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Slow Invariant Manifolds

for Reactive-Diffusive Systems

Joshua D. MengersJoseph M. Powers

Department of Aerospace and Mechanical Engineering

University of Notre Dame

7th International Congress on Industrial and Applied Mathematics

Vancouver, British Columbia, CanadaJuly 21, 2011

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 1 / 26

Outline

1 Motivation and Background

2 Model

3 ResultsOxygen DissociationZel’dovich Mechanism

4 Conclusions

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 2 / 26

Motivation and Background

Reactive systems induce awide range of spatial andtemporal scales, andsubsequently severe stiffness

DNS resolves all ranges ofcontinuum physical scalespresent

Under-resolved simulationsattempt to account for missedphysical phenomena withmodeling

Fully resolved simulations areexpensive to compute

Direct numerical

simulation

Large eddy

simulation

Reynolds-averaged

simulation

Molecular

dynamics

Quantum

Mechanics

10-12

10-12

10-9

10-6

10-3

10-9

10-6

10-3

LENGTH SCALE (m)

TIM

E S

CA

LE

(s)

Kinetic

Monte Carlo

“Research needs for future internal combustionengines,” Physics Today, Nov. 2008, pp 47-52.

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 3 / 26

Motivation and Background

Manifold methods providepotential savings

Most methods are for spatiallyhomogeneous systems

We employ the slow invariantmanifold (SIM) model ofAl-Khateeb, et al.(2009, Journal of Chemical Physics)

SIM

z3 Fast

Slow

Fast

Slow

z1

z2

We adjust for the dynamics ofdiffusion in the presence ofweak spatial heterogeneity

This is valid when diffusion isfast relative to reaction, i.e.thin regions of flames

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 4 / 26

Assumptions

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

Ideal mixture

Calorically perfect

Ideal gases

Negligible advection

Constant specific heat

Single constant massdiffusivity

Constant thermalconductivity

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 5 / 26

Balance Laws

Evolution of species and energy

ρ∂Yi

∂t+

∂jmi

∂x= Miωi(Yn, T ), for i, n ∈ [1, N ]

ρ∂h

∂t+

∂jq

∂x= 0

Boundary conditions

∂Yi

∂x

∣∣∣∣x=0

=∂Yi

∂x

∣∣∣∣x=ℓ

= 0, for i ∈ [1, N ]

∂T

∂x

∣∣∣∣x=0

=∂T

∂x

∣∣∣∣x=ℓ

= 0

Initial conditions

Yi(x, t = 0) = Yi(x), for i ∈ [1, N ]

T (x, t = 0) = T (x)

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 6 / 26

Constitutive Equations

Simple diffusive flux terms

jmi = −ρD

∂Yi

∂x, for i ∈ [1, N ]

jq = −k∂T

∂x+

N∑

i=1

hfi jm

i

Caloric equation of state

h =

N∑

i=1

Yi

(

cPi(T − T o) + hfi

)

Ideal gas equation of state

P =ρRT

∑Ni=1

Mi

Yi

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 7 / 26

Constitutive Equations

Molar production rate

ωi =

J∑

j=1

νijrj , for i ∈ [1, N ]

rj = kj

(N∏

i=1

(ρYi

Mi

)ν′

ij

−1

Kcj

N∏

i=1

(ρYi

Mi

)ν′′

ij

)

, for j ∈ [1, J ]

kj = ajTβj exp

(−Ej

RT

)

, for j ∈ [1, J ]

Kcj = exp

(

−∑N

i=1 goi νij

RT

)

, for j ∈ [1, J ]

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 8 / 26

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]

In adiabatic systems, when the Lewis number is unity

∂

∂t

(

cP (T − T o) +

N∑

i=1

hfi Yi

)

︸ ︷︷ ︸

h

= D∂2

∂x2

(

cP (T − T o) +

N∑

i=1

hfi Yi

)

