A high-order numerical algorithm for DNS of low-Mach-number reactive flows with detailed chemistry and quasi-spectral accuracy

A novel and efficient algorithm is presented in this paper to deal with DNS of turbulent reacting flows under the low-Mach-number assumption, with detailed chemistry and a quasi-spectral accuracy. The temporal integration of the equations relies on an operating-split strategy, where chemical reactions are solved implicitly with a stiff solver and the convection-diffusion operators are solved with a Runge-Kutta-Chebyshev method. The spatial discretisation is performed with high-order compact schemes, and a FFT based constant-coefficient spectral solver is employed to solve a variable-coefficient Poisson equation. The numerical implementation takes advantage of the 2DECOMP&FFT libraries developed by 1, which are based on a pencil decomposition method of the domain and are proven to be computationally very efficient. An enhanced pressure-correction method is proposed to speed up the achievement of machine precision accuracy. It is demonstrated that a second-order accuracy is reached in time, while the spatial accuracy ranges from fourth-order to sixth-order depending on the set of imposed boundary conditions. The software developed to implement the present algorithm is called HOLOMAC, and its numerical efficiency opens the way to deal with DNS of reacting flows to understand complex turbulent and chemical phenomena in flames.

[1]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[2]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[3]  I. Orlanski.  A Simple Boundary Condition for Unbounded Hyperbolic Flows , 1976 .

[4]  G. Volpe.  Performance of compressible flow codes at low Mach numbers , 1993 .

[5]  Willem Hundsdorfer,et al.  RKC time-stepping for advection-diffusion-reaction problems , 2004 .

[6]  J. Shuen,et al.  A coupled implicit method for chemical non-equilibrium flows at all speeds , 1993 .

[7]  Danesh K. Tafti,et al.  A time-accurate variable property algorithm for calculating flows with large temperature variations , 2008 .

[8]  P. S. Wyckoff,et al.  A Semi-implicit Numerical Scheme for Reacting Flow , 1998 .

[9]  Franck Nicoud,et al.  Conservative High-Order Finite-Difference Schemes for Low-Mach Number Flows , 2000 .

[10]  Ning Li,et al.  Incompact3d: A powerful tool to tackle turbulence problems with up to O(105) computational cores , 2011 .

[11]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[12]  Pierre Degond,et al.  An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations , 2011, J. Comput. Phys..

[13]  Sylvain Laizet,et al.  High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy , 2009, J. Comput. Phys..

[14]  Dominique Thévenin,et al.  Comparison of direct numerical simulations of turbulent flames using compressible or low‐Mach number formulations , 2002 .

[15]  Miguel R. Visbal,et al.  High-Order Schemes for Navier-Stokes Equations: Algorithm and Implementation Into FDL3DI , 1998 .

[16]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[17]  Dohyung Lee.  The Design of Local Navier-Stokes Preconditioning for Compressible Flow , 1998 .

[18]  T. Magin,et al.  Transport algorithms for partially ionized and unmagnetized plasmas , 2004 .

[19]  Alexandre Ern,et al.  Fast and accurate multicomponent transport property evaluation , 1995 .

[20]  Miltiadis V. Papalexandris,et al.  Time-accurate calculation of variable density flows with strong temperature gradients and combustion , 2006, J. Comput. Phys..

[21]  Cosmin Safta,et al.  A high-order low-Mach number AMR construction for chemically reacting flows , 2010, J. Comput. Phys..

[22]  P. Wesseling,et al.  A conservative pressure-correction method for flow at all speeds , 2003 .

[23]  P. Gresho.  Incompressible Fluid Dynamics: Some Fundamental Formulation Issues , 1991 .

[24]  T. Poinsot,et al.  A two-step chemical scheme for kerosene–air premixed flames , 2010 .

[25]  James J. Riley,et al.  Direct numerical simulations of a reacting mixing layer with chemical heat release , 1985 .

