Wildland Fire Modelling including Turbulence and Fire spotting

Transcription

1 Wildland Fire Modelling including Turbulence and Fire spotting Inderpreet Kaur 1 in collaboration with Andrea Mentrelli 1,2, Frederic Bosseur 3, Jean Baptiste Filippi 3, Gianni Pagnini 1,4 1 BCAM, Bilbao, Basque Country Spain 2 Department of Mathematics & AM 2, University of Bologna, Italy 3 SPE CNRS/University of Corsica, Corte, Corsica France 4 Ikerbasque, Bilbao, Basque Country Spain Inderpreet KAUR Modelling random processes in wildfires 1 / 37

2 Introduction Introduction Wildland fires are one of the natural phenomena that can cause hazardous situations for people and property. They also constitute a very serious environmental problem. Inderpreet KAUR Modelling random processes in wildfires 2 / 37

3 Introduction Random Character of Wildland Fire TURBULENCE Moisture and Heat release from fuel combustion alters local thermal structure of the lower Atmospheric Boundary layer (ABL) and induces turbulent circulation in vicinity of fire. Vertical scale : 1-1 m; Horizontal Scale : 1-1 m ( Sullivan 213, Intl. J. Wildland Fire) Inderpreet KAUR Modelling random processes in wildfires 3 / 37

4 Introduction Random Character of Wildland Fire FIRE-SPOTTING Behaviour of a fire producing sparks or embers that are carried by the wind and start new spot fires beyond the zone of direct ignition by the main fire The random effects cause a rapid increase in the rate of spread of the fire. New isolated fires can develop across fire break zones like rivers/roads. Vertical scale : 1-3 m; Horizontal Scale : 1-1 m ( Sullivan 213, Intl. J. Wildland Fire) Inderpreet KAUR Modelling random processes in wildfires 4 / 37

5 Model Description Notation Model Description Notation Let Γ be a simple closed curve, or an ensemble of simple non-intersecting closed curves, representing a propagating interface in two dimensions. Let S R 2 be the domain of interest. The subset of the domain S corresponding to the region Ω enclosed by Γ may be conveniently identified as the region selected by the indicator function I Ω : S [, + [ {, 1} defined as 1, x Ω, I Ω (x, t) = (1), elsewhere. Inderpreet KAUR Modelling random processes in wildfires 5 / 37

6 Model Description Notation Model Description Notation Let Γ be a simple closed curve, or an ensemble of simple non-intersecting closed curves, representing a propagating interface in two dimensions. Let S R 2 be the domain of interest. The subset of the domain S corresponding to the region Ω enclosed by Γ may be conveniently identified as the region selected by the indicator function I Ω : S [, + [ {, 1} defined as 1, x Ω, I Ω (x, t) = (1), elsewhere. Inderpreet KAUR Modelling random processes in wildfires 5 / 37

7 Model Description Random Front Formulation Random Front Formulation Let X ω (t, x ) = x(t, x ) + χ ω + ξ ω be the ω-realization of a random trajectory driven by the random noises χ and ξ corresponding to turbulence and fire spotting, respectively. For every realization, the initial condition is X ω (, x ) = x. Average turbulent fluctuations: χ =. Fire spotting is assumed to be independent of turbulence and to be a downwind phenomenon, then, in the leeward sector only, it is stated: ξ ω = l ω n U, where l is the landing distance from the main fireline such that l > and n U is the mean wind direction. Inderpreet KAUR Modelling random processes in wildfires 6 / 37

8 Model Description Random Front Formulation Random Front Formulation The trajectory of an active burning point can be marked out in terms of δ- function. Using the property of δ-function: g(x) = g(x) δ(x x)dx, the evolution in time of the ω-realization of a random front contour γ ω (x, t) is given by γ ω (x, t) = γ(x ) δ(x X ω (t, x )) dx, (2) S which in terms of the random indicator I Ω ω(x, t) reads I Ω ω(x, t) = I Ω (x ) δ(x X ω (t, x )) dx S = δ(x X ω (t, x )) dx Ω = δ(x X ω (t, x)) dx, (3) where J = dx dx Ω(t) = 1 is assumed. Inderpreet KAUR Modelling random processes in wildfires 7 / 37

