Atlas: A Study of Transition in Rayleigh-Benard Convection using a Karhunen-Loeve Basis by Hakan I. Tarman

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International Conference on Mathematical Modeling and Scientific Computing
April 2-6, 2001
Middle East Technical University and Selcuk University
Ankara and Konya, Turkey

Organizers
F. Bornemann (Munich University of Tecnology, Germany), H. Bulgak (Selcuk University, Konya, Turkey), V. Ganzha (Munich University of Technology, Germany), B. Karasozen (METU, Ankara, Turkey), A. Sinan (Selcuk University, Konya, Turkey), C. Zenger (Munich University of Technology, Germany)

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A Study of Transition in Rayleigh-Benard Convection using a Karhunen-Loeve Basis
by
Hakan I. Tarman
Engineering Sciences Department, METU, Turkey

Rayleigh-Benard convection problem is the instability of a Boussinesq fluid layer on an infinite horizontal plane heated from below and cooled from above in the presence of gravity in the vertical pointing downward. The phenomena is governed by Boussinesq equations which consist of a system of five partial differential equations including continuity equation, three momentum equations and thermal equation. The dynamics is governed by three non-dimensional parameters, namely, Rayleigh number (Ra), Prandtl number (Pr) and the aspect ratio (A) of the spatial domain. The transition as Ra gradually increased beyond the pure conductive state has been extensively studied in literature both numerically and experimentally. These works reveal a picture of transition extending from the steady two-dimensional roll solution to periodic and quasi-periodic time dependence and to irregular chaotic behavior as Ra further increased. In numerous studies, various observations into the physical manifestations, the properties and the nature of these transitions were made.

Karhunen-Loeve (K-L) basis is an empirical basis in nature that can be computed from an experimentally or numerically generated database representative of the underlying physical phenomena. Since the basis is specific to the phenomena in consideration, it provides an optimal parametrization of the database (data compression) and an optimal representation of the dynamics of the phenomena.

In this work, Boussinesq equations are numerically integrated using a pseudo-spectral method on a grid and in time at the selected reference parameter values of , where is the critical Rayleigh number at which convective motion first sets in, and with horizontally periodic and vertically stress-free boundary conditions. The resulting numerical database is used to generate the K-L basis separately for the mechanical and the thermal components of the flow which, in turn, they are used to reduce the governing system of equations into a model of amplitude equations through a Galerkin procedure. In generation of the K-L basis elements, the symmetry of the governing system of equations and the geometry of the spatial domain are fully exploited resulting in a data enlargement and sharper basis elements. It is shown that the K-L basis elements each carry a physical character of the flow and obey a certain symmetry grouping which provided the grounds for a rational truncation scheme. The truncated model amplitude equations are then numerically integrated in time for a range of Ra covering the transition regime. It is shown that the known dynamics of the flow in the transition regime is completely captured by this relatively low dimensional model amplitude equations and further, the seemingly disparate results in literature are shown to be embodied in the solution of these model equations.

Date received: February 21, 2001