Beale The investigator undertakes four projects related to incompressible fluid flow. The first continues work on numerical methods for the motion of inviscid fluid interfaces, such as water waves. In previous work, convergent methods of boundary integral type have been developed, using a new approach for the analysis of fluid interfaces. In this project, methods of more general applicability are designed that still maintain full numerical stability. In a related project, improved analytical understanding is sought for existence and regularity in exact solutions of the fully nonlinear equations of water waves. Instabilities in exact vortex rings with swirl are studied, comparing predictions from a short-wave asymptotic analysis with direct simulation using vortex methods, in which computational elements of vorticity move with the flow. These two very different approaches should allow detailed treatment of this test case for time-dependent, inviscid three-dimensional flow. Finally, the approximation of the Navier-Stokes equations of viscous flow by fractional time steps are investigated, in which inviscid flow alternates with linear viscosity, and artificial boundary layers are generated. Fluid flow is important in diverse applications, including aircraft design, industrial processes, and the motion of the oceans and atmosphere. Often quantitative prediction is difficult because of the development of small scales in the flows -- for instance, local disturbances in the general behavior of the flows. One role for mathematical analysis is the design of improved numerical methods. For complicated equations, such as those of inviscid fluid interfaces, numerical instabilities have been difficult to avoid. Carefully designed methods, such as those developed here, could be used for better predictions of the behavior of water waves. Basic analytical understanding of solutions is often closely related to the analysis of numerical approximations. The com parison of two methods for predicting instabilities in fluid flow, short-wave analysis and direct simulation by vortex methods, should help to establish the realm of validity of each approach. The use of the two together for a quantitative study of exact solutions, such as vortex rings with swirl, is advantageous for better understanding of nonlinear dynamics in realistic three-dimensional flow. The fractional step approximation of the Navier-Stokes equations replaces the full evolution by two parts of more special character. Some numerical methods take advantage of this dual structure. Better understanding of such approximations, especially the boundary behavior, could suggest more accurate versions of these numerical schemes.
