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International Journal of Wildland Fire, 2005, 14, 141–151

Numerical study of a crown fire spreading toward a fuel break using a multiphase physical model Jean-Luc DupuyA,C and Dominique MorvanB A INRA

Unité de Recherches Forestiéres Méditerranéennes, Equipe de Prévention des Incendies de Forêt, 20 Avenue Antonio Vivaldi, 84000 Avignon, France. B Université de la Méditerranée, UNIMECA 60 rue Joliot Curie Technopôle de Château Gombert, 13453 Marseille cedex 13, France. C Corresponding author. Telephone: +33 4 90135939; fax: +33 4 90135959; email: [email protected]

Abstract. The propagation of a wildfire through a Mediterranean pine stand was simulated using a multiphase physical model of fire behaviour. The heterogeneous character of the vegetation was taken into account using families of solid particles, i.e. the solid phases (foliage, twigs, grass). The thermal decomposition of the solid fuel by drying and pyrolysis, and the combustion of chars were considered, as well as the radiative and convective heat transfer between the gas and the vegetation. In the gaseous phase, turbulence was modelled using a two transport equations model (RNG k–ε) and the rate of combustion, which was assumed to be controlled by the turbulent mixing of fuel and oxygen, was calculated using an eddy dissipation concept. The radiation transfer equation, which includes absorption and emission of both the gas–soot mixture and the vegetation, was solved to calculate the contribution of radiation to the energy balance equations. Numerical solutions were calculated in a two-dimensional domain (vertical plane). Results showed the ability of this approach to simulate the propagation of a crown fire and to test the efficiency of a fuel break with success. The effects of the terrain slope were also tested. Some effects on fire behaviour of vortices resulting from the interaction of the wind flow with the canopy layer are shown. Additional keywords: forest fire spread.

Introduction Fuel breaks consist of areas where the structure of the natural vegetation has been modified, in particular where the amount of shrubs has been reduced. In the French Mediterranean region, these managed areas are an important component of the policy of forest fire prevention (Rigolot and Costa 2000). Networks of fuel breaks are created inside forests to divide large areas where fire hazard is high into smaller compartments. More generally, wildland–urban interfaces where the vegetation is managed to reduce the danger for inhabitants, may also be regarded as fuel breaks. Usually, a fuel break is not expected to stop the fire itself, but firefighters are expected to fight the fire on it under acceptable safety conditions. Forest managers often maintain trees on fuel breaks. Therefore a main question is to determine the level of reduction of surface fuel and the level of tree pruning necessary to avoid the vertical transition of fire from the ground level to the tree crowns. Indeed, crowning fires are generally recognised as the most dangerous and the most difficult to control (Van Wagner 1977; Scott and Reinhardt 2001; Stocks et al. 2004). © IAWF 2005

The present paper describes a numerical study of the behaviour of a crown fire propagating through a natural forest fuel, representative of a Mediterranean pine stand, towards a fuel break. For this we ran simulations of a physical model of forest fire behaviour, which considers the interaction between the ambient air flow, the vegetation and the fire, and accounts for the thermal degradation of the fuel, the radiation of both the fuel and the gas–soot mixture, and the combustion processes. Our goal is to illustrate the use of this model for examination of fire behaviour in different structures of vegetation and to show how the model can help the management of fuel on fuel breaks. The mathematical formulation of this model has been described in detail elsewhere (Morvan and Dupuy 2001, 2004); therefore, here we mainly describe the physics of the model. Basic equations are reported in Appendix 1. For computational cost reasons, we calculated the numerical solutions of the model in a two-dimensional configuration (vertical plane). We recognise that this assumption is a crude approximation of the phenomena encountered in a real wildfire where three-dimensional effects are observed. Nevertheless, this approach allowed us to consider the detailed 10.1071/WF04028

