bluuebunny/crossref_metadata_2025 huggingface dataset:
The interaction between swirling and recirculating velocity components in unsteady, inviscid flow
Authors: Davidson P. A.
Published: April 2006

In this paper we consider the transient evolution of a swirling, recirculating flow in a truncated cylinder. In particular, we consider an initial time period during which the evolution of the flow is controlled by inertia. Such flows exhibit a mutual interaction between the swirl and the poloidal recirculation, whereby any axial gradient in swirl alters the recirculation, which, in turn, redistributes the swirl. This interaction may be visualized as a flexing of the poloidal vortex lines, the best known example of which is the inertial wave. Physical arguments and numerical experiments suggest that, typically, a strong, oscillatory recirculation will develop. We examine the exchange of energy between the swirl and recirculation, and show that the direction of transfer depends on the relative signs of ψ and ηuθ/ηz. In addition, there is a limit to the amount of energy that may be exchanged, since conservation of angular momentum imposes a lower bound on the kinetic energy of the swirl. The characteristic reversal time for the recirculation is estimated by considering the history of fluid particles on the endwalls. Its magnitude depends on the relative strengths of the swirl and recirculation. When the recirculation is large, the reversal time exceeds the turn-over time for a poloidal eddy and, consequently, the vortex lines accumulate at the stagnation points on the endwalls. This leads to accelerated local diffusion on the axis. An elementary one-parameter model is proposed for these nonlinear oscillations. In the limit of very weak recirculation, this model is consistent with the exact solution for inertial waves, while for strong recirculation, it confirms that the reversal time is greater than the turn-over time, and that the vortex lines accumulate on the axis.
Nonlinear Faraday resonance in a box with a square base
Authors: Nagata M.
Published: April 2006

Surface wave motions in a container with a square base, which is subject to a vertical oscillation, are considered when the amplitude of the oscillation is small and the frequency of the oscillation is close to twice the natural frequency of the system. Subcritical wave motions are found for single modes as well as mixed modes. Here, single modes are described by either one of the two horizontal coordinates whereas mixed modes depend on both coordinates. It is found that in some subcritical region a stable single mode and a stable mixed mode coexist, producing complex basins of attraction.
A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry
Authors: Bodonyi R. J. , Welch W. J. C. , Duck P. W. , Tadjfar M.
Published: April 2006

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.
Inertia effects on the motion of long slender bodies
Authors: Khayat R. E. , Cox R. G.
Published: April 2006

A solid long slender body with curved centreline is held at rest in a fluid undergoing a uniform flow. Assuming that the Reynolds number Re based on body length is fixed, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio κ of the cross-sectional radius to body length. In the limit of large Re, this result is no longer valid and an asymptotic expansion in κRe is necessary. A uniformly valid solution is obtained from these two expansions. The total force and torque acting on a body with a straight centreline are explicitly determined. The limiting cases of small and large Re are studied in detail.
Experiments on mixing due to chaotic advection in a cavity
Authors: Leong C. W. , Ottino J. M.
Published: April 2006

Chaotic mixing of fluids in slow flows is ubiquitous but incompletely understood. However, relatively simple experiments provide a wealth of information regarding mixing mechanisms and indicate the need for complementary theoretical developments in dynamical systems. In this work we presnt a versatile cavity flow apparatus, capable of producing a variety of two-dimensional velocity fields, and use it to conduct a detailed experimental study of mixing in low-Reynolds-number flows. Since the goal is detailed understanding, only two time-periodic co-rotating flows induced by wall motions are considered: one continuous and the other discontinuous. Both types of flows produce exponential growth of intermaterial area, as expected from chaotic flows, and a mixture of islands and chaotic regions. A procedure for identifying periodic points and determining their movements is presented as well as how to make meaningful comparisons between periodic flows. We observe that periodic points move very much as a planetary system; planets (hyperbolic points) have moons (elliptic points) with twice the period of the planets; furthermore the spatial arrangement of periodic points becomes symmetric at regular time intervals. Detailed analyses reveal complex behaviour: birth, bifurcation, and collapse of islands; formation and periodic motion of coherent structures, such as islands and large-scale folds. However, the richness and complexity of the results obtained indicate that these two-dimensional time-periodic systems are far from completely understood and that other wall motions might deserve a similar level of scrutiny.
The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres
Authors: Barnocky Guy , Davis Robert H.
Published: April 2006

