Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity

1990 ◽  
Vol 217 ◽  
pp. 519-527 ◽  
Author(s):  
M. Nagata
Keyword(s):  
Couette Flow ◽  
Rotation Rate ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Narrow Gap ◽  
Plane Couette Flow ◽  
Bifurcation Problem ◽  
Bifurcation From Infinity ◽  
Flow Bifurcation ◽  
Average Rotation

Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

scholarly journals A note on the mirror-symmetric coherent structure in plane Couette flow

2013 ◽  
Vol 727 ◽  
Author(s):  
M. Nagata
Keyword(s):  
Couette Flow ◽  
Coherent Structure ◽  
Rotation Rate ◽  
Symmetric Solution ◽  
Three Dimensional ◽  
Plane Couette Flow ◽  
System Rotation ◽  
The Way

AbstractWe note that the mirror-symmetric solution in plane Couette flow, found recently by Gibson, Halcrow & Cvitanović (J. Fluid Mech., vol. 611, 2009, pp. 107–130) and Itano & Generalis (Phys. Rev. Lett., vol. 102, 2009, p. 114501), belongs to the solution group classified as ‘ribbon’ in rotating-plane Couette flow (RPCF). It represents a subcritical (in terms of the system rotation) solution at zero rotation rate on the three-dimensional tertiary flow branch which bifurcates from thesecondstreamwise-independent flow in RPCF. The way of its appearance is similar to that of the Nagata solution (J. Fluid Mech., vol. 217, 1990, pp. 519–527), which lies on the subcritical three-dimensional tertiary flow branch bifurcating from thefirststreamwise-independent flow in RPCF.

Finite-amplitude equilibrium states in plane Couette flow

1997 ◽  
Vol 342 ◽  
pp. 159-177 ◽  
Author(s):  
A. CHERHABILI ◽  
U. EHRENSTEIN
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Equilibrium States ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Plane Couette Flow

A numerical bifurcation study in plane Couette flow is performed by computing successive finite-amplitude equilibrium states, solutions of the Navier–Stokes equations. Plane Couette flow being linearly stable for all Reynolds numbers, first two-dimensional equilibrium states are computed by extending nonlinear travelling waves in plane Poiseuille flow through the Poiseuille–Couette flow family to the plane Couette flow limit. The resulting nonlinear states are stationary with a spatially localized structure; they are subject to two-dimensional and three-dimensional secondary disturbances. Three-dimensional disturbances dominate the dynamics and three-dimensional stationary equilibrium states bifurcating at criticality on the two-dimensional equilibrium surface are computed. These nonlinear states, periodic in the spanwise direction and spatially localized in the streamwise direction, are computed for Reynolds numbers (based on half the velocity difference between the walls and channel half-width) close to 1000. While a possible relationship between the computed solutions and experimentally observed coherent structures in turbulent plane Couette flow has to be assessed, the present findings reinforce the idea that subcritical transition may be related to the existence of finite-amplitude states which are (unstable) fixed points in a dynamical systems formulation of the Navier–Stokes system.

scholarly journals Equilibrium and travelling-wave solutions of plane Couette flow

2009 ◽  
Vol 638 ◽  
pp. 243-266 ◽  
Author(s):  
J. F. GIBSON ◽  
J. HALCROW ◽  
P. CVITANOVIĆ
Keyword(s):  
Couette Flow ◽  
Three Dimensional ◽  
Physical Space ◽  
Space Velocity ◽  
Travelling Wave ◽  
Isotropy Group ◽  
Travelling Wave Solutions ◽  
Plane Couette Flow ◽  
Equilibrium Solutions ◽  
Wave Solutions

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

A resonance mechanism in plane Couette flow

1980 ◽  
Vol 98 (1) ◽  
pp. 149-159 ◽  
Author(s):  
L. HÅKan Gustavsson ◽  
Lennart S. Hultgren
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Velocity Component ◽  
Three Dimensional ◽  
Phase Speed ◽  
Maximum Amplitude ◽  
Streamwise Velocity ◽  
Plane Couette Flow ◽  
Mean Velocity ◽  
Streamwise Velocity Component

The temporal evolution of small three-dimensional disturbances on viscous flows between parallel walls is studied. The initial-value problem is formally solved by using Fourier–Laplace transform techniques. The streamwise velocity component is obtained as the solution of a forced problem. As a consequence of the three-dimensionality, a resonant response is possible, leading to algebraic growth for small times. It occurs when the eigenvalues of the Orr–Sommerfeld equation coincide with the eigenvalues of the homogeneous operator for the streamwise velocity component. The resonance has been investigated numerically for plane Couette flow. The phase speed of the resonant waves equals the average mean velocity. The wavenumber combination that leads to the largest amplitude corresponds to structures highly elongated in the streamwise direction. The maximum amplitude, and the time to reach this maximum, scale with the Reynolds number. The aspect ratio of the most rapidly growing wave increases with the Reynolds number, with its spanwise wavelength approaching a constant value of about 3 channel heights.

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.

Finite-Amplitude Perturbation in Plane Couette Flow

1994 ◽  
Vol 28 (4) ◽  
pp. 225-230 ◽  
Author(s):  
O Dauchot ◽  
F Daviaud
Keyword(s):  
Couette Flow ◽  
Finite Amplitude ◽  
Plane Couette Flow

New results in rotating Hagen–Poiseuille flow

2000 ◽  
Vol 417 ◽  
pp. 103-126 ◽  
Author(s):  
D. R. BARNES ◽  
R. R. KERSWELL
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Rotation Rates ◽  
Supercritical Hopf Bifurcation

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

On wavy instabilities of the Taylor-vortex flow between corotating cylinders

1988 ◽  
Vol 188 ◽  
pp. 585-598 ◽  
Author(s):  
Masato Nagata
Keyword(s):  
Vortex Flow ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Taylor Vortex ◽  
Axisymmetric Solution ◽  
Boundary Flow ◽  
Taylor Vortex Flow ◽  
Average Rotation ◽  
The One ◽  
Outflow Boundary

At least four wavy instabilities are found numerically by analysing the linear stability of Taylor-vortex flow (TVF) in the limit of a small gap between two concentric cylinders which rotate differentially in the same direction. Two of the wavy instabilities, including the one leading to conventional wavy vortex (WVF), have the same axial wavelength as TVF at the onset of instability, while the other two are characterized by subharmonic modes with axial wavelengths twice as long as those of TVF. The two subharmonic instabilities appear to correspond to the wavy-inflow-boundary flow (WIB) and the wavy-outflow-boundary flow (WOB) observed in the experiment of Andereck, Liu & Swinney (1986). The phase velocities, measured in the rotating frame of reference, of all the wavy instabilities are non-zero at the onset except that the phase velocity of WVF vanishes in the region where the average rotation rate Ω of the cylinders is small. By using this simple bifurcation property of WVF for small Ω, time-independent finite-amplitude non-axisymmetric solution branches bifurcating from TVF are followed numerically. The most interesting findings are that some of the solution branches cross the line Ω = 0, producing three-dimensional nonlinear solutions in plane Couette flow.

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