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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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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