Finite-amplitude instability of parallel shear flows

1967 ◽  
Vol 27 (3) ◽  
pp. 465-492 ◽  
Author(s):  
W. C. Reynolds ◽  
Merle C. Potter
Keyword(s):  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Expansion Method ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Combined Flow ◽  
Formal Expansion ◽  
Parallel Shear Flows

A formal expansion method for analysis of the non-linear development of an oblique wave in a parallel flow is presented. The present approach constitutes an extension and modification of the method of Stuart and Watson. Results are obtained for plane Poiseuille flow, and for a combination of plane Poiseuille and plane Couette flow. The Poiseuille flow exhibits finite-amplitude subcritical instability, and relatively weak but finite disturbances markedly reduce the critical Reynolds number. The combined flow, which becomes stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently great, remains unstable to finite disturbances.

The effect of external forcing on the stability of plane Poiseuille flow

1978 ◽  
Vol 359 (1699) ◽  
pp. 453-478 ◽  
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Stable Solution ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Plane Poiseuille Flow ◽  
Critical Reynolds Numbers ◽  
Threshold Amplitude ◽  
The Stability

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude Є of this forcing is taken to be small. The most dangerous modes of forcing are identified and it is found in general the critical Reynolds number is changed by O (Є) 2 . However, we identify two particular modes of forcing which give rise to decrements of order Є 2/3 and Є in the critical Reynolds number. Some types of forcing are found to generate sub critical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable subcritical limit cycle solution from its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (1951). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

An experimental investigation of the stability of plane Poiseuille flow

1975 ◽  
Vol 72 (4) ◽  
pp. 731-751 ◽  
Author(s):  
M. Nishioka ◽  
S. Iid A ◽  
Y. Ichikawa
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Rectangular Cross Section ◽  
Reynolds Numbers ◽  
Plane Poiseuille Flow ◽  
Background Turbulence ◽  
Sinusoidal Disturbance ◽  
The Stability ◽  
Long Channel

Stability experiments were made on plane Poiseuille flow generated in a long channel of a rectangular cross-section with a width-to-depth ratio of 27·4. By reducing the background turbulence down to a level of 0·05 %, we succeeded in maintaining the flow laminar at Reynolds numbers up to 8000, which is much larger than the critical Reynolds number of the linear theory, about 6000. The downstream development of the sinusoidal disturbance introduced by the vibrating ribbon technique was studied in detail at various frequencies in the range of Reynolds number from 3000 to 7500. This paper presents the experimental results and clarifies the linear stability, the nonlinear subcritical instability and the breakdown leading to the transition.

Difference in Critical Reynolds Number Between Hagen-Poiseuille and Plane Poiseuille Flows

2001 ◽  
Author(s):  
Hidesada Kanda
Keyword(s):  
Experimental Data ◽  
Reynolds Number ◽  
Poiseuille Flow ◽  
Average Velocity ◽  
Critical Reynolds Number ◽  
Inlet Section ◽  
Parallel Plates ◽  
Plane Poiseuille Flow ◽  
Contraction Ratio ◽  
Significant Difference

Abstract For plane Poiseuille flow, results of previous investigations were studied, focusing on experimental data on the critical Reynolds number, the entrance length, and the transition length. Consequently, concerning the natural transition, it was confirmed from the experimental data that (i) the transition occurs in the entrance region, (ii) the critical Reynolds number increases as the contraction ratio in the inlet section increases, and (iii) the minimum critical Reynolds number is obtained when the contraction ratio is the smallest or one, and there is no-shaped entrance or straight parallel plates. Its value exists in the neighborhood of 1300, based on the channel height and the average velocity. Although, for Hagen-Poiseuille flow, the minimum critical Reynolds number is approximately 2000, based on the pipe diameter and the average velocity, there seems to be no significant difference in the transition from laminar to turbulent flow between Hagen-Poiseuille flow and plane Poiseuille flow.

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 Subcritical transition in channel flows

2002 ◽  
Vol 451 ◽  
pp. 35-97 ◽  
Author(s):  
S. JONATHAN CHAPMAN
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Domain Of Attraction ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Laminar Flows ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equations ◽  
Laminar State

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as Rγ say, with γ < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact γ <−1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations γ =−1 and −7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations γ =−5/4 and −7/4 Here, through a formal asymptotic analysis of the Navier–Stokes equations, it is found that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations γ =−1 and −5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

Spin-down in rotating Hagen–Poiseuille flow: a simple criterion to detect the onset of absolute instabilities

