Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve

1967 ◽  
Vol 29 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Chaim L. Pekeris ◽  
Boris Shkoller
Keyword(s):  
Periodic Solutions ◽  
Linear Theory ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Neutral Curve ◽  
Lower Branch ◽  
Plane Poiseuille Flow ◽  
Periodic Disturbances ◽  
Existence Of Periodic Solutions ◽  
The Stability

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

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.

Numerical Computation of the Stability of Plane Poiseuille Flow

1978 ◽  
Vol 45 (1) ◽  
pp. 13-18 ◽  
Author(s):  
L. Wolf ◽  
Z. Lavan ◽  
H. J. Nielsen
Keyword(s):  
Stream Function ◽  
Poiseuille Flow ◽  
Numerical Technique ◽  
Finite Amplitude ◽  
Wave Number ◽  
Plane Poiseuille Flow ◽  
Governing Equations ◽  
The Stability ◽  
Successive Over Relaxation ◽  
Small Disturbances

The hydrodynamic stability of plane Poiseuille flow to infinitesimal and finite amplitude disturbances is investigated using a direct numerical technique. The governing equations are cast in terms of vorticity and stream function using second-order central differences in space. The vorticity equation is used to advance the vorticity values in time and successive over-relaxation is used to solve the stream function equation. Two programs were prepared, one for the linearized and the other for the complete disturbance equations. Results obtained by solving the linearized equations agree well with existing solutions for small disturbances. The nonlinear calculations reveal that the behavior of a disturbance depends on the amplitude and on the wave number. The behavior at wave numbers below and above the linear critical wave number is drastically different.

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.

scholarly journals A numerical method for studying Hoof bifurcation of plane Poiseuille flow

1993 ◽  
Vol 15 (2) ◽  
pp. 42-49
Author(s):  
Tran Van Tran
Keyword(s):  
Numerical Method ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Simple Algorithm ◽  
Neutral Curve ◽  
Lower Branch ◽  
Numerical Determination ◽  
Plane Poiseuille Flow ◽  
The One

The Hoof bifurcation of plane Poiseuille flows is studied numerically. On the base of the rigorous theory worked out by Joseph and Stringer [1], a detailed effective and simple algorithm for numerical determination of the type of the above mentioned bifurcation is proposed. The obtained result shows that the bifurcation at the lower branch of the linear stability neutral curve is supercritical meantime at the upper branch the one is subscritical.

Finite-Amplitude Elastic Instability of Plane-Poiseuille Flow of Viscoelastic Fluids

1999 ◽  
Vol 67 (4) ◽  
pp. 834-837 ◽  
Author(s):  
R. E. Khayat ◽  
N. Ashrafi
Keyword(s):  
Poiseuille Flow ◽  
Nonlinear Dynamical System ◽  
Elastic Stability ◽  
Finite Amplitude ◽  
Base Flow ◽  
Galerkin Projection ◽  
Plane Poiseuille Flow ◽  
Nonlinear Dynamical ◽  
Stability And Bifurcation ◽  
The Stability

The purely elastic stability and bifurcation of the one-dimensional plane Poiseuille flow is determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. The problem is reduced to a nonlinear dynamical system using the Galerkin projection method. It is shown that elastic normal stress effects can be solely responsible for the destabilization of the base (Poiseuille) flow. It is found that the stability and bifurcation picture is dramatically influenced by the solvent-to-solute viscosity ratio, ε. As the flow deviates from the Newtonian limit and ε decreases below a critical value, the base flow loses its stability. Two static bifurcations emerge at two critical Weissenberg numbers, forming a closed diagram that widens as the level of elasticity increases. [S0021-8936(00)00703-0]

The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow

1969 ◽  
Vol 39 (3) ◽  
pp. 629-639 ◽  
Author(s):  
C. L. Pekeris ◽  
B. Shkoller
Keyword(s):  
Power Series ◽  
Perturbation Method ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Neutral Curve ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Periodic Perturbations ◽  
Neutral Curves ◽  
The Mean

It is shown that there exist undamped solutions for perturbations of finite amplitude of plane Poiseuille flow, which are periodic in the direction of the axis of the channel. The shift in the ‘neutral curve’ as a function of the amplitude λ* of the disturbance is shown in figure 2. The solution is obtained by a perturbation method in which the eigenfunctions and the eigenvalue c are expanded in power series of the amplitude λ, as shown in (14), (15), (16) and (17). Near the neutral curve for a finite amplitude disturbance, the curvature of the mean flow shows a tendency to become negative (figure 5).

On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients

1983 ◽  
Vol 133 ◽  
pp. 179-206 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu
Keyword(s):  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Plane Poiseuille Flow ◽  
Supercritical Region ◽  
Subcritical Region ◽  
Poor Convergence ◽  
The Stability ◽  
Good Agreement ◽  
False Problem

In this work a study has been made of the Stuart (1960)–Watson (1960) formalism as applied to plane Poiseuille flow. In particular, the higher-order Landau coefficients have been calculated for the Reynolds & Potter (1967) method and for the Watson (1960) method. The results have been used to study the convergence of the Stuart–Landau series. A convergence curve in the (α, R)-plane has been obtained by using suitable Domb–Sykes plots. In the region of poor convergence of the series, and also in a part of the divergent region of the series, it has been found that the Shanks (1955) method, using the em1 transformation, serves as a very effective way of finding the proper sum of the series, or of finding the proper antilimit of the series. The results for the velocity calculations at R = 5000 are in very good agreement with Herbert's (1977) Fourier-truncation method using N = 4. The Watson method and the Reynolds & Potter method have also been compared inthe subcritical and supercritical regions. It is found in the supercritical region that there is not much difference in the results by the ‘true problem’ of Watson and the ‘false problem’ of Reynolds & Potter when the respective series in both methods are summed by the Shanks method. This fact could possibly be capitalized upon in the subcritical region, where the Watson method is difficult to apply.

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