The stability of a thermally radiating stratified shear layer, including self-absorption

1974 ◽  
Vol 64 (1) ◽  
pp. 65-83 ◽  
Author(s):  
Joseph J. Dudis
Keyword(s):  
Shear Layer ◽  
Optical Depth ◽  
Richardson Number ◽  
Stability Boundary ◽  
Lower Troposphere ◽  
Neutral Stability ◽  
Viscous Effects ◽  
The Earth ◽  
Critical Richardson Number ◽  
The Stability

A linear stability analysis is applied to a stably stratified, thermally radiating shear layer. The grey Milne–Eddington approximation is employed as a radiation model. In contrast to a previously reported optically thin analysis, no inviscid instability exists, in the limit of vanishing horizontal wavenumber, for this selfabsorbing model. The inviscid neutral-stability boundary (Richardson number us. dimensionless wavenumber) for the Milne–Eddington approximation converges to the optically thick limit as the optical depth of the shear layer is increased. As the optical depth of the shear layer is decreased, the inviscid Milne–Eddington neutral-stability boundary approaches the optically thin limit, although not uniformly in the wavenumber. For fixed mean velocity gradient and fluid properties, the inviscid critical Richardson number approaches infinity as the optical depth of the shear layer approaches zero. Viscous effects neutralize this radiative destabilization, and the critical Richardson number eventually returns to zero as the optical depth continues to decrease. A shearlayer thickness exists for which the viscous critical Richardson number is a maximum. For shear depths greater than this thickness, self-absorption effects increase the stability; and for shear depths less than this thickness, viscous effects increase the stability. Results of the analysis are applied to the atmospheres of Venus and the earth. A critical Richardson number somewhat above the non-radiating value of 3 (although below the previously reported optically thin value) is found for the lower troposphere of the earth. No substantial effect is found for the earth's lower stratosphere or for the 100 km level above Venus.

On the stability of a heterogeneous shear layer subject to a body force

1961 ◽  
Vol 11 (2) ◽  
pp. 284-290 ◽  
Author(s):  
J. Menkes
Keyword(s):  
Shear Layer ◽  
Richardson Number ◽  
Body Force ◽  
Density Variation ◽  
The Body ◽  
Neutral Stability ◽  
Horizontal Shear ◽  
Critical Richardson Number ◽  
The Stability ◽  
Small Disturbances

The effects of density variation and body force on the stability of a heterogeneous horizontal shear layer are investigated. The density is assumed to decrease exponentially with height, and the body force is assumed to be derivable from a potential; the velocity distribution in the shear layer is taken to be U(y) = tanh y. The method of small disturbances is employed to obtain a family of neutral stability curves depending on the choice of the Richardson number. It is demonstrated, furthermore, that the value of the critical Richardson number depends on the magnitude of the non-dimensional density gradient.

An experiment on the stability of small disturbances in a stratified free shear layer

1972 ◽  
Vol 52 (3) ◽  
pp. 499-528 ◽  
Author(s):  
R. S. Scotti ◽  
G. M. Corcos
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Test Section ◽  
Richardson Number ◽  
Molecular Diffusion ◽  
Free Shear Layer ◽  
Horizontal Beam ◽  
Theoretical Predictions ◽  
Critical Richardson Number ◽  
The Stability

A statically stable stratified free shear layer was formed within the test section of a wind tunnel by merging two uniform streams of air after uniformly heating the top stream. The two streams were accelerated side by side in a contraction section. The resulting sheared thermocline thickened gradually as a result of molecular diffusion and was characterized by nearly self-similar temperature (odd), velocity (odd) and Richardson number (even) profiles. The minimum Richardson numberJ0could be adjusted over the range 0·07 ≥J0≥ 0·76; the Reynolds number Re varied between 30 and 70. Small periodic disturbances were introduced upstream of the test section by a fine wire oscillating in the thermocline. The wire generated a narrow horizontal beam of internal waves, which propagated downstream and remained confined within the thermocline. The growth or decay of these waves was observed in the test section. The results confirm the existence of a critical Richardson number the value of which is in plausible agreement with theoretical predictions (J0≅ 0·22 for the Reynolds number of the experiment). The growth rate is a function of the wavenumber and is somewhat different from that computed for the same Reynolds and Richardson numbers, but the calculation assumed velocity and density profiles which were also somewhat different.

