Calculation of Slow Invariant Manifolds for Reactive Systems Ashraf - PowerPoint PPT Presentation

Calculation of Slow Invariant Manifolds for Reactive Systems Ashraf N. Al-Khateeb Joseph M. Powers Samuel Paolucci Department of Aerospace and Mechanical Engineering Andrew J. Sommese Jeffery A. Diller Department of Mathematics University of Notre Dame, Notre Dame, Indiana 47 th AIAA Aerospace Science Meeting Orlando, Florida 8 January 2009

Outline • Introduction • Slow Invariant Manifold (SIM) • Method of Construction • Illustration Using Model Problem • Application to Hydrogen-Air Reactive System • Summary

Introduction Motivation and background • Detailed kinetics are essential for accurate modeling of real systems. • Reactive flow systems admit multi-scale solutions. • Severe stiffness arises in detailed gas-phase kinetics modeling. • Computational cost for reactive flow simulations increases with the spatio-temporal scales’ range, the number of species, and the number of reactions. • Manifold methods provide a potential for computational saving.

Partial review of manifold construction in reactive systems • ILDM, CSP , and ICE-PIC are approximations of the system’s slow invariant manifold. • MEPT, RCCE, and similar methods are based on minimizing a thermodynamic potential function. • Iterative methods may not converge. • Davis and Skodje, 1999, present a technique to construct the 1-D SIM based on global phase analysis. • Creta et al. and Giona et al. , 2006, extend the technique to slightly higher dimensional reactive systems.

Long-term objective Create an efficient algorithm that reduces the computational cost for simulating reactive flows based on a reduction in the stiffness and dimension of the composition phase space. Immediate objective Construct 1-D SIMs for dynamical system arising from modeling unsteady spatially homogenous closed reactive systems.

Slow Invariant Manifold (SIM) • The composition phase space for closed spatially homogeneous reactive system: d z z ∈ R 3 . dt = f ( z ) , Phase space Fast modes trajectory Phase space trajectory 1-D manifold Slow modes 2-D manifold 0-D manifold ( i.e. equilibrium point)

• An invariant manifold is defined as a subset S ⊂ R N − L − Q if for any solution z ( t ) , z ( t 0 ) ∈ S , implies that for any t f > t 0 , z ( t ) ∈ S for all t ∈ [ t 0 , t f ] . • Not all invariant manifolds are attracting. • SIMs describe the asymptotic structure of the invariant attracting trajectories. • Attractiveness of a SIM increases as the system’s stiffness in- creases. • On a SIM, only slow modes are active. • SIMs can be constructed by identifying all critical points, finite and infinite, and connecting relevant ones via heteroclinic orbits.

Mathematical Model For a mixture of mass m confined in volume V containing N species composed of L elements that undergo J reversible reactions, dn i dt = V ˙ ω i , i = 1 , . . . , N, where, N ! J N ” ν ′ ” ν ′′ “ n i ij − 1 “ n i X Y Y ij ω i ˙ = ν ij k j , i = 1 , . . . , N, K c V V j j =1 i =1 i =1 „ − E j « A j T β j exp = , j = 1 , . . . , J, k j ℜ T „ p o « P N ! i =1 ν ij P N µ o i =1 ¯ i ν ij K c = exp , j = 1 , . . . , J. − j ℜ T ℜ T

System reduction • In chemical reactions, the total number of moles of each element is conserved, N N � � φ li n ∗ i = φ li n i , l = 1 , . . . , L. i =1 i =1 • Additional Q constraints can arise in special cases. • The reactive system is recast as an autonomous dynamical system, dz i dt = f i ( z 1 , . . . , z N − L − Q ) , i = 1 , . . . , N − L − Q, where, R N → R N − L − Q � � L : � � z = L ( n ) .