A theoretical description of the low-Reynolds-number collision and rebound of two rigid or elastic spheres separated by a thin layer of viscous fluid with pressure-dependent physical properties is presented. It has previously been shown by Davis et al. (1986) that the hydrodynamic pressure which builds up in the thin fluid layer must become large enough to elastically deform the spheres near the axis of symmetry, if they are to rebound subsequent to colliding. Under these extreme pressures, however, it is expected that the fluid may also compress and that its viscosity may increase by several orders of magnitude. It is shown that these pressure-dependent effects may significantly alter the minimum separation reached during approach of the spheres, as well as the maximum separation and relative velocity attained during rebound of the spheres. In particular, the pressure buildup during the collision process is predicted to become sufficiently large under some conditions so that the corresponding viscosity increase causes the fluid in the gap between the colliding spheres to behave nearly as a solid and to limit the close approach of the opposing surfaces. Also, the storage of energy via the compression of the fluid in the gap allows rigid spheres to bounce as this energy is released subsequent to their collision. However, it is found that this rebound is very weak relative to that which is predicted for elastic spheres.
Growth and translation of a liquid-vapour compound drop in a second liquid. Part 2. Heat transfer
Authors: Vuong S. T. , Sadhal S. S.
Published: April 2006

The present work is a comprehensive theoretical study of the heat transfer associated with a 3-singlet compound drop that is growing because of change of phase. The geometry is the same as in Part 1, i.e. a vapour bubble partially surrounded by its own liquid in another immiscible liquid. The attempt here is to gain fundamental understanding of the transport processes that take place in connection with direct-contact heat exchange. The fluid dynamics associated with its growth and translation is treated in Part 1. Here, that flow field solution is used to obtain the temperature field and hence the evaporation rate. The energy equation for the system consisting of a single compound drop is solved numerically by finite-difference methods. The results give the complete time history of evaporation of the drop. In addition, useful quantities such as the Nusselt number are given and compared with existing experimental data. Most of the results have good agreement with experimental data.
The transition from two-dimensional to three-dimensional planforms in infinite-Prandtl-number thermal convection
Authors: Travis Bryan , Olson Peter , Schubert Gerald
Published: April 2006

The stability of two-dimensional thermal convection in an infinite-Prandtl-number fluid layer with zero-stress boundaries is investigated using numerical calculations in three-dimensional rectangles. At low Rayleigh numbers (Ra< 20000) calculations of the stability of two-dimensional rolls to cross-roll disturbances are in agreement with the predictions of Bolton & Busse for a fluid with a large but finite Prandtl number. Within the range 2 × 104 2.22 (aspect ratio < 1.41) are stable solutions. Two-dimensional rolls with basic wavenumber α < 1.96 (aspect ratio > 1.6) are time dependent forRa> 4 × 104. For every case in which the initial condition was a time-dependent large-aspect-ratio roll, two-dimensional convection was found to be unstable to three-dimensional convection. Time-dependent rolls are replaced by either bimodal or knot convection in cases where the horizontal dimensions of the rectangular box are less than twice the depth. The bimodal planforms are steady states forRa[les ] 105, but one case atRa= 5 × 105exhibits time dependence in the form of pulsating knots. Calculations atRa= 105in larger domains resulted in fully three-dimensional cellular planforms. A steady-state square planform was obtained in a 2.4 × 2.4 × 1 rectangular box. started from random initial conditions. Calculations in a 3 × 3 × 1 box produced steady hexagonal cells when started from random initial conditions, and a rectangular planform when started from a two-dimensional roll. An hexagonal planform started in a 3.5 × 3.5 × 1 box atRa= 105exhibited oscillatory time dependence, including boundary-layer instabilities and pulsating plumes. Thus, the stable planform in three-dimensional convection is sensitive to the size of the rectangular domain and the initial conditions. The sensitivity of heat transfer to planform variations is less than 10%.
Numerical simulation of liquid-metal MHD flows in rectangular ducts
Authors: Sterl A.
Published: April 2006

To design self-cooled liquid metal blankets for fusion reactors, one must know about the behaviour of MHD flows at high Hartmann numbers. In this work, finite difference codes are used to investigate the influence of Hartmann number M, interaction parameter N, wall conductance ratio c, and changing magnetic field, respectively, on the flow.

As liquid-metal MHD flows are characterized by thin boundary layers, resolution of these layers is the limiting issue. Hartmann numbers up to 103 are reached in the two-dimensional case of fully developed flow, while in three-dimensional flows the limit is 102. However, the calculations reveal the main features of MHD flows at large M. They are governed by electric currents induced in the fluid. Knowing the paths of these currents makes it possible to predict the flow structure.

Results are shown for two-dimensional flows in a square duct at different Hartmann numbers and wall conductivities. While the Hartmann number governs the thickness of the boundary layers, the wall conductivities are responsible for the pressure losses and the structure of the flows. The most distinct feature is the side layers where the velocities can exceed those at the centre by orders of magnitude.

