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


Turbulence Sources in Wildland Fire Front
Propagation (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.


Propagation (2)

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


Propagation (2)
... 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 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.


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


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


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.


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.


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).


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)


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)


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.

Turbulent Premixed Combustion (1)

week ending
PRL 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
ing the main role in important industrial issues such as sense, change from reactant to product when their average


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 .



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)


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



Fire front evolution equation (1)

∂ϕeff ∂p
= dx + · (V p) dx . (11)
∂t Ω(t) ∂t Ω(t)



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)).

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.


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



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



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


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)


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



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


Comparison

randomized level-ser
level-set
1

0.8

0.6

0.4

0.2

0
-10 -5 0 5 10



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.



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



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


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)
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 .

Figures (2)
a) b)
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)
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 .

Figures (3)
a) b)
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)
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 .

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.


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.


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).


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”.


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