9 Model Description Random Front Formulation Random Front Formulation Let ϕ e (x, t) : S [, + [ [, 1] be an effective indicator and it may be defined as ϕ e (x, t) = I Ω ω(x, t) = δ(x X ω (t, x))dx Ω(t) = δ(x X ω (t, x)) dx = Ω(t) Ω(t) f (x; t x) dx, (4) where f (x; t x) = δ(x X ω (t, x)) is the probability density function (PDF) of fluctuations of the fireline perimeter around the deterministic part contour Γ(t) = {x S γ(x, t) = γ }. Inderpreet KAUR Modelling random processes in wildfires 8 / 37

10 Burning Criteria Model Description Random Front Formulation Function ϕ e (x, t) continuosly ranges from to 1. Q: How to mark a point as burned? A: For example, points such that ϕ e (x, t) >.5 can be marked as burned, i.e. Ω e (t) = {x : ϕ e (x, t) >.5}. Inderpreet KAUR Modelling random processes in wildfires 9 / 37

11 Burning Criteria Model Description Random Front Formulation Function ϕ e (x, t) continuosly ranges from to 1. Q: How to mark a point as burned? A: For example, points such that ϕ e (x, t) >.5 can be marked as burned, i.e. Ω e (t) = {x : ϕ e (x, t) >.5}. Q: How to take into account the effects due to turbulence and fire spotting which act ahead the fireline? A: For example, with a model for a heating-before-burning law. Inderpreet KAUR Modelling random processes in wildfires 9 / 37

12 Model Description Random Front Formulation The Model: Heating-before-burning Law The presence and persistence of ϕ e (x, t) can be understood as an accumulation in time of the igniting heat and, since the amount of heat ψ(x, t) is proportional to the increasing of the fuel temperature T (x, t), it follows ψ(x, t) = t ϕ e (x, s) ds τ T (x, t) T (x, ) T ign T (x, ), (5) where τ is an ignition delay time-scale and T ign is the ignition temperature. The evolution of T (x, t) is T t ϕ e (x, t) T ign T (x, ) τ, T T ign. (6) Hence, when for example ψ(x, t) = 1, it is stated that I Ω (x, t) = 1 and an ignition occurs in (x, t) starting a new fire. Inderpreet KAUR Modelling random processes in wildfires 1 / 37

13 Model Description Random Front Formulation The Model: Heating-before-burning Law Electrical resistance analogy can be considered in order to the estimate the ignition delay τ, which is due to the combined actions by heat transfer and fire spotting. Because of this two pathways of ignition, heating and firebrand act as resistances in parallel giving 1 τ = 1 τ h + 1 τ f = τ h + τ f τ h τ f. (7) Inderpreet KAUR Modelling random processes in wildfires 11 / 37

14 Model for Turbulence Model Description Model for Random Processes : Turbulence A Gaussian distribution is chosen as the most simple model for turbulent dispersion of hot air 1 G(x x; t) = { 2π σ 2 (t) exp (x x)2 + (y y) 2 } 2 σ 2, (8) (t) σ 2 (t) = (x x) 2 = (y y) 2 = 2 D t. (9) Inderpreet KAUR Modelling random processes in wildfires 12 / 37

15 Model Description Model for Random Processes : Fire-spotting Model for Random Processes : Fire-spotting A numerical study by Sardoy et al., Combust. Flame, 28 proposed that the firebrands follow a log-normal distribution q(l; t) = } 1 (ln l µ(t))2 exp { 2π s(t) l 2 s 2, (1) (t) where µ(t) = ln l and s(t) = (ln l µ(t)) 2 are the mean and the standard deviation of ln l. Inderpreet KAUR Modelling random processes in wildfires 13 / 37

16 Model Description The f (x; t x) probability density function Model for Random Processes : Fire-spotting The PDF of the sum of two independent random variables is given by the convolution of the corresponding PDFs. Then in the leeward sector, i.e. θ < π/2 with θ the angle between n and n U, f (x; t x) = otherwise in the windward sector G(x x l n U ; t) q(l; t) dl, (11) f (x; t x) = G(x x; t). (12) Inderpreet KAUR Modelling random processes in wildfires 14 / 37

17 Approaches to Fire Modelling Approaches to Fire Modelling Numerical simulation of wildfires is a two fold process Fire Spread Model : Rate of Spread Model provides the velocity of the front driven by the fuel characteristics, atmospheric conditions and topography Rothermel model (Rothermel, 1971) Balbi Model (Balbi et al., 25) Fire-front tracking Method : Simulates the advancement of the fire-front. Level Set Method (Mandel et al., 211) Discrete EVents System Specification (DEVS; Filippi et al., 21) Huygens Principle (Rios et al., 214; Anderson et al., 1971) Both models are developed independently of each other. Inderpreet KAUR Modelling random processes in wildfires 15 / 37

