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A bifurcation study of mixed-convection heat transfer in horizontal ducts

Published online by Cambridge University Press:  26 April 2006

K. Nandakumar  and
H. J. Weinitschke
K. Nandakumar
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton. Alberta, Canada T6G 2G6
H. J. Weinitschke
Affiliation:
Institut für Angewandte Matheniatik, Universität Erlangen — Nürnberg, 8520 Erlangen, Federal Republic of Germany
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Abstract

The bifurcation structure of two-dimensional, pressure-driven flows through a horizontal, rectangular duet that is heated with a uniform flux in the axial direction and a uniform temperature around the periphery is examined. The solution structure of the flow in a square duct is determined for Grashof numbers (Gr) in the range of 0 to 106 using an arclength continuation scheme. The structure is much more complicated than reported earlier by Nandakumar, Masliyah & Law (1985). The primary branch with two limit points and a hysteresis behaviour between the two-and four-cell flow structure that was computed by Nandakumar et al. is confirmed. An additional symmetric solution branch, which is disconnected from the primary branch (or rather connected via an asymmetric solution branch), is found. This has a two-cell flow structure at one end, a four-cell flow structure at the other, and three limit points are located on the path. Two asymmetric solution branches emanating from symmetry-breaking bifurcation points are also found for a square duct. Thus a much richer solution structure is found with up to five solutions over certain ranges of Or. A determination of linear stability indicates that all two-dimensional solutions develop some form of unstable mode by the time Gr is increased to about 220000. In particular, the four-cell becomes unstable to asymmetric perturbations. The paths of the singular points are tracked with respect to variation in the aspect ratio using the fold-following algorithm. Transcritical points are found at aspect ratios of 1.408 and 1.456 respectively for Prandtl numbers Pr = 0.73 and 5. Above these aspect ratios the four-cell solution is no longer on the primary branch. Some of the fold curves are connected in such a way as to form a tilted cusp. When the channel cross-section is tilted even slightly (1°) with respect to the gravity vector, the bifurcation points unfold and the two-cell solution evolves smoothly as the Grashof number is increased. The four-cell solutions then become genuinely disconnected from the primary branch. The uniqueness range in Grashof number increases with increasing tilt, decreasing aspect ratio and decreasing Prandtl number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 231 , October 1991 , pp. 157 - 187
Copyright
© 1991 Cambridge University Press

