1. Dimension Reduction of Combustion Chemistry Zhuyin (Laniu) Ren S. B. Pope Mar. 29 th, 2005

2. Outline Motivation Advantage of dimension reduction combined with ISAT in CFD simulation Dimension reduction of combustion chemistry Pre-image curve method

3. Motivation Knowledge of detailed mechanism Continually increasing in accuracy and scope 50 – 1000 species in detailed description Use in computations of combustion DNS, LES, PDF and other approaches Need general methodology to Reduce the computational cost Retain accuracy and adequate detail Reduce the computational cost by using dimension reduction combined with ISAT

4. CFD/Full Chemistry/Direct Integration Schematic of DI with full representation used in the reaction fractional step of a CFD computation. Computationally prohibitive with large detailed mechanism (Both CPU/Memory)

5. CFD/ Full Chemistry/ISAT Schematic of ISAT with full representation used in the reaction fractional step of a CFD computation. Computationally much less expensive (100-1000 fast than previous approach) Memory issue!

6. CFD/Dimension Reduction (via Species Reconstruction)/ISAT Schematic of ISAT with dimension reduction ( via species reduction and reconstruction ) used in the reaction fractional step of a CFD computation. Use only the major species both in CFD and ISAT (Both speed up the CFD part and ISAT retrieval; solve memory issue ) ISAT allows more expensive procedure to do the dimension reduction Error in dimension reduction (in species reconstruction part) is crucial to this methodology. Develop more accurate method to do the dimension reduction!!

7. Represent combustion chemistry in terms of reduced composition r (n r ) (e.g. major species ) instead of the full composition φ(n φ ) (e.g. full species) Dimension Reduction of combustion chemistry 1: Unimportant species and reactions in the detailed mechanism 2: Large range of chemical time scales

8. Dimension Reduction of combustion chemistry PRF: Primary Reference Fuel; mixture of Heptanes and Octane ---1034 species; 4236 reactions Great potential for dimension reduction --1d-5 sec --1d-6 sec --1d-7 sec

9. Existing Dimension Reduction methods Skeletal mechanism  1 Quasi-steady state assumptions (QSSA)  2 Intrinsic low-dimensional manifolds (ILDM)  2 Rate-controlled constrained equilibrium (RCCE)  2 Trajectory-generated low-dimensional manifold  2 Flamelet generated manifolds (FGM)  2 Roussel & Fraser algorithm (RF)  2 Pre-Image Curves  2 General procedure of dimension reduction of combustion chemistry: Step 1: Detailed mechanism => Skeletal mechanism (Small detailed mechanism) Step 2: Further reduction based on time scale analysis

10. Evaluation of existing methods X Quasi-steady state assumptions (QSSA) ---difficult to choose QSSA species X Intrinsic low-dimensional manifolds (ILDM) X Rate-controlled constrained equilibrium (RCCE) ---difficult to choose constraints X Trajectory-generated low-dimensional manifold X Flamelet generated manifolds (FGM) X Roussel & Fraser algorithm (RF) Pre-Image Curves ---Easy to choose the reduced composition while maintain the accuracy ---Reconstructed composition is independent of the reduced composition, fully determined by the chemical kinetics Accuracy (very important) Easy to generate and use

11. Dimension Reduction (Geometric Picture) 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)

12. 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 reduced composition r 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)

13. Pre-Image Curves 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. Pre-Image Curves (Current Status) Have identified the tangent space of the pre-image manifold Have formulated different ways to construct the pre-image curves Have understood the method itself to some extent Plan to do a PaSR simulation to solve the embarrassment

15. Dimension reduction by Pre-Image Curves method Trajectories colored by dimensionReconstructed 1-dimension manifold (blue dashed line)

16. 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 species at different temperatures during autoignition.

17. Conclusion Dimension reduction of combustion chemistry is crucial and promising in numerical simulation of combustion process with complicated fuels. Developing new method with great accuracy Combining dimension reduction with ISAT in simulations