be associated to the unsteady behavior of the fire: the Kelvin-Helmholtz instability resulting from the shearing between the wind and the vegetation and the thermo-convective instability resulting from the difference of density between the plume and the ambient air.

The Strouhal number (obtained experimentally) associated with the Kelvin-Helmholtz instability (characterized by a frequency F KH ) above a canopy (longitudinal mode) [START_REF] Kaimal | Atmospheric Boundary Layer Flows[END_REF]) defined using the wind velocity at the top of the canopy (U H ) is equal to 0.15, taking into account that for a logarithmic wind profile, U H = 0.62 × U 10 (U 10 : 10 m open wind velocity), the Strouhal number defined using this reference wind velocity can be written as:

S t = F KH × H U 10 = 0.15 × 0.62 = 0.093 (7)
The characteristic frequency (F B ) associated with the puffing instability of the fire plume can be evaluated using the same expression highlighted for a pool fire [START_REF] Cox | Combustion Fundamentals of Fire[END_REF][START_REF] Hamins | Characteristics of pool fire burning[END_REF](g = acceleration of gravitation):

F B = (0.5 ± 0.04) g D Fire 1/2 (8)
In this expression, the fire depth (D Fire ) has been evaluated as the product of the fire residence time by the rate of spread (ROS) [START_REF] Morvan | Numerical study of the effect of fuel moisture content (FMC) upon the propagation of a surface fire on a flat terrain[END_REF]. As detailed in a previous publication [START_REF] Morvan | Numerical study of the effect of fuel moisture content (FMC) upon the propagation of a surface fire on a flat terrain[END_REF], the fire residence time was determined from the time history of the gas temperature recorded 0.25 m above the ground level. Compared to an evaluation of this quantity at the ground level, the main advantage of this procedure is to separate clearly the gas heating coming from the flaming combustion from the part in contact with the hot embers. A consequence of that is that the values found using this method are a little bit smaller than some common values reported in the literature.

To fix some typical values, for H = 0.7 m, U 10 = 2m/s and 20 m/s, we obtain the set of values shown in Table 2.

In order to clarify the role played by these two mechanisms of instability upon the fire dynamics, a spectral analysis (FFT) has been carried out on the fire intensity signal (time variations), in selecting an interval of time for which the fire had reached a quasisteady behavior (see Figure 5). The Strouhal number formed using two sets of reference scales (related to inertial and buoyancy forces) versus the inverse of the convective Byram number has been reported in Figures 6 and7. The two major conclusions that we can extract from these two curves are: r When the frequency is reduced using the inertial scales, for large values of 1/N C the corresponding Strouhal number converged toward a value nearly equal to 0.093.

r When the frequency is reduced using the buoyancy scales, for small values of 1/N C the corresponding Strouhal number converged toward a value nearly equal to 0.5.

As indicated previously these two domains of variation of the convective Byram number (N C ), corresponds to wind driven and plume dominated fires, respectively. These results confirm the concept developed previously concerning the behavior of fires, in including also the unsteady part, i.e., wind driven fires are piloted by shear interactions between the wind flow and the canopy, whereas plume dominated fires are governed by thermo-convective instabilities between the plume and the ambient atmosphere. The trend highlighted in Figure 6 for wind driven fires shows that the frequency characterizing the fire intensity signal increases when the wind speed increases. In the same manner, for plume dominated fires, the results shown in Figure 7 show that the same frequency must decrease if the depth of the vegetation layer increases. These results are in agreement with experimental results reported by [START_REF] Finney | A study of flame spread in engineering cardboard fuelbeds. Part I: Correlations and observations[END_REF]. The spectral analysis of temperature signals collected in these experiments had shown that the curve Strouhal number versus Froude number, did not match perfectly those observed for pool fire. Consequently, one can deduce from this observation (confirmed in the present numerical study) that for the great majority of situations the behavior of fires was governed under the combined action of two forces, namely, the inertia (wind) and the buoyancy (plume), and not from the action of a single one (except in extreme situations, very weak and very strong wind conditions). The shear flow resulting from the interaction of the wind flow with the vegetation, promoted the development of Kelvin-Helmholtz instability. Then the buoyancy resulting from the gradient of temperature between the thermal plume and the ambient atmosphere induced a lift effect in deviating these structures vertically [START_REF] Finney | A study of flame spread in engineering cardboard fuelbeds. Part I: Correlations and observations[END_REF].

