A LAGRANGIAN APPROACH FORREACTING FRONT PROPAGATION
IN TURBULENT FLOWS
Gianni PAGNINI
Awarded IKERBASQUE Research Fellow
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.1
Eulerian approachEulerian approach:the flux is studied in a fixed point, e.g.,anemometer. Experimentally “easy” and,generally, used in engineering applications.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.2
Lagrangian approachLagrangian approach:the flux is studied along fluid element trajectories.Experimentally very difficult and used forexample for pollutant dispersion analysis.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.3
Single step chemical reaction
reactants → products + heat
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.4
Assumptions
− homogeneous, isotropic and stationaryturbulence,
− zero mean velocity field,
− the motion of reactants is identical to that ofproducts,
− any molecular effect is neglected.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.5
The idea
The mixture is intended to be an ensemble ofparticles.
The reactant particles along their turbulent trajec-
tory may collide with the reacting interface located
in Lf(t) and they turn into product.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.6
The idea
Let a reactant particle, with initial position locatedahead the reacting front in x0, turn to product attime t+ when it is in x(t+) = Lf(t
+).
This means that the chemical transformation of
such particle in x at instant t+ is dependent on
the particle displacement x − x0.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.7
The ideastatistically:,the chemical transformation is dependent on the variance
of particle displacement, i.e., σ2(t) =1
3〈(x − x0)
2〉,
a reactant particle is marked as product when its averageposition collides with the reacting front. Since azero-average velocity field is considered it holds 〈x〉 = x0,
the product particle makes to increase the average fraction
of product mass according to its probability density function.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.8
Definition (1)
Let the average fraction of the mass of productsc(x, t) (0 ≤ c ≤ 1), named also average progressvariable, be defined as
c(x, t) =
∫
Ω(t)
p(x; t|x0) dx0 ,(1)
where p(x; t|x0) is the probability density function
of particle displacement and Ω(t) is the volume
bounded by the moving reacting front.Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.9
Reynolds transport theorem
d
dt
∫
V
Ψ(x, t) dV =
∫
V
∂Ψ
∂tdV +
∫
S
Ψu · nS dS ,
divergence theorem∫
S
Ψu · nS dS =
∫
V
∇ · (uΨ) dV ,
d
dt
∫
V
Ψ(x, t) dV =
∫
V
∂Ψ
∂tdV +
∫
V
∇ · (uΨ) dV .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.10
Reacting front velocityThe evolution in time of the domain Ω(t) is drivenby the the volume consumption rate of reactantsu(x, t)
u(x, t) = U(κ, t) n , U(κ, 0) = 0 ,(2)
where n(x, t) = −∇c/||∇c|| is the outward normalto Ω(t) and κ(x, t) = ∇ · n/2 denotes the localmean curvature of Ω(t) and
Lf(t) = L0 +
∫ t
0
U dτ .(3)
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.11
Transport equation (1)
∂c
∂t=
∫
Ω(t)
∂p
∂tdx0+
∫
Ω(t)
∇x0·[u(x0, t)p(x; t|x0)] dx0 ,
∂p
∂t= Ex[p] ,
∂c
∂t= Ex[c] +
∫
Ω(t)
∇x0· [u(x0, t)p(x; t|x0)] dx0 .(4)
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.12
Definition (2)
c(x, t) =
∫
Rd
p(x; t|x0)φ(x0, t) dx0 , d = 1, 2, 3 ,(5)
∂φ
∂t= U(x, t)||∇φ|| , φ(x, t) =
1 , if x ∈ Ω(t) ,
0 , otherwise ,
u(x, t) = U(x, t) n , n = − ∇c
||∇c|| = − ∇φ
||∇φ|| .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.13
Transport equation (2)
∂c
∂t=
∫
Rd
∂p
∂tφ(x0, t) dx0 +
∫
Rd
p(x; t|x0)∂φ
∂tdx0
=
∫
Rd
Ex[p]φ(x0, t) dx0
+
∫
Rd
p(x; t|x0)U(x0, t)||∇x0φ|| dx0
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.14
Transport equation (2)∂c
∂t= Ex
[∫
Rd
p(x; t|x0)φ(x0, t)
]dx0
−∫
Rd
p(x; t|x0)u(x0, t) · ∇x0φ dx0
by using integration by partZ
Rd
p(x; t|x0)u(x0, t) · ∇x0φ dx0 = −
Z
Rd
∇x0· [u(x0, t)p(x; t|x0)]φ(x0, t) dx0 ,
∂c
∂t= Ex[c]+
∫
Ω(t)
∇x0·[u(x0, t)p(x; t|x0)] dx0 .(6)
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.15
Non-diffusive caseWhen turbulent diffusion is neglected, particlesare frozen and their motion do not depends ontime p → δ(x − x0), equation (6) reduces to
∂c
∂t= U(κ, t) ||∇c|| ,(7)
which is the same Hamilton–Jacobi equationadopted in the level-set method for trackinginterfaces.
