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)=V_{max}.cos(p.N_{b}x/x_{f }) cos(p.(y-y_{f})/y_{f})
with
V_{max} : maximum of the initial velocity | |
N_{b} : number of rolls expected |
Nb=3
To have 3 rolls , we took V_{max} 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 V_{max} 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)=V_{o}.cos(p.N_{b}.i
/_{N }) sin(p.
j /_{M})
U(j)=U_{o}.cos(p._{
}i /_{N }) sin(p.
N_{b}. j
/_{M})
with
V_{o} , U_{o} : amplitudes | |
N_{b} : 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.