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physics of a fire and to include the heterogeneous nature of the forest fuel, which is composed of a number of families of different physical properties (foliage and twigs of different species, dead or live fuel). Linn et al. (2002) showed that three-dimensional simulations of a physics-based model (FIRETEC/HIGRAD system) can be used to study the interaction between a natural fire and the atmospheric flow. The large size of resolved volumes (several m3 for the smallest, Linn et al. 2002) with respect to the physical scales of fire mechanisms lead to the use of a probability density function approach to describe average properties of the medium. The same model was used recently in a fuel made of individual trees preserving the discontinuous nature of the vegetation (Linn et al. 2003). However, in these three-dimensional simulations, important physical processes like radiation transfer, thermal degradation of the fuel and combustion, were still crudely approximated and the heterogeneous nature of the fuel was not completely rendered. The multiphase physical model The model we used in the present study is based on a multiphase formulation of the conservation equations (mass, momentum, energy) and of the radiation transfer equation in a semi-transparent medium (Larini et al. 1998).A set of balance equations governs the evolution of the coupled system formed by the vegetation and the surrounding gas (see Appendix 1). The heterogeneous nature of the vegetation is described using a set of solid fuel families, each characterised by its physical properties, such as the density of the dry material, the moisture content and the surface area-to-volume ratio. Each fuel family is a solid phase in the multiphase approach. In real vegetation, the moisture content may range between 5 and 15% for a dead fine fuel and 100% or more for a living fuel. Therefore only a heterogeneous representation of the vegetation can account for the effect of such variability of the fuel moisture content on fire behaviour. The vegetation located in front of the burning fuel receives a part of the energy released by this combustion, and thus goes into a thermal degradation process (drying and pyrolysis of solid fuel), followed by the combustion of the remaining chars. The evolution of the state of the vegetation along these processes was calculated from the resolution of mass balance equations verified by the three components of the solid fuel (dry material, water and char). The kinetics of the drying process was modelled simply as an isothermal process occurring at the boiling temperature of water (373K). The kinetics of the pyrolysis of dry material and of char combustion were modelled based on Arrhenius type laws (Grishin 1997) and on measurements of the thermal degradation of Mediterranean fuels in a furnace (Morvan and Dupuy 2004). A number of studies have shown that pyrolysis products of wood are mainly composed of a mixture of carbon monoxide and carbon dioxide (Di Blasi 1993; Grishin 1997; Klose et al.

J.-L. Dupuy and D. Morvan

2000; Di Blasi et al. 2001). As proposed by Grishin (1997) we modelled the combustion reaction in the gaseous phase using the combustion of carbon monoxide with oxygen. For the turbulent flames observed in wildland fires, the rate of the combustion reaction is mainly limited by the turbulent mixing between the gaseous fuel and the oxidizer. Consequently, the rate of reaction was evaluated using the eddy dissipation concept turbulent combustion model proposed by Magnussen and Hjertager (1976). The turbulent transport was calculated using a two transport equation re-normalisation group (RNG) k–ε model developed initially by Yakhot and Orszag (1986). This approach of turbulence modelling is a cheap method in terms of computational time (two-dimensional simulations) and captures the dynamics of large eddies near the fire front and in the plume. As mentioned by a number of authors (e.g. Pagni and Peterson 1973; Baines 1990; Pitts 1991), two modes of heat transfer contribute to the heating of the vegetation located ahead of the fire front: the radiation from flames and embers and the convective exchange with hot gases. Of course, cooling of the vegetation is also due to these two modes of heat transfer. A number of physical models of fire spread in a forest fuel bed have been derived based on different approximations of the contribution of radiation and convection in the energy balance of the fuel, and often radiation is the only process of fuel heating considered in these models (see the reviews of Catchpole and DeMestre 1986 and Weber 1991). In the present multiphase model, both radiation and convection are included in the energy balance equations of the gaseous phase and of each solid phase (Appendix 1). Radiation from flames of forest fuels (luminous flames) is mainly due to the presence of soot particles (Tien and Lee 1982).As suggested by Grishin (1997) and because the soot production is not well known in natural fires, we assumed that the soot production rate was a constant fraction (5%) of the rate of solid fuel pyrolysis. The soot oxidation process was modelled using an Arrhenius type law and the thermophoretic effect was also included in the balance equation for soot volume fraction. The set of coupled transport equations governing the flow in the gas phase was solved numerically using a high order finite-volume method proposed by Leonard and Mokhtari (1990), which combines accuracy (this scheme introduces no numerical diffusion) and numerical stability. Balance equations for the solid phases were integrated in time using a Runge-Kutta algorithm (4th order). The radiation intensity field was calculated using a discrete ordinate method (see Siegel and Howell 1992), which involves solving the radiation transfer equation for a finite number of optical paths and estimating the resulting irradiance using a numerical quadrature. For these calculations we used 40 discrete directions (S8 method in two dimensions of space). This modelling approach was tested against experimental fires propagating through a homogeneous pine needle fuel bed (Porterie et al. 2000; Morvan and Dupuy 2001).

