Symmetric Disturbances Research Articles

The effect of horizontal shear flow on geostrophic adjustment in a barotropic fluid

The effect of a basic horizontal shear flow on the linear geostrophic adjustment process in an unbounded barotropic fluid is investigated. It is shown that the basic flow is absolutely stable to axially symmetric transverse disturbances and that energy is not abstracted from the basic flow through the action of Reynolds stresses. A potential vorticity equation is derived and solved numerically in order to determine the relative amount of initial energy which is partitioned to geostrophic kinetic and available potential energy, as a function of the initial current width. It is also shown that, in contradistinction to the adjustment process in an atmosphere with a basic state at rest, a significant portion of geostrophic energy resides in the low wave-number part of the energy spectrum. This latter feature is most noticeable when a large gradient of the basic flow exists. DOI: 10.1111/j.2153-3490.1969.tb00428.x

The stability of steady and time-dependent plane Poiseuille flow

The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.

Stability of Parallel Flow between Concentric Cylinders

The linear stability of parallel flow in a concentric annulus to infinitesimal, axially symmetric disturbances is considered. First, the Poiseuille flow in annular cylinders is studied with the ratio k of the outer to inner cylinder as a parameter. The critical Reynolds number is a monotone function of this radius ratio, increasing without bound as k → ∞ (Hagen-Poiseuille flow) from the plane Poiseuille flow limit (k = 1). Second, a one-parameter family of skewed (variable viscosity) flows in a fixed annulus is studied. The neutral curves for many of these skewed profiles have a second minimum, which for sufficiently skewed profiles, gives the lowest value of the Reynolds number. These two minima are associated with two very different distributions of Reynolds stress. Both distributions are such that on part of the channel the energy is transferred from the disturbance motion to the basic motion and both can be explained by analysis of the Reynolds stress jump condition and the known structure of Reynolds stress near a rigid wall.

Dynamic Behavior of a Foil in the Presence of a Lubricating Film

Equations for the oscillations of a foil over a lubricating fluid film are derived and are simplified by a small parameter expansion. A few particular cases are discussed, and a linearized solution is obtained for the case of a massless, perfectly flexible foil moving at a speed U over an incompressible film. The solution reveals the interesting phenomenon that small disturbances in the film thickness, as well as symmetrical large disturbances, propagate at a speed U/2.

Propagation of an isothermal shock in stellar envelopes

We have studied in this paper the propagation of an isothermal shock in the radiative envelopes of the Bosman-Crespin model for a hot star and Boury’s model for a giant star. A spherically symmetric disturbance is supposed to be originated at or outside the surface of the convective core. We have used Whitham’s rule to study the variation in the shock strength and the shock velocity after modifying it for inclusion of pressure, energy and flux of radiation. We find the shock increases in strength as it propagates through the envelopes of decreasing density, pressure and temperature. The velocity of the shock decreases for very weak initial shock strengths, for intermediate initial shock strength it first decreases and then increases, while for large initial shock strength, it always increases. This aspect of the problem throws some light on the stability of the models under consideration.

Radiative and Photochemical Processes in Mesospheric Dynamics. Part III, Stability of a Zonal Vortex at Mid-Latitudes to Axially Symmetric Disturbances

The stability of a baroclinic, axially symmetric vortex on an f-plane to axially symmetric disturbances is investigated. It is found that with photochemistry and radiative transfer acting, such disturbances are unstable regardless of the value of the Richardson number. The growth rates under conditions relevant to the mesosphere are, however, very small.

On the Stability of the Baroclinic Circular Vortex: a Sufficient Criterion for Instability

The stability of baroclinic circular vortices in an incompressible and inviscid fluid for axially symmetric disturbances is discussed, and the well-known necessary condition for instability is shown to be also a sufficient one in the absence of friction. The customary approach to stability problems by means of the eigen-solution with a separate, time-dependent factor is not generally applicable because eigen-solutions do not necessarily exist in this problem. Instead, the perturbation equation for meridional motion is considered as an initial value problem, and the stability of a given circular vortex is determined from consideration of the total kinetic energy of meridional motion. That is, the vortex is considered stable if the total kinetic energy of the perturbation is always bounded for any initial disturbance; and the vortex is considered unstable if there exists at least one set of initial data for which the total kinetic energy of the perturbation would increase with time beyond limit. With the use of integral relations which are derived from the perturbation equation, the stability of the vortex is discussed without actually constructing solutions of the initial value problem. The method and results are valid in the case of variable coefficients, including the so-called “mixed hyperbolic-elliptic” case.