Method of Construction • For isothermal reactive systems, reaction speeds depend on combinations of polynomials of z . • The set of equilibria of the full reaction network is complex: { z e ∈ C N − L − Q | f ( z e ) = 0 } . • The set consists of several different dimensional components and contains finite and infinite equilibria. • A 1-D SIM has a maximum of two branches that connect the unique physical critical point (a sink) to two equilibria. • These equilibria are identified by their special dynamical char- acter: their eigenvalue spectrum typically contains only one unstable direction.

Sketch of SIM construction R 3 SIM R 1 R 2

Projective space • One-to-one mapping of the composition space, R N − L − Q → R N − L − Q , 1 Z k = , k ∈ { 1 , . . . , N − L − Q } , z k z i Z i = , i � = k, i = 1 , . . . , N − L − Q. z k • This maps equilibria located at infinity into a finite domain. • To deal with the time singularity, we add the transformation dt dτ = ( Z k ) d − 1 , where d is the highest polynomial degree of f ( z ) .

Computational strategy • We use the Bertini software (based on a homotopy continu- ation numerical technique) to compute the system’s equilibria up to any desired accuracy. • Thermodynamic data is obtained from Chemkin-II . • The SIM heteroclinic orbits are obtained by numerical integration of the species evolution equations using a computationally inex- pensive scheme. • Computation time is typically less than 1 minute on a 2 . 16 GHz Mac Pro machine.

Simple Hydrogen-Oxygen Mechanism • The kinetic model is adopted from Michael, 1992, Prog. Energy Combust. Sci. 18 (4), p. 327. • The mechanism consists of J = 8 bimolecular elementary reactions involving N = 6 species { H, H 2 , O, O 2 , OH, H 2 O } and L = 2 elements { H, O } . In addition, since the total number of moles is constant, Q = 1 . Subsequently, z ∈ R 3 . • The system is spatially homogenous with isothermal and iso- choric conditions, T = 1200 K, V = 10 3 cm 3 . • Selected species are i = { 1 , 2 , 3 } = { H 2 , O, O 2 } . i = 10 − 3 mol . • Initial number of moles of all species are n ∗

Reactive system evolution -3 × 10 25 H [ ] 20 mol/g H O 2 O 2 15 n /m 10 i O 5 H 2 OH -10 -8 -6 -4 10 10 10 10 t s [ ]

Dynamical system dz 1  3 . 45 × 10 4 − 1 . 68 × 10 11 z 1 − 3 . 47 × 10 16 z 2 = 1  dt    − 1 . 35 × 10 10 z 2 + 6 . 27 × 10 16 z 1 z 2 + 1 . 40 × 10 10 z 3     +1 . 11 × 10 17 z 1 z 3 − 1 . 35 × 10 16 z 2 z 3 − 2 . 04 × 10 16 z 2  3 ,        dz 2  7 . 69 × 10 5 + 2 . 66 × 10 11 z 1 + 2 . 25 × 10 16 z 2  =  1  dt  ≡ f ( z ) . − 1 . 29 × 10 12 z 2 − 2 . 47 × 10 17 z 1 z 2  +3 . 91 × 10 17 z 2 2 − 8 . 66 × 10 11 z 3 − 1 . 51 × 10 17 z 1 z 3     7 . 49 × 10 17 z 2 z 3 + 2 . 46 × 10 17 z 2  3 ,        dz 3  6 . 84 × 10 11 z 2 + 1 . 37 × 10 17 z 1 z 2 − 2 . 74 × 10 17 z 2  =  2  dt   10 − 6 − z 1 + z 3  − 4 . 10 × 10 17 z 2 z 3 − 2 . 24 × 10 15 z 3 ` ´ , 