Sethian & Smereka, Ann. Rev. Fluid Mech. 35, 341 (2003).
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.16
Reacting rate and turbulence
In the considered model the front progressionand the turbulent diffusion are mutually relatedsince, when molecular processes are neglected,the reacting front is solely fueled and carried bythe turbulent dynamics of the reactingenvironment.Since the motion of particles is both forward andbackward and that of the reacting front is solelyforward, a double time is necessery to σ(t) tohave the same expansion rate of the reacting front
Lf(t).Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.17
Lf Expansion rate
δLf(t) = Lf(t) − L0 ,
δLf(t + τ) = r(τ) δLf (t) , r(0) = 1 ,
δLf(t + τ) − δLf(t) = δLf(t)[r(τ) − 1] ,
1
δLflimτ→0
Lf(t + τ) − Lf(t)
τ= lim
τ→0
r(τ) − r(0)
τ,
1
Lf − L0
dLf
dt=
dr
dt
∣∣∣∣0
.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.18
σ Expansion rate
σ(t + 2τ) = r(τ)σ(t) , r(0) = 1 ,
σ(t + 2τ) − σ(t) = σ(t)[r(τ) − 1] ,
1
σlimτ→0
σ(t + 2τ) − σ(t)
2τ=
1
2limτ→0
r(τ) − r(0)
τ,
1
σ
dσ
dt=
1
2
dr
dt
∣∣∣∣0
.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.19
Lf-σ Relationship
In this mechanism, the expansion rate of thefront and that of the root mean square of theparticle displacement are related by
1
σ
dσ
dt=
1
2 (Lf − L0)
dLf
dt.(8)
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.20
Turbulent dispersion model
Non-Markovian model
∂p
∂t= D(t)∇2p , p(x; 0|x0) = δ(x − x0) ,(9)
where
D(t) =1
2
dσ2
dt, σ2(t) =
1
3〈(x − x0)
2〉 ,
and
D(t) =⇒ Deq = 〈u′2〉TL , t ≫ TL .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.21
c-Transport equationChanging Ex → D(t)∇2, formula (6) becomes
∂c
∂t= D(t)∇2c +
∫
Ω(t)
u · ∇x0p dx0(10)
+
∫
Ω(t)
p
∂U∂κ
∇x0κ · n + 2U(κ, t)κ(x0)
dx0 .
Then the evolution of c(x, t) is determined by:the turbulent diffusion,the chemical progression (expressed by turbulent diffusion (8)),
the front mean curvature.Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.22
U-D Relationship
In terms of U(t) =dLf
dtand D(t) =
1
2
dσ2
dt,
identities (8) yields∫ t
0 U(τ) dτ
U(t)
D(t)∫ t
0 D(τ) dτ= 1 ,
and it follows that∫ t
0
[D(t)U(τ) − U(t)D(τ)] dτ = 0 ,
for any t ≥ 0.Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.23
U-D Relationship
As a consequence of the increasing monotonicityof both U and D, since they are non-negativefunctions and U(0) = D(0) = 0, the followingequality holds
D(τ)
U(τ)=
D(t)
U(t), 0 ≤ τ ≤ t ,
so that, when 0 ≤ τ ≤ t, the ratio D(τ)/U(τ) must
be constant.Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.24
U-D Relationship
For t → ∞ both D(t) and U(t) are bounded,D(∞) = Deq and U(∞) = Ueq, therefore such aconstant is equal to
D(t)
U(t)=
Deq
Ueq= λ , t ≥ 0 .(11)
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.25
Lf equation of motion
Combining definition of Lf(t) (3) and (11) gives
Lf(t) = L0 +
∫ t
0
U(τ) dτ = L0 +1
λ
∫ t
0
D(τ) dτ ,
finally
Lf(t) = L0 +σ2(t)
2λ.(12)
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.26
Application in premixed combustion
Gianni PAGNINI & Ernesto BONOMI,Lagrangian Formulation of Turbulent PremixedCombustion.
