part summarizes some information on the theory and the
mathematical modeling of the Kelvin Helmholtz instabilities. In this
part we also put some Links to others web sites on
instabilities and some pictures of phenomena in action:
the billow clouds.
A-1 Time wave
In most physical case a simpler expression for s can be used (both fluids with the same density, no surface tension or gravity):
We call these type of waves time waves
Click here to see details of the theory. We quote the the lecture: Astronomy 202: Astrophysical Gas Dynamics, by James R. Graham, Astronomy Department, UC, Berkeley. The complete paper also describing other instabilities can be viewed at: http://astron.berkeley.edu/~jrg/ay202/lectures/lectures.html
When the initial perturbation is not spatial but over time (like the small oscillation of the wing edge), it can be shown that we have the same phenomenon with a slightly modified form. This time the evolution can be written as:
This type of wave that grow
with the distance from the perturbation are called spatial waves.
B) Links and pictures
http://www-sccm.stanford.edu/Students/witting/kh.html: A good page on the Kelvin Helmholtz instabilities.
http://www.iecn.u-nancy.fr/~sonnen/kh2.html: A page on how to generate Kelvin Helmholtz instability for real.
http://www.amath.washington.edu/~rjl/clawpack/euler/sester/KH.html: A nice animation of the Kelvin Helmholtz instability.
http://www.cita.utoronto.ca/~armitage/gallery/kelvin_helmholtz.html: another animation of the Kelvin Helmholtz instability.
http://www.iecn.u-nancy.fr/~sonnen/kh2.html: a 3D animation of the Kelvin Helmholtz instability in a plasma
http://www.itsc.com/movies/index.htm: ITSC Fluids Movie Archive, a web site with a lot of movies of flows and instabilities
http://www.met.rdg.ac.uk/radar/research/kh/: A good page on the billow clouds