The instability of oscillatory plane Poiseuille flow

1982 ◽  
Vol 116 ◽  
pp. 91-114 ◽  
Author(s):  
Christian H. Von Kerczek
Keyword(s):  
Steady Flow ◽  
Poiseuille Flow ◽  
Growth Rates ◽  
Oscillatory Flow ◽  
Amplitude Ratio ◽  
Perturbation Technique ◽  
Wave Number ◽  
Plane Poiseuille Flow ◽  
Velocity Amplitude ◽  
Very High

The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.

The stability of steady and time-dependent plane Poiseuille flow

1968 ◽  
Vol 34 (1) ◽  
pp. 177-205 ◽  
Author(s):  
Chester E. Grosch ◽  
Harold Salwen
Keyword(s):  
Reynolds Number ◽  
Steady Flow ◽  
Poiseuille Flow ◽  
Time Dependent ◽  
Wave Number ◽  
Neutral Stability ◽  
Neutral Stability Curve ◽  
Plane Poiseuille Flow ◽  
Stability Curve ◽  
Wave Numbers

The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.

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 stability of plane Poiseuille flow between flexible walls

1972 ◽  
Vol 51 (2) ◽  
pp. 403-416 ◽  
Author(s):  
C. H. Green ◽  
C. H. Ellen
Keyword(s):  
Approximate Solution ◽  
Poiseuille Flow ◽  
Stability Boundary ◽  
Rigid Wall ◽  
Wave Number ◽  
Plane Poiseuille Flow ◽  
Computational Evaluation ◽  
Flexible Walls ◽  
Tollmien Schlichting ◽  
The Stability

This paper examines the linear stability of antisymmetric disturbances in incompressible plane Poiseuille flow between identical flexible walls which undergo transverse displacements. Using a variational approach, an approximate solution of the problem is formulated in a form suitable for computational evaluation of the (complex) wave speeds of the system. A feature of this formulation is that the varying boundary conditions (and the Orr-Sommerfeld equation) are satisfied only in the mean; this reduces the labour involved in determining the approximate solution for a variety of wall conditions without increasing the difficulty of obtaining solutions to a given accuracy. In this paper the symmetric stream function distribution across the channel is represented by a series of cosines whose coefficients are determined by the variational solution. Comparisons with previous work, both for the flexible-wall and rigid-wall problems, show that the method gives results as accurate as those obtained previously by other methods while new results, for flexible walls, indicate the presence of a higher wave-number stability boundary which joins the distorted Tollmien-Schlichting stability boundary at lower wave-numbers. In some cases this upper unstable region, which is characterized by large amplification rates, may determine the critical Reynolds number of the system.

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.

scholarly journals A New Approach for Study the Electrohydrodynamic Oscillatory Flow Through a Porous Medium in a Heating Compliant Channel

2020 ◽  
Vol 25 (3) ◽  
pp. 30-44
Author(s):  
M.H. Haroun
Keyword(s):  
Potential Function ◽  
Oscillatory Flow ◽  
Amplitude Ratio ◽  
Electrical Potential ◽  
Fluid Pressure ◽  
Wave Number ◽  
Moving Frame ◽  
Secondary Wave ◽  
Regular Perturbation ◽  
New Approach

AbstractThe governing equations of an electrohydrodynamic oscillatory flow were simplified, using appropriate nondimensional quantities and the conversion relationships between fixed and moving frame coordinates. The obtained system of equations is solved analytically by using the regular perturbation method with a small wave number. In this study, modified non-dimensional quantities were used that made fluid pressure in the resulting equations dependent on both axial and vertical coordinates. The current study is more realistic and general than the previous studies in which the fluid pressure is considered functional only in the axial coordinate. A new approach enabled the author to find an analytical form of fluid pressure while previous studies have not been able to find it but have found only the pressure gradient. Analytical expressions for the stream function, electrical potential function and temperature distribution are obtained. The results show that the electrical potential function decreases by the increase of the Prandtl number, secondary wave amplitude ratio and width of the channel.

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