A. Solan

Stability of confined swirling flow with and without vortex breakdown

A numerical investigation of steady states, their stability, onset of oscillatory instability, and slightly supercritical unsteady regimes of an axisymmetric swirling flow of a Newtonian incompressible fluid in a closed circular cylinder with a rotating lid is presented for aspect ratio (height/radius) 1 ≤ γ ≤ 3.5. Various criteria for the appearance of vortex breakdown are discussed. It is shown that vortex breakdown takes place in this system not as a result of instability but as a continuous evolution of the stationary meridional flow with increasing Reynolds number. The dependence of the critical Reynolds number Re cr and frequency of oscillations ω cr on the aspect ratio of the cylinder γ is obtained. It is found that the neutral curve Re cr (γ) and the curve ω cr (γ) consist of three successive continuous segments corresponding to different modes of the dominant perturbation. The calculated critical parameters are in good agreement with the available experimental and numerical data for γ < 3. It is shown that the onset of the oscillatory instability does not depend on the existence of a separation bubble in the subcritical steady state. By means of a weakly nonlinear analysis it is shown that the axisymmetric oscillatory instability sets in as a result of a supercritical Hopf bifurcation for each segment of the neutral curve. A weakly nonlinear asymptotic approximation of slightly supercritical flows is carried out. The results of the weakly nonlinear analysis are verified by direct numerical solution of the unsteady Navier-Stokes equation using the finite volume method. The analysis of the supercritical flow field for aspect ratio less than 1.75, for which no steady vortex breakdown is found, shows the existence of an oscillatory vortex breakdown which develops as a result of the oscillatory instability.

Three-dimensional instability of axisymmetric flow in a rotating lid{cylinder enclosure

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k . The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.

Steady states and oscillatory instability of swirling flow in a cylinder with rotating top and bottom

In this study we present a numerical investigation of steady states, onset of oscillatory instability, and slightly supercritical oscillatory states of an axisymmetric swirling flow of a Newtonian incompressible fluid in a cylinder, with independently rotating top and bottom. The first part of the study is devoted to the influence of co‐ and counter‐rotation of the bottom on the steady vortex breakdown, which takes place in the well‐known problem of flow in a cylinder with a rotating top. It is shown that weak counter‐rotation of the bottom may suppress the vortex breakdown. Stronger counter‐rotation may induce a stable steady vortex breakdown at relatively large Reynolds numbers for which a vortex breakdown does not appear in the case of the stationary bottom. Weak corotation may promote the vortex breakdown at lower Reynolds numbers than in the cylinder with the stationary bottom. Stronger corotation leads to the detachment of the recirculation zone from the axis and the formation of an additional vorte...

The separation of flow past a cylinder in a rotating system

The flow past a cylinder bounded by parallel planes in a rotating frame is treated in terms of a nonlinear Stewartson layer. It is shown that separation is strongly dependent on the ratio of the Rossby number to the square root of the Ekman number, covering the whole range from fully attached flow to the classical non-rotating separated flow past a cylinder. The results are in agreement with published experimental observations.

Effect of axial magnetic field on three-dimensional instability of natural convection in a vertical Bridgman growth configuration

A study of the effect of an externally imposed magnetic field on the axisymmetry-breaking instability of an axisymmetric convective flow, associated with crystal growth from bulk of melt, is presented. Convection in a vertical cylinder with a parabolic temperature profile on the sidewall is considered as a representative model. A parametric study of the dependence of the critical Grashof number Grcr on the Hartmann number Ha for fixed values of the Prandtl number (Pr=0.015) and the aspect ratio of the cylinder (A=height/radius=1, 2 and 3) is carried out. The stability diagram Grcr(Ha) correspondingto the axisymmetric }three-dimensional transition for increasingvalues of the axial magnetic field is obtained. The calculations are done using the spectral Galerkin method allowing an effective and accurate three-dimensional stability analysis. It is shown that at relatively small values of Ha the axisymmetric flow tends to be oscillatory unstable. After the magnitude of the magnetic field (Ha) exceeds a certain value the instability switches to a steady bifurcation caused by the Rayleigh–B! mechanism. # 2001 Elsevier Science B.V. All rights reserved.

