Stability of Poiseuille Flow in Elastic Tubes

1977 ◽  
Vol 44 (1) ◽  
pp. 18-24 ◽  
Author(s):  
W. Midvidy ◽  
W. T. Rouleau
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Tube Wall ◽  
The Other ◽  
Wave Number ◽  
Rotatory Inertia ◽  
Elastic Tubes ◽  
Temporal And Spatial ◽  
Axisymmetric Disturbances

This paper presents a theoretical analysis of the temporal and spatial stability of Poiseuille flow in elastic tubes to infinitesimal axisymmetric disturbances. A cylindrical shell model which includes the effects of transverse shear and rotatory inertia is employed for the tube wall. The characteristic equation of the system is solved numerically and two sets of modes are obtained; one set has eigenvalues that are independent of the properties and dimensions of the tube wall, while the other set has eigenvalues that depend on the tube parameters. One mode of the “tube-dependent” set is shown to have a critical Reynolds number that depends on the elastic properties and dimensions of the tube and either wave number or frequency of the disturbance.

Linear instability of annular Poiseuille flow

2008 ◽  
Vol 610 ◽  
pp. 391-406 ◽  
Author(s):  
C. J. HEATON
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Circular Cylinders ◽  
Flow Profile ◽  
Mean Flow ◽  
Annular Pipe ◽  
Axisymmetric Disturbances

The linear stability of flow along an annular pipe formed by two coaxial circular cylinders is considered. We find that the flow is unstable above a critical Reynolds number for all 0 < η ≤ 1, where η is the ratio between the radii of the inner and outer cylinders. This contradicts a recent claim that the flow is stable at all Reynolds numbers for radius ratio η less than a finite critical value. We find that non-axisymmetric disturbances become stable at all Reynolds numbers for η < 0.11686215, and we are able to study this ‘bifurcation from infinity’ asymptotically. However, axisymmetric disturbances remain unstable, with critical Reynolds number tending to infinity as η → 0. A second asymptotic analysis is performed to show that the critical Reynolds number Rec ∝ η−1 log(η−1) as η → 0, with the form of the mean flow profile causing the appearance of the logarithm. The stability of Hagen–Poiseuille flow (η = 0) at all Reynolds numbers is therefore interpreted as a limit result, and there are no annular pipe flows which share this stability.

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.

Stability of Nonparallel Developing Flow in a Pipe to Nonaxisymmetric Disturbances

1983 ◽  
Vol 50 (1) ◽  
pp. 210-214 ◽  
Author(s):  
V. K. Garg
Keyword(s):  
Experimental Data ◽  
Critical Reynolds Number ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Main Flow ◽  
Wave Number ◽  
Developing Flow ◽  
Azimuthal Wave Number ◽  
Flow Velocity Profile ◽  
Axisymmetric Disturbances

Linear spatial stability of the nonparallel developing flow in a rigid circular pipe has been studied at several axial locations for nonaxisymmetric disturbances. The main flow velocity profile is obtained by Hornbeck’s finite-difference method assuming uniform flow at entry to the pipe. The method of multiple scales is used to account for all the nonparallel effects. It is found that the nonparallel developing flow is most unstable to nonaxisymmetric disturbances with azimuthal wave number n equal to unity. Axisymmetric disturbances are, however, more unstable than nonaxisymmetric disturbances with n ≥ 2 except in the near-entry region. The results show that the parallel flow theory overpredicts the critical Reynolds number by as much as 136.5 percent in the near entry region for the n = 1 disturbance. The present results compare well with the available experimental data.

scholarly journals Non-modal stability in sliding Couette flow

2012 ◽  
Vol 710 ◽  
pp. 505-544 ◽  
Author(s):  
R. Liu ◽  
Q. S. Liu
Keyword(s):  
Reynolds Number ◽  
Modal Analysis ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Initial Conditions ◽  
Streamwise Vortex ◽  
Optimal Time ◽  
Energy Growth ◽  
Wide Range ◽  
Axisymmetric Disturbances