Stability of laminar-jet flames

A self-excited flame oscillation of low frequency appears often, when laminar fuel jets burn in open air. The flame oscillation is axially symmetrical and has no connection with the size of combustion chambers and the length of fuel pipes. The present paper reports the results of the study of this phenomenon on the basis of the hypothesis that the flame oscillation is caused by the instability of the laminar-jet flows. The instability of axially symmetrical parallel flows of inviscid gases is studied in the case of axially symmetrical disturbances, and the criteria for the instability are derived. A detailed study based on the instability theory is conducted on an oscillating city-gas laminar-jet flame, measuring its mean-velocity and mean-temperature fields. As the results of the study, it is shown that the observed facts, such as the occurrence of the oscillation during combustion, the aspects of streamlines of the flame, and the frequency of the oscillation, can be explained reasonably with the instability theory. Thus, it is concluded that the hypothesis that the flame oscillation of this type is caused by the instability of the laminar jet flows is reasonable.

Instability of a viscous fluid of variable density in a magnetic field

The instability to small two-dimensional disturbances of an electrically conducting fluid of variable density is investigated. The viscous fluid is bounded between two vertical parallel planes normal to which a magnetic field of constant intensity is applied. Significant parameters upon which the behaviour of the Rayleigh number at neutral stability depends are the Hartmann number M and the wave-number α which is associated with a periodic disturbance with periodicity in the unbounded horizontal direction.The solution may be sought by considering basic disturbances which are either symmetric or antisymmetric about the median plane parallel to the boundary planes. It is found that for a given magnetic field strength the critical Rayleigh number governing stability is associated with an antisymmetric disturbance of zero wave-number. The least stable symmetric disturbance which arises when the wave-number is zero is less easily excited. This trend is seen again in the purely hydrodynamic case (M = 0) where, corresponding to a finite wave-numbe value, the more unstable mode at neutral stability is found to be an antisymmetric one.The most unstable situation occurs when both the Hartmann number and the wave number zero. In this case the result of Wooding (1960) that the minimum critical Rayleigh number is zero and is associated with a symmetric disturbance is reobtained.

Stability of Nonrotationally Symmetric Disturbances for Inviscid Flow between Rotating Cylinders

The stability of an inviscid fluid between two concentric rotating cylinders to nonrotationally symmetric disturbances is investigated. The limiting form of the disturbance equations for small gap-to-radius ratios is derived. Computations of the growth rate of unstable disturbances are carried out for the case in which the cylinders rotate in the same direction. It is found that the growth rate of rotationally symmetric disturbances is greater than that of nonrotationally symmetric disturbances.

Stability of a Revolving Fluid with Variable Density in the Presence of a Circular Magnetic Field

The stability of an electrically conductive and incompressible fluid between coaxial rotating cylinders in the presence of a radial density gradient, as well as a circular magnetic field, is studied. The effects of gravity and of the heat generated through viscous and magnetic dissipation is neglected. An eigenvalue problem is formulated for rotationally symmetrical disturbances. Detailed calculations are made for the case of small spacing between the two cylinders. Three dimensionless parameters, each of which contains only one of the three (momentum, thermal, and magnetic) diffusivities, are defined. Parametric equations for surfaces of marginal stability are given, and several planes cutting such surfaces are drawn for the discussions of the roles played by the diffusivities. It is found that both the momentum diffusivity and the magnetic diffusivity are capable of playing the dual role (i.e., the role of stabilization and destabilization simultaneously) in the problem.

The barotropic stability of the mean winds in the atmosphere

This paper considers the stability of a barotropic current on a beta earth. The motion is assumed to be horizontal, non-divergent and barotropic. The current is taken to be of the formU(y)=Asech2by+B. The perturbations are required to approach zero asyapproaches ± ∞. We introduce the non-dimensional wave-numberland a parameter χ, which is a measure of the rotation effect. χ is inversely proportional to β.There are only two kinds of perturbations: symmetric disturbances (those with maximum amplitude aty= 0) and antisymmetric disturbances (those with zero amplitude aty= 0). We find the neutral curve in the (χ,l2)-plane for both types of disturbances. The rates of amplification in the immediate vicinity of the neutral curves are also found. It is seen that the beta effect, which is due to the earth's rotation, tends to stabilize the current. For the symmetric disturbances we find a band of unstable wavelengths when χ > 1/2; and for large χ the estimated curve of the maximum value of the imaginary part of the phase velocity is asymptotic to the lower branch of the neutral curve. The antisymmetric disturbances are more stable than the symmetric disturbances.

