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TECHNICAL PAPERS: Natural and Mixed Convection

Transition in Convective Flows Heated Internally

[+] Author and Article Information
Masato Nagata

Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Japan

Sotos Generalis

School of Engineering and Applied Sciences, Division of Chemical Engineering and Applied Chemistry, Aston University, U.K.e-mail: s.c.generalis@aston.ac.uk

J. Heat Transfer 124(4), 635-642 (Jul 16, 2002) (8 pages) doi:10.1115/1.1470169 History: Received July 13, 2001; Revised November 21, 2001; Online July 16, 2002
Copyright © 2002 by ASME
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References

Wilkie, D., and Fisher, S. A., 1961, “Natural Convection in a Liquid Containing a Distributed Heat Source,” International Heat Transfer Conference, Paper 119, University of Colorado, Boulder, CO, pp. 995–1002.
Generalis, S., and Nagata, M., 2001, “Transition in Homogeneously Heated Inclined Convective Plane-Parallel Shear Flows,” in preparation.
Busse,  F. H., 1967, “On the Stability of Two-Dimensional Convection in a Layer Heated From Below,” J. Math. Phys., 46, pp. 149–160.
Clever,  R. M., and Busse,  F. H., 1975, “Transition to Time-Dependent Convection,” J. Fluid Mech., 65, pp. 625–645.
Clever,  R. M., and Busse,  F. H., 1995, “Tertiary and Quartenary Solutions for Convection in a Vertical Fluid Layer Heated From the Side,” Chaos, Solitons Fractals, 5, pp. 1795–1803.
Nagata,  M., and Busse,  F. H., 1983, “Three-Dimensional Tertiary Motions in a Plane Shear Layer,” J. Fluid Mech., 135, pp. 1–26.
Bergholz,  R. F., 1977, “Instability of Steady Natural Convection in a Vertical Fluid Layer,” J. Fluid Mech., 84, pp. 743–768.
Ehrenstein,  U., and Koch,  W., 1991, “Three-Dimensional Wavelike Equilibrium States in Plane Poiseuille Flow,” J. Fluid Mech., 228, pp. 111–148.
Wall,  D. P., and Nagata,  M., 2000, “Nonlinear Equilibrium Solutions for the Channel Flow of Fluid With Temperature-Dependent Viscosity,” J. Fluid Mech., 406, pp. 1–26.
Gershuni, G. Z., and Zhukhovitskii, E. M., 1976, Convective Stability of Incompressible Fluids, Keterpress Enterprises, Jerusalem, translated from the Russian by D. Lowish.
Chandrasekhar, S., 1961, Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York.
Schmitt, B. J., and Wahl, W. von, 1992, “Decomposition of Solenoidal Fluids into Poloidal Fields, Toroidal Fields and the Mean Flow. Applications to the Boussinesq-Equations,” in The Navier-Stokes Equations II—Theory and Numerical Methods, J. G. Heywood, K. Masuda, R. Rautmann, and S. A. Solonnikov, eds., Springer Lecture Notes in Mathematics, pp. 291–305.

Figures

Grahic Jump Location
Linear stability curves in the Gr, α plane for various values of R, as indicated
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Phase velocity curves as functions of the wavenumber α for various values of R, as indicated
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L2-norm curves as functions of Gr for various values of R and the wavenumber α, as indicated
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Phase velocity curves as functions of Gr for α=1.247,R=0. Linear analysis predictions are given by the solid line, while the dashed curve represents the phase velocity values for the nonlinear periodic equilibrium state bifurcating from the laminar state at Grc=2906.3637.
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The stream function of (a) the velocity fluctuations ∂ϕ/∂x,(b) the disturbance ∂ϕ/∂x+∫−1zǓdz, and (c) the total flow, ∂ϕ/∂x+∫−1zU⁁dz, for the secondary state α=1.247, Gr=7000,R=0
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Total mean flow (U⁁) profile for various Grashof numbers and for a fixed wavenumber α=1.62 and Reynolds number R=−500, Gr=1550, 1600, 1650, and 1800, as indicated. B represents the basic flow contribution.
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Stability of transverse traveling waves for the case of R=0. The stable region is bounded by the Eckhaus and Hopf (labeled by o (Oscillatory)) curves. Various Eckhaus curves are presented depending on the maximum growth obtained for various values of the parameter d: (a)d=0.00001,(b)d=0.006,(c)d=0.01, and (d)d=0.02.L represents the linear stability curve.
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Real part of the leading eigenvalue σ1r as a function of d for α=1.247, Gr=2908, and for fixed values of the parameter b as indicated
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Stability of transverse traveling waves for the case R=−500. The stable region is bounded by the Eckhaus and Hopf (labeled by o (Oscillatory)) curves. Various Eckhaus curves are presented depending on the maximum growth observed for various values of the parameter d: (a)d≈α/10,(b)d=0.1,(c)d=0.08, and (d)d=0.04.L represents the linear stability curve.

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