by
Olivier Antibi ( mfn01 )
Arnaud Delaunay ( mfn05 )
Note:
All of the following graphs are from CFD software : Fluent Inc
We're about to tackle the stability of a fluid placed between two infinite horizontal plane sheets, in which a difference of temperature is kept. Usually, the solution of this problem is a linear distribution of the temperature, but under particular conditions, this heavy fluid can be the cradle of a thermalconvective movement as shown below. Thus, two kinds of solution can be found, an unconditionally steady solution and a conditionally one.
This is the standard solution of Fourier's equation: .
It's obtained if . Physically speaking, Archimede's force keep the system stable, because it makes a molecule get back to his initial position and avoid it to follow thermal flux. The simulation below describes the linear field of temperature illustrating this case. Practically speaking the same situation can be observed if T_{2} is higher than T_{1} but for small differences of temperature. This results were obtained with a standard mesh 20 x 10 with a length of 20 x 10 mm. It's important to note that parallel yaxes boundary conditions are periodic as if the xaxes length was infinite.
We are now in the case of , and after many experiments the flow turned to be unstable after a precise difference of temperature. In fact, the observed contra rotary rolls embodies the concept of RayleighBenard instability. Here, the Archimede's force destabilises the system but viscous drag and forces due to thermal diffusion are unable to compensate it.
The number which is used to characterise if a situation is stable or not is called the Rayleigh number which is the ratio between Archimede's force and viscous drag :
where :
a is the thermal expansion coefficient
g is the gravity
a is the distance between sheets
dt is the difference of temperature between sheets.
ath is the thermal diffusivity
Nu is the dynamic viscosity of the fluid
Theoretically speaking, the critical value of this number, before which the flow is stable, is :
If , the situation may look like the case below if the flow remains laminar.
and the temperature profile :
We study several laminar steady flows and we increase the difference of temperature till the flow becomes unstable. We try to find in our experiments the same value as predicted in the theory. In order to manage to know if the flow is stable or not we can use several methods. The first is obviously to display velocity vectors and to evaluate if there's a movement or not. This method is quite quick but still imprecise. We preferred using a more precise method linked to the analysis of maximum velocity magnitude or Nusselt value vs. Rayleigh number.
As we can see, with this two drawing illustrating the transition between before and after the critical Rayleigh number, the difference between velocity profiles are significative and can be found relatively easily. Nevertheless this method is a bit unprecise.
As predicted by the theory, the velocity profile have a huge discontinuity.
Influence of mesh's precision
The method is the same as employed to determine the critical Rayleigh number. We select a kind of mesh and try to find its influence on the precision on the critical Rayleigh number. We studied two kind of meshes : regular and irregular ones.


Regular 10 x 5 
Regular 20 x 10 


Regular 40 x 20 
Irregular 40 x 30 
The results found for each mesh are summed up in the array below :
Mesh 
Ra_{c} 
Regular mesh 10 x 5 
1688 
Regular mesh 20 x 10 
1720 
Regular mesh 40 x 20 
1929 
Irregular mesh 40 x 30 
1970 
Here's given few examples of mesh influence on the precision of the solution taken at Ra_{c}
10 x 5 mesh 
20 x 10 mesh 
40 x 30 mesh 
To conclude with mesh influence, we can say that the critical Rayleigh number decrease with the precision of the mesh. What's striking is that with a relatively precise mesh we should obtain a critical Rayleigh number sensibly equal to the theoretical one. To illustrate this example we can show last year promotion results obtained with Phoenics ( Cf. RayleighBenard synthesis ):
Mesh 80 x 80 35 x 40 
Ra_{c} 1708 1680 
20 x 20 
1640 
The differences between the two results may come from a different treatment in Phoenics and Fluent Inc.
Influence of initial conditions
Initial conditions are very important to well understand Rayleigh Benard Instability and more precisely how you can obtain more than two contra rotary rolls. We try to implement a classic condition which allow us to initialise how many rolls as we want. The condition deal with y velocity and the expression of the function is :
where A, B, C, D are four constants . It's important to know that as D can e consider as a wave number, it characterises the number of rolls in the initial condition. An example an initial condition velocity contours is drawn below with D=4.
To implement initial conditions in Fluent we need to patch the fluid with the last function after having initialised it. The snapshot of this procedure is summed up :
With such an initialisation, we manage to obtain three rolls converged solution with a Rayleigh number of 4019. This solution is stable :
We also manage to obtain four rolls but the solution wasn't
3D mesh 20 x 10 x 10
We first need to realise a prebfc 3D mesh as drawn above.
The result is in good agreement with theory imported from plane model. It was simulated with a difference of temperature of 1° so at Ra=8038.
This hands on turns to be interesting and formative. We have improved our initial knowledge about the classic Rayleigh Benard phenomenon. It embodies a good overview of its complexity.
Theoretically speaking, it's a quite complex phenomenon which implies the help of numerical computations. The numerical model can be not too far from the theory ( 4% in some cases ) but it has to be taken into account different parameters which could interfere with the solution : the precision of the mesh, the type of the flow ( laminar, turbulent ),... If not, the results obtained can have no physical sense and this is our goal to be able to detect it.
Different studies could have been added to this handson but the results they did weren't enough convincing. We tried two main ways without success :
Influence of turbulent models which didn't because of the closeness of the deadline
Other initial conditions to obtain other figures of convection :