Non-parallel flow corrections for the stability of shear flows

1973 ◽  
Vol 59 (3) ◽  
pp. 571-591 ◽  
Author(s):  
Chi-Hai Ling ◽  
W. C. Reynolds
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Parallel Flow ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Blasius Boundary Layer ◽  
Stability Of Shear Flows ◽  
Parallel Shear Flow ◽  
The Stability ◽  
Small Disturbances

The proper corrections for non-parallel flow to the eigenvalues for small disturbances on a nearly parallel shear flow have been determined through a perturbation about the parallel flow solutions. The resulting shifts in the neutral stability curves have been calculated for the Blasius boundary layer, for the two-dimensional jet, and for the two-dimensional flat-plate wake.

The upper branch stability of the Blasius boundary layer, including non-parallel flow effects

1981 ◽  
Vol 375 (1760) ◽  
pp. 65-92 ◽  
Keyword(s):  
Boundary Layer ◽  
Parallel Flow ◽  
Neutral Stability ◽  
Relative Order ◽  
Numerical Work ◽  
Blasius Boundary Layer ◽  
Stability Structure ◽  
Tollmien Schlichting ◽  
Flow Effects ◽  
The Stability

The stability of the Blasius boundary layer is studied theoretically, with the aim of fixing the character of the upper branch of the neutral stability curve(s) and its dependence on non-parallel flow effects. Unlike most previous studies this work has a rational basis since, throughout, we consider the linear stability structure for asymptotically large Reynolds numbers ( Re ). The structure is five-zoned and quite complicated, more so than the structure (discussed in Smith (1979 a )) governing the lower branch stability properties, but nevertheless it lends itself to the systematic determination of the neutral frequency and of the influence of non-parallelism. The four leading terms in the asymptotic expansion of the neutral frequency are determined and then the non-parallel flow effects are considered. The latter are shown to be of relative order Re -3/10 in general, much larger than the relative order Re -1/2 suggested by the parallel flow approximations used extensively in the literature. The cause of this discrepancy lies partly in the relatively large wavelength of the Tollmien-Schlichting modes but, more especially, in a ‘transmission feature’, associated with the stability structure and brought about by the major determining role played by the small curvature of the boundary layer profile at the critical layer. This transmission feature enables even quite small effects in the disturbance velocity field to produce a much more profound effect in the neutral stability criteria. The results of this study are not inconsistent overall with previous numerical work but they do tend to suggest that linear non-parallel flow stability theory may well explain most of the related experimental observations, even near the critical Reynolds number.

On the behaviour of small disturbances in plane Couette flow with a temperature gradient

1965 ◽  
Vol 286 (1404) ◽  
pp. 117-128 ◽  
Keyword(s):  
Rayleigh Number ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Plane Couette Flow ◽  
Infinitesimal Disturbance ◽  
Prandtl Numbers ◽  
The Stability ◽  
Small Disturbances

The stability of plane Couette flow with a heated lower plate is considered with respect to a two-dimensional infinitesimal disturbance. The eigenvalues are found with the aid of a digital computer as the latent roots of a matrix. Neutral stability curves for various Prandtl numbers at Reynolds numbers up to 150 are obtained by a second method. It is found that the principle of the exchange of stabilities does not hold for this problem. With the aid of Squire’s transformation the conclusion is drawn that all fluids will become unstable at the same value of the Rayleigh number irrespective of whether shear is present or not.

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.

The stability of filamentary vorticity in two-dimensional geophysical vortex-dynamics models

1991 ◽  
Vol 231 ◽  
pp. 575-598 ◽  
Author(s):  
D. W. Waugh ◽  
D. G. Dritschel
Keyword(s):  
Green Function ◽  
Shear Flow ◽  
Stream Function ◽  
Potential Vorticity ◽  
Two Dimensional ◽  
Interaction Range ◽  
Inversion Operator ◽  
The Green Function ◽  
The Stability ◽  
Background Shear

The linear stability of filaments or strips of ‘potential’ vorticity in a background shear flow is investigated for a class of two-dimensional, inviscid, non-divergent models having a linear inversion relation between stream function and potential vorticity. In general, the potential vorticity is not simply the Laplacian of the stream function – the case which has received the greatest attention historically. More general inversion relationships between stream function and potential vorticity are geophysically motivated and give an impression of how certain classic results, such as the stability of strips of vorticity, hold under more general circumstances.In all models, a strip of potential vorticity is unstable in the absence of a background shear flow. Imposing a shear flow that reverses the total shear across the strip, however, brings about stability, independent of the Green-function inversion operator that links the stream function to the potential vorticity. But, if the Green-function inversion operator has a sufficiently short interaction range, the strip can also be stabilized by shear having the same sense as the shear of the strip. Such stabilization by ‘co-operative’ shear does not occur when the inversion operator is the inverse Laplacian. Nonlinear calculations presented show that there is only slight disruption to the strip for substantially less adverse shear than necessary for linear stability, while for co-operative shear, there is major disruption to the strip. It is significant that the potential vorticity of the imposed flow necessary to create shear of a given value increases dramatically as the interaction range of the inversion operator decreases, making shear stabilization increasingly less likely. This implies an increased propensity for filaments to ‘roll-up’ into small vortices as the interaction range decreases, a finding consistent with many numerical calculations performed using the quasi-geostrophic model.

