A 6th order staggered compact finite difference method for the incompressible Navier-Stokes and scalar transport equations

In a previous paper we have developed a staggered compact finite difference method for the compressible Navier-Stokes equations. In this paper we will extend this method to the case of incompressible Navier-Stokes equations. In an incompressible flow conservation of mass is ensured by the well known pressure correction method [7,21]. The advection and diffusion terms are discretized with 6th order spatial accuracy. The discrete Poisson equation, which has to be solved in the pressure correction step, has the same spatial accuracy as the advection and diffusion operators. The equations are integrated in time with a third order Adams-Bashforth method. Results are presented for a 1D advection-diffusion equation, a 2D lid driven cavity at a Reynolds number of 1000 and 10,000 and finally a 3D fully developed turbulent duct flow at a bulk Reynolds number of 5400. In all cases the methods show excellent agreement with analytical and other numerical and experimental work.

[1]  C. W. Gear,et al.  Numerical initial value problem~ in ordinary differential eqttations , 1971 .

[2]  Asmund Huser,et al.  Direct numerical simulation of turbulent flow in a square duct , 1992 .

[3]  B. J. Boersma,et al.  A staggered compact finite difference formulation for the compressible Navier-Stokes equations , 2005 .

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

[5]  M. Niederschulte.  Turbulent flow through a rectangular channel , 1989 .

[6]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[7]  Miguel R. Visbal,et al.  On the use of higher-order finite-difference schemes on curvilinear and deforming meshes , 2002 .

[8]  Per Lötstedt,et al.  High order accurate solution of the incompressible Navier-Stokes equations , 2005 .

[9]  S. Gavrilakis,et al.  Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct , 1992, Journal of Fluid Mechanics.

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

[11]  E. Erturk,et al.  Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers , 2004, ArXiv.

[12]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[13]  E. Erturk,et al.  Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers , 2004, ArXiv.

[14]  Peter Moore,et al.  Simulation and measurement of flow generated noise , 2007, J. Comput. Phys..

[15]  Javier Jiménez,et al.  A high-resolution code for turbulent boundary layers , 2009, J. Comput. Phys..

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

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

[18]  Joel H. Ferziger,et al.  Computational methods for fluid dynamics , 1996 .

[19]  O. Botella,et al.  BENCHMARK SPECTRAL RESULTS ON THE LID-DRIVEN CAVITY FLOW , 1998 .

[20]  Cornelis Vuik,et al.  GMRESR: a family of nested GMRES methods , 1994, Numer. Linear Algebra Appl..