arXiv subjects

Bruno Eckhardt

Publications and source records attributed to Bruno Eckhardt.

At least 37 records · Page 2Linked to original sources

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

Heat transport in Rayleigh-Benard convection and angular momentum transport in Taylor-Couette flow: a comparative study

Rayleigh-Benard 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-Benard convection in air at Rayleigh number Ra=1e7 and Taylor-Couette flow at shear Reynolds number Re_S=2e4 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.

physics.flu-dyn↗

arXiv2016

Marginally stable and turbulent boundary layers in low-curvature Taylor-Couette flow

Marginal stability arguments are used to describe the rotation-number dependence of torque in Taylor-Couette (TC) flow for radius ratios $η\geq 0.9$ and shear Reynolds number $Re_S=2\times 10^4$. With an approximate representation of the mean profile by piecewise linear functions, characterized by the boundary-layer thicknesses at the inner and outer cylinder and the angular momentum in the center, profiles and torques are extracted from the requirement that the boundary layers represent marginally stable TC subsystems and that the torque at the inner and outer cylinder coincide. This model then explains the broad shoulder in the torque as a function of rotation number near $R_Ω\approx 0.2$. For rotation numbers $R_Ω< 0.07$ the TC stability conditions predict boundary layers in which shear Reynolds numbers are very large. Assuming that the TC instability is bypassed by some shear instability, a second maximum in torque appears, in very good agreement with numerical simulations. The results show that, despite the shortcomings of marginal stability theory in other cases, it can explain quantitatively the non-monotonic torque variation with rotation number for both the broad maximum as well as the narrow maximum.

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↗

arXiv2016

Edge states as mediators of bypass transition in boundary-layer flows

The concept of edge state is investigated in the asymptotic suction boundary layer in relation with the receptivity process to noisy perturbations and the nucleation of turbulent spots. Edge tracking is first performed numerically, without imposing any discrete symmetry, in a large computational domain allowing for full spatial localisation of the perturbation velocity. The edge state is a three-dimensional localised structure recurrently characterised by a single low-speed streak that experiences erratic bursts and planar shifts. This recurrent streaky structure is then compared with predecessors of individual spot nucleation events, triggered by non-localised initial noise. The present results suggest a nonlinear picture, rooted in dynamical systems theory, of the nucleation process of turbulent spots in boundary-layer flows, in which the localised edge state play the role of state-space mediator.

physics.flu-dyn↗

arXiv2016

Bypass transition and spot nucleation in boundary layers

The spatio-temporal aspects of the transition to turbulence are considered in the case of a boundary layer flow developing above a flat plate exposed to free-stream turbulence. Combining results on the receptivity to free-stream turbulence with the nonlinear concept of a transition threshold, a physically motivated model suggests a spatial distribution of spot nucleation events. To describe the evolution of turbulent spots a probabilistic cellular automaton is introduced, with all parameters directly fitted from numerical simulations of the boundary layer. The nucleation rates are then combined with the cellular automaton model, yielding excellent quantitative agreement with the statistical characteristics for different free-stream turbulence levels. We thus show how the recent theoretical progress on transitional wall-bounded flows can be extended to the much wider class of spatially developing boundary-layer flows.

physics.flu-dyn↗

arXiv2015

Order-disorder transitions in a sheared many body system

Motivated by experiments on sheared suspensions that show a transition between ordered and disordered phases, we here study the long-time behavior of a sheared and overdamped 2-d system of particles interacting by repulsive forces. As a function of interaction strength and shear rate we find transitions between phases with vanishing and large single-particle diffusion. In the phases with vanishing single-particle diffusion, the system evolves towards regular lattices, usually on very slow time scales. Different lattices can be approached, depending on interaction strength and forcing amplitude. The disordered state appears in parameter regions where the regular lattices are unstable. Correlation functions between the particles reveal the formation of shear bands. In contrast to single particle densities, the spatially resolved two-particle correlation functions vary with time and allow to determine the phase within a period. As in the case of the suspensions, motion in the state with low diffusivity is essentially reversible, whereas in the state with strong diffusion it is not.

