Axisymmetric Disturbances Research Articles

Strong Stability of Impulsively Initiated Couette Flow for Both Axisymmetric and Nonaxisymmetric Disturbances

A viscous, incompressible fluid is enclosed between two infinitely long coaxial cylinders. The entire system is assumed to be rotating initially as a rigid body with angular velocity Ω. At time t = 0, the outer cylinder is impulsively stopped. The resulting unsteady Couette flow is subject to centrifugal instabilities. Energy theory (strong stability) calculations have been performed for both axisymmetric and nonaxisymmetric disturbances to verify that the most dangerous mode is axisymmetric. The results include global stability bounds, lower bounds on the onset time, and/or upper bounds on the decay times, and are compared to previous marginal stability results for the same basic state.

Motions in a Bose condensate. IV. Axisymmetric solitary waves

For pt.III see ibid., vol.7, p.260 (1974). Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 rho kappa 3/c2 and minimum energy about 0.145 rho kappa 3/c, where rho is density, c is the speed of sound and kappa is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation kappa ; as p to infinity its radius omega approximately (p/ pi kappa )1/2 becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p to infinity ; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be delta E/ delta p, the derivative being taken along the family.

On the linear stability of spiral flow between rotating cylinders

A numerical study is made of the effects of both axisymmetric and non-axisymmetric disturbances on the stability of spiral flow between rotating cylinders. If we letΩ1andΩ2be the angular speeds of the inner and outer cylinders, andR1andR2be their respective radii, then for fixed values ofη=R1/R2andμ=Ω2/Ω1, the onset of instability depends on both the Taylor numberTand the axial Reynolds numberR. HereRis based on the gap width between the cylinders and the average axial velocity of the basic flow, whileTis based on the average angular speeds of the cylinders. Using the compound matrix method, we have computed the complete stability boundary in theR,T-plane for axisymmetric disturbances withη= 0.95 andμ= 0. We find that, for sufficiently high Reynolds numbers, there are two distinct axisymmetric modes corresponding to the usual shear and rotational instabilities. We have also obtained the stability boundaries for non-axisymmetric disturbances for R ≼ 6000 forη= 0.95 and 0.77 withμ= 0. These last results are found to be in substantial agreement with the experimental observations of Snyder (1962, 1965), Nagib (1972) and Mavec (1973) in the low and moderate axial Reynolds number régimes.

Stability of developing flow in an annulus. part II. Non-axisymmetric disturbances

Abstract Spatial stability of the developing axial flow in a concentric annulus against infinitesimal axisymmetric disturbances is described. The mainflow velocity field is determined by means of a finite difference technique and the eigenvalue problem is solved using the fourth order Runge-Kutta method along with a selective application of the Gram-Schmidt orthonormalization technique. Two unstable modes, termed as outer and inner wall modes (OWM and IWM) are found to exist for this problem. These modes behave differently in the inlet region with respect to the annulus diameter ratio. For all diameter ratios except those close to unity the OWM is more unstable than the IWM in the near entry region. However, as the flow develops hydrodynamically, these modes reverse roles so that the fully developed axial flow in the annulus has only one unstable mode (the IWM) for all diameter ratios. Variation of critical Reynolds number, frequency and wavenumber for both the modes in the inlet region is displayed for a wide range of diameter ratios.

Stability of Nonparallel Developing Flow in an Annulus to Asymmetric Disturbances

Linear spatial stability of the nonparallel developing flow in a concentric annulus shows that the asymmetric disturbance with an azimuthal wave number equal to unity is more unstable than the axisymmetric disturbance at all axial locations. Also, in the near entry region, the critical Reynolds number corresponding to the parallel flow theory is as much as three times that due to the nonparallel theory for some values of the annular diameter ratio.

Current-induced instabilities in rotating hydromagnetic flows between concentric cylinders

The stability of hydromagnetic flow produced in a thin annular region between concentric cylinders by the interaction of a superimposed radial current and an axial magnetic field is examined both when the bounding cylinders are stationary and when rotating. After deriving the small gap approximation equations for the steady-state one-dimensional azimuthal flow and for the corresponding induced electric potential, the equations governing the growth of infinitesimal axisymmetric disturbances are obtained and solved at neutral stability by use of the Galerkin method. Stability results for the nonrotating cylinder case are shown to be expressible in terms of existing results for the stability of hydromagnetic flow produced by an azimuthal pressure gradient. It is found, in our thin film approximation, that the radial current must generally exceed a certain critical value before the one-dimensional flow breaks down into a three-dimensional pattern. Rotation of the outer cylinder enhances stability while rotation of the inner cylinder produces a greater tendency toward instability as reflected by lower Dean numbers at the onset of secondary flow.

