A correction function method to solve incompressible fluid flows to high accuracy with immersed geometries

Abstract

Numerical simulations of incompressible viscous flows in realistic configurations are increasingly important in many scientific and engineering fields. In Aeronautics, for instance, relatively cheap numerical computations replace costly hours of wind tunnel investigations in the early design stages of new aircraft. However, standard methods to obtain numerical solutions over complex geometries require sophisticated meshing techniques and intensive human interaction. In contrast, "immersed methods" incorporate complex boundaries and/or interfaces into regular meshes (Cartesian meshes or simple triangulations). Hence, immersed methods simplify the task of mesh generation and are of great interest in the study of incompressible viscous flows. The objective of this thesis is to advance current immersed methods by formulations that yield highly accurate discretizations without compromising computational efficiency. This is achieved by introducing a new type of immersed method, the correction function method. This new method is based on the concept of a correction function that provides smooth extensions of the solution across boundaries and/or interfaces, such that standard (accurate and efficient) discretizations of the governing equations remain valid everywhere in the computational domain. Furthermore, the key concept behind the correction function method is the introduction of the correction functions as solutions to partial differential equations, which are defined locally around the immersed boundaries and interfaces. Then, we can solve these equations to any desired order of accuracy, resulting in high accuracy methods. Specifically, in this thesis the correction function method is implemented to 4th order of accuracy in the context of Poisson's equation, the heat equation, and the nonlinear convection advection diffusion in 2D. Then, these techniques are combined to solve the incompressible Navier-Stokes equations, which govern the dynamics of incompressible viscous flows.