Mirror-symmetric exact coherent states in plane Poiseuille flow

2013 ◽  
Vol 735 ◽  
Author(s):  
M. Nagata ◽  
K. Deguchi
Keyword(s):  
Couette Flow ◽  
Poiseuille Flow ◽  
Mirror Symmetry ◽  
Coherent States ◽  
Travelling Wave ◽  
Homotopy Continuation ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Saddle Node Bifurcation ◽  
Saddle Node

AbstractTwo new families of exact coherent states are found in plane Poiseuille flow. They are obtained from the stationary and the travelling-wave mirror-symmetric solutions in plane Couette flow by a homotopy continuation. They are characterized by the mirror symmetry inherited from those continued solutions in plane Couette flow. The first family arises from a saddle-node bifurcation and the second family bifurcates by breaking the top–bottom symmetry of the first family. We find that both families exist below the minimum saddle-node-point Reynolds number known to date (Waleffe, Phys. Fluids, vol. 15, 2003, pp. 1517–1534).

scholarly journals Patterns in Wall-Bounded Shear Flows

2020 ◽  
Vol 52 (1) ◽  
pp. 343-367 ◽  
Author(s):  
Laurette S. Tuckerman ◽  
Matthew Chantry ◽  
Dwight Barkley
Keyword(s):  
Numerical Simulations ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Pipe Flow ◽  
Shear Flows ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Flow Focusing ◽  
Annular Pipe ◽  
Flow Plane

Experiments and numerical simulations have shown that turbulence in transitional wall-bounded shear flows frequently takes the form of long oblique bands if the domains are sufficiently large to accommodate them. These turbulent bands have been observed in plane Couette flow, plane Poiseuille flow, counter-rotating Taylor–Couette flow, torsional Couette flow, and annular pipe flow. At their upper Reynolds number threshold, laminar regions carve out gaps in otherwise uniform turbulence, ultimately forming regular turbulent–laminar patterns with a large spatial wavelength. At the lower threshold, isolated turbulent bands sparsely populate otherwise laminar domains, and complete laminarization takes place via their disappearance. We review results for plane Couette flow, plane Poiseuille flow, and free-slip Waleffe flow, focusing on thresholds, wavelengths, and mean flows, with many of the results coming from numerical simulations in tilted rectangular domains that form the minimal flow unit for the turbulent–laminar bands.

Instability due to viscosity stratification

1967 ◽  
Vol 27 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Neutral Mode ◽  
Usual Theory

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

scholarly journals Forced Linear Shear Flows with Rotation: Rotating Couette–Poiseuille Flow, Its Stability, and Astrophysical Implications

2021 ◽  
Vol 922 (2) ◽  
pp. 161
Author(s):  
Subham Ghosh ◽  
Banibrata Mukhopadhyay
Keyword(s):  
Shear Flow ◽  
Couette Flow ◽  
Accretion Disk ◽  
Poiseuille Flow ◽  
Small Region ◽  
Transition To Turbulence ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Plane Shear ◽  
Linear Shear

Abstract We explore the effect of forcing on the linear shear flow or plane Couette flow, which is also the background flow in the very small region of the Keplerian accretion disk. We show that depending on the strength of forcing and boundary conditions suitable for the systems under consideration, the background plane shear flow, and hence the accretion disk velocity profile, is modified into parabolic flow, which is a plane Poiseuille flow or Couette–Poiseuille flow, depending on the frame of reference. In the presence of rotation, the plane Poiseuille flow becomes unstable at a smaller Reynolds number under pure vertical as well as three-dimensional perturbations. Hence, while rotation stabilizes the plane Couette flow, the same destabilizes the plane Poiseuille flow faster and hence the forced local accretion disk. Depending on the various factors, when the local linear shear flow becomes a Poiseuille flow in the shearing box due to the presence of extra force, the flow becomes unstable even for Keplerian rotation, and hence turbulence will ensue. This helps to resolve the long-standing problem of subcritical transition to turbulence in hydrodynamic accretion disks and the laboratory plane Couette flow.

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.

