combustion dynamics in steady compressible flows · (shapiro) canadian snowshoe hare (c.h. smith)...

Combustion dynamics insteady compressible flows

S. Berti 1, D. Vergni 2, A. Vulpiani 3

1LSP, Université J. Fourier Grenoble I and CNRS, France

2Istituto Applicazioni del Calcolo (IAC) - CNR, Roma, Italy

3Dip. Fisica, CNISM and INFN, Università “La Sapienza”, Roma, Italy

Nice, France 16/2/2009

Outline

1 Reaction-transport systems

2 Ignition dynamics in compressible flow

3 Numerical results

4 Conclusions

Berti Ignition reactions in compressible flow Nice 2009 2 / 24

Outline



3 Numerical results

4 Conclusions


Front propagation

biologicalpopulationdynamics

clorophyll/plankton(O. Naval Research)

bacterial colonies(Shapiro)

canadian snowshoe hare(C.H. Smith)

chemicalreactions

BZ (Winfree) BZ (Steinbock)

flamepropagation

DNS of flames (Vladimirova) forest fire (USA) firemen (USA)

[J.D. Murray, “Mathematical biology”, Springer]


Front propagationbiologicalpopulationdynamics

clorophyll/plankton(O. Naval Research)

bacterial colonies(Shapiro)

canadian snowshoe hare(C.H. Smith)

chemicalreactions

BZ (Winfree) BZ (Steinbock)

flamepropagation

DNS of flames (Vladimirova) forest fire (USA) firemen (USA)

[J.D. Murray, “Mathematical biology”, Springer]


Advection-Reaction-Diffusion

passive limit −→ advection-reaction-diffusion equation(no back-reaction of reactants on the velocity field)

∂tθ + ∇ · (u θ) = D0∇2θ + 1τ f (θ)

θ(x , t) ∈ [0, 1] fractional concentration of products, normalizedtemperatureu = u(x , t) given velocity fieldD0 molecular diffusivityf (θ) reaction term with characteristic time τ


Reaction dynamicsu = 0 ⇒ travelling-front solutions θ(x , t) = θ(x ± v0t)

pulled f (θ) = θ(1− θ) (FKPP - autocatalytic)

pushed f (θ) = (1− θ)e−θcθ (Arrhenius)

f (θ) = 0 for θ ≤ θc (ignition)


The role of advection

vf

enhancement of front propagationspeed with respect to the case u = 0vf > v0

quenching of the reactive processdue to the combined action ofadvection and diffusion if ` ≤ `c

`←→

localized i.c.t0

t1

t2 t3

−→ what is the critical size `c = `c(U, D0, τ, θc)?


Outline



3 Numerical results

4 Conclusions


Localization approximation (1)1D stationary compressible velocity field: u(x) = U0 sin

(πxL

)(U0 > 0)

the Lagrangian equation x = u(x) has...

(S) stable fixed points: x = ±L,±3L, . . . ,±(2n − 1)L, . . .

(U) unstable fixed points: x = 0,±2L, . . . ,±2nL, . . .

localization of θ aroundxn = (2n − 1)Ln = 0,±1,±2, . . .

random walk among these points⇒ lattice model


Localization approximation (2)

we introduce:

θn(t) =

∫ xn+L

xn−Lθ(x , t)dx =

∫ xn+δ

xn−δ

θ(x , t)dx where δ � L

master equation for θn(t):

θn(t + ∆t) =∑

j

P(∆t)j→n θj(t)

P(∆t)n→n = 1− 2W∆t P(∆t)

n→n−1 = P(∆t)n→n+1 = W∆t

W = W (U0, D0) Brownian particle’s escape rate


Kramers formula

∂tθ = ∂x (f ′ θ) + D0∂2x θ FPE

f (x) = −∫

dx u(x) potential

in the low-noise limit D0B � 1:

W ≡ rK = 12π

√f ′′(x0)|f ′′(xB)|e−

f (xB )−f (x0)

D0

f (x) = U0Lπ cos

(πxL

)B = 2U0L

π D0 → 0

x0 = (2n − 1)L xB = (2n + 1)L =⇒ ln W (U0, D0) ∼ −U0D0

[H. Risken “The Fokker-Planck equation”, Springer; J.P. Sethna “Entropy, order parameters, and complexity”, OUP]


Reactive mapIn presence of reaction: θn(t + ∆t) = G∆t

(∑j P(∆t)

j→n θj(t))

For the ignition-type nonlinearity:

G∆t(θ) =

{θ 0 ≤ θ ≤ θc

θ + (θ − θc)(1− θ)∆tτ

θc < θ ≤ 1

threshold value: θ ≤ θc ⇒ no reaction

[Mancinelli et al., Physica D 185, 175 (2003)]


Outline



3 Numerical results

4 Conclusions


Simulation settingsNumerical integration of the lattice reactive model with:

lattice spacing ∆x = 1

lattice size Lx ≤ 4 · 104

time step ∆t ≤ 10−2

Localized initial condition θn(0) = Θn:

Θn =

{1 for |n| ≤ `

20 for |n| > `

2

Analysis:

Q(t) =+∞∑

n=−∞θn(t) total burnt area (ideal fronts)

vf (t) =1∆t

+∞∑n=−∞

[θn(t + ∆t)− θn(t)] front speed


Quenching phenomenapropagation =⇒ Q(t) ' 2vf t for large t

θc = 0.6

W = 0.1

τ = 50

` = 18

` = 16

` = 14

This way it is possible to determine a critical length `c separating thetwo regimes:

