9303779 E The investigator studies two problems in fluid flow. (1). Statistical behavior of two dimensional turbulence. There are two major problems in this area. The first is concerned with the spatial (scaling) behavior and the second the temporal scaling behavior. Classical theory attempts to answer these questions via Kolmogorov-like arguments. Recently this classical picture is challenged by a combination of very detailed numerical and analytical results of E and Majda. The investigator continues this study to establish an alternative picture for 2D turbulence. (2). Mathematical and numerical problems for incompressible flows in the presence of boundaries. In particular, the investigator studies the separation of boundary layers and the approximation to Navier-Stokes equation before the separation. Recently he has shown that in the presence of an adverse pressure gradient the steady boundary layers have to separate and separate self-similarly. Now he takes up the unsteady Prandtl equation, examining its validity in describing the boundary layers in the zero-viscosity limit. Previous results of Fife et. al. are restricted to the class of solutions satisfying some uniform bounds (independent of viscosity). The existence of such a class is in doubt. On the numerical side, the investigator studies numerical methods with vorticity boundary conditions through the introduction of a simple model problem that captures the essence of the full problem and is explicitly computable. Such a program has already been carried out for the projection method by E and Liu. One can view both problems as preparation for studying turbulence. In reality, most turbulence is generated at the fluid-solid boundary through friction. Large-scale computing offers another dimension for studying such problems. But the efficient exploitation of this new dimension also requires powerful numerical methods. At the present time, even the basic issues in formulating a nu merical method are unsettled whenever boundaries are present. Insight can be improved drastically if simple representative models are introduced. This motivates the investigator's study and design of numerical methods. Understanding of turbulence is a central problem in applied mathematics, with broad scientific and technological ramifications. ***
