Compact fourth-order finite volume method for numerical solutions of Navier-Stokes equations on staggered grids

The development of a compact fourth-order finite volume method for solutions of the Navier-Stokes equations on staggered grids is presented. A special attention is given to the conservation laws on momentum control volumes. A higher-order divergence-free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes. Three forms of the nonlinear correction for staggered grids are proposed and studied. The accuracy of each approximation is assessed comparatively in Fourier space. The importance of higher-order approximations of pressure is discussed and numerically demonstrated. Fourth-order accuracy of the complete scheme is illustrated by the doubly-periodic shear layer and the instability of plane-channel flow. The efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations. The proposed scheme is highly accurate and efficient. At the same level of accuracy, the fourth-order scheme can be ten times faster than the second-order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.

[1]  E. Krause,et al.  A comparison of second- and sixth-order methods for large-eddy simulations , 2002 .

[2]  Joseph Mathew,et al.  A high‐resolution scheme for low Mach number flows , 2004 .

[3]  Parviz Moin,et al.  Direct simulations of turbulent flow using finite-difference schemes , 1991 .

[4]  Pierre Sagaut,et al.  Large-eddy simulation for acoustics , 2007 .

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

[6]  Marcelo H. Kobayashi,et al.  A fourth-order-accurate finite volume compact method for the incompressible Navier-Stokes solutions , 2001 .

[7]  S. Ghosal.  An Analysis of Numerical Errors in Large-Eddy Simulations of Turbulence , 1996 .

[8]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[9]  Joel H. Ferziger,et al.  A robust high-order compact method for large eddy simulation , 2003 .

[10]  F. Denaro,et al.  On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions , 2003 .

[11]  Ayodeji O. Demuren,et al.  HIGHER-ORDER COMPACT SCHEMES FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS, PART I: THEORETICAL DEVELOPMENT , 2001 .

[12]  John Kim,et al.  DIRECT NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOWS UP TO RE=590 , 1999 .

[13]  F. Mashayek,et al.  Evaluation of a Fourth-Order Finite-Volume Compact Scheme for LES with Explicit Filtering , 2005 .

[14]  J. Kim,et al.  Optimized Compact Finite Difference Schemes with Maximum Resolution , 1996 .

[15]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[16]  Marcelo H. Kobayashi,et al.  Regular Article: On a Class of Padé Finite Volume Methods , 1999 .

[17]  Martine Baelmans,et al.  A finite volume formulation of compact central schemes on arbitrary structured grids , 2004 .

[18]  Arpiruk Hokpunna,et al.  Compact Fourth-order scheme for Numerical Simulations of Navier-Stokes Equations , 2009 .

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

[20]  R. Knikker,et al.  Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows , 2009 .

[21]  Neil D. Sandham,et al.  Wall Pressure and Shear Stress Spectra from Direct Simulations of Channel Flow , 2006 .

[22]  D. Gottlieb,et al.  The Stability of Numerical Boundary Treatments for Compact High-Order Finite-Difference Schemes , 1993 .

[23]  J. Williamson.  Low-storage Runge-Kutta schemes , 1980 .

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

[25]  Olga Shishkina,et al.  A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains , 2007 .

[26]  Ayodeji O. Demuren,et al.  Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows , 1998 .

[27]  P. Moin,et al.  DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research , 1998 .

[28]  Pierre Sagaut.  Large-Eddy Simulation for Acoustics: Theoretical Background: Large-Eddy Simulation , 2007 .

[29]  Harold S. Stone,et al.  An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations , 1973, JACM.

[30]  Joseph Mathew,et al.  Direct numerical simulation of turbulent spots , 2001 .

[31]  M. Piller,et al.  Finite-volume compact schemes on staggered grids , 2004 .

[32]  Datta V. Gaitonde,et al.  Optimized Compact-Difference-Based Finite-Volume Schemes for Linear Wave Phenomena , 1997 .

[33]  M. Minion,et al.  Performance of Under-resolved Two-Dimensional Incompressible Flow , 1995 .

[34]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[35]  Mujeeb R. Malik,et al.  A spectral collocation method for the Navier-Stokes equations , 1985 .

[36]  Giuseppe Passoni,et al.  A spectral–finite difference solution of the Navier–Stokes equations in three dimensions , 1998 .

[37]  P. Moin,et al.  Turbulence statistics in fully developed channel flow at low Reynolds number , 1987, Journal of Fluid Mechanics.