An experimental study of the instability of a stably stratified free shear layer

1975 ◽  
Vol 71 (3) ◽  
pp. 563-575 ◽  
Author(s):  
Yu-Hwa Wang
Keyword(s):  
Flow Field ◽  
Shear Layer ◽  
Richardson Number ◽  
Open Channel ◽  
Salt Water ◽  
Stability Boundary ◽  
Water Channel ◽  
Free Shear Layer ◽  
Circulating Water ◽  
The Stability

A stably stratified free shear layer is created in a continuously circulating water channel in the laboratory. Two streams of salt water of different concentrations are brought together at the entrance to the open channel and a layered uniform flow field with a distinct sharp interface is produced in the test section. The maximum density difference between the two layers is Δρx = 0·0065ρw, where ρw is the density of water. The velocity of each layer can be adjusted at will to create free shear across the interface. At the end of the open channel, a mechanical device to separate the layers for recirculation is provided. The resulting flow field has a viscous region approximately 15 times larger than the scale of the salinity diffusion layer. Visual observations are made with hydrogen bubbles and dye traces. Interfacial waves are initiated by artificial excitation. The perturbation frequencies range from 0·476 to 10·40Hz. The measured wavelengths range from 0·46 to 3·02 cm. Damped waves as well as growing waves are observed at various exciting frequencies. Velocity profiles and instantaneous velocities are measured by a hot-film anemometer designed for use in salt water. Experimental values of the Richardson number, the dominant parameter characterizing the instability process, range from 1·23 to 14·45. The stability boundary is determined experimentally. Comparisons with Hazel's numerical results and the earlier results of Scotti & Corcos for low values of the Richardson number are also made.

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 On the stability of waves in classically neutral flows

2020 ◽  
Vol 85 (2) ◽  
pp. 309-340
Author(s):  
Colin Huber ◽  
Meaghan Hoitt ◽  
Nathaniel S Barlow ◽  
Nicole Hill ◽  
Kimberlee Keithley ◽  
...  
Keyword(s):  
Stability Boundary ◽  
Fractional Power ◽  
Single Mode ◽  
Fourier Integral ◽  
Linear Stability Theory ◽  
System Response ◽  
Neutral Stability ◽  
Complex Frequency ◽  
The Stability ◽  
Dispersive Flows

Abstract This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx{{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance, $k$ is a real wavenumber and $\omega (k)$ is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically unstable dispersive flows. Phys. Rev. Fluids, 1, 073604) demonstrates that when Im$[\omega (k)]=0$ for all $k$, it is possible for a system response to grow or damp algebraically as $h\approx{{t}^{s}}$ where $s$ is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im$[\omega (k)]=0$ for a single mode (i.e. for one value of $k$) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined.

Numerical studies of the stability of inviscid stratified shear flows

1972 ◽  
Vol 51 (1) ◽  
pp. 39-61 ◽  
Author(s):  
Philip Hazel
Keyword(s):  
Richardson Number ◽  
Shear Flows ◽  
Stability Boundary ◽  
Phase Speed ◽  
Neutral Curve ◽  
Free Shear Layer ◽  
Density Profiles ◽  
Complex Phase ◽  
Numerical Studies ◽  
The Stability

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters,Ja ‘typical’ Richardson number, α the (real) wavenumber andcthe complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first findscgiven α andJ, the second finds α andJwhenc≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes,but not always, the stability boundary in the α,Jplane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

Destabilization of a stratified shear layer by ambient turbulence

2015 ◽  
Vol 771 ◽  
pp. 1-15 ◽  
Author(s):  
Lin Li ◽  
W. D. Smyth ◽  
S. A. Thorpe
Keyword(s):  
Shear Layer ◽  
Eddy Viscosity ◽  
Stability Boundary ◽  
Neutral Stability ◽  
Vorticity Distribution ◽  
Fluid Properties ◽  
Parameter Ranges ◽  
Ambient Turbulence ◽  
Small Eddy ◽  
Varying Viscosity

A small eddy viscosity or mass diffusivity that varies with height has been found to have unexpected effects on the Kelvin–Helmholtz (KH) instability of a stably stratified shear layer near the neutral stability boundary. In particular, varying viscosity can increase the growth rate of the instability in contrast to the effect of uniform viscosity. Here, these results are extended to parameter ranges relevant in many geophysical and engineering contexts. We find that linearization of the viscous terms based on the assumption of weak viscosity/diffusivity is valid for non-dimensional values (inverse Reynolds number) up to ${\sim}10^{-2}$. Decreasing the Richardson number far below its critical value $1/4$ can change, or even reverse, the effects of eddy viscosity and diffusivity. A primary goal is to explain the unexpected destabilization by viscosity. Varying viscosity affects vorticity (and other fluid properties) in a manner identical to advection with an advecting velocity equal to minus the gradient of viscosity. Destabilization occurs when this viscous ‘advection’ reinforces the vorticity distribution of a growing mode.

