**Development
of the instability**

In order to have a better idea of the Kelvin-Helmholtz instability, we studied its development versus the time.

To realize this study, we choosed the following parameters:

__Beater:__ frequency 5Hz, amplitude: 5mm.

__Inlet velocities:__ upper fluid: 3m/s, lower fluid: 1m/s.

__Patch and initial conditions:__ the domain of calculation is
patched, in order to have the good initial conditions. The
velocities are set for the upper and lower fluid respectively to 3
and 1m/s, and the zones of the fluids are defined, by means of the
volume fraction of fuel oil liquid (1 for the oil, 0 for the water).

The results were saved with a timestep of 0.4s, which corresponds to 2 oscillations. The calculations were continued until the periodic stationnary flow was reached.

The periodic stationnary flow was reached at the time of 2s.

In order to vizualize the phenomenon, we drew the evolution of the interface versus the time.

At the beginning, the beater begins to make the surface oscillate. We can observe that the oscillation is convected and begins to grow and to be distorded (t=0.4s).

t = 0s

t = 0.4s

For the times 0.8 to 1.6s, the first oscillation is growing and deformed a lot: it is due to the adaptation of the flow to the perturbations. This first oscillation has therefore nothing to do with the Kelvin-Helmholtz instability, only the following oscillations have to be taken in account..

t = 0.8s

t = 1.2s

From 1.6s, we can observe that the oscillations are stabilized and that they grow from the inlet to the point where they begin to deferlete. We observe also great vortexs, which are similar to the experimental observations.

t = 1.6s

t = 2.0s

We can observe 5 oscillations along the mesh. The wave length is therefore approximatively 0.4 m, which corresponds to the theoretical formula: where c = 2 m/s and f = 5 Hz in the case.

The theoretical study proved that the vortexs are generated by the vorticity. We drew therefore the vorticity magnitude in the periodic stationnary flow (t=2s). The negative vorticity is shown by the nuances of green: the stronger vorticity is shown by darker green. We placed on the picture the profile of the vortexs, in order to compare the value of the vorticity and the quality of the flow.

We can observe that the zone were the vortexs deferlete is characterized by a stronger (» -4) negative vorticity magnitude. It permits to say that our calculation respects the theory.

We also drew the Y-velocity magnitude, to analyze the influence of the Y-velocity on the development of the vortexs. In this case, we also placed the profile of the vortexs on the picture. The negative velocity is represented in blue and the positive in yellow or red.

The range of value of the Y-velocity is very little: between -1.7E-01 and +1.6E-01.

We can see that the regions which correspond to an elevation of the interface have a positive Y-velocity, and that the regions which correspond to a lowering of the interface level have a negative Y-velocity.

This illustrate the influence of the vorticity (coupled with the X and Y-velocity) on the generation of vortexs.