Research Papers

Combined Spectral-Perturbation Approach for Systematic Mode Selection in Thermal Convection

[+] Author and Article Information
Roger E. Khayat

e-mail: rkhayat@uwo.ca Department of Mechanical and Materials
University of Western Ontario,
London, ON N6A 5B9, Canada

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received April 22, 2012; final manuscript received January 28, 2013; published online September 23, 2013. Assoc. Editor: Sujoy Kumar Saha.

J. Heat Transfer 135(11), 111007 (Sep 23, 2013) (12 pages) Paper No: HT-12-1181; doi: 10.1115/1.4024612 History: Received April 22, 2012; Revised January 28, 2013

A nonlinear spectral approach is proposed to simulate the post critical convective state for thermogravitational instability in a Newtonian fluid layer heated from below. The spectral methodology consists of expanding the flow and temperature fields periodically along the layer, and using orthonormal shape functions in the transverse direction. The Galerkin projection is then implemented to generate the equations for the expansion coefficients. Since most of the interesting bifurcation picture is close to criticality, a perturbation approach is developed to solve the nonlinear spectral system in the weakly post critical range. To leading order, the Lorenz model is recovered. The problem is also solved using amplitude equations for comparison. The similarity and difference among the three models are emphasized.

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Bénard, H., 1900, “Les Tourbillons Cellulaires dans une Nappe Liquide,” Rev. Gen. Sci. Pures Appl., 11, pp. 1261–1271.
Dubios, M., and Berge, P., 1978, “Experimental Study of Velocity Field in Rayleigh-Bénard Convection,” J. Fluid Mech., 85, pp. 641–653. [CrossRef]
Busse, F. H., and Clever, R. M., 1979, “Instabilities of Convection Rolls in a Fluid of Moderate Prandtl Number,” J. Fluid Mech., 91(2), pp. 319–335. [CrossRef]
Aurnou, J. M., and Olson, P. L., 2001, “Experiments on Rayleigh-Bénard Convection, Magnetoconvection and Rotating Magnetoconvection in Liquid Gallium,” J. Fluid Mech., 430, pp. 283–307. [CrossRef]
Chiffaudel, A., Fauve, S., and Perrin, B., 1987, “Viscous and Inertial Convection at Low Prandtl Number: Experimental Study,” Europhys. Lett., 4, pp. 555–560. [CrossRef]
Kek, V., and Muller, U., 1993, “Low Prandtl Number Convection in Layers Heated From Below,” Int. J. Heat Mass Transfer, 36(11), pp. 2795–2804. [CrossRef]
Cioni, S., Ciliberto, S., and Sommeria, J., 1996, “Experimental Study of High-Rayleigh-Number Convection in Mercury and Water,” Dyn. Atmos. Oceans, 24, pp. 117–127. [CrossRef]
LordRayleigh, O. M. F. R. S., 1916, “On Convection Currents in a Horizontal Layer of Fluid, When the Higher Temperature is on the Under Side,” Philos. Mag. Suppl., 32(192), pp. 529–546. [CrossRef]
Kuo, H. L., 1961, “Solution of the Non-Linear Equations of Cellular Convection and Heat Transport,” J. Fluid Mech., 10, pp. 611–630. [CrossRef]
Lorenz, E. N., 1963, “Deterministic Nonperiodic Flow,” J. Atmos. Sci., 20, pp. 130–141. [CrossRef]
Curry, J. H., 1978, “A Generalized Lorenz System,” Commun. Math. Phys., 60, pp. 193–204. [CrossRef]
Hohenberg, P. C., and Swift, J. B., 1987, “Hexagons and Rolls in Periodically Modulated Rayliegh-Bénard Convection,” Phys. Rev. A, 35(9), pp. 3855–3873. [CrossRef] [PubMed]
Malkus, W. V. R., and Veronis, G., 1958, “Finite Amplitude Cellularconvection,” J. Fluid Mech., 4, pp. 225–260. [CrossRef]
Khayat, R. E., 1994, “Chaos and Overstability in the Thermal Convection of Viscoelastic Fluids,” J. Non-Newtonian Fluid Mech., 53, pp. 227–255. [CrossRef]
Khayat, R. E., 1995, “Non-Linear Overstability in the Thermal Convection of Viscoelastic Fluids,” J. Non-Newtonian Fluid Mech., 58, pp. 331–356. [CrossRef]
Khayat, R. E., 1996, “Chaos in the Thermal Convection of Weakly Shear-Thinning Fluids,” J. Non-Newtonian Fluid Mech., 63, pp. 153–178. [CrossRef]
