Exact coherent structures in stably stratified plane Couette flow

The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds–Richardson number ($Re$–$Ri_{b}$) space for a fluid of unit Prandtl number $(Pr=1)$ using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge tracking – EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (J. Fluid Mech., vol. 745, 2014, pp. 25–61) – and found to connect with two-dimensional convective roll solutions when tracked to negative $Ri_{b}$ (the Rayleigh–Bénard problem with shear). Both these states and Nagata’s (J. Fluid Mech., vol. 217, 1990, pp. 519–527) original exact solution feel the presence of stable stratification when $Ri_{b}=O(Re^{-2})$ or equivalently when the Rayleigh number $Ra:=-Ri_{b}Re^{2}Pr=O(1)$. This is confirmed via a stratified extension of the vortex wave interaction theory of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at $Ri_{b}=O(Re^{-2/3})$. This corresponds to a stratified version of the boundary region equations regime of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58–85). Increasing the stratification further appears to lead to a third, ultimate regime where $Ri_{b}=O(1)$ in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar–turbulent boundary in the ($Re$–$Ri_{b}$) plane are briefly discussed.

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Disruption of states by a stable stratification

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.596 ◽

2015 ◽

Vol 784 ◽

pp. 548-564 ◽

Cited By ~ 10

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 $\mathit{Ri}_{B}=O(\mathit{Re}^{-2})$.

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Dynamic mode decomposition analysis of coherent structures in rotating plane Couette flow

Journal of Physics Conference Series ◽

10.1088/1742-6596/1522/1/012012 ◽

2020 ◽

Vol 1522 ◽

pp. 012012

Author(s):

Francesco Sacco ◽

Rodolfo Ostilla-Mónico ◽

Roberto Verzicco

Keyword(s):

Couette Flow ◽

Coherent Structures ◽

Decomposition Analysis ◽

Dynamic Mode Decomposition ◽

Dynamic Mode ◽

Plane Couette Flow ◽

Mode Decomposition

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Various coherent structures in roughened plane Couette flow

The Proceedings of the Fluids engineering conference ◽

10.1299/jsmefed.2016.1012 ◽

2016 ◽

Vol 2016 (0) ◽

pp. 1012

Author(s):

Takahiro ISHIDA ◽

Geert BRETHOUWER ◽

Yohann DUGUET ◽

Takahiro TSUKAHARA

Keyword(s):

Couette Flow ◽

Coherent Structures ◽

Plane Couette Flow

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Fractal sets of neutral curves for stably stratified plane Couette flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.377 ◽

2019 ◽

Vol 872 ◽

pp. 697-728 ◽

Cited By ~ 1

Author(s):

Jonathan J. Healey

Keyword(s):

Couette Flow ◽

Density Profile ◽

Basic Density ◽

Stable Stratification ◽

Plane Couette Flow ◽

Homogeneous Fluid ◽

Longitudinal Vortices ◽

Fractal Sets ◽

Neutral Curves ◽

Wave Angle

The linear stability of plane Couette flow is investigated when the plates are horizontal, and the fluid is stably stratified with a cubic basic density profile. The disturbances are treated as inviscid and diffusion of the density field is neglected. Previous studies have shown that this density profile can develop multiple neutral curves, despite the stable stratification, and the fact that plane Couette flow of homogeneous fluid is stable. It is shown that when the neutral curves are plotted with wave angle on one axis, and location of the density inflexion point on the other axis, they produce a self-similar fractal pattern. The repetition on smaller and smaller scales occurs in the limit when the waves are highly oblique, i.e. longitudinal vortices almost aligned with the flow; the corresponding limit for two-dimensional waves is that of strong buoyancy/weak flow. The fractal set of neutral curves also represents a fractal of bifurcation points at which nonlinear solutions can be continued from the trivial state, and these may be helpful for understanding turbulent states. This may be the first example of a fractal generated by a linear ordinary differential equation.

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

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2007.2132 ◽

2007 ◽

Vol 366 (1868) ◽

pp. 1297-1315 ◽

Cited By ~ 54

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.

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Bioconvection under uniform shear: linear stability analysis

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.604 ◽

2013 ◽

Vol 738 ◽

pp. 522-562 ◽

Cited By ~ 20

Author(s):

Yongyun Hwang ◽

T. J. Pedley

Keyword(s):

Stability Analysis ◽

Shear Rate ◽

Linear Stability ◽

Couette Flow ◽

Linear Stability Analysis ◽

Plane Couette Flow ◽

Shear Rates ◽

Uniform Shear ◽

Rayleigh Benard Convection ◽

Rayleigh Benard

AbstractThe role of uniform shear in bioconvective instability in a shallow suspension of swimming gyrotactic cells is studied using linear stability analysis. The shear is introduced by applying a plane Couette flow, and it significantly disturbs gravitaxis of the cell. The unstably stratified basic state of the cell concentration is gradually relieved as the shear rate is increased, and it even becomes stably stratified at very large shear rates. Stability of the basic state is significantly changed. The instability at high wavenumbers is drastically damped out with the shear rate, while that at low wavenumbers is destabilized. However, at very large shear rates, the latter is also suppressed. The most unstable mode is found as a pair of streamwise uniform rolls aligned with the shear, analogous to Rayleigh–Bénard convection in plane Couette flow. To understand these findings, the physical mechanism of the bioconvective instability is reexamined with several sets of numerical experiments. It is shown that the bioconvective instability in a shallow suspension originates from three different physical processes: gravitational overturning, gyrotaxis of the cell and negative cross-diffusion flux. The first mechanism is found to rule the behaviour of low-wavenumber instability whereas the last two mechanisms are mainly associated with high-wavenumber instability. With the increase of the shear rate, the former is enhanced, thereby leading to destabilization at low wavenumbers, whereas the latter two mechanisms are significantly suppressed. For streamwise varying perturbations, shear with sufficiently large rates is also found to play a stabilizing role as in Rayleigh–Bénard convection. However, at small shear rates, it destabilizes these perturbations through the mechanism of overstability discussed by Hill, Pedley and Kessler (J. Fluid Mech., vol. 208, 1989, pp. 509–543). Finally, the present findings are compared with a recent experiment by Croze, Ashraf and Bees (Phys. Biol., vol. 7, 2010, 046001) and they are in qualitative agreement.

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Recurrent motions within plane Couette turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112007005459 ◽

2007 ◽

Vol 580 ◽

pp. 339-358 ◽

Cited By ~ 163

Author(s):

D. VISWANATH

Keyword(s):

Periodic Solutions ◽

Couette Flow ◽

Boundary Layers ◽

Coherent Structures ◽

Three Dimensional ◽

Turbulent Boundary Layers ◽

Transition To Turbulence ◽

Good Resolution ◽

Plane Couette Flow ◽

Near Wall

The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low-speed fluid away from the wall, has been studied experimentally, theoretically and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. Five new three-dimensional solutions of turbulent plane Couette flow are produced, one of which is periodic while the other four are relative periodic. Each of these five solutions demonstrates the breakup and re-formation of near-wall coherent structures. Four of our solutions are periodic, but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic, but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. It is argued that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the breakup and re-formation of coherent structures. A new periodic solution of plane Couette flow is also computed that could be related to transition to turbulence.The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. This computationally demanding requirement is addressed with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas – namely the Newton–Krylov iteration and the locally constrained optimal hook step. Each of the six solutions is accompanied by an error estimate.Dynamical principles are discussed that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier–Stokes equation.

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