Phys. Rev. Lett. 107, 044503 (2011).
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.27
Zimont-type equationWhen the normal n to the front is assumedconstant, then the mean curvature κ is zero and(10) becomes
∂c
∂t= D(t)∇2c + U(t) ||∇c|| ,(13)
that in the asymptotic regime t ≫ TL is theso-called Zimont equation, a paradigmatic modelfor turbulent premixed combustion (ANSYS/FLUENT).
Zimont, Exp. Thermal Fluid Sci. 21, 179 (2000).
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.28
Combustion processes
Non-premixed combustion:Fuel and oxidizer enter the reaction zone indistinct streams.
Premixed combustion:
Fuel and oxidizer are mixed at the molecular level
prior to ignition. Combustion occurs as a flame
front propagating into the unburnt reactants.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.29
Why to study premixed combustion?
The engineering challenge:Turbulent premixed combustion plays the mainrole in important industrial issues as energyproduction and engine design.
The environmental challenge:
Turbulent premixed combustion is characterized
by an ultra-low NOx production and very low par-
ticulate, so to meet future emission standards.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.30
Theoretical validation of (11) and (12)
Formula (11) states that Ueq =Deq
λ
and (12) that Lf(t) = L0 +σ2(t)
2λ,
Borghi, Prog. Energy Combust. Sci. 14, 245 (1988): Uplanar =Deq
λ,
Biagioli, Combust. Theory Model. 10, 389 (2006): x0 = x1 + 2σ2
F
RF,
where x is the axial coordinate, σF and RF are the flame thickness and the radius of
curvature, respectively.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.31
Experimental validation of (12)
Figure 1: Experimental validation of the propagation law (12). Lines are the
plots of the analytic σ =
s
2Deqt
»
1 +TL
t
`
e−t/TL − 1´
–
and the predicted flame front
positions Lf . The values λ = 5.1mm and λ = 8.2mm correspond to two differentequivalence ratios F = 0.68 and F = 0.56, respectively.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.32
Exact solution of Zimont modelWhen κ = 0, Zimont equation reduces to aone-dimensional problem along the normaldirection to the flame front and the solution is
c(x, t) =1
2
Erfc
[x − LR(t)√
2σ(t)
]− Erfc
[x + LL(t)√
2 σ(t)
],
where Erfc is the complementary error function,while LR and LL are the two coordinates of thetwo moving flame front sides:
dLR
dt=
dLL
dt= U(t) , Ω(t) = [−LL(t); +LR(t)] .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.33
Quenching
The combustion and turbulent diffusion budget is
U(t)L0
D(t)=
L0
λ,
if L0 = L0c = λ the budget is equal to 1 so that
when L0 ≪ L0c = λ the flame quenches, other-
wise when L0 ≫ L0c = λ the flame not quenches.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.34
Plots set up
σ2 = 2Deq t
[1 +
TL
t
(e−t/TL − 1
)],
〈u′2〉 = 1 , TL = 1 and λ = 0.1 ,
L0 = 0 , L0 = 0.05 , L0 = 0.1 ,
L0 = 0.5 , L0 = 1 L0 = 5 .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.35
c(x, t)-Profile
0
0.2
0.4
0.6
0.8
1
-300 -200 -100 0 100 200 300
aver
age
pro
gre
ss v
aria
ble
x/λ
