Localized states in the transition to turbulence in plane Poiseuille flow and thermal boundary layers
This dissertation numerically investigates the transition to turbulence and occurring localized structures in plane Poiseuille flow and the asymptotic suction boundary layer over a heated plate. Calculations show that the laminar profiles of both flows are linearly unstable. Nevertheless, in both...
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|Zusammenfassung:||This dissertation numerically investigates the transition to turbulence and occurring localized structures in plane Poiseuille flow and the asymptotic suction boundary layer over a heated plate. Calculations show that the laminar profiles of both flows are linearly unstable. Nevertheless, in both cases long-living, but transient turbulence can be observed in the subcritical range. In the two systems a multitude of exact solutions of the Navier-Stokes equations is identified using specially tailored numerical methods. A detailed analysis shows that for both systems the origin of subcritical turbulence can be traced back to a bifurcation cascade that starts at one of those exact solutions. The bifurcation cascade creates a chaotic attractor which is transformed into a chaotic saddle by a so-called boundary crisis bifurcation. This chaotic saddle shows exponentially distributed lifetimes. A second type of crisis bifurcations, so-called interior crisis bifurcations, which are studied in detail for plane Poiseuille flow, contributes to an increase of the attractor and its temporal complexity. For the asymptotic suction boundary layer over a heated plate, the exact solution that is the starting point for the bifurcation cascade is directly connected to the instability of the laminar state. For plane Poiseuille flow there is no such connection causing the coexistence of two transition mechanisms to turbulence. These are the bypass transition and the Tollmien-Schlichting transition. The state space structure underlying the coexistence of the two transition scenarios is explored by two-dimensional projections of the state space that use exact solutions. The slices show that below the critical Reynolds number the state space consists of three parts. In one of these parts, bypass transition is located and in a second one a immediate return to the laminar states can be observed. In the third part, which is quite small and grows with increasing Reynolds number, initial conditions undergo Tollmien-Schichting transition to turbulence. In the state space, all three regions lie close to each other and can be distinguished by the time needed to reach the turbulent state. Above the critical Reynolds number, only the regions containing bypass and Tollmine-Schlichting transition can be found. There, both regions are separated by the stable manifold of a special exact solution of the system, the bypass edge state. In both studied systems, various spatially localized exact solutions exist. For plane Poiseuille flow, the instabilities of the spatially extended Tollmien-Schlichting wave are analyzed. This allows to identify periodic orbits that consist of streamwise localized packages of Tollmien-Schlichting waves. Furthermore, by using the technique of edge-tracking, which allows to follow trajectories on the boundary between the laminar and the transient turbulent part of the state space, a streamwise localized three-dimensional periodic orbit can be found. The streamwise length of this periodic orbit grows approximately linear with the Reynolds number. The result show further that the orbit is created at high Reynolds number in an subcritical long-wavelength instability of a streamwise extended traveling wave. By tacking the orbit to wider computational domains, a relative periodic orbit that is localized in both direction parallel to the plates can be found. The streamwise localized state shows a decay of the velocity components that over a wider range is in a good agreement with an exponential decay. However, the results from the doubly-localized states, as well as simulations of turbulent spots, show that for large distances the decay of the velocity components follows a power law. For the asymptotic suction boundary layer over a heated plate, it is also possible to identify different exact solutions that are localized in one or two directions parallel to the plate. For one of the doubly-localized states it is shown that it is created in two simultaneous long-wavelength instabilities of a spatially extended equilibrium solution. By studying the instabilities of these localized exact solutions one recognizes that plume-like dynamics appear along their unstable directions. Thus, plume motion in this flow can be interpreted as moving along the unstable direction of an exact solution. A systematic analysis of the plume shows that for a long time its tip moves with a constant speed that increases with the Rayleigh number. To study the asymptotic suction boundary layer over a heated plate numerically, the Channelflow-code (www.channelflow.org) was enhanced. In addition to the parallel thermal boundary layer, the developed code allows to simulate Rayleigh-Bénard-, Poiseuille-Rayleigh-Bénard- and Couette-Rayleigh-Bénard flow.|