Computations using the preconditioning BI‐CGSTAB algorithm in chemical non‐equilibrium problems

A robust method for solving the chemical non-equilibrium Navier-Stokes equations, including all of the species conservation and energy production equations, is developed. The algorithm is embodied in a fully coupled, implicit, large block structure. Van Leer flux splitting for inviscid terms and central differencing for viscous terms in the explicit operators are applied in the numerical algorithm. The fully-coupled system is solved implicitly and the bi-conjugate gradient stable (Bi-CGSTAB) method with a preconditioner of incomplete lower-upper (LU)-factorization (ILU) is used for solving large block structure and diagonal dominate matrix equations. The computations are performed for the hypersonic inflow over blunt bodies including half cylinder, double ellipse and blunt nose. The adaptive grid constructed by moving grid method is employed to capture the shock location. Computational results in the present study are compared with other calculated data and exhibit good agreement. Convergence histories of the mean flow variables and species equations demonstrate that the fast convergent rate can be achieved by the preconditioned Bi-CGSTAB method

[1]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[2]  C. Wilke.  A Viscosity Equation for Gas Mixtures , 1950 .

[3]  V. Venkatakrishnan.  Preconditioned conjugate gradient methods for the compressible Navier-Stokes equations , 1990 .

[4]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[5]  Seokkwan Yoon,et al.  Fully coupled implicit method for thermochemical nonequilibrium air at suborbital flight speeds , 1991 .

[6]  Ching-Chang Chieng,et al.  Variants of Biconjugate Gradient Method for Compressible Navier-Stokes Solver , 1995 .

[7]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[8]  J. Steger,et al.  Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods , 1981 .

[9]  W. L. Jones,et al.  Theoretical and measured electron-density distributions for the ram vehicle at high altitudes. , 1972 .

[10]  Seokkwan Yoon,et al.  Numerical study of chemically reacting flows using a lower-upper symmetric successive overrelaxation scheme , 1989 .

[11]  J. Shuen,et al.  Upwind differencing and LU factorization for chemical non-equilibrium Navier-Stokes equations , 1992 .

[12]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[13]  Harry A. Dwyer,et al.  Adaptive Grid Method for Problems in Fluid Mechanics and Heat Transfer , 1980 .

[14]  Bernard Grossman,et al.  Analysis of flux-split algorithms for Euler's equations with real gases , 1989 .

[15]  J. Meijerink,et al.  Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems , 1981 .

[16]  W. K. Anderson,et al.  Comparison of Finite Volume Flux Vector Splittings for the Euler Equations , 1985 .

[17]  H. Dwyer.  Grid adaption for problems in fluid dynamics , 1984 .

[18]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[19]  T. J. Coakley,et al.  Turbulence modeling methods for the compressible Navier-Stokes equations , 1983 .

[20]  Bernard Grossman,et al.  Flux-split algorithms for flows with non-equilibrium chemistry and vibrational relaxation , 1990 .

[21]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[22]  Y. Liu,et al.  Nonequilibrium flow computations. I. an analysis of numerical formulations of conversation laws , 1989 .

[23]  C. Chieng,et al.  Comparison of variants of the bi‐conjugate gradient method for compressible Navier‐Stokes solver with second‐moment closure , 1995 .

[24]  P. A. Gnoffo,et al.  Point-implicit relaxation strategies for viscous, hypersonic flows , 1989 .

[25]  R. Maccormack.  Current status of numerical solutions of the Navier-Stokes equations , 1985 .

[26]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[27]  C. Park,et al.  On convergence of computation of chemically reacting flows , 1985 .

[28]  Marcel Vinokur,et al.  Generalized Flux-Vector splitting and Roe average for an equilibrium real gas , 1990 .

[29]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[30]  Meng-Sing Liou,et al.  Splitting of inviscid fluxes for real gases , 1990 .

[31]  Meng-Sing Liou,et al.  Inviscid flux-splitting algorithms for real gases with non-equilibrium chemistry , 1990 .

[32]  Shigeru Obayashi,et al.  Numerical simulation of underexpanded plumes using upwind algorithms , 1988 .

[33]  An efficient, explicit finite-rate algorithm to compute flows in chemical nonequilibrium , 1989 .