︸ ︷︷ ︸

h

If initially spatially homogeneous, these PDEs can be integrated

N∑

i=1

ϕli

Yi

Mi

=

N∑

i=1

ϕli

Yi

Mi

, for l ∈ [1, L]

cP (T − T o) +

N∑

i=1

hfi Yi = cP (T − T o) +

N∑

i=1

hfi Yi

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 9 / 26

Reduced Variables

Transform to specific mole concentrations

zi =Yi

Mi

, for i ∈ [1, N − L]

Evolution of remaining L species and temperature are coupled tothese reduced variables by the algebraic constraints

∂zi

∂t=

ω(zn, T )

ρ+ D

∂2zi

∂x2, for i, n ∈ [1, N − L]

T =

T , if isothermal

h −∑N

i=1 zi(zn)hfi

∑Ni=1 zi(zn)cPi

+ T o, if adiabatic

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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 diffusionoperator that match boundary conditions

φm(x) = cos(mπx

ℓ

)

, for m ∈ [0,∞)

Finite system of ODEs for amplitude evolution are recovered bytaking the inner product with φn, and truncated at M

dzi,m

dt=

〈φm, ωi (∑

∞

m=0 zi,nφn)〉

〈φm, φm〉−

m2π2D

ℓ2zi,m,

for i ∈ [1, N − L],and m ∈ [0,M ]

Diffusion time scale identified, τD =ℓ2

π2D

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Oxygen Dissociation

O + O + M ⇌ O2 + M

N = 2 species

J = 1 reactions

L = 1 constraints

N − L = 1 reduced variablesz = zO

Isochoric,ρ = 1.6 × 10−4 g/cm3

Isothermal, T = 5000 K

Partial differential equation governing evolution

∂z

∂t= 249.84130 − 74734.78 z2 − 172406.48 z3 + D

∂2z

∂x2

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Oxygen Dissociation

O + O + M ⇌ O2 + M

N = 2 species

J = 1 reactions

L = 1 constraints

N − L = 1 reduced variablesz = zO

Isochoric,ρ = 1.6 × 10−4 g/cm3

Isothermal, T = 5000 K

Spatially homogeneous evolution equation

dz0

dt= 249.84130 − 74734.78 z2

0 − 172406.48 z30

-0.4 -0.3 -0.2 -0.1 0.1

-1500

-1000

-500

500

z(m

ol/g

)

ω (mol/g/s)

R1 R2 R3

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 12 / 26

Diffusion-Correction – Galerkin Projection

One spatial mode (M = 1) evolution equation

dz0

dt= 249.84130 − 74734.78

(

z20 +

z21

2

)

− 172406.48

(

z30 +

3z0z21

2

)

dz1

dt= −74734.78 (2z0z1) − 172406.48

(

3z20z1 +

3z31

4

)

−π2D

ℓ2z1

Spatially homogeneousevolution when z1 = 0

Spatially homogeneousequilibria retained

Eigenvalues about theseequilibria are modified

λ1 = λ0 −π2D

ℓ2

0.2 0.5 1 2

10-5

10-4

0.001

ℓ (mm)

τ (s)

λ0 R3λ0 R2

λ1 R3λ1 R2

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Bifurcation

Change in sign of modified eigenvalue, λ1 = λ0 −π2D

ℓ2, identifies a

critical length where SIM origin changes character

Bifurcation occurs at R2

equilibrium

π2D

ℓ2= λ0 = 7321.5 s−1

ℓ = 1.04 mm

Diffusion-corrected SIM originshifts to bifurcated branches

1.035 1.040 1.045 1.050 1.055 1.060 1.065- 0.03

- 0.02

- 0.01

0.00

0.01

0.02

0.03

ℓ (mm)

z 1(m

ol/g

)

