Numerical simulation of two-dimensional laminar mixed-convection in a lid-driven cavity using the mixed finite element consistent splitting scheme

International Journal of Numerical Methods for Heat and Fluid Flow

Volumn 17, Issue 1, 2007, Pages 46-93

  Wong, Jeff C F ^a  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Convection; Finite element analysis; Programming and algorithm theory

Indexed keywords

ALGORITHMS; COMPUTER PROGRAMMING; FINITE ELEMENT METHOD; LAMINAR FLOW; PRANDTL NUMBER; REYNOLDS NUMBER;

NUMERICAL ALGORITHMS; RICHARDSON NUMBER;

MIXED CONVECTION;

ALGORITHMS; COMPUTER PROGRAMMING; FINITE ELEMENT METHOD; LAMINAR FLOW; MIXED CONVECTION; PRANDTL NUMBER; REYNOLDS NUMBER;

EID: 33845748291     PISSN: 09615539     EISSN: None     Source Type: Journal    
DOI: 10.1108/09615530710716081     Document Type: Article

References (51)

1. Amirthaganesh, S. and Raghunandan, B.N. (1998), "Mixed convection in a square cavity - some new results", Computational Fluid Dynamics Journal, Vol. 6 No. 4, pp. 43-51.
2. Auteri, F. and Parolini, N. (2002), "Numerical investigation of the first instabilities in the differentially heated 8:1 cavity", Int. J. Numer. Methods Fluids, Vol. 40, pp. 1121-32.
3. Aydin, O. (1999), "Aiding and opposing mechanisms of the mixed convection problems in a shear- and buoyancy-driven cavity", Int. Commun. Heat Mass Transfer, Vol. 26 No. 7, pp. 1019-28.
4. Bermúdez, B. and Nicolás, A. (1999), "An operator splitting numerical scheme for thermal/isothermal incompressible viscous flows", Int. J. Numer. Methods Fluids, Vol. 29, pp. 397-410.
5. Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin.
6. Cha, C.K. and Jaluria, Y. (1984), "Recirculating mixed convection flow for energy extraction", Int. J. Heat Mass Transfer, Vol. 27 No. 10, pp. 1801-12.
7. Chamkha, A.J. (2002), "Hydromagnetic combined convection flow in a vertical lid-driven cavity with internal heat generation or absorption", Numer. Heat Transfer - Part A: Applications, Vol. 41, pp. 529-46.
8. Chorin, A.J. (1968), "Numerical solution of the Navier-Stokes equations", Math. Comput., Vol. 22, pp. 745-62.
9. Chorin, A.J. (1969), "On the convergence of discrete approximations to the Navier-Stokes equations", Math. Comput., Vol. 23, pp. 341-53.
10. Chu, C.K. (1978), "Numerical methods in fluid dynamics", Advances in Applied Mechanics, Vol. 18, pp. 285-331.
11. Comini, G. and Del Giudice, S. (1982), "Finite-element solution of the incompressible Navier-Stokes equations", Numer. Heat Transfer, Vol. 5, pp. 463-78.
12. Cuvelier, C., Segal, A. and Van Steenhoven, A.A. (1986), Finite Element Methods and Navier-Stokes Equations, D. Reidel Publishing, Dordrecht.
13. Donea, J., Giuliani, S., Laval, H. and Quartapelle, L. (1981), "Solution of the unsteady Navier-Stokes equations by a finite-element projection method", in Taylor, C. and Morgan, K. (Eds), Recent Advances in Numerical methods in Fluids, Vol. 2, Prineridge, Swansea, pp. 97-132.
14. Ethier, C.R. and Steinman, D.A. (1994), "Exact fully 3D Navier-Stokes solutions for benchmarking", Int. J. Numer. Methods Fluids, Vol. 19, pp. 369-75.
15. Gervais, J.J., Lemelin, D. and Pierre, R. (1997), "Some experiments with stability analysis of discrete incompressible flows in the lid-driven cavity", Int. J. Numer. Methods Fluids, Vol. 24, pp. 477-92.
16. Girault, V. and Raviart, P. (1981), Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, New York, NY.
17. Goda, K. (1979), "A multistep technique for the with implicit difference schemes for calculating two- or three-dimensional cavity flows", J. Comput. Physics, Vol. 30, pp. 76-95.
18. Gresho, P., Lee, R.L. and Sani, R. (1980), "On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions", in Taylor, C. and Morgan, K. (Eds), Recent Advances in Numerical methods in Fluids, Vol. 1, Prineridge, Swansea, pp. 27-79.
19. Gresho, P., Sani, R. and Engelman, M.S. (1998), Incompressible Flow and the Finite Element Method Advection-Diffusion and Isothermal Laminar Flow, Wiley, Chichester.
20. Guermond, J.L. and Shen, J. (2003), "A new class of truly consistent splitting schemes for incompressible flows", J. Comp. Physics, Vol. 192, pp. 262-76.
21. Guermond, J.L., Minev, P. and Shen, J. (2006), "An overview of projection methods for incompressible flows", Comput. Methods Appl. Mech. Eng., Vol. 195, pp. 6011-45.
22. Henshaw, W.D. and Petersson, N.A. (2003), "A split-step scheme for the incompressible Navier-Stokes equations", in Hafez, M.M. (Ed.), Numerical Solutions of Incompressible Flows, World Scientific Publishing Co., Singapore, pp. 108-25.
23. Hsu, T.H., Hsu, P.T. and Chen, C.K. (1995), "Thermal convection of micropolar fluids in a lid-driven cavity", Int. Commun. Heat Mass Transfer, Vol. 22 No. 2, pp. 189-200.
24. Ideriah, F.J. (1980), "Prediction of turbulent cavity flow driven by buoyancy and shear", J. Mech. Engng Sci., Vol. 22 No. 6, pp. 287-95.
