Collaborative Research: Numerical approaches for incompressible viscous flows with high order accuracy up to the boundary

这个项目的研究重点是开发、分析和实施有效的策略来解决高精度到边界的不可压缩粘性流动问题，特别是精确计算边界处的应力。为此，考虑了含时N-S方程的压力-泊松方程(PPE)重构。这些重新公式与原始方程等价，但它们给出了流体压力的显式边界条件。因此，PPE重构的数值离散不会受到传统投影技术引起的某些问题的影响，例如数值边界层和边界应力的不准确。这个项目的目标是利用这一基本优势，利用有限元、有限差分和无网格质点方法等各种技术，在一般区域内开发有效的、高精度的不可压缩粘性流体流动的实现。此外，实际压力泊松方程的数值逼近是数值线性代数社区感兴趣的丰富问题的来源，本项目涉及到与该领域的合作者的相互作用。在许多科学和工程应用中，流体和固体边界处力和应力的准确和高效的计算是至关重要的。例如，飞机机翼、机动车辆和风力涡轮机的设计，以及分层流体中的沉积和生物运动的模拟，边界力(以升力和阻力的形式)是关键的量。调查人员正在研究新的方法和办法的实施，以便能够高度准确地计算这些边界力。该项目涉及计算流体力学的发展，包括关于不可压缩流动方程的数学结构的理论方面，以及在有效求解大型方程组中出现的基本问题。研究生的参与和培养是该项目的重要组成部分。

The research in this project focuses on the development, analysis, and implementation of efficient strategies to solve incompressible viscous flow problems with high order accuracy up to the boundary, with specific emphasis on the accurate calculation of stresses at the boundaries. To that end, Pressure Poisson Equation (PPE) reformulations of the time-dependent Navier-Stokes equations are considered. These reformulations are equivalent to the original equation, however, they yield explicit boundary conditions for the fluid pressure. As a consequence, numerical discretizations of PPE reformulations do not suffer from certain problems that traditional projection techniques incur, such as numerical boundary layers and inaccuracies in stresses at boundaries. The goal of this project is the exploitation of this fundamental advantage to develop effective, and high order accurate, implementations for incompressible viscous fluid flows in general domains, using various techniques such as finite elements, finite differences, and meshfree particle methods.Moreover, the numerical approximation of the actual Pressure Poisson Equations is a rich source of questions of interest to the numerical linear algebra community, and this project involves interactions with collaborators from that area.In many applications in science and engineering, the accurate and efficient computation of forces and stresses at boundaries between fluids and solids is of crucial importance. Examples in which boundary forces (in the form of lift and drag) are key quantities of interest are the design of airplane wings, motor vehicles, and wind turbines, as well as the simulation of sedimentation in stratified fluids and bio-locomotion. The investigators are researching new methodologies and implementations of approaches that allow for a highly accurate computation of these boundary forces. This project relates developments in computational fluid dynamics with both theoretical aspects regarding the mathematical structure of the equations of incompressible flows, as well as fundamental questions that arise in the effective solution of large systems of equations. The involvement and training of graduate students is an important component of this project.