Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-20T07:16:33.591Z Has data issue: false hasContentIssue false
  • English
  • Français

The linear stability of mixed convection in a vertical channel flow

Published online by Cambridge University Press:  26 April 2006

Yen-Cho Chen  and
J. N. Chung
Yen-Cho Chen
Affiliation:
Oriental Institute of Technology, Panchiao, Taiwan, R.O.C.
J. N. Chung
Affiliation:
School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164-2920, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study, the linear stability of mixed-convection flow in a vertical channel is investigated for both buoyancy-assisted and -opposed conditions. The disturbance momentum and energy equations were solved by the Galerkin method. In addition to the case with a zero heat flux perturbation boundary condition, we also examined the zero temperature perturbation boundary condition. In general, the mixed-convection flow is strongly destabilized by the heat transfer and therefore the fully developed heated flow is very unstable and very difficult to maintain in nature. For buoyancy-assisted flow, the two-dimensional disturbances dominate, while for buoyancy-opposed flow, the Rayleigh–Taylor instability prevails for zero heat flux perturbation boundary condition, and for the zero temperature perturbation on the boundaries the two-dimensional disturbances dominate except at lower Reynolds numbers where the Rayleigh–Taylor instability dominates again. The instability characteristics of buoyancy-assisted flow are found to be strongly dependent on the Prandtl number whereas the Prandtl number is a weak parameter for buoyancy-opposed flow. Also the least-stable disturbances are nearly one-dimensional for liquids and heavy oils at high Reynolds numbers in buoyancy-assisted flows.

From an energy budget analysis, we found that the thermal–buoyant instability is the dominant type for buoyancy-assisted flow. In buoyancy-opposed flow, under the zero temperature perturbation boundary condition the Rayleigh–Taylor instability dominates for low-Reynolds-number flow and then the thermal–shear instability takes over for the higher Reynolds numbers whereas the Rayleigh–Taylor instability dominates solely for the zero heat flux perturbation boundary condition. It is found that the instability characteristics for some cases of channel flow in this study are significantly different from previous results for heated annulus and pipe flows. Based on the distinctly different wave speed characteristics and disturbance amplification rates, we offer some suggestions regarding the totally different laminar–turbulent transition patterns for buoyancy-assisted and -opposed flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 325 , 25 October 1996 , pp. 29 - 51
Copyright
© 1996 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

Busse, F. H. 1981 Transition to turbulence in Rayleigh—Benard convection. In Hydrodynamic Instabilities and Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), chap. 5. Springer.
Davis, P. J. & Rabinowitz, P. 1984 Methods of Numerical Integration. Academic Press.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
El-Genk, M. S. & Rao, D. V. 1990 Buoyancy induced instability of laminar flows in vertical annuli-I. flow visualization and heat transfer experiments. Intl J. Heat Mass Transfer 33, 2145–2159.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Gebhart, B., Jaluria, Y., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy-Induced Flows and Transport. Hemisphere.
Hanratty, T. J., Rosen, E. M. & Kabel, R. L. 1958 Effect of heat transfer on the flow field at low Reynolds numbers in vertical tubes. Ind. Engng Chem. 50, 815–820.Google Scholar
Hart, J. E. 1971 Stability of the flow in a differentially heated inclined box. J. Fluid Mech. 47, 547–576.Google Scholar
Kemeny, G. A. & Somers, E. V. 1962 Combined free and forced convection in vertical circular tubes — experiments with water and oil. Trans. ASME C: J. Heat Transfer 108, 339–346.Google Scholar
Maitra, D. & Raju, K. S. 1975 Combined free and forced convection laminar heat transfer in a vertical annulus. Trans. ASME C: J. Heat Transfer 97, 135–137.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr—Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Ostrach, S. 1954 Combined natural and forced-convection laminar flow and heat transfer of fluids with and without heat sources in channels with linearly varying wall temperatures. NACA TN 3141.Google Scholar
Rogers, B. B., Moulic, S. G. & Yao, L. S. 1993 Finite-amplitude instability of mixed convection. J. Fluid Mech. 254, 229–250.Google Scholar
Rogers, B. B. & Yao, L. S. 1993a Finite-amplitude instability mixed-convection in a heated vertical pipe. Intl J. Heat Mass Transfer 36, 2305–2315.Google Scholar
Rogers, B. B. & Yao, L. S. 1993b The importance of Prandtl number in mixed-convection instability. Trans. ASME C: J. Heat Transfer 115, 482–486.Google Scholar
Scheele, G. F. & Hanratty, T. J. 1962 Effect of natural convection on stability of flow in a vertical pipe. J. Fluid Mech. 140, 244–256.Google Scholar
Scheele, G. F., Rosen, E. M. & Hanratty, T. J. 1960 Effect of natural convection on transition to turbulence in vertical pipes. Can J. Chem. Engng 38, 67–73.Google Scholar
Singer, B. A., Ferziger, J. H. & Read, H. L. 1989 Numerical simulations of transition in oscillatory plane channel flow. J. Fluid Mech. 208, 45–66.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18, 513–528.Google Scholar
Tao, L. N. 1960 On combined free and forced convection in channel. Trans. ASME C: J. Heat Transfer 82, 233–238.Google Scholar
Yao, L. S. 1987a Is fully developed and non-isothermal flow possible in a vertical pipe? Intl J. Heat Mass Transfer 30, 707–715.Google Scholar
Yao, L. S. 1987b Linear Stability analysis for opposing mixed convection in a vertical pipe. Intl J. Heat Mass Transfer 30, 810–811.Google Scholar
Yao, L. S. & Rogers, B. B. 1989a The linear stability of mixed convection in a vertical annulus. J. Fluid Mech. 201, 279–298.Google Scholar
Yao, L. S. & Rogers, B. B. 1989b Mixed convection in a annulus of large aspect ratio. Trans. ASME C: J. Heat Transfer 111, 683–689.Google Scholar
Yao, L. S. & Rogers, B. B. 1992 Finite-amplitude instability of non-isothermal flow in a vertical annulus. Proc. R. Soc. Lond. A 437, 267–290.Google Scholar