Sviluppi modellistici sulla propagazione degli incendi boschivi

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

Sviluppi modellisticisulla propagazione degli incendi boschivi

Gianni PAGNINIBorsista RAS

PO Sardegna FSE 2007-2013 sulla L.R. 7/2007“Promozione della ricerca scientifica e dell’innovazione tecnologica in Sardegna”


Introduction


Turbulence Sources in Wildland Fire FrontPropagation (1)

Wildland fire propagates at the ground level and then it isdependent on the dynamics of the Atmospheric BoundaryLayer, whose flow is turbulent in nature.



Moreover, in this atmospheric layer the turbulence is amplifiedby the forcing due to the fire-atmosphere coupling



... and by the appearing of the fire-induced flow.


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 thefire motion to take into account turbulence.

Accounting for the effect of turbulence on the fire propagationimproves the usefulness of the operational models and therebyincreases the firefighting safety and in general theefficiency of the fire suppression and control efforts.


The Level-Set Method (1)

Let Γ(t) = (x(s, t), y(s, t)) be a parameterized evolvinginterface.

Let ϕ(x, t) be a function such that the level-set ϕ = constantcorresponds to the evolving front Γ(t). Then the equation forthe evolution of ϕ corresponding to the motion of the interfaceΓ(t) is given by

Dϕ

Dt= 0 . (1)


The Level-Set Method (2)

Dϕ

Dt=

∂ϕ

∂t+

dxdt

· ∇ϕ = 0 . (2)

dxdt

= V(x, t) = V(x, t) n , n = − ∇ϕ

||∇ϕ||, (3)

∂ϕ

∂t= V(x, t) ||∇ϕ|| . (4)


The Rate of Spread (1)

The rate of spread is the fire front velocity, firstly estabilished byRothermel (1972) as an operative approximation of atheoretically 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.


The Rate of Spread (2)

V0 =IR ξ

ρb ε Qign, (6)

IR: reaction intensityξ: propagation flux ratio, the proportion of IR transferred tounburned fuelsρb: oven dry bulk densityε: effective heating number, the proportion of fuel that is heatedbefore ignition occursQign: heat of pre-ignition.


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 spreadV(x, t).

Moreover, let ϕ(x, t) be the function with values 1 and 0 suchthat 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 unburnedarea, i.e., x 6∈ Ω(t).


The Model: Random Front (1)

Let Xω(t , x0) = x(t , x0) + σω be the ω-realization of a randomtrajectory driven by the noise σ, with average value〈Xω(t , x0)〉 = x(t , x0) and the same fixed initial conditionXω(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)


The Model: Random Front (2)

Since the trajectory x(t , x0) is time-reversible, the Jacobian of

the transformation x(t , x0) = Ft(x0) isdx0

dx= 1, then formula

(7) becomes

ϕω(x, t) =

∫ϕ(x, t) δ(x− Xω(t , x)) dx . (8)


The Model: Randomized LS-Method

Finally, after averaging, the effective fire front contour emergesto 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

=

∫Ω(t)

p(x; t |x) dx = ϕeff (x, t) , (9)

where p(x; t |x) = p(x− x; t) is the probability density function(PDF) of the turbulent dispersion of the hot flow particles withaverage position x.


Turbulent Premixed Combustion (1)

Lagrangian Formulation of Turbulent Premixed Combustion

Gianni Pagnini and Ernesto BonomiCRS4, 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.!i

Turbulent premixed combustion is a challenging scien-tific field involving nonequilibrium phenomena and play-ing the main role in important industrial issues such asenergy production and engine design.

A Lagrangian point of view is here adopted, leading to adescription of turbulent premixed combustion which takesinto account, for all elapsed times, the turbulent dispersion,the volume consumption rate of reactants, and the flamemean curvature. The proposed approach generalizes andunifies classical literature approaches that are based on theso-called level-set method [1] or are based on the Zimontbalance equation, originally hinted at by Prudnikov [3],also known as Turbulent Flame Closure model [2].Moreover, the proposed formulation has the striking prop-erty to be compatible with every type of geometry and flowin an easier and more versatile way than previous ap-proaches, and it emerges to be easily modifiable to includemore detailed and correct physics. It is worth recalling thatthe Zimont equation was introduced on the basis of experi-mental observations and that a great effort has been under-taken to give a deeper theoretical foundation to it [2,4–6].The present Lagrangian formulation constitutes a reliabletheoretical support for the Zimont combustion model.

