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.

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.

Experimental and Theoretical Investigation of the Supersonic Boundary Layer Stability on the Swept Wing

2008 ◽  
Vol 3 (3) ◽  
pp. 34-38
Author(s):  
Sergey A. Gaponov ◽  
Yuri G. Yermolaev ◽  
Aleksandr D. Kosinov ◽  
Nikolay V. Semionov ◽  
Boris V. Smorodsky
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Supersonic Boundary Layer ◽  
Mean Flow ◽  
Swept Wing ◽  
Wing Model ◽  
Dimensional Boundary ◽  
The Stability ◽  
Good Agreement

Theoretical and an experimental research results of the disturbances development in a swept wing boundary layer are presented at Mach number М = 2. In experiments development of natural and small amplitude controllable disturbances downstream was studied. Experiments were carried out on a swept wing model with a lenticular profile at a zero attack angle. The swept angle of a leading edge was 40°. Wave parameters of moving disturbances were determined. In frames of the linear theory and an approach of the local self-similar mean flow the stability of a compressible three-dimensional boundary layer is studied. Good agreement of the theory with experimental results for transversal scales of unstable vertices of the secondary flow was obtained. However the calculated amplification rates differ from measured values considerably. This disagreement is explained by the nonlinear processes observed in experiment

scholarly journals Stability of the laminar boundary layer in a streamwise corner

1984 ◽  
Vol 393 (1804) ◽  
pp. 101-116 ◽  
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Problem ◽  
Viscous Incompressible Flow ◽  
Boundary Layer Region ◽  
Dimensional Boundary ◽  
The Mean ◽  
Layer Region ◽  
The Stability ◽  
Infinite Boundary Value Problem ◽  
Layer Problem

This work examines the stability of viscous, incompressible flow along a streamwise corner, often called the corner boundary-layer problem. The semi-infinite boundary value problem satisfied by small-amplitude disturbances in the ‘blending boundary layer’ region is obtained. The mean secondary flow induced by the corner exhibits a flow reversal in this region. Uniformly valid ‘first approximations’ to solutions of the governing differ­ential equations are derived. Uniformity at infinity is achieved by a suitable choice of the large parameter and use of an appropriate Langer variable. Approximations to solutions of balanced type have a phase shift across the critical layer which is associated with instabilities in the case of two-dimensional boundary layer profiles.

The stability of spatially periodic flows

1981 ◽  
Vol 108 ◽  
pp. 461-474 ◽  
Author(s):  
D. N. Beaumont
Keyword(s):  
Velocity Profile ◽  
Phase Speed ◽  
Neutral Curve ◽  
Long Waves ◽  
Inviscid Fluid ◽  
Periodic Flows ◽  
A Value ◽  
Parallel Flows ◽  
Spatially Periodic ◽  
The Stability

The stability characteristics for spatially periodic parallel flows of an incompressible fluid (both inviscid and viscous) are studied. A general formula for the determination of the stability characteristics of periodic flows to long waves is obtained, and applied to approximate numerically the stability curves for the sinusoidal velocity profile. The neutral curve for the sinusoidal velocity profile is obtained analytically. The stability of two broken-line velocity profiles in an inviscid fluid is studied and the results are used to describe the overall pattern for the sinusoidal velocity profile in the case of long waves. In an inviscid fluid it is found that all periodic flows (other than the trivial flow in which the basic velocity is constant) are unstable to long waves with a value of the phase speed determined by simple integrals of the basic flow. In a viscous fluid it is found that the sinusoidal velocity profile is very unstable with the inviscid solution being a good approximation to the solution of the viscous problem when the value of the Reynolds number is greater than about 20.

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.

