Solving Nonlinear Systems Inside Implicit Time Integration Schemes for Unsteady Viscous Flows

Abstract

Historically, the computation of steady flows has been at the forefront of the development of computational fluid dynamics (CFD). This began with the design of rockets and the computation of the bow shock at supersonic speeds and continued with the aerodynamic design of airplanes at transonic cruising speed [14]. Only in the last decade, increasing focus has been put on unsteady flows, which are more difficult to compute. This has several reasons. First of all, computing power has increased dramatically and for 5,000 Euro it is now possible to obtain a machine that is able to compute about a minute of realtime simulation of a nontrivial unsteady three dimensional flow in a day. As a consequence, ever more nonmilitary companies are able to employ numerical simulations as a standard tool for product development, opening up a large number of additional applications. Examples are the computation of tunnel fires [4], flow around wind turbines [29], fluid-structure-interaction like flutter [10], flows inside nuclear reactors [25], wildfires [24], hurricanes and unsteady weather phenomenas [23], gas quenching [20] and many others. More computing capacities will open up further possibilities in the next decade, which suggests that the improvement of numerical methods for unsteady flows should start in earnest now. Finally, the existing methods for the computation of steady states, while certainly not at the end of their development, have matured, making the consideration of unsteady flows interesting for a larger group of scientists. In this article, we will focus on the computation of laminar viscous flows, as modelled by the Navier-Stokes equations.

This work was supported by the DFG as part of the collaborative research area SFB TRR 30.