Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T15:36:33.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation phenomena and cellular-pattern evolution in mixed-convection heat transfer

Published online by Cambridge University Press:  21 April 2006

L. Fung ,
K. Nandakumar  and
J. H. Masliyah
L. Fung
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada
K. Nandakumar
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada
J. H. Masliyah
Affiliation:
Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

This investigation is concerned with the numerical calculation of multiple solutions for a mixed-convection flow problem in horizontal rectangular ducts. The numerical results are interpreted in terms of recent observations by Benjamin (1978a) on the bifurcation phenomena for a bounded incompressible fluid. The observed mutations of cellular flows are discussed in terms of dynamic interchange processes. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number Gr and aspect ratio γ, which is delimited by stability boundaries. Such a stability map has been generated for each type of cellular flow by a series of numerical experiments. Once these boundaries are crossed one cellular flow mutates into another via a certain dynamical process. Although the nature of the singular points on this map have not been determined precisely, a plausible general structure of the cellular-flow exchange process emerges from this map with several features in common with the Taylor-Couette flow. The primary modes appear to exchange roles via the formation of tilted cusp. Other salient features such as primary-mode hysteresis and quasi-critical range for cellular development appear to be present. However no anomolous modes have been observed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 177 , April 1987 , pp. 339 - 357
Copyright
© 1987 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comp. Phys. 1, 119–143.Google Scholar
Behringer, R. P. & Ahlers, G. 1982 Heat transport and temporal evolution of fluid flow near the Rayleigh–Bénard instability in cylindrical containers. J. Fluid Mech. 125, 219–258.Google Scholar
Bénard, M. 1901 Les troubillions cellulaires dans une nappe liquide transportant de la chaleur par convection en regime permanent. Ann. Chim. Phys. 23, 62–144.Google Scholar
Benjamin, T. B. 1976 Application of Leray–Schauder degree theory to problems of hydrodynamic stability. Math. Proc. Camb. Phil. Soc., 79, 373–392.Google Scholar
Benjamin, T. B. 1978a Bifurcation phenomena in steady flow of a viscous liquid. I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Benjamin, T. B. 1978b Bifurcation phenomena in steady flow of a viscous liquid. II. Experiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Benjamin, T. B. 1978c Application of generic bifurcation theory in fluid mechanics. In Contemporary Developments in Continuum Mechanics and Partial Differential Equations (ed. G. M. de la Penha & L. A. J. Madeiros). North-Holland.
Benjamin, T. B. & Mullin, T. 1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 337, 221–249.Google Scholar
Benjamin, T. B. & Mullin, T. 1982 Notes on the multiplicity of flows in the Taylor experiment. J. Fluid Mech. 121, 219–230.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, 59–66Google 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
Cliffe, K. A. 1983 Numerical calculation of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233.Google Scholar
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
Cliffe, K. A. & Winters, K. H. 1984 A numerical study of the cusp catastrophe for Bénard convection in tilted cavities. J. Comp. Phys. 54, 531–534.Google Scholar
Coutier, J. P. & Greif, R. 1985 An investigation of laminar mixed convection inside a horizontal tube with isothermal wall conditions. Intl J. Heat Mass Transfer 28, 1293–1305.Google Scholar
Daniels, P. G. 1977 The effect of distant sidewalls on the transition to finite amplitude Bénard convection. Proc. R. Soc. Lond. A 358, 173–197.Google Scholar
Daniels, P. G. 1981 The effect of distant sidewalls on the evolution and stability of finite amplitude Rayleigh-Bénard convection. Proc. R. Soc. Lond. A 378, 539–566.Google Scholar
Daniels, P. G. 1984 Roll-pattern evolution in finite amplitude Rayleigh-Bénard convection in a two-dimensional fluid layer bounded by distant sidewalls. J. Fluid Mech. 143, 125–152.Google Scholar
Drazin, P. G. 1975 On the effects of sidewalls on Bénard convection. Z. Angew. Math. Phys. 27, 239–243.Google Scholar
