Our 2D study is limited to stationary case. As fluid,
we choose water.

__I - Model elaboration with FLUENT__

__- Geometry :__

We decide to use a rectangular geometry as shown in this
figure:

__- Meshing :__

Due to simple geometry, we can take a Cartesian mesh.
We choose a meshing with 80 nodes in horizontal segment and 40 nodes in
vertical segment in order to optimize results ( reduce CPU time to
obtain acceptable results precision) . So square cells are generated :

__- Boundaries conditions :__

We choose symmetric on the left and on the right
boundaries so that rolls should be completely in the domain and we might
simulate infinite domain in x direction (horizontal direction) with adjacent
rolls. Don't forget our study is bidimensional; so, variation in domain's
perpendicular direction is null . Consequently, We take effect of
wall in account only on (x,z) planes.

__Precision of computation:__

In stationary case, FLUENT allows to resolve numerically
equations with SIMPLE method (with first order precision)

__Convergence's behavior:__

As shown in this picture, residuals grow slowly
until approximately 200 iterations, then go down rapidly:

__First results :__

Below, an example of pure conduction is presented (for
DT= 0.1 K). So linear evolution of temperature is obtained:

With DT= 0.2K, the convection is established. So non linear
temperature evolution takes place.

__II - Critical Rayleigh Computation :__

__- Use of Vmax and Nu number__

Close to the threshold, phenomenon is linear; a theoretical
study allows to determinate an analytical expression of maximum velocity
:

Theoretical results are compared with numerical
ones as shown below:

We can see results are relatively good, but not really
satisfactory. It is normal insofar as FLUENT is a non- specialized industrial
code. The convection starts for Ra=1795.

The same remark as curve of Vmax is done. Here, we represent
Nu in function of Ra. Nu=1 signifies heat transfer's mode is conductive;
when Nu grows up, convection mode is established.

__III - Influence of meshing__

Two grid dimensions were tested. In the first one,
the domain was covered by 20x40 cells. Where in second one, 40x80
cells were used. Without increasing convergence criterion precision, results
with big cells were found nearest to theoretical ones.