Francois DABIREAU
Sebastien MASSART
3DGEOMETRY
2.4. Critical number of Rayleigh
2.5. Example of 3D geometry : a piece of cylinder
2.6. Relation between Nusselt and Rayleigh numbers
2.7. Determination of the critical number of Rayleigh depending on the factor radius/height
Let's consider a viscous fluid which can transfer heat contained between two plane surfaces which are at different temperatures.
We will try to find what kind of motion can appear.
If TUp>TDown , then there is no motion and the temperature distribution is linear.
If TUp<TDown, then 2D stationary contrarotative rolls appear.
1.3. Mecanisms and criteria of instability.
Let's take the example of a fluide which is heat by the down plane.
There are three different forces which act on particules :
Archimède force which is an instability source :
Drag force which is a stability force
Thermical diffusion which plays the same role as drag force
The balance between this three forces introduces the stability
condition on the Rayleigh number :
We choose to illustrate the Rayleigh Benard instability in a spherical box.
The dimensions of our cylindrer are a diameter of 4 cm and a hight of 1cm.


We wanted to know the influence of the mesh because there are many difficulty to adapt a grid to a circular aera.
First we used a BFC mesh without adjusting mapped nodes.
Then we tried to specify the weight of the eight corners of the initial cube.
We began with increasing the weight, and then with decreasing it.
Finally we choosen an unstructured mesh.
Normal BFC mesh 
BFC mesh constrict in the boundary 
BFC mesh constrict in the center 
Unstructured mesh 
We have done the same experience for each mesh at Ra=8021.7.
This mesh does not converge. So we have to play whith underrelaxation
So we can obtain the following results :
Xplane velocity 
X=Yplane velocity 
Z=h/2plane velocity 
Z=h/2plane temperature 
2.3.2 BFC mesh constrict in the boundary
The problem of this mesh is exactly the same as the last case.
Xplane velocity 
X=Yplane velocity 
Z=h/2plane velocity 
Z=h/2plane temperature 
2.3.3 BFC mesh constrict in the center
This mesh is the most adapted because the solution can converge without any particular condition.
Xplane velocity 
X=Yplane velocity 
Z=h/2plane velocity 
Z=h/2plane temperature 
In fact the results are non axisymetric because of the BFC mesh which is based on a tranformation of a cube .
This mesh can resolve the problem of the last meshs.
Xplane velocity 
X=Yplane velocity 
Z=h/2plane velocity 
Z=h/2plane temperature 
This mesh make the solution converge but it does not resolve the problem of an axisymetric results.
We tried to determine the relation that exists between the Rayleigh number and the geometrical factor radius/hight in the case of the cylinder.
A experimental graph gives a relation we wanted to find with Fluent.
First, we determined the value of the critical Rayleigh for h=2, the factor used in the report (0.02/0.010).
We found Rc~1765. This value is not so different of the theorical 1708...
The following pictures are the result of many different simulations for many different values of the temperature gradient.
You can easily notice the value of the temperature gradient significant : the gradient of temperature becomes non linear.
h=2 dT=0.2oC
h=2 dT=0.21oC
h=2 dT=0.22oC
h=2 dT=0.3oC
h=2 dT=0.35oC
2.5. Example of 3D geometry : a piece of cylinder.
The aim is to observe the reaction of FLUENT with such a geometry.
We will be able to compare with the results we have in the case of the whole cylinder.
(the caracteristics are developped in the precedent parts of this report).
You can discover on the first picture the geometry of the "system" used for the simulation .
This is a simple part of a cylinder.
It is made of six surfaces, in order to simplify the construction.
This first picture comes from PreBFC.
You can see on these two others pictures the mesh of the geometry.
The particularity of this mesh : there is only one cell in angledirection (the Y direction in rectangular representation).
2.5.3. Results of some different simulations:
This image gives you a vision of the 3D results you can obtain with some degrees between the two plane surfaces.
TEMPERATURE vectors in a 2oC simulation:
We find nearly the same results as the whole cylinder. There are two contrarotative rolls really significant.
This result is the proof of a realistic simulation according to the second part of this report.
VELOCITY in a 2oC simulation:
VELOCITY in a 2^{o}C simulation:
VELOCITY vectors in a 5oC simulation:
With an increase of the difference of temperature, we observe a modification of the structure of the rolls.
In conclusion of these simulations with a part of a cylinder, we can say it is simple to simulate the RayleighBenard instabilities with FLUENT, in many different geometries.
And the most important results are the same : we can easily observe the contrarotative rolls...
2.6 Relation between Nusselt and Rayleigh numbers
We wanted to know if the relation between the Nusselt number and the Rayleigh number remains valid in this particular geometry.
This last image show that we have to separate two cases:
Low number of Rayleigh
High number of Rayleigh
2.7. Determination of the critical number of Rayleigh depending on the factor radius/hight
DIMENSION FACTOR (Radius/Height)  CRITICAL DELTA T  CRITICAL RAYLEIGH 
2 
0.22 
Rac~1765 
4 
0.0... 
SIMULATIONS RESULTS
Conclusion : in fact, the results are not precise because of the mesh which was not adapted to the geometrical dimensions.
Others simulations must be done to really evaluate the dependance of the Rayleigh with the geometrical caracteristics in the case of a cylinder...
Because of the time it takes to simulate the different situations, it has not been possible before the deadline, but it would be very interesting to verify the experimental results.
It seems that Fluent can modelise 3D instabilities in a geometry more complexe than a simple cube.
However, the structured mesh necessitates an adapted geometry. Indeed we observed deformations of the rolls on the boundaries walls.
A larger geometry could resolve this problem.
Moreover, the mesh must be modified to make the solution converge.
This kind of study opens a large view of research.