18 The Level-Set Method Approaches to Fire Modelling Level Set Method Let γ(x, t) be a function such that a certain isoline γ(x, t) = γ = constant corresponds to the interface Γ(t). Then the motion of the interface Γ(t) is determined by the evolution equation of the isoline Dγ Dt = γ t + dx γ =, (13) dt dx dt = V(x, t) = V(x, t) n, n = γ γ, (14) γ t = V(x, t) γ, (15) where V(x, t) is the Rate of Spread (ROS) of the fireline. A huge literature exists for its theoretical and phenomenological determination. Inderpreet KAUR Modelling random processes in wildfires 16 / 37

19 Approaches to Fire Modelling Fire Propagation Methods Level Set Method (Mandel et al., 211 ) Level Set Method Eulerian time-driven technique Represents the burning region on a simple cartesian grid. Temporal resolution constrained by the global time step restriction imposed by the Courant-Friedrichs-Levy (CFL; t < x/v max ) criteria. Direction of propagation is given by the normal to the front. Inderpreet KAUR Modelling random processes in wildfires 17 / 37

20 Approaches to Fire Modelling Fire Propagation Methods ForeFire (Filippi et al., 29) ForeFire Lagrangian specification of the fire interface. Fire interface is discretized by a set of points/markers. No global time step is defined. Each marker moves asynchronously according to its speed and direction function and the CFL condition ( t < x/v max ) is applied locally. The asynchronous movement is managed by Discrete EVent systems Specification (DEVS). Inderpreet KAUR Modelling random processes in wildfires 18 / 37

21 Approaches to Fire Modelling Fire Propagation Methods ForeFire (Filippi et al., 29) ForeFire The maximum distance covered by each marker is fixed, Quantum Distance ( q). Details smaller than this distance might not be considered. Perimeter Resolution ( c) restricts the maximum distance allowed between the two markers. Measure of c is used to decompose/coalesce the markers. The choice of q and c is dependent on the type of problem. The advection scheme is first order Euler scheme in space: x n+1 i t n+1 i = x n i + q n b i (16) = t n i + q v b i The direction of propagation is given by the bisector of the angle made by each marker with its immediate neighbours. (17) Inderpreet KAUR Modelling random processes in wildfires 19 / 37

22 Simulation Setup Simulation Setup The formulation of random processes in both Eulerain and Lagragian front tracking methods is studied. It is tried to parametrize the models in an identical set-up. The grid size in LSM is chosen to be 2 m in x and y direction. q = 4m, c = 18m for zero wind; and q =.3m, c = 8m in the presence of wind. To avoid instability in the presence of wind, q is chosen to be of a much higher resolution than the wind data (2 m 2 m in this setup). No particular type of vegetation is defined and work with pre-defined value of ROS. ROS is assumed to be constant throughout the terrian. These are simplified and idealised test cases and no attempt is made to choose the parameters for a realistic setup Inderpreet KAUR Modelling random processes in wildfires 2 / 37

23 Results LSM & DEVS (without random phenomena) (a) 4 35 (b) Evolution in time of the fire perimeter in no wind conditions with an initial circular profile of radius R = 3 m and ROS =.5 ms 1 by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 21 / 37

24 LSM & DEVS Results (a) (b) Evolution in time of the fire perimeter in no wind conditions with an initial circular profile of radius R = 3 m and ROS =.5 ms 1 in presence of a firebreak line by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 22 / 37

25 Results LSM & DEVS (without random phenomena) (a) (b) Evolution in time of the fire perimeter in no wind conditions with an initial circular profile of radius R = 3 m and ROS =.3 ms 1 ;.2 ms 1 ;.1 ms 1 in the upper left, upper right and lower quadrants respectively, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 23 / 37

26 Results LSM & DEVS (without random phenomena) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m, ROS =.3 U n and mean wind U = 3 ms 1 by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 24 / 37

27 Results LSM & DEVS (without random phenomena) (a) (b) Evolution in time of the fire perimeter with an initial square profile of side R = 6 m, ROS =.3 U n and mean wind U = 3 ms 1 by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 25 / 37

28 Results LSM & DEVS (without random phenomena) With prescription of head, flank and rear ROS: { ɛ o(1 + a U cos ROS = n θ), if θ π 2 ɛ o(α + (1 α) sin θ ), if θ > π, (Mallet et al., Comput. Math. Appl., 29) 2 Inderpreet KAUR Modelling random processes in wildfires 26 / 37

