Stability analysis of natural convection in porous cavities through integral transforms

Programa de Engenharia Metalúrgica e de Materiais, EE, COPPE, Universidade Federal do Rio de Janeiro, Cidade Universitária, Cx. Postal 68503, Rio de Janeiro, RJ, 21945-970, Brazil
International Journal of Heat and Mass Transfer (Impact Factor: 2.38). 03/2002; 45(6):1185-1195. DOI: 10.1016/S0017-9310(01)00231-9


The onset of convection and chaos related to natural convection inside a porous cavity heated from below is investigated using the generalized integral transform technique (GITT). This eigenfunction expansion approach generates an ordinary differential system that is adequately truncated in order to be handled by linear stability analysis (LSA) as well as in full nonlinear form through the Mathematica software system built-in solvers. Lorenz's system is generated from the transformed equations by using the steady-state solution to scale the potentials. Systems with higher truncation orders are solved in order to obtain more accurate results for the Rayleigh number at onset of convection, and the influence of aspect ratio and Rayleigh number on the cell pattern transition from n to n+2 cells (n=1,3,5,…) is analyzed from both local and average Nusselt number behaviors. The qualitative dependence of the Rayleigh number at onset of chaos on the transient behavior and aspect ratio is presented for a low dimensional system (Lorenz equations) and its convergence behavior for increasing expansion orders is investigated.

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Available from: Leonardo Alves, Feb 02, 2015
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    • "There is a vast literature about this subject, including papers and books that become impossible to mention all of them. But, for instance, we mention the works of [31, 32, 33, 34], etc. In this chapter, we restrict our attention to the linear problem, because for nonlinear problem the linear result is iteratively used after the linearization of nonlinear transformed equation. "

    Full-text · Chapter · Jan 2015
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    • "The main feature of this method relies on the following steps: stepwise approximation of the eddy diffusivity and wind speed, the Laplace transform application to the advection-diffusion equation, semi-analytical solution of the set of linear ordinary equation resulting for the Laplace transform application and construction of the pollutant concentration by the numerical Laplace transform inversion. The GITT is a well-known hybrid method that had solved a wide class of direct and inverse problems mainly in the area of heat transfer and fluid mechanics (Cotta, 1993; Cotta and Mikhailov, 1997; Cheroto et al., 1999; Alves et al., 2002; Magno et al., 2002; Neto et al., 2002 and Cotta et al., 2003). "
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    ABSTRACT: We present a three-dimensional solution of the steady-state advection-diffusion equation considering a vertically inhomogeneous planetary boundary layer (PBL). We reach this goal applying the generalized integral transform technique (GITT), a hybrid method that had solved a wide class of direct and inverse problems mainly in the area of heat transfer and fluid mechanics. The transformed problem is solved by the advection-diffusion multilayer model (ADMM) method, a semi-analytical solution based on a discretization of the PBL in sub-layers where the advection-diffusion equation is solved by the Laplace transform technique. Numerical simulations are presented and the performances of the solution are compared against field experiments data.
    Full-text · Article · Sep 2006 · Atmospheric Environment
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    ABSTRACT: Ce travail concerne la convection naturelle au sein d'un système fluide-poreux en couche horizontale. On présente l'analyse de stabilité linéaire des modèles à un et deux domaines, avec diffusion visqueuse dans le milieu poreux. Nos résultats sont comparés avec ceux du modèle à deux domaines utilisant la formulation de Darcy. Un bon accord est observé entre les résultats des modèles à deux domaines, ce qui indique que le terme de Brinkman joue un rôle secondaire dans la stabilité. On montre que le modèle à un domaine peut conduire à des résultats sensiblement différents lorsque la transition entre fluide et le milieu poreux est décrite par une discontinuité des propriétés. Il faut alors modifier la formulation en effectuant la différentiation au sens des distributions. Ainsi, le modèle à un domaine conduit aux mêmes seuils de stabilité que les formulations à deux domaines. L'influence des paramètres caractéristiques sur la stabilité des systèmes thermique et thermosolutal est discutée.
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