On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance

1972 ◽  
Vol 326 (1566) ◽  
pp. 289-313 ◽  
Keyword(s):  
Poiseuille Flow ◽  
Nonlinear Response ◽  
Nonlinear Evolution ◽  
Numerical Study ◽  
Parallel Flow ◽  
Main Stream ◽  
Plane Poiseuille Flow ◽  
Plane Parallel ◽  
All Solutions ◽  
Infinitesimal Disturbance

The differential equation governing the nonlinear evolution of an initial centred infinitesimal disturbance to a marginally unstable plane parallel flow was obtained by Stewartson & Stuart (1971) and some of its properties elucidated by Hocking, Stewartson & Stuart (1972). Of especial interest is the final localized burst of the solution which occurs when all the coefficients of the equation are real and the first Landau constant is positive. In plane Poiseuille flow, however, the standard example of plane parallel flow, these coefficients are complex and in the present paper an analytic and numerical study is made of the evolution of the solution when they are permitted to take general values. It is found that if the real part 8r of the first Landau constant is positive it is possible to have either a burst or a solution which remains finite for all time depending on the values of the other coefficients. In addition when a burst occurs it can take on two different structures. If δ r < 0 all solutions remain finite but the amplitude of the oscillation does not tend to a limit if the imaginary part δ of the first Landau constant is large enough. For the particular example of plane Poiseuille flow, skewed disturbances burst only if they are inclined to the main stream at an angle exceeding about 56°.

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.

The stability of Poiseuille flow modulated at high frequencies

1975 ◽  
Vol 344 (1639) ◽  
pp. 453-464 ◽  
Keyword(s):  
Frequency Modulation ◽  
Poiseuille Flow ◽  
High Frequency ◽  
Parallel Flow ◽  
Plane Poiseuille Flow ◽  
High Frequencies ◽  
The Stability

The effect of high frequency modulation on the stability of plane Poiseuille flow is considered. It is shown how the stability characteristics of this flow can be completely determined from those of the unmodulated flow. It is found that modulation destabilizes the flow. The method can be used to investigate the stability of any parallel or nearly parallel flow modulated at high frequencies.

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.

On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow

1974 ◽  
Vol 63 (3) ◽  
pp. 529-536 ◽  
Author(s):  
A. Davey ◽  
L. M. Hocking ◽  
K. Stewartson
Keyword(s):  
Pressure Gradient ◽  
Poiseuille Flow ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Governing Equations ◽  
Amplitude Function ◽  
Coupled Partial Differential Equations

The equations governing the nonlinear development of a centred three-dimensional disturbance to plane parallel flow at slightly supercritical Reynolds numbers are obtained, In contrast to the corresponding equation for two-dimensional disturbances, two slowly varying functions are needed to describe the development: the amplitude function and a function related to the secular pressure gradient produced by the disturbance. These two functions satisfy a pair of coupled partial differential equations. The equations derived in Hocking, Stewartson & Stuart (1972) are shown to be incorrect, Some of the properties of the governing equations are discussed briefly.

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