scholarly journals The stability of thermally stratified plane Poiseuille flow

1968 ◽  
Vol 33 (1) ◽  
pp. 21-32 ◽  
Author(s):  
K. S. Gage ◽  
W. H. Reid
Keyword(s):  
Poiseuille Flow ◽  
Richardson Number ◽  
Three Dimensional ◽  
Complete Analysis ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Plane Poiseuille Flow ◽  
Spiral Flow ◽  
Tollmien Schlichting ◽  
The Stability

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Bénard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

scholarly journals On the stability of plane Couette–Poiseuille flow with uniform crossflow

2010 ◽  
Vol 656 ◽  
pp. 417-447 ◽  
Author(s):  
ANIRBAN GUHA ◽  
IAN A. FRIGAARD
Keyword(s):  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Base Flow ◽  
Two Dimensional ◽  
Wall Velocity ◽  
Resonant Interactions ◽  
Flow Stabilization ◽  
Tollmien Schlichting ◽  
The Stability

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

On the stability of Poiseuille flow of a Bingham fluid

1994 ◽  
Vol 263 ◽  
pp. 133-150 ◽  
Author(s):  
I. A. Frigaard ◽  
S. D. Howison ◽  
I. J. Sobey
Keyword(s):  
Yield Stress ◽  
Poiseuille Flow ◽  
Linear Instability ◽  
Bingham Fluid ◽  
Additional Parameter ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Bingham Number ◽  
The Stability ◽  
Drilling Muds

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

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.

scholarly journals Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

2016 ◽  
Vol 791 ◽  
pp. 97-121 ◽  
Author(s):  
L. J. Dempsey ◽  
K. Deguchi ◽  
P. Hall ◽  
A. G. Walton
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Spanwise Direction ◽  
Plane Poiseuille Flow ◽  
Tollmien Schlichting

Strongly nonlinear three-dimensional interactions between a roll–streak structure and a Tollmien–Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett et al. (J. Fluid Mech., vol. 223, 1991, pp. 475–495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier–Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

1985 ◽  
Vol 402 (1823) ◽  
pp. 299-322 ◽  
Keyword(s):  
Poiseuille Flow ◽  
Plane Waves ◽  
Three Dimensional ◽  
Amplitude Equation ◽  
Bounded Solutions ◽  
Plane Poiseuille Flow ◽  
Centre Manifold ◽  
Nonlinear Parabolic ◽  
The Stability ◽  
Centre Manifold Theorem

The stability conditions of plane waves against three-dimensional perturbations in plane Poiseuille flow, as described by a dispersive cubically nonlinear complex-amplitude equation, under perturbations quasi-periodic in two of the space dimensions are investigated. It is found that if the parameters satisfy certain conditions, a wave is totally stable. These conditions are an extension of those given for the lower dimensional case by J. T. Stuart and R. C. DiPrima ( Proc R. Soc. Lond . A 362, 27-41 (1978)). The centre manifold theorem is then used to investigate the nature of the solutions bifurcating from a marginally unstable plane wave. Hopf bifurcations occur in the 1, 2 or 3 perturbing sidebands that are neutrally stable to the unperturbed wave and can give rise to limit cycles or tori.

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.

Instabilities of plane Poiseuille flow with a streamwise system rotation

2008 ◽  
Vol 603 ◽  
pp. 189-206 ◽  
Author(s):  
S. MASUDA ◽  
S. FUKUDA ◽  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Secondary Flow ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
System Rotation ◽  
Spiral Vortex

We analyse the stability of plane Poiseuille flow with a streamwise system rotation. It is found that the instability due to two-dimensional perturbations, which sets in at the well-known critical Reynolds number, Rc = 5772.2, for the non-rotating case, is delayed as the rotation is increased from zero, showing a stabilizing effect of rotation. As the rotation is increased further, however, the laminar flow becomes most unstable to perturbations which are three-dimensional. The critical Reynolds number due to three-dimensional perturbations at this higher rotation case is many orders of magnitude less than the corresponding value due to two-dimensional perturbations. We also perform a nonlinear analysis on a bifurcating three-dimensional secondary flow. The secondary flow exhibits a spiral vortex structure propagating in the streamwise direction. It is confirmed that an antisymmetric mean flow in the spanwise direction is generated in the secondary flow.

Acoustic receptivity and evolution of two-dimensional and oblique disturbances in a Blasius boundary layer

2001 ◽  
Vol 432 ◽  
pp. 69-90 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Acoustic Wave ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Surface Waviness ◽  
S Wave ◽  
Boundary Layer Instability ◽  
Blasius Boundary Layer ◽  
Tollmien Schlichting

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.

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