In this mini-symposium, we discuss emerging numerical techniques in the application of dynamical systems theory to problems in fluid dynamics, or other high-dimensional systems. Novel methods and software for the computation and continuation of steady, periodic and connecting solutions are presented, and specific issues associated with the high-dimensional nature of the systems are highlighted. Applications include dynamics of bubbles, planar shear flow and pipe flow. The first talk will provide a context and will include a short introduction of the session.

SPEAKERS
Lennaert van Veen Faculty of Science, UOIT	Matrix-free computation of 2D unstable manifolds In this presentation, I will provide a brief introduction to the application of dynamical systems theory to fluid dynamics. As an example, I will discuss an algorithm for the computation of 2D invariant manifolds based on a covering of the manifold by orbit segments which are solutions to an under-determined boundary value problem. I show how this algorithm can be combined with multiple shooting and Newton-Krylov techniques. The resulting scalable algorithm comes with an exact convergence results for the subspace iteration. We demonstrate our approach by computing a cycle-to-cycle homoclinic orbit in a well-resolved simulation of plane Couette turbulence.
John F. Gibson Dept. Mathematics and Statistics, University of New Hampshire	Channelflow: a high-level software system for numerical research in wall-bounded shear flows Research in computational fluid dynamics is often hindered by the complexity of algorithms and the impenetrability of inherited codes. Channelflow is a C++ software system whose principal aim is to lower this barrier to entry by providing a high-level, Matlab-like language for numerical research in wall-bounded channel flows. In channelflow, CFD codes are short scripts that can be rapidly developed and easily understood. This talk will give an overview of the channelflow libraries, present some examples of channelflow programming, and demonstrate some of channelflow's flexible command-line utilities for dynamical-systems computations.
Andrew L. Hazel Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester	Multiple states of bubble propagation in axially-uniform tubes We use a combination of physical experiments and numerical continuation methods to examine the bifurcation structure associated with the propagation of long air bubbles in tubes of rectangular cross section. A unique, centred solution exists for all such tubes, but the introduction of an axially-uniform, centred constriction can lead to symmetry-broken (or localised) solutions above a critical flow rate. Regions of bistability are found for sufficiently severe constrictions and increasing the constriction width leads to oscillations between the symmetric and localised states. We investigate the physical mechanisms that lead to these oscillations and their connection to a global bifurcation scenario.
Edward Hall School of Mathematical Sciences, University of Nottingham	Discontinuous Galerkin methods for bifurcation phenomena in the flow through open systems (slides of talk) In the past, studies of bifurcation phenomena of flow in a cylindrical pipe with a sudden expansion have proven inconclusive. In a recent study we sought to exploit the O(2)-symmetric properties of the problem, thus making it tractable by reducing a 3-dimensional problem to a series of 2-dimensional ones. In this talk we will advocate the use of a discontinuous Galerkin method for the numerical solution of the incompressible Navier-Stokes equations and develop goal-oriented error estimation techniques and an hp-adaptive strategy to ensure the accurate location of any bifurcation points. We then apply the method to the flow in a suddenly expanding pipe.