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 (Notre Dame) Bifurcations in SIMs January 10, 2012 1 / 24
Outline 1 Motivation and background 2 Model 3 Oxygen dissociation reaction mechanism 4 Results Spatially homogeneous Reaction-diffusion 5 Conclusions J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 2 / 24
Motivation and Background Detailed kinetics are essential for Reynolds-averaged accurate modeling of reactive systems simulation -3 Large eddy 10 simulation Reactive systems induce a wide range TIME SCALE (s) of spatial and temporal scales, and Direct numerical -6 simulation 10 subsequently severe stiffness occurs -9 Kinetic 10 Monte Carlo The spatial and temporal scales are Molecular dynamics coupled by the underlying physics of -12 10 Quantum Mechanics the problem -12 -9 -6 -3 10 10 10 10 LENGTH SCALE (m) “Research needs for future internal Verification of a simulation’s accuracy combustion engines,” Physics Today, Nov. 2008, pp. 47–52. requires resolution of all scales 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. J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 3 / 24
Motivation and Background Manifold methods are typically Fast z 3 spatially homogeneous, yet most Fast engineering applications require spatial variation. Slow Diffusion is often modeled with a Slow correction to the spatially z 2 z 1 homogeneous methods in the long wavelength limit. 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. J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 4 / 24
Assumptions Model a system of N species reacting in J reactions with diffusion in one spatial dimension Ideal mixture Ideal gases Isochoric Isothermal Negligible advection Single constant mass diffusivity J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 5 / 24
Balance Laws Evolution of species ∂t + ∂j m ρ∂Y i i ∂x = M i ˙ ω i ( Y n , T ) , for i, n ∈ [1 , N ] Boundary conditions � � ∂Y i = ∂Y i � � = 0 , for i ∈ [1 , N ] � � ∂x ∂x � � x =0 x = ℓ Initial conditions Y i ( x, t = 0) = ˜ Y i ( x ) , for i ∈ [1 , N ] J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 6 / 24
Constitutive Equations Fick’s law of diffusion = − ρ D ∂Y i j m ∂x , for i ∈ [1 , N ] i Ideal gas equation of state N Y i � P = ρ ¯ R T M i i =1 J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 7 / 24
Constitutive Equations Molar production rate J � ω i ˙ = ν ij r j , for i ∈ [1 , N ] j =1 � N ij � � ρY i � ν ′ N � ρY i � ν ′′ − 1 � ij � r j = k j , for j ∈ [1 , J ] K c M i M i j i =1 i =1 � − ¯ � E j a j T β j exp k j = , for j ∈ [1 , J ] ¯ R T � � − � N g o i =1 ¯ i ν ij K c = exp , for j ∈ [1 , J ] j ¯ R T J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 8 / 24
Generalized Shvab-Zel’dovich Certain linear combinations of molar production rate sum to zero, � N � N � � = D ∂ 2 ∂ Y i Y i � � ϕ li ϕ li , for l ∈ [1 , L ] ∂x 2 ∂t M i M i i =1 i =1 Some evolution PDEs can be integrated to yield algebraic constraints if these quantities are Initially spatially homogeneous, and Not perturbed at the boundaries, N N ˜ Y i Y i � � ϕ li = ϕ li , for l ∈ [1 , L ] M i M i i =1 i =1 J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 9 / 24
Reduced Variables The L algebraic constraints can be used to reduce N PDEs to N − L PDEs Transform to reduced variables: specific mole concentrations z i = Y i , for i ∈ [1 , N − L ] M i Evolution of remaining L species are coupled to these reduced variables by the algebraic constraints + D ∂ 2 z i ∂z i ∂t = ˙ ω i ( z n , T ) ∂x 2 , for i, n ∈ [1 , N − L ] ρ J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 10 / 24
