2-D Simulation of the Kelvin-Helmholtz Instability

What, exactly, is an instability, anyway?

   An Instability is an unstable response to a small perturbation.  For an instability to occur, a system must have a source of free energy.  Once this small perturbation to the initial equilibrium is applied, the instability will grow in amplitude in non-equilibrium stages---fed by the sources of free energy.

A bit about the Kelvin-Hemholtz Instability

   The free energy for the Kelvin-Helmholtz is provided by the kinetic energy of the anti-parallel velocity components across a plane boundary:

   An ordinary inviscid fluid is alway KH unstable in the presence of a velocity shear, and its growth rate--for the simple case with uniform density--is given by , where is the difference of the velocities across the shear flow layer and is the wave number.  This indicates that for short wave lengths (small ) the instability grows quickly, and that large velocity shear increases the growth rate.  When two shear fluid streams are adjacent to each other, the flow can be perturbed by even infintessimal perturbations.  Fluid Kelvin-Helmholtz vortices can always be seen when a velocity shear is present: in uprising cigarette smoke; at the intersection of two rivers (i.e. the Chena and Tanana river in Alaska); on the ocean and in lakes; in a coffeee cup when one pours cream into it; and in cloud formations:

"Cat's eyes"---photo credited to Kim Witting

Also, airplane wings can "stall" when the critical angle of attack is exceeded--due to turbulence produced by the Kelvin-Helmholtz instability.

The Simulation.

The simulation was serial, written in FORTRAN (visualizations routines written in IDL), and used a "leapfrog" (Dufort-Frankel) fitting scheme for the main integration:

A Lax-Wendroff scheme is used on output cycles---to fill in the "gaps."

   When surface tension was desired between the two sheared layers, it was calculated from the [rate of] curvature of the boundary---the location of which was kept track of numerically.

Results

   For the "production run," two inviscid, compressible fluids were chosen.  (The parameter file could be easily adjusted to account for these changes.)

Conclusions

   The simulation was able to very nicely reproduce--at least qualitatively--a Kelvin-Helmholtz instability.

   Unfortunately, I was not able to create/reproduce the ?self-similar?, smaller, Kelvin-Helmholtz vortex along the boundary---and this was one of the things for which I was looking.  I honestly don't know if this was due to a limitation in the resolution of the code--or if this model could not produce it.  This leaves some potential further investigation. . . .

Credit where credit is due:

   The original simulation was written by Dr. Antonius Otto and Dr. Katariina Nykyri.  The original IDL visualization routines were also written by Dr. Katariina Nykyri---with some sections possibly having been written by Dr. Antonius Otto.