This grant and a companion grant in DMS (DMS-9706951) will support an interdisciplinary group drawn from the mathematics and engineering communities to study dynamic control and parametric excitations in hydrodynamic systems and to examine the interplay between parametric resonance and stabilization/destabilization of transitions/bifurcations, through an integrated program of analysis, computation and experiment. The diverse combination of the investigators' expertise offers an exceptional opportunity to develop a broadly based and innovative research program of research in dynamics and control in hydrodynamics. Two basic flow sets have been chosen in order to develop a general understanding of the complex hydrodynamics and control strategies. The two flows are variations on Taylor-Couette flow and on vortex breakdown flow. These generic flows are attractive because the degree of complexity can be precisely controlled by relatively simple changes in the flow state or boundary conditions, and at the same time they possess rich dynamics which are representative of a wide class of general hydrodynamic systems that may respond to dynamic control in a complex manner. In particular, these classes of flows may respond resonantly to the parametric excitation of the applied control. An investigation of the behavior of these generic flows will help to form a better understanding of more general time-dependent hydrodynamic systems. The interplay between dynamic control mechanisms and parametric resonance will be investigated using the (linear) Floquet theory as a first step in understanding the dynamics at the point of transition or bifurcation. Flows beyond the validity of the Floquet analysis will be studied using three different nonlinear computational approaches, each with distinct advantages and limitations. Their combined implementation is capable of resolving a wide range of problems and addressing distinct issues within each problem. The results from the experiment will be used to ref ine the analysis and control implementation. The impact of this research will not only be of a fundamental nature, resulting in a deeper understanding of the complex spatio-temporal dynamics of hydrodynamic systems, but it will also make a significant impact on several problems of practical importance, such as drag reduction and reduced fatigue due to aeroacoustic structural resonances in aeronautics, and effective control of transitions and instabilities in the materials and chemical processing industries. The new fundamental and practical insights on control dynamics of complex systems anticipated from this research can provide US industry with a competitive edge.
