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.

scholarly journals Neutral stability calculations for boundary-layer flows

1980 ◽  
Vol 23 (2) ◽  
pp. 241 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Atul Padhye
Keyword(s):  
Boundary Layer ◽  
Neutral Stability ◽  
Boundary Layer Flows

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.

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 almost parallel boundary layer flows

1982 ◽  
Vol 10 (4) ◽  
pp. 223-241 ◽  
Author(s):  
Th.L. van Stijn ◽  
A.I. van de Vooren
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flows ◽  
The Stability

Kelvin-Helmholtz instability of a dusty gas

1965 ◽  
Vol 61 (2) ◽  
pp. 569-571 ◽  
Author(s):  
D. H. Michael
Keyword(s):  
Small Volume ◽  
Volume Concentration ◽  
Dusty Gas ◽  
Number Density ◽  
Plane Poiseuille Flow ◽  
Small Particles ◽  
Helmholtz Instability ◽  
Parallel Flows ◽  
The Stability ◽  
Small Disturbances

In a recent paper ((2)) Saffman gave a representation of the flow of a dusty gas in which the dust was described in terms of a number density of small particles with very small volume concentration but appreciable mass concentration. In the same paper the problem of the stability of plane parallel flows to small disturbances was formulated, and subsequently the author ((1)) gave some approximate results for the stability of plane Poiseuille flow.

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.

On the stability of the asymptotic suction boundary-layer profile

1965 ◽  
Vol 23 (4) ◽  
pp. 715-735 ◽  
Author(s):  
T. H. Hughes ◽  
W. H. Reid
Keyword(s):  
Exact Solution ◽  
Stability Problem ◽  
Adjoint Problem ◽  
Analytical Continuation ◽  
Neutral Stability ◽  
The Stability ◽  
Asymptotic Suction ◽  
Linear Stability Problem ◽  
Original Treatment ◽  
General Method

This paper presents a discussion of some aspects of the linear stability problem for the asymptotic suction profile. An exact solution of the inviscid equation is first obtained in terms of the usual hypergeometric function and its analytical continuation. This exact solution provides both a corrected version of an earlier treatment by Freeman and an independent check on the more general method suggested for solving the inviscid equation numerically. Various approximations to the characteristic equation, and hence to the curve of neutral stability, are then considered. In particular, it is found that, in a consistent asymptotic treatment of the related adjoint problem, at least one viscous correction to the singular inviscid solution must be considered. Based on the present results for the adjoint problem, it is suggested that Tollmien's original treatment of the viscous corrections must be slightly modified.

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