scholarly journals Dynamics of spatially localized states in transitional plane Couette flow

2019 ◽  
Vol 867 ◽  
pp. 414-437 ◽  
Author(s):  
Anton Pershin ◽  
Cédric Beaume ◽  
Steven M. Tobias
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Initial Conditions ◽  
Reynolds Numbers ◽  
Localized States ◽  
Transition To Turbulence ◽  
Dynamical Structure ◽  
Plane Couette Flow ◽  
Homoclinic Snaking ◽  
Chaotic Transients

Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider et al., Phys. Rev. Lett., vol. 104, 2010, 104501). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ($Re<175$) to transitional ones ($Re\approx 325$), these spatially localized states traverse various regimes where their relaminarization time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterized in this paper for a $4\unicode[STIX]{x03C0}$-periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarizing. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarizing. Very long chaotic transients ($t>10^{4}$) can be observed without difficulty for relatively low values of the Reynolds number ($Re\approx 250$).

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals Doubly localized states in plane Couette flow

2014 ◽  
Vol 758 ◽  
pp. 1-4 ◽  
Author(s):  
Bruno Eckhardt
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Linear Instability ◽  
Localized States ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Plane Couette Flow ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

AbstractMuch of our understanding of the transition to turbulence in flows without a linear instability came with the discovery and characterization of fully three-dimensional solutions to the Navier–Stokes equation. The first examples in plane Couette flow were periodic in both spanwise and streamwise directions, and could explain the transitions in small domains only. The presence of localized turbulent spots in larger domains, the spatiotemporal decoherence on larger scales and the ability to trigger turbulence with pointwise perturbations require solutions that are localized in both directions, like the one presented by Brand & Gibson (J. Fluid Mech., vol. 750, 2014, R3). They describe a steady solution of the Navier–Stokes equations and characterize in unprecedented detail, including an analytic computation of its localization properties. The study opens up new ways to describe localized turbulent patches.

Dynamical systems and the transition to turbulence in linearly stable shear flows

2007 ◽  
Vol 366 (1868) ◽  
pp. 1297-1315 ◽  
Author(s):  
Bruno Eckhardt ◽  
Holger Faisst ◽  
Armin Schmiegel ◽  
Tobias M Schneider
Keyword(s):  
Couette Flow ◽  
Coherent Structures ◽  
Pipe Flow ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Transition To Turbulence ◽  
Plane Couette Flow ◽  
Numerical Studies ◽  
Lifetime Studies ◽  
Transition Scenario

Plane Couette flow and pressure-driven pipe flow are two examples of flows where turbulence sets in while the laminar profile is still linearly stable. Experiments and numerical studies have shown that the transition has features compatible with the formation of a strange saddle rather than an attractor. In particular, the transition depends sensitively on initial conditions and the turbulent state is not persistent but has an exponential distribution of lifetimes. Embedded within the turbulent dynamics are coherent structures, which transiently show up in the temporal evolution of the turbulent flow. Here we summarize the evidence for this transition scenario in these two flows, with an emphasis on lifetime studies in the case of plane Couette flow and on the coherent structures in pipe flow.

scholarly journals Turbulent plane Couette flow at moderately high Reynolds number

2014 ◽  
Vol 751 ◽  
Author(s):  
V. Avsarkisov ◽  
S. Hoyas ◽  
M. Oberlack ◽  
J. P. García-Galache
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Wall Layer ◽  
Wall Region ◽  
Plane Couette Flow ◽  
Turbulent Plane ◽  
High Reynolds ◽  
Near Wall

AbstractA new set of numerical simulations of turbulent plane Couette flow in a large box of dimension ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}20\pi h,\, 2h,\, 6\pi h$) at Reynolds number $(\mathit{Re}_{\tau }) =125$, 180, 250 and 550 is described and compared with simulations at lower Reynolds numbers, Poiseuille flows and experiments. The simulations present a logarithmic near-wall layer and are used to verify and revise previously known results. It is confirmed that the fluctuation intensities in the streamwise and spanwise directions do not scale well in wall units. The scaling failure occurs both near to and away from the wall. On the contrary, the wall-normal intensity scales in inner units in the near-wall region and in outer units in the core region. The spectral ridge found by Hoyas & Jiménez (Phys. Fluids, vol. 18, 2003, 011702) for the turbulent Poiseuille flow can also be seen in the present flow. Away from the wall, very large-scale motions are found spanning through all the length of the channel. The statistics of these simulations can be downloaded from the webpage of the Chair of Fluid Dynamics.

scholarly journals Disruption of states by a stable stratification

2015 ◽  
Vol 784 ◽  
pp. 548-564 ◽  
Author(s):  
T. S. Eaves ◽  
C. P. Caulfield
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Coherent Structures ◽  
Initial Conditions ◽  
Edge State ◽  
Stable Stratification ◽  
Plane Couette Flow ◽  
Moderate Reynolds Number ◽  
The Waves ◽  
Laminar State

We identify ‘minimal seeds’ for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density $E_{c}$ that trigger turbulence from the laminar state, in stratified plane Couette flow, the flow between two horizontal plates of separation $2H$, moving with relative velocity $2{\rm\Delta}U$, across which a constant density difference $2{\rm\Delta}{\it\rho}$ from a reference density ${\it\rho}_{r}$ is maintained. To find minimal seeds, we use the ‘direct-adjoint-looping’ (DAL) method for finding nonlinear optimal perturbations that optimise the time-averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system’s ‘edge state’. Therefore, the trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (Phys. Fluids, vol. 9, 1997, pp. 883–900), or equivalently finite Reynolds number solutions of the vortex–wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666). The stratified coherent states we identify at moderate Reynolds number display an altered form from their unstratified counterparts for bulk Richardson numbers $\mathit{Ri}_{B}=g{\rm\Delta}{\it\rho}H/({\it\rho}_{r}{\rm\Delta}U^{2})=O(\mathit{Re}^{-1})$, and exhibit chaotic motion for larger $\mathit{Ri}_{B}$. We demonstrate that at hith Reynolds number the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when $\mathit{Ri}_{B}=O(\mathit{Re}^{-2})$.

scholarly journals On the linear stability of compressible plane Couette flow

1994 ◽  
Vol 258 ◽  
pp. 131-165 ◽  
Author(s):  
Peter W. Duck ◽  
Gordon Erlebacher ◽  
M. Yousuff Hussaini
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Small Amplitude ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Plane Couette Flow ◽  
Moving Wall ◽  
Type Boundary ◽  
The Stability ◽  
Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

scholarly journals The linear instability of the stratified plane Couette flow

2018 ◽  
Vol 853 ◽  
pp. 205-234 ◽  
Author(s):  
Giulio Facchini ◽  
Benjamin Favier ◽  
Patrice Le Gal ◽  
Meng Wang ◽  
Michael Le Bars
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Couette Flow ◽  
Vertical Direction ◽  
Stability Diagram ◽  
Linear Instability ◽  
Plane Couette Flow ◽  
Horizontal Shear ◽  
The Stability

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

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