Tobias Kreilos

We measure the response of cylindrical shells to poking and identify a stability landscape, which fully characterizes the stability of perfect shells and imperfect ones in the case where a single defect dominates. We show that the landscape of stability is independent of the loading protocol and the poker geometry. Our results suggest that the complex stability of shells reduces to a low dimensional description. Tracking ridges and valleys of this landscape defines a natural phase-space coordinates for describing the stability of shells.

We compute nonlinear force equilibrium solutions for a clamped thin cylindrical shell under axial compression. The equilibrium solutions are dynamically unstable and located on the stability boundary of the unbuckled state. A fully localized single dimple deformation is identified as the edge state —the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. These fully nonlinear solutions embedded in the stability boundary of the unbuckled state constitute critical shape deformations. The solutions may thus be a step towards explaining when the buckling and subsequent collapse of an axially loaded cylinder shell is triggered.

The spatiotemporal aspects of the transition to turbulence are considered in the case of a boundary-layer flow developing above a flat plate exposed to free-stream turbulence. Combining results on the receptivity to free-stream turbulence with the nonlinear concept of a transition threshold, a physically motivated model suggests a spatial distribution of spot nucleation events. To describe the evolution of turbulent spots a probabilistic cellular automaton is introduced, with all parameters directly obtained from numerical simulations of the boundary layer. The nucleation rates are then combined with the cellular automaton model, yielding excellent quantitative agreement with the statistical characteristics for different free-stream turbulence levels. We thus show how the recent theoretical progress on transitional wall-bounded flows can be extended to the much wider class of spatially developing boundary-layer flows.

The concept of edge states is investigated in the asymptotic suction boundary layer in relation to the receptivity process to noisy perturbations and the nucleation of turbulent spots. Edge tracking is first performed numerically, without imposing any discrete symmetry, in a large computational domain allowing for full spatial localisation of the perturbation velocity. The edge state is a three-dimensional localised structure recurrently characterised by a single low-speed streak that experiences erratic bursts and planar shifts. This recurrent streaky structure is then compared with predecessors of individual spot nucleation events, triggered by non-localised initial noise. The present results suggest a nonlinear picture, rooted in dynamical systems theory, of the nucleation process of turbulent spots in boundary-layer flows, in which the localised edge state plays the role of state-space mediator.

We present two spanwise-localized travelling-wave solutions in the asymptotic suction boundary layer, obtained by continuation of solutions of plane Couette flow. One of the solutions has the vortical structures located close to the wall, similar to spanwise-localized edge states previously found for this system. The vortical structures of the second solution are located in the free stream far above the laminar boundary layer and are supported by a secondary shear gradient that is created by a large-scale low-speed streak. The dynamically relevant eigenmodes of this solution are concentrated in the free stream, and the departure into turbulence from this solution evolves in the free stream towards the walls. For invariant solutions in free-stream turbulence, this solution thus shows that the source of energy of the vortical structures can be a dynamical structure of the solution itself, instead of the laminar boundary layer.

Turbulent-laminar banded patterns in plane Poiseuille flow are studied via direct numerical simulations in a tilted and translating computational domain using a parallel version of the pseudospectral code Channelflow. 3D visualizations via the streamwise vorticity of an instantaneous and a time-averaged pattern are presented, as well as 2D visualizations of the average velocity field and the turbulent kinetic energy. Simulations for 2300 ⩾ Re_b ⩾ 700 show the gradual development from uniform turbulence to a pattern with wavelength 20 half-gaps at Re_b ≈ 1900, to a pattern with wavelength 40 at Re_b ≈ 1300 and finally to laminar flow at Re_b ≈ 800. These transitions are tracked quantitatively via diagnostics using the amplitude and phase of the Fourier transform and its probability distribution. The propagation velocity of the pattern is approximately that of the mean flux and is a decreasing function of Reynolds number. Examination of the time-averaged flow shows that a turbulent band is associated with two counter-rotating cells stacked in the cross-channel direction and that the turbulence is highly concentrated near the walls. Near the wall, the Reynolds stress force accelerates the fluid through a turbulent band while viscosity decelerates it; advection by the laminar profile acts in both directions. In the center, the Reynolds stress force decelerates the fluid through a turbulent band while advection by the laminar profile accelerates it. These characteristics are compared with those of turbulent-laminar banded patterns in plane Couette flow.

Parallel shear flows have continuous symmetries of translation in the downstream and spanwise directions. As a consequence, flow states that differ in their spanwise or downstream location but are otherwise identical are dynamically equivalent. In the case of travelling waves, this trivial degree of freedom can be removed by going to a frame of reference that moves with the state, thereby turning the travelling wave in the laboratory frame into a fixed point in the comoving frame of reference. We here discuss a general approach, the method of comoving frames, by which the symmetry related motions can also be removed for more complicated and dynamically active states and demonstrate its application for several examples. For flow states in the asymptotic suction boundary layer (ASBL) we show that in the case of the long-period oscillatory edge state we can find local phase speeds which remove the fast oscillations and reveal the slow vortex dynamics underlying the burst phenomenon. For spanwise translating states we show that the method removes the drift but not the dynamical events that cause the big spanwise displacement. For a turbulent case we apply the method to the spanwise shifts and find slow components that are correlated over very long times. Calculations for plane Poiseuille flow show that the long correlations in the transverse motions are not specific to the ASBL.

We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar-turbulent separatrix is investigated in a spanwise-extended domain that allows for robust localisation of all edge states. We work at fixed Reynolds number and study the edge states as a function of the streamwise period. We demonstrate the complex spatio-temporal dynamics of these localised states, which exhibits multistability and undergoes complex bifurcations leading from periodic to chaotic regimes. It is argued that in all regimes the dynamics restricted to the edge is essentially low-dimensional and non-extensive.

We study the stability of coherent structures in plane Couette flow against long-wavelength perturbations in wide domains that cover several pairs of coherent structures. For one and two pairs of vortices, the states retain the stability properties of the small domains, but for three pairs new unstable modes are found. They are shown to be connected to bifurcations that break the translational symmetry and drive the coherent structures from the spanwise extended state to a modulated one that is a precursor to spanwise localized states. Tracking the stability of the orbits as functions of the spanwise wave length reveals a rich variety of additional bifurcations.

In linearly stable shear flows, turbulence spontaneously decays with a characteristic lifetime that varies with Reynolds number. The lifetime sharply increases with Reynolds number so that a possible divergence marking the transition to sustained turbulence at a critical point has been discussed. We present a mechanism by which the lifetimes increase: in the system's state space, turbulent motion is supported by a chaotic saddle. Inside this saddle a locally attracting periodic orbit is created and undergoes a traditional bifurcation sequence generating chaos. The formed new `` turbulent bubble'' is initially an attractor supporting persistent chaotic dynamics. Soon after its creation, it collides with its own boundary, by which it becomes leaky and dynamically connected with the surrounding structures. The complexity of the chaotic saddle that supports transient turbulence hence increases by incorporating the remnant of a new bubble. As a a result, the time it takes for a trajectory to leave the saddle and decay to the laminar state is increased. We demonstrate this phenomenon in plane Couette flow and show that characteristic lifetimes vary nonsmoothly and nonmonotonically with Reynolds number.