[26]  C. F. Curtiss,et al.  Molecular Theory Of Gases And Liquids , 1954 .

[27]  N. Peters,et al.  Temperature cross-over and non-thermal runaway at two-stage ignition of n-heptane , 2002 .

[28]  Tianfeng Lu,et al.  Structure of a spatially developing turbulent lean methane–air Bunsen flame , 2007 .

[29]  F. Nicoud.  Numerical study of a channel flow with variable properties , 2022 .

[30]  R. J. Kee,et al.  Chemkin-II : A Fortran Chemical Kinetics Package for the Analysis of Gas Phase Chemical Kinetics , 1991 .

[31]  Xue-Song Bai,et al.  An improved high-order scheme for DNS of low Mach number turbulent reacting flows based on stiff chemistry solver , 2012, J. Comput. Phys..

[32]  Habib N. Najm,et al.  Regular Article: A Semi-implicit Numerical Scheme for Reacting Flow , 1999 .

[33]  Heinz Pitsch,et al.  High order conservative finite difference scheme for variable density low Mach number turbulent flows , 2007, J. Comput. Phys..

[34]  William H. Raymond,et al.  A radiation boundary condition for multi‐dimensional flows , 1984 .

[35]  M S Day,et al.  Numerical simulation of laminar reacting flows with complex chemistry , 2000 .

[36]  Andrew W. Cook,et al.  Direct Numerical Simulation of a Turbulent Reactive Plume on a Parallel Computer , 1996 .

[37]  F. Ducros,et al.  A thickened flame model for large eddy simulations of turbulent premixed combustion , 2000 .

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

[39]  Lars-Erik Eriksson,et al.  Acoustic source terms for the linearized euler equations in conservative form , 2005 .

[40]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[41]  G. D. Byrne,et al.  VODE: a variable-coefficient ODE solver , 1989 .

[42]  Marc Massot,et al.  Time–space adaptive numerical methods for the simulation of combustion fronts , 2013 .

[43]  J. Abraham,et al.  Influence of heat release and turbulence on scalar dissipation rate in autoigniting n-heptane/air mixtures , 2012 .

[44]  Krishnan Mahesh,et al.  A numerical method for DNS/LES of turbulent reacting flows , 2007, J. Comput. Phys..

[45]  H. Pitsch.  LARGE-EDDY SIMULATION OF TURBULENT COMBUSTION , 2006 .

[46]  Thierry Poinsot,et al.  Large Eddy Simulations of gaseous flames in gas turbine combustion chambers , 2012 .

[47]  R. Knikker.  A comparative study of high‐order variable‐property segregated algorithms for unsteady low Mach number flows , 2011 .

[48]  Robert J. Kee,et al.  CHEMKIN-III: A FORTRAN chemical kinetics package for the analysis of gas-phase chemical and plasma kinetics , 1996 .

[49]  Antonino Ferrante,et al.  A fast pressure-correction method for incompressible two-fluid flows , 2014, J. Comput. Phys..

[50]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[51]  C. Merkle,et al.  Dual time-stepping and preconditioning for unsteady computations , 1995 .

[52]  F. Nicoud,et al.  Mixed acoustic–entropy combustion instabilities in gas turbines , 2014, Journal of Fluid Mechanics.

[53]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[54]  Habib N. Najm,et al.  Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport , 2005, J. Sci. Comput..

[55]  Ning Li,et al.  2DECOMP&FFT - A Highly Scalable 2D Decomposition Library and FFT Interface , 2010 .

[56]  James A. Sethian,et al.  THE DERIVATION AND NUMERICAL SOLUTION OF THE EQUATIONS FOR ZERO MACH NUMBER COMBUSTION , 1985 .

[57]  V. Giovangigli.  Multicomponent flow modeling , 1999 .

[58]  Jie Shen,et al.  A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows , 2014, J. Comput. Phys..

[59]  E. Dick,et al.  Mach-uniformity through the coupled pressure and temperature correction algorithm , 2005 .