cond-mat.soft↗

arXiv2015

Turbulence transition in the asymptotic suction boundary layer

We study the transition to turbulence in the asymptotic suction boundary layer (ASBL) by direct numerical simulation. Tracking the motion of trajectories intermediate between laminar and turbulent states we can identify the invariant object inside the laminar-turbulent boundary, the edge state. In small domains, the flow behaves like a travelling wave over short time intervals. On longer times one notes that the energy shows strong bursts at regular time intervals. During the bursts the streak structure is lost, but it reforms, translated in the spanwise direction by half the domain size. Varying the suction velocity allows to embed the flow into a family of flows that interpolate between plane Couette flow and the ASBL. Near the plane Couette limit, the edge state is a travelling wave. Increasing the suction, the travelling wave and a symmetry-related copy of it undergo a saddle-node infinite-period (SNIPER) bifurcation that leads to bursting and discrete-symmetry shifts. In wider domains, the structures localize in the spanwise direction, and the flow in the active region is similar to the one in small domains. There are still periodic bursts at which the flow structures are shifted, but the shift-distance is no longer connected to a discrete symmetry of the flow geometry. Two different states are found by edge tracking techniques, one where structures are shifted to the same side at every burst and one where they are alternatingly shifted to the left and to the right.

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

Momentum transport in Taylor-Couette flow with vanishing curvature

We numerically study turbulent Taylor-Couette flow (TCF) between two independently rotating cylinders and the transition to rotating plane Couette flow (RPCF) in the limit of infinite radii. By using the shear Reynolds number $Re_S$ and rotation number $R_Ω$ as dimensionless parameters, the transition from TCF to RPCF can be studied continuously without singularities. Already for radius ratios $η\geq0.9$ we find that the simulation results for various radius ratios and for RPCF collapse as a function of $R_Ω$, indicating a turbulent behaviour common to both systems. We observe this agreement in the torque, mean momentum transport, mean profiles, and turbulent fluctuations. Moreover, the central profiles in TCF and RPCF for $R_Ω>0$ are found to conform with inviscid neutral stability. Intermittent bursts that have been observed in the outer boundary layer and have been linked to the formation of a torque maximum for counter-rotation are shown to disappear as $η\rightarrow 1$. The corresponding torque maximum disappears as well. Instead, two new maxima of different origin appear for $η\geq0.9$ and RPCF, a broad and a narrow one, in contrast to the results for smaller $η$. The broad maximum at $R_Ω=0.2$ is connected with a strong vortical flow and can be reproduced by streamwise invariant simulations. The narrow maximum at $R_Ω=0.02$ only emerges with increasing $Re_S$ and is accompanied by an efficient and correlated momentum transport by the mean flow. Since the narrow maximum is of larger amplitude for $Re_S=2\cdot10^4$, our simulations suggest that it will dominate at even higher $Re_S$.

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↗

arXiv2015

Direct and noisy transitions in a model shear flow

The transition to turbulence in flows where the laminar profile is linearly stable requires perturbations of finite amplitude. "Optimal" perturbations are distinguished as extrema of certain functionals, and different functionals give different optima. We here discuss the phase space structure of a 2-d simplified model of the transition to turbulence and discuss optimal perturbations with respect to three criteria: energy of the initial condition, energy dissipation of the initial condition and amplitude of noise in a stochastic transition. We find that the states that trigger the transition are different in the three cases, but show the same scaling with Reynolds number.

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

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 torque for a given Re is a non-monotonic function of rotation number, Ro. For low-to-moderate turbulent Reynolds numbers we find a maximum that is associated with flow fields that are dominated by downstream vortices and calculations of 2-d vortices capture the maximum also quantitatively. For higher shear 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 3-d structure and cannot be captured by 2-d vortex studies. As the Reynolds number increases, this maximum becomes stronger and eventually overtakes the one associated with the 2-d flow state.

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

Groebner basis methods for stationary solutions of a low-dimensional model for a shear flow

We use Groebner basis methods to extract all stationary solutions for the 9-mode shear flow model that is described in Moehlis et al, New J. Phys. 6, 54 (2004). Using rational approximations to irrational wave numbers and algebraic manipulation techniques we reduce the problem of determining all stationary states to finding roots of a polynomial of order 30. The coefficients differ by 30 powers of 10 so that algorithms for extended precision are needed to extract the roots reliably. We find that there are eight stationary solutions consisting of two distinct states that each appear in four symmetry-related phases. We discuss extensions of these results for other flows.

physics.flu-dyn↗