Mathematical Sciences: Computational Study of Vortex Sheet Motion

研究人员将开发改进的数值方法来模拟流体流动，重点是涡片模型。该方法基于涡量法，其中拉格朗日点携带涡量。工作假设是普朗特尔的思想，即涡旋片是微粘性流动中自由剪切层的渐近模型。本文的第一个任务是发展一种快速求和方法来研究开尔文-亥姆霍兹问题中的螺旋标度性质。其他项目是模拟具有现实边界条件的分流板流动，以及平面和轴对称涡旋板卷起的比较。研究者将用纳维-斯托克斯方程的有限差分解和现有的实验室实验来比较涡旋模拟。一个关键的问题是获得精确的数值解，以便对各种建模假设的有效性作出精确的陈述。流体动力学与气体（如空气）和液体（如水）的运动有关。一个关键的目标是了解作用于流体和浸入固体表面之间的力。例如，通过机翼的气流产生了支撑飞机飞行的升力。流体流动在许多科学技术领域都很重要，从内燃机到天气预报再到血液流动。描述流体运动的数学方程很难求解，因此研究者正在开发用于高速计算机的数值算法。目标是提供洞察流体运动的物理，并创建建模工具，以帮助工程师解决设计问题。

The investigator will develop improved numerical methods for simulating fluid flow, with emphasis on the vortex sheet model. The approach is based on the vortex method, in which Lagrangian points carry the vorticity. The working hypothesis is Prandtl's idea that the vortex sheet is an asymptotic model for a free shear layer in slightly viscous flow. The first task in this pro- posal is to develop a fast summation method to study spiral scal- ing properties in the Kelvin-Helmholtz problem. Other projects are a simulation of splitter plate flow with realistic boundary conditions, and a comparison of planar and axisymmetric vortex sheet roll-up. The investigator will compare vortex simulations with finite-difference solutions of the Navier-Stokes equations and with the available laboratory experiments. A key concern is to obtain accurate numerical solutions in order to make precise statements about the validity of the various modelling assump- tions. Fluid dynamics is concerned with the motion of gas (e.g. air) and liquid (e.g. water). A key goal is to understand the forces which act between the fluid and an immersed solid surface. For example, air flow past the wing is responsible for the lift force which supports an aircraft in flight. Fluid flow is important in many areas of science and technology, from internal combustion engines to weather prediction to blood flow. The mathematical equations which describe fluid motion are difficult to solve, and hence the investigator is developing numerical algorithms for use on high speed computers. The goal is to provide insight into the physics of fluid motion, and to create modelling tools to help engineers solve design problems.