II. 2D unsteady instabilities.

*I.1 Fluid initially immobile :*

The number of rolls we obtain depend on the number of rolls imposed by the perturbation but it depend also on the Rayleigh number.

The rolls are circles and contrarotatif**.**

With Rayleigh =3000 we get 2 rolls. (cf. fig. 1)

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__fig.1 velocity Vectos.__

__fig. static Temperature __

*I.2 sinusoidal Initialisation.*

If we impose a perturbation of the form A*sin(k*pi*x/L)*sin(2*pi*y/d). with k the number of rolls In the steady case we don't get always the same number of rolls. This can be explained by the fact that the proper modes amplified have a k value itch is a multiple of k0. Notice that we can compute the k value with the following formula : k = (2p /L)*Nb

L is the length of plaque. Here it's set equal to 2cm.

Nb is the number of rolls. The geometry dimensions impose a dominant mode with two rolls.

The final number of rolls depend on the initial number of rolls imposed and on the Rayleigh number. We have tested several initial values of k. for example with 4 rolls we get two rolls for all Raleigh number greater than critical Rayleigh number. Generally, for a pair values of k we get two rolls all the time in the final velocity vectors field.

In order to get three rolls in final velocity vectors field, we should take a impair values of k, and for specific zone of Rayleigh number and specific amplitude of perturbations. But this case is not stable. Effectively, if we increase the number of iterations we go back to the first case (two rolls).

In the other hand, we don't obtain one roll even if we have tested different initial conditions and different Rayleigh number.

*Explication *: in addition to the initial perturbation
introduced, the code generate his own numerical perturbations that cover
a large game of k values. The mode corresponding to k = 2 will be amplified
more than the others.

Starting with a perturbation of 4 rolls. In each case we get a different number of rolls.

In this part we attempt to study instabilities time evolution, especially the passage from 4 rolls to 2 ones. To do this we track the transition phase. We notice that is transition phase, each 2 rolls are merged together and form at the end only one roll. The two rolls nucleus join each other.

The instabilities represented by convector rolls are a primitive form of these ones.

2D instabilities are a first level. But for greater values of Rayleigh the flow reach a superior level in instabilities. It's the 3D instabilities.

For computational time raisons, with PREBFC we construct a grid in tree dimension, we have chosen a box with these dimensions.

Height : 1cm

Depth : 8cm

Width : 6cm

__Convergence problems:__

So as to have relatively quick convergence we don't choose a very thin grid, and we try several values of the relaxation parameter. The grid size used is 40*20*40. The relaxation parameter is set equal to 0.2.

With Rayleigh number at 5000, and with 4 rolls initialisation we require this results:

__fig.4 vector velocity before 100 iteration.__

__fig.5 x velocity before 100 iteration.__

__fig.6 y velocity before 100 iteration.__

If the Rayleigh number keep sensibly greater than Rayleigh critical number, we don't see instability in the Z direction. The 4 rolls configuration don't change and the flow keep 2D. If we continue to increase Ra the results are hazardous and there isn't a structured geometry. The system become very sensible to weak perturbations and unsteady. In this case we should use an unsteady turbulent model.

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