3D Numerical Simulation of Rayleigh-Bénard Convection in a Cylindrical Container - Selected Topics of Computational and Experimental Fluid Mechanics - page 355

Subtracting Eq. ( 6 ) from ( 5 ) we obtain:

∇

2 ʔ t

u n + 1 u ∗ − p n + 1

2

u ∗ − u n + 1

= + P r ∇ ( ) . (7)

3

u n + 1 Taking the divergence of Eq. ( 7 ) and using the fact that ∇ · = 0 we obtain

the equation for the pressure:

3

2 p n + 1

u ∗ −

2

u ∗ ).

∇ =

t ∇ ·

P r ∇

( ∇ ·

(8)

2 ʔ

2.3 Initial and Boundary Conditions

As the initial conditions we consider a temperature distribution with a constant gra-
dient d , so that the pressure is hydrostatic and the fluid at rest. For the boundary

conditions we impose a non-slip condition at the solid walls for the velocity field,
whereas the temperature is set to a constant (in agreement with the imposed tem-
perature gradient) in both the lower and upper walls. For the remaining boundaries
(this includes velocity at r

ʔ T /

0) Neumann and/or Dirichlet boundary conditions are
imposed. The Neumann conditions mean that the normal derivative of

=

ˆ vanishes at

r = 0, and the Dirichlet condition implies that ˆ( r = 0 ) = 0.

3 Results

The numerical simulations were carried out with a constant Prandtl number ( P r =

0 7), which correspond to air and other gases, and some Rayleigh number above the

critical one Ra

.

= 1,708. To what concerns the geometry, we use either a cylindrical

container (0 r ext ).

Figure 1 shows the vertical component of the velocity in the container midplane.
Each red region corresponds to positive vertical velocity, while the blue regions
correspond to negative velocities. So, we can empirically establish that the number
of rolls is twice the number of red regions (or blue regions). Figure 1 a corresponds to
the steady state for R a =

< r < r ext ) or an annular domain ( r int < r <

10. We can see eight concentric circular
rolls. This figure is similar to the experimental result reported by Charru ( 2007 ).
Figure 1 b corresponds to the steady state for R a

2,000 and r ext =

= 2,500 and the annular domain,

with r int = 1 and r ext = 5. We can count fourteen convective rolls, but they are not

concentric.
Figure 2 shows the results of our simulations for R a

= 2,500 and a cylindrical

container with r ext 10 at different times. At the beginning the pattern is always

concentric but after a while (see Fig. 2 aat t

=

8), the concentric rolls begin to
deform. We can see how the concentric circular pattern starts to ripple. In Fig. 2 b,

= 13 .