On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow

1960 ◽  
Vol 9 (3) ◽  
pp. 371-389 ◽  
Author(s):  
J. Watson
Keyword(s):  
Couette Flow ◽  
Poiseuille Flow ◽  
Complete Solution ◽  
Equations Of Motion ◽  
Plane Poiseuille Flow ◽  
Wave Disturbances ◽  
Parallel Flows ◽  
Infinitesimal Disturbance ◽  
Non Linear Mechanics ◽  
Finite Disturbance

In Part 1 by Stuart (1960), a study was made of the growth of an unstable infinitesimal disturbance, or the decay of a finite disturbance through a stable infinitesimal disturbance to zero, in plane Poiseuille flow, and that paper gave the most important terms in a solution of the equations of motion. The greater part of the present paper is concerned with a re-formulation of this problem which readily yields the complete solution. By the same method a solution for Couette flow is obtained. This solution is only a formal one for the present because the conditions imposed in deriving the solution may not be valid for Couette flow; this flow is believed to be stable to infinitesimal disturbances of the type considered.

scholarly journals Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

2014 ◽  
Vol 761 ◽  
pp. 348-359 ◽  
Author(s):  
Stefan Zammert ◽  
Bruno Eckhardt
Keyword(s):  
Periodic Orbits ◽  
Poiseuille Flow ◽  
Coherent Structures ◽  
Wave Structure ◽  
Edge State ◽  
Travelling Wave ◽  
Spanwise Direction ◽  
Plane Poiseuille Flow ◽  
Streamwise Direction ◽  
Extended States

AbstractWe 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 localisation 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 the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number $\mathit{Re}$, but the downstream exponent is essentially independent of $\mathit{Re}$. 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.

Stability of plane Couette flow of a granular material

1998 ◽  
Vol 377 ◽  
pp. 99-136 ◽  
Author(s):  
MEHEBOOB ALAM ◽  
PRABHU R. NOTT
Keyword(s):  
Stability Analysis ◽  
Granular Material ◽  
Bulk Density ◽  
Couette Flow ◽  
Flow Direction ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Plane Couette Flow ◽  
Wall Properties ◽  
Wave Instabilities

This paper presents a linear stability analysis of plane Couette flow of a granular material using a kinetic-theory-based model for the rheology of the medium. The stability analysis, restricted to two-dimensional disturbances, is carried out for three illustrative sets of grain and wall properties which correspond to the walls being perfectly adiabatic, and sources and sinks of fluctuational energy. When the walls are not adiabatic and the Couette gap H is sufficiently large, the base state of steady fully developed flow consists of a slowly deforming ‘plug’ layer where the bulk density is close to that of maximum packing and a rapidly shearing layer where the bulk density is considerably lower. The plug is adjacent to the wall when the latter acts as a sink of energy and is centred at the symmetry axis when it acts as a source of energy. For each set of properties, stability is determined for a range of H and the mean solids fraction [barvee ]. For a given value of [barvee ], the flow is stable if H is sufficiently small; as H increases it is susceptible to instabilities in the form of cross-stream layering waves with no variation in the flow direction, and stationary and travelling waves with variation in the flow and gradient directions. The layering instability prevails over a substantial range of H and [barvee ] for all sets of wall properties. However, it grows far slower than the strong stationary and travelling wave instabilities which become active at larger H. When the walls act as energy sinks, the strong travelling wave instability is absent altogether, and instead there are relatively slow growing long-wave instabilities. For the case of adiabatic walls there is another stationary instability for dilute flows when the grain collisions are quasi-elastic; these modes become stable when grain collisions are perfectly elastic or very inelastic. Instability of all modes is driven by the inelasticity of grain collisions.

Hydrodynamic Stability

1983 ◽  
Vol 50 (4b) ◽  
pp. 983-991 ◽  
Author(s):  
R. C. DiPrima ◽  
J. T. Stuart
Keyword(s):  
Couette Flow ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Plane Poiseuille Flow ◽  
Concentric Cylinders ◽  
The Stability

Theoretical and experimental developments for the stability and transition of plane Poiseuille flow and for Couette flow between rotating concentric cylinders are reviewed. The paper concludes with brief comments on the stability of Hagen-Poiseuille flow in a pipe and brief comments on the stability of slowly varying flows.

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