Kelvin-Helmhotz instability

Principle of Kelvin-Helmholtz instability
Mathematical theory : determination of the domain of stability
Manifestations of Kelvin-Helmholtz instability

Principle of Kelvin-Helmholtz instability

    Consider the basic flow of incompressible inviscid fluids (1) and (2) in two horizontal parallel infinite streams of different velocities U1 and U2 and densities $ \rho_1$ and $ \rho_2$, 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 Kelvin-Helmholtz 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 Kelvin-Helmholtz instability.


Example of Kelvin-Helmholtz instability


Click here for animation of simulation
Click here for animation of experiment
 

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Mathematical theory : determination of the domain of stability

    This flow can be remplaced by the superposition of a global translation of velocity$ \left( U_{1}+U_{2} \right)$ and of a symetric flow from an horizontal plane y=0 with velocity$ \pm \frac{U}{2}$($ U=U_{1}-U_{2}$) .

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 Navier-Stokes equations in each fluid. So, for incompressible inviscid fluids we have the Euler model: 

$\displaystyle \frac{\partial U_{j}}{\partial x_{j}}=0 $

$\displaystyle \rho \frac{\partial U_{i}}{\partial t}=\rho \left( \frac{\partial...... U_{j}}{\partial x_{i}} \right)=\frac{\partial P}{\partial x_{i}}+\rho F_{i} $

We look for a solution using potentials $ \Phi_{1}$ and $ \Phi_{2}$ associated to velocities $ v_{1}$ and $ v_{2}$:

$\displaystyle v_{1}=grad \left [ \frac{Ux}{2} + \Phi_{1}(x,y,t) \right] $

$\displaystyle v_{2}=grad \left [ -\frac{Ux}{2} + \Phi_{2}(x,y,t) \right] $

    We consider we have a linear approximation which means magnitude of perturbations $ \xi(x,t)$ 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: 

$\displaystyle [v_{1\bot}]_{y=\xi} = [v_{2\bot}]_{y=\xi} \cong \frac{\partial \xi}{\partial t} $

    Projecting velocity of each fluid onto the normal of the interface we have now (i=1 or 2):

$\displaystyle v_{iy} \cong \frac{\partial \xi}{\partial t} + v_{ix} \frac{\partial \xi}{\partial x} $

    By retaining only terms of first order in perturbation $ \xi$, $ \Phi_{1}$ and$ \Phi_{2}$, relations above become:

$\displaystyle v_{1y}=\frac{\partial \Phi_{1}}{\partial y} = \frac{\partial \xi}{\partial t} + \frac{1}{2} \frac{\partial \xi}{\partial x} $

$\displaystyle v_{2y}=\frac{\partial \Phi_{2}}{\partial y} = \frac{\partial \xi}{\partial t} + \frac{1}{2} \frac{\partial \xi}{\partial x} $






The case when surface tension and difference of density are absent ($ \rho_{1}=\rho_{2}=\rho$):

    In this case, there is continuity of pressure by crossing the interface:

$\displaystyle P_{1}(x,\xi,t)=P_{2}(x,\xi,t) $

    With Bernoulli equation, we can write for each fluid:

$\displaystyle P_{i}+\rho \frac{\partial \Phi}{\partial t} + \rho gy+\frac{1}{2} \rhov^{2}_{i}=Cte $

    With these two equations, we obtain for terms of first order:

$\displaystyle \left( \frac{\partial \Phi_{1}}{\partial t} \right)_{y=0} + \frac......_{y=0} - \frac{U}{2}\left(\frac{\partial \Phi_{2}}{\partial x} \right)_{y=0} $

    We look for solutions with the following form:

$\displaystyle \xi=A e^{ikx+\sigma t} $

$\displaystyle \Phi_{1}=B_{1} e^{ikx-ky+\sigma t} $

$\displaystyle \Phi_{2}=B_{2} e^{ikx+ky+\sigma t} $

with $ \sigma$ 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,$ B_{1}$ and $ B_{2}$. The condition of compatibility is obtain by saying the determinant of the matrix of the system equal to zero so :


$\displaystyle \sigma = \pm k \frac{U}{2} = \pm \frac{2 \pi}{\lambda} \frac{U}{2} $

which is the relation of dispersion for wave. Like$ Re(\sigma)$ 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: 

$\displaystyle (P_{1})_{y=\xi}=(P_{2})_{y=\xi} + \gamma \frac{\partial^{2} \xi}{\partial^{2} x}=(P_{2})_{y=\xi}-\gamma k^{2} \xi $

$ \gamma$ is the value of surface tension between the two fluids.

    In Bernoulli equation hydrostatic term$ \rho_{i}g\xi$ (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: 

$\displaystyle 4\rho_{1}\rho_{2} \left( \frac{U}{2(\rho_{1}+\rho_{2})^{2}} \righ......frac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}}+\frac{\gamma k}{\rho_{2}+\rho_{1}} $






with$ c_{0min}=\sqrt{\frac{4\gamma g (\rho_{2}-\rho_{1})}{(\rho_{1}+\rho_{2})^{2}}}$ and$ k_{c}=\sqrt{\frac{g(\rho_{2}-rho{1})}{\gamma}}$.
  


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Manifestations of Kelvin-Helmholtz instability

    In indrustry like in Nature, a lot of physical phenomenons present Kelvin-Helmholtz instability. We are going to give you some typical examples.
 

    In domain of physics, for example in aerodynamics, vortices of Kelvin-Helmholtz 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 Kelvin-Helmholtz 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 Kelvin-Helmholtz instability which develops during time.
 
 

Click here for animation





    We find also Kelvin-Helmholtz 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 Kelvin-Helmholtz 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.
 
 

Scheme of a solid propergol booster of Ariane 5 
Simulation of the instability inside the last segment of the booster 

Click here for animation





    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".
 
 

Hiroshige Utagawa "Vortices in the Konaruto stream"
 Vincent Van Gogh "La Nuit Etoilee" 

    An other painting which is very known and which represents Kelvin-Helmholtz 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 Kelvin-Helmholtz instability in clouds 


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