2016 ◽  
Vol 793 ◽  
pp. 316-334 ◽  
Author(s):  
A. Miranda-Barea ◽  
C. Fabrellas-García ◽  
L. Parras ◽  
C. del Pino
Keyword(s):  
Poiseuille Flow ◽  
Wave Packets ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Gradual Transition ◽  
Initial State ◽  
Mode Shift ◽  
Through Flow ◽  
Reliable Procedure ◽  
Swirl Parameter

We conduct experiments in a circular pipe with rotating Hagen–Poiseuille flow (RHPF) to which we apply spin-down or impulsive spin-down to rest, in order to analyse the threshold between convective and absolute instabilities through flow visualisations in the inlet region of the pipe. For a constant value of the Reynolds number,$Re$, the finite-amplitude wave packets generated by the arbitrary perturbation that results by reducing the swirl parameter, propagate upstream or downstream depending on the initial value of the swirl parameter,$L_{0}$. In fact, the main characteristic of the flow is that the velocity front of these wave packets changes from negative to positive when absolutely unstable modes are present in the initial state. The experimental results show that spin-down becomes a precise, reliable procedure to detect the onset of absolute instabilities. In addition, we give evidence of a gradual transition for Reynolds numbers ranging from 300 to 500 where a mode shift from$n=-1$to$n=-2$appears in the absolutely unstable region.

Stability of plane Poiseuille flow to periodic disturbances of finite amplitude

1969 ◽  
Vol 39 (3) ◽  
pp. 611-627 ◽  
Author(s):  
C. L. Pekeris ◽  
B. Shkoller
Keyword(s):  
Poiseuille Flow ◽  
Subsequent Development ◽  
Critical Value ◽  
Finite Amplitude ◽  
Higher Order ◽  
Initial Disturbance ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Linear Ordinary Differential Equations ◽  
Sommerfeld Equation

A disturbance of finite amplitude λ, which is periodic in the direction of the axis of the channel, is superimposed on plane Poiseuille flow, and the subsequent development of the disturbance is studied. The disturbance is represented by an expansion in the eigenfunctions of the Orr-Sommerfeld equation with coefficients which are functions of the time, and an accurate numerical solution of the truncated system of non-linear ordinary differential equations for the coefficients is obtained.It is found that even for Reynolds numbers R less than the critical value Rc, the flow breaks down when λ exceeds a critical value λc(R). This is shown in figure 11 for the case when the initial disturbance is represented by the first mode of the Orr-Sommerfeld equation. The development of this type of disturbance is illustrated in figures 1, 3 and 13 and, for the case of a higher-order mode initial disturbance, in figure 14. Near the time of breakdown, the curvature of the modified mean flow changes sign (figure 15), but a disturbance may die down even after a reversal in the sign of the curvature has taken place (see figure 2).The stability of plane Poiseuille flow to disturbances of finite amplitude is affected by the characteristics of the higher-order modes of the Orr-Sommerfeld equation. As shown in figures 4, 10, and 12, and in figures 5, 6, and 7, these modes are either of a ‘boundary type’, characteristic of the region near the wall, or of an ‘interior type’, characteristic of the centre of the channel. The modes in the transition zone, where the two types merge, are easily amplified through mutual constructive interference, even though individually they have high damping coefficients. It is these transition modes which are mainly responsible for the breakdown through finite amplitude effects.

Hydrodynamic stability in plane Poiseuille flow with finite amplitude disturbances

1972 ◽  
Vol 51 (4) ◽  
pp. 687-704 ◽  
Author(s):  
W. D. George ◽  
J. D. Hellums
Keyword(s):  
Reynolds Number ◽  
Linear Theory ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Classical Problem ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Plane Poiseuille Flow ◽  
Navier Stokes Equation ◽  
Numerical Process

A general method for studying two-dimensional problems in hydrodynamic stability is presented and applied to the classical problem of predicting instability in plane Poiseuille flow. The disturbance stream function is expanded in a Fourier series in the axial space dimension which, on substitution into the Navier-Stokes equation, leads to a system of parabolic partial differential equations in the coefficient functions. An efficient, stable and accurate numerical method is presented for solving these equations. It is demonstrated that the numerical process is capable of accurate reproduction of known results from the linear theory of hydrodynamic stability.Disturbances that are stable according to linear theory are shown to become unstable with the addition of finite amplitude effects. This seems to be the first work of quantitative value for disturbances of moderate and larger amplitudes. A relationship between critical amplitude and Reynolds number is reported, the form of which indicates the existence of an absolute critical Reynolds number below which an arbitrary disturbance cannot be made unstable, no matter how large its initial amplitude. The critical curve shows significantly less effect of amplitude than do those obtained by earlier workers.

Sign in / Sign up

Export Citation Format

Share Document