Stagnation Points in Vortex Flows
H. Aref,* M. Broeens (Technical Univ. of Denmark)
National Science Foundation, CTS 93-11545

We consider the problem of locating stagnation points in the flow produced by a system of N interacting point vortices in 2-D. There is a general solution, due to a theorem by Siebeck (1864), that the stagnation points are the foci of a certain plane curve of class N -1 that has all lines connecting vortices pairwise as tangents. The theorem uses ideas from algebraic geometry. We have considered in detail the case N =3, for which Siebeck's curve is a conic. The issue of invariance of the topology of streamlines during the motion is being explored.

This research investigates mathematically the propagation of a surface wave over a curved elastic shell and demonstrates that it will couple from the exterior surface to the interior one, propagate along it, and ultimately return to the exterior. Thus, it can sense damage to an interior surface and carry that information to the exterior. Curved structures such as pipelines must be nondestructively inspected for damage at their interior surface while in service. Ultrasonic inspection using this wave provides one way to carry this out. This research also studies the scattering experienced by this wave when it strikes a fatigue crack or encounters a patch of corrosive roughness, both indicators of damage.

Fatigue cracks and diffusion bonds are two common objects of nondestructive inspection. Both are characterized by extensive areas of rough, partially contacting surfaces, where the characteristic sizes are on the order of a compressional wavelength or less at a frequency of 10 MHz. Studies of scattering of a scanned, antiplane, focused beam from model interfaces consisting of closely spaced cracks, periodic arrays of cylinders with arbitrary members missing, and from grazing sinusoids have been completed. It has been shown that there are many cases where the measured signal does not accurately represent the scatterers at the interface. Work on 3-D calculations has been started.

Numerical simulation of turbulence has generally relied on highly accurate numerical methods known as spectral methods. However, spectral methods are extremely limited in the geometries in which they can be used and this (among other things) has limited the application of turbulence simulation. A new class of numerical methods is being developed that retains much of the high accuracy of spectral methods, but which is considerably more flexible. The methods are similar to finite-element methods, but use spline expansions, which are more accurate than finite elements, but less flexible. They can be viewed as a compromise between the extreme accuracy of spectral methods and the great flexibility of finite-element methods.

A new nonlinear front-evolution equation for the propagation of the detonation shock has been derived for gaseous explosive mixtures through the use of weak curvature, slow evolution, and large activation energy asymptotic methods. The equation is the analog of the Kuramoto-Shivashinsky equation in laminar combustion theory but for detonations. The new equation exhibits solutions that mimic cellular detonation and in a different parameter range exhibits one-dimensional pulsations. The properties of this equation are being systematically explored and crafted into a new theory for detonation.