Finite-dimensional dynamics and chaos in fluid flows

We present a discussion of the role of finite-dimensional dynamics and chaos in interpreting nonlinear fluid mechanical motion. The discussion will be restricted to two examples of fluid flows which have been studied by the author and which appear to be understandable in terms of ideas based in modern thinking in dynamical systems. The specific examples we have chosen are the flow between concentric rotating cylinders commonly called the Taylor-Couette problem and the flow through a nominally two dimensional sudden symmetric expansion. The first of these is a so called ‘closed flow’ problem and the second is an example of an ‘open flow’ where disturbances can grow as they are carried down stream. The aim of the present article is to focus attention on the practicalities of applying the abstract concepts of finite-dimensional dynamics to the experimental study of fluid flows. We will show how a careful consideration of the important symmetries of these problems can lead to the uncovering of structurally stable local organising centres for the global low-dimensional dynamical behaviour.