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Sviluppi modellistici sulla propagazione degli incendi boschivi
 

Sviluppi modellistici sulla propagazione degli incendi boschivi

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By Gianni Pagnini

By Gianni Pagnini
Percorso A Collana Seminari CRS4 2012

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    Sviluppi modellistici sulla propagazione degli incendi boschivi Sviluppi modellistici sulla propagazione degli incendi boschivi Presentation Transcript

    • 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