bluuebunny/crossref_metadata_2025 huggingface dataset:
Propagating Holmboe waves at the interface between two immiscible fluids
Authors: Pouliquen O. , Chomaz J. M. , Huerre P.
Published: April 2006

The instability of the surface of discontinuity separating two streams of immiscible constant-density fluids is studied experimentally and theoretically near onset when surface tension effects are significant. Following Thorpe's original idea, a tube filled with two immiscible fluids is tilted at an angle and returned to its horizontal position to produce a nearly constant velocity difference between both streams that can be varied continuously across threshold. In order to control the wavenumber near onset, the flow is spatially forced by periodically distributing small obstacles on the upper side of the tank. When the kinematic viscosities of each fluid are nearly equal, ones observes two counter-propagating waves of equal amplitude, which cannot be explained from a vortex sheet model. A linear stability analysis of a density discontinuity embedded within a piecewise-linear velocity profile demonstrates that such waves are Holmboe modes associated with the diffusive layers above and below the interface. Good agreement is obtained between the measured and predicted values of the critical velocity difference, propagation velocity and growth factors of the waves. The instability analysis of the asymmetric velocity profile reveals that the breaking of reflectional symmetry gives rise to a single propagating wave near onset. When the kinematic viscosities of each fluid differ, the first destabilized wave is observed to propagate in the same direction as the less-viscous fluid, in agreement with the theoretical results, and the dominant direction of propagation can be manipulated by adjusting the viscosities accordingly.
Acoustic radiation pressure on a compressible sphere in a viscous fluid
Authors: Doinikov Alexander A.
Published: April 2006

The acoustic radiation pressure exerted by a plane — progressive or standing — sound wave on a compressible sphere suspended freely in a viscous fluid is calculated. In deriving the general expression for the radiation pressure, it is supposed that the radius of the sphere is arbitrary. Two limiting cases of interest are then considered. In the first of these, it is assumed that the sound wavelength is much larger than the radius of the sphere which is, in turn, much larger than the viscous wavelength, it being supposed that this condition is satisfied both outside and inside the sphere. In the second case, the situation is investigated when the radius of the sphere is small compared with the viscous wavelength which is, in turn, much smaller than the sound wavelength, it being supposed that this condition is satisfied, as before, both outside and inside the sphere. It is shown that in both cases the expressions for the radiation pressure are drastically different from the well-known expressions for the radiation pressure in a perfect fluid: the calculation of the radiation pressure from the formulae obtained for a perfect fluid in the cases when the effect of viscosity is not negligible gives both quantitatively and qualitatively wrong results.
Multipolar vortices in two-dimensional incompressible flows
Authors: Morel Yves G. , Carton Xavier J.
Published: April 2006

In a two-dimensional incompressible fluid, the barotropic instability of isolated circular vortices can lead to multipole formation. The multipoles we study here are composed of a core vortex surrounded by two or more identical satellite vortices, of opposite-sign vorticity to the core, and the total circulation is zero. First, we present the generation of multipoles from unstable piecewise-constant monopoles perturbed on a monochromatic azimuthal mode. The stationary multipoles formed by this nonlinear evolution retain the same energy, circulation and angular momentum as the original monopoles, but possess a lower enstrophy. These multipolar steady states are then compared to multipolar equilibria of the Euler equation, obtained either analytically by a perturbation expansion or numerically via a relaxation algorithm. Finally the stability of these equilibria is studied. Quadrupoles (one core vortex bound to three satellites) prove relatively robust, whether initially perturbed or not, and resist severe permanent deformations (mode-2 shears or strains of amplitude up to 0.1ζ(max). Amplification of the mode-3 deformation proves more destructive. More complex multipoles degenerate in less than a turnover period into end-products of a lesser complexity, via vortex splitting, pairing or merging. We use the conservation of integral properties to classify the large variety of instability mechanisms along physical guidelines. To conclude, we synthetize the connections between these various vortex forms.
The life-cycle of tripoles in two-dimensional incompressible flows
Authors: Carton Xavier , Legras Bernard
Published: April 2006

The mechanisms of coherent tripole formation from unstable shielded circular vortices are analysed in the context of two-dimensional incompressible flows. Three stages are identified during the transformation process: the linear growth of the initial normal mode perturbation, its nonlinear amplification and the finite-amplitude saturation under the tripolar form. We give a geometrical discussion of the mutual influence of the core vortex and of the satellites generated from the shield. The role of the angular momentum in determining the finite amplitude saturation is demonstrated using a simple elliptical model of the core vortex associated with two point-vortex satellites. The long-time asymmetric breaking of the tripole into a dipole and a monopole is shown to be driven by the erosion of the core vortex by stripping and diffusion. Finally the influence of bottom topography on tripole formation is considered, providing a rich phenomenology when the height of the topography is varied.
Nonlinear modulations of solitary waves
Authors: Pedersen Geir
Published: April 2006

The leading optical approximation to a slowly varying solitary crest on constant depth is the plane soliton solution with the local values of amplitude and orientation substituted. This leads to two nonlinear hyperbolic equations for the local amplitude and inclination of the crest that have been reported by several authors and predict the formation of progressive wave jumps, or shocks, from any initial perturbation of the crest. In comparison to numerical solutions of the Boussinesq equations we find that this optical approximation fails to reproduce essential properties of the crest dynamics, in particular that the crest modulations are damped and that well-defined wave jumps do not necessarily evolve. One purpose of the present work is to include such features in an amended optical approximation.

