1. KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association Institut für Technische Thermodynamik www.kit.edu Coping with complexity: Model Reduction for the Simulation of Turbulent Reacting flows V. Bykov, U. Maas (Karlsruhe Institute of Technology) V. Goldshtein (Ben Gurion University)

2. Institut für Technische Thermodynamik 2 Overview Introduction Manifold-Based Concepts for Model Reduction Dimension reduction for reaction/diffusion systems ImplementationConclusions

3. Institut für Technische Thermodynamik 3 equation for the scalar field filtered or averaged Problems: extremely high dimension of the system! non-linear chemical source terms strong coupling of chemistry with molecular transport stiffness of the governing equation system On which level of accuracy does this equation system have to be solved? Reduce the dimension of the governing equation system! Note: Chemistry has to be analyzed in the context of a reacting flow! convectionchemistrytransport Conservation Equations

4. Institut für Technische Thermodynamik 4 describe temporal evolution of the species concentrations in chemical reactions needed for modeling reacting flows species conservation equations averaged species conservation equations FDF/PDF-transport equation source terms are functions of the thermokinetic state concept of elementary reactions Chemical Source Terms

5. Institut für Technische Thermodynamik 5 Points of View detailed chemistry equation for the scalar field comprises n s + 2 equations Warnatz, Maas, Dibble: Combustion 2001 detailed and accurate, but enormous computational effort enormous amount of unimportant information infinitely fast chemistry equation for the scalar field reduces to an equation system for h, p, c i all species concentrations and the temperature are known as funcions of these variables

6. Institut für Technische Thermodynamik 6 Stiff chemical kinetics as well as molecular transport processes cause the existence of attractors in composition space ILDMs of higher hydrocarbons (Maas & Pope 1992, Blasenbrey & Maas 2000) Correlation analysis of DNS-Data (Maas & Thevenin 1998) Observation:

7. Institut für Technische Thermodynamik 7 Decomposition of Motions Decomposition into very slow, intermediate and fast subspaces convectionchemistrytransport diffusion-convection equation for quasi conserved variables evolution along the LDM ILDM-equations

8. Institut für Technische Thermodynamik 8 Low-Dimensional Manifold Concepts QSSA (Bodenstein 1913) Set right hand side for qss species to zero ILDM (Maas & Pope 1992) Use eigenspace decomposition of Jacobian GQL (Bykov et al. 2007) Use eigenspace decomposition of global quasilinearization matrix system equationmanifold equation

9. Institut für Technische Thermodynamik 9 the system is transformed into fast/slow subsystems fast subsystem: slow subsystem: Projection of the state space of the CO-H 2 -O 2 system Reduction - decomposition of motions

10. Institut für Technische Thermodynamik 10 red mesh: ILDM, green mesh: manifold, symbols: reference points blue curve: detailed system solution, cyan curve: fast subsystem solution magenta curves: detailed stationary system solution of flat flames Bykov, Goldshtein, Maas 2007 GQL application

11. Institut für Technische Thermodynamik 11 Red curve: detailed solution green mesh: 2D GQL manifold red cubes: reference set, Spheres: reduced solution GQL for an Ignition Problem Temperature dependence of the ignition delay time Circles: reduced model (m s = 14) red dashed curve: detailed model (m d =31)

12. Institut für Technische Thermodynamik 12 REDIMs for Le = 1 and equal diffusivities Bykov and Maas 1997 Stationary solution gives the invariant manifold, is an estimate for grad Stationary solution gives the invariant manifold, is an estimate for grad KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) Evolution of a manifold according to reaction and diffusion Reaction-Diffusion-Manifolds (REDIM) (Bykov & Maas 2007)

13. Institut für Technische Thermodynamik 13 Principle of the Evolution equation mixing lineequilibrium curve

14. Institut für Technische Thermodynamik 14 Principle of the Evolution equation mixing line equilibrium curve

15. Institut für Technische Thermodynamik 15 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) Evolution equation for the manifold Basic Procedure: formulate initial guess specify boundary conditions estimate the gradient solve the evolution equation (PDE) Extension to detailed transport

16. Institut für Technische Thermodynamik 16 Comparison ILDM-REDIM Premixed syngas/air system Left: red mesh: ILDM, green mesh: REDIM Right: reaction rate of CO2, mesh: domain of existence of the 2D ILDM

17. Institut für Technische Thermodynamik 17 It has been shown (Bykov & Maas 2007) that a good estimate gets more and more unimportant for increasing dimension In this work: use gradients from typical flamelets KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) Estimation of the gradient

18. Institut für Technische Thermodynamik 18 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) Results: Non-Premixed Syngas Flame symbols: reduced solution; curves: detailed solution green:Le=1, equal diffusivities blue:detailed transport, no thermal diffusion red:detailed transport very good gradient estimates used from flamelets (cf. Bykov & Maas 2008)

19. Institut für Technische Thermodynamik 19 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) symbols: reduced solution; curves: detailed solution green:Le=1, equal diffusivities blue:detailed transport, no thermal diffusion red:detailed transport very good gradient estimates used from flamelets (cf. Bykov & Maas 2008) Results: Stoichiometric Premixed Syngas Flame

20. Institut für Technische Thermodynamik 20 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) Results: Stoichiometric Premixed Syngas Flame symbols: reduced solution; curves: detailed solution green:Le=1, equal diffusivities blue:detailed transport, no thermal diffusion red:detailed transport very good gradient estimates used from flamelets (cf. Bykov & Maas 2008)

21. Institut für Technische Thermodynamik 21 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH) 2-D Manifold for a Non-Premixed Syngas Flame stoichiometric syngas-air flat flame, detailed transport curves: detailed solution, mesh: REDIM Left: starting guess (linear interpolation between flamelets) Right: REDIM

22. Institut für Technische Thermodynamik 22 Attracting Properties of the REDIM 2D REDIM (mesh) and convergence of an unsteady flame (cyan lines) towards the REDIM For simpolicity: use visualization to monitor the movement towards the manifold.

23. Institut für Technische Thermodynamik 23 Implementation ILDM GQL REDIM interpolation mass momentum energy reduced variables reactiontransport CFD-code reduced states

24. Institut für Technische Thermodynamik 24 Example: LES of a premixed flame Large eddy simulation and experimental studies of turbulent premixed combustion near extinction P. Wang, F. Zieker, R. Schießl, N. Platova, J. Fröhlich, U. Maas European Combustion Meeting 2011 Scatter plot of temperature vs. hydrogen mass fraction. = 0.71 at one time step, calculated from LES resolved values. Instantaneous contours of temperature, red line: Z H =0.7. An event of local extinction is seen around x/R=8, r/R=1.

25. Institut für Technische Thermodynamik 25 Conclusions Efficient methods for kinetic model reduction and its subsequent implementation in reacting flow calculations have been presented. GQL and ILDM allow an efficient decoupling of fast chemical processes The slow chemistry domain can be treated efficiently by the REDIM (REaction-DIffusion-Manifold, REduction of the DIMension)- method. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.