Locus of roots near R2

Bold branches are saddles; dashed branch is source

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 14 / 26

Poincare Sphere

Map variables into a space where infinity is on the unit circle

η0 =αz0

√

1 + α2z20 + α2z2

1

η1 =αz1

√

1 + α2z20 + α2z2

1

ℓ = 0.334 mm

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

SIM

η0

η1

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 15 / 26

Poincare Sphere

Map variables into a space where infinity is on the unit circle

η0 =αz0

√

1 + α2z20 + α2z2

1

η1 =αz1

√

1 + α2z20 + α2z2

1

ℓ = 1.05 mm

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

SIM

η0

η1

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 15 / 26

Poincare Sphere

Map variables into a space where infinity is on the unit circle

η0 =αz0

√

1 + α2z20 + α2z2

1

η1 =αz1

√

1 + α2z20 + α2z2

1

ℓ = 3.34 mm

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

SIM

η0

η1

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 15 / 26

Zel’dovich Mechanism – Isothermal

N + NO ⇌ N2 + O

N + O2 ⇌ NO + O

N = 5 species

J = 2 reactions

L = 3 constraints

N − L = 2 reduced variablesz1 = zNO, z2 = zN

Isochoric, ρ = 1.2002 g/cm3

Isothermal, T = 4000 K

Bimolecular, isobaric,P = 1.6629 × 106 dyne/cm2 =1.64 atm

Partial differential equation governing evolution

∂z1

∂t= 250−9.97×104z1+1.16×107z2−3.22×109z1z2+6.99×108z2

2 + D∂2z1

∂x2

∂z2

∂t= 250+8.47×104z1−1.17×107z2−1.84×109z1z2−6.98×108z2

2 + D∂2z2

∂x2

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 16 / 26

Zel’dovich Mechanism – Isothermal

N + NO ⇌ N2 + O

N + O2 ⇌ NO + O

N = 5 species

J = 2 reactions

L = 3 constraints

N − L = 2 reduced variablesz1 = zNO, z2 = zN

Isochoric, ρ = 1.2002 g/cm3

Isothermal, T = 4000 K

Bimolecular, isobaric,P = 1.6629 × 106 dyne/cm2 =1.64 atm

Spatially homogeneous evolution equations – second order polynomials.

dz1,0

dt= 250−9.97×104z1,0+1.16×107z2,0−3.22×109z1,0z2,0+6.99×108z2

2,0

dz2,0

dt= 250+8.47×104z1,0−1.17×107z2,0−1.84×109z1,0z2,0−6.98×108z2

2,0

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 16 / 26

Spatially Homogeneous Isothermal Phase Space

Identify equilibria

Characterize equilibriaby eigenvalues of theirJacobian matrix

Classify time scales asfast and slow

Identify SIM as aheteroclinic orbit fromsaddle to sink

SIM

−0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

z1 (mol/g)

z 2(m

ol/g

)

R3R2

R1

I1

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 17 / 26

Spatially Homogeneous Isothermal Phase Space

Identify equilibria

Characterize equilibriaby eigenvalues of theirJacobian matrix

Classify time scales asfast and slow

Identify SIM as aheteroclinic orbit fromsaddle to sink

SIM

−0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

z1 (mol/g)

z 2(m

ol/g

)

R3R2

R1

I1

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 17 / 26

Spatially Homogeneous Isothermal Phase Space

Identify equilibria

Characterize equilibriaby eigenvalues of theirJacobian matrix

Classify time scales asfast and slow

Identify SIM as aheteroclinic orbit fromsaddle to sink

−4 −3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x 10−5

SIM

x 10−3

z1 (mol/g)

z 2(m

ol/g

)

R3

R2

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 17 / 26

Diffusion-Correction – Galerkin Projection

First diffusion mode addsmodified time scale

Positive eigenvalue identifiescritical length scale

139.46 139.47 139.48 139.49 139.50

- 2

- 1

0

1

2

x 10−5 Locus of roots near R2

z 1,1

(mol

/g)

ℓ (µm)

10 20 50 100 200 500 1000

10-8

10-7

10-6

10-5

ℓ (µm)

τ (s)

λi,0 R3λi,0 R2

λi,1 R3λi,1 R2

Bifurcation occurs at thislength scale

Let us examine a length belowthis critical length scale,ℓ = 17 µm

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 18 / 26

Diffusion-Correction Isothermal Phase Space

−2

2

x 10−3

−4

0

4

x 10−5

0

4

8

x 10−4

SIM

z1,0(mol/g

)z2,0 (mol/g)

z 1,1

(mol

/g)

