Axisymmetric Disturbances Research Articles

Numerical study of axisymmetric vortex breakdown in an annulus

The steady axisymmetric Navier-Stokes equations describing the flow in a cylindrical container and in an annulus with one or two rotating endwalls were solved by the continuation method. The unsteady flow regime was obtained from steady solution by following the temporal development of a small axisymmetric disturbance introduced into the initial conditions for either the full or the linearized system of equations. By variation of the boundary condition for vorticity at the axis the transition of the flow in a conventional container to that in an annulus with infinitesimal inner radius was continuously followed; the flow in the latter case has been found to differ only insignificantly from that in the container. By further continution in the radii ratio (the ratio between the inner and the outer radii), flow patterns in an annulus were obtained. An annulus with the stress-free (slip) boundary condition at the inner wall was also considered in order to examine the influence of viscous effects on the formation of separation bubble. While for the solid inner cylinder the recirculation region turned out to persist even at very large values of the radii ratio, the stress-free condition imposed at this boundary leads to vanishing of the separation bubble already at relatively small values of this parameter. The transition to unsteady motion (which, in application to a cylindrical container, was shown by Tsitverblit [14] to be due to the supercritical Hopf bifurcation) was studied in comparison with the results for a cylindrical container.

Vortical structure in forced unsteady circular jet: simulation by 3D vortex method

Three-dimensional vortex-method simulations are made of a round jet forced by axisymmetric, helical and multiple disturbances. The multiple disturbances are combinations of two helical modes rotating in the opposite direction. The present simulations demonstrate the evolution of large-scale vortex structures in the forced jet. Results of the simulations are in reasonable agreement with available experimental results.

Magnetogravitodynamic instability of a cylindrical fluid jet acting upon a general, varying magnetic field

The instability of a fluid cylinder under the action of self-gravitating, inertia, and electromagnetic forces is developed. A general eigenvalue relation is derived studied analytically and the results are confirmed numerically. The jet is gravitationally unstable for small axisymmetric disturbances and stable for all other disturbances. The electromagnetic force has stabilizing and destabilizing influences not only in the axisymmetric mode but also in the non-axisymmetric disturbances. This is due to the fact that the pervading magnetic field is non-uniform. The vacuum field attains its maximum destabilizing influence if the perturbed and unperturbed vacuum fields are orthogonal whether the disturbance is axisymmetric or not. The Lorentz force for some states has the character of enlarging the gravitational unstable domain and decreasing the stable regions in all possible modes of perturbations. In other states the Lorentz force is totally stabilizing but no magnetic field, however strong, can suppress the gravitational instability because the gravitational instability of sufficiently long waves will persist.

General Local Stability Criteria for Stratified, Weakly Magnetized Rotating Systems

We consider the local stability of laminar differential rotation in a weakly magnetized plasma to adiabatic axisymmetric disturbances. Our assumptions are considerably less restrictive than those of previous analyses. The angular velocity need not have cylindrical isorotational surfaces, and {ital a} {ital priori} convective stability is not assumed. In an unmagnetized gas, stability under these circumstances would be governed by the Ho/iland criteria. These criteria change qualitatively when even a weak magnetic field is present. In a magnetized plasma, the new criteria for stability have the same general form exhibited by the purely hydrodynamical case, but with all angular momentum gradients replaced by angular velocity gradients. Our findings are in accord with those of Papaloizou & Szuszkiewicz, who, using very different methods, considered the stability of iso-rotational field configurations. The results presented here thus point very generally to angular velocity gradients, not angular momentum gradients, as the fundamental stability discriminants in rotating astrophysical plasmas. {copyright} {ital 1995 The American Astronomical Society.}

The effect of a radial temperature gradient on the stability of Couette flow: Non-axisymmetric modes analysis

The effect of a radial temperature gradient on the character of the critical disturbances responsible for the onset of instability of Couette flow is examined. The stability equations with respect to non-axisymmetric disturbances are derived and solved numerically for the narrow gap case. For counter-rotating cylinders, we found that the critical non-axisymmetric disturbances of the isothermal flow become axisymmetric when the outer cylinder temperature θ 2 is higher than the inner cylinder temperature θ 1 by a certain critical value. For the stationary outer cylinder, the critical axisymmetric disturbances become non-axisymmetric when θ 2 is less than θ 1 by a certain critical value. These critical values are calculated for fluids with different Prandtl numbers.

