Coping with complexity: Model Reduction for the Simulation of Turbulent Reacting flows

1. Coping with complexity: Model Reduction for the Simulation of Turbulent Reacting flows V. Bykov, U. Maas (Karlsruhe Institute of Technology) V. Goldsh‘tein (Ben Gurion University)

2. Introduction Manifold-Based Concepts for Model Reduction Dimension reduction for reaction/diffusion systems Implementation Conclusions Overview

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! chemistry convection transport Conservation Equations

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. detailed chemistry equation for the scalar field comprises ns + 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, ci all species concentrations and the temperature are known as funcions of these variables Points of View

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

7. chemistry convection transport diffusion-convection equation for “quasi conserved” variables evolution along the LDM ILDM-equations Decomposition of Motions Decomposition into “very slow, intermediate and fast subspaces”

8. Low-Dimensional Manifold Concepts system equation manifold equation 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

9. Reduction - decomposition of motions the system is transformed into fast/slow subsystems slow subsystem: fast subsystem: Projection of the state space of the CO-H2-O2 system

10. GQL application 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

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

12. Evolution of a manifold according to reaction and diffusion Reaction-Diffusion-Manifolds (REDIM) (Bykov & Maas 2007) KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

13. equilibrium curve mixing line Principle of the Evolution equation

14. equilibrium curve mixing line Principle of the Evolution equation

15. 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 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

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

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 Estimation of the gradient KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

18. 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) KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

19. 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) KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

20. 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) KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

21. 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 KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universität Karlsruhe (TH)

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

23. reduced states ILDM GQL REDIM CFD-code reduced variables mass momentum reaction transport energy interpolation Implementation

24. Example: LES of a premixed flame 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: ZH=0.7. An event of local extinction is seen around x/R=8, r/R=1. 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

25. 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. Conclusions