On Turbulence Transition in Shear Flows

The onset of turbulence in shear flows like pipe flow or plane Couette flow, for which the laminar profile is linearly stable for all Reynolds numbers, has remained a puzzle for many years. Immense progress towards the understanding of the underlying physical mechanisms has been made by the application of ideas from dynamical systems theory. In particular, the search for exact solutions of the Navier-Stokes equations, like fixed points and periodic orbits, allowed a better grasp of the turbulent dynamics as these exact coherent structures are underlying the turbulent motion in state space. Direct numerical simulations (DNS) of the Navier-Stokes equations are challenging, even for simple geometries, as many spatial and temporal scales need to be resolved. Therefore, there exists a need to develop efficient, low-dimensional models to further explore the features of the transition to turbulence. The most elementary models consist of a vortex and a streak plus a nonlinearity. These are the basic ingredients for the transient amplification of perturbations and the self-sustaining process, which in turn cause the transition. We compare deterministic and noise-induced transitions for such a low-dimensional model and we find qualitatively different transition states for the different scenarios. A promising ansatz are quasilinear approximations, for which the nonlinearities of the Navier-Stokes equations are restricted to a small set which is sufficient to maintain turbulent dynamics. For this purpose, the velocity fields are decomposed into two groups of modes, and only certain couplings between the groups are kept. In particular, all self-interactions within the second group are neglected, except for the ones that map to the first group, thereby introducing a feedback from the second to the first group. We implement quasilinear approximations into DNS of plane Couette flow and analyze these models from a dynamical systems perspective, i.e. we investigate their underlying state space structure. For the streamwise quasilinear approximation, the first and the second group describe the flow field without and with variation in downstream direction, respectively. In the equations of the second group, the self-interactions between streamwise varying modes are neglected. A detailed comparison between the approximation and the full system is possible for the exact coherent structures in plane Couette flow. From the continuation of known fixed points of the full nonlinear system to the streamwise quasilinear system we observe qualitatively similar velocity fields and mean profiles. The bifurcation diagrams of the states and in particular the bifurcation points are captured well by the approximation. Furthermore, we were able to follow a bifurcation cascade starting at an exact coherent structure and leading to the formation of a local chaotic attractor, by analogy with the fully nonlinear system. An interesting property of the streamwise quasilinear model is that the energy spectra of the quasilinear states contain a few elements only. Even though the set of active modes is considerably reduced, many features of the full system can be found within the streamwise quasilinear approximation. With increasing Reynolds number, further modes can be activated by unstable eigenvectors and bifurcations. When the additional modes subsequently emerge, their amplitudes show intermittent behaviour. In a generalized quasilinear setting we can systematically interpolate between the quasilinear approximation and the fully nonlinear system by increasing the set of modes contained in the first group. This leads to a quantitative improvement of the results compared to the full system, but the reduction in the number of active modes is lost. The results show that the quasilinear approximation allows to systematically deduce from the Navier-Stokes equations simplified models that share the characteristics of the full system, and that should be useful for further analytical representations of the dynamics of transitional shear flows