Multiphase physical model of crown fire spread


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The same approach has been used to study the propagation of a fire through heterogeneous vegetation (Mediterranean shrub). A realistic interaction between the ambient flow and the fire plume was found; in particular it was shown how this interaction characterised by a Froude number influences the behaviour of a fire (Morvan et al. 2002; Morvan and Dupuy 2004). Numerical simulations The multiphase physical model was applied to study the propagation of a wildfire through a vegetation layer representative of a Mediterranean forest stand. The understorey was composed of a mixture of shrub (Quercus coccifera, QC) and grass (Brachypodium ramosum, BR) and the canopy was composed of pine crowns (Pinus halepensis, PH). These species are well represented in the Mediterranean regions of southern Europe. The two-dimensional resolved domain was 200 m long (x-direction) and 50 m high (z-direction). The area between x = 0 and x = 25 m was free of vegetation in order to start the fire (ignition) far from the left boundary (x = 0 m) of the resolved domain and to force the fire to spread from the left to the right. The grid mesh was refined along the xdirection in the combustion zone and in the vicinity of the fire. Because the fire front moved from the left to the right, this refined part of the mesh was attached to the fire front. In this refined area, at the canopy level, the typical size of cells was 25 × 25 cm. The height of cells was much lower at the ground level (2.5 cm) and much larger at the top of the domain (∼1 m). Out of the refined area, the length of the cells in the x-direction was 1 m. As shown below, in simulated crown fires, heating and even ignition of the top of the canopy may occur very far from the main fire front. Thus, in order to capture these events with enough accuracy, we were forced to maintain rather fine cells (1 m length in the x-direction) even far from the main fire front. Running a simulation in these conditions took ∼48 h on a PC equipped with a Pentium 4 (2 GHz) processor. It is only very recently that we improved the computational method (based on the use of two separate grids for solid phases and gaseous phase computations) and the up-dated computer program is still undergoing tests. We can indicate that the same simulation would now take ∼12 h on the same hardware configuration. The vertical profile of wind at the left boundary of the domain was given by a power law: z 1/7 U (z) = UH , H where z is the height above the ground and UH the wind velocity given at height H. The calculations were performed with a wind speed equal to 50 km h−1 , a value given at 10 m above the ground.According to the expertise of firefighters and forest managers of the south of France, this wind speed, associated with dry summer conditions, leads to a very severe level of danger.

Fig. 1. Spatial distribution of biomass (solid fuel density field) of a Mediterranean pine forest (graduations along horizontal and vertical axis are in m). Fuel density includes the water content.

Crown fire behaviour In the untreated forest stand, the vegetation was structured as follows: the depth of the shrub layer was equal to 1 m, the height of grass was equal to 0.25 m, and the tree crowns were between 2 and 12 m in height. Each species covered only a part of the horizontal surface: 70% for the shrub, 50% for the grass and 80% for the trees (Fig. 1).According to the expertise of forest managers, these values for a Pinus halepensis stand are representative of dense vegetation (Rigolot 2002). The fire was ignited in the shrub and in the grass, introducing a transient heat source term in the solid fuel energy equation (at x = 25 m). Experimental studies (Call and Albini 1997) have shown that, during the propagation of a wildfire through a living fuel (moisture content of ∼100%), the main part of the burned fuel (>65%) had a thickness less than 6 mm. Consequently for the present study, the description of the vegetation was limited to the thinnest solid fuel particles, introducing six solid fuel families: one for the grass (BR), three for the shrub (QC) and two for the trees (PH) (for this kind of pine, all twigs have a diameter greater than 2 mm).The physical properties of fuel families are reported in Table 1. Surface area-to-volume ratio and density of fuel particles (QC and PH families) were measured on control samples used for experimental purposes in INRA’s facilities. The values of fuel moisture content were assigned based on measurements of moisture content of vegetation samples (foliage) collected for risk-assessment purposes and they are representative of summer conditions. As usually observed, the foliage of shrub species has a lower moisture content than the tree foliage. The values of volume fraction of shrubs (QC) were measured according to a method described in Cohen et al. (2003) and the values of volume fraction of grass (BR) were deduced from biomass sampling. Values of volume fraction for trees (PH) were mainly assumed. Figure 2 shows three images of the temperature field calculated in the gaseous phase, after the ignition step. First, a surface fire propagated in shrubs and grass (Fig. 2a). When the fire covered a distance of ∼25 m from the ignition area, a transition of the surface fire to the canopy base occurred (Fig. 2b). We plotted the successive fields of radiation and

