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# The Direct Computation of Time-Periodic Solutions of PDEs With Applications to Fluid Dynamics

[Thesis]. Manchester, UK: The University of Manchester; 2020.

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## Abstract

At a sufficiently large Reynolds number, the flow around a stationary cylinder results in the formation of the famous von Karman vortex street - a time-periodic 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 so-called "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 Navier-Stokes equations? Analysing changes in the wake pattern requires the computation of the time-periodic solution. Near bifurcations, the computation of time-periodic solutions with the classical time-evolution approach can be extremely slow because transients take a long time to decay. Moreover, if the time-periodic flow becomes unstable, it is impossible to obtain with this approach. To tackle this issue, we adopt a finite-element based space-time approach that allows us to directly compute time-periodic solutions, bypassing the computation of transients and allowing for the computation of unstable time-periodic 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 space-time system. The solution strategy makes the cost of directly computing the time-periodic comparable to time-integration over a single period. We apply the newly developed methodology to compute the time-periodic flow past an oscillating cylinder for a Reynolds number of 100 and demonstrate that the transition from the so-called 2S wake pattern to the P+S wake pattern arises through a combination of both mechanism (i) and (ii); a spatio-temporal symmetry-breaking subcritical pitchfork bifurcation of the time-periodic solution at A=A_{P1} (i.e. scenario (ii)) leads to the creation of a time-periodic 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 spatio-temporal symmetry-breaking subcritical pitchfork bifurcation of the 2S solution, past which the 2S solution becomes stable again.

## Keyword(s)

Block preconditioning; Finite-element method; Flow past an oscillating cylinder; Fluid mechanics; Forced unsteady heat equation; Iterative solvers; Navier-Stokes equations; Scientific Computing; Space-time finite-element method; Spatio-temporal symmetry breaking; Symmetry-breaking bifurcation; Time-periodic solutions; Topological fluid mechanics

## Bibliographic metadata

- Time-periodic solutions
- Flow past an oscillating cylinder
- Spatio-temporal symmetry breaking
- Space-time finite-element method
- Scientific Computing
- Navier-Stokes equations
- Forced unsteady heat equation
- Fluid mechanics
- Finite-element method
- Iterative solvers
- Block preconditioning
- Topological fluid mechanics
- Symmetry-breaking bifurcation