**Meshes
description :**

We used two kind of rectangular meshes, the first one is a 20x40 nodes mesh and second is a 50x100 nodes mesh. We also used a square mesh.

We
studied the mesh influence with the Rayleigh critical number. For each
mesh we determined the Rayleigh critical number. We obtained interesting
results.

*The
20x40 mesh :* it measures
0.02 m long and 0.01 m high.

*The
50x100 mesh :* it
measures 0.02 m long and 0.01 m high.

*The
square 40x40 mesh :*
it measures 0.02 m long.

**Simulation
results :**

We
present here the results of the steady calculation for a case where the
Rayleigh number is equal to 2500. Hence we have a
of 0.16 K for the rectangular box and of 0.02 K for the square box (the
square box is 2 time as high as the rectangular one so the temperature
has to 8=2^{3} time higher).

**The
rectangular box :**

*20x40
mesh :*

*Velocity vectors :*

*Temperature contours :*

*50x100
mesh :*

*Velocity vectors :*

*Temperature contours :*

When
we compare the results of the simulations with the two different rectangular
meshes, we notice that the velocity amplitude are different. They have
the same shape, they decrease and increase at the same place but they don't
have the same amplitude, the ratio between the amplitudes is around 5 but
it doesn't seem to be constant. We think that this difference between the
results of the simulations is related to the problems of convergence of
the calculus on the refined grid.

**The
square box ***(40x40
mesh)*** :**

*Velocity vectors :*

*Temperature contour :*

The calculus on the square
grid leeds to a single roll whose size is twice the size of the rolls in
the rectangular box. This shows that the rolls are the taller they can
be, they use the whole height of the box they're in. The convergence of
those calculus was "harder" than with the rectangular grid (20x40) because
of a much lower temperature for the same Rayleigh number (
ratio of 8 because the distance between the two walls is in a ratio of
2) hence the heat flux between the two walls is much smaller too.