Model reduction for reacting flow applications

A model reduction approach based on Galerkin projection, proper orthogonal decomposition (POD), and the discrete empirical interpolation method (DEIM) is developed for chemically reacting flow applications. Such applications are challenging for model reduction due to the strong coupling between fluid dynamics and chemical kinetics, a wide range of temporal and spatial scales, highly nonlinear chemical kinetics, and long simulation run-times. In our approach, the POD technique combined with Galerkin projection reduces the dimension of the state (unknown chemical concentrations over the spatial domain), while the DEIM approximates the nonlinear chemical source term. The combined method provides an efficient offline–online solution strategy that enables rapid solution of the reduced-order models. Application of the approach to an ignition model of a premixed H2/O2/Ar mixture with 19 reversible chemical reactions and 9 species leads to reduced-order models with state dimension several orders of magnitude smaller than the original system. For example, a reduced-order model with state dimension of 60 accurately approximates a full model with a dimension of 91,809. This accelerates the simulation of the chemical kinetics by more than two orders of magnitude. When combined with the full-order flow solver, this results in a reduction of the overall computational time by a factor of approximately 10. The reduced-order models are used to analyse the sensitivity of outputs of interest with respect to uncertain input parameters describing the reaction kinetics.

[1]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[2]  T. Poinsot.  Boundary conditions for direct simulations of compressible viscous flows , 1992 .

[3]  Ulrich Maas,et al.  Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space , 1992 .

[4]  Juan Du,et al.  Non-linear model reduction for the Navier-Stokes equations using residual DEIM method , 2014, J. Comput. Phys..

[5]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[6]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[7]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[8]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[9]  Paul Van Dooren,et al.  Model Reduction of Large-Scale Systems Rational Krylov Versus Balancing Techniques , 1999 .

[10]  Yiqing Shen,et al.  High order conservative differencing for viscous terms and the application to vortex-induced vibration flows , 2008, J. Comput. Phys..

[11]  Razvan Stefanescu,et al.  POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model , 2012, J. Comput. Phys..

[12]  A. F. Adams,et al.  The Survey , 2021, Dyslexia in Higher Education.

[13]  David J. Lucia,et al.  Reduced order modeling of a two-dimensional flow with moving shocks , 2003 .

[14]  Nancy J. Brown,et al.  Mechanism reduction via principal component analysis , 1997 .

[15]  J. Boris,et al.  A numerical technique for solving stiff ordinary differential equations associated with the chemical kinetics of reactive-flow problems , 1977 .

[16]  T. Young,et al.  CHEMEQ - A Subroutine for Solving Stiff Ordinary Differential Equations , 1980 .

[17]  M. W. Chase.  NIST-JANAF thermochemical tables , 1998 .

[18]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[19]  Peter Orbanz.  Probability Theory II , 2011 .

[20]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[21]  Gregory J. Wilson,et al.  Modeling supersonic combustion using a fully-implicit numerical method , 1990 .

[22]  Ulrich Maas,et al.  Laminar flame calculations using simplified chemical kinetics based on intrinsic low-dimensional manifolds , 1994 .

[23]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics ∗ , 1992 .

[24]  Luc Vervisch,et al.  Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows , 2008, J. Comput. Phys..

[25]  Dimitris A. Goussis,et al.  Asymptotic Solution of Stiff PDEs with the CSP Method: The Reaction Diffusion Equation , 1998, SIAM J. Sci. Comput..

[26]  J. D. Ramshaw.  Partial chemical equilibrium in fluid dynamics , 1980 .

[27]  Ulrich Maas,et al.  Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds , 1992 .

[28]  Siep Weiland,et al.  Missing Point Estimation in Models Described by Proper Orthogonal Decomposition , 2004, IEEE Transactions on Automatic Control.

[29]  Marco Amabili,et al.  Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method , 2006 .

[30]  Danny C. Sorensen,et al.  The Sylvester equation and approximate balanced reduction , 2002 .

[31]  S. Lam,et al.  The CSP method for simplifying kinetics , 1994 .

[32]  Gecheng Zha,et al.  Implicit WENO Scheme and High Order Viscous Formulas for Compressible Flows , 2007 .

[33]  Robert Maccormack.  Modeling supersonic combustion using a fully-implicit numerical method , 1990 .

[34]  L. Sirovich.  Turbulence and the dynamics of coherent structures. III. Dynamics and scaling , 1987 .

[35]  Roland W. Freund,et al.  Efficient linear circuit analysis by Pade´ approximation via the Lanczos process , 1994, EURO-DAC '94.

[36]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[37]  T. Poinsot,et al.  Theoretical and numerical combustion , 2001 .

[38]  D. Sorensen,et al.  A Survey of Model Reduction Methods for Large-Scale Systems , 2000 .

[39]  E. Grimme,et al.  Pade approximation of large-scale dynamic systems with Lanczos methods , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[40]  Forman A. Williams,et al.  The asymptotic structure of stoichiometric methaneair flames , 1987 .

[41]  Lawrence Sirovich,et al.  Karhunen–Loève procedure for gappy data , 1995 .

[42]  Maria Vittoria Salvetti,et al.  Low-dimensional modelling of a confined three-dimensional wake flow , 2006, Journal of Fluid Mechanics.

[43]  George Em Karniadakis,et al.  A low-dimensional model for simulating three-dimensional cylinder flow , 2002, Journal of Fluid Mechanics.

[44]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[45]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[46]  Sanford Gordon,et al.  Computer program for calculation of complex chemical equilibrium compositions , 1972 .

[47]  Michel Loève,et al.  Probability Theory I , 1977 .

[48]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics , 1993 .

[49]  D. A. Lavis.  A Survey of Models , 2015 .