Sviluppi modellistici sulla propagazione degli incendi boschivi

Sviluppi modellistici sulla propagazione degli incendi boschivi Gianni PAGNINI Borsista RAS PO Sardegna FSE 2007-2013 sulla L.R. 7/2007“Promozione della ricerca scientiﬁca e dell’innovazione tecnologica in Sardegna” Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Introduction Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Turbulence Sources in Wildland Fire FrontPropagation (1) Wildland ﬁre propagates at the ground level and then it is dependent on the dynamics of the Atmospheric Boundary Layer, whose ﬂow is turbulent in nature. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Turbulence Sources in Wildland Fire FrontPropagation (2) Moreover, in this atmospheric layer the turbulence is ampliﬁed by the forcing due to the ﬁre-atmosphere coupling Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Turbulence Sources in Wildland Fire FrontPropagation (2) ... and by the appearing of the ﬁre-induced ﬂow. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Importance of Turbulence Modelling in Wildland Fire As a consequence of the turbulent transport of the hot air mass, that can pre-heat and then ignite the area ahead the fire, the fire front position becomes random. Hence, it is of paramount importance for the prediction of the fire motion to take into account turbulence. Accounting for the effect of turbulence on the fire propagation improves the usefulness of the operational models and thereby increases the firefighting safety and in general the efficiency of the fire suppression and control efforts. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Level-Set Method (1) Let Γ(t) = (x(s, t), y(s, t)) be a parameterized evolving interface. Let ϕ(x, t) be a function such that the level-set ϕ = constant corresponds to the evolving front Γ(t). Then the equation for the evolution of ϕ corresponding to the motion of the interface Γ(t) is given by Dϕ = 0. (1) Dt Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Level-Set Method (2) Dϕ ∂ϕ dx = + · ϕ = 0. (2) Dt ∂t dt dx ϕ = V(x, t) = V(x, t) n , n=− , (3) dt || ϕ|| ∂ϕ = V(x, t) || ϕ|| . (4) ∂t Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Rate of Spread (1) The rate of spread is the ﬁre front velocity, ﬁrstly estabilished by Rothermel (1972) as an operative approximation of a theoretically based formula due to Frandsen (1971), V(x, t) = V0 (1 + fW + fS ) , (5) where V0 is the spread rate in the absence of wind, fW is the wind factor and fS is the slope factor. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Rate of Spread (2) IR ξ V0 = , (6) ρb ε Qign IR : reaction intensity ξ: propagation ﬂux ratio, the proportion of IR transferred to unburned fuels ρb : oven dry bulk density ε: effective heating number, the proportion of fuel that is heated before ignition occurs Qign : heat of pre-ignition. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Deterministic Front Let x(t, x0 ) be a deterministic trajectory with initial condition x0 , i.e., x(0, x0 ) = x0 , and driven solely by the rate of spread V(x, t). Moreover, let ϕ(x, t) be the function with values 1 and 0 such that ϕ(x, t) = 1 markes the burned area Ω(t), i.e., Ω(t) = {x, t : ϕ(x, t) = 1}, and ϕ(x, t) = 0 markes the unburned area, i.e., x ∈ Ω(t). Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Random Front (1) Let Xω (t, x0 ) = x(t, x0 ) + σ ω be the ω-realization of a random trajectory driven by the noise σ, with average value Xω (t, x0 ) = x(t, x0 ) and the same ﬁxed initial condition Xω (0, x0 ) = x0 in all realizations. Hence, the ω-realization of the ﬁre line contour follows to be ϕω (x, t) = ϕ(x0 , 0) δ(x − Xω (t, x0 )) dx0 . (7) Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Random Front (2) Since the trajectory x(t, x0 ) is time-reversible, the Jacobian of dx0 the transformation x(t, x0 ) = Ft (x0 ) is = 1, then formula dx (7) becomes ϕω (x, t) = ϕ(x, t) δ(x − Xω (t, x)) dx . (8) Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Randomized LS-Method Finally, after averaging, the effective ﬁre front contour emerges to be determined as ϕω (x, t) = ϕ(x, t) δ(x − Xω (t, x)) dx = ϕ(x, t) δ(x − Xω (t, x)) dx = ϕ(x, t) p(x; t|x) dx = p(x; t|x) dx = ϕeff (x, t) , (9) Ω(t) where p(x; t|x) = p(x − x; t) is the probability density function (PDF) of the turbulent dispersion of the hot ﬂow particles with average position x. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Turbulent Premixed Combustion (1) week endingPRL 107, 044503 (2011) PHYSICAL REVIEW LETTERS 22 JULY 2011 Lagrangian Formulation of Turbulent Premixed Combustion Gianni Pagnini and Ernesto Bonomi CRS4, Polaris Building 1, 09010 Pula, Italy (Received 4 November 2010; published 21 July 2011) The Lagrangian point of view is adopted to study turbulent premixed combustion. The evolution of the volume fraction of combustion products is established by the Reynolds transport theorem. It emerges that the burned-mass fraction is led by the turbulent particle motion, by the flame front velocity, and by the mean curvature of the flame front. A physical requirement connecting particle turbulent dispersion and flame front velocity is obtained from equating the expansion rates of the flame front progression and of the unburned particles spread. The resulting description compares favorably with experimental data. In the case of a zero- curvature flame, with a non-Markovian parabolic model for turbulent dispersion, the formulation yields the Zimont equation extended to all elapsed times and fully determined by turbulence characteristics. The exact solution of the extended Zimont equation is calculated and analyzed to bring out different regimes. DOI: 10.1103/PhysRevLett.107.044503 PACS numbers: 47.70.Pq, 05.20.Jj, 47.27.Ài Turbulent premixed combustion is a challenging scien- In this Letter, the fresh mixture is intended to be a popu-tific field involving nonequilibrium phenomena and play- lation of particles in turbulent motion that, in a statistical Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012ing the main role in important industrial issues such as sense, change from reactant to product when their average