The three-dimensional results are also for a square duct. The main interest here is to investigate the redistribution of the fluid in a region where the magnetic field changes. Large axial currents are induced leading to the ‘M-shaped’ velocity profiles characteristic of MHD flow. So-called Flow Channel Inserts (FCI), of great interest in blanket design, are investigated. They serve to decouple the load carrying wall from the currents in the fluid. The calculations show that the FCI is indeed a suitable measure to reduce the pressure losses in the blanket.
Autorotation of rectangular plates
Authors: Skews B. W.
Published: April 2006

A series of tests have been conducted on the autorotation of plates of rectangular section and thickness to chord ratios of from 0.1 to 1.0. The major difference from previous work was that the plates spanned the full width of the wind tunnel, i.e. the flow was essentially two-dimensional. The results show major differences from predictions of infinite aspect ratio plates inferred from finite aspect ratio tests as given by Iversen (1979). Moreover, they give excellent correlation with the two-dimensional numerical solution of Lugt (1980) for a 10% thick plate. In contrast to previous results which indicate no autorotation for square cylinders (t/c = 1.0), it is found that autorotation is easily achieved. A new result is that tipspeed ratios are found to be independent of thickness ratio, at approximately the Lugt value over the full range of thickness ratios. Drag coefficients are found to be independent of thickness ratio. There appears to be a critical thickness ratio above and below which the lift coefficients are constant, but are of different magnitude.
Capillary–gravity Kelvin–Helmholtz waves close to resonance
Authors: Bontozoglou V. , Hanratty T. J.
Published: April 2006

Capillary–gravity waves of permanent form at the interface between two unbounded fluids in relative motion are considered. The range of wavelengths for an internal resonance with the second harmonic and a period-doubling bifurcation are found to depend on the current speed. The Kelvin–Helmholtz instability of short waves becomes strongly subcritical near resonance. It is speculated that this instability is needed to trigger a period-doubling bifurcation. This notion is used to explain the development of waves at short fetch and the initiation of liquid slugs for gas–liquid flow in a horizontal pipe.
Radial migration of a single particle in a pore by the resistive pulse and the pressure reversal technique
Authors: Berge Lars Inge
Published: April 2006

Radial migration (particle motion transverse to streamlines) in a system which combines frequent entry, straight pore, and exit regions were investigated experimentally for small particle and pore sizes. Pore diameters were less than 30 μm and typical particle to pore diameter ratios were about 0.25. Our new modification of the resistive pulse technique based on pressure reversal, extends this experimental technique to also include single-particle flow dynamics. By pressure drive, a particle in an electrolyte enters a current-carrying pore and an increase in resistance proportional to the particle volume is detected. When the particle exits the pore, the pressure can be reversed such that the particle re-enters the pore. Detailed studies of particle flow properties in a size range relevant to flow in porous media is now possible. The emphasis in this investigation is on radial migration. The effect of particle sedimentation has been negligible, while particle diffusion becomes significant for submicron particles. The measured evolution of the transit time as the particle migrates compares well with an empirical relationship for the migration velocity first proposed by Segré & Silberberg (1962 a, b) and later verified by Ishii & Hasimoto (1980). Entrance and exit effects do not seem to be important for long pores, the results scale very nicely when the pore length is changed.
Heating a salinity gradient from a vertical sidewall: nonlinear theory
Authors: Kerr Oliver S.
Published: April 2006

When a body of fluid with a vertical salinity gradient is heated from a vertical sidewall instabilities are sometimes observed. The linear stability of this basic state has been investigated by Kerr (1989). This linear theory predicts the onset of instability well when compared with experimental results; however, the form of the observed nonlinear instabilities does not coincide with the linear predictions (cf. Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986; Tanny & Tsinober 1988). In this paper we investigate some of the nonlinear aspects of the problem. A weakly nonlinear analysis reveals that the bifurcation into instability is subcritical, and that the initial trend along this branch of solutions is towards the co-rotating cells observed in experiments. The heating levels for which instabilities are absent are investigated by the use of energy stability analysis. This yields a weak result for arbitrary disturbances, showing that disturbances will decay for sufficiently low wall heating. This bound is greatly strengthened by imposing a vertical periodicity on the lengthscale proposed by Chen et al.
Harbour excitations by incident wave groups
Authors: Wu Jiang-Kang , Liu Philip L.-F.
Published: April 2006

By using the multiple-scales perturbation method, analytical solutions are obtained for the second-order low-frequency oscillations inside a rectangular harbour excited by incident wave groups. The water depth is a constant. The width of the harbour entrance is of the same order of magnitude as the wavelength of incident carrier (short) waves, but small in comparison with the wavelength of the wave envelope. Because of the modulations in the wave envelope, a second-order long wave is locked in with the wave envelope and propagates with the speed of the group velocity. Outside the harbour, locked long waves also exist in the reflected wave groups, but not in the radiated wave groups. Inside the harbour, the analytical expressions for the locked long waves are obtained. Owing to the discontinuity of the locked long waves across the harbour mouth, second-order free long waves are generated. The free long waves propagate with a speed of (gh)½ inside and outside the harbour. The free long waves inside the harbour may be resonated in a low-frequency range which is relevant to the harbour resonance.
The instability of a moving viscous drop
Authors: Pozrikidis C.
Published: April 2006

