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.

Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex

2007 ◽  
Vol 576 ◽  
pp. 325-348 ◽  
Author(s):  
C. J. HEATON
Keyword(s):  
Continuous Spectrum ◽  
Single Point ◽  
Stability Boundary ◽  
Infinite Family ◽  
Neutral Curve ◽  
Swirling Flows ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Numerical Studies ◽  
The Stability

We identify a family of centre-mode disturbances to inviscid swirling flows such as jets, wakes and other vortices. The centre modes form an infinite family of modes, increasingly concentrated near to the symmetry axis of the mean flow, and whose frequencies accumulate to a single point in the complex plane. This asymptotic accumulation allows analytical progress to be made, including a theoretical stability boundary, inO(1) parameter regimes. The modes are located close to the continuous spectrum of the linearized Euler equations, and the theory is closely related to that of the continuous spectrum. We illustrate our analysis with the inviscid Batchelor vortex, defined by swirl parameterq. We show that the inviscid instabilities found in previous numerical studies are in fact the first members of an infinite set of centre modes of the type we describe. We investigate the inviscid neutral curve, and find good agreement of the neutral curve predicted by the analysis with the results of numerical computations. We find that the unstable region is larger than previously reported. In particular, the value ofqabove which the inviscid vortex stabilizes is significantly larger than previously reported and in agreement with a long-standing theoretical prediction.

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.

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.

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.

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 large-amplitude geostrophic flows in a two-layer fluid: the case of ‘strong’ beta-effect

1995 ◽  
Vol 284 ◽  
pp. 137-158 ◽  
Author(s):  
E. S. Benilov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Stability Boundary ◽  
Phase Speed ◽  
Stability Theorems ◽  
Beta Effect ◽  
Zonal Velocity ◽  
Special Cases ◽  
Geostrophic Flows ◽  
The Stability

This paper examines the stability of two-layer geostrophic flows with large displacement of the interface and strong β-effect. Attention is focused on flows with non-monotonic interface profiles which are not covered by the Rayleigh-style stability theorems proved by Benilov (1992a, b) and Benilov & Cushman-Roisin (1994). For such flows the coefficient of the highest derivative in the corresponding boundary-value problem vanishes at the point where the depth profile has an extremum. Although this singularity is similar to a critical level, it cannot be regularized by the simplistic introduction of infinitesimal viscosity through the assumption that the phase speed of the disturbance is complex. In order to regularize the singularity properly, one should consider the problem within the framework of the original ageostrophic viscous equations and, having obtained the boundary-value problem for harmonic disturbance, take the limit Rossby number → 0, viscosity → 0.The results obtained analytically and (for special cases) numerically indicate that the stability of flows with non-monotonic profiles strongly depends on the depth of the upper layer. If the upper layer is ‘thick’ (i.e. if the average depth H1 of the upper layer is of the order of the total depth of the fluid H0), the stability boundary-value problem does not have any solutions at all, which means stability (however, this stability is structurally unstable, and the flow, generally speaking, can be made weakly unstable by any small effect such as external forcing, viscosity, or ageostrophic corrections). In the case of ‘thin’ upper layer (H1/H0 [lsim ] Ro), the order of the singularity changes and all non-monotonic flows are unstable regardless of their profiles. It is also demonstrated that thin-upper-layer flows do not have to be non-monotonic to be unstable: if u–βR20 (where u is the zonal velocity, β is the β-parameter, and R0 is the deformation radius) changes sign somewhere in the flow, the stability boundary-value problem has another singular point which leads to instability.