Linear stability analysis of thermocapillary convection in the floating zone

The stability of thermocapillary convection in the floating zone for molybdenum (Mo) and lithium niobate (LiNbO 3) has been studied using a linear theory. The input power induced by the heat source is assumed to be a Gaussian distribution. The steady, axisymmetric basic state is solved numerically using a finite difference method, employing the boundary-fitted curvilinear coordinate system. The linear disturbance equations for axisymmetric disturbances, which are partial differential equations, are treated by a mixed finite difference spectral method. The critical zone lengths and the critical input powers of Mo for axisymmetric disturbances are predicted. The thermocapillary instability of the LiNbO 3 melt may not appear before the capillary instability (originating from the gas-melt interface) sets in.

Linear stability analysis of thermocapillary convection in liquid bridges using a mixed finite difference‐spectral method

A linear stability analysis has been employed to investigate the thermocapillary instability occurring in a nonisothermal liquid bridge. The steady, axisymmetric basic state was solved numerically using a finite difference method. A mixed finite difference‐spectral method, combining the advantages of both methods, was then used to reduce the linear disturbance equations to an eigenvalue problem. The critical Marangoni numbers for axisymmetric disturbances are predicted for small Prandtl numbers and various aspect ratios. The effect of surface heat transfer is also investigated. The present results are compared with energy‐theory results and with the results of other experiments.

Phase-Transition Theory of Instabilities. II. Fourth-Harmonic Bifurcations and lambda -Transitions

view Abstract Citations (8) References (55) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Phase-Transition Theory of Instabilities. II. Fourth-Harmonic Bifurcations and lambda -Transitions Christodoulou, Dimitris M. ; Kazanas, Demosthenes ; Shlosman, Isaac ; Tohline, Joel E. Abstract We use a free-energy minimization approach to describe in simple and clear physical terms the secular and dynamical instabilities as well as the bifurcations along equilibrium sequences of rotating, self-gravitating fluid systems. Our approach is fully nonlinear and stems from the Ginzburg-Landau theory of phase transitions. In this paper, we examine fourth-harmonic axisymmetric disturbances in Maclaurin spheroids and fourth-harmonic nonaxisymmetric disturbances in Jacobi ellipsoids. These two cases are very similar in the framework of phase transitions. It has been conjectured (Hachisu & Eriguchi 1983) that third-order phase transitions, manifested as smooth bifurcations in the angular momentum-rotation frequency plane, may occur on the Maclaurin sequence at the bifurcation point of the axisymmetric one-ring sequence and on the Jacobi sequence at the bifurcation point of the dumbbell-binary sequence. We show that these transitions are forbidden when viscosity maintains uniform rotation. The uniformly rotating one-ring/dumbbell equilibria close to each bifurcation point and their neighboring uniformly rotating nonequilibrium states have higher free energies than the Maclaurin/Jacobi equilibria of the same mass and angular momentum. These high-energy states act as free-energy barriers preventing the transition of spheroids/ellipsoids from their local minima to the free-energy minima that exist on the low rotation frequency branch of the one-ring/binary sequence. At a critical point, the two minima of the free-energy function are equal, signaling the appearance of a first-order phase transition. This transition can take place beyond the critical point only nonlinearly if the applied perturbations contribute enough energy to send the system over the top of the barrier (and if, in addition, viscosity maintains uniform rotation). In the angular momentum-rotation frequency plane, the one-ring and dumbbell-binary sequences have the shape of an "inverted S" and two corresponding turning points each. Because of this shape, the free-energy barrier disappears suddenly past the higher turning point, leaving the spheroid/ellipsoid on a saddle point but also causing a "catastrophe" by permitting a "secular" transition toward a one-ring/binary minimum energy state. This transition appears as a typical second-order phase transition, although there is no associated sequence bifurcating at the transition point (cf. Christodoulou et al. 1995a). Irrespective of whether a nonlinear first-order phase transition occurs between the critical point and the higher turning point or an apparent second-order phase transition occurs beyond the higher turning point, the result is fission (i.e., "spontaneous breaking" of the topology) of the original object on a secular timescale: the Maclaurin spheroid becomes a uniformly rotating axisymmetric torus, and the Jacobi ellipsoid becomes a binary. The presence of viscosity is crucial since angular momentum needs to be redistributed for uniform rotation to be maintained. We strongly suspect that the "secular catastrophe" is the dynamical analog of the notorious λ-transition of liquid 4He because it appears as a "second-order" phase transition with infinite "specific heat" at the point where the free-energy barrier disappears suddenly. This transition is not an elementary catastrophe. In contrast to this case, a "dynamical catastrophe" takes place from the bifurcation point to the lower branch of the Maclaurin toroid sequence because all conservation laws are automatically satisfied between the two equilibrium states. Furthermore, the free-energy barrier disappears gradually, and this transition is part of the elementary cusp catastrophe. This type of "λ-transition" is the dynamical analog of the Bose-Einstein condensation of an ideal Bose gas. The phase transitions of the dynamical systems are briefly discussed in relation to previous numerical simulations of the formation and evolution of protostellar systems. Some technical discussions concerning related results obtained from linear stability analyses, the breaking of topology, and the nonlinear theories of structural stability and catastrophic morphogenesis are included in an appendix. Publication: The Astrophysical Journal Pub Date: June 1995 DOI: 10.1086/175807 arXiv: arXiv:astro-ph/9504062 Bibcode: 1995ApJ...446..485C Keywords: GALAXIES: KINEMATICS AND DYNAMICS; GALAXIES: STRUCTURE; INSTABILITIES; STARS: FORMATION; STARS: ROTATION; Astrophysics E-Print: 34 pages, postscript, compressed,uuencoded. 7 figures available in postscript, compressed form by anonymous ftp from asta.pa.uky.edu (cd /shlosman/paper2 mget *.ps.Z). To appear in ApJ full text sources arXiv | ADS | Related Materials (3) Part 1: 1995ApJ...446..472C Part 3: 1995ApJ...446..500C Part 4: 1995ApJ...446..510C

