Stability Landscape of Shell Buckling.

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.

Increasing lifetimes and the growing saddles of shear flow turbulence.

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.

Complexity of localised coherent structures in a boundary-layer flow.

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.

Periodic orbits near onset of chaos in plane Couette flow.

We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo a periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show that the ones that exist fall into the universal sequence described by Metropolis, Stein and Stein for unimodal maps. The periodic orbits cover much of the turbulent dynamics in that their temporal evolution overlaps with turbulent motions when projected onto a plane spanned by energy production and dissipation.

Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes.

Wall-bounded shear flows transitioning to turbulence may self-organize into alternating turbulent and laminar regions forming a stripe pattern with non-trivial oblique orientation. Different experiments and flow simulations identify oblique stripe patterns as the preferred solution of the well-known Navier-Stokes equations, but the origin of stripes and their oblique orientation remains unexplained. In concluding his lectures, Feynman highlights the unexplained stripe pattern hidden in the solution space of the Navier-Stokes equations as an example demonstrating the need for improved theoretical tools to analyze the fluid flow equations. Here we exploit dynamical systems methods and demonstrate the existence of an exact equilibrium solution of the fully nonlinear 3D Navier-Stokes equations that resembles oblique stripe patterns in plane Couette flow. The stripe equilibrium emerges from the well-studied Nagata equilibrium and exists only for a limited range of pattern angles. This suggests a mechanism selecting the non-trivial oblique orientation angle of turbulent-laminar stripes.

Fully localized post-buckling states of cylindrical shells under axial compression.

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.

Long-wavelength instability of coherent structures in plane Couette flow.

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.