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Singularity Formation in 3-D Vortex SheetsOptimization
|Speaker: ||Tom Hou, Caltech|
|Location: ||202 Wellman|
|Start time: ||Fri, Dec 8 2000, 4:10PM|
We study singularity formation of 3-D vortex sheets using a new approach.
First, we derive a leading order approximation to the boundary integral
equation governing the 3-D vortex sheet. This leading order equation captures
the most singular contribution of the integral equation. Moreover, after
applying a transformation to the physical variables, we found that this
leading order 3-D vortex sheet equation de-generates into a two-dimensional
vortex sheet equation in the direction of the tangential velocity jump.
This rather surprising result confirms that the tangential velocity jump
is the physical driving force of the vortex sheet singularities. We also
show that the singularity type of the three-dimensional problem is similar
to that of the two-dimensional problem. Furthermore by using the abstract
Cauchy-Kowalewski theorem, we prove the long time existence of 3-D vortex
sheets for analytic initial data. The existence time can be arbitrarily
close to the singularity time predicted by the leading order equation when
the initial condition is near equilibrium. A generalized Moore's
approximation to 3-D vortex sheets is introduced. Detailed numerical study
will be provided to support the analytic results, and to reveal the generic
form of the three-dimensional nature of the vortex sheet singularity.