Symmetry-preserving discretization of Navier-Stokes equations on collocated unstructured grids

A fully-conservative discretization is presented in this paper. The same principles followed by Verstappen and Veldman (2003) 3] are generalized for unstructured meshes. Here, a collocated-mesh scheme is preferred over a staggered one due to its simpler form for such meshes. The basic idea behind this approach remains the same: mimicking the crucial symmetry properties of the underlying differential operators, i.e., the convective operator is approximated by a skew-symmetric matrix and the diffusive operator by a symmetric, positive-definite matrix. A novel approach to eliminate the checkerboard spurious modes without introducing any non-physical dissipation is proposed. To do so, a fully-conservative regularization of the convective term is used. The supraconvergence of the method is numerically showed and the treatment of boundary conditions is discussed. Finally, the new discretization method is successfully tested for a buoyancy-driven turbulent flow in a differentially heated cavity.

[1]  N. N. Yanenko.  Application of the Method of Fractional Steps to Boundary Value Problems for Laplace’s and Poisson’s Equations , 1971 .

[2]  Bernardus J. Geurts,et al.  Interacting errors in large-eddy simulation: a review of recent developments , 2006 .

[3]  Robert L. Street,et al.  A comparison of staggered and non-staggered grid Navier--Stokes solutions for the 8:1 cavity natural convection flow , 2005 .

[4]  Darryl D. Holm,et al.  On a Leray–α model of turbulence , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  A. Arakawa.  Computational design for long-term numerical integration of the equations of fluid motion: two-dimen , 1997 .

[6]  Roel Verstappen.  On restraining the production of small scales of motion in a turbulent channel flow , 2008 .

[7]  Pierre Sagaut,et al.  Comparison of some Lie-symmetry-based integrators , 2011, J. Comput. Phys..

[8]  Shashank,et al.  A co-located incompressible Navier-Stokes solver with exact mass, momentum and kinetic energy conservation in the inviscid limit , 2010, J. Comput. Phys..

[9]  Thomas S. Lund,et al.  Kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow , 2006, J. Comput. Phys..

[10]  Hervé Jeanmart,et al.  Exact Expansions for Filtered-Scales Modelling with a Wide Class of LES Filters , 1999 .

[11]  P. Moin,et al.  Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow , 1998 .

[12]  Sílvia Barbeiro,et al.  Supraconvergent cell-centered scheme for two dimensional elliptic problems , 2009 .

[13]  J. Blair Perot,et al.  A discrete calculus analysis of the Keller Box scheme and a generalization of the method to arbitrary meshes , 2007, J. Comput. Phys..

[14]  Jean-Luc Guermond,et al.  Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows , 2004 .

[15]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[16]  Pierre Sagaut,et al.  Discrete filters for large eddy simulation , 1999 .

[17]  P. Moin,et al.  A numerical method for large-eddy simulation in complex geometries , 2004 .

[18]  Lars Davidson,et al.  A PRESSURE CORRECTION METHOD FOR UNSTRUCTURED MESHES WITH ARBITRARY CONTROL VOLUMES , 1996 .

[19]  Maarten van Reeuwijk,et al.  Leray-α simulations of wall-bounded turbulent flows , 2009 .

[20]  Pramod K. Subbareddy,et al.  A fully discrete, kinetic energy consistent finite-volume scheme for compressible flows , 2009, J. Comput. Phys..

[21]  Yong Jung Kim.  A MATHEMATICAL INTRODUCTION TO FLUID MECHANICS , 2008 .

[22]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[23]  Akio Arakawa,et al.  Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model , 1977 .

[24]  Johan Meyers,et al.  A computational error-assessment of central finite-volume discretizations in large-eddy simulation using a Smagorinsky model , 2007, J. Comput. Phys..

[25]  F. Xavier Trias,et al.  A simple approach to discretize the viscous term with spatially varying (eddy-)viscosity , 2013, J. Comput. Phys..

[26]  A. Veldman,et al.  Symmetry-preserving discretization of turbulent flow , 2003 .

[27]  Gabriel Wittum,et al.  On the influence of different stabilisation methods for the incompressible Navier-Stokes equations , 2007, J. Comput. Phys..

[28]  Robert L. Street,et al.  A divergence free fractional-step method for the Navier{Stokes equations on non-staggered grids , 2010 .

[29]  Oleg V. Vasilyev.  High Order Finite Difference Schemes on Non-uniform Meshes with Good Conservation Properties , 2000 .

[30]  Yohei Morinishi,et al.  Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows , 2010, J. Comput. Phys..

[31]  L. Weinstein,et al.  The legacy. , 2004, Journal of gerontological nursing.

[32]  A. Veldman,et al.  Spectro-Consistent Discretization of Navier-Stokes: a Challenge to RANS and LES , 1998 .

[33]  B. Perot.  Conservation Properties of Unstructured Staggered Mesh Schemes , 2000 .

[34]  F. X. Trias,et al.  A Self-Adaptive Strategy for the Time Integration of Navier-Stokes Equations , 2011 .

[35]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

[36]  M. Shashkov,et al.  Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆ , 1997 .

[37]  Maarten van Reeuwijk,et al.  A mimetic mass, momentum and energy conserving discretization for the shallow water equations , 2011 .

[38]  Mikhail Shashkov,et al.  Solving Diffusion Equations with Rough Coefficients in Rough Grids , 1996 .

[39]  M. Shashkov,et al.  The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods , 1999 .

[40]  F. X. Trias,et al.  On the construction of discrete filters for symmetry-preserving regularization models , 2011 .

[41]  J. Blair Perot,et al.  Discrete calculus methods for diffusion , 2007, J. Comput. Phys..

[42]  Jason E. Hicken,et al.  A shift transformation for fully conservative methods: turbulence simulation on complex, unstructured grids , 2005 .

[43]  Darryl D. Holm,et al.  Regularization modeling for large-eddy simulation , 2002, nlin/0206026.

[44]  F. X. Trias,et al.  Direct numerical simulations of two- and three-dimensional turbulent natural convection flows in a differentially heated cavity of aspect ratio 4 , 2007, Journal of Fluid Mechanics.

[45]  Benjamin Sanderse,et al.  Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations , 2013, J. Comput. Phys..

[46]  U. Frisch.  Turbulence: The Legacy of A. N. Kolmogorov , 1996 .

[47]  J. Kan.  A second-order accurate pressure correction scheme for viscous incompressible flow , 1986 .

[48]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .

[49]  Arthur Veldman,et al.  Symmetry-preserving upwind discretization of convection on non-uniform grids , 2008 .

[50]  Lars Davidson,et al.  Natural convection boundary layer in a 5:1 cavity , 2007 .

[51]  H. B. Keller.  A New Difference Scheme for Parabolic Problems , 1971 .

[52]  Thomas A. Manteuffel,et al.  The numerical solution of second-order boundary value problems on nonuniform meshes , 1986 .

[53]  D. Schmidt,et al.  Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics , 2002 .

[54]  A. Veldman,et al.  Playing with nonuniform grids , 1992 .

[55]  F. Xavier Trias,et al.  Parallel direct Poisson solver for discretisations with one Fourier diagonalisable direction , 2011, J. Comput. Phys..

[56]  J. Blair Perot,et al.  Discrete Conservation Properties of Unstructured Mesh Schemes , 2011 .

[57]  Darryl D. Holm,et al.  Leray and LANS-α of turbulent mixing , 2006 .

[58]  Arthur E. P. Veldman,et al.  Mass, momentum and energy conserving (MaMEC) discretizations on general grids for the compressible Euler and shallow water equations , 2012, J. Comput. Phys..