The deformation of a moving spherical viscous drop subject to axisymmetric perturbations is considered. The problem is formulated using two different variations of the boundary integral method for Stokes flow, one due to Rallison & Acrivos, and the other based on an interfacial distribution of Stokeslets. An iterative method for solving the resulting Fredholm integral equations of the second kind is developed, and is implemented for the case of axisymmetric motion. It is shown that in the absence of surface tension, a moving spherical drop is unstable. Prolate perturbations cause the ejection of a tail from the rear of the drop, and the entrainment of a thin filament of ambient fluid into the drop. Oblate perturbations cause the drop to develop into a nearly steady ring. The viscosity ratio plays an important role in determining the timescale and the detailed pattern of deformation. Filamentation of the drop emerges as a persistent but secondary mechanism of evolution for both prolate and oblate perturbations. Surface tension is not capable of suppressing the growth of perturbations of sufficiently large amplitude, but is capable of preventing filamentation.
Time-depeiident and time-averaged turbulence structure near the nose of a wing-body junction
Authors: Devenport William J. , Simpson Roger L.
Published: April 2006

The behaviour of a turbulent boundary layer on a flat surface as it encounters the nose of a cylindrical wing mounted normal to that surface is being investigated. A three-component laser anemometer has been developed to measure this highly turbulent three-dimensional flow. Measurements of all the non-zero mean-velocity and Reynolds-stress components have been made with this instrument in the plane of symmetry upstream of the wing. These data have been used to estimate some of the component terms of the turbulence kinetic energy equation. Histograms of velocity fluctuations and short-time cross-correlations between the laser anemometer and a hot-wire probe have also been measured in the plane of symmetry. In all, these results reveal much of the time-dependent and time-averaged turbulence structure of the flow here.

Separation occurs in the plane of symmetry because of the adverse pressure gradient imposed by the wing. In the time mean the resulting separated flow consists of two fairly distinct regions: a thin upstream region characterized by low mean backflow velocities and a relatively thick downstream region dominated by the intense recirculation of the mean junction vortex. In the upstream region the turbulence stresses develop in a manner qualitatively similar to those of a two-dimensional boundary layer separating in an adverse pressure gradient. In the vicinity of the junction vortex, though, the turbulence stresses are much greater and reach’ values many times larger than those normally observed in turbulent flows. These large stresses are associated with bimodal (double-peaked) histograms of velocity fluctuations produced by a velocity variation that is bistable. These observations are consistent with large-scale low-frequency unsteadiness of the instantaneous flow structure associated with the junction vortex. This unsteadiness seems to be produced by fluctuations in the momentum and vorticity of fluid from the outer part of the boundary layer which is recirculated as it impinges on the leading edge of the wing. Though we would expect these fluctuations to be produced by coherent structures in the boundary layer, frequencies of the large-scale unsteadiness are substantially lower than the passage frequency of such structures. It therefore seems that only a fraction of the turbulent structures are recirculated in this way.
The decay of turbulence in thermally stratified flow
Authors: Lienhard V J. H. , Van Atta C. W.
Published: April 2006

The decay of grid-generated turbulence in the presence of strong thermal stratification is studied in a continuously stratified, open-loop wind tunnel at Brunt–Väisälä frequencies up to 2.5s−1. The data include one-point statistical measurements through moments of fourth order and associated power- and cross-spectra. Cross-channel phase measurements are used to analyse the scales of correlation of velocity and temperature. The present data are considerably more coherent than previous salt-stratified data, and the structural form of stratified turbulence is thus more clearly manifested. No internal wave effects are observed at any stage of the decay. Stratified turbulence is found to be a two-scale process dominated by buoyancy forces at large scales of motion and dissipative effects at small scales. The two-scale structure is used to develop universal buoyancy scalings for the decay of the vertical heat flux, the scalar variance, and the molecular dissipation rates, and, in particular, for the vertical velocity decay. Velocity and temperature spectra satisfy universal equilibrium scaling at high wavenumbers, but show buoyancy effects at small wavenumbers. The flow remains isotropic at high wavenumbers over the entire range of turbulent decay studied. Cospectral and phase data are used to validate the two-scale model of the turbulence. The flow may show large-scale restratification while active turbulence persists at smaller scales, so that the vanishing of the vertical transport does not represent extinction of turbulent motion. Additionally, an original universal equilibrium scaling is developed for the cross-spectrum. Lengthscale evolution is measured, and the overturning and buoyancy lengthscales (associated with potential and kinetic energy, respectively) are found to characterize flow development. The role of the Prandtl number is assessed by comparison to previous works, and the Prandtl number is found to have a significant influence upon stratified turbulence evolution.
Numerical simulation of interactions between Görtler vortices and Tollmien–Schlichting waves
Authors: Malik M. R. , Hussaini M. Y.
Published: April 2006

