Number of convection rolls  I . Number of convection rolls function of the Rayleigh number

The stability curve summarises this influence . We can obtain it by simulating differents Rayleigh number. The stability is maximum for two rolls; so without imposing the solution with IC, the convection will always be back to a two rolls state. The first is theoretical because we did not succeed in having just one roll in a rectangular box. For the last point, we have to be carefull because the rate of flow can be turbulrnt. II . Number of convection rolls function of the initial conditions

We will study to initial conditions sets . We can do this in Fluent with :

Set 1

In order to see the dependance of the number of rolls on the initial conditions, we use the foolowing velocity functions :

V(x,y)=Vmax.cos(p.Nbx/xf ) cos(p.(y-yf)/yf)

with

 Vmax : maximum of the initial velocity Nb : number of rolls expected Nb=3
To have 3 rolls , we took Vmax around 10-5. As for two rolls, we searched a critic Rayleigh number. Before this threshold, the convection always came down to two rolls. This number is about 2851 for these IC. We can see that for this critic state that the rolls are differents.

Nb=3
In this case, we took Vmax about 10-4. The critic rayleigh number is about 8900 but we are no longer sure that the rate of flow is laminar.

Set 2

An other idea to choose the initial conditions is directly imposing the number of rolls in the cells. The velocity field becomes :

V(i)=Vo.cos(p.Nb.i /N ) sin(p. j /M)
U(j)=Uo.cos(p. i /N ) sin(p. Nb. j /M)

with

 Vo , Uo : amplitudes Nb : number of rolls expected i , j : suffix for x and y N , M : number of nodes in the two dircections

But this condition is not suffisant to impose the number of rolls. We have to choose the fitting critic Rayleigh number .

 Ra = 2015 Ra = 3403 Ra = 9275 At this point , we are not sure to have still a laminar flow. III . Number of convection rolls function of the geometry

For the same conditions, if we double the lenght of the box we can see taht the number of rolls doubles.