Figure 2: Progress variable profile at times t = 0.2TL, 0.4TL, 0.6TL, 0.8TL, 1TL,2TL and 3TL with L0/λ = 0.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.36
c(x, t)-Profile
0
0.2
0.4
0.6
0.8
1
-30 -20 -10 0 10 20 30
aver
age
pro
gre
ss v
aria
ble
x/λ
Figure 3: Progress variable profile at times t = 0.01TL, 0.1TL, 0.3TL, 0.5TL and0.7TL with L0/λ = 0.5.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.37
c(x, t)-Profile
0
0.2
0.4
0.6
0.8
1
-30 -20 -10 0 10 20 30
aver
age
pro
gre
ss v
aria
ble
x/λ
Figure 4: Progress variable profile at times t = 0.01TL, 0.1TL, 0.3TL, 0.5TL and0.7TL with L0/λ = 1.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.38
c(x, t)-Profile
0
0.2
0.4
0.6
0.8
1
-30 -20 -10 0 10 20 30
aver
age
pro
gre
ss v
aria
ble
x/λ
Figure 5: Progress variable profile at times t = 0.01TL, 0.1TL, 0.3TL, 0.5TL and0.7TL with L0/λ = 5.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.39
c(x, t)-Profile
0
0.2
0.4
0.6
0.8
1
-30 -20 -10 0 10 20 30
aver
age
pro
gre
ss v
aria
ble
x/λ
Figure 6: Progress variable profile at times t = 0.01TL, 0.1TL, 0.3TL, 0.5TL and0.7TL with L0/λ = 10.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.40
c(x, t)-Profile
0
0.2
0.4
0.6
0.8
1
-100 -50 0 50 100
aver
age
pro
gre
ss v
aria
ble
x/λ
Figure 7: Progress variable profile at times t = 0.01TL, 0.1TL, 0.3TL, 0.5TL and0.7TL with L0/λ = 50.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.41
c(x, t)-Profile
Figure 8: Comprehensive plot of progress variable profile.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.42
Formulation for the non-planar case
The modulus of the expansion velocity fieldbecomes
U(x, t) = Uplanar(x, t)(1 −M(x, t)) ,
where M(x, t) is the Markstein curvature factor
M(x, t) = λ∇ · n = 2λ κ(x, t) ,
Vervisch et al., Physics Fluids 7, 2496 (1995).
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.43
Formulation for the non-planar case
c(x, t) =
∫
Rd
p(x; t|x0)φ(x0, t) dx0 ,(14)
∂p
∂t= Ex[p] ,
∂φ
∂t= Uplanar(x, t)[1−M(x, t)]||∇φ|| ,
φ(x, t) =
1 , if x ∈ Ω(t) ,
0 , otherwise ,
u(x, t) = Uplanar[1−M] n , n = − ∇c
||∇c|| = − ∇φ
||∇φ|| .Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.44
Formulation for the non-planar case
c(x, t) =
∫
Rd
p(x; t|x0)φ(x0, t) dx0 ,(15)
∂p
∂t= D(t)∇2p ,
∂φ
∂t=
D(t)
λ[1 − 2λ κ(x, t)]||∇φ|| .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.45
Classical reaction-diffusion equations
Comparison with classical reaction-diffusionequations
∂c
∂t= D∇2c + f(c) ,
f(c) = c(1 − c) , Fisher − KPP ,
f(c) = cα(1 − c) , Zeldovich ,
f(c) = c(1 − c)(c − µ) , 0 ≤ µ ≤ 1 , FitzHugh − Nagumo.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.46
Variable density formulation
ρ(x, t)c(x, t) =
∫
Ω(t)
p(x; t|x0)ρ(x0) dx0 ,(16)
τ∂2p
∂t2+
∂p
∂t= Ex[p] ,
Acoustic noise effects (Lighthill equation)
∂2ρ
∂t2− c2
0∇2ρ =∂2T ij
∂xi∂xj.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.47
Application in wildland fire
Gianni PAGNINI & Luca MASSIDDA,The randomized level-set method to modelturbulence effects in wildland fire propagation,
in D. Spano, V. Bacciu, M. Salis, C. Sirca (Eds.):
Modelling Fire Behaviour and Risk, Proceedings
of the International Conference on Fire Behaviour
and Risk, Alghero, Italy, 4–6 October (2011), pp.
