The recent theoretical discovery of families of travelling wave solutions in pipe flow (Faisst & Eckhardt 2003;Hof et al. 2004) at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these travelling waves in transitionary flows. Working within a periodic pipe of 5 D (diameters) length, we find that travelling waves with low wall shear stresses (lower branch solutions) are on a surface which separates initial conditions which uneventfully relaminarise and those which lead to a turbulent evolution. Evidence for recurrent travelling wave visits is found in both 5 D and 10 D long periodic pipes but only for those travelling waves with low-to-intermediate wall shear stress and for less than about 10% of the time in turbulent flow. Given this, it seems unlikely that the mean turbulent properties such as wall shear stress can be predicted as an expansion over the travelling waves in which their individual properties are appropriately weighted. Rather, further dynamical structures such as periodic orbits need to be isolated and included in any such expansion.

By means of a high-frequency analysis it is shown that a Rayleigh instability is possible within the interactive boundary-layer formulation. This instability reflects the tendency of large-Reynolds-number flows to be unstable. For most, but not all, pressure-displacement relations both Rayleigh's and Fjørtoft's theorems hold, although Fjørtoft's criterion is not a sufficient condition for instability. However, for two pressure-displacement relations neither theorem could be proved, and for one of these, unstable flows exist which are free of inflexion points. Analytically, the existence of this instability may result in a finite-time singularity, while numerically the presence of Rayleigh modes often leads to accuracy problems which cannot be overcome by simple grid refinement. A test integration resulted in the generation of small grid-dependent eddies. It is suggested that the instability may be a possible cause of the eddy splitting observed in experiments on unsteady flows through distorted channels. This Rayleigh instability is also possible within the ‘inverse’ boundary-layer formulation, but is absent from classical boundary-layer problems.

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The development of the boundary layer along a long thin cylinder aligned with the flow is considered. Numerical solutions are presented and compared with previous asymptotic results. Very near the leading edge the flow is given by the Blasius solution for a flat plate. However, there is soon a significant deviation from Blasius flow, with a thinner boundary layer and higher wall shear stress. Linear normal mode stability of the flow is investigated. It is found that for Reynolds numbers less than a critical value of 1060 the flow is unconditionally stable. Also, axisymmetric modes are only the fourth least stable modes for this problem, with the first three three-dimensional modes all having a lower critical Reynolds number. Further, for Reynolds numbers above the critical value, the flow is unstable only for a finite distance, and returns to stability sufficiently far downstream.