The effect of stable thermal stratification on the stability of viscous parallel flows

1971 ◽  
Vol 47 (1) ◽  
pp. 1-20 ◽  
Author(s):  
K. S. Gage
Keyword(s):  
Boundary Layer ◽  
Richardson Number ◽  
Thermal Stratification ◽  
Neutral Stability ◽  
Plane Poiseuille Flow ◽  
Boundary Layer Flows ◽  
A Value ◽  
Parallel Flows ◽  
The Stability ◽  
Asymptotic Suction

A unified linear viscous stability theory is developed for a certain class of stratified parallel channel and boundary-layer flows with Prandtl number equal to unity. Results are presented for plane Poiseuille flow and the asymptotic suction boundary-layer profile, which show that the asymptotic behaviour of both branches of the curve of neutral stability has a universal character. For velocity profiles without inflexion points it is found that a mode of instability disappears as η, the local Richardson number evaluated at the critical point, approaches 0.0554 from below. Calculations for Grohne's inflexion-point profile show both major and minor curves of neutral stability for 0 < η [les ] 0.0554; for\[ 0.0554 < \eta < 0.0773 \]there is only a single curve of neutral stability; and, for η > 0.0773, the curves of neutral stability become closed, with complete stabilization being achieved for a value of η of about 0·107.

Instability in Boundary-Layer Free Convection along an inclined Plate

1972 ◽  
Vol 1 (4) ◽  
pp. 197-204 ◽  
Author(s):  
J.B. Lee ◽  
G.S.H. Lock
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Plane Surface ◽  
Convection Flow ◽  
Neutral Stability ◽  
Inclined Plate ◽  
Free Convection Flow ◽  
Roll Vortices ◽  
The Stability ◽  
Rayleigh Numbers

This paper gives theoretical consideration to the problem of the stability of laminar, boundary-layer, free-convection flow of air along a long, inclined plane surface heated isothermally. The analysis considers two forms of small disturbance: a two-dimensional wave disturbance, and a set of longitudinal roll vortices. Development of the appropriate disturbance equations is followed by their numerical solution. The effect of inclination on the neutral stability curves for both disturbance forms is presented graphically along with a comparison of the critical Rayleigh numbers obtained from both disturbance forms.

Linear and nonlinear receptivity of the boundary layer in transonic flows

2015 ◽  
Vol 786 ◽  
pp. 154-189 ◽  
Author(s):  
A. I. Ruban ◽  
T. Bernots ◽  
M. A. Kravtsova
Keyword(s):  
Boundary Layer ◽  
Acoustic Noise ◽  
Neutral Curve ◽  
Wing Surface ◽  
Roughness Height ◽  
Stokes Layer ◽  
Instability Modes ◽  
Direct Numerical Method ◽  
Tollmien Schlichting ◽  
The Stability

In this paper we analyse the process of the generation of Tollmien–Schlichting waves in a laminar boundary layer on an aircraft wing in the transonic flow regime. We assume that the boundary layer is exposed to a weak acoustic noise. As it penetrates the boundary layer, the Stokes layer forms on the wing surface. We further assume that the boundary layer encounters a local roughness on the wing surface in the form of a gap, step or hump. The interaction of the unsteady perturbations in the Stokes layer with steady perturbations produced by the wall roughness is shown to lead to the formation of the Tollmien–Schlichting wave behind the roughness. The ability of the flow in the boundary layer to convert ‘external perturbations’ into instability modes is termed the receptivity of the boundary layer. In this paper we first develop the linear receptivity theory. Assuming the Reynolds number to be large, we use the transonic version of the viscous–inviscid interaction theory that is known to describe the stability of the boundary layer on the lower branch of the neutral curve. The linear receptivity theory holds when the acoustic noise level is weak, and the roughness height is small. In this case we were able to deduce an analytic formula for the amplitude of the generated Tollmien–Schlichting wave. In the second part of the paper we lift the restriction on the roughness height, which allows us to study the flows with local separation regions. A new ‘direct’ numerical method has been developed for this purpose. We performed the calculations for different values of the Kármán–Guderley parameter, and found that the flow separation leads to a significant enhancement of the receptivity process.

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.

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