The problem of nonlinear development of Gortler vortices and interaction with Tollmien–Schlichting waves is considered within the framework of incompressible Navier–Stokes equations which are solved by a Fourier–Chebyshev spectral method., It is shown that two-dimensional waves can be excited in the flow modulated by Gortler vortices. Owing to nonlinear effects, this interaction further leads to the development of oblique waves with spanwise wavelength equal to the Görtler vortex wavelength. Interaction is also considered of oblique waves with spanwise wavelength twice that of Gortler vortices.
The stability of elliptical vortices in an external straining flow
Authors: Dritschel David G.
Published: April 2006

Subject to uniform strain, an elliptical patch of vorticity in an in viscid, incompressible, two-dimensional fluid generally rotates or nutates and extends or compresses while retaining a precisely elliptical shape (the Kida solutions). This result is of interest because the uniform strain idealizes the leading-order distortional influence of distant vortices in a flow with many vortices. Because of the unsteady motion of the distant vortices, both the strain rate and the rotation rate of the strain axes typically vary with time. In the special case that the strain rate and rotation rate are steady, and when the strain rate is not too large, periodic motion of an elliptical vortex is possible. Larger strain rates lead to indefinite extension of the vortex.

Uniform strain, however, only approximately mimics the effect of distant vortices. The local variations- in the strain field around a vortex disturb the vortex, preventing it from retaining a simple, elliptical shape. These disturbances may amplify because of instabilities. In this paper, we examine the stability of periodic elliptical motion to small boundary disturbances, for the case of steady, uniform strain and rotation rate, first by linear Floquet theory and then by direct, high-resolution, nonlinear numerical integrations. It is discovered that a significant portion of the periodic solutions are linearly unstable. Instability can occur even when the strain rate is arbitrarily small and the basic motion arbitrarily close to circular. Extended nonlinear calculations exhibit recurrence, in some cases, and attrition of the vortex by repeated wave amplification, steepening, and breaking in others.
Periodic flows through curved tubes: the effect of the frequency parameter
Authors: Hamakiotes Costas C. , Berger Stanley A.
Published: April 2006

In a previous paper we reported on the effect of Dean number, κm, on the fully developed region of periodic flows through curved tubes. In this paper we again consider a sinusoldally varying volumetric flow rate in a curved pipe of arbitrary curvature ratio, δ, and investigate the effect of frequency parameter α, and Reynolds number Rem on the flow. Specifically, we report on the flow-field development for the range 7.5 [les ] α [les ] 25, and 50 [les ] Rem [les ] 450. The results, obtained by numerical integration of the full Navier–Stokes equations, reveal a number of characteristics of the flow previously unreported. For low values of Rem the secondary flow consists of a single vortex (Dean-type motion) in the half-cross-section at all times and for all values of α studied. For higher Rem we observe inward ‘centrifuging’ (Lyne-type motion) at the centre. This motion always occurs during the accelerating period of the volumetric flow rate. It appears at lower α for higher Rem and, for the given Rem at which it appears, it occurs at earlier times in the cycle for lower a. A striking feature is observed for α = 15 for the range 315 [les ] Rem [les ] 400: period tripling. The flow field varies periodically with time for the duration of three volumetric-flow-rate cycles then repeats for the subsequent three cycles, and so on. The computed axial pressure gradient also varies periodically with time but with the same period as the volumetric flow rate.
Streaming from a sphere due to a pulsating source
Authors: Amin Norsarahaida , Riley N.
Published: April 2006

The steady streaming outside the Stokes shear-wave layer, which forms on the surface of a sphere when placed close to an oscillatory point source, is considered. Particular attention is devoted to the case of high streaming Reynolds-number flow. Thin circular jets, analogous to the plane jets known to occur in two-dimensional flow, are predicted and visualized by means of a simple experiment.
Frictional–collisional equations of motion for participate flows and their application to chutes
Authors: Johnson P. C. , Nott P. , Jackson R.
Published: April 2006

