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.

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.

On the non-parallel flow stability of the Blasius boundary layer

1979 ◽  
Vol 366 (1724) ◽  
pp. 91-109 ◽  
Keyword(s):  
Boundary Layer ◽  
Flow Stability ◽  
Neutral Curve ◽  
Layer Growth ◽  
Parallel Flow ◽  
Large Reynolds Number ◽  
Blasius Boundary Layer ◽  
Leading Term ◽  
Flow Effects ◽  
Boundary Layer Growth

The influence of boundary layer growth on the flow stability of the Blasius boundary layer is analysed on a rational, large Reynolds number, basis, for small disturbances of fixed frequency. The parallel-flow solution forms the leading term and the non-parallel flow effects emerge in a consistent fashion from the asymptotic expansions. Compared with previous, successive approximation, procedures, the theoretical neutral curve obtained here is much more affected by the non-parallel effects and consequently shows somewhat improved agreement with experimental observations, even though the previous and the present approaches (both of which calculate only a finite number of terms) would be identical if taken to infinitely many terms.

On the stability of a laminar incompressible boundary layer over a flexible surface

1962 ◽  
Vol 13 (4) ◽  
pp. 609-632 ◽  
Author(s):  
Marten T. Landahl
Keyword(s):  
Boundary Layer ◽  
Phase Speed ◽  
Travelling Wave ◽  
Wave Number ◽  
Neutral Stability ◽  
Class A ◽  
Flexible Surface ◽  
Flexible Wall ◽  
Tollmien Schlichting ◽  
The Stability

The stability of small two-dimensional travelling-wave disturbances in an incompressible laminar boundary layer over a flexible surface is considered. By first determining the wall admittance required to maintain a wave of given wave-number and phase speed, a characteristic equation is deduced which, in the limit of zero wall flexibility, reduces to that occurring in the ordinary stability theory of Tollmien and Schlichting. The equation obtained represents a slight and probably insignificant improvement upon that given recently by Benjamin (1960). Graphical methods are developed to determine the curve of neutral stability, as well as to identify the various modes of instability classified by Benjamin as ‘Class A’, ‘Class B’, and ‘Kelvin-Helmholtz’ instability, respectively. Also, a method is devised whereby the optimum combination of surface effective mass, wave speed, and damping required to stabilize any given unstable Tollmien-Schlichting wave can be determined by a simple geometrical construction in the complex wall-admittance plane.What is believed to be a complete physical explanation for the influence of an infinite flexible wall on boundary-layer stability is presented. In particular, the effect of damping in the wall is discussed at some length. The seemingly paradoxical result that damping destabilizes class A waves (i.e. waves of the Tollmien-Schlichting type) is explained by considering the related problem of flutter of an infinite panel in incompressible potential flow, for which damping has the same qualitative effect. It is shown that the class A waves are associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only make the wave amplitude increase to compensate for the lowered energy level. The Kelvin-Helmholtz type of instability will occur when the effective stiffness of the panel is too low to withstand, for all values of the phase speed, the pressure forces induced on the wavy wall.The numerical examples presented show that the increase in the critical Reynolds number that can be achieved with a wall of moderate flexibility is modest, and that some other explanation for the experimentally observed effects of a flexible wall on the friction drag must be considered.

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.

Transition of streamwise streaks in zero-pressure-gradient boundary layers

2002 ◽  
Vol 472 ◽  
pp. 229-261 ◽  
Author(s):  
LUCA BRANDT ◽  
DAN S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
High Speed ◽  
Plate Boundary ◽  
Transition Process ◽  
Free Stream Turbulence ◽  
Tollmien Schlichting ◽  
Streamwise Streaks ◽  
The Stability ◽  
Optimal Disturbances

