Multigrid solution of Euler equations using high-resolution NVD differencing scheme for unstructured meshes

High-speed compressible flows in a nozzle and over an airfoil have been computed by solving the Euler equations using the GAMMA differencing scheme for spatial discretisation on unstructured grid. This high-resolution technique originally developed in a segregated algorithm has now been modified for the present coupled solver by introducing a non-linear blending factor to enhance its convergence property while preserving the Total Variation Diminishing (TVD) criterion. The efficiencies of the GAMMA scheme and its variant have been augmented with a novel agglomeration multigrid scheme to accelerate the convergence to steady state using the multi-stage Runge-Kutta time-marching technique.

[1]  Dimitri J. Mavriplis,et al.  Multigrid solution of the Navier-Stokes equations on triangular meshes , 1989 .

[2]  Timothy G. Trucano,et al.  Verification and validation. , 2005 .

[3]  N. Ron-Ho,et al.  A Multiple-Grid Scheme for Solving the Euler Equations , 1982 .

[4]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[5]  V. Vatsa,et al.  development of a multigrid code for 3-D Navier-Stokes equations and its application to a grid-refinement study , 1990 .

[6]  Hrvoje Jasak,et al.  Error analysis and estimation for the finite volume method with applications to fluid flows , 1996 .

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

[8]  A. Gosman,et al.  High resolution NVD differencing scheme for arbitrarily unstructured meshes , 1999 .

[9]  D. Mavriplis.  Multigrid solution of the two-dimensional Euler equations on unstructured triangular meshes , 1987 .

[10]  Stuart Connell,et al.  A 3D unstructured adaptive multigrid scheme for the Euler equations , 1993 .

[11]  Valerio Caleffi,et al.  Finite volume method for simulating extreme flood events in natural channels , 2003 .

[12]  R. F. Warming,et al.  Upwind Second-Order Difference Schemes and Applications in Aerodynamic Flows , 1976 .

[13]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[14]  Alain Dervieux,et al.  Unstructured multigridding by volume agglomeration: Current status , 1992 .

[15]  Branislav Basara,et al.  BOUNDED CONVECTION SCHEMES FOR UNSTRUCTURED GRIDS , 2001 .

[16]  J. Peraire,et al.  A 3D finite element multigrid solver for the Euler equations , 1992 .

[17]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[18]  A. M. Binnie,et al.  An electrical detector of condensation in high-velocity steam , 1942, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[19]  David W. Zingg,et al.  Development of an agglomeration multigrid algorithm with directional coarsening , 1999 .

[20]  Dimitri J. Mavriplis,et al.  AGGLOMERATION MULTIGRID FOR THE THREE-DIMENSIONAL EULER EQUATIONS , 1994 .

[21]  E. Tadmor,et al.  Third order nonoscillatory central scheme for hyperbolic conservation laws , 1998 .

[22]  Dimitri J. Mavriplis,et al.  Agglomeration multigrid for two-dimensional viscous flows , 1995 .

[23]  P. Gaskell,et al.  Curvature‐compensated convective transport: SMART, A new boundedness‐ preserving transport algorithm , 1988 .

[24]  Hou Ping-li,et al.  Refinement of the convective boundedness criterion of Gaskell and Lau , 2003 .

[25]  B. P. Leonard,et al.  A stable and accurate convective modelling procedure based on quadratic upstream interpolation , 1990 .

[26]  Edward Ng,et al.  Higher-order Upwind Distribution-formula Scheme for Structured and Unstructured Adaptive Flow Solvers , 2003 .