Model Reduction for Reaction-Diffusion Systems: Bifurcations in Slow - - PowerPoint PPT Presentation

Model Reduction for Reaction-Diffusion Systems: Bifurcations in Slow Invariant Manifolds

Joshua D. Mengers Joseph M. Powers

Department of Aerospace and Mechanical Engineering University of Notre Dame

50th AIAA Aerospace Sciences Meeting Nashville, Tennessee January 10, 2012

• J. Powers

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Outline

1 Motivation and background 2 Model 3 Oxygen dissociation reaction mechanism 4 Results

Spatially homogeneous Reaction-diffusion

5 Conclusions

• J. Powers

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Motivation and Background

Detailed kinetics are essential for accurate modeling of reactive systems Reactive systems induce a wide range

• f spatial and temporal scales, and

subsequently severe stiffness occurs The spatial and temporal scales are coupled by the underlying physics of the problem Verification of a simulation’s accuracy requires resolution of all scales

Direct numerical simulation Large eddy simulation Reynolds-averaged simulation Kinetic Monte Carlo Molecular dynamics Quantum Mechanics

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LENGTH SCALE (m) TIME SCALE (s)

“Research needs for future internal combustion engines,” Physics Today, Nov. 2008, pp. 47–52.

The computational cost for reactive flow simulations increases with the range of scales present, the number of reactions and species, and the size of the spatial domain. Manifold methods provide a potential for computational savings.

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Motivation and Background

Manifold methods are typically spatially homogeneous, yet most engineering applications require spatial variation. Diffusion is often modeled with a correction to the spatially homogeneous methods in the long wavelength limit.

z3 Fast Slow Fast Slow z1 z2

However, for thin regions of flames, reaction is fast relative to diffusion, and the short wavelength limit is more appropriate. Al-Khateeb, et al. 2009, Journal of Chemical Physics, provides details on construction of spatially homogeneous SIMs.

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Assumptions

Model a system of N species reacting in J reactions with diffusion in

• ne spatial dimension

Ideal mixture Ideal gases Isochoric Isothermal Negligible advection Single constant mass diffusivity

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Balance Laws

Evolution of species ρ∂Yi ∂t + ∂jm

i

∂x = Mi ˙ ωi(Yn, T), for i, n ∈ [1, N] Boundary conditions ∂Yi ∂x

• x=0

= ∂Yi ∂x

• x=ℓ

= 0, for i ∈ [1, N] Initial conditions Yi(x, t = 0) = ˜ Yi(x), for i ∈ [1, N]

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Constitutive Equations

Fick’s law of diffusion jm

i

= −ρD∂Yi ∂x , for i ∈ [1, N] Ideal gas equation of state P = ρ ¯ RT

N

• i=1

Yi Mi

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Constitutive Equations

Molar production rate ˙ ωi =

J

• j=1

νijrj, for i ∈ [1, N] rj = kj N

• i=1

ρYi Mi ν′

ij

− 1 Kc

j N

• i=1

ρYi Mi ν′′

ij

, for j ∈ [1, J] kj = ajT βj exp − ¯ Ej ¯ RT

• ,

for j ∈ [1, J] Kc

j

= exp

• − N

i=1 ¯

go

i νij

¯ RT

• ,

for j ∈ [1, J]

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Generalized Shvab-Zel’dovich

Certain linear combinations of molar production rate sum to zero, ∂ ∂t N

• i=1

ϕli Yi Mi

• = D ∂2

∂x2 N

• i=1

ϕli Yi Mi

• ,

for l ∈ [1, L] Some evolution PDEs can be integrated to yield algebraic constraints if these quantities are

Initially spatially homogeneous, and Not perturbed at the boundaries,

N

• i=1

ϕli Yi Mi =

N

• i=1

ϕli ˜ Yi Mi , for l ∈ [1, L]

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Reduced Variables

The L algebraic constraints can be used to reduce N PDEs to N − L PDEs Transform to reduced variables: specific mole concentrations zi = Yi Mi , for i ∈ [1, N − L] Evolution of remaining L species are coupled to these reduced variables by the algebraic constraints ∂zi ∂t = ˙ ωi(zn, T) ρ + D∂2zi ∂x2 , for i, n ∈ [1, N − L]

