Principle of KelvinHelmholtz instability
Mathematical theory : determination of the domain
of stability
Manifestations of KelvinHelmholtz instability
Principle of KelvinHelmholtz instability
Consider the basic flow of incompressible inviscid fluids (1) and (2) in two horizontal parallel infinite streams of different velocities U_{1} and U_{2} and densities and , the faster stream above the other. The two fluids are unmiscible.
The horizontal boundary, corresponding to a sharp
difference of velocity in the fluid is a shearing layer. In this layer,
vorticity is approximatively uniform while it is equal to zero each side
outside of the layer as velocities are uniform. So a shearing layer appears
as a vortex sheet inside an irrotationnal flow.
Evolution of the interface  3D vortex of KelvinHelmholtz instability 
Like it is drawn of the figure above, an external perturbation may give an oscillation of the vortex sheet. Pressure in concavities is higher than pressure in convexities so the amplitude of the oscillation grows up and the upper part of the sheet is carried by upper fluid instead the lower part of the sheet is carried by lower fluid. So a tautening of the front occurs and there is a phenomenon of rolling up of the interface with a direction corresponding to the vorticity direction of the mixing layer (here positive direction for the figure).
On the figure above, you have an example of KelvinHelmholtz instability.
Example of KelvinHelmholtz instability


Mathematical theory : determination of the domain of stability
This flow can be remplaced by the superposition of a global translation of velocity and of a symetric flow from an horizontal plane y=0 with velocity() .
Scheme of the flow
The first instability which is developping is a bidimensional perturbation and it is characterized by the height of the interface above the plane y=0 and which is noted .
We have to solve classic 2D NavierStokes equations in each fluid. So, for incompressible inviscid fluids we have the Euler model:
We look for a solution using potentials
and
associated to velocities
and :
We consider we have a linear approximation which means magnitude of perturbations is little considering their wave length.
The velocities of fluids perpendicular to the interface must be equal for each fluid and equal to the velocity of the interface:
Projecting velocity of each fluid onto the normal
of the interface we have now (i=1 or 2):
By retaining only terms of first order in perturbation ,
and,
relations above become:
The case when surface tension and difference of density are absent ():
In this case, there is continuity of pressure by
crossing the interface:
With Bernoulli equation, we can write for each fluid:
With these two equations, we obtain for terms of
first order:
We look for solutions with the following form:
with which is the rate of development of perturbation and k the wave length.
By injecting these solutions in last equations we obtain a linear homogeneous system with the unknown variables A, and . The condition of compatibility is obtain by saying the determinant of the matrix of the system equal to zero so :
which is the relation of dispersion for wave. Like
can be positive,we can say that there's always an instable state for this
kind of flow, and amplification of perturbation is more important when
wave lengths are little.
The case when surface tension and difference of density are not absent:
Phenomenons of difference of density and surface
tension are known as to stabilize the flow.
Now, at the interface we don't have equality of pressure because of
surface tension:
In Bernoulli equation hydrostatic term (i=1 or 2) must be introduced.
Like in the last section we have a condition of stability by saying the determinant of the matrix of the system is equal to zero:
with
and.
Manifestations of KelvinHelmholtz instability
In indrustry like in Nature, a lot of physical phenomenons
present KelvinHelmholtz instability. We are going to give you some typical
examples.
In domain of physics, for example in aerodynamics, vortices of KelvinHelmholtz develop behind vehicles such as trains or break shape cars. Effectively, behind these vehicles, there is a big difference of level and the air flow meets a zone where there is no velocity. In fact, it is a flow behind stair and with certain conditions of flow KelvinHelmholtz instability can appear in this zone.
So, a lot of simulations have been done about flow behind stair to give more informations.
Simulations of flow behind stair for two different times
In this simulation, we can see this king of
flow for two different times. It shows KelvinHelmholtz instability which
develops during time.
We find also KelvinHelmholtz instability due to
flow behind stair in the atmosphere where spiral vortices can appear in
downstream direction of a mountain. It is very dangerous for people who
practise flying wing because they can be lied flat on the mountain side
if they don't consider this phenomenon.
In motors of planes or spatial engines like boosters
of Ariane 5 there may be also KelvinHelmholtz instabilities.
In motors, vortices assure a good and necessary
mixing of combustible with air. In this case, instabilities are important
for the smooth working of motors.
But in some physical problems, instabilities must
be avoided. For example, in solid propergol booster of Ariane 5, if frequency
of vortices is the same than frequency of acoustic wave developping inside
the booster there is a phenomenon of aeroacoustic coupling which can lead
to the deterioration of the motor.



In Nature, this kind of instability is also visible
in oceans or rivers. For example, it is the case at the "Cape of the Hague",
in the Channel in France, where there are structures behind the cape. In
rivers, these structures can appear when there is a sudden widening of
the river bed. This phenomenon is not new and it has been represented by
Hiroshige Utagawa in his painting "Vortices in the Konaruto stream".


Vincent Van Gogh "La Nuit Etoilee" 
An other painting which is very known and which represents
KelvinHelmholtz instability is the Van Gogh one called "La Nuit Etoilee"
where the artist paints the exceptional phenomenon of instability in clouds.
This phenomenon is very rare and exceptional but people can see it some
days like the photographies below prouve it.

Photographies of KelvinHelmholtz instability in clouds