Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AN IMPLICIT NUMERICAL SCHEME FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON CURVILINEAR GRIDS

  • Fayyaz, Hassan (Department of Mathematics COMSATS Institute of Information Technology) ;
  • Shah, Abdullah (Department of Mathematics COMSATS Institute of Information Technology)
  • Received : 2017.04.18
  • Accepted : 2018.02.01
  • Published : 2018.05.31
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Abstract

This article deals with implementation of a high-order finite difference scheme for numerical solution of the incompressible Navier-Stokes equations on curvilinear grids. The numerical scheme is based on pseudo-compressibility approach. A fifth-order upwind compact scheme is used to approximate the inviscid fluxes while the discretization of metric and viscous terms is accomplished using sixth-order central compact scheme. An implicit Euler method is used for discretization of the pseudo-time derivative to obtain the steady-state solution. The resulting block tridiagonal matrix system is solved by approximate factorization based alternating direction implicit scheme (AF-ADI) which consists of an alternate sweep in each direction for every pseudo-time step. The convergence and efficiency of the method are evaluated by solving some 2D benchmark problems. Finally, computed results are compared with numerical results in the literature and a good agreement is observed.

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References

  1. C. Azwadi, A. Rajab, and S. Sofianuddin, Four sided lid-driven cavity flow using time splitting method of Adam Bashforth scheme, Int. J. of Autom and Mech. Eng. (IJAME) 9 (2014), 1501-1510. https://doi.org/10.15282/ijame.9.2014.2.0124
  2. R. Beam and R. F. Warming, An implicit scheme for the compressible Navier-Stokes equations, AIAA J. 16 (1978), 393-402. https://doi.org/10.2514/3.60901
  3. A. J. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967), 12-26. https://doi.org/10.1016/0021-9991(67)90037-X
  4. J. A. Ekaterinaris, High-order accurate numerical solutions of incompressible flows with the artificial compressibility method, Internat. J. Numer. Methods Fluids 45 (2004), no. 11, 1187-1207. https://doi.org/10.1002/fld.727
  5. E. Erturk and B. Dursun, Numerical solution of 2D steady incompressible flow in a driven skewed cavity flow at high Reynolds numbers, Z. Angew Math. Mech. 87 (2007), no. 5, 377-392. https://doi.org/10.1002/zamm.200610322
  6. H. Fadel, M. Agouzoul, and P. K. Jimack, High-order finite difference schemes for incompressible flows, Internat. J. Numer. Methods Fluids 65 (2011), no. 9, 1050-1070. https://doi.org/10.1002/fld.2228
  7. D. Fu and Y. Ma, A high order accurate difference scheme for complex flow fields, J. Comput. Phys. 134 (1997), no. 1, 1-15. https://doi.org/10.1006/jcph.1996.5492
  8. D. Fu, Y. Ma, and T. Kobayashi, Nonphysical oscillations in numerical solutions: reason and improvement, CFD J. 4 (1996), no. 4, 427-450.
  9. U. Ghia, K. N. Ghia, and C. T. Shin, High Re solutions for incompressible Navier-Stokes equations, J. Comput. Phys. 49 (1983), 387-411.
  10. N. Gregory and C. L. O'Reilly, Low speed aerodynamic characteristics of NACA 0012 airfoil section, including the effects of upper surface roughness simulation hoarfrost, Aero Report 1308, National Physics Laboratory, Teddington, U.K., 1970.
  11. J. C. Kalita, A. K. Dass, and N. Nidhi, An efficient transient Navier-Stokes solver on compact nonuniform space grids, J. Comput. Appl. Math. 214 (2008), no. 1, 148-162. https://doi.org/10.1016/j.cam.2007.02.021
  12. L. I. G. Kovasznay, Laminar flow behind two-dimensional grid, Proc. Cambridge Philos. Soc. 44 (1948), 58-62. https://doi.org/10.1017/S0305004100023999
  13. S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), no. 1, 16-42. https://doi.org/10.1016/0021-9991(92)90324-R
  14. H. Luo and T. R. Bewley, On the contravariant form of the Navier-Stokes equations in time-dependent curvilinear coordinate systems, J. Comput. Phys. 199 (2004), no. 1, 355-375. https://doi.org/10.1016/j.jcp.2004.02.012
  15. R. Nayak, S. Batachariya, and I. Pop, Numerical study of mixed convection and entropy generation of Cu-Water nano fluid in a differentially heated skewed enclosure, Int. J. Heat and Mass Transfer 85 (2015), 620-634. https://doi.org/10.1016/j.ijheatmasstransfer.2015.01.116
  16. A. D. Perumal and K. A. Dass, Simulation of incompressible flows in two-sided lid-driven square cavities. Part II - LBM, CFD Letters 2(2010), no. 1, 25-38.
  17. P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), no. 2, 357-372. https://doi.org/10.1016/0021-9991(81)90128-5
  18. S. E. Rogers and D. Kwak, An upwind differencing scheme for the incompressible Navier-Stokes equations, Appl. Numer. Math. 8 (1991), no. 1, 43-64. https://doi.org/10.1016/0168-9274(91)90097-J
  19. A. Shah, H. Guo, and L. Yuan, A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations, Commun. Comput. Phys. 5 (2009), no. 2-4, 712-729.
  20. A. Shah, L. Yuan, and A. Khan, Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations, Appl. Math. Comput. 215 (2010), no. 9, 3201-3213. https://doi.org/10.1016/j.amc.2009.10.001
  21. M. R. Visbal and D. V. Gaitonde, On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, J. Comput. Phys. 181 (2002), no. 1, 155-185. https://doi.org/10.1006/jcph.2002.7117