The immersed boundary method for the (2D) incompressible Navier-Stokes equations

Immersed Boundary Methods (IBMs) are a class of methods in Computational Fluid Dynamics where the grids do not conform to the shape of the body. Instead they employ Cartesian meshes and alternative ways to incorporate the boundary conditions in the (discrete) governing equations. The simple grids and data structure are very well suited to handle complex geometries and moving boundaries. The ma in objective of this project was to investigate Immersed Boundary Methods through literature study, brief analysis and numerical experiments, to gain experience and knowledge on the topic and to lay the foundations for practical use of these methods in future research. The approach that was taken to meet the objectives can be split into three parts: a literature study, a simple 1D channel-flow study and a 2D steady Navier-Stokes study. The literature study presents the basic IBM techniques and a brief historical overview, followed by a discussion on some important properties of IBMs. Based on a structured classification of existing methods, a choice is made on the type of Immersed Boundary Methods to be explored in the 1D numerical study. The 1D study makes use of the Poiseuille low problem as a test case, since it has an analytical solution which allows us to calculate the absolute error made by the developed IBMs. The study of three new and two existing 1D methods and their derived variants reveals their accuracy on different grids and shows that this accuracy can be substantially affected by the position of an immersed boundary with respect to the neighboring grid points. The construction of a steady 2D Navier-Stokes code provided an opportunity to test some of the findings from the 1D numerical study in a higher dimension. A lot of effort was put in constructing the pre-processor, which creates the cartesian grid and determines the intersections of the grid lines with the immersed boundary. Additional parameters are defined to create a data structure that allows the IBMs to deal with immersed bodies effectively. The current pre-processor can handle most body shapes fully automatically. Thin, wedge-like shapes (e.g. airfoil trailing edges) still need a little bit of hand-coding. Three IBMs are successfully implemented in an existing 2D first-order finite volume code for the Navier-Stokes equations. These Immersed Boundary Methods are tested on three test cases: a backward-facing step flow, a circular cylinder flow in a channel and a multi-element airfoil flow. The results show that Immersed Boundary Methods are able to treat different boundaries in a satisfying manner. The qualitative aspects of the flows are captured well. Moreover, the grid generation is very straightforward and fast, even for the multi-element airfoil. The recommendations include suggestions on improving the pre-processor, on speeding up the steady solution method and on transforming the present code into an unsteady solver