First results.
1) Ra < Rac.

We first tried to work on a case where the Rayleigh number was smaller than the critical Rayleigh number.
We chose to take dT = 0.100, while dTc is around 0.140.
This first computation's goal was to check if FLUENT was able to predict that no motion should appear and that the temprature field should be linear.

1) The convergence

The picture below shows that convergence is very difficult for sub-critical cases.

In fact, as the expected velocities are very small, the residuals are of the same order than the fluid velocity, bringing very large mistakes on the x and y velocities.
Theorically, the smaller dT is, the smaller the velocities are, and the worse the convergence is.
We can guess by now that this lack of convergence will be one of the major problem to determine the critical Rayleigh number.

2) The velocity field.

The next picture shows the velocity field for the same computation:

We can see that the velocity magnitude is very low (10-7 m/s), which confirms that we are in a sub-critical motion.
We will see further that in a super-critical flow this velocity can be 1000 times larger.
This property seems to be a good criterion to decide if we are above or below the critical point.

3) The temperature field.

What is represented is the dimentionless temprature Tadim=(T-Tcold)/(Thot-Tcold).
The tempearture field is linear, as expected for a sub-critical flow.
We can also remark that in every sub-critical motion, the temperature in the horizontal medium plane is equal to the average temperature Ta=(Thot - Tcold)/2.

2) Ra > Rac.

Then we tried to compute a case where the Rayleigh number was greater than the critical Rayleigh number.
We took dT = 0.150, while dTc is around 0.140.
This computation's goal was to see if the results we obtained were qualitatively correct (if rolls appeared) and to obtain first conclusions on the convergence, on the velocity and the temperature field, etc...

    1) The convergence.

    As shown below, the congergence history has four main steps:


    1. During the first 15 iterations,  the residuals decrease all together: FLUENT uses a very dissipative scheme, to ensure the solution to converge.
    2. Then, from the 15th to the 40th iteration, the continuity residual increase a little bit, because FLUENT is now using the scheme we asked him to use and which is much less dissipative: the convergence is more difficult.
    3. After that, all the residuals decrease together again, until the 80th iterations.
    4. Finally, the residuals oscillates around a constant value: the solution is converged.

    The main interest of this graph is that it is the same one for all super-critical computations, and that it could be used to determine the critical Rayleigh number.

    We will see further that the greater e=(dT-dTc)/dTc is, the faster the solution converges (if we are still in the 2D steady motion zone, i.e. if the flow is not 3D or unsteady), but as long as dT is greater than dTc, the convergence history has still the same four-steps evolution.
    Nevertheless, we will also see that when dT is too close from dTc, these four steps are more and more difficult to see.

    2) The Velocity field.

    We can see below that, as expected, we obtain two contra rotative rolls (actually, we are very close from the critical Rayleigh number, and this mode is the only one that can be observed):

    The speed of the fluid is of the order of 0.1 mm/s, showing that we are in a super-critical motion.
    It also appears that, despite of the weak resolution of the mesh, the results seem to be correct and that confirms the fact that a rough mesh gives good results in a very short time (our simulation took around 10 seconds for 150 iterations).

    3) The temperature field.

    The next picture clearly shows that a macroscopical convection flow has appeared because the temperature field is no more linear, as it was in a sub-critical motion.


    As above, we have represented the dimentionless temperature Tadim=(T-Tcold)/(Thot-Tcold).
    We can guess from this picture that there is a flow stream going up in the middle of the box, bringing heat from the lower wall to the upper one.