Significance of Splitting Method in Non-linear Grid system for the Solution of Navier-Stokes Equation

Authors: M. Zamani, O. Kahar

Abstract:

Solution to unsteady Navier-Stokes equation by Splitting method in physical orthogonal algebraic curvilinear coordinate system, also termed 'Non-linear grid system' is presented. The linear terms in Navier-Stokes equation are solved by Crank- Nicholson method while the non-linear term is solved by the second order Adams-Bashforth method. This work is meant to bring together the advantage of Splitting method as pressure-velocity solver of higher efficiency with the advantage of consuming Non-linear grid system which produce more accurate results in relatively equal number of grid points as compared to Cartesian grid. The validation of Splitting method as a solution of Navier-Stokes equation in Nonlinear grid system is done by comparison with the benchmark results for lid driven cavity flow by Ghia and some case studies including Backward Facing Step Flow Problem.

Keywords: Navier-Stokes, 'Non-linear grid system', Splitting method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334806

References:

[1] G. Karniadakis, M. Israeli, and S. Orszag, 1991, "High-order splitting methods for the incompressible Navier-Stokes equations," Journal of Computational Physics, 97, pp, 414-443.
[2] U. Ghia, K. N. Ghia and C. T. Shin, 1982, "High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method," Journal of Computational Physics, 48, 387-411.
[3] J. C. Tannehill, D. A. Anderson, R. H. Pletcher, 1997, Computational Fluid Mechanics and Heat Transfer. Taylor and Francis Publisher, New York.
[4] S. K. Choi, H. Y. Nam, Y. B. Lee and M. Cho (1993), "An Efficient Three-Dimensional Calculation Procedure for Incompressible Flows in Complex Geometries", Numerical Heat Transfer, Part B, 23, 387-400.
[5] I. Demirdzic and M. Peric (1990), "Finite Volume Method for Prediction of Fluid Flow in Arbitrary Shaped Domains with Moving Boundaries", International Journal for Numerical Methods in Fluids, 10, 771-790.
[6] P. N. Childs, J. A. Shaw, A. J. Peace and J. M. Georgala (1992), "SAUNA: A System for Grid Generation and Flow Simulation using hybrid/Structured/Unstructured Grids", in Computational Fluid Dynamics,Proceedings of the 1st European CFD Conference, Volume 2, 875-882.
[7] S. V. Patankar (1980), Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York.
[8] R. Courant, E. Isaacson and M. Rees (1952), "On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Difference", Communications in Pure and Applied Mathematics, 5, 243-255.
[9] D. B. Spalding (1972), "A Novel Finite Difference Formulation for Differential Expressions Involving both First and Second Derivatives", International Journal for Numerical Methods in Engineering, 4, 551- 559.
[10] S. V. Patankar (1979), "A Calculation Procedure for Two Dimensional Elliptic Situations", Numerical Heat Transfer, 2.
[11] O. Kahar (2004), "Multiple Steady solutions and bifurcations in the Symmetric Driven Cavity"., Universiti Teknologi Malaysia.