Zhuyin (Laniu) Ren Steve Pope June 24 th , 2005

1. Dimension Reduction of Combustion Chemistry Species reconstruction by Constrained Equilibrium Pre-Image Curve method (CE-PIC) Zhuyin (Laniu) Ren Steve Pope June 24th, 2005

2. Outline • Background and Introduction • CE-PIC method • Results

3. Dimension Reduction of combustion chemistry • In combustion, the detailed chemical mechanism involves 50 – 1,000 species and 100-8,000 reactions • In a numerical simulation, one often considers the following ODE system • Dimension Reduction: Represent combustion chemistry in terms of nrreduced compositions r instead of by the nφ full compositionφ • Species reconstruction: Given a reduced composition r, provide a procedure to determine the corresponding full composition on the manifold φm

4. Demonstration of CE-PIC method • (H2/O system): • Three Elements: H, O, N • Six Species : H, O, H2, OH, H2O, N2 • (Note: no O2) • Fixed T, p; Fixed elements • 3 degrees of freedom • Choose H2 and O as the represented species

5. CE-PIC Method: Definition of the manifold • Invariant manifold with constrained-equilibrium edge • Reduced variables r; mapping from full composition • Realizable region in r; convex polytope • Consider point on boundary of realizable region in r • Feasible region in full space; mapped to boundary point • Constrained equilibrium composition; maximum entropy • Manifold: generated by trajectories from the constrained-equilibrium composition corresponding to each boundary point in r

6. 1) CEM manifold

7. 2) CEM Boundary / Trajectory generated manifold:invariant manifold with constrained equilibrium edge From CEM boundary, we can construct a trajectory generated manifold

8. Criteria for dimension-reduction manifolds • Existence, uniqueness and smoothness • Ease of parameterization / independence of parameterization • Attracting • Invariance • Realizability and consistency of conservation laws • Nested hierarchy • Ease of identification??

9. CE-PIC method • How to perform species reconstruction? • Given a reduced composition r, how to identify the corresponding point on the manifold • Consider: • The pre-image manifold (for the given r) • The constrained equilibrium manifold (for all r) • Their intersection – the constrained equilibrium pre-image curve • Follow this pre-image curve to the boundary to identify the constrained-equilibrium pre-image point on the boundary • From this boundary point, follow a trajectory forward until it intersects the feasible region

10. 3) CE-PIC method • Given a reduced composition, we construct a pre-image curve, which is the intersection between the pre-image manifold and CEM manifold. The reconstructed point is taken to be the intersection between the feasible region and the trajectory, which is from the ending point of the pre-image curve (it is on the CEM boundary)

11. 3) CE-PIC method

12. Summary • Identified an ideal manifold which is the trajectory generated manifold from the CEM boundaryInvariant manifold with constrained equilibrium edgeConstruct the CE-PIC method to perform the species reconstruction locally=> Work on the numerical implementation