Measurements of the relation between mass hold-up and flow rate have been made for glass beads in fully developed flow down an inclined chute, over the whole range of inclinations for which such flows are possible. Velocity profiles in the flowing material have also been measured. For a given inclination it is found that two different flow regimes may exist for each value of the flow rate in a certain interval. One is an ‘energetic’ flow, and is produced when the particles are dropped into the chute from a height, while the other is relatively quiescent and occurs when entry to the chute is regulated by a gate. At some values of the inclination jumps in the flow pattern occur between these branches, and it is even possible for both branches to coexist in the same chute, separated by a shock. A theoretical treatment of chute flow has been based on a rheological model of the material which takes into account both collisional and fractional mechanisms for generating stress. Its predictions include most aspects of the observed behaviour, but quantitative comparison of theory and experiment is difficult because of the uncertain values of some parameters appearing in the theory.
Axisymmetric creeping motion of drops through circular tubes
Authors: Martinez M. J. , Udell K. S.
Published: April 2006

The axisymmetric creeping motion of a neutrally buoyant deformable drop flowing through a circular tube is analysed with a boundary integral equation method. The fluids are immiscible, incompressible, and the bulk flow rate is constant. The drop to suspending fluid viscosity ratio is arbitrary and the drop radius varies from 0.5 to 1.15 tube radii. The effects of the capillary number, viscosity ratio, and drop size on the deformation, the drop speed, and the additional pressure loss are examined.

Drops with radius ratios less than 0.7 are insensitive to substantial variation in capillary number and viscosity ratio, and computed values of drop speed and extra pressure loss are in excellent agreement with small deformation theories (Hestroni et al. 1970; Hyman & Skalak 1972a). For this drop size range, significant deformation will result only for Ca > 0.25. The onset of a re-entrant cavity is predicted at the trailing end of the drop for Ca ≈ 0.75. Drop speed and meniscus shape become independent of drop size for radius ratios as small as 1.10. The extra pressure loss can be positive or negative depending mainly on the viscosity ratio, however a relatively inviscid drop can cause a positive extra pressure loss when capillary forces are significant. Computed values for extra pressure loss and drop speed are in good agreement with the experimental data of Ho & Leal (1975) for drops of sizes comparable with the tube radius.
Attenuation and dispersion of sound in bubbly fluids via the Kramers—Kronig relations
Authors: Temkin S.
Published: April 2006

Sound propagation in a dilute bubble–liquid mixture is studied by means of the Kramers–Kronig relationships, which relate the real and imaginary parts of the general susceptibility of a linear medium. These relationships are adopted for the case of acoustic waves, where they become coupled integral equations. A simple but approximate procedure is used to obtain from these equations the phase speed of sound waves for the case when the attenuation coefficient is independently known. The procedure can be used to obtain the speed of propagation of sound waves in acoustic media having internal dissipation, but is here applied only to fluids containing radially pulsating bubbles. Approximate results for the speed of propagation and for the attenuation per wavelength are obtained for this case on the basis of a first-order estimate for the attenuation coefficient. These results are the same as those derived previously on the basis of model equations for bubbly liquids. They therefore provide additional support for those equations, while indicating some of their limitations.
Parallel flows with Soret effect in tilted cylinders
Authors: Jacqmin David
Published: April 2006

Henry & Roux have recently conducted extensive numerical studies on the interaction of Soret separation with convection in cylindrical geometry. Many of their solutions exhibit parallel flow away from endwalls. We show that their parallel flow results can be matched by closed-form solutions. We find that solutions are non-unique in some parameter regions. Disappearance of one branch of solutions correlates with a sudden transition of Henry & Roux's results from a separated to a well-mixed flow.
Spreading and instability of a viscous fluid sheet
Authors: Hocking L. M.
Published: April 2006

Experiments by Huppert (1982) have demonstrated that a finite volume of fluid placed on an inclined plane will elongate into a thin sheet of fluid as it slides down the plane. If the fluid is initially placed uniformly across the plane, the sheet retains its two-dimensionality for some time, but when it has become sufficiently long and thin, the leading edge develops a spanwise instability. A similarity solution for this motion was derived by Huppert, without taking account of the edge regions where surface tension is important. When these regions are examined, it is found that the conditions at the edges can be satisfied, but only when the singularity associated with the moving contact line is removed. When the sheet is sufficiently elongated, the profile of the free surface shows an upward bulge near the leading edge. It is suggested that the observed instability of the shape of the leading edge is a result of the dynamics of the fluid in this bulge. The related problem of a ridge of fluid sliding down the plane is examined and found to be linearly unstable. The spanwise lengthscale of this instability is, however, dependent on the width of the channel occupied by the fluid, which is at variance with the observed nature of the instability. Preliminary numerical solutions for the nonlinear development of a small disturbance to the position of a straight leading edge show the incipient development of a finger of fluid with a width that does not depend on the channel size, in accordance with the observations.
A note on integral properties of periodic gravity waves in the case of a non-zero mean Eulerian velocity
Authors: Klopman G.
Published: April 2006

