Homoclinic bifurcation and switching of edge state in plane Couette flow.

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.

Steady thermal convection representing the ultimate scaling.

Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier-Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance [Formula: see text] and a constant temperature difference [Formula: see text]. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. [Formula: see text] (Jiménez et al. 2001 J. Fluid Mech., 442, 89-117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton-Krylov iteration up to the Rayleigh number [Formula: see text] for the Prandtl number [Formula: see text], the horizontal period [Formula: see text] and the permeability parameter [Formula: see text]-[Formula: see text], [Formula: see text] being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady 'wall-bounded' thermal convection. The ultimate scaling [Formula: see text] has been observed for [Formula: see text] at high [Formula: see text], where [Formula: see text] is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity [Formula: see text] as well as intense temperature variation of [Formula: see text] even in the bulk region. As a consequence, the wall-to-wall heat flux scales with [Formula: see text] independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Time-Periodic Inertial Range Dynamics.

A wide class of physical systems exhibit scale invariance. While the statistical properties of such behavior can often be investigated by theoretical and experimental means, its dynamics are notoriously hard to parse. We investigate scale-invariant dynamics through an unstable periodic orbit. This orbit coexists with turbulence of an incompressible fluid and yields a significant Kolmogorov energy spectrum. We identify events of intense energy transfer across spatial scales and relate them to vortical dynamics. The results support a recently proposed mechanism for turbulent energy transfer.

Homoclinic tangle on the edge of shear turbulence.

Experiments and simulations lend mounting evidence for the edge state hypothesis on subcritical transition to turbulence, which asserts that simple states of fluid motion mediate between laminar and turbulent shear flow as their stable manifolds separate the two in state space. In this Letter we describe flows homoclinic to a time-periodic edge state that display the essential properties of turbulent bursting. During a burst, vortical structures and the associated energy dissipation are highly localized near the wall, in contrast with the familiar regeneration cycle.