Direct numerical simulation of instabilities in a two-dimensional near-critical fluid layer heated from below

2001 ◽  
Vol 442 ◽  
pp. 119-140 ◽  
Author(s):  
S. AMIROUDINE ◽  
P. BONTOUX ◽  
P. LARROUDÉ ◽  
B. GILLY ◽  
B. ZAPPOLI
Keyword(s):  
Boundary Layer ◽  
Critical Point ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Vertical Direction ◽  
Single Layer ◽  
Bulk Temperature ◽  
Two Dimensional ◽  
Piston Effect ◽  
The Stability

An analysis of the hydrodynamic stability of a fluid near its near critical point – initially at rest and in thermodynamic equilibrium – is considered in the Rayleigh–Bénard configuration, i.e. heated from below. The geometry is a two-dimensional square cavity and the top and bottom walls are maintained at constant temperatures while the sidewalls are insulated. Owing to the homogeneous thermo-acoustic heating (piston effect), the thermal field exhibits a very specific structure in the vertical direction. A very thin hot thermal boundary layer is formed at the bottom, then a homogeneously heated bulk settles in the core at a lower temperature; at the top, a cooler boundary layer forms in order to continuously match the bulk temperature with the colder temperature of the upper wall. We analyse the stability of the two boundary layers by numerically solving the Navier–Stokes equations appropriate for a van der Waals' gas slightly above its critical point. A finite-volume method is used together with an acoustic filtering procedure. The onset of the instabilities in the two different layers is discussed with respect to the results of the theoretical stability analyses available in the literature and stability diagrams are derived. By accounting for the piston effect the present results can be put within the framework of the stability analysis of Gitterman and Steinberg for a single layer subjected to a uniform, steady temperature gradient.

Higher eigenstates in boundary-layer stability theory

1976 ◽  
Vol 77 (1) ◽  
pp. 81-104 ◽  
Author(s):  
D. Corner ◽  
D. J. R. Houston ◽  
M. A. S. Ross
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stability Theory ◽  
Free Stream ◽  
Angular Frequency ◽  
Boundary Layer Stability ◽  
Fundamental State ◽  
Blasius Boundary Layer ◽  
The Stability ◽  
Sommerfeld Equation

Using the Orr-Sommerfeld equation with the wavenumber as the eigenvalue, a search for higher eigenstates in the stability theory of the Blasius boundary layer has revealed the existence of a number of viscous states in addition to the long established fundamental state. The viscous states are discrete, belong to two series, and are all heavily damped in space. Within the limits of the investigation the number of viscous states existing in the layer increases as the Reynolds number and the angular frequency of the perturbation increase. It is suggested that the viscous eigenstates may be responsible for the excitation of some boundary-layer disturbances by disturbances in the free stream.

The effect of a shear flow on convection in a layer heated non-uniformly from below

1985 ◽  
Vol 154 ◽  
pp. 303-319 ◽  
Author(s):  
I. C. Walton
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Fluid Layer ◽  
Heated Layer ◽  
Vertical Temperature ◽  
Onset Of Convection ◽  
A Value ◽  
Blasius Boundary Layer ◽  
Convection Problem

In an earlier paper (Walton 1982) we showed that, when a fluid layer is heated non-uniformly from below in such a way that the vertical temperature difference maintained across the layer is a slowly varying monotonic function of a horizontal coordinate x, then convection occurs for x > xc, where xc is the point where the local Rayleigh number is equal to the critical value for a uniformly heated layer. Furthermore, the amplitude of the convection increases smoothly from exponentially small values for x [Lt ] xc and asymptotes to a value given by Stuart–Watson theory for x [Gt ] xc.At the present time no solutions of this kind are available for a class of problems in which the onset of instability is heavily influenced by a shear flow (e.g. Görtler vortices in a boundary layer on a curved wall, convection in a heated Blasius boundary layer). In a first step to bridge the gap between these problems and in order to elucidate the difficulties associated with the presence of a shear flow, we investigate the effect of a (weak) shear flow on our earlier convection problem. We show that the onset of convection is delayed and that it appears more suddenly, but still smoothly. The role of horizontal diffusion is shown to be of paramount importance in enabling a solution of this kind to be found, and the implications of these results for instabilities in higher-speed shear flows are discussed.

Shear flow instability in a conducting viscous fluid

1973 ◽  
Vol 57 (3) ◽  
pp. 481-490
Author(s):  
B. Roberts
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Viscous Fluid ◽  
Flow Instability ◽  
Approximate Solutions ◽  
Viscous Fluids ◽  
Neutral Stability ◽  
Parallel Magnetic Field ◽  
The Stability ◽  
Shear Flow Instability

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

The interactive dynamics of flow and directional solidification in a Hele-Shaw cell Part 1. Experimental investigation of parallel shear flow

2002 ◽  
Vol 470 ◽  
pp. 247-268 ◽  
Author(s):  
M. ZHANG ◽  
T. MAXWORTHY
Keyword(s):  
Shear Flow ◽  
Directional Solidification ◽  
Flow Direction ◽  
Velocity Ratio ◽  
Parallel Flow ◽  
Tilting Angle ◽  
Specimen Material ◽  
Parallel Shear Flow ◽  
Crystal Cells ◽  
Hele Shaw Cell

It is recognized that flow in the melt can have a profound influence on the dynamics of a solidifying interface and hence on the quality of the solidified material. To better understand the effect of fluid flow on the interface morphological stability and on the cellular and dendritic growth, directional solidification experiments were carried out in a horizontally placed Hele-Shaw cell with and without externally imposed parallel shear flow. The specimen material used was SCN–1.0 Wt% acetone. The experiment shows that the transient parallel flow has a stabilizing effect on the planar interface by damping the existing initial perturbations. The left–right symmetry of crystal cells was broken by the parallel flow, with cells tilting toward the incoming flow direction. The tilting angle increased with the velocity ratio. The secondary dendrites were found to either not appear or appear much later on the downstream side of the crystal cells. The wavelengths of the initial perturbations and of the cellular interface were insensitive to the imposed flow.

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