Theoretical Study and Resources

 This 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) Theory     A-1 Time wave If we consider two horizontal parallel streams of two different fluids moving at different speeds, and if consider an initial perturbation (step 1 on the graphic) that modifies the interface so that the elevation is sinusoidal, it can be shown that the perturbation  will follow this evolution: with : We can notice that in some case that will lead to an amplification of the perturbation: an instability (step 2). The real part can be defined as the growth rate. The condition is that s must have a positive real part: This condition tells us that if that the chance of instability will increase with the speed gap between phase. On the other side the gravity has a stabilization effect for long wavelength (when the wave number k is small) and that surface tension stabilizes short wavelength (when k is big). In most physical case a simpler expression for s can be used (both fluids with the same density, no surface tension or gravity): The further evoltion of the instability (step 3) in particular the rolling up process can not be described by these equations. 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   A-2 Spatial wave 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: with This type of wave that grow with the distance from the perturbation are called spatial waves.     B) Links and pictures   B-1 Links http://www.atmos.washington.edu/gfd_exp/exp_e/index.htm: Atmosphere and Ocean in a Laboratory. A web site on various instabilities in the ocean and the atmosphere. Gives a first look at the Kelvin Helmholtz instabilities. 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   B-2 Pictures     These pictures were taken from various locations on the net. Copyright to respective owner. Please email us if you don't want your picture to appear here.

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