Compact Fourth-order scheme for Numerical Simulations of Navier-Stokes Equations

In this dissertation, a compact fourth-order scheme for the solution of the Navier-Stokes equations on staggered grids is developed. The fourth order is ensured by a correction of the non-linear terms and a novel interpolation of the mass fluxes onto the momentum cells. The scheme is parallelised by a new interface splitting algorithm. An approximative projection method allows for an efficient solution of the pressure Poisson equation without loosing accuracy. The accuracy and the efficiency of the parallel compact fourth-order scheme is evaluated in laminar test cases and a direct numerical simulation of turbulent channel flow up to Re τ =950. The proposed scheme is highly scalable and can deliver accurate predictions of the first- and second-order statistics using the grid spacing twice coarser than the usual recommended values.

[1]  Jay C. Hardin,et al.  ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA) , 1995 .

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

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

[4]  Michael Manhart,et al.  A zonal grid algorithm for DNS of turbulent boundary layers , 2004 .

[5]  P. Moin,et al.  Direct Finite-Difference Simulations Of Turbulent Flow , 1991 .

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

[7]  Xian-He Sun.  Application and Accuracy of the Parallel Diagonal Dominant Algorithm , 1995, Parallel Comput..

[8]  Josep-Lluís Larriba-Pey,et al.  Spike Algorithm with savings for strictly diagonal dominant tridiagonal systems , 1993, Microprocess. Microprogramming.

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

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

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

[12]  David J. Kuck,et al.  On Stable Parallel Linear System Solvers , 1978, JACM.

[13]  R. Adrian,et al.  Very large-scale motion in the outer layer , 1999 .

[14]  Lionel M. Ni,et al.  Parallel algorithms for solution of tridiagonal systems on multicomputers , 1989, ICS '89.

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

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

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

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

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

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

[21]  Reinhard Nabben,et al.  Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices , 1999, SIAM J. Matrix Anal. Appl..

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

[23]  H. H. Wang,et al.  A Parallel Method for Tridiagonal Equations , 1981, TOMS.

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

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

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

[27]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .