INFLUENCE OF THE NUMERICAL SCHEMES

 

  First and Second order  scheme:

In order to compare the influence of first and second order schemes on the calculations, we have run some simulations with two different kind of meshes.
 
Resolution with first order scheme (5*11 mesh)
Resolution with second order scheme (5*11  mesh)

 
 
Resolution with first order scheme (15*31 mesh)
Resolution with second order scheme (15*31 mesh)

We have used first and second order schemes for flow and energy equations because they are coupled in the problem studied.
As expected, two main points can be observed. First of all, it is clear that second order scheme leads to a longer convergence time. On the other hand, this kind of scheme gives more accurate results. For example, for the critical Rayleigh number, we obtain values closer from the theorical critical Rayleigh number.
Moreover, as it is underlined in the Fluent help, it is better to use a second order scheme when the flow doesn't remain parallel to the grid as it is the case here.
 

SIMPLE and SIMPLEC:

The case we ran here is a case with a Rayleigh number of 3692 and 465 nodes (15*31). As expected, in the two cases, we obtain some convective rolls as we are quite above the critical Rayleigh number.
Concerning the number of iterations necessary to converge, it is of 185 for SIMPLE and 164 for SIMPLEC. Considering the relatively low number of iterations required, the difference between the two cases is quite important.
 

Resolution with SIMPLE (465 nodes and Ra=3692)
Resolution with SIMPLEC (465 nodes and Ra=3692)

 

This case had been run with the same Rayleigh number but with a mesh of 2592 nodes. The solution converges in 936 iterations with SIMPLE and in 1016 with SIMPLEC.
 
Resolution with SIMPLE (2592 nodes and Ra=3692)
Resolution with SIMPLEC (2592 nodes and Ra=3692)

 

The following case is run with a Rayleigh number of 1723 and with a higher number of nodes (2592 nodes). It is clear that the number of iterations is much more important in this case. With the SIMPLE scheme, the solution converge in 384 iterations whereas with the SIMPLEC scheme, it converges in 345 iterations.
 
Resolution with SIMPLE (2592 nodes and Ra=1723)
Resolution with SIMPLEC (2592 nodes and Ra=1723)

 
 
 
Number of points
Ra
Number of iterations with SIMPLE
Number of iterations with SIMPLEC
N=465
3692
185
164
N=2592
3692
936
1016
N=2592
1723
385
345

What can be noticed here is that SIMPLEC seems to give better results considering convergence speed-up when the Rayleigh number doesn't get too high. This is related to a remark made in Fluent's help that for relatively uncomplicated problems in which convergence is limited by the pressure-velocity coupling, you can often obtain a converged solution more quickly using SIMPLEC.
However, when the Rayleigh number gets higher, ie the flow tends to be less stable, SIMPLEC converges more slowly. The difference is not very important but it marks a changing considering the others results.