On the inviscid instability of a circular jet with external flow

The influence of an external flow velocity on the instability of a circular jet has been investigated by means of the inviscid linearized stability theory. The instability properties of spatially growing axisymmetric and first-order azimuthal disturbances show that the external flow inhibits the instability of the circular jet, but increases the unstable frequency range. Similarity considerations lead to the result that, in a first approximation, the disturbed flow field is independent of the external flow velocity, if the axial co-ordinate is contracted by a suitably chosen stretching factor and if the disturbance frequency is reduced by the same factor. It is concluded that the large-scale structure of jet turbulence is modified in the same manner by the external flow.

Finite-amplitude stability of axisymmetric pipe flow

The stability of pipe flow to axisymmetric disturbances is studied by direct numerical simulation of the incompressible Navier-Stokes equations. There is no evidence of finite-amplitude equilibria at any of the wavenumber/Reynolds number combinations investigated, with all perturbations decaying on a time scale much shorter than the diffusive (viscous) time scale. In particular, decay is obtained where amplitude-expansion perturbation techniques predict equilibria, indicating that these methods are not valid away from the neutral curve of linear stability theory.

Observations of the onset of flow instabilities in constricted tubes

A general stability boundary is determined which establishes the minimum Reynolds number Ri and percent area ratio for the onset of flow instabilities. Both nonaxisymmetrix and axisymmetric disturbances were observed. The existance of a finite extent turbulence region is also described.

Stability of developing flow in an annulus part I. Axisymmetric disturbances

Spatial stability of the developing axial flow in a concentric annulus against infinitesimal axisymmetric disturbances is described. The mainflow velocity field is determined by means of a finite difference technique and the eigenvalue problem is solved using the fourth order Runge-Kutta method along with a selective application of the Gram-Schmidt orthonormalization technique. Two unstable modes, termed as outer and inner wall modes (OWM and IWM) are found to exist for this problem. These modes behave differently in the inlet region with respect to the annulus diameter ratio. For all diameter ratios except those close to unity the OWM is more unstable than the IWM in the near entry region. However, as the flow develops hydrodynamically, these modes reverse roles so that the fully developed axial flow in the annulus has only one unstable mode (the IWM) for all diameter ratios. Variation of critical Reynolds number, frequency and wavenumber for both the modes in the inlet region is displayed for a wide range of diameter ratios.

Magnetohydrodynamic Taylor vortex flow under a transverse pressure gradient

Electromagnetically driven flows between infinite conducting cylinders subjected to a constant axial magnetic field B0 are considered. The linear stability problem for axisymmetric stationary disturbances is treated. In contrast with Chandrasekhar’s conjecture, the present numerical results indicate that the critical wave-number tends to infinity with the magnetic field. The nonlinear state of the flow, slighly above transition, is investigated by means of Stuart’s energy approach. Further theoretical results are compared with experiment. At criticality, excellent agreement is found between linear stability theory and experiment. Above transition, only the shape of the experimental curves is correctly predicted by Stuart’s energy method.

Do eddington–sweet circulations exist?

Abstract The initial value problem for a rotating baroclinic star is considered. Arguments are derived suggesting that the steady meridional circulations first postulated by Vogt (1925) and Eddington (1925) do not exist and that instead a state of a purely zonal velocity field is approached from rather arbitrary initial conditions. By extending previous criteria for stability with respect to axisymmetric disturbances to the case of non-vanishing Prandtl number, it is shown that the purely zonal velocity field may be stable in special cases. In realistic situations, it tends to be unstable, but the preferred instability is likely to introduce nonaxisymmetric toroidal eddies rather than meridional circulations.

The Stability of a Natural Convection Flow above a Point Heat Source

Linear stability theory is applied to laminar natural convection plumes arising from a point heat source for Prandtl numbers of 2 and 0.7 (air). The axisymmetric steady plume which has a boundary-layer structure is used as the basic flow. The stability equations which include the effect of velocity-temperature disturbance coupling are numerically solved for both axisymmetric and non-axisymmetric disturbances. The axisymmetric mode is found to be stable. The non-axisymmetric mode n =1 is found to be unstable but n =2 stable, where n is the azimuthal wave-number of the non-axisymmetric disturbance.

