Particle Simulations of Vortex Sheet Motion

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The investigator will develop particle simulation techniques for vortex sheet motion in two- and three-dimensional fluid flows. Vortex sheets are weak solutions of the ideal fluid equations in which the vorticity is concentrated on a surface and they are widely used in fluid dynamics to represent thin shear layers in slightly viscous flow. Vortex sheet simulations encounter severe difficulties due to Kelvin-Helmholtz instability, singularity formation, and chaotic dynamics, but the investigator has shown that these difficulties can be overcome by regularizing the Biot-Savart kernel in the equation governing the motion of material points on the sheet surface. The present project extends the capability of these regularized particle simulations by developing an improved treecode algorithm for evaluating the Biot-Savart integral, and adaptive quadrature and particle insertion schemes to maintain resolution as the sheet rolls up. In contrast to other approaches using spherical harmonics, the present treecode algorithm uses Taylor approximations in Cartesian coordinates which provides more flexibility and enables the method to be applied to nonharmonic functions such as the regularized Biot-Savart kernel. The quadrature and insertion schemes to be developed will make essential use of the Lagrangian sheet parameterization. The investigator will perform particle simulations of vortex ring dynamics and will compare the results to laboratory experiments and direct numerical solutions of the Navier-Stokes equations. Other topics to be studied include high precision computation of spiral formation in the Kelvin-Helmholtz problem, and extension of the present techniques to density-stratified flow. Computer simulation is a well established tool in basic and applied research in all areas of science and engineering. For example, automobile manufacturers, as well as pharmaceutical companies, routinely use computer simulations in the design of new products. A successful computer simulation relies on several components: a mathematical model of the physical problem, numerical algorithms for implementing the model on a computer, and computer hardware to perform the simulation. The present project focuses on the first two components by developing better models and algorithms for the computer simulation of fluid flows which are dominated by vortices. Vortices in air or water are usually invisible, but they can exert strong forces on nearby solid structures. One example is the trailing vortex wake behind an airplane which is responsible for the lift of the airplane, but also poses a hazard for nearby aircraft. In crowded urban airports with few runways, it is often necessary for an airplane to wait several minutes before taking off, to ensure that the wake of the preceding aircraft has dissipated to a safe level. The present investigation will contribute better algorithms for simulating the trailing wake which can be used by aeronautical engineers in designing methods to enhance wake dissipation and reduce takeoff delays. The algorithms developed in this project are also applicable to evaluating electrostatic forces at the molecular level, a generic computational problem with many potential applications in areas such as chemistry and plasma physics.