3D Numerical Simulation of Rayleigh-Bénard Convection in a Cylindrical Container - Selected Topics of Computational and Experimental Fluid Mechanics - page 352

When Bénard began to study this problem, he decided to keep free the upper
surface. He observed the formation of hexagonal cells [Bénard-Marangoni problem,
(Bénard 1900 ; Guyon et al. 2001 ; Bodenschatz et al. 2000 )]. When Rayleigh studied
this problem, he assumed that all boundaries of the fluid layer are solid walls. He
observed the formation of rolls instead of hexagonal cells [Rayleigh-Bénard prob-
lem, hereafter referred to as the R-B problem (Chandrasekhar 1961 ; Rayleigh 1916 ;
Guyon et al. 2001 ; Bodenschatz et al. 2000 )].
In this work we solve numerically the equations governing the R-B convection in
a cylindrical container (3D). This problem has been studied previously (Getling and
Brausch 2003 ). In most cases the geometry was a rectangular box (Valencia 2005 ;
Ternik et al. 2013 ), or cylindrical one (Tagawa et al. 2003 ;Lietal. 2012 ; Paul et al.
2003 ). The numerical simulation was done by using a Fourier spectral method (Boyd
2000 ;Peyret 2002 ) for the angular coordinate and finite differences for the radial and
vertical coordinates. In addition, we use a semi-implicit Adams-Bashforth second
order scheme. The choice of a Fourier spectral method was made on the basis that
it is easy to run the code in parallel provided the Adams-Bashforth scheme is used.
Under this condition we have for each Fourier mode a system of equation decoupled
from other modes. Finally the Navier-Stokes equations are solved with the projection
method.
This paper is organized as follows. In Sect. 2 , we expose the methodology used
in the numerical study, including the differential equations for the R-B convection.
In Sect. 3 , we show some numerical results and a comparison with cases reported in
the literature. In the last section, we draw some conclusions.

2 Methodology

For the study of convection of a liquid heated from below we need to solve the
three components of the Navier-Stokes equations along with the continuity and the
energy equations. In order to compare with other results, we need to write down the
equations in dimensionless form. To this end we use

d as a representative velocity,
the characteristic length is the height of the container d and the characteristic time
is d 2

ʺ/

is the thermal diffusivity coefficient. In addition, we assume that
the density depends on temperature. We use the Boussinesq hypothesis according
to which the density is left constant except in the bouyancy term. The equations to
solve are:

/ʺ , where ʺ

∇ ∗ · u ∗ = 0

u ∗
∂

∂

2

u ∗ · ∇ ∗ ) u ∗ =−[ g ∗ − T ∗ − T 0 ) ] k − ∇ ∗ P ∗ + P r ∇ ∗ u ∗

t ∗ + (

R a P r (

T ∗
∂

∂ 2

u ∗ · ∇ ∗ T ∗ =∇ ∗ T ∗ ,

+

t ∗