Peltier, W. R., 1989, Mantle Convection: Plate Tectonics and Global Dynamics, Gordon and Breach Science Publishers, New York.
Davies, G. F., 2011, Mantle Convection for Geologists, Cambridge University, Cambridge, UK.
Schubert, G., Turcotte, D. L., and Olson, P., 2001, Mantle Convection in the Earth and the Planets, Cambridge University, Cambridge, UK.
Cardin, P., and Olson, P., 1994, “Chaotic Thermal Convection in a Rapidly Rotating Spherical Shell: Consequences for Flow in the Outer Core,” Phys. Earth Planet. Inter., 82, pp. 235–259. [CrossRef]
Hartmann, D. L., Moy, L. A., and Fu, Q., 2001, “Tropical Convection and the Energy Balance at the Top of the Atmosphere,” J. Climate, 14, pp. 4495–4511. [CrossRef]
Marshall, J., and Schott, F., 1999, “Open-Ocean Convection: Observations, Theory, and Models,” Rev. Geophys., 37, pp. 1–64. [CrossRef]
Rahmstorf, S., 2000, “The Thermohaline Ocean Circulation: A System With Dangerous Thresholds,” Clim. Change, 46, pp. 247–256. [CrossRef]
Downey, J. P., and Pojman, J. A., 2001, “Polymer Processing in Microgravity: An Overview,” Polymer Research in Microgravity, American Chemical Society Symposium Series No. 793, Washington, DC, pp. 2–15.
Li, M., and Xu, S., 2000, “Kumacheva E. Convection in Polymeric Fluids Subjected to Vertical Temperature Gradients,” Macromolecules, 33, pp. 4972–4978. [CrossRef]
Orbán, M., Kurin-Csöregi, K., Zhabotinsky, A. M., and Epstein, I. R., 1999, “Pattern Formation During Polymerization of Acrylamide in the Presence of Sulfide Ions,” J. Phys. Chem. B, 103, pp. 36–40. [CrossRef]
Hansen, C. M., and Pierce, P. E., 1973, “Cellular Convection in Polymer Coatings-An Assessment,” Ind. Eng. Chem. Prod. Res. Dev., 12(1), pp. 67–70. [CrossRef]
Kitano, M., and Shiojiri, M., 1997, “Bénard Convection ZnO/Resin Lacquer Coating—A New Approach to Electrostatic Dissipative Coating,” Powder Technol., 93, pp. 267–273. [CrossRef]
Sakurai, S., Tanaka, K., and Nomura, S., 1993, “Two-Dimensional Undulation Pattern on Free Surface of Polymer Film Cast From Solution,” Polymer, 34, pp. 1089–1092. [CrossRef]
Mitov, Z., and Kumacheva, E., 1998, “Convection-Induced Patterns in Phase-Separating Polymeric Fluids,” Phys. Rev. Lett., 81, pp. 3427–3430. [CrossRef]
Li, L., Sosnowski, S., Kumacheva, E., and Winnik, M. A., 1996, “Coalescence at the Surface of a Polymer Blend as Studied by Laser Confocal Fluorescence Microscopy,” Langmuir, 12, pp. 2141–2144. [CrossRef]
Sunkara, H. B., Penn, B. G., Frazier, D. O., and Ramachandran, N., 1998, “Lattice Dynamics of Colloidal Crystals During Photopolymerization of Acrylic Monomer Matrix,” J. Mat. Sci., 33, pp. 887–894. [CrossRef]
Cunningham, M. F., O'Driscoll, K. F., and Mahabadi, H. K., 1991, “Bulk Polymerization in Tubular Reactors I. Experimental Observations on Fouling,” Can. J. Chem. Eng., 69, pp. 630–638. [CrossRef]
Pojman, J. A., and McCardle, T. W., 2000, “Functionally Gradient Polymeric Materials,” U.S. Patent No. 6,057,406.
Nguyen, N. T., and Wereley, S. T., 2006, Fundamentals and Applications of Microfluidics, Artech House, Boston, MA.
Krishnan, M., Uhaz, V. M., and Burns, M. A., 2002, “PCR in a Rayleigh–Bénard Convection Cell,” Science, 298, p. 793. [CrossRef] [PubMed]
Hwang, H. J., Kim, J. H., and Jeong, K., 2009, “Method and Apparatus for Amplification of Nucleic Acid Sequences by Using Thermal Convection,” U.S. Patent No. 7,628,961.
Braun, F., 2004, “PCR by Thermal Convection,” Mod. Phys. Lett. B, 16, pp. 775–784. [CrossRef]
Busse, F. H., 1985, “Transition to Turbulence in Rayleigh-Bénard Convection,” Topics in Applied Physics, Hydrodynamic Instabilities, and Transition to Turbulence, Vol. 45, Springer, New York, pp. 97–137.
Behringer, R. P., 1985, “Rayliegh-Bénard Convection and Turbulence in Liquid Helium,” Rev. Mod. Phys., 57, pp. 657–688. [CrossRef]
Lohse, D., and Xia, K. Q., 2010, “Small-Scale Properties of Turbulent Rayleigh-Bénard Convection,” Annu. Rev. Fluid Mech., 42, pp. 335–364. [CrossRef]