Convection in Conditionally Unstable Atmosphere

The analysis has shown that if the stratification is unstable for ascending motion over an infinite area and stable for descending motion, and small random perturbations of various scales are introduced, the final disturbance evolved will have the dimension of a cumulus cloud. The stable stratification in the descending region has the effect of increasing the critical lapse-rate and narrowing the ascending region and making the descending motion widespread, so that the centers of the ascending regions are further apart. The distance estimated from the theoretical analysis agrees favorably with those obtained from satellite observations. The distributions of the various perturbation quantities pertaining to a circular symmetrical disturbance have been obtained. They show many resemblances to the corresponding distributions in tropical storms. Since the main source of energy of the mature tropical storms is the latent heat of condensation, it is really tempting to attribute the formation of these large-scale disturbances directly to convection in conditionally unstable air. However, the analysis shows that some organizing mechanisms are needed in order to channel the release of the energy of unstable stratification into a large scale circulation. DOI: 10.1111/j.2153-3490.1961.tb00107.x

Stability of Nonrotationally Symmetric Disturbances for Viscous Flow between Rotating Cylinders

The stability of a viscous fluid between two concentric rotating cylinders to nonrotationally symmetric disturbances is investigated. It is assumed that the cylinders are rotating in the same direction and that the spacing between the cylinders is small. It is found that the critical Reynolds number increases slightly as the ``number of waves'' in the azimuthal direction in the assumed form of the disturbance increases.

Note on the Sufficient Conditions for the Stability of Axially Symmetric Flows to Axially Symmetric Disturbances

Note on the Sufficient Conditions for the Stability of Axially Symmetric Flows to Axially Symmetric Disturbances

Water Waves Produced by Explosions

Explicit formulas are found for the height of the water waves produced by an arbitrary axially symmetric initial disturbance in water of finite depth. The initial disturbance can be either a distribution of impulse or an elevation or depression of the surface or a combination of these. Upper bounds on the wave height are given to be used when the approximate size of the initial disturbance is known, but not its detailed shape. These results are applied to determine the wave resulting from the explosion of a typical large bomb above the surface. The corresponding formulas for waves in a one-dimensional channel are also derived and are compared with experimental results.

An experimental investigation of the stability of Poiseuille flow

An axially symmetric laminar flow of air was established in a long smooth pipe. This flow was steady up to Reynolds numbers of about 20,000, the capacity of the system. Small, nearly axially, symmetric disturbances were superimposed by longitudinally oscillating a thin sleeve adjacent to the inner wall of the pipe. Hot-wire anemometer measurements consisting of radial and longitudinal traverses were made downstream of the sleeve. These measurements indicated that within the Reynolds number range investigated (up to 13,000), the flow is stable to small disturbances. In general, the radial distribution of disturbance amplitudes was not independent of distance downstream; while the disturbances, as generated, exhibited imperfect axial symmetry, the non-symmetric part decayed more rapidly than the symmetric part. Results were interpreted in such a way that rates of propagation and rates of decay of the disturbances could be compared with those given by a recent theoretical stability analysis. It was found that the rates of decay are predicted fairly satisfactorily by the theory; however, the rates of propagation are not. In addition, it was found that transition to turbulent flow occurs whenever the amplitude of the disturbance exceeds a threshold value which decreases with increasing Reynolds number. Due to the departures from axial symmetry in the amplitude of the disturbance, it was not possible to obtain a quantitative measure of the threshold. A mathematical idealization of the disturbances, believed to be more akin to experimental perturbations than the classical model used in small-perturbation analyses, is proposed.

On the stability of fully developed flow in a pipe

The stability of infinitesimal axially symmetric disturbances in fully developed pipe flow is examined anew. The classical eigenvalue problem is treated in part by asymptotic methods and leads to an algebraic relation between the eigenvalue c, the disturbance wavelength 2π/α, and the Reynolds number. Examination of the limiting cases of this relation reveals the existence of two families of characteristic numbers, the value of which tends to unity and to zero as the Reynolds number increases without bounds. For the latter, a more accurate solution is required and given. It is found that all eigenvalues yield stable solutions and that for a given wave number and Reynolds number only a finite number of eigenvalues exists.The limitations of the analysis are discussed in the light of a recent experimental study of the same problem.

The Hydromagnetic Oscillations of Twisted Magnetic Fields. II.

view Abstract Citations (10) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Hydromagnetic Oscillations of Twisted Magnetic Fields. II. Trehan, S. K. ; Reid, W. H. Abstract The stability of twisted magnetic fields imbedded in a conducting fluid which extends to infinity is investigated when fluid motions are present along the lines of force. It is found that this configuration is always stable for cylindrically symmetric disturbances. The periods of oscillation are obtained by a variational method and are found not to differ greatly from the corresponding results for twisted magnetic fields confined to a cylinder of finite radius. Publication: The Astrophysical Journal Pub Date: March 1958 DOI: 10.1086/146473 Bibcode: 1958ApJ...127..454T full text sources ADS | Related Materials (2) Part 1: 1958ApJ...127..446T Part 3: 1973A&A....22..133G

The growth of disturbances in a flame-generated shear region

Results are presented of an experimental and theoretical investigation of the growth of transverse velocity disturbances in the shear region caused by a flame in a duct. In the theoretical stability analysis, a flow field arising from a flame in a duct was analyzed. The flow was neutrally stable to symmetric disturbances and unstable to antisymmetric ones. In the experimental part of the program disturbances of various frequencies were imposed on a flame stabilized in a duct, and the effects were measured by shadow photography and photomultiplier-probe surveys.