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

该项目的研究重点是开发，分析和实施有效的策略，以解决不可压缩粘性流动问题，具有高精度的边界，特别强调边界处应力的精确计算。为此，压力泊松方程（PPE）的时间相关的Navier-Stokes方程的重新考虑。这些重新表述是等效的原始方程，但是，他们产生明确的边界条件的流体压力。因此，数值离散的PPE重新不遭受某些问题，传统的投影技术招致，如数值边界层和边界处的应力不准确。本项目的目标是利用这一基本优势，开发有效的，高阶精度，实现不可压缩粘性流体流动在一般域，使用各种技术，如有限元，有限差分和无网格粒子方法。此外，实际压力泊松方程的数值近似是一个丰富的来源的问题感兴趣的数值线性代数社区，在科学和工程的许多应用中，流体和固体边界处的力和应力的精确和有效的计算是至关重要的。边界力（以升力和阻力的形式）是感兴趣的关键量的例子是飞机机翼，机动车辆和风力涡轮机的设计，以及分层流体和生物运动中沉积的模拟。研究人员正在研究新的方法和实现的方法，允许这些边界力的高度准确的计算。该项目涉及计算流体动力学的发展与不可压缩流方程的数学结构的理论方面，以及在大型方程组的有效解中出现的基本问题。研究生的参与和培训是该项目的一个重要组成部分。

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.