Matrix-based techniques for (flow-)transition studies

Abstract

Numerical flow simulations are playing a more and more mportant role not only in the scientific field but also in the industrial area, where an experimental practice is either difficult or expensive to realize. Numerical simulation methods have been studied intensively in the past decades to achieve good accuracy as well as low turnaround times. There is no numerical technique which is generally suitable for every problem. Usually, there are specific techniques for a given type of problem. For instance, a preconditioner is created depending on the structure and properties of the matrix arising from the system. The motion of a fluid is governed by the Navier-Stokes equations, which can model different types of fluid flows. For example, the fluid air in the modeling of the earth atmosphere is considered as compressible, while in the simulation of ocean tides, the fluid water is regarded as incompressible. Analytical solutions of the Navier-Stokes equations only exist for a limited number of cases, hence, for general cases, computing efficiently numerical solutions for these equations is essential. We focused on steady-state problems related to incompressible flows. There are basically two approaches to find steady states. The first is a timeintegration method which just finds it by integrating to the steady state. The second is to solve the nonlinear system directly by Newton’s method. The goal here is to get an insight into what kind of problems can be solved readily with a continuation method and which of the approaches, time integration or direct solution, is favorable. For the latter we developed our own parallel continuation program, which is also presented in this thesis.