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Subcritical Transition in Shear Flows

Fernando Mellibovsky, Ph.D.
Universitat Politècnica de Catalunya, June 19th. 2008
Understanding transition to turbulence in shear flows, even for rather simple fluid systems, is a major challenge that has attracted the attention of the scientific community for over a century. Its technological implications are far-reaching, very especially in the case of aeronautics, for which shear flows are of outstanding importance. The focus has been set here on subcritical transition of wall-bounded shear flows and, in particular, of pipe flow (pressure-driven flow along a circular pipe).
This work has aimed at providing a deeper understanding of the mechanisms that are responsible for transition bypassing linear instability of the pipe basic flow. To this end, two complementary research approaches have been undertaken.
The first approach has consisted in a direct characterisation of the basin of attraction of the stable basic flow. The critical threshold beyond which finite amplitude perturbations are capable of bringing about transition has been investigated and scaling laws describing how the basin of attraction of the laminar profile shrinks with increasing flow speed have been provided for different types of perturbations. Very good agreement with recent accurate pipe flow experiments has been obtained.
However, simple characterisation of the critical threshold does not provide, on its own, much insight on what the mechanisms behind transition are. A second approach, consisting in a direct exploration of the phase map from a dynamical systems point of view, has acquired great momentum in the very recent past. As a result, new states disconnected from the basic flow have been identified. These solutions, which take the form of periodic travelling waves in pipe flow, have been computed and their implications in transition and in developed turbulence assessed. Some of them arise from a purely theoretical course of action. Their relevance in developed turbulence has been positively established both experimentally and numerically in the literature, but their alleged role in transition has not been clarified. In the present work, new solutions have been found within a chaotic state that resides within the critical threshold and seems to govern transition. Because they naturally dwell in this chaotic saddle, their relevance to transition seems to be beyond any doubt.
The chaotic state and the solutions found, however, correspond to short pipe global transition, where no intermittency phenomena is ever observed. Transition to localised structures typical of long pipes, such as puffs or slugs, seems instead to be governed by a localised chaotic state of about the same characteristic length of the turbulent structures the basin of attraction of which it bounds. No simple travelling-wave-type solutions have been identified within the chaotic localised state. The relationship between the short wavelength periodic states and experimental transition to localised long structures remains an open problem that should be the object of future work.