A LAGRANGIAN APPROACH FOR REACTING FRONT … · A LAGRANGIAN APPROACH FOR REACTING FRONT PROPAGATION IN TURBULENT FLOWS Gianni PAGNINI [email protected] Awarded IKERBASQUE Research

A LAGRANGIAN APPROACH FORREACTING FRONT PROPAGATION

IN TURBULENT FLOWS

Gianni PAGNINI

[email protected]

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.


Lagrangian approachLagrangian approach:the flux is studied along fluid element trajectories.Experimentally very difficult and used forexample for pollutant dispersion analysis.


Single step chemical reaction

reactants → products + heat


Assumptions

− homogeneous, isotropic and stationaryturbulence,

− zero mean velocity field,

− the motion of reactants is identical to that ofproducts,

− any molecular effect is neglected.


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.


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.


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.


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 .


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)


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)


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|| .


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


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)


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).


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

.


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

.


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)


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 .


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)


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)


Application in premixed combustion

Gianni PAGNINI & Ernesto BONOMI,Lagrangian Formulation of Turbulent PremixedCombustion.

Phys. Rev. Lett. 107, 044503 (2011).


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).


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.


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.


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.


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.


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)] .


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.


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 .


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.


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.


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.


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



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



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



c(x, t)-Profile

Figure 8: Comprehensive plot of progress variable profile.


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).



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


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|| .


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.


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.


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.



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.


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.



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.


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 .


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 .


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 .


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.



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.



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 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.


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.


Acknowledgements

Many Thanks toIKERBASQUE and BCAM

and You!


A LAGRANGIAN APPROACH FOR REACTING FRONT … · A LAGRANGIAN APPROACH FOR REACTING FRONT PROPAGATION IN TURBULENT FLOWS Gianni PAGNINI [email protected] Awarded IKERBASQUE Research

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