Interchange instability in and accretion disc with a poloidal magnetic field

We investigate the stability to nonaxisymmetric perturbations of an accretion disc in which a poloidal magnetic field provides part of the radial support against gravity. Interchange instability due to radial gradients in the magnetic field are strongly stabilized by the shear flow in the disc. For smooth field distributions this instability is restricted to discs in which the magnetic energy is comparable to the gravitational energy. An incompressible model for the instability akin to the Boussinesq approximation for convection is given which predicts the behaviour of the instability accurately. Global axisymmetric disturbances are also considered and found to be stable for a certain class of models. The results indicate that accretion discs may be able to support poloidal fields which are strong enough to suppress other forms of magnetic instability. These strong and stable field distributions are likely to be well suited for the magnetic acceleration of jets and winds.

Effect of a time-periodic axial shear flow upon the onset of Taylor vortices.

The onset of instability in the presence of an oscillatory axial flow between concentric rotating cylinders to both axisymmetric and nonaxisymmetric disturbances is investigated on a linear basis for a narrow-gap geometry. We consider two types of axial oscillation: one is due to an oscillatory pressure gradient, and the other is due to an oscillation of the inner cylinder. For small Reynolds numbers, the results are obtained for axisymmetric disturbances by expansion in terms of the amplitude of the oscillatory flow. For finite Reynolds numbers (Re\ensuremath{\le}100), the governing equations are solved for general disturbances by a Galerkin expansion with time-dependent coefficients, and the stability boundaries of the system are determined by use of Floquet theory. The time-modulated axial flow, in general, stabilizes significantly axisymmetric disturbances except for some cases of counter-rotation with an oscillation of the inner cylinder. No subharmonic critical solutions appear for Re\ensuremath{\le}100. For Re\ensuremath{\le}100, axisymmetric disturbances are found to be more unstable than nonaxisymmetric ones when the outer cylinder is stationary, in contrast to the steady Poiseuille flow case for Re\ensuremath{\gtrsim}30. Nonaxisymmetric disturbances exhibit phase locking and jumps in the response frequency.