Turbulent Premixed Combustion (2) In premixed combustion all reactants are intimately mixed at the molecular level before the combustion is started, while in non-premixed combustion the fuel and the oxidant must be mixed before than combustion can take place. Premixed Combustion process can be described as the following one-step irreversible chemical reaction Fresh Gas → Burned Gas + Heat . Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Turbulent Premixed Combustion (3) Premixed combustion is describe by a single non-dimensional variable named progress variable c(x, t), wich represents the burned mass fraction and it is deﬁned statistically deﬁned in the Lagrangian approach as c(x, t) = p(x; t|x) dx . (10) Ω(t) Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Fire Front Propagation (1) Reynolds transport theorem d ∂Ψ Ψ(x, t) dV = dV + ˆ Ψ u · nS dS , dt V (t) V (t) ∂t S Divergence theorem ˆ Ψ u · nS dS = · (u Ψ) dV . S V Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Fire Front Propagation (2) Fire front evolution equation (1) ∂ϕeff ∂p = dx + · (V p) dx . (11) ∂t Ω(t) ∂t Ω(t) Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Fire Front Propagation (3) Fire front evolution equation (2) Mean front curvature: κ(x, t) = · n/2, ∂ϕeff ∂p = dx + V· xp dx ∂t Ω(t) ∂t Ω(t) ∂V + p xκ ˆ · n + 2 V(κ, t)κ(x) dx . Ω(t) ∂κ The ﬁreline propagation is driven by: i) turbulent dispersion (i.e., p(x − x; t)), ii) rate of spread (i.e., V(x, t)), iii) mean front curvature (i.e., κ(x, t)). Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Non-turbulent Limit If turbulence is negligible, hot air particles deterministically move, i.e., p → δ(x − x), and the ﬁre front propagation equation becomes ∂ϕeff = V(x, t) || ϕeff || , (12) ∂t which is the Hamilton–Jacobi equation corresponding to the Level-Set Method. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Heating-before-burning Law (1) The model is completed by a law for the ignition due to the pre-heating by the hot air mass. Let T (x, t) be the temperature ﬁeld and Tign the ignition value. The most simple law for the temperature growing, when T (x, t) ≤ Tign , is ∂T (x, t) Tign − T (x, 0) = , (13) ∂t τ so that t T (x, t) = T (x, 0) + Tign − T (x, 0) . (14) τ Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Heating-before-burning Law (2) In the proposed model, the heating in an unburned point x i.e., 0 < ϕeff (x, t) ≤ 0.5, is due to the presence of hot air. Formula (13) changes according to ∂T (x, t) Tign − T (x, 0) = ϕeff (x, t) , T ≤ Tign . (15) ∂t τ Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

