Commun. Comput. Phys., 11 (2012), pp. 99-113.

Higher-Order Compact Scheme for the Incompressible Navier-Stokes Equations in Spherical Geometry

T. V. S. Sekhar 1*, B. Hema Sundar Raju 1, Y. V. S. S. Sanyasiraju 2

1 Department of Mathematics, Pondicherry Engineering College, Puducherry-605 014, India.
2 Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India.

Received 17 October 2010; Accepted (in revised version) 3 March 2011
Available online 5 September 2011


A higher-order compact scheme on the nine point 2-D stencil is developed for the steady stream-function vorticity form of the incompressible Navier-Stokes (N-S) equations in spherical polar coordinates, which was used earlier only for the cartesian and cylindrical geometries. The steady, incompressible, viscous and axially symmetric flow past a sphere is used as a model problem. The non-linearity in the N-S equations is handled in a comprehensive manner avoiding complications in calculations. The scheme is combined with the multigrid method to enhance the convergence rate. The solutions are obtained over a non-uniform grid generated using the transformation r=e^\xi while maintaining a uniform grid in the computational plane. The superiority of the higher order compact scheme is clearly illustrated in comparison with upwind scheme and defect correction technique at high Reynolds numbers by taking a large domain. This is a pioneering effort, because for the first time, the fourth order accurate solutions for the problem of viscous flow past a sphere are presented here. The drag coefficient and surface pressures are calculated and compared with available experimental and theoretical results. It is observed that these values simulated over coarser grids using the present scheme are more accurate when compared to other conventional schemes. It has also been observed that the flow separation initially occurred at Re = 21.

AMS subject classifications: 76D05, 35Q35, 65N06

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Key words: Fourth order compact scheme, Navier-stokes equations, spherical polar coordinates, drag coefficient.

*Corresponding author.
Email: (T. V. S. Sekhar), (B. H. S. Raju), (Y. V. S. S. Sanyasiraju)

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