AbstractThe problem of an incompressible flow between two coaxial cylinders with radii$a$and$b$subjected to a sliding motion of the inner cylinder in the axial direction is considered. The energy stability and the non-modal stability have been investigated for both axisymmetric and non-axisymmetric disturbances. For the non-modal stability, we focus on two problems: response to external excitations and response to initial conditions. The former is studied by examining the$\epsilon $-pseudospectrum, and the latter by examining the energy growth function$G(t)$. Unlike the results of the modal analysis in which the stability of sliding Couette flow is determined by axisymmetric disturbances, the energy analysis shows that a non-axisymmetric disturbance has a critical energy Reynolds number for all radius ratios$\eta = a/ b$. The results for non-modal stability show that rather large transient growth occurs over a wide range of azimuthal wavenumber$n$and streamwise wavenumber$\ensuremath{\alpha} $, even though the Reynolds number is far below its critical value. For the problem of response to external excitations, the response is sensitive to low-frequency external excitations. For all values of the radius ratio, the maximum response is achieved by non-axisymmetric and streamwise-independent disturbances when the frequency of external forcing$\omega = 0$. For the problem of response to initial conditions, the optimal disturbance is in the form of helical streaks at low Reynolds numbers. With the increase of$\mathit{Re}$, the optimal disturbance becomes very close to straight streaks. For each$\eta $, the maximum energy growth of streamwise-independent disturbances is of the order of${\mathit{Re}}^{2} $, and the optimal time is of the order of$\mathit{Re}$. This relation is qualitatively similar to that for plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow. Direct numerical simulations are applied to investigate the transition of the streamwise vortex (SV) scenario at$\mathit{Re}= 1000$and 1500 for various$\eta $. The initial disturbances are the optimal streamwise vortices predicted by the non-modal analysis. We studied the streak breakdown phase of the SV scenarios by examining the instability of streaks. Moreover, we have investigated the sustainment of the energy of disturbances in the SV scenario.

Stability of fluid flow in a flexible tube to non-axisymmetric disturbances

2000 ◽  
Vol 407 ◽  
pp. 291-314 ◽  
Author(s):  
V. SHANKAR ◽  
V. KUMARAN
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Phase Velocity ◽  
Poiseuille Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Flexible Tube ◽  
Moderate Reynolds Number ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

On the stability of flow in an elliptic pipe which is nearly circular

1978 ◽  
Vol 87 (2) ◽  
pp. 233-241 ◽  
Author(s):  
A. Davey
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Major Axis ◽  
Reynolds Numbers ◽  
Minor Axis ◽  
Critical Reynolds Numbers ◽  
The Stability

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

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.

Effect of tube elasticity on the stability of Poiseuille flow

1977 ◽  
Vol 81 (4) ◽  
pp. 625-640 ◽  
Author(s):  
Vijay K. Garg
Keyword(s):  
Poiseuille Flow ◽  
Bessel Functions ◽  
Critical Reynolds Number ◽  
Elastic Tube ◽  
Dimensionless Frequency ◽  
Tube Material ◽  
Rigid Tube ◽  
Neutral Curves ◽  
The Stability ◽  
Axisymmetric Disturbances

The effect of tube elasticity on the stability of Poiseuille flow to infinitesimal axisymmetric disturbances is investigated. The disturbance equations for the fluid are solved numerically while those for the arbitrarily thick tube are solved analytically in terms of Bessel functions of complex argument. It is shown that an elastic tube can cause instability of Poiseuille flow, unlike a rigid tube, in which the flow is always stable. Neutral curves are presented for various values of the tube parameters. It is found that the critical Reynolds number varies almost as the square root of the Young's modulus of the tube material while the critical dimensionless frequency is almost invariant, being about 1·1 for the cases studied.

Experimental Determination of the Critical Reynolds Number for Pulsating Poiseuille Flow

1966 ◽  
Vol 88 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Turgut Sarpkaya
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Pulsating Flow ◽  
Mean Velocity ◽  
Cross Sectional ◽  
Pressure Transducers ◽  
Mean Pressure ◽  
The Stability

The stability of fully developed Poiseuille flow pulsating under a harmonically and a nonharmonically varying pressure gradient was studied experimentally. The characteristics of turbulent plugs were determined for both steady and pulsating flow by means of pressure transducers. It was found that (a) for oscillating, stable Poiseuille flow, the phase angles determined experimentally agree well with those determined theoretically; (b) for the same mean pressure gradient, pulsating flow is more stable than the corresponding steady Poiseuille flow; (c) in pulsating flow, the presence of one or more inflection points is necessary but not sufficient for instability; and (d) the curves of the critical Reynolds number versus the relative amplitude of the periodic component of the cross-sectional mean velocity reach their maximum when at least one inflection ring continues to exist a time period 53 percent of the period of oscillation.

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