Monin, A. S., and Yaglom, A. M., 1971, Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1, Dover, New York.
Normand, C., Pomeau, Y., and Velarde, M. G., 1977, “Convective Instability: A Physicst's Approach,” Rev. Mod. Phys., 49(3), pp. 581–624. [CrossRef]
Chandrasekhar, S., 1961, Hydrodynamics and Hydromagnetic Stability, Dover, New York.
Drazin, P. G., and Reid, W. H., 1981, Hydrodynamic Stability, Cambridge University, Cambridge, UK.
Cross, M. C., and Hohenberg, P. C., 1993, “Pattern Formation Outside Equilibrium,” Rev. Mod. Phys., 65, pp. 851–1112. [CrossRef]
Getling, A. V., 1998, Rayliegh-Bénard Convection, Structures and Dynamics, World Scientific, Singapore.
Schlueter, A., Lortz, D., and Busse, F. H., 1965, “On the Stability of Steady Finite Amplitude Convection,” J. Fluid Mech., 23, pp. 129–144. [CrossRef]
Newell, A. C., and Whitehead, J. A., 1969, “Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech., 38, pp. 279–303. [CrossRef]
Segel, L. A., 1969, “Distant Side-Walls Cause Slow Amplitude Modulation of Cellular Convection,” J. Fluid Mech., 38, pp. 203–224. [CrossRef]
Parmentier, P. M., Reginer, V. C., and Lebon, G., 1996, “Nonlinear Analysis of Coupled Gravitational and Capillary Thermoconvection in Thin Fluid Layers,” Phys. Rev. E, 54(1), pp. 411–423. [CrossRef]
Parmentier, P., Lebon, G., and Reginer, V., 2000, “Weakly Nonlinear Analysis of Bénard-Marangoni Instability in Viscoelastic Fluids,” J. Non-Newtonian Fluid Mech., 89, pp. 63–95. [CrossRef]
Koschmieder, E. L., 1993, Bénard Cells and Taylor Vortices, Cambridge University, Cambridge, UK.
Bodenschatz, E., Pesch, W., and Ahlers, G., 2000, “Recent Developments in Rayliegh Bénard Convection,” Annu. Rev. Fluid Mech., 32, pp. 709–778. [CrossRef]
Tritton, D. J., 1988, Physical Fluid Dynamics, Oxford Science Publications, Oxford, UK.
Saltzman, B., 1962, “Finite Amplitude Free Convection as an Initial Value Problem,” J. Atmos. Sci., 19, pp. 329–341. [CrossRef]
Vikhansky, A., 2009, “Thermal Convection of a Viscoplastic Fluid With High Rayleigh and Bingham Numbers,” Phys. Fluids, 21, pp. 1–7. [CrossRef]
Albaalbaki, B., and Khayat, R. E., 2012, “A Comparative Study on Low-Order, Amplitude-Equation and Perturbation Approaches in Thermal Convection,” Int. J. Numer. Methods Fluids, Int. J. Numer. Meth. Fluids, 69, pp. 1762–1785. [CrossRef]
Li, Z., and Khayat, R. E., 2005, “Finite-Amplitude Rayleigh-Bénard Convection and Pattern Selection for Viscoelastic Fluids,” J. Fluid Mech., 529, pp. 221–251. [CrossRef]
Albaalbaki, B., and Khayat, R. E., 2011, “Pattern Selection in the Thermal Convection of Non-Newtonian Fluids,” J. Fluid Mech., 668, pp. 500–550. [CrossRef]
McLaughlin, J., 1976, “Successive Bifurcations Leading to Stochastic Behavior,” J. Stat. Phys., 15(4), pp. 307–326. [CrossRef]
Aceves, A., Adachinara, H., Jones, C., Lerman, J. C., McLaughlin, D. W., Moloney, J. V., and Newell, A. C., 1986, “Chaos and Coherent Structures in Partila Differential Equations,” Physica D Nonlinear Phenom., 18, pp. 85–112. [CrossRef]
Curry, J. H., Herring, J. R., Loncaric, J., and Orszag, S. A., 1984, “Order and Disorder in Two-and Three-Dimensional Bénard Convection,” J. Fluid Mech., 147, pp. 1–38. [CrossRef]
Goldhirsch, I., Pelz, R. B., and Orszag, S. A., 1989, “Numerical Simulation of Thermal Convection in a Two-Dimensional Finite Box,” J. Fluid Mech., 199, pp. 1–28. [CrossRef]
Krishnamurti, R., 1970, “On the Transition to Turbulent Convection. Part 1. The Transition From Two- to Three-Dimensional Flow,” J. Fluid Mech., 42(2), pp. 295–307. [CrossRef]
Berge, P., Pomeau, Y., and Vidal, C., 1984, Order Within Chaos: Towards a Deterministic Approach to Turbulence, Wiley-Interscience Publication, NewYork.
Gelfgat, A. Y., Bar-Yoseph, P. Z., and Solan, A., 2000, “Axisymmetric Breaking Instabilities of Natural Convection in a Vertical Bridgman Growth Configuration,” J. Cryst. Growth, 220, pp. 316–325. [CrossRef]
Eckhaus, W., 1965, Studies in Nonlinear Stability Theory, Springer, Berlin, Germany.