The process of turbulent premixed combustion is mainlycharacterized by flame propagation towards the unburnedregion and turbulent dispersion of the resultant productparticles. The combustion process is described by a singledimensionless scalar observable, denoted as averageprogress variable, 0 " c#x; t$ " 1, and representing theburned-mass fraction, i.e., the fraction of burned particleswhich are located in x at time t. The value c#x; t$ % 1describes the presence of only products and the valuec#x; t$ % 0 describes the presence of only reactants. Toavoid unnecessary mathematical difficulties, we considera constant-density mixture and a zero-mean turbulent ve-locity field. Molecular diffusion is also neglected.

In this Letter, the fresh mixture is intended to be a popu-lation of particles in turbulent motion that, in a statisticalsense, change from reactant to product when their averagepositions are hit by the flame. Let!#t$ be the portion of spacesurrounded by the flame surface; then those particles withaverage position hxi 2 !#t$ are marked as burned particles.The occurrence in x at time t of a particle transit is describedby a probability density function (PDF). Let p#x; tjx0$ be thePDF associated with a particle displacement where x0 is theinitial condition of a Lagrangian trajectory and, without lossof generality, let t % 0 be the ignition instant. With theassumption that particle trajectories are not affected by thechemical transformation, the average progress variablec#x; t$ turns out to be defined as the superposition of PDFsof burned particles, i.e., those p#x; tjx0$ with hxi 2 !#t$.For a zero-average velocity field, the particle average positionis hxi % x0 and then

c#x; t$ %Z!#t$

p#x; tjx0$dx0: (1)

The evolution law for the progress variable c#x; t$ is obtainedapplying Reynolds transport theorem to (1) which gives

@c

@t%

Z!#t$

@p

@tdx0 &

Z!#t$

rx0' (u#x0; t$p#x; tjx0$)dx0;

(2)

whererx0is the gradient with respect to x0 and u#x; t$ is the

expansion velocity field of !#t$.Let the turbulent dispersion be represented by the gen-

eral evolution equation

@p

@t% Ex(p); p#x; 0jx0$ % !#x! x0$; (3)

where the spatial operator Ex(') includes the particledisplacement statistics such as the variance "2#t$ % hkx!x0k2i=3.

PRL 107, 044503 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending22 JULY 2011

0031-9007=11=107(4)=044503(4) 044503-1 ! 2011 American Physical Society



In premixed combustion all reactants are intimately mixed at themolecular level before the combustion is started, while innon-premixed combustion the fuel and the oxidant must bemixed before than combustion can take place.

Premixed Combustion process can be described as thefollowing one-step irreversible chemical reaction

Fresh Gas → Burned Gas + Heat .



Premixed combustion is describe by a single non-dimensionalvariable named progress variable c(x, t), wich represents theburned mass fraction and it is defined statistically defined in theLagrangian approach as

c(x, t) =

∫Ω(t)

p(x; t |x) dx . (10)


Fire Front Propagation (1)

Reynolds transport theorem

ddt

∫V (t)

Ψ(x, t) dV =

∫V (t)

∂Ψ

∂tdV +

∫S

Ψ u · nS dS ,

Divergence theorem

∫S

Ψ u · nS dS =

∫V∇ · (u Ψ) dV .



Fire front evolution equation (1)

∂ϕeff

∂t=

∫Ω(t)

∂p∂t

dx +

∫Ω(t)

∇ · (V p) dx . (11)



Fire front evolution equation (2)

Mean front curvature: κ(x, t) = ∇ · n/2,

∂ϕeff

∂t=

∫Ω(t)

∂p∂t

dx +

∫Ω(t)

V · ∇xp dx

+

∫Ω(t)

p

∂V∂κ

∇xκ · n + 2V(κ, t)κ(x)

dx .

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)).


Non-turbulent Limit

If turbulence is negligible, hot air particles deterministicallymove, i.e., p → δ(x− x),and the fire front propagation equation becomes

∂ϕeff

∂t= V(x, t) ||∇ϕeff || , (12)

which is the Hamilton–Jacobi equation corresponding to theLevel-Set Method.


The Model: Heating-before-burning Law (1)

The model is completed by a law for the ignition due to thepre-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, whenT (x, t) ≤ Tign, is

∂T (x, t)∂t

=Tign − T (x, 0)

τ, (13)

so that

T (x, t) = T (x, 0) +[Tign − T (x, 0)

] tτ

. (14)



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)∂t

= ϕeff (x, t)Tign − T (x, 0)

τ, T ≤ Tign . (15)



For a given characteristic ignition delay τ , the time ofheating-before-burning ∆t is such that it holds

τ =

∫ ∆t

0ϕeff (x, ξ) dξ . (16)


Turbulent Dispersion ModelThe most simple model for turbulent dispersion of hot air massaround the average fireline is

∂p∂t

= D∇2p , p(x− x; 0) = δ(x− x0) , (17)

and then

p(x− x; t) =1

2πσ2 exp−(x − x)2 + (y − y)2

2 σ2(t)

