Discretisation of diffusive fluxes on hybrid grids

The main approaches of discretising the viscous operator of fluid flow on hybrid meshes are analysed for accuracy, consistence, monotonicity and sensitivity to mesh quality. As none of these approaches is fully satisfactory, a novel method using an approximated finite-element approach is presented and analysed. The methods are compared for the linear heat equation and the Navier-Stokes equations. While the novel approximated finite-element method performs significantly better for the linear heat equation, a stabilised edge-based method performs equally well for the considered test-cases for the Navier-Stokes equations.

[1]  Daniil Svyatskiy,et al.  Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes , 2009, J. Comput. Phys..

[2]  Pierre Moinier,et al.  Edge-Based Multigrid and Preconditioning for Hybrid Grids , 1999 .

[3]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[4]  L. Fezoui,et al.  A class of implicit upwind schemes for Euler simulations with unstructured meshes , 1989 .

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

[6]  James J. McGuirk,et al.  Finite Volume Discretization Aspects for Viscous Flows on Mixed Unstructured Grids , 1999 .

[7]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[8]  Mengping Zhang,et al.  AN ANALYSIS OF THREE DIFFERENT FORMULATIONS OF THE DISCONTINUOUS GALERKIN METHOD FOR DIFFUSION EQUATIONS , 2003 .

[9]  T. Barth.  Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations , 1994 .

[10]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[11]  Martin Galle.  Unstructured viscous flow solution using adaptive hybrid grids , 1995 .

[12]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[13]  Claus-Dieter Munz,et al.  An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations , 2008, J. Comput. Phys..

[14]  Dartzi Pan,et al.  Upwind finite-volume Navier-Stokes computations on unstructured triangular meshes , 1993 .

[15]  K. Morton,et al.  Cell vertex algorithms for the compressible Navier-Stokes equations , 1993 .

[16]  Jérôme Breil,et al.  A cell-centered diffusion scheme on two-dimensional unstructured meshes , 2007, J. Comput. Phys..

[17]  Brian Launder,et al.  Numerical methods in laminar and turbulent flow , 1983 .

[18]  G Geymonat.  Partial Differential Equations of Hyperbolic Type and Applications , 1987 .

[19]  W. Coirier.  An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations. Ph.D. Thesis - Michigan Univ. , 1994 .

[20]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[21]  V. Venkatakrishnan,et al.  A UNIFIED MULTIGRID SOLVER FOR THE NAVIER-STOKES EQUATIONS ON MIXED ELEMENT MESHES , 1995 .

[22]  Philip C. E. Jorgenson,et al.  A mixed volume grid approach for the Euler and Navier-Stokes equations , 1996 .

[23]  D. Schwamborn,et al.  The DLR-F5 Wing Test Case, Contribution to "EUROVAL - An European Initiative on Validation of CFD Codes" , 1993 .

[24]  Claus-Dieter Munz,et al.  A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes , 2007, J. Comput. Phys..

[25]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[26]  Michael Rudgyard,et al.  Steady and Unsteady Flow Simulations Using the Hybrid Flow Solver AVBP , 1999 .

[27]  Sukumar Chakravarthy,et al.  A 'grid-transparent' methodology for CFD , 1997 .

[28]  Jacques Periaux,et al.  Recent improvements in galerkin and upwind Euler solvers and application to 3-D transonic flow in aircraft design , 1989 .

[29]  Valérie Auffray.  Étude comparative de schémas numériques pour la modélisation de phénomènes diffusifs sur maillages multiéléments , 2007 .

[30]  Bram Van Leer,et al.  Discontinuous Galerkin for Diffusion , 2005 .

[31]  A. Khawaja,et al.  Implementation of adaptive hybrid grids for 3-D turbulent flows , 1996 .

[32]  Arthur Rizzi,et al.  Reynolds Stress Transport Modeling of Transonic Flow Around the RAE2822 Airfoil , 1994 .

[33]  Bijan Mohammadi,et al.  Fluid dynamics computation with NSC2KE : an user-guide : release 1.0 , 1994 .

[34]  R. Eymard,et al.  Finite volume approximation of elliptic problems and convergence of an approximate gradient , 2001 .

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

[36]  Alain Dervieux,et al.  Steady Euler simulations using unstructured meshes , 1985 .

[37]  Bijan Mohammadi,et al.  COMPLEX TURBULENT COMPRESSIBLE FLOW COMPUTATION USING A TWO-LAYER APPROACH , 1992 .