Spatio-temporal dynamics in pipe flow

When fluid flows through a channel, pipe or duct, there are two basic forms of motion: \nsmooth laminar flow and disordered turbulent motion. The transition between these two \nstates is a fundamental and open problem which has been studied for over 125 years. What \nhas received far less attention are the intermittent dynamics which possess qualities of \nboth turbulent and laminar regimes. The purpose of this thesis is therefore to investigate \nlarge-scale intermittent states through extensive numerical simulations in the hopes of further \nunderstanding the transition to turbulence in pipe flow. \nWe begin by reviewing the spectral-element code Semtex which is used to perform the \nsimulations. We discuss modifications to this code to impose a constant flowrate to the flow \nthrough a pipe and to improve the computational efficiency on certain multicore architectures. \nWe then move on to examine the reverse transition from turbulence to laminar flow in a long, \n125 diameter periodic pipe, which unlike the forward transition does not depend on finiteamplitude \nperturbations to the flow and thus captures the natural dynamics contained within \nthe transition. The Reynolds number Re is reduced from Re = 2,800 to Re = 2,250 over \na long timescale, and by investigating the resultant spatio-temporal dynamics we discover \nthat the transition can be characterised by three fundamentally different states separated by \ntwo Reynolds numbers. Below Rec <= 2,300, turbulence takes the form of equilibrium puffs \nwhich eventually decay. Above Rei = 2,600, flow remains uniformly turbulent throughout \nthe domain. Between these two values, the dynamics are an intermitent mixture of both \nturbulent and laminar regimes which take the form of unsteady alternating laminar-turbulent \nbands. \nFinally, we concentrate on finding a more exact value for Rec, which marks the onset \nof sustained turbulence in pipe flow. We examine the process through which isolated \nturbulent puffs split and find that, like decay, this process is stochastic and memoryless. \nBy drawing comparisons with other simple stochastically driven systems – in particular, \ndirected percolation – we compare the timescales for decay and splitting, and ascertain that \nRec = 2,040 +- 10.