1. Dimension Reduction of Combustion Chemistry using Pre-Image Curves Zhuyin (laniu) Ren October 18 th, 2004

2. Background and Motivation Knowledge of detailed mechanism 50 – 1000 species in detailed description Continually increasing in accuracy and scope Use in computations of combustion DNS, LES, PDF and other approaches Need general methodology to Reduce the computational cost Retain accuracy and adequate detail Governing Equations Dimension Reduction

3. Dimension Reduction (Time scales in Chemical Kinetics) (Maas & Pope 1992) Initial point Trajectory Equilibrium point

4. Dimension Reduction (Assumption) The very fast time scales in chemical kinetics correspond to equilibrium processes With a time scale of the order of the physical time scales, all the compositions in a chemically reacting flow will lie on a low-dimensional attracting manifold in the full composition space

5. Dimension Reduction (Approach) Represent combustion chemistry in terms of reduced composition r (n r ) instead of the full composition φ (n φ ) Impose n u = n φ -n r conditions which determine the manifold φ m ; ----i.e., given a reduced composition r, provide a procedure to determine the corresponding full composition on the manifold φ m (Species reconstruction) Assume the existence of a low-dimensional attracting manifold in the full composition space

6. Dimension Reduction (Geometric Picture) Reduced composition r={r 1, r 2,…, r nr } (n r < n Φ ) given by the reduction process: r=B T φ Represented subspace B : the subspace spanned by the columns of B; Unrepresented subspace U = B┴ Feasible region F (r): the union of all realizable, feasible compositions (satisfying B T φ =r ) Species reconstruction is to select from the feasible region the particular composition which is deemed to be most likely to occur in a reactive flow

7. Dimension Reduction (Geometric Picture)

8. Quasi-steady state assumptions (QSSA) Each column of the specified n φ ×n r matrix B corresponds to the unit vector in the direction of one of the slow species (major species) Assume n u species (associated with fast processes) are in steady state with their net chemical production rates being set to zero Global in composition space. And QSSA assumption is poor in some region of the composition space Smoothness? hard to choose the QSSA species

9. Intrinsic low-dimensional manifolds (ILDM) Let The construction of the manifold is independent of matrix B The fast subspace varies in the full composition space With finite scale separation, the ILDM approximate the slow attracting manifold with first order of accuracy O(τ nr+1 /τ nr ) Existence? Smoothness? hard to parameterize

10. Rate-Controlled Constrained-Equilibrium (RCCE) Assume the complex chemical system evolves through a sequence of constrained-equilibrium states, determined by the instantaneous values of n r constraints r imposed by slow rate-limiting reactions B matrix (species, element and general linear constraints on species) Good mathematical properties RCCE relies on the time scale separations. But it is based on thermodynamics. Hard to choose the constraint matrix B

11. Pre-Image Curves (Ideas) Use the fact that trajectories will be attracted to the low dimension attracting manifold Identify the corresponding composition point at the attracting manifold as the reconstructed composition. (Identify the attracting manifold) The reconstructed composition (manifold construction) is independent of the matrix B Give the reduced composition r, construct a curve (Pre-image curve) in the full composition space (the trajectories starting from this curve will have the same reduced composition at some positive time)

12. Pre-Image Curves (1) For the reaction fractional step, homogenous, adiabatic, isobaric system; n s species, full composition φ (t)={ φ 1, φ 2,…, φ n φ } (species specific moles and enthalpy, so n φ = n s +1) Reaction mapping R( φ, t): solution to governing ODE after time t, starting from the initial condition φ Pre-image point of r: a composition φ satisfying B T R( φ, t) =r for some positive t given a reduced composition r Pre-image manifold of r, M P (r): the union of all pre-image points of r, (n φ – n r +1)-dimensional inertial manifold

13. Pre-Image Curves (2) Sketch of reaction trajectories in the pre-image manifold M P. Assumption: there is an attracting manifold (black line) Ideally, species reconstruction should identify point “A” A good approximation to point “A” can being obtained by following the reaction trajectory from a point such as “I ” A suitable initial point “I ” is achieved by generating a curve C in the pre-image manifold from a starting feasible point, denoted by “O” How to generate the Pre-Image Curves?

14. Methods to generate Pre-Image Curves Minimum Curvature Pre-Image Curves (MCPIC) (Implemented) Attracting Manifold Pre-Image Curves (AMPIC) (In progress)

15. Demonstration of Minimum Curvature Pre-image Curves Autoignition of methane GRI 1.2 (4 elements, 31 species and 175 reactions) Adiabatic, isobaric and mass fractions of the 4 elements remained fixed, so composition has 31-4=27 degrees of freedom during the autoignition process. T ini =1500K; N 2 (71.5), O 2 (19), CH 4 (9.5), CO 2 (3), H 2 O(2) in relative volume units; atmospheric pressure throughout. Given B, the reduced composition along the trajectory is r=B T φ DI For every r, species reconstruction using Pre-Image Curves reconstructs the full compositionφ R (r) Compareφ R (r) with the corresponding accurate result φ DI

16. Minimum Curvature Pre-image Curves Performance - Comparison with QSSA and RCCE QSSA: Q 10, Q 12 RCCE: R 4, R 6 Pre-image curve: B 4, B 6 Normalized errors in Pre-Image Curve are less than those in RCCE and QSSA Normalized error in reconstructed composition at different temperatures during autoignition.

17. Minimum Curvature Pre-image Curves Performance T DI =1852.6K; r=B T φ DI Solid red : B 6 Dashed red: B 4 Blue: DI The compositionφ M (s) (mapped from composition along Pre-Image curve) approaches an asymptote. φ R is taken to be this asymptote φ M (s) converges to DI results φ DI.

18. Minimum Curvature Pre-image Curves Performance -inertial property Angle between the reaction rate S( Φ R ) and the tangent space of the manifold M R. The reconstructed manifold M R is inertial (to a good approximation)

19. Construction of the attracting-manifold pre-image curve Identification of the tangent plane of the Pre-image manifold Identification of the “maximally compressive” subspace

20. The sensitivity matrix is defined as Construction of the attracting-manifold pre-image curve -----“maximally compressive” subspace The initial infinitesimal ball is mapped to an ellipsoid The initial ball is squashed to a low dimensional object, and this low dimensional object aligns with the attracting manifold The “maximally compressive” subspace of the initial ball is that spanned by the last n u =n φ -n r columns of V A The “maximally compressive” subspace corresponds to the local fast subspace at the initial point

21. Construction of the attracting-manifold pre-image curve -----Tangent space of the pre-image manifold (1) 1) 2) 3) 4) 5) Let Eq. 2) becomes

22. Construction of the attracting-manifold pre-image curve -----Tangent space of the pre-image manifold (2) Thus the columns of X are orthonormal tangent vectors of the pre-image manifold. The final tangent vector is determined by The set of n u +1 vectors [ X w] forms an orthonormal basis for the tangent space of the pre-image manifold W T X is zero, 5) 6)

23. Construction of the attracting-manifold pre-image curve (method 2) FF T S is the component of S in the “maximally compressive” directions. XX T (FF T )S is projection in the n u -dimensionalτ=const. tangent space Finally the governing equation is Therefore

24. Demonstration of the attracting-manifold pre-image curve

25. Future Work Investigate and implement the above new methods of generating pre-image curves, and automatic ways to determine optimal choice of B Investigate the boundary region, and cold temperature region A computationally-efficient implementation of the new method will be combined with ISAT for application to the simulation of turbulent combustion.