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3D Numerical Simulation of Rayleigh-Bénard
Convection in a Cylindrical Container
N.Y. Sánchez Torres, E.J. López Sánchez, S. Hernández Zapata
and G. Ruiz Chavarría
Abstract
The heat transport by natural convection is a central mechanism in
the explanation of many natural phenomena. Despite many existing work on the
Rayleigh-Bénard convection, often the phenomenon is studied by making a two-
dimensional approach or using a rectangular container. In this work, we solve numer-
ically the Navier-Stokes, continuity and energy equations in cylindrical coordinates.
To this end a finite difference scheme is used for the time and spatial coordinates
r
and
z
, whereas a Fourier spectral method is used for the angular coordinate. The
advantage of this procedure is that it can be easily parallelized. The numerical results
include the formation of concentric rolls and other patterns, which are compared with
experimental results reported in the literature.
1 Introduction
An important problem in Fluid Mechanics is the study of convection. In this paper
we focus our attention on a liquid layer heated from below and initially at rest.
Depending on a dimensionless parameter, that is, the Rayleigh number (
R
a
), the
final state can be a pattern of cells or even the transition to a turbulent state. The
stability theory, the experiments and numerical simulations agree in that the critical
Rayleigh number is 1708 (Chandrasekhar
1961
; Rayleigh
1916
; Guyon et al.
2001
;
Bodenschatz et al.
2000
). Below this value the fluid remains at rest. Analytical results
on this problem have been obtained within the framework of hydrodynamic stability,
which provides information about critical values of dimensionless parameters and
the wavenumber of the most unstable perturbation. Often the stability analysis is
performed assuming small two-dimensional disturbances so that nonlinear terms are
neglected (Drazin and Reid
1981
; Chandrasekhar
1961
). The factors influencing the
growth of instabilities are viscosity, bouyancy and the surface tension if the upper
boundary is a free surface.
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