In a natural circular loop, the thermal convection demonstrates various spatial patterns and temporal instabilities. Problem consists in determining them with respects to thermal boundary conditions. To this end a multiple scales analysis is applied which resembles the inherent characteristic of the pattern formation in the Rayleigh-Be´nard convection. A three-dimensional nonlinear model is proposed by incorporating the flow modes derived along the analysis. The differences of thermal boundary condition are reflected by a coefficient δ. For small δ, numerical solution to the model shows that only temporal instability exists and Lorenz chaos is possible, otherwise, for large values both spatial and temporal instabilities occur leading to cellular flow and intermittency chaos. The model predicted some additional phenomena opening for experimental observation. It seems significant that this study proposes an algorithm for the control of flow stability and distribution by varying the thermal boundary condition.
Issue Section:
Natural and Mixed Convection
1.
Greif
, R.
, 1988
, “Natural Circulation Loops
,” ASME J. Heat Transfer
, 110
, pp. 1243
–1258
.2.
Zvirin
, Y.
, 1981
, “A Review of Natural Circulation Loops in Pressurized Water Reactors and Other Systems
,” Nucl. Eng. Des.
, 67
, pp. 203
–225
.3.
Lorenz
, E. N.
, 1963
, “Deterministic Nonperiodic Flow
,” J. Atmos. Sci.
, 20
, pp. 130
–141
.4.
Keller
, J. B.
, 1966
, “Periodic Oscillations in a Model of Thermal Convection
,” J. Fluid Mech.
, 26
, pp. 599
–606
.5.
Welander
, P.
, 1967
, “On the Oscillatory Instability of a Differentially Heated Fluid Loop
,” J. Fluid Mech.
, 29
, pp. 17
–30
.6.
Malkus, W. R. V., 1972, “Non-Periodic Convection at High and Low Prandtl Number,” mem. Soc. R. Sci. Liege., 4kk, pp. 125–128.
7.
Creveling
, H. F.
, Depaz
, J. F.
, Baladi
, J. Y.
, and Schoenhals
, R. J.
, 1975
, “Stability Characteristics of a Single-Phase Convection Loop
,” J. Fluid Mech.
, 67
, pp. 65
–84
.8.
Gorman
, M.
, Widmann
, P. J.
, and Robbins
, K. A.
, 1986
, “Nonlinear Dynamics of Convection Loop: A Quantitative Comparison of Experiment With Theory
,” Physica D
, 19D
, pp. 255
–267
.9.
Ehrhard
, P.
, and Muller
, U.
, 1990
, “Dynamical Behavior of Natural Convection in a Single-Phase Loop
,” J. Fluid Mech.
, 217
, pp. 487
–518
.10.
Yorke
, J. A.
, Yorke
, E. D.
, and Mallet-Paret
, J.
, 1987
, “Lorenz-Like Chaos in a Partial Differential Equation for a Heated Fluid Loop
,” Physica D
, 24D
, pp. 279
–292
.11.
Jiang
, Y. Y.
, Shoji
, M.
, and Naruse
, M.
, 2002
, “Boundary Condition Effects on Flow Stability in a Toroidal Thermosyphon
,” Int. J. Heat Fluid Flow
, 23
(1
), pp. 81
–91
.12.
Wang
, Y.
, Singer
, J.
, and Bau
, H. H.
, 1992
, “Controlling Chaos in a Thermal Convection Loop
,” J. Fluid Mech.
, 237
, pp. 479
–498
.13.
Yuen
, P. K.
, and Bau
, H. H.
, 1996
, “Rendering a Subcritical Hopf Bifurcation Supercritical
,” J. Fluid Mech.
, 317
, pp. 91
–109
.14.
Yuen
, P. K.
, and Bau
, H. H.
, 1999
, “Optimal and Adaptive Control of Chaotic Convection-Theory and Experiments
,” Phys. Fluids
, 11
(6
), pp. 1435
–1448
.15.
Sano
, O.
, 1991
, “Cellular Structure in a Natural Convection Loop and Its Chaotic Behavior, I. Experiment
,” Fluid Dyn. Res.
, 8
, pp. 189
–204
.16.
Sano
, O.
, 1991
, “Cellular Structure in a Natural Convection Loop and Its Chaotic Behavior, II. Theory
,” Fluid Dyn. Res.
, 8
, pp. 205
–220
.17.
Damerell
, P. S.
, and Schoenhals
, R. J.
, 1979
, “Flow in a Toroidal Thermosyphon With Angular Displancement of Heated and Cooled Sections
,” ASME J. Heat Transfer
, 101
, pp. 672
–676
.18.
Lavine
, A. S.
, Grief
, R.
, and Humphrey
, J. A. C.
, 1986
, “Three-Dimensional Analysis of Natural Convection in a Toroidal Loop: Effect of Tilt Angle
,” ASME J. Heat Transfer
, 108
, pp. 796
–805
.19.
Stern
, C. H.
, Greif
, R.
, and Humphrey
, J.
, 1988
, “An Experimental-Study of Natural-Convection in a Toroidal Loop
,” ASME J. Heat Transfer
, 107
, pp. 877
–884
.20.
Cross
, M. C.
, and Hohenberg
, P. C.
, 1993
, “Pattern Formation out of Equilibrium
,” Rev. Mod. Phys.
, 65
(3
), pp. 851
–1112
.21.
Haken, H., 1983, Synergetics, An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology, Springer-Verlag, Berlin, Chap. 5.
22.
Landau, L. D., and Lifshitz, E. M., 1958, Statistical Physics, Addison Wesley, Reading, MA, Chap. XIV.
23.
Newell
, A. L.
, Passot
, T.
, and Lega
, J.
, 1993
, “Order Parameter Equations for Patterns
,” Annu. Rev. Fluid Mech.
, 25
, pp. 399
–453
.24.
Nicolis, G., and Prigogine, I., 1977, Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order through Fluctuations, Wiley, New York.
25.
Newell
, A. L.
, and Whitehead
, J. A.
, 1969
, “Finite Bandwidth, Finite Amplitude Convection
,” J. Fluid Mech.
, 38
, pp. 279
–303
.26.
Segel
, L. A.
, 1969
, “Distant Side-Walls Cause Slow Amplitude Modulation of Cellular Convection
,” J. Fluid Mech.
, 38
, pp. 203
–224
.27.
Pomeau
, Y.
, and Manneville
, P.
, 1979
, “Stability and Fluctuations of a Spatially Periodic Convective Flow
,” J. Phys. Lett.
, 40
, pp. 609
–612
.28.
Newell
, A. L.
, Passot
, T.
, and Souli
, M.
, 1990
, “The Phase Diffusion and Mean Drift Equations for Convection at Finite Rayleigh Numbers in Large Containers
,” J. Fluid Mech.
, 220
, pp. 187
–252
.29.
Cross
, M. C.
, 1980
, “Derivation of the Amplitude Equation at the Rayleigh-Benard Instability
,” Phys. Fluids
, 23
(9
), pp. 1727
–1731
.30.
Swift
, J.
, and Hohenberg
, P. C.
, 1977
, “Hydrodynamic Fluctuations at the Convective Instability
,” Phys. Rev. A
, 15
(1
), pp. 319
–328
.31.
Thompson, J. M. T., and Stewart, H. B., 1993, Nonlinear Dynamics and Chaos, John Wiley and Sons Ltd., New York, pp. 212–227.
Copyright © 2003
by ASME
You do not currently have access to this content.
