Numerical continuation for periodic pipe flow with finite element method

Abstract

This thesis is concerned with the continuation theory of incompressible periodic pipe flow. For describing the dynamics of incompressible fluids we use the incompressible Navier-Stokes equation. For a better understanding of it we'll look at its derivation. For a long time now the consensus has been that the laminar solution is linearly stable for all Reynolds numbers. The original idea of this thesis was to adapt a numerical continuation procedure to see if it is possible to jump from the laminar solution branch onto a turbulent one, as it happens in practical experiments. Therefore we inspect the different numerical methods that are used in this procedure. Especially we look at a preconditioner for the linearized problem as the matrix given by the finite element method, using Hood-Taylor elements, becomes less well conditioned as the Reynolds number increases. Prompted by this we look at the convection-diffusion equation and the streamline diffusion discretisation to be able to use it in a multigrid method. To motivate the use of the continuation procedure we look at bifurcation theory, with Fredholm operators and Crandall-Rabinowitz' theorem. We also take a short look at the Allen-Cahn equation to test if the algorithm is correctly defined.