Instability driven by boundary inflow across shear: a way to circumvent Rayleigh's stability criterion in accretion disks?

We investigate the 2D instability recently discussed by Gallet et al. (2010) and Ilin \& Morgulis (2013) which arises when a radial crossflow is imposed on a centrifugally-stable swirling flow. By finding a simpler rectilinear example of the instability - a sheared half plane, the minimal ingredients for the instability are identified and the destabilizing/stabilizing effect of inflow/outflow boundaries clarified. The instability - christened `boundary inflow instability' here - is of critical layer type where this layer is either at the inflow wall and the growth rate is $O(\sqrt{\eta})$ (as found by Ilin \& Morgulis 2013), or in the interior of the flow and the growth rate is $O(\eta \log 1/\eta)$ where $\eta$ measures the (small) inflow-to-tangential-flow ratio. The instability is robust to changes in the rotation profile even to those which are very Rayleigh-stable and the addition of further physics such as viscosity, 3-dimensionality and compressibility but is sensitive to the boundary condition imposed on the tangential velocity field at the inflow boundary. Providing the vorticity is not fixed at the inflow boundary, the instability seems generic and operates by the inflow advecting vorticity present at the boundary across the interior shear. Both the primary bifurcation to 2D states and secondary bifurcations to 3D states are found to be supercritical. Assuming an accretion flow driven by molecular viscosity only so $\eta=O(Re^{-1})$, the instability is not immediately relevant for accretion disks since the critical threshold is $O(Re^{-2/3})$ and the inflow boundary conditions are more likely to be stress-free than non-slip. However, the analysis presented here does highlight the potential for mass entering a disk to disrupt the orbiting flow if this mass flux possesses vorticity.Comment: 44 pages, 14 figure

Asymmetric, helical and mirror-symmetric travelling waves in pipe flow

New families of three-dimensional nonlinear travelling waves are discovered in pipe flow. In contrast to known waves (Faisst & Eckhardt Phys. Rev. Lett. 91, 224502 (2003), Wedin & Kerswell, J. Fluid Mech. 508, 333 (2004)), they possess no rotational symmetry and exist at much lower Reynolds numbers. Particularly striking is an `asymmetric mode' which has one slow streak sandwiched between two fast streaks located preferentially to one side of the pipe. This family originates in a pitchfork bifurcation from a mirror-symmetric travelling wave which can be traced down to a Reynolds number of 773. Helical and non-helical rotating waves are also found emphasizing the richness of phase space even at these very low Reynolds numbers. The delay in Reynolds number from when the laminar state ceases to be a global attractor to turbulent transition is then even larger than previously thought.Comment: 4 pages, 7 figures (2 at low resolution

Spatiotemporal dynamics in 2D Kolmogorov flow over large domains

Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes equations with a sinusoidal body force - is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimicks the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynamics are observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially-localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) PDEs based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions - a kink and antikink - which connect two steady one-dimensional flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour

Coherent structures in localised and global pipe turbulence

The recent discovery of unstable travelling waves (TWs) in pipe flow has been hailed as a significant breakthrough with the hope that they populate the turbulent attractor. We confirm the existence of coherent states with internal fast and slow streaks commensurate in both structure and energy with known TWs using numerical simulations in a long pipe. These only occur, however, within less energetic regions of (localized) `puff' turbulence at low Reynolds numbers (Re=2000-2400), and not at all in (homogeneous) `slug' turbulence at Re=2800. This strongly suggests that all currently known TWs sit in an intermediate region of phase space between the laminar and turbulent states rather than being embedded within the turbulent attractor itself. New coherent fast streak states with strongly decelerated cores appear to populate the turbulent attractor instead.Comment: As accepted for PRL. 4 pages, 6 figures. Alterations to figs. 4,5. Significant changes to tex

Reply to Comment on 'Critical behaviour in the relaminarization of localized turbulence in pipe flow'

This is a Reply to Comment arXiv:0707.2642 by Hof et al. on Letter arXiv:physics/0608292 which was subsequently published in Phys Rev Lett, 98, 014501 (2007). In our letter it was reported that in pipe flow the median time $\tau$ for relaminarisation of localised turbulent disturbances closely follows the scaling $\tau\sim 1/(Re_c-Re)$. This conclusion was based on data from collections of 40 to 60 independent simulations at each of six different Reynolds numbers, Re. In the Comment, Hof et al. estimate $\tau$ differently for the point at lowest Re. Although this point is the most uncertain, it forms the basis for their assertion that the data might then fit an exponential scaling $\tau\sim \exp(A Re)$, for some constant A, supporting Hof et al. (2006) Nature, 443, 59. The most certain point (at largest Re) does not fit their conclusion and is rejected. We clarify why their argument for rejecting this point is flawed. The median $\tau$ is estimated from the distribution of observations, and it is shown that the correct part of the distribution is used. The data is sufficiently well determined to show that the exponential scaling cannot be fit to the data over this range of Re, whereas the $\tau\sim 1/(Re_c-Re)$ fit is excellent, indicating critical behaviour and supporting experiments by Peixinho & Mullin 2006.Comment: 1 page, 1 figur