Control of Turbulence as a Chaotic System
R. D. Moser,* P. L. Boyland (Univ. of Florida), M. Heath, V. Lopez
National Science Foundation, CTS-9729189

One of the difficulties with attempting active control of turbulent flows is that the required control inputs are often large, so large that the cost of the control can exceed the gains. A new strategy for controlling turbulence using very small control inputs in being pursued. It is based on the observation that as a chaotic system, turbulence is very sensitive to perturbations; thus carefully chosen very small control inputs can produce order-one effects. The strategy being pursued is to determine unstable periodic solutions to the turbulent flow problem that have desirable features (in this case low drag) and apply control to stabilize these solutions.

Finite-Time Blowup in Semilinear Parabolic Equations
M. Short*
University of Illinois

Many real physical processes are modeled mathematically by parabolic equations describing the interaction of diffusion and some forcing function, representing, say, chemical reaction. Many of these equations have finite-time singularities, characteristic of a rapid change in the physical behavior of the system. Examples include heterogeneous reaction-diffusion, porous-medium reaction-diffusion, population dynamics, and water droplet formation problems. Knowledge of the spatial structure of blowup is a prerequisite to understanding the physical dynamics associated with such systems. In the current project, we are exploring a new idea of weak diffusion limit asymptotics, which enables us to find exact analytical solutions to many of the systems described above.

Propagation and Scattering of Coupled Surface Waves in Curved Elastic Structures
J. G. Harris*
U.S. Air Force, F49620-96-1-0190

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 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. I am working on the in-plane equations and am close to completion.

Theory of Detonation Instability
D. S. Stewart,* M. Short
U.S. Air Force Office of Scientific Research

Continuation of the project exploits relevant asymptotic limits for the standard detonation model to construct a wholly analytic theory of detonation stability. New theories for weak-heat release detonation are being considered.

Ultrasonic Inspection of Thin Coatings Using Surface Waves
J. G. Harris,* G. I. Block
National Science Foundation, CMS 98-12820

Thin films and coatings are used as thermal barriers and to increase the hardness of surfaces, especially engine parts. The goal of our work is to develop analytical expressions that describe the time delay, and transmission loss or attenuation of ultrasonic surface waves, propagating in the coatings, caused by their thinning, flaking or cracking. The coatings used as thermal barriers are such that their wavespeed steadily increases with depth until it matches that of the substrate. The coatings used for wear resistance are very thin and very fast. The defects found in both types of coating are slivers of defective material at many length scales oriented parallel to the surface.

Weakly Nonlinear Evolution Equations for Galloping and Cellular Detonations
M. Short,* D. S. Stewart
U.S. Air Force Office of Scientific Research, F49620-96-1-0260

Gaseous detonation waves propagating in a tube are highly unstable because of nonlinear coupling between exothermic heat release from the reaction and the hydrodynamic motion of the gas. Such detonations either pulsate in a periodic fashion or form a sequence of fish-scale-like patterns on the walls of the tube. Using a multiscale asymptotic approach, we are deriving weakly nonlinear evolution equations for the one-dimensional pulsating and two-dimensional cellular forms of detonation instability. Our aim is to understand the physical dynamics of the chemical-hydrodynamic interactions which cause the instability.