25. Imberger, J. and Hamblin, P.F. (1982), "Dynamics of lakes, reservoirs, and cooling ponds", Ann. Rev. Fluid Mech., Vol. 14, pp. 153-87.
26. Iwatsu, R. and Hyun, J.M. (1995), "Three-dimensional driven-cavity flows with a vertical temperature gradient", Int. J. Heat Mass Transfer, Vol. 38 No. 18, pp. 3319-28.
27. Iwatsu, R., Hyun, J.M. and Kuwahara, K. (1993), "Mixed convection in a driven cavity with a stable vertical temperature-gradient", Int. J. Heat Mass Transfer, Vol. 36 No. 6, pp. 1601-8.
28. Kellogg, O.D. (1929), Foundations of Potential Theory, Springer, Berlin.
29. Ladyžhenskaya, O.A. (1963), Mathematical Problems in the Dynamics of a Viscous Incompressible Flow, Gordon & Breach, New York, NY.
30. Ladyžhenskaya, O.A. (1975), "Mathematical analysis of Navier-Stokes equations for incompressible liquids", Ann. Rev. Fluid Mech., Vol. 7, pp. 249-72.
31. Minev, P.D., Van De Vosse, F.N., Timmermans, L.J.P. and Van Steenhoven, A.A. (1996), "A second order splitting algorithm for thermally-driven flow problems", Int. J. Num. Meth. Heat Fluid Flow, Vol. 6 No. 2, pp. 51-60.
32. Moallemi, M.K. and Jang, K.S. (1992), "Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity", Int. J. Heat Mass Transfer, Vol. 35 No. 8, pp. 1881-92.
33. Morzynski, M. and Popiel, C.O. (1988), "Laminar heat transfer in a two-dimensional cavity covered by a moving wall", Numer. Heat Transfer, Vol. 13, pp. 265-73.
34. Nicolás, A. and Bermúdez, B. (2005), "2D thermal/isothermal incompressible viscous flows", Int. J. Numer. Methods Fluids, Vol. 48, pp. 349-66.
35. Oosthuizen, P.H. and Paul, J.T. (1985), "Mixed convective heat transfer in a cavity", Fundamental Forced and Mixed Convection, Proceedings of the 23rd National Heat Transfer Conference, August 4-7, Denver, Colorado, HTD-vol. pp. 159-69.
36. Pilkington, L.A.B. (1969), "Review lecture: the float glass process", Proc. Roy. Soc. Lond. A., Vol. 314, pp. 1-25.
37. Prasad, A.K. and Koseff, J.R. (1989), "Combined forced and natural convection heat transfer in a deep lid-driven cavity flow", Heat Transfer in Convective Flows, ASME, New York, NY, HTD-107, pp. 155-62.
38. Quartapelle, L. (1994), Numerical Solutions of the Incompressible Navier-Stokes Equations, Birkhäuser, Basel.
39. Quarteroni, A. and Valli, A. (1994), Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, Vol. 23, Springer, Berlin.
40. Rosenhead, L. (1963), "Introduction - boundary layer theory", in Rosenhead, L. (Ed.), Laminar Boundary Layers: An Account of the Development, Structure, and Stability of Laminar Boundary Layers in Incompressible Fluids, Clarendon Press, Oxford, pp. 46-109.
41. Shen, J. (1992), "On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations", Numer. Math., Vol. 62 No. 1, pp. 49-73.
42. Soulaïmani, A., Fortin, M., Ouellet, Y., Dhatt, G. and Bertrand, F. (1987), "Simple continuous pressure elements for two- and three-dimensional incompressible flows", Comput. Methods Appl. Mech. Eng., Vol. 62, pp. 47-69.
43. Tang, L.Q. and Tsang, T.H. (1995), "Transient solutions by a least-squares finite-element method and Jacobi-conjugate gradient technique", Numer. Heat Transfer, Part B: Fundamentals, Vol. 28, pp. 183-98.
44. Témam, R. (1969a), "Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires I", Arch. Rational Mech. Anal., Vol. 32, (in French), pp. 135-53.
45. Témam, R. (1969b), "Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II", Arch. Rational Mech. Anal., Vol. 33, (in French), pp. 377-85.
46. Timmermans, L.J.P., Minev, P.D. and Van De Vosse, F.N. (1996), "An approximate projection scheme for incompressible flow using spectral elements", Int. J. Numer. Methods Fluids, Vol. 22, pp. 673-88.
47. Torrance, K., Davis, R.W., Eike, K., Gill, P., Gutman, D., Hsui, A., Lyons, S. and Zein, H. (1972), "Cavity flows driven by buoyancy and shear", J. Fluid Mech., Vol. 51, pp. 221-31.
48. Turek, S. (1996), "A comparative study of time-stepping techniques for the incompressible Navier-Stokes equations: from fully implicit non-linear schemes to semi-implicit projection methods", Int. J. Numer. Methods Fluids, Vol. 22, pp. 987-1011.
49. Wong, J.C.F. (2005), "Numerical simulation of two-dimensional laminar mixed-convection in a lid-driven cavity using the mixed finite element consistent splitting scheme: influence of Richardson numbers", paper presented at the 20 min contributed talk in International Conference on Scientific Computing (ICSC05), Nanjing, June 4-8.
50. Wong, J.C.F. and Chan, M.K.H. (2006), "A consistent splitting scheme for unsteady incompressible viscous flows I. Dirichlet boundary condition and applications", Int. J. Numer. Methods Fluids, Vol. 51, pp. 385-424.
51. Saad, Y. (2005), "SPARSKIT: a basic tool-kit for sparse matrix computations (Version 2)", available at: www-users.cs.umn.edu/ ∼ saad/software/SPARSKIT/sparskit.html.