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.