After studying the development of the instability versus the time, we decided to realize a stability study.
We fixed the frequency and the amplitude of the beater and also the velocity of the lower fluid and make variate the difference of velocity between the two fluids.
The aim is to determine the difference of velocity which allow the appearance of the instability.
To realize this study, we choosed the following parameters:
Beater: frequency 5Hz, amplitude: 5mm.
Inlet velocities: upper fluid: variable (from 1.3 to 5m/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 to their values, 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 when the periodic stationnary flow was reached (after 1.6 or 2.0s)
The simulation was realized for the following differences of velocities DU = 0.3, 0.4, 0.5, 0.6, 0.7, 1.0, 1.2, 1.5, 2.0, 4.0 m/s. For each case, we drew the profile of the interface for analysis.
DU<0.5 m/s
When DU<0.5 m/s, we can observe that the flow is stable. The oscillations generated by the beater decrease indeed along the domain.
DU = 0.3m/s
DU = 0.4m/s
In these cases, we can say that the flow begins to become unstable. The oscillations are stronger, and we can observe a beginning of breaking on the interface.
DU = 0.5 m/s
DU = 0.6 m/s
DU = 0.7 m/s
DU = 1.0 m/s
This permits to evaluate the stability limit to 0.7 m/s.
DU > 1.0 m/s
After the limit of 1 m/s, the flow is clearly unstable. We observe a growing of the oscillations due to the beater and strong breakings when the differential velocity increases.
DU = 1.2 m/s
DU = 1.5 m/s
DU = 2.0 m/s
DU = 4.0 m/s
The amplitude of the vortexs is function of the differential velocity: when it increases, the amplitude increases.
Moreover, we can notice that the shear appears earlier with smaller differential velocities. This phenomenon is shown by the break of the vortexs.
For the 5 Hz frequency, we can therefore evaluate the stability limit and summarize our results in the following table:
DU (m/s) |
DU<0.7 |
0.7<DU<1 |
1< DU |
Quality |
Stable |
Limit of stability |
Unstable |
The limit of stability can be evaluated to a DU = 0.7 m/s. The corresponding wave velocity is 1.35 m/s. We can notice that this velocity is much higher than the values calculated in the theoretical study: it is due to the different approach of the phenomenon. The study that we realized is a frquential one.