Linear analysis of the stability of particle-laden stratified shear layers

2014 ◽  
Vol 92 (2) ◽  
pp. 103-115 ◽  
Author(s):  
Ehsan Khavasi ◽  
Bahar Firoozabadi ◽  
Hossein Afshin
Keyword(s):  
Layer Thickness ◽  
Piecewise Linear ◽  
Stability Boundary ◽  
Shear Layers ◽  
Unstable Flow ◽  
Density Profiles ◽  
Background Density ◽  
Layer Flow ◽  
Bed Slope ◽  
The Stability

Hydrodynamic instabilities at the interface of stratified shear layers could occur in various modes and have an important role in the mixing process. In this work, the linear stability analysis in the temporal framework is used to study the stability characteristics of a particle-laden stratified two-layer flow for two different background density profiles: smooth (hyperbolic tangent) and piecewise linear. The effect of parameters, such as bed slope, viscosity, and particle size, on the stability is also considered. The pseudospectral collocation method employing Chebyshev polynomials is used to solve two coupled eigenvalue equations. Based on the results, there are some differences in the stability characteristics of the two density profiles. In the case of R = 1 (R is the ratio of the shear layer thickness to the density layer thickness), the stability boundary in smooth profile is the transition from the unstable flow (where the dominant unstable mode is Kelvin–Helmholtz) to the stable one where in the piecewise linear profile this boundary is the transition from Kelvin–Helmholtz to the Holmboe mode. It is also shown that the unstable region increases with the bed slope and unstable modes amplify as the bed slope increases. For R = 5 the flow does not become stable by increasing the stratification in nonzero bed slope, and in some wavenumbers the Kelvin–Helmholtz and Holmboe modes coexist. In addition, by increasing the bed slope the growth rate of the Holmboe mode and the range of its existence decrease. As expected, the viscosity makes the current more stable, and for large values of the viscosity (small Reynolds number) the flow becomes stable at long waves (small wave numbers) for all bulk Richardson numbers. Existence of small particles does not change the instability characteristics so much, however, large particles make the flow more unstable.

The rapid-rotation limit of the neutral curve for Taylor–Couette flow

2016 ◽  
Vol 808 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Couette Flow ◽  
Asymptotic Theory ◽  
Stability Boundary ◽  
Outer Cylinder ◽  
Stability Result ◽  
Neutral Curve ◽  
Navier Stokes ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

An asymptotic theory is developed for the linear stability curve of rapidly rotating Taylor–Couette flow. The analytic curve obtained by the theory excellently explains the limiting Navier–Stokes stability result for general disturbances. When the cylinders are corotating, the asymptotic theory describes the gap between the neutral curve and the Rayleigh stability criterion. For the case when the cylinders are counter-rotating, it is found that, along the stability boundary, the Reynolds number based on the inner cylinder speed is proportional to that based on the outer cylinder speed to the power of $3/5$.

The stability of inviscid flows over passive compliant walls

1987 ◽  
Vol 183 ◽  
pp. 265-292 ◽  
Author(s):  
K. S. Yeo ◽  
A. P. Dowling
Keyword(s):  
Phase Velocity ◽  
Shear Flows ◽  
Temporal Stability ◽  
Sufficient Conditions ◽  
Marginal Stability ◽  
Inviscid Flows ◽  
Numerical Studies ◽  
Parallel Flows ◽  
Compliant Walls ◽  
The Stability

The linear temporal stability of incompressible semi-bounded inviscid parallel flows over passive compliant walls is studied. It is shown that some of the well-known classical results for inviscid parallel flows with rigid boundaries can, in fact, be extended in modified form to passive compliant walls. These include a result of Rayleigh (1880) which shows that the real part of the phase velocity of a non-neutral disturbance must lie within the range of the velocity distribution; the semi-circle theorem of Howard (1961) and a result of Høiland (1953) which places a bound on the temporal amplification rates of unstable disturbances. The bounds on the phase velocity and the temporal amplification rates of unstable two-dimensional disturbances provide useful guides for numerical studies.The results are valid for a large class of passive compliant walls. This generality is achieved through a variational-Lagrangian formulation of the essential dynamics of wall motion. A general treatment of the marginal stability of thin shear flows over general passive compliant walls is given. It represents a generalization of the analysis given by Benjamin (1963) for membrane and plate surfaces. Sufficient conditions for the stability of thin shear flows over passive compliant walls are deduced. The applications of the stability criteria to simple cases of compliant wall are described to illustrate the use and the effectiveness of these criteria.

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.

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