The Model: Heating-before-burning Law (3) For a given characteristic ignition delay τ , the time of heating-before-burning ∆t is such that it holds ∆t τ= ϕeff (x, ξ) dξ . (16) 0 Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Turbulent Dispersion Model The most simple model for turbulent dispersion of hot air mass around the average ﬁreline is ∂p 2 =D p, p(x − x; 0) = δ(x − x0 ) , (17) ∂t and then 1 (x − x)2 + (y − y)2 p(x − x; t) = exp − , (18) 2πσ 2 2 σ 2 (t) where σ 2 (t) is the particle displacement variance related to the turbulent diffusion coefﬁcient D, i.e., σ 2 (t) = (x − x)2 = (y − y )2 = 2 D t . (19) Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Analytic Results (1) Plane ﬁre front ˆ When the normal to the front n is constant, then the curvature κ is null. The ﬁreline propagation is driven by ∂ϕeff 2 =D ϕeff + V(t) || ϕeff || . (20) ∂t Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Analytic Results (2) The exact solution of (20) is 1 x − LR (t) x − LL (t) ϕeff (x, t) = Erfc √ − Erfc √ , (21) 2 2 Dt 2 Dt where Erfc is the complementary Error function, LR and LL are the right and left fronts, respectively, deﬁned as dLR dL = − L = V(t) , Ω(t) = [LL (t); LR (t)] . dt dt Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Comparison randomized level-ser level-set 1 0.8 0.6 0.4 0.2 0 -10 -5 0 5 10 Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Analytic Results (2) If V = constant then the Level-Set ﬁre front position is Lf = L0 + Vt and it holds 1 L0 + V t 1 ϕeff (LR , t) = 1 − Erfc √ < , 0 < t < ∞, (22) 2 Dt 2 hence the “cold” isoline ϕeff = 1/2 is slower than the Level-Set contour. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Analytic Results (3) The “hot” isoline ϕeff = 1/2 can be faster than the Level-Set contour. In fact, the elapsed time ∆t neccessery to meet condition (16) ∆t τ= ϕeff (x, t) dt , (23) 0 can be shorter than the elapsed time δt such that δt ∂ϕeff 1 ϕeff (x, δt) = ϕeff (x, 0) + dt = . (24) 0 ∂t 2 Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Analytic Results (4) In particular, with the initial condition   1 , x ∈ Ω(0) ϕeff (x, 0) = , (25) 0 , x ∈ Ω(0)  the “hot” isoline ϕeff = 1/2 is faster than the Level-Set contour when ∂ϕeff ϕ < eff , (26) ∂t 2τ so that ∆t ∆t ∂ϕeff ϕeff 1 dt < dt = . (27) 0 ∂t 0 2τ 2 Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Figures (1) a) b) Time [min] Time [min] 100 1e+03 100 800 800 600 600 400 80 400 80 200 200 60 60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 c) d) Time [min] Time [min] 100 600 100 500 400 400 200 300 80 80 200 100 60 60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Evolution in time of the ﬁre line contour, when τ = 10 [min], for the level-set method a) and for the randomized level-set method with increasing turbulence: b) D = 25 [ft]2 [min]−1 , c) D = 100 [ft]2 [min]−1 , d) D = 225 [ft]2 [min]−1 . Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Figures (2) a) b) Time [min] Time [min] 100 1e+03 100 1e+03 800 800 600 600 80 400 80 400 200 200 60 60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 c) d) Time [min] Time [min] 100 1e+03 100 1e+03 800 800 600 600 80 400 80 400 200 200 60 60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Evolution in time of the ﬁre line contour, when τ = 50 [min], for the level-set method a) and for the randomized level-set method with increasing turbulence: b) D = 25 [ft]2 [min]−1 , c) D = 100 [ft]2 [min]−1 , d) D = 225 [ft]2 [min]−1 . Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Figures (3) a) b) Time [min] Time [min] 100 1e+03 100 1e+03 800 800 600 600 80 400 80 400 200 200 60 60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 c) d) Time [min] Time [min] 100 1e+03 100 1e+03 800 800 600 600 80 400 80 400 200 200 60 60 40 40 20 20 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Evolution in time of the ﬁre line contour in the presence of a ﬁrebreak, when τ = 100 [min], for the level-set method a) and for the randomized level-set method with increasing turbulence: b) D = 25 [ft]2 [min]−1 , c) D = 100 [ft]2 [min]−1 , d) D = 225 [ft]2 [min]−1 . Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Conclusions (1) A new formulation for modelling the wildland fire front propagation is proposed. It includes smallscale processes driven by the turbulence generated by the Atmospheric Boundary Layer dynamics and by the fire-induced flow. It is based on the randomization of the level-set method for tracking fire line contour by considering a distribution of the contour according to the PDF of the turbulent displacement of hot air particles. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Conclusions (2) This formulation is emerged to be suitable, more than the ordinary level-set approach, to model the following two dangerous situations: i) the increasing of the rate of spread of the fire line as a consequence of the pre-heating of zones ahead the fire front by the hot air mass ii) the overcoming of a breakfire by the fire because of the diffusion of the hot air behind it, which is for the level-set method a failed task because in the firebreak zone the rate of spread is null since the absence of fuel. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

References W.H. Frandsen, Fire spread through porous fuels from the conservation of energy. Combust. Flame 16, 9–16 (1971). R.C. Rothermel, A mathematical model for predicting fire spread in wildland fires. USDA Forest Service, Research Paper INT–115, (1972). J.A. Sethian & P. Smereka, Level Set Methods for fluid interfaces. Ann. Rev. Fluid Mech. 35, 341–372 (2003). G. Pagnini & E. Bonomi, Lagrangian formulation of turbulent premixed combustion. Phys. Rev. Lett. 107, 044503 (2011). Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Acknowledgements Regione Autonoma della Sardegna PO Sardegna FSE 2007-2013 sulla L.R. 7/2007 “Promozione della ricerca scientiﬁca e dell’innovazione tecnologica in Sardegna”. Collana di Seminari per la Valorizzazione dei Risultati della Ricerca al CRS4, 22 marzo 2012

Sviluppi modellistici sulla propagazione degli incendi boschivi

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Sviluppi modellistici sulla propagazione degli incendi boschivi Presentation Transcript