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Table 1. Fuel family QC (leaves) QC (twigs 0–2 mm Ø) QC (twigs 2–6 mm Ø) BR PH (needles) PH (twigs 2–6 mm Ø)

Physical properties of fuel families

Mean volume fraction

Density of dry material (kg/m3 )

Surface area-to-volume ratio (m2 /m3 )

Moisture content (%)

0.0010 0.0003 0.0003 0.0010 0.0003 0.0002

820 900 900 400 850 900

6000 3000 1000 10 000 10 000 1000

70 70 70 15 100 100

Fig. 3. Fields of solid fuel density (a), foliage moisture content (b) and gas temperature and velocity (c), calculated at the same time during the propagation of a crown fire. Solid fuel density includes the water content. Fig. 2. Images of the gas temperature fields calculated at (a) 5 s, (b) 25 s and (c) 40 s of a crown fire spreading in a Mediterranean pine forest.

convection heat flux received by the solid fuel about the time of transition, focusing on the points where ignition of crowns took place. We found that the base of tree crowns was mainly heated and then ignited through convection heat transfer; the convection heating represented more than 90% of the total heat input at this location. Finally, the overall canopy was ignited and we observed the propagation of an active crown fire (Fig. 2c). The biomass field (solid fuel density), the foliage moisture content field and the gas temperature and velocity fields are reported at the same time in Fig. 3, as the regime of crown

fire spread reached a quasi-steady state. If we look at both the shape of the interface between dried fuel and moist fuel (fuel moisture field) and the shape of the area of hot gases (gas temperature field), we can infer that the flow of hot gases due to the combustion of shrubs and grass (surface fire) was strongly inclined from the vertical and ‘provided’ energy to the crown fire. On the contrary, from ∼4–5 m above the ground to the top of the crowns (the region of crowning), the flow of hot gases was roughly vertical. The examination of gas velocity field confirmed these observations. Therefore, following the well-known classification of Van Wagner (1977) (see also Scott and Reinhardt 2001), we can deduce that the crown fire was an active but dependent crown fire: the existence of the crown fire depends on the presence of the surface fire, which



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Fig. 4. Time evolution of the fire front position during the propagation of a crown fire.

supplies an amount of energy necessary to support the combustion of tree crowns. In addition, at the top of the canopy the thermal degradation of the vegetation appeared several metres ahead of the main fire front (foliage moisture field): flames were leant towards the top of the canopy by the wind and caused the drying and the ignition of the top of the crowns. According to the plot of the total bulk density of fuel, the combustion of the surface fuel was complete, whereas only about half of the biomass available on the crown level was burned at this time. When looking at the results for the two families of fuel particles in the canopy, we noticed that the needles were completely burned and reduced to ashes, but a significant part of the twigs remained as unburned chars. We attribute this effect to the difference of thickness of the two kinds of particles, needles and twigs, resulting in very different surface area-to-volume ratios and thus in different rates of heat and mass transfer. Conversely, the shrub twigs, which have the same diameter as the tree twigs, were completely burnt in the understorey: this could be due to a lower cooling of fuel by the ambient wind in the understorey and also to the fact that shrub twigs had a lower moisture content than tree twigs. Figure 4 shows the time evolution of the position of the main front of the crown fire. We tracked the position of the fire front by looking at the successive isotherms of the solid phases and retaining the maximum value of the position x where the isotherm is achieved. We usually used the isotherms 500K (pyrolysis), but isotherms 700K (end of pyrolysis) or isotherms 1100K (char combustion) generally gave the same results with a short time shift. In order to track the main fire front, ignoring hot pockets that may appear more or less far ahead at the top of the tree crowns (most advanced points, discussed below), we retained only points located in

Fig. 5. Spatial distribution of biomass (solid fuel density field) resulting from fuel treatment #1 (a) and fuel treatment #2 (b). Fuel density includes the water content.