Non-axisymmetric subcritical bifurcations in viscoelastic Taylor–Couette flow

Recently, an account of the linear and nonlinear analysis of the viscoelastic Taylor–Couette flow between independently rotating cylinders against axisymmetric disturbances was presented (Avgousti & Beris 1993 a ). However, more recent linear stability analysis has shown that for the range of geometric and kinematic parameters studied and for high enough values of flow elasticity, the critical instabilities are caused by non-axisymmetric, time-dependent disturbances (Avgousti & Beris 1993 b ). In this work, we calculate the bifurcating families corresponding to each one of the two possible non-axisymmetric patterns emerging at the point of criticality, namely the spirals and ribbons and determine their stability. It is shown that for a narrow gap size, for upper convected Maxwell and Oldroyd-B fluids, at least one of the non-axisymmetric families bifurcates subcritically. This result, in conjunction with the theoretical analysis of Hopf bifurcation in presence of symmetries (Golubitsky et al . 1988), implies that neither of the bifurcating families is stable. Consequently, there is a finite transition corresponding to infinitesimal changes of the flow parameters in the vicinity of the Hopf bifurcation point. Although a change in the ratio of the Deborah and Reynolds numbers has not been found to qualitatively affect this behaviour, calculations with a wider gap size have shown that both bifurcating families become supercritical. There, a Ginzburg–Landau analysis shows that the ribbons are the stable pattern. This behaviour is qualitatively similar to that seen for the newtonian fluid, but for counterrotating cylinders, albeit there, spirals have been found to be stable (Golubitsky & Langford 1988).

The existence of critical Reynolds numbers in pipe entrance flows subjected to infinitesimal axisymmetric disturbances

A modified Chebyshev operational Tau approach was implemented to establish temporal eigenvalues for the axisymmetric Sexl stability equation, transformed to remove the singularity at the origin. Its application to the pipe entrance flow stability problem resulted in the determination of a well-defined maximum axial position—relatively close to the pipe inlet—which possesses a critical Reynolds number of infinity. Thus for the first time a whole range of pipe entrance velocity profile shapes is established—of which the Hagen–Poiseuille limit is only one—for which critical Reynolds numbers do not exist. When combined with the fact that nonparallel effects disappear at the last linearly unstable station, this result is significant because it shows that the greater proportion of the pipe entrance requires finite amplitude disturbances to destabilize the flow.

Thermomechanical and magnetohydrodynamic stability of elongating plastic jets

In this study the stability characteristics of uniformly elongating plastic jets exposed to axial electric currents are investigated. The objective of this study is to expand the results of previous analyses by Littlefield [‘‘The effect of electromagnetic fields on the stability of a uniformly elongating plastic jet,’’ Phys. Fluids A 2, 2240 (1990); ‘‘Finite conductivity effects on the MHD instabilities in uniformly elongating plastic jets,’’ ibid. 3, 166 (1991); ‘‘Enhancement of stability in uniformly elongating plastic jets with electromagnetic fields,’’ ibid. 3, 2927 (1991)] to include high levels of electric current, where thermal energy effects must be included. Coupling of the magnetohydrodynamic and thermal characteristics of the flow is accomplished through the variation of mechanical, thermal, and electrical properties with temperature. Phase change effects are also considered. The jet is assumed incompressible and perfectly plastic, with the Levy–von Mises criterion imposed to limit the effective stress. Solutions to the appropriate base flow are subjected to small axisymmetric disturbances, and linear perturbation theory is employed to determine the time evolution of these disturbances. Perturbations that grow the fastest in magnitude as time progresses are identified as the most unstable. Results of the analysis indicate that thermal effects can dramatically alter both the base and perturbed flow fields, as well as the growth rate of perturbations.

Comments on the numerical investigation of Rayleigh and Marangoni convection in a vertical circular cylinder

Convection in a cylindrical container was simulated with a three-dimensional, time-dependent code. For the case of purely Rayleigh convection, a completely rigid cylinder with adiabatic vertical walls and conducting horizontal walls was considered. The calculations showed that the individual velocity components could be in a transient state, while the total heat transfer was steady. This occurred in cases where the maximum azimuthal component of velocity was very small in magnitude, in comparison to the other two components, and this component decreased in time. On the other hand, the total kinetic energy along with the heat transfer reached a steady value. Consequently the present results have been shown to be at variance with the calculations of Neumann [J. Fluid Mech. 214, 559 (1990)]. Marangoni convection was modeled with a free flat surface on the upper side, assuming the superimposed second layer to be passive. The numerically obtained critical Marangoni numbers and flow patterns were compared favorably to earlier results from linearized stability. In addition, flow structural changes for supercritical Marangoni numbers were illustrated. Interestingly, axisymmetric disturbances led to nonsymmetric bifurcation diagrams, but three-dimensional disturbance calculations led to symmetry in the bifurcation plots. Another very interesting result was the observed transition from three-dimensional to two-dimensional patterns as the Marangoni number was increased. Large computational requirements precluded a detailed parametric study. The special case of Prandtl number equal to 6.7 (corresponding to water), and in the case of Marangoni convection, a surface Biot number of unity was assumed.

