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A Finite Difference Calculation Procedure for the Navier-Stokes Equations on a Staggered Curvilinear Grid

Authors: R. M. Barron, B. Zogheib


A new numerical method for solving the twodimensional, steady, incompressible, viscous flow equations on a Curvilinear staggered grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well-established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity-pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation in a manner similar to the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results obtained from the present method are based on the first-order upwind scheme for the convective terms, but the methodology can easily be modified to accommodate higher order differencing schemes.

Keywords: Curvilinear, finite difference, finite volume, SIMPLE.

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[1] T. F. Miller and F. W. Schmidt, "Use of a pressure-weighted interpolation method for the solution of the incompressible Navier- Stokes equations on a non-staggered grid system," Numerical Heat Transfer , vol. 14, pp. 213-233, 1988.
[2] G. D. Thiart, "Finite difference schemes for the numerical solution of fluid flow and heat transfer problems on nonstaggered grids," Numerical Heat Transfer, vol. 17, Part B, pp. 43-62, 1990.
[3] I. E. Barton and R. Kirby, "Finite difference scheme for the solution of fluid flow problems on non-staggered grids," International Journal for Numerical Methods in Fluids, vol. 33, pp. 939-959, 2000.
[4] Z. Tian and Y. Ge, "A fourth-order compact finite difference scheme for the steady stream function-vorticity formulation of the Navier- Stokes/Boussinesq equations," International Journal for Numerical Methods in Fluids, vol. 41, pp. 495-518, 2003.
[5] M. Ben-Artzi, J. P. Croisille, and D. Fishelov, " Trachtenberg S. A purecompact scheme for the streamfunction formulation of Navier-Stokes equations," Journal of Computational Physics, vol. 205, pp. 640-664, 2005.
[6] A. K. De and K. Eswaran, "A high-order accurate method for twodimensional incompressible viscous flows," International Journal for Numerical Methods in Fluids, vol. 53, pp. 1613-1628, 2007.
[7] B. Zhu, "Finite volume solution of the Navier-Stokes equations in velocity-vorticity formulation," International Journal for Numerical Methods in Fluids, vol. 48, pp. 607-629, 2005.
[8] F. H. Harlow and J.E. Walsh, "Numerical Calculations of Timedependent Viscous Incompressible Flow of Fluid with Free Surface," Physics of Fluids, vol. 8, pp. 2182-2189, 1965.
[9] S. Patankar, "Numerical Heat Transfer and Fluid Flow", Hemisphere, New York, 1980.
[10] S. V. Patankar and D.B. Spalding, "A Calculation Procedure for Heat and Mass Transfer in Three Dimensional Parabolic Flows," Int. J. Heat and Mass Transfer, vol. 15, pp. 1787-1806, 1972.
[11] W. R. Briley and H. McDonald, "Analysis and Computation of Viscous Subsonic Primary and Secondary Flows," AIAA Paper 79-1453, 1979.
[12] K. N. Ghia and J.S. Sokhey, "Laminar Incompressible Viscous Flow in Curved Ducts of Rectangular Cross Sections," J. of Fluids Engineering, vol. 99, pp. 640-645, 1977.
[13] A. J. Chorin, "A Numerical Method for Solving Incompressible Viscous Flow Problems", J. of Computational Physics, vol. 2, 12-26, 1967.
[14] J. Kim and P. Moin, "Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations," J. of Computational Physics, vol. 59, pp. 308-323, 1985.
[15] C. W. Hirt, B. D. Nichols and N.C. Romero, "SOLA - A Numerical Solution Algorithm for Transient Fluid Flow," Los Alamos Scientific Laboratory Report LA-5852, 1975.
[16] C. H. Tia and Y. Zhao, "A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows," International Journal for Numerical Methods in Fluids, vol.46, pp. 59- 84, 2004.
[17] S. Patankar, "A Calculation Procedure For Two-Dimensional Elliptic Situations," Numerical Heat Transfer, vol. 4, pp. 409-425, 1981.
[18] B. Zogheib and R. Barron, "Velocity-Pressure Coupling in Finite Difference Formulations for the Navier-Stokes Equations," Accepted for publication, International Journal for Numerical Methods in Fluids, to be published.
[19] B. P. Leonard, "A stable and accurate convective modelling procedure based on quadratic upstream interpolation," Computational Methods in Applied Mechanical Engineering, vol. 19, pp. 59-98, 1979.
[20] P. H. Gaskell and A. K. C Lau, "Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm," International Journal for Numerical Methods in Fluids, vol. 8, pp. 617- 641. 1988.
[21] A. Varonos and G. Bergeles, "Development and assessment of a variable-order non-oscillatory scheme for convection term discretization," International Journal for Numerical Methods in Fluids, vol. 26, pp. 1-16, 1998.
[22] Alves MA, Oliveira PJ, Pinho FT. A convergent and universally bounded interpolation scheme for the treatment of advection. International Journal for Numerical Methods in Fluids, vol. 21, pp. 47- 75, 2003.
[23] K. A. Hoffmann and S. T. Chiang, "Computational Fluid Dynamics," vol. 2, Wichita, KS, 1998.
[24] K. A. Cliffe, C. P. Jackson and A. C. Greenfield, "Finite Element Solutions for Flow in a Symmetric Channel with a Smooth Expansion," AERE-R, 10608, UK, 1982.
[25] M. Napolitano and P. Orlandi, "Laminar Flow in Complex Geometry: A Comparison," International Journal for Numerical Methods in Fluids, vol. 5, pp. 667-683, 1985.
[26] P. S. Carson, "Computation of Laminar Viscous Flows Using Von Mises Coordinates," Ph.D. Thesis, University of Windsor, Canada, 1994.
[27] A. K. Rostagi, "Hydrodynamics and Mass Transport in Pipelines Perturbed by Welded Joint," DNV Technical Report No. 82.0152, 1982.