
MINISYMPOSIA 
Numerical
bifurcation techniques for applications in fluid dynamics
Organized by
Greg Lewis Faculty of Science, UOIT
Lennaert van Veen (UOIT)

In
this minisymposium, we discuss emerging numerical techniques in the
application of dynamical systems theory to problems in fluid dynamics,
or other highdimensional systems. Novel methods and software for the
computation and continuation of steady, periodic and connecting solutions
are presented, and specific issues associated with the highdimensional
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

Matrixfree
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 underdetermined boundary value problem.
I show how this algorithm can be combined with multiple shooting
and NewtonKrylov techniques. The resulting scalable algorithm comes
with an exact convergence results for the subspace iteration. We
demonstrate our approach by computing a cycletocycle homoclinic
orbit in a wellresolved simulation of plane Couette turbulence.

John
F. Gibson
Dept. Mathematics and Statistics, University of New Hampshire 
Channelflow:
a highlevel software system for numerical research in wallbounded
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 highlevel, Matlablike language
for numerical research in wallbounded 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 commandline utilities for dynamicalsystems
computations. 
Andrew
L. Hazel
Manchester Centre for Nonlinear Dynamics and School of Mathematics,
University of Manchester 
Multiple
states of bubble propagation in axiallyuniform 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 axiallyuniform, centred constriction can lead to symmetrybroken
(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 3dimensional problem
to a series of 2dimensional ones. In this talk we will advocate
the use of a discontinuous Galerkin method for the numerical solution
of the incompressible NavierStokes equations and develop goaloriented
error estimation techniques and an hpadaptive strategy to ensure
the accurate location of any bifurcation points. We then apply the
method to the flow in a suddenly expanding pipe. 
