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The case without advection
Our firt tests consisted in simulating the flow between water and fueloil
liquid (water under fueloilliquid) 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, firstorder 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.

Figure 4.1: Kelvin Helmholtz instabilities
obtained
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.

Figure 4.2: Kelvin's cat's eyes' pattern
of streamlines
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).

Figure 4.3: Numerical cat's eyes
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).

Figure 4.4: Total pressure
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.

Figure 4.5: Total pressure versus x (1),
versus y (2)
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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Stephanie Terrade
Julien Delbove
20001106