Galerkin Reduction to ODEs Assume a spectral decomposition ∞ � z i ( x, t ) = z i,m ( t ) φ m ( x ) , for i ∈ [1 , N − L ] m =0 Orthogonal basis functions, φ m ( x ), are eigenfunctions of diffusive operator that match boundary conditions ∂ 2 φ m = − µ 2 m φ m ∂x 2 Complete orthogonal basis, � mπx � φ m ( x ) = cos , for m ∈ [0 , ∞ ) ℓ J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 11 / 24
Galerkin Reduction to ODEs �� ∞ � � ∞ � ∞ � � ω i ˙ n =0 z ˆ n φ ˆ + D ∂ 2 ∂ n � � ˆ i, ˆ z i,n φ n = z i,n φ n ∂x 2 ∂t ρ n =0 n =0 Finite system of ODEs for amplitude evolution are recovered by taking the inner product with φ m , and truncated at M � �� ∞ � � φ m , ˙ ω i n =0 z ˆ n φ ˆ /ρ dz i,m n for i ∈ [1 , N − L ] , ˆ i, ˆ −D µ 2 = m z i,m , and m ∈ [0 , M ] dt � φ m , φ m � � �� � ˙ Ω i,m Projection modifies reaction eigenvalues, λ i,m = λ 0 ,m − D µ 2 m 1 Diffusion time scales defined as τ D ,m ≡ µ 2 m D J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 12 / 24
Example Problem Oxygen dissociation reaction: O 2 + M ⇌ O + O + M N = 2 species Isochoric, J = 1 reaction ρ = 1 . 6 × 10 − 4 g/cm 2 L = 1 constraint Isothermal, N − L = 1 reduced variable T = 5000 K z = Y O M O J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 13 / 24
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 � � � � g 2 dz 249 . 8 mol � g � z 2 − 7 . 473 × 10 4 1 . 724 × 10 5 z 3 dt = − mol 2 s g s mol s � �� � ˙ Ω J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 14 / 24
SIM Construction Identify equilibria Characterize equilibria by eigenvalues of their Jacobian ˙ Ω ( mol/g/s ) matrix (slopes) z ( mol/g ) 500 J ij = ∂ ˙ Ω i R 2 R 3 R 1 ∂z j - 0.4 - 0.3 - 0.2 - 0.1 SIM 0.1 - 500 Reaction time scale is the - 1000 reciprocal of the eigenvalue - 1500 τ R = | λ | − 1 SIM is a heteroclinic orbit from R 2 to R 1 J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 15 / 24
Spatially Homogeneous Evolution Use z to reconstruct mass Evolution of Species 1.00 fractions of O and O 2 Y O 0.70 0.50 Only one time scale present τ R ∼ 10 − 4 s Y 0.30 0.20 Y O 2 0.15 Time scale corresponds to 0.10 reciprocal of equilibrium 10 -7 10 -5 0.001 0.1 eigenvalue t ( s ) J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 16 / 24
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 � � � � � 0 + z 2 dz 0 249 . 8 mol � g 7 . 473 × 10 4 z 2 1 = − dt g s mol s 2 � � � � g 2 0 + 3 z 0 z 2 1 . 724 × 10 5 z 3 1 − mol 2 s 2 dz 1 � g � 7 . 473 × 10 4 = − 2 z 0 z 1 dt mol s � � � � g 2 0 z 1 + 3 z 3 − π 2 D 1 . 724 × 10 5 3 z 2 1 − ℓ 2 z 1 mol 2 s 4 J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 17 / 24
Local Timescales 0.001 Time-scale coupling between reaction and diffusion 10 - 4 1 = 1 + 1 τ ( s ) λ 0 R 1 τ C τ R τ D λ 0 R 2 λ 1 R 1 10 - 5 λ 1 R 2 2 R 3 is a sink; diffusion keeps it π 2 D ℓ 2 1 ℓ c stable 0.02 0.05 0.1 0.2 ℓ ( cm ) R 2 is a source, diffusion changes its stability Critical wavelength, ℓ c , where stable diffusion time-scale is equal to unstable reaction time-scale J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 18 / 24
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 ? ℓ = 0 . 0334 cm 1.0 z 0 η 0 = � 0.8 M − 1 O + z 2 0 + z 2 1 z 1 0.6 η 1 η 1 = � M − 1 O + z 2 0 + z 2 0.4 1 0.2 R 3 SIM 0.0 R 2 R 1 -1.0 -0.5 0.0 0.5 1.0 η 0 J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 19 / 24
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 ? ℓ = 0 . 105 cm 1.0 z 0 η 0 = � 0.8 M − 1 O + z 2 0 + z 2 1 z 1 0.6 η 1 η 1 = � M − 1 O + z 2 0 + z 2 0.4 1 S I M 0.2 R 3 0.0 -1.0 R 2 -0.5 0.0 0.5 R 1 1.0 η 0 J. Powers (Notre Dame) Bifurcations in SIMs January 10, 2012 19 / 24
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