Theory of the Kelvin-Helmholtz instability
| Modelization
| Conditions
of stability |
Stability curve
|
Modelization of the flow
We can modelize the Kelvin-Helmholtz
instability by the flow of two fluids with different densities and
axial velocities in a rectangular channel. We replace this flow by
the superposition of two flows: a global translation with the
velocity U and a symetrical flow with a velocity of
.
This modelization corresponds to a spatial study of stability.
Boundary conditions at the interface
On the interface, the two normal
velocities must be equal to the velocity of the interface:
(1)
There is also a condition with the
pressures:
(2)
UP
Condition of stability
After some transformations and the writing of the compatibility relation between the equations, we obtained the following stability conditions:
This condition can be written in another way:
(3)
Supplementary conditions
The Reynolds number of the flow must be superior than 1: Re>>1
The first instability which develops (the studied case here) is bidimensionnal.
Another sufficient condition for instability is to put the heaviest fluid in the top layer. As it is not very physical and to avoid simulations problems, we do not use this condition in our calculations.
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Minima
The condition (3) permits to determine
the minimum condition for the appearance of the instability: the
minimum wave propagation velocity is
and the corresponding wave number is
.
Stability curve
With the condition of stability and the chosen fluids for our simulations (see Parameters), we can draw the theoretical curve of stability versus the wave number (and thus versus the wave length). We obtain the following curve:
This curve corresponds to the awaited results for a spatial study.
Stability conditions for a test case
For the chosen fluids, we calculate the stability conditions for a chosen wave length of 0.4 m.
With the formula (3), we obtain that U must be higher than 0.226 m/s to observe the instability.
Other way to obtain the instability curve
Another way to determine the instability curve is to place a beater at the inlet of the channel with the two fluids. The frequency and the amplitude of the beater can be fixed to desired values. Then, for a fixed frequency the point of unstability can be found by means of variations of the velocity of the upper fluid. For many frequencies, it is then possible to obtain the stability curve, in function of the frequency.
This study cannot be compared with the study above: it is indeed the other part of the stability theory, the frequencial study, for convective flows. In this case, the limit of stability is different from the limit given by the spatial theory, except for little mean velocities.
In our work, we will use this type of analysis, with a basis velocity for the lower fluid of 1 m/s.
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