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Kelvin Helmholtz Instability
Marcus Vit (Spring 2021)
Overview
An instability is when a pertubation in a fluid will continuously grow. A Kelvin Helmholtz instability is a special case which can occur in a system consisting of either two fluids with different densities or a single fluid with two distint layers. In both cases there needs to be a velocity difference between the layers to for the instability to take place. Let the subscript 1 and 2 represent the upper layer and lower layers of the fluids respectively
\[k\rho_1\rho_2(U_1 - U_2)^2 > g(\rho_2^2 - \rho_1^2)\]If this inequality is satisfied then the layers will become unstable. Since $k = 2\pi/\lambda$ it can be seen that there will always be a $\lambda$ small enough to make the inequality true if there is a difference between the densities and the velocities of the layers.
The inequality has some other implications as well. If $\rho_1 = \rho_2$ the the left side of the inequality will be zero so if there is a velocity difference the inequality is always true thus the fluid layers will always be unstable. This is referred to as a vortex sheet. It can also be seen that if there is no velocity difference ($U_1 = U_2$) then the right hand side of the inequality is always zero. If this is true then the inequality is never true so the fluid is always stable.
Below are two examples of what Kelvin Helmholtz Instability looks like in a cloud
Apparatus
- Salt water
- Fresh water
- Tank (preferably a long sealable one)
- Dye
Procedure
Start pouring the dyed salt water into the tank. Add the fresh water to the tank very slowly as the experiment won’t work if the fluid layers mix (the boundary should be as distinct as possible). The next step is creating a velocity difference between the layers. This can be done by tilting the tank such that there is a significant velocity difference or by lifting the tank and dropping it.
I was using a short, unsealed tank so my photos demonstrate the dropping method while a youtube video will demonstate the method with the sealed tank.
Results
This is one where I dropped the tank. It can be seen that there are small bumps trialing the surface wave which are the beginning of the instability. I think that they didn’t have time to fully form before the surface wave rebounded.
This picture is from a second after the above picture when the rebounded surface wave distrubs the instabilities which just results in turbulence
This a demo where the fluids used were the same but a long, sealed tank was used. The results are much clearer than the one I attempted.
https://www.youtube.com/watch?v=mf_143gkKSQ
Discussion
From the youtube video I think its clear why having a long sealable tank is preferred. A long sealed tank only needs to be tilted as this will lead to a sufficent shear between the fluid layers while having the added benefit of not adding a surface wave to the system if the tank is completely full. If an unsealed tank is used then dropping the tank to create the shear is the only real option which will form a surface wave that will disturb the instability.
Hints
I found that it was easier to get a good boundary if the fresh water was added while the tank was tilted so that there was a small area without the salt water.
The dye I used seemed like it had particulate in it which almost made it seem like the salt water had dirt in it rather than dye. I’d suggest using a darker, preferably “smoother” dye than I did.