the middle part of the crowns. We saw two regimes of propagation (Fig. 4): the average rate of spread was 1.1 m/s during the first 100 s of propagation; then the rate of spread went down to 0.5 m/s. At the beginning of the fire propagation (edge of the pine stand), the wind rushed into the vegetation layer and pushed the fire to the right of the domain. During the first 100 s of propagation, the flow of air inside the canopy layer was slowed down by the presence of unburned fuel (twigs), but remained mainly directed towards the right of the domain (positive x-direction), pushing the fire. At the same time horizontal transverse roll vortices (Haines 1982; Haines and Smith 1987) were observed to form just above the canopy and, at ∼100 s, a large vortex located behind the fire caused a reverse flow from ground level to ∼7 m height (mainly oriented towards the negative x-direction). Successive vortices then formed behind the fire, intermittently causing a type of reverse flow, which finally slowed down the fire propagation. An example of such vortices can be seen in Fig. 3 (gas temperature and velocity field). Effects of a fuel break on fire behaviour To study the effects of fuel reduction on wildfire behaviour, two additional numerical simulations were performed after modification of the vegetation between x = 125 m and x = 200 m (fuel break). Two techniques of fuel reduction were tested (Fig. 5): (1) a reduction of the shrub layer to 5 cm height and tree pruning up to 3 m height (treatment #1); and (2) a reduction of the shrub layer to 35 cm height only, but removal of trees to decrease the canopy cover to 25% of the area and tree pruning of the remaining trees up to 3 m height

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Fig. 6. Images of the gas temperature field calculated at (a) 80 s, (b) 115 s and (c) 155 s of a crown fire reaching a fuel break (fuel treatment #1).

Fig. 7. Images of the gas temperature field calculated at (a) 100 s, (b) 115 s and (c) 235 s of a crown fire reaching a fuel break (fuel treatment #2).

(treatment #2). In the French Mediterranean region, forest managers consider that the safety requirements are fulfilled when the space volume occupied by the surface fuel is below 1000 m3 /ha (empirical criteria). In contrast, when the surface fuel volume reaches a value above 2500 m3 /ha, they consider that the probability of observing a transition of the fire from the surface fuel to the tree crowns increases significantly and hence the safety requirements are no longer fulfilled (Rigolot 2002; Dureau 2003). For the first treatment tested in the present paper (#1), the surface fuel volume is reduced to 400 m3 /ha, and for the second treatment (#2) this parameter is reduced to only 2800 m3 /ha. Figure 6 shows images of the temperature field in the gas phase during propagation of the crown fire towards a fuel break created following treatment #1. Figure 6a shows the active crown fire propagating through the untreated zone. In Fig. 6, a strong reduction of the burning zone was observed as the fire reached the edge of the fuel break. Finally, the fire transitioned to a surface fire (Fig. 6c). For these conditions, the propagation of the fire through the canopy could not be maintained in absence of a surface fire. Indeed, by reducing the depth of the shrubs in the fuel break, the biomass was also

reduced and the heat released by the surface fire was no longer sufficient to sustain fire propagation through the canopy. Figure 7 shows the results when fuel treatment #2 was applied. Although the reduction of the surface fuel was less severe than in treatment #1, in the same manner the burning zone was strongly reduced as the fire reached the edge of the fuel break (Fig. 7b), and finally the crown fire transitioned to a surface fire. However, a vertical transition of the surface fire, which continued to propagate, to a tree crown (torching) was observed in the fuel break (Fig. 7c). According to these two simulations, one can conclude that treatment #2 was less efficient than treatment #1 in terms of safety. Figure 8 shows the effect of fuel treatment #1 (Fig. 8a) and of the fuel treatment #2 (Fig. 8b) on the fire intensity when the time evolutions of this variable are compared. Fire intensity was calculated as the sum of the heat release rate due to the combustion in the gaseous phase (flames) and the heat release rate due to the combustion of chars (embers). At the beginning of the fire propagation, a continuous and fast rise of intensity occurred, to reach a peak of ∼70 MW/m after the ignition of crowns. Then the time-averaged intensity gently decreased and, with no fuel treatment, the intensity



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Fig. 9. Images of the gas temperature (a) and solid fuel density (b) fields of an up-slope crown fire for a 15◦ slope angle. Solid fuel density includes the water content.

