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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Format:  Doctoral Thesis 
Language:  English 
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PhilippsUniversität Marburg
2015

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Summary:  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 longliving, but transient turbulence can be observed in the subcritical range.
In the two systems a multitude of exact solutions of the NavierStokes 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 socalled boundary crisis bifurcation.
This chaotic saddle shows exponentially distributed lifetimes. A second type of crisis bifurcations,
socalled 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 TollmienSchlichting transition.
The state space structure underlying the coexistence of the two transition scenarios is explored
by twodimensional 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 TollmienSchichting 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 TollmineSchlichting 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 TollmienSchlichting
wave are analyzed. This allows to identify periodic orbits that consist of streamwise localized packages of TollmienSchlichting waves.
Furthermore, by using the technique of edgetracking, which allows to follow trajectories on the boundary between
the laminar and the transient turbulent part of the state space, a streamwise localized threedimensional 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 longwavelength 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 doublylocalized 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 doublylocalized states it is shown that it is created in two simultaneous longwavelength instabilities of a spatially extended equilibrium solution.
By studying the instabilities of these localized exact solutions one recognizes that plumelike
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 Channelflowcode (www.channelflow.org) was enhanced.
In addition to the parallel thermal boundary layer, the developed code allows to simulate RayleighBénard, PoiseuilleRayleighBénard and CouetteRayleighBénard flow. 

Physical Description:  163 Pages 
DOI:  10.17192/z2015.0463 