The Stability of Pipe Entrance Flows Subjected to Axisymmetric Disturbances

This paper reexamines an important unresolved problem in fluid mechanics—the discrepancy between measurements and predictions of stability in pipe entrance flows. Whereas measured critical Reynolds numbers are relatively insensitive to velocity profile shape in the streamwise direction, the theoretical results indicate a rapid increase, both as the equilibrium profile is approached, and toward the inlet. The current work uses the displacement thickness based Reynolds number as a rational basis on which to compare new stability predictions obtained by means of the Q-Z algorithm, with existing theoretical results. Although the present data are shown to be the only that are consistent with the classical parallel boundary layer limit towards the inlet, they still deviate increasingly with axial distance from the only available experimental results. By examining pipe inlet stability data in relation to boundary layer measurements and predictions, the work effectively questions the commonly held belief that streamwise variations of flow alone are responsible for these deviations, suggesting that the finite amplitude nature of the applied disturbances is the most likely cause.

Wave packets, resonant interactions and soliton formation in inlet pipe flow

The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds numberR→ ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.

On Rayleigh’s Stability Criterion for Couette Flow

Abstract The stability problem of a stationary circular flow of an ideal fluid between two coaxial cylinders is considered. It is shown that Rayleigh's circulation criterion is necessary and sufficient for the total kinetic energy of an axisymmetric disturbance to be bounded in time.

Nonlinear Displacement of the Geostrophic Velocity Jet Created by Mass Imbalance

Nonlinear geostrophic adjustment to a mass imbalance is investigated analytically, focusing on horizontal displacement of the equilibrium velocity jet. For no initially imposed horizontal length scale, the jet displacement is the difference between the deformation radii of the two initial states; as the mass imbalance increases, the displacement approaches the deformation radius of the dominant state. For a disturbance of finite width, the jet displacement decreases as the initial width decreases, but disturbances of relatively small width can still produce appreciable displacements. For an axisymmetric disturbance, geometric dispersion reduces the jet displacement. A two-layer model creates two jets, the displacements of the lower and upper jets being, respectively, slightly larger and considerably smaller than that predicted by the one-layer model. For stratified fluids, these mass differences represent horizontal changes in stratification. An example of jet displacement resulting from horizontal stratification differences occurs at a surface temperature discontinuity, where reduced stratification produced by vertical mixing over a warmer surface creates jets in the boundary layer and in the free atmosphere: when the boundary layer becomes well mixed, the jet displacements approach a mesoscale deformation radius that depends upon the free atmosphere stratification and the boundary-layer depth.

On the linear stability of cellular spiral Couette flow

The stability of flow in an annulus capped at both ends and bounded by a fixed outer cylinder and a spiraling inner rod is determined in the context of linear theory. For infinite aspect ratio and constant fluid properties the problem is governed by a Taylor number Ta, an axial Reynolds number Re, and the radius ratio η. Linear stability is tested with respect to both axisymmetric (n=0) and nonaxisymmetric (n≠0) disturbances for η=0.2, 0.4, 0.6, 0.8 over the range 0≤Re≤2000. The evolution of axial wave number, axial phase speed, and spiral inclination angle with increasing Re at each η through all mode transitions to an asymptotic state n=N is reported. It is found that N=2, 4, 7, 19 for η=0.2, 0.4, 0.6, 0.8, respectively, and that the asymptotic state is reached at Re≂2000 for all radius ratios. The asymptotic state is characterized by critical Taylor numbers and frequencies that approach constant values while critical wave numbers fall off as Re−1. In this same limit the counter-rotating vortex pairs align themselves axially within the annulus. A conjecture is made on the influence of η on stability in the high Reynolds number limit.

The effect of magnetic fields on the stability of a cylindrical flow with mass and heat transfer

The effect of axial and radial magnetic fields on the Kelvin-Helmholtz stability of a cylindrical interface between the vapor and liquid phases of a fluid is studied when the vapor is hotter than the liquid and the two phases are enclosed between two cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. Both axisymmetric and asymmetric disturbances are considered. The linear dispersion relations are obtained and discussed. It is found that a uniform axial magnetic field has a stabilizing effect on the interface, while the effect of a radial magnetic field depends strongly on the choice of some physical parameters of the system. It is also found that the instability criterion is independent of heat and mass transfer coefficient, but it is different fromthat in the same problem without heat and mass transfer. Finally, the heat and mass transfer has a destabilizing influence on the system.