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Galerkin Reduction to ODEs

Assume a spectral decomposition zi(x, t) =

∞

• m=0

zi,m(t)φm(x), for i ∈ [1, N − L] Orthogonal basis functions, φm(x), are eigenfunctions of diffusive

• perator that match boundary conditions

∂2φm ∂x2 = −µ2

mφm

Complete orthogonal basis, φm(x) = cos mπx ℓ

• ,

for m ∈ [0, ∞)

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Galerkin Reduction to ODEs

∂ ∂t ∞

• n=0

zi,nφn

• =

˙ ωi ∞

ˆ n=0 zˆ i,ˆ nφˆ n

• ρ

+ D ∂2 ∂x2 ∞

• n=0

zi,nφn

• Finite system of ODEs for amplitude evolution are recovered by

taking the inner product with φm, and truncated at M dzi,m dt =

• φm, ˙

ωi ∞

ˆ n=0 zˆ i,ˆ nφˆ n

• /ρ
• φm, φm
• ˙

Ωi,m

−Dµ2

mzi,m,

for i ∈ [1, N − L], and m ∈ [0, M] Projection modifies reaction eigenvalues, λi,m = λ0,m − Dµ2

m

Diffusion time scales defined as τD,m ≡ 1 µ2

mD

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Example Problem

Oxygen dissociation reaction:

O2 + M ⇌ O + O + M N = 2 species J = 1 reaction L = 1 constraint N − L = 1 reduced variable z = YO

MO

Isochoric, ρ = 1.6 × 10−4 g/cm2 Isothermal, T = 5000 K

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Spatially Homogeneous System

For domain lengths small enough that diffusion is much faster than reaction Galerkin truncation at M = 0 is appropriate Spatially homogeneous system is recovered dz dt =

• 249.8mol

g s

• −
• 7.473 × 104

g mol s

• z2 −
• 1.724 × 105

g2 mol2 s

• z3
• ˙

Ω

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SIM Construction

Identify equilibria Characterize equilibria by eigenvalues of their Jacobian matrix (slopes) Jij = ∂ ˙ Ωi ∂zj Reaction time scale is the reciprocal of the eigenvalue τR = |λ|−1 SIM is a heteroclinic orbit from R2 to R1

• 0.4
• 0.3
• 0.2
• 0.1

0.1

• 1500
• 1000
• 500

500

SIM

R3 R2 R1 ˙ Ω (mol/g/s) z (mol/g)

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Spatially Homogeneous Evolution

Use z to reconstruct mass fractions of O and O2 Only one time scale present τR ∼ 10−4 s Time scale corresponds to reciprocal of equilibrium eigenvalue

10 -7 10 -5 0.001 0.1 0.10 1.00 0.50 0.20 0.30 0.15 0.70 YO2 YO t (s) Y

Evolution of Species

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Reaction-Diffusion System

For larger domain lengths where diffusion is not much faster than reaction Additional terms in Galerkin projection are retained We examine the truncation at M = 1 dz0 dt =

• 249.8mol

g s

• −
• 7.473 × 104

g mol s z2

0 + z2 1

2

• −
• 1.724 × 105

g2 mol2 s z3

0 + 3z0z2 1

2

• dz1

dt = −

• 7.473 × 104

g mol s

• 2z0z1

−

• 1.724 × 105

g2 mol2 s 3z2

0z1 + 3z3 1

4

• − π2D

ℓ2 z1

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Local Timescales

Time-scale coupling between reaction and diffusion 1 τC = 1 τR + 1 τD R3 is a sink; diffusion keeps it stable

0.02 0.05 0.1 0.2 10- 5 10- 4 0.001 2 1

λ0 R1 λ0 R2 λ1 R1 λ1 R2

π2D ℓ2

τ (s) ℓ (cm)

ℓc

R2 is a source, diffusion changes its stability Critical wavelength, ℓc, where stable diffusion time-scale is equal to unstable reaction time-scale

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Poincar´ e Sphere

Map variables into a space where infinity is on the unit circle We can see the dynamics of the entire system What changes occur in the SIM as we very ℓ? η0 = z0

