Today, I attended a session organized by Christian Kharif on water-waves. Three of the speakers were colleagues: Vishal, Vera, and Chris. While quite a few topics were discuss, I personally found the conversations that happened between talks the most useful. I discussed various projects with various permutations of people. The following contains a brief summary of some of the topics/projects/questions discussed.
What’s so special about 1.68?
During Chris’ talk on particle paths with linear shear flow, he made the observation that the particle paths on the free surface close for very special combinations of the wave number and the constant vorticity parameter . Specifically, when when $\latex k = 1$, the particle paths at the interface become closed trajectories as the average over the fast phase becomes zero.
I feel that I’ve come across this number in my studies on stability with constant vorticity. So far, the best I’ve been able to find is that it could be where the pressure becomes non-monotonic. I should be able to check this.
What happened to the stagnation point?
In Vera’s talk, she noted that the limiting wave for constant vorticity in the large vorticity limit approaches the limiting Crapper wave. But, if the Crapper wave is irrotational, and there’s a the stagnation point is within the fluid domain for the constant vorticity solution (when ), what happened to the stagnation point? Did they collide at the surface? If so, what happens to that stagnation point at one limits to the other? How are surface tension and vorticity alike? And for the self-intersecting solutions, what’s going on with the stream-functions and pseudo velocity potential inside the bulk of the fluid?
What about instability when there is a pressure sink?
This goes back to the stability project with Pat with constant vorticity. When there is a pressure sink in the fluid, how can that be stable? Future ideas include looking back at the normal derivative of the pressure at the free surface. The problem is that I believe it will be impossible to make this term zero for small amplitude waves. But then, what about the stability question? And why do things seem to break down in terms of the AFM formulation?
What about the rigid lid with piecewise linear shear and uniform density?
Go back and ask Vishal about this. It seems he wanted some calculations regarding this scenario; I don’t seem to recall his question. I’ll need to go back and ask.
Per our conversation, Vishal is most interested in the stability properties as they relate to the sign and magnitude of the shear in each layer.
Introducing an Observer This is related to the project with Chris. Chris is using an Ensemble Kalman filter to take measurements from a few buoys to reconstruct the wave profile. Unfortunately, his Kalman filter doesn’t seem to be converging. This could potentially be due to how he’s handing the data observations and may be improved by introducing a simple Luenberger observer. This should be relatively straight forward and shouldn’t take too long to implement. I’ll try to follow up with this sooner rather than later.
In the end, the above wasn’t necessary – there was a typo in the code!