` < `c ⇒ θ(x , t →∞)→ 0 quenching` > `c ⇒ θ(x , t →∞)→ 1 propagation


Dimensional scalingrelevant parameters: τ , W , θc

Measure of the critical size `c:

(A) τ = const, W variable

(B) W = const, τ variable

`c = F (θc)√

W τ

Continuum limit: Lx � ∆x = 1 ⇒ pure reaction-diffusion with D = W∆x2 = W

W ≡ D, τ , θc ⇒ [`c ] = [Dτ ]1/2[F (θc)] = [W τ ]1/2


Dimensional scalingrelevant parameters: τ , W , θc

Measure of the critical size `c:

(A) τ = const, W variable

(B) W = const, τ variable`c = F (θc)

√W τ

Continuum limit: Lx � ∆x = 1 ⇒ pure reaction-diffusion with D = W∆x2 = W

W ≡ D, τ , θc ⇒ [`c ] = [Dτ ]1/2[F (θc)] = [W τ ]1/2


Dependence on the threshold concentration (1)

ansatz based on the physical hypothesis:

(i) F (θc) ≥ 0 monotonically increasing with θc ∈ [0, 1]

(ii) F (θc)→ 0 when θc → 0

(iii) F (θc)→∞ when θc → 1

(iv) F (θc) is analytic for θc 6= 1

⇒ F (z) = a0 +a−1

1− z+

a−2

(1− z)2 + ... =∞∑

k=0

a−k

(1− z)k Laurent exp. (z=1)

from the numerics...

limz→1(1− z)αF (z) = 0 for α ≥ 3 =⇒ F (z) = a0 +a−11−z +

a−2(1−z)2

imposing (ii): F (z = 0) = 0 =⇒ F (z) = a0z

1−z

(1 +

a−2a0

11−z

)


Dependence on the threshold concentration (1)

ansatz based on the physical hypothesis:

(i) F (θc) ≥ 0 monotonically increasing with θc ∈ [0, 1]

(ii) F (θc)→ 0 when θc → 0

(iii) F (θc)→∞ when θc → 1

(iv) F (θc) is analytic for θc 6= 1

⇒ F (z) = a0 +a−1

1− z+

a−2

(1− z)2 + ... =∞∑

k=0

a−k

(1− z)k Laurent exp. (z=1)

from the numerics...

limz→1(1− z)αF (z) = 0 for α ≥ 3 =⇒ F (z) = a0 +a−11−z +

a−2(1−z)2

imposing (ii): F (z = 0) = 0 =⇒ F (z) = a0z

1−z

(1 +

a−2a0

11−z

)Berti Ignition reactions in compressible flow Nice 2009 17 / 24

Dependence on the threshold concentration (2)F (z) = a0

z1−z

(1 +

a−2a0

11−z

)

a0 ' 4.39

a−2 ' 0.17

experiments A (τ = 10, τ = 50) and B (W = 0.1, W = 0.25)


Front thickness

θ(x < 0, t0) = 1 (reservoir of burnt material) ⇒ front propagation

ξ ∝√

W τ characteristic front width

FKPP: θ(x , t) ∼ e−x−v0 t

ξ0 for x & v0t

where v0 = 2√

D0/τ and ξ0 =√

D0τ


Different length scales

in the present case: θ(x , t) ∝ e−x − vf t

ξ ⇒ measure of ξ

ξ/√

Wτ

F (θc )

ξ0 (FKPP)

`c 6= ξ ⇒ quenching is not simply related to usual features of the front


Outline



3 Numerical results

4 Conclusions


ConclusionsWe studied the quenching phenomenon of ignition-type reaction dynamics ina steady compressible flow.

1 Localization approximation ⇒ lattice model and efficient numericalimplementation.

2 Single critical length-scale `c :

` < `c ⇒ θ(x , t →∞)→ 0 quenching` > `c ⇒ θ(x , t →∞)→ 1 propagation

3 Dimensional scaling: `c = F (θc)√

W τ , with F (θc) characterized by apole of order α = 2 in θc = 1.

4 Agreement with results in different configurations: pure reaction-diffusionsystems [Zlatoš 2005], and reactive transport systems in two-dimensionalincompressible velocity fields [Constantin et al. 2001,2003].


Comparisons

2D ARD system with ∇ · u = 0 and τ � τA

a) shear flow (U) ⇒ `c ∼ U

∼√

Deff τ

b) cellular flow (U) ⇒ `c ∼ U1/4

∼√

Deff τ

... we observe `c ∼√

W τ ; W = W (U0)

τ � τA =⇒ ∂tθ = Deff∇2θ + 1τ f (θ) ; Deff = Deff (U)

a) shear flow Deff ∼ U2

b) cellular flow Deff ∼ U1/2

[Constantin et al., Commun. Pure Appl. Math. 54, 1320 (2001); Vladimiriova et al., Combust. Theory Model. 7, 487 (2003)]


Comparisons

2D ARD system with ∇ · u = 0 and τ � τA

a) shear flow (U) ⇒ `c ∼ U ∼√

Deff τ

b) cellular flow (U) ⇒ `c ∼ U1/4 ∼√

Deff τ

... we observe `c ∼√

W τ ; W = W (U0)

τ � τA =⇒ ∂tθ = Deff∇2θ + 1τ f (θ) ; Deff = Deff (U)

a) shear flow Deff ∼ U2

b) cellular flow Deff ∼ U1/2

[Constantin et al., Commun. Pure Appl. Math. 54, 1320 (2001); Vladimiriova et al., Combust. Theory Model. 7, 487 (2003)]


combustion dynamics in steady compressible flows · (shapiro) canadian snowshoe hare (c.h. smith)...

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