Multiple states, stability and bifurcations of natural convection in a rectangular cavity with partially heated vertical walls

The multiplicity, stability and bifurcations of low-Prandtl-number steady natural convection in a two-dimensional rectangular cavity with partially and symmetrically heated vertical walls are studied numerically. The problem represents a simple model of a set-up in which the height of the heating element is less than the height of the molten zone. The calculations are carried out by the global spectral Galerkin method. Linear stability analysis with respect to two-dimensional perturbations, a weakly nonlinear approximation of slightly supercritical states and the arclength path-continuation technique are implemented. The symmetry-breaking and Hopf bifurcations of the flow are studied for aspect ratio (height/length) varying from 1 to 6. It is found that, with increasing Grashof number, the flow undergoes a series of turning-point bifurcations. Folding of the solution branches leads to a multiplicity of steady (and, possibly, oscillatory) states that sometimes reaches more than a dozen distinct steady solutions. The stability of each branch is studied separately. Stability and bifurcation diagrams, patterns of steady and oscillatory flows, and patterns of the most dangerous perturbations are reported. Separated stable steady-state branches are found at certain values of the governing parameters. The appearance of the complicated multiplicity is explained by the development of the stably and unstably stratified regions, where the damping and the Rayleigh–B´ enard instability mechanisms compete with the primary buoyancy force localized near the heated parts of the vertical boundaries. The study is carried out for a low-Prandtl-number fluid with Pr =0 .021. It is shown that the observed phenomena also occur at larger Prandtl numbers, which is illustrated for Pr = 10. Similar three-dimensional instabilities that occur in a cylinder with a partially heated sidewall are discussed.

Metal drilling with a CO2 laser beam. I. Theory

The initiation and evolution of a ‘‘keyhole’’ produced with a CO2 laser beam in a metal plate is analyzed, accounting for the following effects either exactly or by estimates of their limiting values: the spatial intensity profile of the beam, the temperature‐dependent absorptivity, attenuation of the beam intensity by the metal vapor, the exothermic reaction in the presence of oxygen, evaporative material removal and flushing of molten drops, heat conduction within the workpiece, phase transitions from solid to liquid and from liquid to gas, and heat losses at the workpiece boundaries. A study of the drilling of aluminum plates shows the relative effect of these parameters on the crater shape and penetration time. Simple criteria for initiation, penetration rate, and shift to spontaneous burning are presented. In a companion paper experimental results are analyzed to validate the formalism and to yield estimates of the interaction parameters for a specific exposure condition.

Vortex breakdown in the polar region between rotating spheres

The flow field in the polar region between a rotating and stationary, co‐ or counter‐rotating sphere was studied numerically by a steady‐state axisymmetric finite‐element code and experimentally by flow visualization using a photochromic dye excited by a UV laser. The experimental results show that at a critical Reynolds number a steady axisymmetric vortex breakdown bubble forms at the polar axis, as predicted by the computation. At higher values of the Reynolds number, and depending on the gap, the eccentricity of the spheres along the axis of rotation, and their rotation ratio, further unsteady structures are observed experimentally, viz., an axially oscillating bubble, or a three‐dimensional irregular structure.

Three-Dimensional Instabilities of Natural Convection Flow in a Vertical Cylinder With Partially Heated Sidewall

The three-dimensional axisymmetry-breaking instability of an axisymmetric convective flow in a vertical cylinder with a partially heated sidewall is studied numerically. The central part of the sidewall is maintained at constant temperature, while its upper and lower parts are thermally insulated. The dependence of the critical Grashof number on the cylinder aspect ratio (A=height/radius) is obtained for a fixed value of the Prandtl number, Pr=0.021, and fixed length of the heated central region, equal to the cylinder radius. Three different modes of the most dangerous three-dimensional perturbations, which replace each other with the variation of the aspect ratio, are found

Taylor vortex flow between eccentric coaxial rotating spheres

The rotationally symmetric, incompressible, spherical Couette flow between two spheres is examined, where the inner one rotates and the outer one is at rest. The investigation was done by means of a finite‐element program that solves the axisymmetric Navier–Stokes equations. Both concentric and eccentric spherical gaps are considered for two different radii ratios of a medium sized gap. The emphasis is laid upon the development of Taylor vortices out of the basic laminar flow, the transition between flow modes and the effect of eccentricities of different magnitudes. In the concentric case, the transition from the basic flow to a flow with two pairs of Taylor vortices is investigated by a steady as well as a transient analysis. The transition from the basic flow to a flow with one pair of Taylor vortices is examined by a steady analysis using a mesh that is asymmetric about the equator and also by introducing a geometrical perturbation in the form of a small eccentricity. A hysteresis for this transition ...