UNSTEADY RESPONSE OF A SURFACE FIRE SUBMITTED TO AN UNSTEADY WIND FLOW

In order to study the response of the coupled system formed by the fire, the vegetation and the ambient air, to a time variation of wind conditions reproducing wind gusts, we have introduced at the inlet boundary a sinusoidal variation of the wind module, as follows:

U X (z) = A × U 10 × Ln z + z 0 z 0 + U 10 sin (2π × F × t) (9)
where U 10 and F designate, respectively, the amplitude and the frequency of wind variations. The present study has been limited to one wind conditions, U 10 = 2m/s and U 10 = 1m/s; the frequency F varied between 0.5 and 3 Hz. This wind condition (U 10 = 2m/s) has been chosen because it represents a situation for which we have found (in quasi steady wind conditions) a relatively high value of the variation of the rate of spread (ROS) versus U 10 dROS dU 10 (see Table 3). As stated previously, the characteristic frequencies associated to this wind condition were: F KH = 0.26 Hz (Kelvin-Helmholtz instability) and F B = 1.3 Hz (Thermo-convective instability) (see Table 2).

In Figures 8 and9, snapshots of the temperature field (gaseous phase) are shown at three different times and for two values of the wind gusts frequency (F = 1 and 0.5 Hz). As shown on these figures (the snapshot were taken exactly at the same time for the two frequencies), the trajectory of the fire front seemed to be not too much affected by the frequency of variations of the inlet wind flow. However, for F = 0.5 Hz (Figure 9), the vertical dynamics of the flame (part located above the vegetation top) was significantly affected and subjected to long wave (low frequency) oscillations, which were not observed for F = 1H z( Figure 8). The amplitude of variations of the inlet wind speed was ranged between 1 and 3 m/s, we have consequently reported in Figure 10 the trajectory of the pyrolysis front (isotherm T S = 500 K in the solid phase) obtained for steady and unsteady (F = 0.5 and 1 Hz) state conditions. For comparison, we have also plot on the same curve the trajectory obtained (in steady state conditions) for U 10 = 1 and 3 m/s. This figure shows that in terms of propagation, for these average wind conditions (<U 10 > = 2m/s), the fire behavior was not much affected by the unsteady character of the inlet wind conditions. Even if the result obtained for F = 0.5 Hz shows a small acceleration of the fire front at the end of the simulation, the trajectories of the fire front obtained in unsteady conditions did not differ significantly from one obtained in fully steady conditions (see Figure 10). The fire intensity signals obtained for these two conditions were far from similar (see Figure 11). While the signal for F = 1 Hz was nearly identical with the one obtained for a steady wind in exhibiting a peak of energy at 3.65 Hz (nearly equal to 3.4 Hz observed in steady conditions), whereas the signal obtained for F = 0.5 Hz was characterized by a peak of energy at 0.24 Hz (nearly equal to the frequency, F KH = 0.26 Hz, of the Kelvin-Helmholtz instability along the streamwise direction). This result suggests that even if the heat transfer between the flame and the vegetation was naturally responsible of the fire behavior, for these wind conditions ( <U 10 > = 2m /s), the part of the flame situated above the top of the canopy (more affected by wind variations) must play a less important role upon the fire propagation than the part situated below the top of the canopy. These results suggest in the following remarks:

r The variations of wind conditions affect mainly the part of the flame located above the top of the vegetation layer.

r The ROS of plume dominated fires is mostly affected by the part of the flame located below the top of the vegetation, which was evidently much less affected by wind variations. r One of the actions of wind gusts was to stimulate the mode associated with the Kelvin- Helmholtz instability (longitudinal mode).

This last remark has been confirmed in regarding the curve representing the evolution of the standard deviation of the fire intensity (normalized by the average fire intensity) versus the ratio F/F KH (see Figure 12). This figure highlights clearly that in exciting the inlet wind flow with a frequency two times larger than the Kelvin-Helmholtz frequency (F/F KH = 2), it can produce an increase by a factor 2 (in relative value) of the fluctuations of fire intensity. This factor 2 comes certainly from the fact that we have compared unsteady wind conditions with empirical results obtained for a steady flow configuration (with the additional possibility that the boundary condition imposed at the inlet flow, allowed the fire front to modify by aspiration the inlet flow). However, the result obtained for 0.25 Hz (F/F KH nearly equal to 1) shows without any doubt that the maximal relative response was obtained for a ratio F/F KH equal to 2 and not 1 as one could expected.

