Title:

The direct computation of timeperiodic solutions of PDEs with applications to fluid dynamics

At a sufficiently large Reynolds number, the flow around a stationary cylinder results in the formation of the famous von Karman vortex street  a timeperiodic flow in which vortices are shed, alternately on either side of the cylinder. When the cylinder performs forced oscillations transverse to the flow direction, the vortex shedding pattern becomes significantly more complex, leading to the formation of socalled "exotic wakes" whose character is controlled by the Reynolds number as well as the dimensionless period and amplitude of the cylinder's motion. In this thesis, we wish to address the following question: do these different patterns arise via (i) a continuous change in vorticity pattern (with quantifiable discrete changes to its topology) in a "complicated" flow or; (ii) via bifurcations of the NavierStokes equations? Analysing changes in the wake pattern requires the computation of the timeperiodic solution. Near bifurcations, the computation of timeperiodic solutions with the classical timeevolution approach can be extremely slow because transients take a long time to decay. Moreover, if the timeperiodic flow becomes unstable, it is impossible to obtain with this approach. To tackle this issue, we adopt a finiteelement based spacetime approach that allows us to directly compute timeperiodic solutions, bypassing the computation of transients and allowing for the computation of unstable timeperiodic solutions. This approach requires the repeated solution of an extremely large system of linear equations, containing tens of millions of degrees of freedom. The application of direct solvers for the solution of this system is prohibitively expensive. To make the solution of this linear system tractable, we develop a fast preconditioner for the iterative Krylov subspace based iterative solution of the spacetime system. The solution strategy makes the cost of directly computing the timeperiodic comparable to timeintegration over a single period. We apply the newly developed methodology to compute the timeperiodic flow past an oscillating cylinder for a Reynolds number of 100 and demonstrate that the transition from the socalled 2S wake pattern to the P+S wake pattern arises through a combination of both mechanism (i) and (ii); a spatiotemporal symmetrybreaking subcritical pitchfork bifurcation of the timeperiodic solution at A=A_{P1} (i.e. scenario (ii)) leads to the creation of a timeperiodic solution that, through a continuous evolution of the vorticity field along the bifurcating branch (i.e. scenario (i)), leads to the formation of the P+S wake mode. For values of A > A_{P1}, the 2S solution still exists but is unstable. Further, for A > A_{P1}, we discover a second spatiotemporal symmetrybreaking subcritical pitchfork bifurcation of the 2S solution, past which the 2S solution becomes stable again.
