Computational and Analytical Problems in Fluid Mechanics

这个项目由三个子项目组成，分别是流体力学中的分析和计算方法。克拉斯尼、博格斯和哈拉贝斯蒂安教授将作为一个小组相互合作。它们将解决以下问题：1.涡片演变的计算。该项目致力于推广所谓的涡团方法来计算涡片演化。一个计划中的应用是确定在周期性扰动的涡片中形成的螺旋的结构。2.局部加密网格上的快速椭圆求解器。这项研究将把区域分解方法与McCormick开发的FAC算法相结合。这些方法将在流体力学应用中出现的区域上的拉普拉斯方程的Neumann问题上进行测试。3.守恒律组的数值分析问题。将考虑小粘性摄动、消失粘性方法和接触不连续的不可压缩极限的问题。在这笔赠款下进行的研究将有可能应用于空气动力学和燃烧研究，并将广泛依赖于与非线性偏微分方程组有关的计算工具。

This project consists of three subprojects on analytical and computational methods in fluid mechanics. Professors Krasny, Borgers and Harabetian will work in collaboration with each other as a group. They will attack the following problems: 1. Computation of vortex sheet evolution. This project is concerned with extending the so-called vortex blob method for computing vortex sheet evolution. One planned application is to determine the structure of the spiral which forms in a periodically perturbed vortex sheet. 2. Fast elliptic solver on locally refined meshes. This research will combine domain decomposition methods with the FAC algorithm developed by McCormick. These methods will be tested on Neumann problems for Laplace's equation on regions arising in fluid dynamics applications. 3. Numerical and analytical problems for systems of conservation laws. Problems with small viscous perturbations, with vanishing viscosity method, and with the incompressible limit of a contact discontinuity are going to be considered. Research performed under this grant will have potential applications to studies of aerodynamics and combustion, and will rely extensively on computational tools related to nonlinear partial differential equations.