CONCLUSION

A set of numerical simulations reproducing the propagation of a surface fire on a flat terrain and through a homogeneous vegetation layer has been performed to identify some phenomena characterizing the behavior of fire in various regimes of propagation, from plume dominated fires to wind driven fires. The analysis of the results concerning the time evolution of the fire intensity has shown that plume dominated fires were subject to larger time variations than wind driven fires, with a consequence that the behavior of this kind of fire was much more difficult to predict that the other one. In terms of frequencies, wind driven fires were clearly dominated by shear effects (Kelvin-Helmholtz instability) and plume dominated fires were driven by thermo-convective instability. Preliminary results obtained for weak wind conditions have highlighted that a maximum impact (in terms of unsteadiness) can be obtained if the inlet wind intensity was excited in using a frequency two times larger than the frequency characterizing the Kelvin-Helmholtz (streamwise mode) instability associated with the shear interaction between the wind flow and the canopy.

ACKNOWLEDGMENT

The author wishes to thank the anonymous reviewers for their constructive remarks and the detailed analysis of the article, which allowed us to clarify some points and to improve significantly the presentation of this study.

Figure 1

 1 Figure 1 Temperature field calculated during the propagation of a surface fire for three wind flow conditions: U 10 = 1m/s, 8 m/s, and 20 m/s (top to bottom).

Figure 2

 2 Figure 2 Ratio fire rate of spread/wind speed (ROS/U 10 ) as a function of the convective Byram number.

Figure 3

 3 Figure 3 Time history of the fireline intensity for two wind conditions (U 10 = 1 and 20 m/s).

Figure 4

 4 Figure 4 Ratio standard deviation/mean value for the fireline intensity as a function of the convective Byram number.

Figure 5

 5 Figure 5 Time evolution (top) and FFT analysis of fireline intensity signal (U 10 = 2m/s).

Figure 6

 6 Figure 6 Evolution of the Strouhal number (inertial scales) as a function of the convective Byram number.

Figure 7

 7 Figure 7 Evolution of the buoyant Strouhal number (plume scales) as a function of the convective Byram number.

Figure 8

 8 Figure 8 Snapshots of the temperature field (62, 63, and 64 s after the ignition) calculated for a surface fire propagating through a grassland under unsteady wind conditions (U 10 = <U 10 >[1 + 0.5 × sin(2π Ft)], <U 10 >= 2m/s, F = 1Hz).

Figure 9

 9 Figure 9 Snapshots of the temperature field (62, 63, and 64 s after the ignition) calculated for a surface fire propagating through a grassland under unsteady wind conditions (U 10 = <U 10 >[1 + 0.5 × sin(2π Ft)], <U 10 >= 2m/s, F = 0.5 Hz).

Figure 10

 10 Figure 10 Time evolution of pyrolysis front (T S = 500 K) calculated for various wind conditions (steady: U 10 = 1, 2, and 3 m/s; unsteady: <U 10 >= 2m/s, < U 10 > = 1m/s, (F = 0.5and1Hz)).

Figure 11

 11 Figure 11 Time evolution of the fireline intensity calculated for two sinusoidal wind speed variations (F = 0.5 Hz and F = 1Hz).

Figure 12

 12 Figure 12 Reduced value of the fireline intensity standard deviation versus reduced wind gusts frequency (<U 10 >= 2m/s).

Table 1

 1 Solid fuel physical properties and external conditions imposed in the present numerical simulations

	Solid fuel density (kg/m 3 )	500
	Volume fraction ×10 3	2
	Fuel moisture content (FMC) (%)	10
	Fuel depth (m)	0.7
	Fuel load (t/ha)	7
	Surface area to volume ratio (m -1 )	4000
	Length of extinction (m)	0.5
	Leaf Area Index (LAI)	2.8
	10 m open wind velocity	1-25 m/s

Table 2

 2 Characteristics frequency of Kelvin-Helmholtz (F KH ), buoyant plume (F B ) instabilities, and fundamental mode (f ) extracted from a FFT analysis of the fireline intensity signal for two values of the wind speed U 10 = 2 and 20 m/s

	U 10 (m/s)	F KH (Hz)	F B (Hz)	f (Hz)
	2	0.26	1.3	3.4
	20	2.66	0.66	5.5

Table 3

 3 Rate of spread (ROS) obtained for average wind conditions (U 10 ) ranged between 1 and 25 m/s

	U 10 (m/s)	1	2	3	4	8	12	15	20	25
	ROS (m/s)	0.46	0.84	1.06	1.18	1.24	1.23	1.10	1.28	2.51

FUNDING

This study was supported by the Agence Nationale de la Recherche (Project No.: ANR-09-COSI-006).


Agnès Le Saux

Vincent Béringue

Human Rezaei

Jan Bohl

Angélique Igel-Egalon

Fabienne Reine

Laetitia Herzog

Agathe Urvoas

Marielle Valerio-Lepiniec

Philippe Minard

Caractérisation de l'hétérogénéité structurale des prions

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