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.
     

    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.