Turbulent patterns in wall-bounded shear flows: invariant solutions and their bifurcations

When laminar shear flows in large wall-bounded domains transition to turbulence, the flow exhibits spatio-temporally chaotic dynamics. Despite its chaotic dynamics, the flow may self-organize into characteristic spatially periodic patterns of unknown origin. To understand how regular patterns emerge in a turbulent flow, a nonlinear theory is needed. In this thesis we apply dynamical systems theory to explain the spatial structure, origin and dynamics of turbulent patterns. To this end, we construct and analyze exact invariant solutions of the 3D nonlinear fluid flow equations that capture the non-trivial spatial structure of the patterns. This approach requires high-performance computational tools, that have been developed as part of this thesis and are now publicly available within the widely used state-of-the-art open source software Channelflow 2.0. Using the developed tools, we explain turbulent patterns in two different wall-bounded shear flows:

In plane Couette flow, the flow between two parallel walls moving in opposite direction, turbulent flow emerges subcritically and may coexist with regions of laminar flow. For specific wall velocities, the turbulent-laminar flow self-organizes into an intricate pattern of periodically alternating laminar and turbulent bands oriented obliquely against the direction of the wall movement. Experiments and simulations have reproduced the oblique stripe pattern for more than 50 years but the pattern characteristics, in particular the wavelength and the oblique orientation, remain to be explained. We present the first unstable equilibrium solution of the fully nonlinear Navier-Stokes equations that captures the flow structure of oblique turbulent-laminar stripes. Using numerical continuation, we show how the stripe equilibrium bifurcates from the well-known Nagata equilibrium via two successive symmetry-breaking bifurcations. Within the subspace of the symmetry that is broken by the second bifurcation, we identify three obliquely patterned periodic orbits embedded in the edge of chaos. The spatial structure of these invariant solutions suggests a nonlinear mechanism by which weakly turbulent Couette flow selects the wavelength and the oblique orientation of turbulent-laminar stripes.

The second studied system is inclined layer convection, a thermally driven shear flow in an inclined channel. Like Rayleigh-Bénard convection, the case of zero inclination, inclined layer convection shows a large variety of different convection patterns when the control parameters are varied. While linear stability analysis has explained the onset of some of these patterns, their spatio-temporally complex dynamics is not well understood. We identify a multitude of invariant solutions underlying previously observed pattern motifs at a Prandtl number of 1.07 and show how the nonlinear time evolution in inclined layer convection follows dynamical connections between invariant solutions. Numerical continuation of stable and unstable invariant solutions under changing thermal driving and inclination angle reveals an extensive network of bifurcating solution branches. The bifurcation structures indicate existence, stability and dynamical connectivity of invariant solutions. We thereby reveal how spatio-temporally complex convection patterns depend on the control parameters in inclined layer convection.