Corrected relationships are presented between integral properties of periodic gravity surface waves in the case of a non-zero mean Eulerian velocity. Also considered is the special case of a reference frame in which the mean wave momentum vanishes.
Knotted vortex filaments in an ideal fluid
Authors: Keener J. P.
Published: April 2006

Knotted closed-curve solutions of the equation of self-induced vortex motion are studied. It is shown that there are invariant torus knots which translate and rotate as rigid bodies. The general motion of ‘small-amplitude’ torus knots and iterated (cabled) torus knots is described and found to be almost periodic in time, and for some, but not all, initial data, the topology of the knot is shown to be invariant.
Dynamic compression and weak shock formation in an inert gas due to fast piston acceleration
Authors: Wang Meng , Kassoy D. R.
Published: April 2006

Unsteady gasdynamic concepts are used to model the piston-driven compression of a confined gas. Perturbation methods, based on the limit of small piston Mach number, are used to construct solutions. The piston Mach number increases smoothly from zero to a maximum value, Mp = O(10−2) during an acoustic time period ta* = O(10−4 s). A linear a coustic field is generated and is represented in terms of an infinite series of Fourier spatial modes. During the longer piston time period tp* = O(10−2 s) the piston moves at constant speed. A multiple-timescale formulation is used to separate the instantaneous acoustic field from the accumulated bulk response of the gas to piston compression. The latter is found to be identical to the classical quasi-static results from equilibrium thermodynamic calculations. Nonlinear effects become important on the piston timescale. Modal interactions are represented by a system of coupled, nonlinear ordinary differential equations for the time-dependent Fourier coefficients. A numerical solution for this system describes the wavefront steepening to form a weak shock and its propagation back and forth repeatedly inside the cylinder.
The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances
Authors: Breuer Kenneth S. , Landahl Marten T.
Published: April 2006

Navier–Stokes calculations were performed to simulate the evolution of a moderate-amplitude localized disturbance in a laminar flat-plate boundary layer. It was found that, in accordance with previous results for linear and weakly nonlinear disturbances, the evolving disturbance consists of two parts: an advective, or transient portion which travels at approximately the local mean velocity, and a dispersive wave portion which grows or decays according to Tollmien–Schlichting instability theory. The advective portion grows much more rapidly than the wave portion, initially linearly in time and, in contrast to the weak-disturbance case, gives rise to two distinct nonlinear effects. The first is a streamwise growth of the disturbed region producing a low-speed streak, bounded in the vertical and spanwise directions by intense shear layers. The second nonlinear effect is the onset of a secondary instability on the vertical shear layer formed as a result of spanwise stretching of the mean vorticity and giving rise to oscillations in the v- and w-components with a substantially smaller spatial scale than that of the initial disturbance. The effect of initial spanwise scale is assessed by calculating the disturbance for three different cases in which the spanwise scale and the initial disturbance amplitude were varied. It was found that the resulting perturbation depends primarily on the initial distribution of v in each plane z = const., but is approximately independent of the spanwise scale.
Flow past cylindrical obstacles on a beta-plane
Authors: Page Michael A. , Johnson E. R.
Published: April 2006

The flow past a cylindrical obstacle in an enclosed channel is examined when the entire configuration is rotating rapidly about an axis which is aligned with that of the obstacle. When viewed from a frame of reference which is rotating with the channel, Coriolis forces dominate and act to constrain the motion to be two-dimensional. The channel is considered to have depth varying linearly across its width, producing effects equivalent to the so-called β-plane approximation and permitting waves to travel away from the obstacle, both upstream and downstream. For the eastward flow considered in this paper, this leads to the formation of a lee-wavetrain downstream of the obstacle and, under some conditions, a region of retarded, or ‘blocked’, flow upstream of the obstacle. The flow regime studied is essentially inviscid, although one form of frictional effect on the flow, introduced through the Ekman layers, is included. The properties of this system are examined numerically and compared with the theoretical predictions from other studies, which are applicable in asymptotic limits of the parameters. In particular, the relevance of ‘Long's model’ solutions is considered.
Numerical study of elliptical modons using a spectral method
Authors: Boyd John P. , Ma Hong
Published: April 2006

We study the relationship between dynamical structure and shape for vortex pairs, now usually named ‘modons’. When the boundary between the exterior irrotational flow and the inner core of non-zero vorticity is a circle, an analytical solution is known. Here, we generalize the circular modons to solitary vortex pairs whose vorticity boundary is an ellipse. We find that as the eccentricity of the ellipse increases, the vorticity becomes concentrated in narrow ridges which run just inside the elliptical vorticity boundary and continue just inside the line of zero vorticity which divides the two vortices. Each vortex becomes increasingly ‘hollow’ in the sense that each contains a broad valley of low vorticity which is completely enclosed by the ridge of high vorticity already described. The relationship between vorticity ζ and streak function Ψ, which is linear for the circular modons, becomes strongly nonlinear for highly eccentric modons, qualitatively resembling ζ ∝ Ψe−λΨ for some constant λ. In this study, we neglect the Earth's rotation, but our method is directly applicable to quasi-geostrophic modons, too. An efficient and simple spectral method for modon problems is provided.
Modelling the pressure–strain correlation of turbulence: an invariant dynamical systems approach
Authors: Speziale Charles G. , Sarkar Sutanu , Gatski Thomas B.
Published: April 2006