Finite equilibria R 1 ≡ ( z e 1 , z e 2 , z e − 5 . 84 × 10 − 2 , 6 . 85 × 10 − 4 , − 3 . 52 × 10 − 4 ´ ` 3 ) = mol/g, (5 . 93 × 10 6 ± i 5 . 10 × 10 5 , − 1 . 18 × 10 6 ) 1 /s, ( λ 1 , λ 2 , λ 3 ) = R 2 ≡ ( z e 1 , z e 2 , z e 4 . 65 × 10 − 2 , 0 , 3 . 49 × 10 − 2 ´ ` 3 ) = mol/g, ( − 1 . 01 × 10 7 , − 3 . 35 × 10 6 , 7 . 93 × 10 5 ) 1 /s, ( λ 1 , λ 2 , λ 3 ) = R 3 ≡ ( z e 1 , z e 2 , z e 3 . 73 × 10 − 3 , 6 . 32 × 10 − 3 , 1 . 61 × 10 − 2 ´ ` 3 ) = mol/g, ( − 1 . 02 × 10 7 , − 1 . 23 × 10 6 , − 4 . 30 × 10 5 ) 1 /s, ( λ 1 , λ 2 , λ 3 ) = R 4 ≡ ( z e 1 , z e 2 , z e 6 . 33 × 10 − 3 , − 1 . 86 × 10 − 3 , 2 . 49 × 10 − 2 ´ ` 3 ) = mol/g, (6 . 88 × 10 6 , 3 . 51 × 10 6 , 1 . 57 × 10 6 ) 1 /s, ( λ 1 , λ 2 , λ 3 ) = R 5 ≡ ( z e 1 , z e 2 , z e 1 . 28 × 10 − 3 , − 5 . 98 × 10 − 2 , 6 . 00 × 10 − 2 ´ ` 3 ) = mol/g, (5 . 65 × 10 7 , 3 . 56 × 10 6 , − 1 . 06 × 10 4 ) 1 /s, ( λ 1 , λ 2 , λ 3 ) = R 6 ≡ ( z e 1 , z e 2 , z e 1 . 43 × 10 − 3 , − 7 . 58 × 10 − 2 , 7 . 08 × 10 − 2 ´ ` 3 ) = mol/g, (7 . 19 × 10 7 , 4 . 47 × 10 6 , 1 . 05 × 10 4 ) 1 /s. ( λ 1 , λ 2 , λ 3 ) =

Infinite equilibria • Employ the projective space mapping with d = 2 and k = 2 : 0 1 0 1 Z − 1 t 2 B C B C Z 1 f 1 ( Z 1 , Z 2 , Z 3 ) − Z 1 f 2 ( Z 1 , Z 2 , Z 3 ) d B C B C = Z 2 ≡ F ( Z ) , 2 · B C B C dτ B C B C Z 2 − Z 2 f 2 ( Z 1 , Z 2 , Z 3 ) B C B C @ A @ A Z 3 f 3 ( Z 1 , Z 2 , Z 3 ) − Z 3 f 2 ( Z 1 , Z 2 , Z 3 ) I 1 ≡ ( Z e 1 , Z e 2 , Z e 3 ) = ( − 9 . 77 , 0 , − 4 . 59) , ( − 5 . 74 × 10 12 ± i 7 . 83 × 10 12 , 6 . 10 × 10 12 ) , ( λ 1 , λ 2 , λ 3 ) = I 2 ≡ ( Z e 1 , Z e 2 , Z e 3 ) = (0 . 60 , 0 , − 0 . 48) , ( − 1 . 19 × 10 13 , 7 . 35 × 10 11 , 6 . 32 × 10 11 ) , ( λ 1 , λ 2 , λ 3 ) = I 3 ≡ ( Z e 1 , Z e 2 , Z e 3 ) = ( − 0 . 01 , 0 , − 0 . 67) , ( − 1 . 12 × 10 13 , − 6 . 50 × 10 11 , 7 . 62 × 10 9 ) . ( λ 1 , λ 2 , λ 3 ) =

The system’s 1-D SIM S I M 0.03 R 2 [ ] z mol/g 0.02 R 3 0.01 3 0 0 0.01 z mol/g 2 -2 0.02 ×10 [ 0.03 0 0.5 1 1.5 2 2.5 3 SIM 3.5 ] 4 4.5 z mol/g [ ] 1 I 3