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References

Allen, M. B., Herrera, I. & Pinder, G. F. 1988 Numerical Modeling in Science and Engineering. John Wiley.
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous fluid, I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Cheng, K. C. & Hwang, G.-J. 1969 Numerical solution for combined free and forced laminar convection in horizontal rectangular channels. Trans. ASME C: J. Heat Transfer 91, 59–66.Google Scholar
Cheng, K. C. & Yuen, F. P. 1987 Flow visualization studies on secondary flow patterns in straight tubes downstream of a 180° bend and in isothermally heated horizontal tubes. Trans. ASME C: J. Heat Transfer 109, 49–54.Google Scholar
Chou, F. C. & Hwang, G.-J. 1984 Combined free and forced laminar convection in horizontal rectangular channels for high Re-Ra. Can. J. Chem. Engng 62, 830–836.Google Scholar
Chu, E., George, A., Liu, J. & Ng, E. 1984 SPARSPAK. Research Rep. CS-84-36. Dept. of Computer Science. University of Waterloo.
Cliffe, K. A. & Mullin, T. 1985 A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech. 153, 243–258.Google Scholar
Dean, W. R. 1927 The stream-line motion of fluid in a curved pipe. Phil. Mag. 5, 673–695.Google Scholar
Dennis, S. C. R. & Ng, M. 1982 Dual solutions for steady laminar flow through a curved tube. Q. J. Mech. Appl. Maths 35, 305–324.Google Scholar
Faris, G. N. & Viskanta, R. 1969 An analysis of combined forced and free convection heat transfer in a horizontal tube. Intl J. Heat Mass Transfer 12, 1295–1309.Google Scholar
Fung, L., Nandakumar, K. & Masliyah, J. H. 1987 Bifurcation phenomena and cellular-pattern evolution in mixed-convection heat transfer. J. Fluid Mech. 177, 339–357.Google Scholar
Holger, W. W. 1988 Applied Numerical Linear Algebra. Prentice Hall.
Hwang, G.-J. & Cheng, K. C. 1970 Boundary vorticity method for convective heat transfer with secondary flow - applications to the combined free and forced convection in horizontal tubes. Heat Transfer 4 Paper XC3.5.Google Scholar
Iqbal, M. & Stachiewicz, J. W. 1966 Influence of tube orientation on combined free and forced laminar convection heat transfer. Trans. ASME C: J. Heat Transfer 88, 109–116.Google Scholar
Jepson, A. & Spence, A. 1985 Folds in solutions of two parameter systems and their calculations. Part I. SIAM J. Numer. Anal. 22, 347–368.Google Scholar
Keller, H. B. 1977 Applications of Bifurcation Theory (ed. P. H. Rabinowitz), p. 359. Academic.
Moore, G. & Spence, A. 1980 The calculation of turning points of non-linear equations. SIAM J. Numer. Anal. 17, 567–576.Google Scholar
Morton, B. R. 1959 Laminar convection in uniformly heated horizontal pipes at low Rayleigh numbers. Q. J. Mech. Appl. Maths 12, 410–420.Google Scholar
Nandakumar, K. & Masliyah, J. H. 1982 Bifurcation in steady laminar flow through curved tubes. J. Fluid Mech. 119, 475–490.Google Scholar
Nandakumar, K., Masliyah, J. H. & Law, H. S. 1985 Bifurcation in steady laminar mixed convection flow in horizontal ducts. J. Fluid Mech. 152, 145–161 (referred to herein as I).Google Scholar
Patankar, S. V, Ramadhyani, S. & Sparrow, E. M. 1978 Effect of circumferentially nonuniform heating on laminar combined convection in a horizontal tube. Trans. ASME C: J. Heat Transfer 100, 63–70.Google Scholar
Ravi Sankar, S., Nandakumar, K. & Masliyah, J. H. 1988 Oscillatory flows in coiled square ducts. Phys. Fluids 31, 1348–1359.Google Scholar
Ravi Sankar, S., Nandakumar, K. & Weinitschke, H. J. 1991 Mixed convection heat transfer in horizontal ducts. Part 2: Development of three-dimensional, streamwise periodic flows. J. Fluid Mech. (to be submitted).Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Hermosa.
Shah, R. C. & London, A. L. 1978 Laminar Flow Forced Convection in Ducts. Academic.
Spence, A. & Werner, B. 1982 Non-simple turning points and cusps. IMA J. Numer. Anal. 2, 413–427.Google Scholar
Van Dyke, M. 1990 Extended Stokes series: laminar flow through a heated horizontal pipe. J. Fluid Mech. 212, 289–308.Google Scholar
Weinitschke, H. J. 1985 On the calculation of limit and bifurcation points in stability problems of elastic shells. Intl J. Solids Structures 21, 79–95.Google Scholar
Weinitschke, H. J., Nandakumar, K. & Ravi Sankar, S. 1990 A bifurcation study of convective heat transfer in porous media. Phys. Fluids A 2, 912–921.Google Scholar
Werner, B. & Spence, A. 1984 The computation of symmetry breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399.Google Scholar
Winters, K. H. 1987 A bifurcation study of laminar flow in a curved tube of rectangular cross-section. J. Fluid Mech. 180, 343–369.Google Scholar
Yang, Z.-H. & Keller, H. B. 1986 Multiple laminar flows through curved pipes. Appl. Numer. Maths. 2, 257–271.Google Scholar
Yang, Z.-H. & Keller, H. B. 1986 A direct method for computing higher order folds. SIAM J. Sci. Statist. Comput. 7, 351–361.Google Scholar