Centrifugal instabilities of circumferential flows in finite cylinders: nonlinear theory

In an earlier paper, Blennerhassett & Hall (1979) investigated the linear stability of the flow between concentric cylinders of finite length. The inner cylinder was taken to rotate, while the outer cylinder was fixed. Furthermore, the end walls rotated such that the flow was purely circumferential. In this paper the finite amplitude development of the unstable disturbances to the flow is considered. It is found that the usual perturbation expansion of nonlinear stability theory must be modified if the cylinders are not infinitely long. The bifurcation to a Taylor vortex flow in finite cylinders is shown to be two-sided. The latter effect is shown to be a direct consequence of the finiteness of the cylinders and by taking the cylinders to be very long, we recover the results obtained previously for the infinite problem. The interaction of the two most dangerous modes of linear theory is also investigated. For certain values of the length of the cylinders the initial finite amplitude Taylor vortex flow is shown to become unstable to another class of axisymmetric disturbances. The effect of perturbing the end conditions towards the no-slip conditions appropriate to most experimental configurations is also discussed. Some discussion of the instability problem in very long cylinders with fixed ends is given.

Nonlinear stability of modulated finite gap Taylor flow

The energy stability analysis for an arbitrary gap time-dependent flow is performed. The solution for the basic flow is obtained assuming that either one or both cylinders rotate with a mean and/or sinusoidal velocity. Arbitrary finite amplitude three-dimensional disturbances are considered with two-dimensional axisymmetric disturbances found to render the smallest eigenvalues. The energy stability limits are found to be strongly dependent on the gap size as well as on the modulation frequency.

Non-parallel flow effects on the stability of film flow down a right circular cone

A global asymptotic solution for the linear stability of this flow with respect to axisymmetric disturbances is developed without invoking the usual quasi-parallel flow assumption. This solution predicts the occurrence of quasi-periodic spatially amplified disturbances relatively near the apex, which ultimately are stabilized further down the cone by the relative increase in viscous forces associated with the progressive thinning of the film. The wave speed and wavelength of these disturbances are found to decrease with increasing distance from the apex.

Taylor vortices and the Goldreich—Schubert instability

Abstract A linear stability analysis is applied to the flow of an incompressible viscous fluid contained between two infinite coaxial rotating cylinders with radial gravity and a stabilising radial temperature gradient. Only fluids of low and moderate Prandtl number, Pr. (Pr≦2), are considered. The Boussinesq and narrow-gap approximations are used to obtain the linearised perturbation equations. The latter are solved numerically both for axisymmetric and non-axisymmetric disturbances and stability boundaries obtained. It is found that when Pr≦0.972 the most critical disturbance is always axisymmetric; when Pr>0.972 mode crossing occurs in a manner which appears to indicate that the onset of instability is favoured by the mode with the smallest wave-number. Overstable modes have not been found to exist when Pr<1. Transformation equations are given that enable existing Couette flow results to be applied to the model discussed here. A dispersion relation is obtained describing the growth rates of instabiliti...

On the MHD‐Instability of Slowly Spreading Jets

AbstractThe problem of inviscid instability of electrically conducting, slowly diverging circular jets in the presence of homogeneous axial magnetic fields is analyzed. Using the method of multiple scales, the study investigates the spatial growth of axisymmetric disturbances in the limit of very low values of the magnetic Reynolds number. It is shown that the effect of the combined action of jet divergence and magnetic field is to pick out disturbances suffering the greatest axial gain. The corresponding Strouhal numbers are reduced by the impact of the field as well as the gains and the regime where the growth can take place.

Observations of the instabilities of a round jet and the effect of cocurrent flow

Experimental observations of a fully developed round jet entering a second, less dense, immiscible liquid reveal that cocurrent flow of the outer fluid can eliminate instabilities. Both helical and axisymmetric disturbances were observed.

Centrifugal instabilities of circumferential flows in finite cylinders: linear theory

The linear stability of circular Couette flow in a finite-length cylindrical annulus is investigated. The basic flow is taken to be circumferential, whilst perturbations to the flow are required to vanish at the ends of the annulus, as well as the bounding cylindrical surfaces. Both axisymmetric and non-axisymmetric disturbances are considered, and the governing partial differential system is solved using an eigenfunction expansion technique; the critical Taylor numbers, T c , for the different modes are then determined as functions of the length of the annulus. It is found that, in a finitelength annulus, T c is always greater than the critical Taylor number for infinitely long cylinders, though no marked increase in T c occurs until the length to gap ratio of the annulus is about six. Also, it is found that the finiteness of the container can lead to disturbance flow fields quite different from those of the infinite problem. In particular, it is found that in finite cylinders adjacent vortices can have the same sense of rotation. Streamline patterns illustrating this phenomenon, and showing the arrangement of the Taylor cells in a finite-length container are given for neutrally stable axisymmetric disturbances.