Zhuyin (laniu) Ren Feb. 1 th , 2005

1. Species reconstruction by attracting-manifold pre-image curve method Zhuyin (laniu) Ren Feb. 1th, 2005

2. Dimension Reduction of combustion chemistry • Dimension Reduction: Represent combustion chemistry in terms of reduced composition r (nr) instead of the full compositionφ(nφ) • Species reconstruction: Given a reduced composition r, provide a procedure to determine the corresponding full composition on the manifold φm

3. Dimension Reduction (Geometric Picture) Reduced composition r={r1, r2,…, rnr} (nr < nΦ) given by the reduction process: r=BTφ Represented subspace B: the subspace spanned by the columns of B; Unrepresented subspace U= B┴ • Feasible region F(r): the union of all realizable, feasiblecompositions (satisfyingBTφ =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

4. 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)

5. Pre-Image Curves (2) • Assumption: there is an attracting manifold (black line) • Ideally, speciesreconstruction 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” Sketch of reaction trajectories in the pre-image manifoldMP. How to generate the Pre-Image Curves?

6. 2) 3) Let 4) Eq. 2) becomes 5) Construction of the attracting-manifold pre-image curve -----Tangent space of the pre-image manifold (1) 1)

7. 5) WTX is zero, 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 6) The set of nu+1 vectors [ X w] forms an orthonormal basis for the tangent space of the pre-image manifold

8. Construction of the attracting-manifold pre-image curve -----“maximally compressive” subspace The sensitivity matrix is defined as • 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 VC • The “maximally compressive” subspace corresponds to the local fast subspace at the initial point

9. Construction of the attracting-manifold pre-image curve 1) 2) αXXT(VCVCT)S, where VCVCTS is the component of S in the “maximally compressive directions Therefore Where dτ and dα are two adjustable Parameters and dα= α dτ .

10. Demonstration H2-O2 system Full composition space -10 dimensional (9 species + Enthalpy). Given fixed elements and enthalpy, the system has 6 degree of freedom.

11. Effects of the two terms in the Pre-image construction Reduced composition : H2O+ 3 Elements +Enthalpy.

12. Auto ignition test –case 1 Reduced composition : H2O+ 3 Elements +Enthalpy. α=0 α=1 Effect of parameter α α=10

13. Reduced composition : H2O+ 3 Elements +Enthalpy. Auto ignition test –case 1

14. Auto ignition test –case 1 Reduced composition : H2O+ 3 Elements +Enthalpy. Reduced composition : OH+ 3 Elements +Enthalpy.

15. Laminar flame test –case 1 Mixing process; Mixing time scale;

16. Laminar flame test –case 1 Reduced composition : H2O+ 3 Elements +Enthalpy.

17. Laminar flame test –case 2 Reduced composition : ?+ 3 Elements +Enthalpy.