R3

R2

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 19 / 26

Diffusion-Correction Isothermal Evolution

10−10

10−8

10−6

10−4

10−20

10−15

10−10

10−5

1 , 0

2 , 0

1 , 1

2 , 1

t (s)

z i,m

(mol

/g)

zzzz 0

8.5

17

10−10

10−8

10−6

10−4

0

5

10

15

x 10−4

t (s)

z NO

(mol

/g)

x (µm)

Two additional fast time scales from diffusion

Spatially inhomogeneous amplitudes decay earlier than eitherreaction time scale

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 20 / 26

Zel’dovich Mechanism – Adiabatic

N + NO ⇌ N2 + O

N + O2 ⇌ NO + O

N = 5 species

J = 2 reactions

L = 3 constraints

N − L = 2 reduced variablesz1 = zNO, z2 = zN

T = T (z1, z2)

Isobaric,P = 1.6629 × 106 dyne/cm2

= 1.64 atm

Adiabatic,h = 9.0376 × 10−10 erg/gchosen to keep chemicalequilibrium at same point

Evolution equations highly nonlinear due to temperature-dependancein exponentials

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 21 / 26

Isothermal Equilibria as a Function of Temperature

Equilibria of adiabatic systemdifficult to identify

Find isothermal equilibria forvarious temperatures

3000 4000 5000 6000 7000 8000 9000 10000

4

6

8

10

12

14

16

x 10−10

h(e

rg/g

)

T (K)

R3

R2

R1

R′

2 R′

10 0.01 0.02

−0.01

0

0.01

z1 (mol/g)

z 2(m

ol/g

)

R3

R2

R1

R′2

R′1

Adiabatic equilibria whereenthalpy constraint is met

Method is not exhaustive

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 22 / 26

Spatially Homogeneous Adiabatic Phase Space

0 0.02 0.04 0.06

−0.02

−0.01

0

0.01

0.02

SIM

z 1(m

ol/g

)

z2 (mol/g)

R3R′2

R′

1

P ′

1

T = 0

Equilibria and dynamics remain similar to isothermal case

SIM is heteroclinic orbit connecting analogous points

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 23 / 26

Spatially Homogeneous Adiabatic Phase Space

0 0.02 0.04 0.06

−0.02

−0.01

0

0.01

0.02

SIM

z 1(m

ol/g

)

z2 (mol/g)

R3R′2

R′

1

P ′

1

T = 0

Equilibria and dynamics remain similar to isothermal case

SIM is heteroclinic orbit connecting analogous points

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 23 / 26

Spatially Homogeneous Adiabatic Phase Space

−6 −4 −2 0 2 4−8

−6

−4

−2

0

2

4x 10

−5

SIM

x 10−3

z 1(m

ol/g

)

z2 (mol/g)

R3

R′2

Equilibria and dynamics remain similar to isothermal case

SIM is heteroclinic orbit connecting analogous points

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 23 / 26

Spatially Homogeneous Adiabatic Evolution

10−10

10−8

10−6

10−4

10−4

10−3

10−2

zNO

zN

zN2

zO

zO2

z(m

ol/g

)

t (s)10

−10

10−8

10−6

10−4

2000

2500

3000

3500

4000

T(K

)

t (s)

Species and temperature evolution exhibit fast and slow timescales, consistent with equilibrium eigenvalues

Adiabatic reactive-diffusive systems have yet to be analyzed

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 24 / 26

Conclusions

The SIM isolates the slowest dynamics, making it ideal for areduction technique

A critical length scale has been identified where the diffusion timescale matches a reaction time scale; at this length a bifurcationoccurs that affects the slow dynamics of the system

For sufficiently short length scales, diffusion time scales are fasterthan reaction time scales, and the system dynamics are dominatedby reaction

When lengths are near or above the critical length, diffusion playsa more important role

Extension of SIM to spatially homogeneous adiabatic systems isshown to be feasible

J. Mengers Notre Dame SIM Reactive-Diffusive July 21, 2011 25 / 26