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Fig. 8. Reduction of fire intensity in a fuel break due to treatment #1 (a) and treatment #2 (b).

of the crown fire reached a representative average value of 30 MW/m. With both fuel treatments, the reduction of fire intensity due to the fuel reduction is clear, but in the case of fuel treatment #2 the torching event caused a fast rise in fire intensity, up to about the same value as the crown fire intensity (30 MW/m). This event had a short duration (less than 10 s) but obviously it could have been an additional danger for a firefighter fighting on the fuel break. Slope effects on fire behaviour The effects of slope were also studied using two slope angles of 15◦ and 30◦ . Figures 9 and 10 show the temperature and solid fuel density fields of the gaseous phase obtained 80 s after ignition, for the 15◦ and 30◦ slope angles. Hot gases were observed very far ahead of the main front at the level

of the upper part of the canopy. The solid fuel density fields show that at least water was evaporated very far from the fire front, at the top of the canopy. We observed by plotting the field of gas velocity vectors that hot gases were blown back down to the tree crowns due to the motion of vortices, which formed less than 100 m from the top of the slope and were ∼10–15 m in diameter. The fields of solid phase variables (PH needles family) revealed that these high gas temperatures actually caused the ignition of the crowns (the temperature of PH needles was above 500K and dry fuel density was reduced). We emphasise that these ignitions far ahead of the fire were due to hot gases, not to fire brands, which are not considered in the model. The length of the domain was not enough to see whether these ignitions would have caused a second propagating fire front or not. But the combustion of the crowns, which started from their top, continued and propagated to the base of the crowns. This suggests that a second fire front could have started from these ignitions. Figure 11 shows the time evolution of the position of the main front for the three simulated crown fires (0◦ , 15◦ and 30◦ slopes). The rates of spread of the two up-slope fires were very similar and were greater than the rate of spread of the no-slope fire. We also plotted the position of the most advanced points of each fire, thus including the ignitions far ahead of the main fire front. These ‘jumps’were very clear for

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From these simulations it appears that the slope of the terrain can dramatically increase the fire danger and could reduce the efficiency of a fuel break. We observed that the slope changed the fire behaviour in such a way that the ‘jumps’ of a crown fire can reach several tens of metres. We expect that perhaps 300 m of propagation or more could be necessary to obtain a representative regime of crown fire propagation for these up-slope cases combined with a strong wind. It might explain in turn why a sharp increase of the crown fire rate of spread was not observed when slope angle went from 0◦ to 15◦ and furthermore from 15◦ to 30◦ . This increase was expected given the common experience that the slope of the terrain greatly influences the rate of spread of a fire. Conclusions

Fig. 10. Images of the gas temperature (a) and solid fuel density (b) fields of an up-slope crown fire for a 30◦ slope angle. Solid fuel density includes the water content. 200 Slope 30º

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Fig. 11. Time evolution of the fire front position of a crown fire for different angles of terrain slope. Most advanced points show fuel ignitions occurring ahead of the main fire front at the top of the tree crowns.

the two up-slope fires and could reach a distance of ∼70 m. In contrast, only short-distance ‘jumps’ appeared for the no-slope fire, of which the progression of the main front is very similar to the progression of the most advanced point.

The propagation of a wildfire through a Mediterranean pine stand was simulated using a multiphase physical model. The numerical results showed the ability of this approach to simulate the fire propagation through an actual forest fuel of a heterogeneous nature, and to test the efficiency of fuel reduction techniques devoted to the reduction of fire danger in a fuel break. The numerical results highlighted the connection between the surface fire and the so-called active crown fire. For the conditions tested (wind and vegetation data) the crown fire could not propagate if it was not supported by a surface fire. Consequently, by reducing the biomass at the ground level in a fuel break, the intensity of the simulated surface fire was reduced and crowning did not continue. We also found that it might be better to maintain the tree density, but reduce the shrub layer drastically, than to clear the stand and to maintain a significant amount of shrubs. It was shown from the simulations of up-slope fires (15◦ and 30◦ angle) how the fire danger can be increased when a strong wind is combined with a terrain slope due to local ignitions of the tree crowns far ahead of the main fire front. The important and complex role of vortices that formed due to the interaction between the wind flow and the vegetation, was also observed in these simulations. Vortices travelling ahead of the fire front may cause ignitions far from it in the case of up-slope fires, increasing the fire danger. In contrast, vortices that were observed to form behind the fire front, due to the interaction with the unburned fuel remaining in tree crowns, were found to prevent the ambient wind from pushing the fire intermittently and, finally, to reduce the average rate of spread of a crown fire. Acknowledgements The authors thank the two anonymous reviewers who greatly contributed to improve the present article. This study has been partially funded by the European Commission in the frame of the FIRESTAR research programme (contract EVG1–2001– 00041).



