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Our firt tests consisted in simulating the flow between water and fuel-oil- liquid (water under fuel-oil-liquid) without surface tension and without advection. This means that the two fluids moving in opposite direction have the same velocity magnitude and so instabilities would not be advected but develop on the spot. To be in the domain of instabilities, we have chosen a velocity of . The calculation was 2D, first-order implicit and unsteady and the time step size was . With this time step, calculations took a lot of time that's why we have used calculations ``in batch''.

The results show very clearly the formation and the development of instabilities during time. We can see that instabilities don't developt indefinitely because there a phenomenon of recovering : after a certain level, structures recover each other. The process runs in the same way again and again.

These numerical instabilities are probably generating by errors due to the grid which is very dense in the center of the domain, position of the interface.

On the following view (Fig 4.1), we can see the structures more precisely.

Theoretically, streamlines must have the famous ``cat's eye'' pattern, as viewed by an observer moving with the wave, like it is shown on Fig 4.2.
In our numerical case, we find also these cat's eyes which is a good point for comparison between thery and numerical study (Fig 4.3).
An other point that we can notice is pressure. Theoretically, in a flow presenting vortices there's a peak of pressure between vortices and a hollow of pressure at the center of each vortex.

Here again our results are in agreement with theory (Fig 4.4).

In the region between 0.25m and 0.5m, where there are four clear vortices, we have drawn the evolution of pressure versus x (Fig 4.5). The result shows the evolution of pressure following if we are between two vortices (high pressure) or inside a vortex (low pressure). The Fig 4.5 shows also the evolution of pressure following a vertical axe x=0.35.
We can notice that pressure is negative.In fact, it is due to the pressure of reference we have chosen and which is not zero in our case but the atmospheric pressure.

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