Critical Rayleigh Benard Computation



 
 
 
 
 
 

Our 2D study is limited to stationary case. As fluid, we choose water.
 
 

I -   Model elaboration  with FLUENT
 
 

- Geometry :

We decide to use a rectangular geometry as shown in this figure:
 
 













- Meshing :

Due to simple geometry, we can take a Cartesian mesh. We choose a meshing with 80 nodes in horizontal segment and 40 nodes in vertical segment in order to optimize results ( reduce CPU time  to obtain acceptable results precision) . So square cells are generated :
 
 









- Boundaries conditions :

We choose  symmetric on the left and on the right boundaries so that rolls should be completely in the domain and we might simulate infinite domain in x direction (horizontal direction) with adjacent rolls. Don't forget our study is bidimensional; so, variation in domain's perpendicular direction is null . Consequently, We take  effect of wall in account only on (x,z) planes.
 
 









Precision of computation:

In stationary case, FLUENT allows to resolve numerically equations with SIMPLE method (with first order precision)
 

Convergence's behavior:

As shown in this picture, residuals  grow slowly until approximately 200 iterations, then go down rapidly:
 
 








First results :
 

Below, an example of pure conduction is presented (for DT= 0.1 K). So linear evolution of temperature is obtained:
 
 


 
 


 
 
 








With DT= 0.2K, the convection is established. So non linear temperature evolution takes place.
 
 


 
 
 








II - Critical Rayleigh Computation :
 
 

- Use of Vmax and Nu number

Close to the threshold, phenomenon is linear; a theoretical study allows to determinate an analytical expression of maximum velocity :
 
 







 Theoretical results are compared with numerical ones as shown below:
 
 






We can see results are relatively good, but not really satisfactory. It is normal insofar as FLUENT is a non- specialized industrial code. The convection starts for Ra=1795.
 
 










The same remark as curve of Vmax is done. Here, we represent Nu in function of Ra. Nu=1 signifies heat transfer's mode is conductive; when Nu grows up,  convection mode is established.
 
 
 
 
 
 
 
 
 

III - Influence of meshing
 
 
 

Two grid dimensions were tested.  In the first one, the domain was covered by 20x40 cells.  Where in second one, 40x80 cells were used. Without increasing convergence criterion precision, results with big cells were found nearest to theoretical ones.
 



 
 
 
 


 
 
 
 
 
 
 


 
 
 
 
 
 
 
 


 
 
 

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