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Appendix 1 d Ms,k,α ρs,k = − dt α

Nomenclature Latin symbols ap B Cp,s,k D Fi,s

gi h hconv

I J k Ms,k,α t T ui xi Y

radiation absorption coefficient of the gas–soot mixture (m−1 ) Stefan-Boltzman constant (W m−2 K−4 ) specific heat of the solid phase k (J kg−1 K−1 ) mass diffusivity (kg m−1 s−1 ) drag forces due to interactions between the solid phases and the gaseous phase (component in the i-direction) (kg m−2 s−2 ) gravity acceleration (component in the idirection) (m s−2 ) enthalpy (J m−3 ) heat transfer coefficient (convection between a solid phase k and the gaseous phase) (W m−2 K−1 ) radiative intensity (W sr−1 m−2 ) irradiance (W m−2 ) turbulent kinetic energy (m2 s−2 ) production of the α chemical species due to thermal degradation of the solid phase k (kg m−3 s−1 ) time (s) temperature (K) gas velocity (component in the i-direction) (m s−1 ) cartesian coordinate in the i-direction (m) mass fraction

Greek symbols α ε ρ σi,j σs,k ω

volume fraction dissipation rate (m2 s−3 ) density (kg m−3 ) stress tensor (kg m−1 s−2 ) surface area-to-volume ratio of the solid phase k (m−1 ) reaction rate (kg m−3 s−1 )

d 1 αs,k = − ωchar dt ρs,k dTs,k ρs,k Cp,s,k = hconv,k αs,k σs,k (Tg − Ts,k ) dt αs,k σs,k + (J − 4BT 4 ) s,k 4 − Ms,k,α hs,k,α .

chemical species gaseous phase solid phase

(A6)

Equations (A1), (A2) and (A3) govern the evolution of the mass fractions of water, dry material and char respectively, resulting from drying, pyrolysis and char combustion processes. The terms on the right hand side of equation (A3) are written assuming that the rate of production of char is a known fraction (υchar ) of the pyrolysis reaction rate and that a part of char is lost in the plume to form soot particles + (υchar = υchar − υsoot · ωchar represents the rate of char combustion. Equation (A4) represents the mass balance of the solid phase. Equation (A5) governs the evolution of the volume fraction, assuming that the reduction of the volume of solid fuel particles results only from char combustion (when the particle undergoes drying and pyrolysis processes, its volume remains constant). The first and second terms on the right hand side of the energy balance equation (A6) represent the contributions resulting from convective exchanges with the surrounding gas and from radiation heat transfer respectively. The third term represents energy transfers due to mass transfers between the solid phase and the gaseous phase. Transport equations of the gaseous phase Basically, the transport equations of the gaseous phase are the Navier-Stokes equations generalised to a compressible, reactive and multiphase flow: Mass:

∂ρg ∂ρg uj = Ms,α + ∂t ∂xj α

Momentum: ∂ρg ui ∂ρg ui uj ∂αg σij + = + ρg gi − Fi,s ∂t ∂xj ∂xj

Balance equations of solid phases The following set of balance equations governs the evolution of each solid phase (or fuel family) k: d H2 O = −ωvap (A1) ρs,k Ys,k dt d wood ρs,k Ys,k = −ωpyr (A2) dt d + − char = υchar ρs,k Ys,k ωpyr − υchar ωchar (A3) dt

(A5)

α

Subscripts α g k

(A4)

Energy: ∂ρg hg ∂ρg uj hg + ∂t ∂xj ∂Tg ∂ k − = hconv,k αs,k σs,k (Tg − Ts,k ) ∂xj ∂xj k Ms,α hs,α + αg ap (J − 4BTg4 ) + α


Chemical species: ∂ρg Yg,α ∂ρg uj Yg,α + ∂t ∂xj ∂Yg,α ∂ Dg,α + ωg,α + Mα,s . = ∂xj ∂xj To take into account the turbulence effects, these basic transport equations are time-averaged using the Favre formalism and a turbulent diffusion model (RNG k–ε) is introduced. Radiation transfer equation The radiation intensity I is calculated solving the following radiation transfer equation: 4 4 dαg I αs,k σs,k σTs,k σT −I . = αg ap −I + π ds π 4 k


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The irradiance J, which appears in the equations of energy balance, is formally calculated at each point through an integration of the radiation intensity over the space solid angle:

4π I dω. J = 0