Particle Simulations of Vortex Sheet Motion

研究人员将开发二维和三维流体流中涡片运动的粒子模拟技术。涡片是涡量集中在一个表面上的理想流体方程的弱解，在流体力学中被广泛用于表示弱粘性流动中的薄剪切层。由于Kelvin-Helmholtz不稳定性、奇点形成和混沌动力学，涡片模拟遇到了严重的困难，但研究人员已经证明，通过将控制片面上物质点运动的方程中的Biot-Savart核正则化可以克服这些困难。本项目扩展了这些正则化粒子模拟的能力，开发了一种改进的树形编码算法来计算Biot-Savart积分，并开发了自适应求积和粒子插入方案以在卷纸时保持分辨率。与其他使用球谐函数的方法相比，本文的树形编码算法使用笛卡尔坐标下的泰勒近似，这提供了更大的灵活性，并使该方法能够应用于非调和函数，如正则化的Biot-Savart核。将要开发的求积和插入格式将主要使用拉格朗日薄片参数化。研究人员将进行涡环动力学的粒子模拟，并将结果与实验室实验和N-S方程的直接数值解进行比较。其他要研究的课题包括开尔文-亥姆霍兹问题中螺旋形成的高精度计算，以及将现有技术扩展到密度分层流动。在所有科学和工程领域的基础和应用研究中，计算机模拟都是一种成熟的工具。例如，汽车制造商和制药公司经常在新产品的设计中使用计算机模拟。成功的计算机模拟依赖于几个组件：物理问题的数学模型、在计算机上实现该模型的数值算法以及执行模拟的计算机硬件。本项目通过开发更好的模型和算法来对以涡流为主的流体流动进行计算机模拟，将重点放在前两个组成部分上。空气或水中的漩涡通常是看不见的，但它们可以对附近的固体结构施加强大的力。一个例子是飞机后面的尾涡，它负责飞机的升力，但也对附近的飞机构成危险。在人满为患、跑道稀少的城市机场，飞机往往需要等待几分钟才能起飞，以确保前一架飞机的尾迹已消散到安全水平。本文的研究将为航空工程师设计增强尾迹耗散和减少起飞延误的方法提供更好的尾迹模拟算法。该项目中开发的算法也适用于在分子水平上评估静电力，这是一个在化学和等离子体物理等领域具有许多潜在应用的通用计算问题。

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.