Simulation
Since there is no analytic solution to the problem, stimulation is an important way to study K-H instability. In Wang etc. 2009, they use WENO scheme to stimulate K-H instability and here is their results.
           figure3_wang
                                                                               figure4_wang
We can clearly see the develop of the instability.

Here is the link to an awesome video for the simulation of K-H instability http://www.youtube.com/watch?v=e5APRW4EIvQ.