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 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.

scholarly journals Basic Flows of Generalized Second Grade Fluids Based on a Sisko Model

2017 ◽  
Vol 22 (4) ◽  
pp. 1019-1033
Author(s):  
A. Walicka
Keyword(s):  
Couette Flow ◽  
Poiseuille Flow ◽  
Equations Of Motion ◽  
Plane Channel ◽  
Second Grade ◽  
Sisko Fluid ◽  
Coaxial Cylinders ◽  
Solutions Of Equations ◽  
Simple Flows ◽  
Second Grade Fluids

Abstract The present investigation is concerned with basic flows of generalized second grade fluids based on a Sisko fluid. After formulation of the general equations of motion three simple flows of viscoplastic fluids of a Sisko type or fluids similar to them are considered. These flows are: Poiseuille flow in a plane channel, Poiseuille flow in a circular pipe and rotating Couette flow between two coaxial cylinders. After presentation the Sisko model one was presented some models of fluids similar to this model. Next it was given the solutions of equations of motion for three flows mentioned above.

An experimental study of oblique transition in plane Poiseuille flow

1998 ◽  
Vol 358 ◽  
pp. 177-202 ◽  
Author(s):  
PER A. ELOFSSON ◽  
P. HENRIK ALFREDSSON
Keyword(s):  
Poiseuille Flow ◽  
High Amplitude ◽  
Linear Wave ◽  
Plane Poiseuille Flow ◽  
Low Velocity ◽  
Oblique Waves ◽  
Wave Disturbances ◽  
Rapid Transition ◽  
Streamwise Streaks ◽  
Transition Scenario

Interactions of oblique waves have recently been investigated theoretically and numerically and found to give rise to rapid transition in flows subcritical to linear wave disturbances. The transition scenario consists of the formation and transient growth of streamwise streaks of high and low velocity and later a rapid growth of high-frequency disturbances leading to breakdown. The present study is the first extensive experimental investigation of oblique transition. The experiments were carried out in a plane Poiseuille flow air channel in which the oblique waves were generated, one at each wall, by vibrating ribbons and the development of the flow was mapped with hot-wire anemometry. The experiments consist both of low- and high-amplitude wave disturbances; in both cases streaky structures are created. For the low-amplitude case these structures decay, whereas for the high amplitude the flow goes towards breakdown. This study has confirmed and extended previous theoretical and numerical results showing that oblique transition may be an important transition scenario.

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).

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.

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.

On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow

1960 ◽  
Vol 9 (3) ◽  
pp. 353-370 ◽  
Author(s):  
J. T. Stuart
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Unstable Equilibrium ◽  
Wave Number ◽  
Neutral Stability ◽  
Plane Poiseuille Flow ◽  
Wide Range ◽  
Non Linear ◽  
Infinitesimal Disturbance

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.

Wave velocity in parallel flows of a viscous fluid

1973 ◽  
Vol 58 (4) ◽  
pp. 703-708 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Viscous Fluid ◽  
Wave Velocity ◽  
Poiseuille Flow ◽  
Plane Poiseuille Flow ◽  
Parallel Flows

It is shown in this note that the velocity of unstable or neutral waves in plane Poiseuille flow or plane Couette-Poiseuille flow, or of axisymmetric waves in Poiseuille flow, stable or unstable, must lie within the range of the velocity of the flow.

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