126–131.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.48
Application in wildland fire
Turbulence is of paramount importance inwildland fire propagation because randomlytransports the hot air mass that can pre-heat andthen ignite the area ahead the fire, therefore thefire front position gets a random character.
This approach allows the simulation of the fireovercoming of a firebreak zone, a situation thatmodels based on the ordinary level-set methodcan not resolve.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.49
Rate of spread
The fire front velocity, firstly estabilished byRothermel (1972), is parameterized as
U(x, t) = U0 (1 + fW + fS) ,(17)
where U0 is the spread rate in the absence of
wind, fW is the wind factor and fS is the slope
factor.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.50
Application in wildland fire
U0 =IR ξ
ρb ε Qign,
IR: reaction intensityξ: propagation flux ratio, the proportion of IR
transferred to unburned fuelsρb: oven dry bulk densityε: effective heating number, the proportion of fuelthat is heated before ignition occursQign: heat of pre-ignition.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.51
Turbulent dispersion model
∂p
∂t= Deq ∇2p , p(x − x; 0) = δ(x − x0) ,
and then
p(x − x; t) =1
2πσ2exp
−(x − x)2 + (y − y)2
2σ2(t)
,
where σ2(t) is the particle displacement variancerelated to the turbulent diffusion coefficient Deq,
σ2(t) = 〈(x − x)2〉 = 〈(y − y)2〉 = 2Deq t .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.52
Heating-before-burning Law
Points with c(x, t) > 0.5 are marked as burned.The model is completed by a law for the ignitiondue to the pre-heating by the hot air mass.
Let T (x, t) be the temperature field and Tign theignition value, temperature is assumed to growaccording to
∂T (x, t)
∂t= c(x, t)
Tign − T (x, 0)
τ, T ≤ Tign .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.53
Heating-before-burning Law
For a given characteristic ignition delay τ , thetime of heating-before-burning ∆t is such that itholds
τ =
∫ ∆t
0
c(x, ξ) dξ .
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.54
Application in wildland firea)
Time [min] 1e+03
800 600 400 200
0 20 40 60 80 100 0
20
40
60
80
100
b)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
100
d)Time [min]
500 400 300 200 100
0 20 40 60 80 100 0
20
40
60
80
100
Figure 9: Evolution in time of the fire line contour, when τ = 10 [min], for thelevel-set method a) and for the randomized level-set method with increasing turbulence:b) Deq = 25 [ft]2[min]−1, c) Deq = 100 [ft]2[min]−1, d) Deq = 225 [ft]2[min]−1.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.55
Application in wildland firea)
Time [min] 1e+03
800 600 400 200
0 20 40 60 80 100 0
20
40
60
80
100
b)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
100
d)Time [min]
1e+03 800 600 400 200
0 20 40 60 80 100 0
20
40
60
80
100
Figure 10: Evolution in time of the fire line contour, when τ = 50 [min], for thelevel-set method a) and for the randomized level-set method with increasing turbulence:b) Deq = 25 [ft]2[min]−1, c) Deq = 100 [ft]2[min]−1, d) Deq = 225 [ft]2[min]−1.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.56
Application in wildland firea)
Time [min] 1e+03
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b)Time [min]
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c)Time [min]
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d)Time [min]
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Figure 11: Evolution in time of the fire line contour in the presence of a fire-break, when τ = 100 [min], for the level-set method a) and for the randomized level-setmethod with increasing turbulence: b) Deq = 25 [ft]2[min]−1, c) Deq = 100 [ft]2[min]−1,d) Deq = 225 [ft]2[min]−1.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.57
Conclusion
The proposed Lagrangian approach generatesan evolution equation that
− when applied to turbulent premixedcombustion generalizes both the literatureapproaches based on the level-set equationand on the Zimont equation,
− when applied to the wildland fire propagationpermits to consider the motion of the hot airand to model fire overcaming a firebreak.
Gianni PAGNINI A Lagrangian approach for reacting front propagation in turbulent flows. BCAM, 19th of July, 2012 – p.58