Bruno Eckhardt

Finite lifetime of turbulence in shear flows

Generally, the motion of fluids is smooth and laminar at low speeds but becomes highly disordered and turbulent as the velocity increases. The transition from laminar to turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states, or it can occur suddenly. Once the transition has taken place, it is generally assumed that, under steady conditions, the turbulent state will persist indefinitely. The flow of a fluid down a straight pipe provides a ubiquitous example of a shear flow undergoing a sudden transition from laminar to turbulent motion. Extensive calculations and experimental studies have shown that, at relatively low flow rates, turbulence in pipes is transient, and is characterized by an exponential distribution of lifetimes. They also suggest that for Reynolds numbers exceeding a critical value the lifetime diverges (that is, becomes infinitely large), marking a change from transient to persistent turbulence. Here we present experimental data and numerical calculations covering more than two decades of lifetimes, showing that the lifetime does not in fact diverge but rather increases exponentially with the Reynolds number. This implies that turbulence in pipes is only a transient event (contrary to the commonly accepted view), and that the turbulent and laminar states remain dynamically connected, suggesting avenues for turbulence control.

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.Graphical abstract

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 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.

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.

Turbulent states in plane Couette flow with rotation

Shearing and rotational forces in fluids can significantly alter the transport of momentum. A numerical investigation was undertaken to study the role of these forces using plane Couette flow subject to rotation about an axis perpendicular to both wall-normal and streamwise directions. Using a set of progressively higher Reynolds numbers up to Re = 5200, we find that the momentum flux, measured by the wall shear stress, for a given Re is a non-monotonic function of rotation number, Ro. For low-to-moderate Reynolds numbers, we find a maximum that is associated with flow fields that are dominated by downstream vortices and calculations of 2D vortices capture the maximum also quantitatively. For higher Reynolds numbers, a second stronger maximum emerges at smaller rotation numbers, closer to non-rotating plane Couette flow. It is carried by flows with a markedly 3D structure and cannot be captured by 2D vortex studies. As the Reynolds number increases, this maximum becomes stronger and eventually overtakes the one associated with the 2D flow state.

Harbingers and latecomers – the order of appearance of exact coherent structures 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 ...

Localization in plane Couette edge dynamics

In linearly stable shear flows including pipe and plane Couette flow finite amplitude perturbations are required to trigger turbulence [1]. Studying lifetimes of perturbations and locating the transition between smooth variations and sensitive dependence on initial conditions we identify the boundary in between laminar and turbulent dynamics [2, 3]. States within this edge of chaos evolve towards a relative attractor, the edge state whose stable set forms the edge. In numerical studies of small domains, spatially periodic edge states have been identified. However, transitional turbulence in spatially extended systems is observed in localized patterns such as slugs and puffs in pipe flow or striped turbulence in Taylor-Couette and plane Couette flow [5, 6] suggesting the existence of localized edge states.

Heat transport in Rayleigh–Bénard convection and angular momentum transport in Taylor–Couette flow: a comparative study

Rayleigh–Bénard convection and Taylor–Couette flow are two canonical flows that have many properties in common. We here compare the two flows in detail for parameter values where the Nusselt numbers, i.e. the thermal transport and the angular momentum transport normalized by the corresponding laminar values, coincide. We study turbulent Rayleigh–Bénard convection in air at Rayleigh number Ra=107 and Taylor–Couette flow at shear Reynolds number ReS=2×104 for two different mean rotation rates but the same Nusselt numbers. For individual pairwise related fields and convective currents, we compare the probability density functions normalized by the corresponding root mean square values and taken at different distances from the wall. We find one rotation number for which there is very good agreement between the mean profiles of the two corresponding quantities temperature and angular momentum. Similarly, there is good agreement between the fluctuations in temperature and velocity components. For the heat and angular momentum currents, there are differences in the fluctuations outside the boundary layers that increase with overall rotation and can be related to differences in the flow structures in the boundary layer and in the bulk. The study extends the similarities between the two flows from global quantities to local quantities and reveals the effects of rotation on the transport. This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

A spotlike edge state in plane Poiseuille flow

J.M. Burgerscentrum, Delft University of Technology, Mekelweg 2, 2628 CD DelftWe study the laminar turbulent boundary in plane Poiseuille flow at Re = 1400 and 2180 using the technique of edgetracking. For large computational domains the attracting state in the laminar-turbulent boundary is localized in spanwise andstreamwise direction and chaotic.

Chaos control applied to coherent states in transitional flows

Chaos control refers to a group of techniques by which an otherwise unstable dynamical state of a system can be maintained by small control forces. We here discuss their application to stabilizing the fixed points in a low dimensional model for shear flows. The simulations demonstrate a prototypical application of chaos control, show that control is almost always possible, and give insights into optimizing the control matrix from a design point of view.