|Ph.D Thesis||Department of Aerospace Engineering|
|Supervisor:||Prof. Cohen Jacob|
|Full Thesis text|
In the initial stage of transition to turbulence finite amplitude perturbations trigger a secondary instability, thus introducing to the flow growing disturbances which lead eventually to transition.
Finite amplitude disturbances appear in the flow in two different ways: they evolve transiently from small initial disturbances, or alternatively they appear as a result of mild distortions in the base-flow.
In pipe Poiseuille flow several studies have shown that the optimal initial disturbances yielding the largest transient growth have the structure of a pair of counter rotating vortices in the pipe cross section plane. They are independent of the axial coordinate, and evolve in time or space into high amplitude streaks. In this study the main features of this mechanism are predicted analytically. The application of this approach is shown to be relevant to other shear flows as well.
In the alternative approach we explore the temporal growth of disturbances developing in the pipe Poiseuille flow, which is modified by an optimal primary axially-independent, axisymmetric and helical finite amplitude deviations. The optimal modification is defined as the primary base-flow deviation, with a specific amplitude norm, that yields the maximum growth rate for the secondary disturbances. Optimal modifications are computed by a variational technique.
The axisymmetric distortions analysis reveals a bifurcation into two different branches of solutions. At high Reynolds numbers one branch tends towards the centerline, whereas the other tends towards the wall. A critical Reynolds number of 1840 is found. Below this value the near-wall solution has the lower energy and above it the centerline solution becomes the less energetic one. This critical Reynolds number is in a remarkable accordance with unexplained experimental observations.
In the case of helical distortions it is found that bifurcations of solutions occur at lower Reynolds numbers. Unlike the axisymmetric case the near-wall solution is less preferable at all Reynolds numbers. The centerline solution requires twice the energy of the axisymmetric solutions to trigger a secondary instability. Although helical deviations require more energy than axisymmetric ones in order to trigger a secondary instability, they are important in a different path of instability: helical distortions are likely to be a result of a transient growth amplification of initial disturbances, and therefore a consequence of smaller perturbations.