A transition scenario initiated by streamwise low- and high-speed streaks in a flat-plate boundary layer is studied. In many shear flows, the perturbations that show the highest potential for transient energy amplification consist of streamwise-aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. In a previous investigation (Andersson et al. 2001), the stability of these streaks in a zero-pressure-gradient boundary layer was studied by means of Floquet theory and numerical simulations. The sinuous instability mode was found to be the most dangerous disturbance. We present here the first simulation of the breakdown to turbulence originating from the sinuous instability of streamwise streaks. The main structures observed during the transition process consist of elongated quasi-streamwise vortices located on the flanks of the low-speed streak. Vortices of alternating sign are overlapping in the streamwise direction in a staggered pattern. The present scenario is compared with transition initiated by Tollmien–Schlichting waves and their secondary instability and by-pass transition initiated by a pair of oblique waves. The relevance of this scenario to transition induced by free-stream turbulence is also discussed.

scholarly journals The mechanics of the Tollmien-Schlichting wave

1996 ◽  
Vol 312 ◽  
pp. 107-124 ◽  
Author(s):  
Peter G. Baines ◽  
Sharan J. Majumdar ◽  
Humio Mitsudera
Keyword(s):  
Boundary Layer ◽  
Outer Part ◽  
Slip Boundary Condition ◽  
Slip Condition ◽  
Linear Solution ◽  
Slip Boundary ◽  
Uniform Shear ◽  
Blasius Boundary Layer ◽  
Partial Mode ◽  
Tollmien Schlichting

We describe a mechanistic picture of the essential dynamical processes in the growing Tollmien-Schlichting wave in a Blasius boundary layer and similar flows. This picture depends on the interaction between two component parts of a disturbance (denoted ‘partial modes’), each of which is a complete linear solution in some idealization of the system. The first component is an inviscid mode propagating on the vorticity gradient of the velocity profile with the free-slip boundary condition, and the second, damped free viscous modes in infinite uniform shear with the no-slip condition. There are two families of these viscous modes, delineated by whether the phase lines of the vorticity at the wall are oriented with or against the shear, and they are manifested as resonances in a forced system. The interaction occurs because an initial ‘inviscid’ disturbance forces a viscous response via the no-slip condition at the wall. This viscous response is large near the resonance associated with the most weakly damped viscous mode, and in the unstable parameter range it has suitable phase at the outer part of the boundary layer to increase the amplitude of the inviscid partial mode by advection.

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.

Hydrodynamic stability of the flow in the laminar boundary layer between parallel streams

1954 ◽  
Vol 50 (1) ◽  
pp. 105-124 ◽  
Author(s):  
R. C. Lock
Keyword(s):  
Boundary Layer ◽  
Surface Tension ◽  
Water Waves ◽  
Horizontal Wind ◽  
Neutral Stability ◽  
Wind Speeds ◽  
Sinusoidal Oscillations ◽  
Parallel Streams ◽  
The Stability ◽  
Steady Laminar

ABSTRACTA method is given for determining the stability of small sinusoidal oscillations in the steady laminar flow of a horizontal wind over the surface of a liquid at rest (with particular reference to the flow of air over water), taking into account viscosity, gravity and surface tension. It is shown that there are two fundamental types of oscillation of the system, which may be called ‘water’ waves and ‘air’ waves, and curves showing the conditions for neutral stability of these two types of wave are given for a range of wind speeds from 100 to 300 cm./sec.

scholarly journals On the long-wave instability of natural-convection boundary layers

1997 ◽  
Vol 335 ◽  
pp. 57-73 ◽  
Author(s):  
P. G. DANIELS ◽  
JOHN C. PATTERSON
Keyword(s):  
Boundary Layer ◽  
Travelling Waves ◽  
Vertical Wall ◽  
Phase Speed ◽  
Neutral Curve ◽  
Neutral Stability ◽  
Layer Width ◽  
Dimensional Boundary ◽  
Prandtl Numbers ◽  
The Stability

This paper considers the stability of the one-dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi-steady approximation is used to analyse the asymptotic form of the lower branch of the neutral curve, corresponding to disturbances of wavelength much greater than the boundary-layer width. This leads to predictions of the critical wavenumber for neutral stability and the maximum phase speed of the travelling waves. Results are obtained for a range of Prandtl numbers and are compared with solutions of the full stability equations and with numerical simulations and experimental observations of cavity flows driven by sudden heating of the sidewalls.

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