The modelling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - appears to yield improved results over the Launder, Reece & Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in five independent homogeneous turbulent flows. However, some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modelling of the pressure–strain correlation. Possible alternative approaches are discussed briefly.
Islands in three-dimensional steady flows
Authors: Hegna C. C. , Bhattacharjee A.
Published: April 2006

We consider the problem of steady Euler flows in a torus. We show that in the absence of a direction of symmetry the solution for the vorticity contains δ-function singularities at the rational surfaces of the torus. We study the effect of a small but finite viscosity on these singularities. The solutions near a rational surface contain cat's eyes or islands, well known in the classical theory of critical layers. When the islands are small, their widths can be computed by a boundary-layer analysis. We show that the islands at neighbouring rational surfaces generally overlap. Thus, steady toroidal flows exhibit a tendency towards Beltramization.
The nonlinear growth of surface-tension-driven instabilities of a thin annular film
Authors: Johnson Mark , Kamm Roger D. , Ho Lee Wing , Shapiro Ascher , Pedley T. J.
Published: April 2006

The stability and initial growth rate of disturbances on an annular film lining a cylindrical tube have been the focus of several previous works. The further development of thsse disturbances as they grow to form stable unduloids or liquid bridges is investigated by means of a thin-film integral model. The model is compared both with perturbation theories for early times, and a numerical solution of the exact equations (NEKTON) for later times. The thin-film model gave results that were in good agreement with solutions of the exact equations. The results show that linear perturbation theory can be used to give good estimates of the times for unduloid and liquid bridge formation. The success of the model derives from the dominant influence of narrow draining regions that feed into the growing unduloid, and these regions remain essentially one-dimensional throughout the growth of the instability.

The model is used to analyse the evolution of the liquid layer lining the small airways of the lung during a single breath. The timescales for formation of unduloids and liquid bridges are found to be short enough for the liquid layer to be in a virtually quasi-equilibrium state throughout the breathing cycle. This conclusion is only tentative, however, because the model assumes that the surface tension of the airway liquid lining does not change with changes in interfacial area despite the known presence of pulmonary surfactant.
Vortex shedding from an impulsively started rotating and translating circular cylinder
Authors: Chang Chien-Cheng , Chern Ruey-Ling
Published: April 2006

A numerical study is made of the flow past an impulsively started rotating and translating circular cylinder using a hybrid vortex method. The Reynolds number (Re) ranges from 103 to 106 while the (counter-rotating) rotating-to-translating speed ratio (α) is increased from 0 to 2. It is found that three basic patterns of vortex shedding can be identified according to the behaviour of the stagnation points associated with the first upper and the first lower vortices. Depending on the parameters Re and α, the rotation may favour the shedding of the first upper vortex, or the first lower vortex (typically at high Reynolds numbers). In a transition region, strong competition for shedding exists between the first two vortices in the form of double transposition of stagnation (closure) points associated with the two vortices. Time variations of lift coefficients characterize different shedding patterns; the cylinder may first experience a substantial maximal downward lift when the first shedding vortex is from the upper wake, or a maximal upward lift otherwise.
Some characteristics of small-scale turbulence in a turbulent duct flow
Authors: Antonia R. A. , Kim J. , Browne L. W. B.
Published: April 2006

The fine-scale structure of turbulence in a fully developed turbulent duct flow is examined by considering the three-dimensional velocity derivative field obtained from direct numerical simulations at two relatively small Reynolds numbers. The magnitudes of all mean-square derivatives (normalized by wall variables) increase with the Reynolds number, the increase being largest at the wall. These magnitudes are not consistent with the assumption of local isotropy except perhaps near the duct centre-line. When the assumption of local isotropy is relaxed to one of local axisymmetry, or invariance with respect to rotation about a coordinate axis (here chosen in the streamwise direction), satisfactory agreement is indicated by the data outside the wall region. Support for axisymmetry is demonstrated by anisotropy invariant maps of the dissipation and vorticity tensors. The departure from axisymmetry does not appear to be affected by the Reynolds number. Expressions are proposed for approximations to the average energy dissipation and components of the mean-square vorticity. These proposals should allow these quantities to be measured accurately, at least in the present flow.