Dufort, E. C. & Frankel, S. P. 1953 Stability conditions in the numerical treatment of parabolic differential equations. Math. Tables and Other Aids to Comps. 7, 135–152.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
Festa, J. F. 1970 A numerical model of a convective cell driven by non-uniform horizontal heating. MS thesis, Massachusetts Institute of Technology.
Foias, C. & Temain, R. 1977 Structure of the set of stationary solutions of the Navier–Stokes equations. Commun. Pure Appl. Maths 30, 149–164Google Scholar
Hall, P. & Walton, I. C. 1977 The smooth transition to a convective regime in a two-dimensional box. Proc. R. Soc. Lond. A 358, 199–221.Google Scholar
Hattori, N. & Kotake, S. 1978 Combined free and forced convection heat transfer for fully developed laminar flow in horizontal tubes (experiments). Bull. JSME 21, 861–868.Google Scholar
Horne, R. N. & O'Sullivan, M. J.1974 Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339–352.Google Scholar
Hwang, G.-J & Cheng, K. C. 1970 Boundary vorticity method for convective heat transfer with secondary flow-application to the combined free and forced laminar convection in horizontal tubes. Heat Transfer, vol. 4, Paper No. NC3–5.
Hwang, G.-J & Liu, C.-L. 1976 An experimental study of convective instability in the thermal entrance region of a horizontal parallel plate channel heated from below. Can. J. Chem. Engng 54, 521–525.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, 109–116Google Scholar
Iqbal, M. & Stachiewicz, J. W. 1967 Variable density effects in combined free and forced convection in inclined tubes. Intl J. Heat Mass Transfer 10, 1625–1629.Google Scholar
Jeffreys, H. 1928 Some cases of instability in fluid motion. Proc. R. Soc. Lond. A 118, 195–208.Google Scholar
Keller, H. B. 1977 Numerical solutions of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), pp. 359–384. Academic.
Lee, Y. & Korpela, S. A. 1983 Multicellular natural convection in a vertical slot. J. Fluid Mech. 126, 91–121.Google Scholar
Linthorst, S. J. M., Schinkel, W. M. M. & Hoogendoorn, C. J. 1980 Natural convection flow in inclined air-filled enclosures of small and moderate aspect ratio. Proc. 2nd Intl Symp. on Flow Visualization, Bochum, West Germany (ed. W. Merzkirch), pp. 93–97.
Low, A. R. 1929 On the criterion for stability of a layer of viscous fluid heated from below. Proc. R. Soc. Lond. A 125, 180–195.Google Scholar
Masliyah, J. H. 1980 On laminar flow in curved semi-circular ducts. J. Fluid Mech. 99, 469–479.Google Scholar
Meyer-Spasche, R. & Keller, H. B. 1985 Some bifurcation diagrams for Taylor vortex flows. Phys. Fluids 28, 1248–1252.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
Mullin, T. 1982 Mutations of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121, 207–218.Google Scholar
Nandakumar, K., Masliyah, J. H. & Law, H.-S. 1985 Bifurcation in steady laminar mixed convection flow in horizontal rectangular tubes. J. Fluid Mecch. 152, 145–161.Google Scholar
Osborne, D. G. & Incropera, F. P. 1985 Laminar mixed convection heat transfer for flow between horizontal parallel plates with asymmetric heating. Intl J. Heat Mass Transfer 28, 207–217.Google Scholar
Patankar, S. B., Ramadhyani, S. & Sparrow, E. M. 1978 Effect of circumferentially non-uniform heating on laminar combined convention in a horizontal tube. Trans. ASMEC: J. Heat Transfer 100, 63–70.Google Scholar
Platten, J. K. & Chavepeyer, G. 1975 An hysteresis loop in the two component Bénard problem. Intl. J. Heat Mass Transfer 18, 1071–1075.Google Scholar
Quon, C. 1977 Free convection in enclosure revisited. Trans. ASME C: J. Heat Transfer 99, 340–342.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the underside. Scientific Papers, vol. 6, pp. 432–446. Cambridge University Press.
Roache, P. J. 1972 Computational Fluid Dynamics. Hermosa.
Schaeffer, D. G. 1980 Qualitative analysis of a model for boundary effects in the Taylor problem. Math. Proc. Camb. Phil. Soc. 87, 307–377.Google Scholar
Winters, K. H. & Brindley, R. C. G. 1984 Multiple solutions for laminar flow in helically-coiled tubes. Harwell Rep. AERE-R 11373.Google Scholar
Wirtz, R. A. & Liu, L. H. 1975 Numerical experiments on the onset of layered convection in a narrow slot containing a stably stratified fluid. Intl J. Heat Mass Transfer 18, 1299–1305.Google Scholar
Yousef, W. W. & Tarasuk, J. D. 1981 An interferometric study of combined free and forced convection in a horizontal isothermal tube. Trans. ASME C: J. Heat Transfer 103, 249–256.Google Scholar