Abstract
The coupled heat conduction/convection problem for a solid cylinder in either a rectangular or a circular enclosure filled with air is solved by an operator-splitting pseudo-time-stepping finite element method, which automatically satisfies the continuity of the interfacial temperature and heat flux. The temperature distribution in the cylinder and in the fluid is obtained showing that the usual practice of prescribing a uniform heat flux boundary condition at the interface may not lead to an accurate solution. From the profile of the local Nusselt number, which is strongly dependent on the Rayleigh number but weakly dependent on the thermal conductivity ratio, it is concluded that most of the heat transfer takes place in the lower half of the cylinder through a convective mode.
Similar content being viewed by others
References
Bishop, E. H.; Carley, C. T.; Powe, R. E. 1968: Natural Convective Oscillatory Flow in Cylindrical Annuli. Int. J. Heat Mass Transfer, 11: 1741–1752
Boussinesq, J. 1903: Theorie Analytique de la Chaleur, Vol. 2, Gauthier-Villars, Paris
Bristeau, M. O.; Glowinski, R.; Periaux, J. 1987: Computer Physics Reports, 6: 73–187
Cho, C. H.; Chang, K. S.; Park, K. H. 1982: Numerical Simulation of Natural Convection in Concentric and Eccentric Horizontal Cylindrical Annuli. J. Heat Transfer, ASME, 104 (11): 624–630
Dean, E.; Glowinski, R.; Li, C. H. 1988: Mathematics Applied to Science, Ed. Goldstein, J., Rosecrans, S. and Sod, G.
De Vahl Davis, G.; Jones, I. P. 1981: Natural Convection in a Square Cavity a Comparison Exercise. Numerical Methods in Thermal Problems, 2: 552–572. Eds. Lewis, Morgan and Schrefler. Pineridge Press
De Vahl Davis, A. 1976: Finite Difference Methods for Natural and Mixed Convection in Enclosures. Proc. 8th Inter. Conf. on Heat Transfer, 1: 101–109
Gray, D. D.; Giorgini, A. 1976: The Validity of the Boussinesq Approximation for Liquid and Gases. Int. J. Heat Mass Transfer. 19: 545–551
Grigull, U.; Hauf, W. 1966: Natural Convection in Horizontal Cylindrical Annuli. 3rd Int. Heat Transfer Conf., Chicago. 182–195
Guj, G., Stella, F. 1995: Natural Convection in Horizontal Eccentric Annuli: Numerical Study. J. Num. Heat Transfer, Part A. 27: 89–105 Handbook of Chemistry and Physics. 1975 56th edition. Ed. R. C.Weast. CRC Press, Ohio
Liu, C. Y.; Mueller, W. K.; Landis, F. 1961: Natural Convection Heat Transfer in long Horizontal Cylindrical Annuli. Int. Developments in Heat Transfer, ASME. 976–984
Kuehn, T. H.; Goldstein, R. J. 1976: An Experimental and Theoretical Study of Natural Convection in the Annulus between Horizontal Concentric Cylinders. J. Fluid Mech. 74: 695–719
Kuehn, T.-H.; Goldstein, R. J. 1978: An Experimental Study of Natural Convection Heat Transfer in Concentric and Eccentric Horizontal Cylindrical Annuli. J. Heat Transfer, ASME, 100 (11): 635–640
Leonardi, E.; Reizes, J. S. 1981a: Convective Flows in Closed Cavities with Variable Fluid Properties. Numerical Methods in Heat Transfer. 387–412. Eds. R. W. Lewis, K. Morgan, O. D. Zienkiewicz, John Wiley and Sons
Leonardi, E.; Reizes, J. S. 1981b: Natural Convection Heat Transfer for Variable Property Fluids using the Boussinesq Approximation. Numerical Methods in Thermal Problems. 2: 978–989. Eds. R. W. Lewis, K. Morgan and B. A. Schrefler, Pineridge Press
Luo, X. L.; Stokes, N.; Mooney, J. 1984: Manual for Fastflo ™ Ver. 2
Malkus, W. V. R. 1989: A Scaling and Expansion of Equations of Motion to Yield the Boussinesq Equations. Notes on the 1969 Summer Study Program in Geophysical Fluid Dynamics at the Woods Hole Oceanographic Institution. 1: 23–28. U. S. National Technical Information Service, PB 189618
Mihaljan, J. M., A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J. 136: 1126–1133
Oberbeck, A. 1879: Uber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infloge von Temperature Differenzen, Ann. Phys. Chem. 7: 271–292
Patankar, S. V. 1978: A numerical Method for Conduction in Composite Materials, Flow in Irregular Geometries and Conjugate Heat Transfer. Proc. 6th Int. Heat Transfer Conf., Toronto. 3: 297–302
Projahn, U.; Rieger, H.; Beer, H. 1981: Numerical Analysis of Laminar Natural Convection between Concentric and Eccentric Cylinders. J. Numerical Heat Transfer. 4: 131–146
Reddy, J. N.; Gartling, D. K. 1994: The Finite Element Method in Heat Transfer and Fluid Dynamics. CRC Press
Rieger, H.; Projahn, U. 1981: Laminar Natural Convection Heat Transfer in a Horizontal Gap, Bounded by an Elliptic and a Circular Cylinder. Numerical Methods in Thermal Problems. 2: 1036–1047. Eds. R. W. Lewis, K. Morgan and B. A. Schrefler, Pineridge Press
Sundén, B. 1993: A Numerical Study of Transient Coupled Conduction-Forced Convection. Numerical Methods in Thermal Problems. 641–651. Eds. R. W. Lewis, J. A. Johnson and R. Smith, Pineridge Press
Sundén, B. 1980a. Conjugated Heat Transfer from Circular Cylinders in Low Reynolds Number Flow. Int. J. Heat Mass Transfer. 23: 1359–1367
Sundén, B. 1980b: A Numerical Study of Coupled Conduction-Mixed Convection. Numerical Methods for Non-Linear Problems. 1: 795–805. Eds. C. Taylor, E. Hinton and D. R. J. Owen, Pineridge Press
Author information
Authors and Affiliations
Additional information
Communicated by Y. Jaluria, 25 April 1996
This research is supported by a Sydney University/CSIRO collaboration grant through a scholarship to Y. Liu. This support is gratefully acknowledged. Fastflo ™ code is provided by Division of Mathematics and Statistics, CSIRO.
Rights and permissions
About this article
Cite this article
Liu, Y., Phan-Thien, N. & Kemp, R. Coupled conduction-convection problem for a cylinder in an enclosure. Computational Mechanics 18, 429–443 (1996). https://doi.org/10.1007/BF00350251
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00350251