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TECHNICAL PAPERS: Porous Media, Particles, and Droplets

Heat Transfer Regimes and Hysteresis in Porous Media Convection

[+] Author and Article Information
Peter Vadasz

Department of Mechanical Engineering, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa

J. Heat Transfer 123(1), 145-156 (Aug 17, 2000) (12 pages) doi:10.1115/1.1336505 History: Received September 13, 1999; Revised August 17, 2000
Copyright © 2001 by ASME
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References

Nield, D. A., and Bejan, A., 1999, Convection in Porous Media, 2nd ed., Springer Verlag, New York.
Bejan, A., 1995, Convection Heat Transfer, 2nd edition, Wiley, New York.
Vadasz,  P., and Olek,  S., 1999, “Weak Turbulence and Chaos for Low Prandtl Number Gravity Driven Convection in Porous Media,” Transport in Porous Media, 37, pp. 69–91.
Lorenz,  E. N., 1963, “Deterministic Non-Periodic Flows,” J. Atmos. Sci., 20, pp. 130–141.
Sparrow, C., 1982, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York.
Vadasz,  P., 1999, “Local and Global Transitions to Chaos and Hysteresis in a Porous Layer Heated from Below,” Transport in Porous Media, 37, pp. 213–245.
Adomian,  G., 1988, “A Review of the Decomposition Method in Applied Mathematics,” J. Math. Anal. Appl., 135, pp. 501–544.
Adomian, G., 1994, Solving Frontier Problems in Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht.
Vadasz,  P., and Olek,  S., 2000, “Convergence and Accuracy of Adomian’s Decomposition Method for the Solution of Lorenz Equations,” Int. J. Heat Mass Transf., 43, pp. 1715–1734.
Elder,  J. W., 1967, “Steady Free Convection in a Porous Medium Heated from Below,” J. Fluid Mech., 27, pp. 609–623.
Braester,  C., and Vadasz,  P., 1993, “The Effect of a Weak Heterogeneity of the Porous Medium on the Natural Convection,” J. Fluid Mech., 254, pp. 345–362.
Vadasz,  P., and Braester,  C., 1992, “The Effect of Imperfectly Insulated Sidewalls on Natural Convection in Porous Media,” Acta Mech., 91, No. 3–4, pp. 215–233.
Bau, H. H., 1994, “An Engineer’s Perspective on Chaos,” Chaos in Heat Transfer and Fluid Dynamics, Arpacci, Hussain, Paollucci and Watts, eds., HTD-Vol. 298, pp. 1–7.
Vadasz,  P., and Olek,  S., 2000, “Route to Chaos for Moderate Prandtl Number Convection in a Porous Layer Heated from Below,” Transport in Porous Media, 41, pp. 211–239.
Wang,  Y., Singer,  J., and Bau,  H. H., 1992, “Controlling Chaos in a Thermal Convection Loop,” J. Fluid Mech., 237, pp. 479–498.
Yuen,  P., and Bau,  H. H., 1996, “Rendering a Subcritical Hopf Bifurcation Supercritical,” J. Fluid Mech., 317, pp. 91–109.
Mladin,  E. C., and Zumbrunnen,  D. A., 1995, “Dependence of Heat Transfer to a Pulsating Stagnation Flow on Pulse Characteristics,” J. Thermophys. Heat Transfer, 9, No. 1, pp. 181–192.

Figures

Grahic Jump Location
A fluid saturated porous layer heated from below
Grahic Jump Location
The bifurcation diagram obtained analytically from Eqs. (101112): (a) X̃ versus R, (b) Ỹ versus R, and (c) Z̃ versus R.
Grahic Jump Location
The critical time as a function of (R−Ro) for six values of initial conditions in terms of ro2. The transition from steady convection to chaos (or backwards) is linked to the existence (disappearance) of this critical time, explaining the mechanism for Hysteresis.
Grahic Jump Location
The computational results for the evolution of X(t) in the time domain for three values of Rayleigh number (in terms of R): (a) X as a function of time for R=23—the solution stabilizes to the fixed point; (b) the inset of Fig. 4(a) detailing the oscillatory decay of the solution; (c) X as a function of time for R=24.9—the solution exhibits chaotic behavior; (d) the inset of Fig. 4(c) detailing the chaotic solution; (e) X as a function of time for R=24.422—the solution is periodic; and (f ) the inset of Fig. 4(e) detailing the periodic solution (data points are connected).
Grahic Jump Location
Transitional sub-critical values of Rayleigh number in terms of Rt/Ro as a function of the initial conditions ro. A comparison between the weak nonlinear solution (— analytical) and the computational results (• computational).
Grahic Jump Location
(a) The impact of the accumulated effect of the variation of the mean Nusselt number as a function of the time range of the integration, τ1; (b) the inset of Fig. 6(a), highlighting the details of the oscillations.
Grahic Jump Location
(a) The variation of the mean Nusselt number as a function of R as obtained by solving the system of Eqs. (141516) in terms of the rescaled variables, via constant initial conditions, forward and backward variation of R, and compared with the analytical relationship, Eq. (23), for sub-transitional values of R; (b) the inset of Fig. 7(a), highlighting the transition from steady convection to chaos, and the corresponding Hysteresis effect.
Grahic Jump Location
(a) The variation of the mean Nusselt number as a function of R as obtained by solving the system of Eqs. (101112), via forward and backward variation of R, and compared with the analytical relationship, Eq. (23), for sub-transitional values of R; (b) the inset of Fig. 8(a), highlighting the transition from steady convection to chaos, and the corresponding Hysteresis effect.

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