arXiv subjects

Stefan Zammert

Publications and source records attributed to Stefan Zammert.

13 recordsLinked to original sources

arXiv2020

Linear feedback control of invariant solutions in channel flow

Considering channel flow at Reynolds numbers below the linear stability threshold of the laminar profile as a generic example system showing a subcritical transition to turbulence connected with the existence of simple invariant solutions, we here discuss issues that arise in the application of linear feedback control of invariant solutions of the Navier-Stokes equations. We focus on the simplest possible problem, that is, travelling waves with one unstable direction. In view of potential experimental applicability we construct a pressure-based feedback strategy and study its effect on the stable, marginal and unstable directions of these solutions in different periodic cells. Even though the original instability can be removed, new instabilities emerge as the feedback procedure affects not only the unstable but also the stable directions. We quantify these adverse effects and discuss their implications for the design of successful control strategies. In order to highlight the challenges that arise in the application of feedback control methods in principle and concerning potential applications in the search for simple invariant solutions of the Navier-Stokes equations in particular, we consider an explicitly constructed analogue to closed-loop linear optimal control that leaves the stable directions unaffected.

physics.flu-dyn↗

arXiv2017

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

The transition to turbulence in plane Couette flow and several other shear flows is connected with saddle node bifurcations in which fully 3-d, nonlinear solutions, so-called exact coherent states (ECS), to the Navier-Stokes equation appear. As the Reynolds number increases, the states undergo secondary bifurcations and their time-evolution becomes increasingly more complex. Their spatial complexity, in contrast, remains limited so that these states cannot contribute to the spatial complexity and cascade to smaller scales expected for higher Reynolds numbers. We here present families of scaling ECS that exist on ever smaller scales as the Reynolds number is increased. We focus in particular on two such families for plane Couette flow, one centered near the midplane and the other close to a wall. We discuss their scaling and localization properties and the bifurcation diagrams. All solutions are localized in the wall-normal direction. In the spanwise and downstream direction, they are either periodic or localized as well. The family of scaling ECS localized near a wall is reminiscent of attached eddies, and indicates how self-similar ECS can contribute to the formation of boundary layer profiles.

physics.flu-dyn↗

arXiv2017

Analysis and modeling of localized invariant solutions in pipe flow

Turbulent spots surrounded by laminar flow are a landmark of transitional shear flows, but the dependence of their kinematic properties on spatial structure is poorly understood. We here investigate this dependence in pipe flow for Reynolds numbers between 1500 and 5000. We compute spatially localized relative periodic orbits in long pipes and show that their upstream and downstream fronts decay exponentially towards the laminar profile. This allows to model the fronts by employing the linearized Navier-Stokes equations, and the resulting model yields the spatial decay rate and the front velocity profiles of the periodic orbits as a function of Reynolds number, azimuthal wave number and propagation speed. In addition, when applied to a localized turbulent puff, the model is shown to accurately approximate the spatial decay rate of its upstream and downstream tails. Our study provides insight into the relationship between the kinematics and spatial structure of localized turbulence and more generally into the physics of localization.

physics.flu-dyn↗

arXiv2017

Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist

Plane Poiseuille flow, the pressure driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien-Schlichting (TS) waves, and another one, the bypass transition, that is triggered with finite amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance $2H$ apart and in a domain of width $2πH$ and length $2πH$ the subcritical instability to TS waves sets in at $Re_{c}=5815$ that extends down to $Re_{TS}\approx4884$. The bypass route becomes available above $Re_E=459$ with the appearance of three-dimensional finite-amplitude traveling waves. The bypass transition covers a large set of finite amplitude perturbations. Below $Re_c$, TS appear for a tiny set of initial conditions that grows with increasing Reynolds number. Above $Re_c$ the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.

physics.flu-dyn↗

arXiv2016

Streamwise decay of localized states in channel 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 side. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)] 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 wave lengths.

physics.flu-dyn↗

arXiv2016

Transition in the asymptotic suction boundary layer over a heated plate

The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wavenumber depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, showing different symmetries. In the subcritical regime, transient turbulent states which are connected to exact coherent states and follow the same transition scenario as found in linearly stable shear flows are identified and analyzed. The study extends the bypass transition scenario from shear flows to thermal boundary layers and highlights the intricate interactions between thermal and shear forces.

physics.flu-dyn↗

arXiv2016

Harbingers and latecomers - The order of appearance of exact coherent structures in plane Poiseuille flow

The transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. In contrast to other shear flows, PPF has a number of different coherent states that are relevant for the transition. We here discuss the different states, 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, the ones for constant pressure drop are always higher. The Tollmien-Schlichting waves bifurcate subcritically from the laminar profile at $Re=5772$ and reach down to $Re=2609$ (at a different optimal wave length). Their localized 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 is spanwise localized and has a critical Reynolds number of $316$. Comparison to plane Couette flow suggests that this is likely the lowest Reynolds number for exact coherent structures in PPF. Streamwise localized versions of this state require higher Reynolds numbers, with the lowest bifurcation occurring near $Re=1018$.

physics.flu-dyn↗

arXiv2015

Bypass transition and subcritical turbulence in plane Poiseuille flow

Plane Poiseuille flow shows turbulence at a Reynolds number that is lower than the critical one for the onset of Tollmien-Schlichting waves. The transition to turbulence follows the same route as the by-pass transition in boundary layers, i.e. finite amplitude perturbations are required and the flow is dominated by downstream vortices and streaks in the transitional regime. In order to relate the phenomenology in plane Poiseuille flow to our previous studies of plane Couette flow (Kreilos & Eckhardt, 2012), we study a symmetric subspace of plane Poiseuille flow in which the bifurcation cascade stands out clearly. By tracing the edge state, which in this system is a travelling wave, and its bifurcations, we can trace the formation of a chaotic attractor, the interior crisis that increase the phase space volume affected by the flow, and the ultimate transition into a chaotic saddle in a crisis bifurcation. After the boundary crisis we can observe transient chaos with exponentially distributed lifetimes.

physics.flu-dyn↗

arXiv2015

Crisis bifurcations in plane Poiseuille flow

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.

physics.flu-dyn↗

arXiv2014

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 localization 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 streamwise direction, the envelope shows exponential localization, with different exponents on the upstream and downstream side. The upstream exponent increases linearly with Reynolds number Re, but the downstream exponent is essentially independent of Re. 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.

physics.flu-dyn↗

arXiv2014

A spotlike edge state in plane Poiseuille flow

We study the laminar turbulent boundary in plane Poiseuille flow at $Re=1400$ and $2180$ using the technique of edge tracking. For large computational domains the attracting state in the laminar-turbulent boundary is localized in spanwise and streamwise direction and chaotic.

physics.flu-dyn↗

arXiv2014

Symmetry related dynamics in parallel shear flows

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.

physics.flu-dyn↗

arXiv2014

Periodically bursting edge states in plane Poiseuille 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 in Reynolds number that all states in the boundary converge to a period orbit with a period of the order of $10^{3}$ time units. The attracting states in these small domains are periodically extended in the spanwise and streamwise direction, but always localized to one side of the channel in the normal direction. In short and wide domains the edge states are localized in the spanwise direction. The periodic motion found in the small domains then induces a large variety of dynamical activity. The findings are very similar to the ones in the asymptotic suction boundary layer.

physics.flu-dyn↗