Introduction.

1) Origin of the problem.

### 1) Physical problem.

Let's consider a viscous fluid between two infinite plane surfaces at different temperatures in a gravity field. We will try to find what kind of motion can appear.

### 2) Condition of motion.

If T1>T2 , then there is no motion and the temperature field is linear.

If T1<T2, then 2D steady contra-rotative rolls can appear if dT=(T2-T1) is greater than a critical value dTc. ### 3) Mecanisms and criteria of instability.

Let's take the example of a fluid that is heated from the bottom. There are three different forces acting on these particules :

• Archimede force which is an instability source :

• • Drag force which is a stability force :

• • Thermical diffusion, which also stabilize the movement, and which is defined through a caracteristic time:

• The balance between this three forces introduces the stability condition on the Rayleigh number :  with  Rac ~ 1708.

The question that comes out naturally is to know if FLUENT is able to predict correctly this critical Rayleigh number.
We are going to show in this report that such a code is quite good for determining this critical Rayleigh number under a certain number of conditions.
We will also provide to the reader some techniques to evaluate this number precisely and efficiently.

## 2) Experimental conditions.

The first thing to do for these kinds of numerical experiments is to specify the conditions of the computations.

1) The geometry of the system.

We studied the flow of a fluid in a small 2D box of size 1cm x 2cm.
The top and the bottom of this box are considered as walls.
The sides are considered as planes of symetry: this is a model for an infinite box of 1cm height.

2) The mesh.

We chose to use at first a small mesh of 5x10=50 cells. This can seem to be very few but our experiments showed that the results we obtained were quite good and the advantage is that calculation time is very short, allowing us to make a lot of iterations to determine precisely the critical Rayleigh number.

3) The physical datas and models.

For this configuration, we must consider a viscous fluid, with of course thermal properties and in laminar flow, and we can neglect the viscous dissipation.  We also chose to use the Boussinesq approximation for modeling the density: We chose to work with water which constants are:

 rho 998.2 kg/m3 mu 10-3 kg/(m.s) nu 10-6 m2/s lambda 0.6 W/(m.K) Cp 4182 J/(kg.K) kappa 10-7 m2/s alpha 1.8*10-4 K-1

Using these parameters we can write :
 Ra=C . dT, with C = 12226 (K-1).

We can also write that, with this fluid, the Rayleigh condtion for motion is :
 dT > dTc=0.1397 K.

4) The solver.

We used the models shown below:  Linear interpolation is the default procedure for computing the face pressure from cell pressures. For problems involving large body forces and high pressure gradients, we may wish to use PRESTO! Typical problems include flows with high swirl numbers, high-Rayleigh-number natural convection, high speed rotating flows and flows in strongly curved domains. This may only be used with qudrilateral and hexahedral meshes.

The other major issue is that we chose to use the QUICK scheme for the momentum and the energy iteartions, which is a very conservative scheme, that is to say that the numerical viscosity is very low.
It is important to notice it because when we made our first  computations, we used a first order upwind scheme which is much more disspative, and our results where completely wrong : the critical Rayleigh number was much too large (around 2300....)

5) The boundary conditions.

As said above, we defined wall conditions for the upper and lower face, and symetry conditions for the sides of the box.
The bottom wall temperature T2 is set to 288K for all the experiments and Ra number variation are obtained by modifying the upper wall temperature T1.

6) The operating conditions.

The reference pressure was kept at it's default value but we had to define the gravity, a reference temperature and a reference density. 7) The initial conditions.

Several types of initialization are proposed by FLUENT. We chose the simplest one which is to initialize the whole fluid at the same average temperature Ta=(T1+T2)/2. 