• M−1

O + z2 0 + z2 1

η1 = z1

• M−1

O + z2 0 + z2 1

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0

SIM

ℓ = 0.0334 cm

R1 R2

R3

η0 η1

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Poincar´ e Sphere

Map variables into a space where infinity is on the unit circle We can see the dynamics of the entire system What changes occur in the SIM as we very ℓ? η0 = z0

• M−1

O + z2 0 + z2 1

η1 = z1

• M−1

O + z2 0 + z2 1

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0

SI M

ℓ = 0.105 cm

R1 R2

R3

η0 η1

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Poincar´ e Sphere

Map variables into a space where infinity is on the unit circle We can see the dynamics of the entire system What changes occur in the SIM as we very ℓ? η0 = z0

• M−1

O + z2 0 + z2 1

η1 = z1

• M−1

O + z2 0 + z2 1

• 1.0
• 0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0

S I M

ℓ = 0.334 cm

R1 R2

R3

η0 η1

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Bifurcation

The change is stability of the combined Fourier mode at the critical wavelength, ℓc, is indicative of a bifurcation Bold branches are saddles; dashed branch is source This bifurcation changes the starting point of the SIM Subsequently, the slow dynamics of the entire system are modified

0.1035 0.1040 0.1045 0.1050 0.1055 0.1060 0.1065

• 0.03
• 0.02
• 0.01

0.00 0.01 0.02 0.03

ℓ (cm) z1 Locus of roots near R2

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Reaction-Diffusion Evolution

Use z0 and z1 to reconstruct spatial distributions of mass fractions of O and O2 For ℓ = 0.0334 cm < ℓc, diffusion is faster than reaction Difficult to segregate into reaction and diffusion contributions

0.0083 0.0167 0.025 0.0334 10

−7

10

−5

10

−3

10

−1

0.1 0.2 0.3 0.5 0.7 1 YO2 YO t (s) x (cm) Y

Evolution of Species

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Reaction-Diffusion Evolution

Project spatial evolution of mass fractions onto Y − t plane to see diffusion time scale. Bold line is spatially homogeneous SIM Two time scales present:

τR ∼ 10−4 s τD = ℓ2 π2D ∼ 10−5 s

Slow dynamics change from reaction to diffusion at ℓc

10 -7 10 -5 0.001 0.1 0.10 1.00 0.50 0.20 0.30 0.15 0.70 YO2 YO t (s) Y

Evolution of Species

ℓ = 0.0334 cm

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Reaction-Diffusion Evolution

Project spatial evolution of mass fractions onto Y − t plane to see diffusion time scale. Bold line is spatially homogeneous SIM Two time scales present:

τR ∼ 10−4 s τD = ℓ2 π2D ∼ 10−4 s

Slow dynamics change from reaction to diffusion at ℓc

10 -7 10 -5 0.001 0.1 0.10 1.00 0.50 0.20 0.30 0.15 0.70 YO2 YO t (s) Y

Evolution of Species

ℓ = 0.105 cm

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Reaction-Diffusion Evolution

Project spatial evolution of mass fractions onto Y − t plane to see diffusion time scale. Bold line is spatially homogeneous SIM Two time scales present:

τR ∼ 10−4 s τD = ℓ2 π2D ∼ 10−3 s

Slow dynamics change from reaction to diffusion at ℓc

10 -7 10 -5 0.001 0.1 0.10 1.00 0.50 0.20 0.30 0.15 0.70 YO2 YO t (s) Y

Evolution of Species

ℓ = 0.334 cm

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Conclusions

The SIM isolates the slowest dynamics, making it ideal for a reduction technique. For sufficiently short length scales, diffusion time scales are faster than reaction time scales, and the system dynamics are dominated by reaction. When lengths are near or above a critical length where the diffusion time scale is on the same order as reaction time scales, diffusion will play a more important role. In the limit of large length scales, a truncation at M = 1 is insufficient, and more terms are required to fully resolve the dynamics.

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Acknowledgments

Partial support provided by NSF Grant No. CBET-0650843 and Notre Dame ACMS Department Fellowship

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