Ph.D Thesis

Ph.D StudentKarp Michael
SubjectTransition to Turbulence in Wall-Bounded Shear Flows
DepartmentDepartment of Aerospace Engineering
Supervisor PROF. Jacob Cohen


In this research several aspects of transition to turbulence in wall-bounded shear flows are investigated. One aspect discusses the formation and evolution of coherent structures, observed in various transitional as well as fully developed wall-bounded turbulent flows. The vortex dynamics is followed using theoretical and numerical methods for the evolution of localized disturbances in homogeneous shear flows, in which the base velocity components are at most linear functions of the spatial coordinates. The other aspect discusses transition from an instability point of view, including linear and nonlinear processes. An analytical model for subcritical transition via the transient growth mechanism is developed. It is shown that maximal energy growth is not essential to obtain transition.

The evolution of coherent structures, such as counter-rotating vortex pairs (CVPs) and hairpin vortices, is studied using a novel analytical-based numerical method for the evolution of localized disturbances in homogeneous shear base flows. CVPs and hairpins are formed for the cases where the streamlines of the base flow are open (hyperbolic and simple shear). The effect of various parameters, such as disturbance initial amplitude and orientation, on the evolution is investigated.

Using insights gained from the evolution of localized disturbances, a minimal element model, capable of following the evolution of packets of hairpins, is developed. Its three components are: simple shear, a CVP having finite streamwise vorticity magnitude and a wavy (in the streamwise direction) spanwise vortex sheet. This combination is inherently unstable due to the wall-normal inflection point formed by the CVP. The results of the unbounded (having no walls) Cartesian model are compared with pipe and channel flow experiments and with a direct numerical simulation of transition in Couette flow. The excellent agreement in all cases demonstrates the universality and robustness of the model.

Finally, an analytical model for transition via secondary instability of CVPs in channel flows is developed. The linear transient growth mechanism is represented analytically by four decaying normal modes and their nonlinear interactions. The model utilizes separation of scales between the slowly evolving base flow and the rapidly evolving secondary disturbance to capture most transition stages using the multiple time scales method. The model predictions are verified by comparison with direct numerical simulations. A parametric investigation enables us to find all possible routes to transition and the optimal parameters for each type of secondary instability. It is shown that the most dangerous secondary disturbances are associated with spanwise wavenumbers which generate the strongest inflection points, i.e. those having maximal shear, rather than with those maximizing the energy gain during the transient growth phase.