, (18)

where σ2(t) is the particle displacement variance related to theturbulent diffusion coefficient D, i.e.,

σ2(t) = 〈(x − x)2〉 = 〈(y − y)2〉 = 2D t . (19)


Analytic Results (1)

Plane fire front

When the normal to the front n is constant, then the curvature κis null. The fireline propagation is driven by

∂ϕeff

∂t= D∇2ϕeff + V(t) ||∇ϕeff || . (20)



The exact solution of (20) is

ϕeff (x , t) =12

Erfc

[x − LR(t)

2√D t

]− Erfc

[x − LL(t)

2√D t

], (21)

where Erfc is the complementary Error function,LR and LL are the right and left fronts, respectively, defined as

dLR

dt= −dLL

dt= V(t) , Ω(t) = [LL(t);LR(t)] .


Comparison

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

randomized level-serlevel-set



If V = constant then

the Level-Set fire front position is Lf = L0 + Vt and it holds

ϕeff (LR, t) =12

1− Erfc

[L0 + V t√

D t

]<

12

, 0 < t < ∞ , (22)

hence the “cold” isoline ϕeff = 1/2 is slower than the Level-Setcontour.



The “hot” isoline ϕeff = 1/2 can be faster than the Level-Setcontour. In fact, the elapsed time ∆t neccessery to meetcondition (16)

τ =

∫ ∆t

0ϕeff (x, t) dt , (23)

can be shorter than the elapsed time δt such that

ϕeff (x, δt) = ϕeff (x, 0) +

∫ δt

0

∂ϕeff

∂tdt =

12

. (24)



In particular, with the initial condition

ϕeff (x, 0) =

1 , x ∈ Ω(0)

0 , x 6∈ Ω(0), (25)

the “hot” isoline ϕeff = 1/2 is faster than the Level-Set contourwhen

∂ϕeff

∂t<

ϕeff

2 τ, (26)

so that ∫ ∆t

0

∂ϕeff

∂tdt <

∫ ∆t

0

ϕeff

2 τdt =

12

. (27)


Figures (1)a) Time [min]

1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100b) Time [min]

800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100

c) Time [min] 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100d) Time [min]

500 400 300 200 100

0 20 40 60 80 100 0

20

40

60

80

100

Evolution in time of the fire line contour, when τ = 10 [min], forthe level-set method a) and for the randomized level-setmethod with increasing turbulence: b) D = 25 [ft]2[min]−1,c) D = 100 [ft]2[min]−1, d) D = 225 [ft]2[min]−1.



1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100b) Time [min]

1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100

c) Time [min] 1e+03

800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100d) Time [min]

1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100

Evolution in time of the fire line contour, when τ = 50 [min], forthe level-set method a) and for the randomized level-setmethod with increasing turbulence: b) D = 25 [ft]2[min]−1,c) D = 100 [ft]2[min]−1, d) D = 225 [ft]2[min]−1.



1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100b) Time [min]

1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100

c) Time [min] 1e+03

800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100d) Time [min]

1e+03 800 600 400 200

0 20 40 60 80 100 0

20

40

60

80

100

Evolution in time of the fire line contour in the presence of afirebreak, when τ = 100 [min], for the level-set method a) andfor 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.


Conclusions (1)

A new formulation for modelling the wildland fire frontpropagation is proposed.

It includes smallscale processes driven by the turbulencegenerated by the Atmospheric Boundary Layer dynamics andby the fire-induced flow.

It is based on the randomization of the level-set method fortracking fire line contour by considering a distribution of thecontour according to the PDF of the turbulent displacement ofhot air particles.


Conclusions (2)

This formulation is emerged to be suitable, more than theordinary level-set approach, to model the following twodangerous situations:

i) the increasing of the rate of spread of the fire line as aconsequence of the pre-heating of zones ahead the fire front bythe hot air mass

ii) the overcoming of a breakfire by the fire because of thediffusion of the hot air behind it, which is for the level-setmethod a failed task because in the firebreak zone the rate ofspread is null since the absence of fuel.


References

W.H. Frandsen, Fire spread through porous fuels from theconservation of energy. Combust. Flame 16, 9–16 (1971).

R.C. Rothermel, A mathematical model for predicting firespread in wildland fires. USDA Forest Service, Research PaperINT–115, (1972).

J.A. Sethian & P. Smereka, Level Set Methods for fluidinterfaces. Ann. Rev. Fluid Mech. 35, 341–372 (2003).

G. Pagnini & E. Bonomi, Lagrangian formulation of turbulentpremixed combustion. Phys. Rev. Lett. 107, 044503 (2011).


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”.

Sviluppi modellistici sulla propagazione degli incendi boschivi

Technology

Transcript of Sviluppi modellistici sulla propagazione degli incendi boschivi