Transition in Pipe Flow Nonlinear Mechanisms

Pipe flow is perhaps the classic problem of fluid dynamics. Its simplicity of form lends itself admirably to experimenters but conceals a wealth of unanswered questions. Despite Reynolds’ conducting his seminal experiments over a century ago, few formal results are known for this flow. Over the years many simplifications have been considered, including linearisation, azimuthal invariance and streamwise independence. It is becoming increasingly accepted that these reductions are an over simplification and the full problem must be considered. In this thesis we begin by looking for exact solutions of the NavierStokes equations in pipe flow. These solutions take advantage of the nonlinearity inherent in the system to sustain themselves against viscous decay. The simplest solutions take the form of travelling waves, steady states which propogate down the pipe at a constant velocity. The travelling waves explored here appear to be of particular importance in two different senses. One particular travelling wave dominates the dynamics on the boundary that separates flows which will relaminarise from those which will lead to a turbulent episode. This travelling wave is central to the transition scenario. We also explore two new families of travelling waves which are particularly elegant form and appear to organise phase space to the extent that all previously known travelling waves appear to be subsidiaries of them. We also explore the first known relative periodic orbit in pipe flow. This represents the next rung up the ladder in terms of complexity of the flow being calculated. In the second half of this thesis we consider the transient growth mechanism. We recognise the limitations of a linear study and attempt to circumvent this by including nonlinearity. This allows for significantly enhanced growth due to the nonlinear interaction of scales. It also provides us with new and powerful techniques opening the door to wider problems.