Dokument

Summary:
The onset of turbulence in channel flow can be observed way below the actual linear instability of the laminar profile \parencite{Carlson1982, Orszag1971}. In this subcritical regime, suitably chosen three-dimensional perturbations lead to transiently growing turbulence, showing a spatio-temporal complex, intermittent character in extended domains \parencite{Lemoult2012,Duguet2013}. This transition is characterized by the global, sudden and simultaneous activity of many degrees of freedom \parencite{Grossmann2000}. It differs from the better understood local character of the in comparison rather slowly evolving linear instability at one particular wavenumber, where a cascade of further instabilities leads to ever more complex dynamics \parencite{Eckhardt2017,Manneville2005,ZammertPHD}. While the linear mechanism of non-normal amplification is one important ingredient of self-amplifying turbulence \parencite{Trefethen1993}, nonlinearity is at least a likewise important characteristic on the way of understanding mechanisms of subcritical turbulence within the framework of more or less simple models \parencite{Waleffe1997}. The particular nonlinearities of these models form the state space \parencite{Eckhardt2006, Dauchot1997}, and their precise form ultimately organizes the dynamics we observe. Dwight Barkley recently developed model for pipe flow, consisting of two 1+1-dimensional, coupled FitzHugh-Nagumo type reaction-advection-diffusion equations \parencite{Barkley2011}. The dynamics in these equations, built on excitability and bistability and originally used in the modeling of the axons of the nervous system, mimics the interplay between transient turbulence, transition to turbulence and relaminarization. A surprising number of phenomena are reproduced in just that transition region of pipe flow, although the model is not derived directly from the original momentum equations. Given the performance of this simple model, the idea arose to derive a similar model consisting of two coupled ordinary differential equations more directly from the Navier-Stokes-equations. The analytically derived equations describe the temporal evolution of the turbulent kinetic energy in the direction of flow and perpendicular thereto. Central to this is that in the case of the volumetric mean in stationary case, a balance is established between the production rate of turbulent kinetic energy, the energy transfer rate between the components and the respective dissipation. In the following, term by term is modeled to finally obtain a closed model with two unknowns that represent the turbulent energies in the direction of flow and perpendicular thereto. On a qualitative level, we will base our approach on dimensional-analytic arguments and consider topological conditions to allow a turbulent fixed point to emerge from a critical Reynolds number in addition to the laminar fixed point. But even quantitatively, the model will finally reproduce some observables from turbulence theory with considerable accuracy. Subsequently, we will extend this model kernel by stochastic terms and mimic the behavior of transient turbulence, as well as a spatial extension of the model in the direction of flow to simulate the propagation of subcritical turbulence.

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