29 Results LSM & DEVS (without random phenomena) With prescription of head, flank and rear ROS: { ɛ o(1 + a U cos ROS = n θ), if θ π 2 ɛ o(α + (1 α) sin θ ), if θ > π, (Mallet et al., Comput. Math. Appl., 29) (a) (b) Evolution in time of the fire perimeter with initial circular profile of radius R = 3 m according to Mallet ROS, where n = 3, U = 3 ms 1, a =.5 sm 1, ɛ =.2, α =.8 and θ is the angle between the line joining a contour point and the center of the initial profile and the mean wind direction, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 26 /

30 Results LSM & DEVS (without random phenomena) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m and { ɛ o(1 + a U cos ROS = n θ), if θ π 2 ɛ o(α + (1 α) sin θ ), if θ > π, (Mallet et al., Comput. Math. Appl., 29) where n = 3, U = 3 ms 1, a =.5 sm 1, ɛ =.2, α =.8 and θ is the angle between the outward normal in a contour point and the mean wind direction, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 27 / 37

31 Results Turbulence and Fire Spotting Set Up Turbulence is modelled with a Gaussian with diffusion coefficient D. The jump-length of embers is modelled with a stationary log-normal distribution with (Perryman et al., Int. J. Wildland Fire, 213): mean value µ = 1.32 I.26 f U.11.2, standard deviation s = 4.95 I.1 f U , where U is the modulus of the mean wind and I f = I + I t with I = 1kWm 1 is the fire intensity and I t =.15 kwm 1 is the tree torching intensity. ROS =.5 ms 1 with no wind ROS =.3 U n with wind. Ignition delay of hot air τ h = 6 s, ignition delay of firebrands τ f = 6 s. Inderpreet KAUR Modelling random processes in wildfires 28 / 37

32 Results LSM & DEVS (with turbulence) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m for zero wind and diffusion coefficient D =.15 m 2 s 1, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 29 / 37

33 Results LSM & DEVS (with turbulence) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m, for zero wind and diffusion coefficient D =.3 m 2 s 1, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 3 / 37

34 Results LSM & DEVS (with turbulence) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m, with a mean wind U = 3 ms 1 and diffusion coefficient D =.15 m 2 s 1, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 31 / 37

35 Results LSM & DEVS (with turbulence and fire spotting) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m, a mean wind U = 3 ms 1 and diffusion coefficient D =.15 m 2 s 1, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 32 / 37

36 Results LSM & DEVS (with turbulence) (a) (b) Evolution in time of the fire perimeter with an initial circular profile of radius R = 3 m in presence of two firebreak zones with a mean wind U = 3 ms 1 and diffusion coefficient D =.7 m 2 s 1, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 33 / 37

37 Results LSM & DEVS (with turbulence and fire spotting) (a) (b) Evolution in time of the fire perimeter with initial circular profile of radius R = 3 m in presence of two firebreak zones with a mean wind U = 3 ms 1 and diffusion coefficient D =.7 m 2 s 1, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b). Inderpreet KAUR Modelling random processes in wildfires 34 / 37

38 LSM & DEVS Results LSM based model LSM based model + Turbulence LSM based model + Turbulence + Fire spotting DEVS based model DEVS based model + Turbulence DEVS based model + Turbulence + Fire spotting Inderpreet KAUR Modelling random processes in wildfires 35 / 37

39 Conclusions Conclusions This formulation emerges to be suitable for both Eulerian (LSM) and Lagrangian (DEVS) approaches to model dangerous situations as: the increase in the ROS as a consequence of the pre-heating of the fuel by the hot air and generation of new fires due to fire spotting phenomenon. the spread of the fire in the direction normal to the mean wind (flanking fire) and opposite to the mean wind direction (backing fire). the propagation of fire across the roads, rivers, firebreak lines etc. where both LSM and DEVS fail due to null ROS in the absence of the ground fuel. DEVS uses the bisector of the angle between the marker and its neighbours as propagation direction of the fire perimeter which generates differences with the LSM. Such differences result in additional flanking fires, which however provide a more realistic fire contour. The LSM and DEVS based models show a variety of behaviours in response to the random phenomena so that it is not possible to provide a unique conclusion about their effects on the models differences. Inderpreet KAUR Modelling random processes in wildfires 36 / 37

40 Conclusions Thank You for your attention! Inderpreet KAUR Modelling random processes in wildfires 37 / 37