On the stability of a shear layer

1959 ◽  
Vol 6 (4) ◽  
pp. 518-522 ◽  
Author(s):  
J. Menkes
Keyword(s):  
Shear Layer ◽  
Transition Layer ◽  
Density Variation ◽  
Neutral Stability ◽  
Neutral Stability Curve ◽  
Horizontal Shear ◽  
Stability Curve ◽  
Wave Numbers ◽  
The Stability ◽  
Small Disturbances

The effects of density variation in the absence of gravity on the stability of a horizontal shear layer between two streams of uniform velocities is investigated. The density is assumed to decrease exponentially with height and the velocity is represented by U(y) = tanh y.The method of small disturbances is employed to obtain the neutral stability curve. It is demonstrated that disturbances with wave-numbers larger than the width of the transition layer are attenuated.Qualitative agreement with experimental evidence is obtained.

Pseudoplastic Flow Between Concentric Rotating Cylinders With Viscous Dissipation

2012 ◽  
Author(s):  
A. Hazbavi ◽  
N. Ashrafi
Keyword(s):  
Heat Flux ◽  
Constant Heat Flux ◽  
Base Flow ◽  
Maximum Temperature ◽  
Exponential Dependence ◽  
Neutral Stability ◽  
Viscous Effects ◽  
Constant Heat ◽  
Highly Correlated ◽  
The Stability

The rotational flow of pseudoplastic fluids between concentric cylinders is examined while dissipation due to viscous effects is taken into account. The viscosity of fluid is simultaneously dependent on shear rate and temperature. Exponential dependence of viscosity on temperature is modeled through Nahme law and the shear dependency is modeled according to the Carreau equation. Hydrodynamically, stick boundary conditions are applied and thermally, both constant temperature and constant heat flux on the exterior of cylinders are considered. The governing motion and energy balance equations are coupled adding complexity to the already highly correlated set of differential equations. Introduction of Nahme number has maintained a nonlinear base flow between the cylinders. As well, the condition of constant heat flux has moved the point of maximum temperature towards the inner cylinder. In the presence of viscous heating, the effect of parameters such as Nahme, Prandtl and Brinkman numbers, material time and pseudoplasticity constant on the stability of the flow is presented in terms of neutral stability curves. The flow parameters along with viscosity maps are given for different scenarios of the flow.

Transition and Mixing in the Shear Layer Produced by Tangential Injection in Supersonic Flow

1971 ◽  
Vol 93 (4) ◽  
pp. 610-618 ◽  
Author(s):  
H. E. Gilreath ◽  
J. A. Schetz
Keyword(s):  
Mach Number ◽  
Supersonic Flow ◽  
Initial Conditions ◽  
Stability Boundary ◽  
Mixing Layer ◽  
Initial Boundary ◽  
Neutral Stability ◽  
Pressure Distributions ◽  
The Stability ◽  
Supersonic Injection

The interaction between a viscous mixing layer induced by tangential injection, and an external supersonic flow field is considered experimentally and analytically. Both subsonic and supersonic injection are investigated. The experiments were performed at freestream Mach numbers of 2.85 and 4.19 using air as the injectant. The principal observations are in the form of spark schlieren photographs, interferograms, and wall pressure distributions. The experiments were arranged to cross Lin’s neutral stability boundary for parallel streams. Transition occurred in all cases, but an increase in stability was noted with either a decrease in the injectant Mach number or an increase in the Mach number of the external flow. Both of these results follow the trends predicted by the stability theory. For the supersonic injection cases, it was found that simple inviscid theory is sufficient to predict the overall interaction pattern between streams, when the ratio of initial boundary layer thickness to the injection slot height is small. However, when the injection is subsonic, the injectant initial conditions in terms of either pressure or Mach number at the slot exit are determined by the downstream viscous-inviscid interaction with the external supersonic flow. A simple one-dimensional theory is applied to this problem to enable prediction of the initial conditions.

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