A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities(Proceedings of the 12th NAL Symposium on Aircraft Computational Aerodynamics)

A flux splitting scheme is proposed for the general nonequilibrium flow equations with an aim at removing numerical dissipation of Van-Leer-type flux-vector splittings on a contact discontinuity. The scheme obtained is also recognized as an improved Advection Upwind Splitting Method (AUSM) where a slight numerical overshoot immediately behind the shock is eliminated. The proposed scheme has favorable properties: high-resolution for contact discontinuities; conservation of enthalpy for steady flows; numerical efficiency; applicability to chemically reacting flows. In fact, for a single contact discontinuity, even if it is moving, this scheme gives the numerical flux of the exact solution of the Riemann problem. Various numerical experiments including that of a thermo-chemical nonequilibrium flow were performed, which indicate no oscillation and robustness of the scheme for shock/expansion waves. A cure for carbuncle phenomenon is discussed as well.

[1]  Philip L. Roe,et al.  Sonic Flux Formulae , 1992, SIAM J. Sci. Comput..

[2]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[3]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[4]  Thomas W. Roberts,et al.  The behavior of flux difference splitting schemes near slowly moving shock waves , 1990 .

[5]  M. Liou.  A generalized procedure for constructing an upwind-based TVD scheme , 1987 .

[6]  Peter A. Gnoffo,et al.  Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium , 1989 .

[7]  J. Quirk.  A Contribution to the Great Riemann Solver Debate , 1994 .

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

[9]  R. Schwane,et al.  ON THE ACCURACY OF UPWIND SCHEMES FOR THE SOLUTION OF THE NAVIER-STOKES EQUATIONS , 1987 .

[10]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[11]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[12]  F. Blottner,et al.  Chemically Reacting Viscous Flow Program for Multi-Component Gas Mixtures. , 1971 .

[13]  M. Liou,et al.  A New Flux Splitting Scheme , 1993 .

[14]  S. Chakravarthy,et al.  The versatility and reliability of Euler solvers based on high-accuracy TVD formulations , 1986 .

[15]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

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

[17]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[18]  S. Osher,et al.  Computing with high-resolution upwind schemes for hyperbolic equations , 1985 .

[19]  M. Liou,et al.  On a new class of flux splittings , 1993 .

[20]  Roger C. Millikan,et al.  Systematics of Vibrational Relaxation , 1963 .

[21]  S. Osher,et al.  Upwind difference schemes for hyperbolic systems of conservation laws , 1982 .

[22]  H. C. Yee,et al.  Linearized form of implicit TVD schemes for the multidimensional Euler and Navier-Stokes equations , 1986 .

[23]  S. Osher,et al.  A new class of high accuracy TVD schemes for hyperbolic conservation laws. [Total Variation Diminishing] , 1985 .

[24]  Shigeru Obayashi,et al.  Practical formulation of a positively conservative scheme , 1994 .

[25]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[26]  C. Park,et al.  Assessment of two-temperature kinetic model for dissociating and weakly-ionizing nitrogen , 1986 .

[27]  Philip L. Roe,et al.  A comparison of numerical flux formulas for the Euler and Navier-Stokes equations , 1987 .

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

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

[30]  Yoko Takakura,et al.  ON THE RECENT DIFFERENCE SCHEMES FOR THE THREE-DIMENSIONAL EULER EQUATIONS , 1987 .