Grahic Jump Location
Fig. 1

Influence of higher-order terms in the perturbation of the streamfunction (a), the x-dependent (b) and x-independent (c) temperature deviation. Here, k = π/2.

Grahic Jump Location
Fig. 2

Dependence of the leading- and higher-order solutions on x for the horizontal velocity component at z = 0 (a) and the vertical velocity component at z = 1/2 (b) for ε = 1 and k = π/2

Grahic Jump Location
Fig. 3

Distributions of the leading-order (a) and higher-order (b) temperature deviation. The dependence on position is shown in (c). Here, ε = 1 and k = π/2.

Grahic Jump Location
Fig. 4

Influence of higher-order terms in the perturbation of the Nusselt number as function of ε (a) and the wavenumber (b). Results based on the Lorenz model are included for reference.

Grahic Jump Location
Fig. 5

Comparison between the spectral/perturbation and amplitude equation methods. Relative difference for the streamfunction (a), the x-dependent (b) and x-independent temperature deviations.

Grahic Jump Location
Fig. 6

Comparison between the spectral/perturbation and amplitude equation methods. Relative difference for the Nusselt number.

Grahic Jump Location
Fig. 7

Error remainder in the vorticity (a) and energy (b) equations when using the spectral/perturbation solution (ε = 0.1)

Grahic Jump Location
Fig. 8

Error remainder in the vorticity (a) and energy (b) equations when using the amplitude equation solution (ε = 0.1)



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