Stefan Zammert

Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller periodically continued domains, by increasing the length to obtain streamwise localisation and then by increasing the width to achieve spanwise localisation. The states maintain the travelling wave structure of the extended states, which is then modulated by a localised envelope on larger scales. In the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number

Periodically bursting edge states in plane Poiseuille flow

\mathit{Re}

Crisis bifurcations in plane Poiseuille flow

, but the downstream exponent is essentially independent of

Comoving frames and symmetry-related motions in parallel shear flows

\mathit{Re}

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

. In the spanwise direction the decay is compatible with a power-law localisation. As the width increases the localised state undergoes further bifurcations which add additional unstable directions, so that the edge state, the relative attractor on the boundary between the laminar and turbulent motions, in the system becomes chaotic.

Streamwise decay of localized states in channel flow

We investigate the laminar-turbulent boundary in plane Poiseuille flow by the method of edge tracking. In short and narrow computational domains we find for a wide range of Reynolds numbers that all states in the boundary converge to a periodic orbit with a period of the order of time units. The attracting states in these small domains are periodically extended in the spanwise and streamwise directions, but always localized to one side of the channel in the normal direction. In wider domains the edge states are localized in the spanwise direction as well. The periodic motion found in the small domains then induces a large variety of dynamical activity that is similar to that found in the asymptotic suction boundary layer.

Transition in the asymptotic suction boundary layer over a heated plate

Many shear flows follow a route to turbulence that has striking similarities to bifurcation scenarios in low-dimensional dynamical systems. Among the bifurcations that appear, crisis bifurcations are important because they cause global transitions between open and closed attractors, or indicate drastic increases in the range of the state space that is covered by the dynamics. We here study exterior and interior crisis bifurcations in direct numerical simulations of transitional plane Poiseuille flow in a mirror-symmetric subspace. We trace the state space dynamics from the appearance of the first three-dimensional exact coherent structures to the transition from an attractor to a chaotic saddle in an exterior crisis. For intermediate Reynolds numbers, the attractor undergoes several interior crises, in which new states appear and intermittent behavior can be observed. The bifurcations contribute to increasing the complexity of the dynamics and to a more dense coverage of state space.

Analysis and modeling of localized invariant solutions in pipe flow

Parallel shear flows come with continuous symmetries of translation in the downstream and spanwise direction. 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 to a fixed point in the comoving frame of reference. We here discuss a general method by which the translational displacements can be removed also for more complicated and dynamically active states and demonstrate its application for several examples. For flows states in the asymptotic suction boundary layer 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 special to the asymptotic suction boundary layer.

A spotlike edge state in plane Poiseuille flow

ABSTRACTThe transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. We here discuss a variety of different structures that are relevant for the transition, compare the critical Reynolds numbers and optimal wavelengths for their appearance, and explore the differences between flows operating at constant mass flux or at constant pressure drop. The Reynolds numbers quoted here are based on the mean flow velocity and refer to constant mass flux. Reynolds numbers based on constant pressure drop are always higher. The Tollmien–Schlichting (TS) waves bifurcate subcritically from the laminar profile at Re = 5772 at wavelength 6.16 and reach down to Re = 2610 at a different optimal wave length of 4.65. Their streamwise localised counter part bifurcates at the even lower value Re = 2334. Three-dimensional exact solutions appear at much lower Reynolds numbers. We describe one exact solutions that has a critical Reynolds number of 316. Streamwise localised ...

Small scale exact coherent structures at large Reynolds numbers in plane Couette flow

Channel flow, the pressure driven flow between parallel plates, has exact coherent structures that show various degrees of localization. For states which are localized in streamwise direction but extended in spanwise direction, we show that they are exponentially localized, with decay constants that are different on the upstream and downstream sides. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)]JFLSA70022-112010.1017/jfm.2014.285 for stationary states to the case of advected structures that is needed here, and derive expressions for the decay in terms of eigenvalues and eigenfunctions of certain second order differential equations. The results are in very good agreement with observations on exact coherent structures of different transversal wavelengths.