We obtain the leading correction to the ‘local soliton’ solution by a multiple scale technique. In addition to a modification to the wave profile the perturbation expansion also yields a diffracted wave system and a celerity that depends on the curvature of the crest. The principle of energy conservation then leads us to a second-order optical approximation consisting of transport equations of mixed hyperbolic/parabolic nature. Under additional assumptions the transport equations can be reduced to the well-known Burgers equation.

Numerical simulations of the Boussinesq equations are performed for modulations on otherwise straight crests and radially converging solitons. The improved optical, or ray, theory reproduces all essential features and agrees closely with the numerical solution in both cases. Contrary to purely hyperbolic optical descriptions the present theory also predicts wave jumps of finite width that are consistent with the triad solution of Miles (1977).

The present work indicates that while sinusoidal waves often are appropriately described by the lowest-order physical optics, higher-order corrections must be expected to be important for single crested waves.
A new vectorial bedload formulation and its application to the time evolution of straight river channels
Authors: Kovacs Agnes , Parker Gary
Published: April 2006

The derivation of a new vectorial bedload formulation for the transport of coarse sediment by fluid flow is presented in the first part of the paper. This relation has been developed for slopes up to the angle of repose both in the streamwise and transverse directions. The pressure distribution is assumed to be hydrostatic. The bed shear stress for the onset of particle motion and mean particle velocity are obtained from the mean force balance on a particle. A new generalized Bagnold hypothesis is introduced to calculate the sediment content of the bedload layer. The new formulation possesses two innovative features. It is fully nonlinear and vectorial in nature, in addition, it behaves smoothly up to the angle of repose.

A mathematical model of the time evolution of straight river channels is presented in the second half of the paper. This study focuses on the evolution process due to bank erosion in the presence of bedload only. The bed and bank material is taken to be coarse, non-cohesive and uniform in size. The sediment continuity and the fluid momentum conservation equations describe the time evolution of the bed topography and flow field. These equations are coupled through the fluid shear stress acting on the bed. This bed shear stress distribution is predicted with the aid of a simple algebraic turbulent closure model. As regards the computation of the sediment flux, the new fully nonlinear vectorial formulation is found to perform well and renders the evolution model fully mechanistic.

The formation of an erosional front in the time development of straight river channels has been so far obscured in physical experiments. Herein, with the help of the new bedload formulation, the existence and migration speed of the front of erosion are inferred from the analysis of the sediment continuity equation.

The model successfully describes the time relaxation of an initially trapezoidal channel toward an equilibrium cross-sectional shape, as evidenced by comparison with experimental data. This equilibrium is characterized by a constant width, vanishing sediment transport in the transverse direction, and a small but non-vanishing streamwise transport rate of bed sediment.
Averaged equations for inviscid disperse two-phase flow
Authors: Zhang D. Z. , Prosperetti A.
Published: April 2006

Averaged equations governing the motion of equal rigid spheres suspended in a potential flow are derived from the equation for the probability distribution. A distinctive feature of this work is the derivation of the disperse-phase momentum equation by averaging the particle equation of motion directly, rather than the microscopic equation for the particle material. This approach is more flexible than the usual one and leads to a simpler and more fundamental description of the particle phase. The model is closed in a systematic way (i.e. with no ad hoc assumptions) in the dilute limit and in the linear limit. One of the closure quantities is related to the difference between the gradient of the average pressure and the average pressure gradient, a well-known problem in the widely used two-fluid engineering models. The present result for this quantity leads to the introduction of a modified added mass coefficient (related to Wallis's ‘exertia’) that remains very nearly constant with changes in the volume fraction and densities of the phases. Statistics of this coefficient are provided and exhibit a rather strong variability of up to 20% among different numerical simulations. A detailed comparison of the present results with those of other investigators is given in § 10.

As a further illustration of the flexibility of the techniques developed in the paper, in Appendix C they are applied to the calculation of the so-called ‘particle stress’ tensor. This derivation is considerably simpler than others available in the literature.
On the stability of vertical double-diffusive interfaces. Part 2. Two parallel interfaces
Authors: Eltayeb I. A. , Loper D. E.
Published: April 2006

This is the second part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material between two parallel interfaces. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The stability of the interfaces is analysed first in the limit that they are close together (compared with the salt-finger lengthscale), then for general spacing. Attention is focused on whether the preferred mode of instability is varicose or sinuous and whether its wavevector is vertical or oblique.

The interfaces are found to be unstable for some wavenumber for all values of the Prandtl number and interface spacing. The preferred mode of instability for closely spaced interfaces is varicose and vertical for Prandtl number less than about 9, sinuous oblique for Prandtl number between 9 and 15 and sinuous vertical for larger Prandtl number. For general spacing each of the four possible modes of instability is preferred for some range of Prandtl number and interface separation, with no clear pattern of preference, except that the sinuous oblique mode is preferred for widely separated interfaces. The growth rate of the preferred mode is largest for interfaces having separations of from 1 to 3 salt-finger lengths.
On coupling between the Poincaré equation and the heat equation
Authors: Zhang Keke
Published: April 2006

It has been suggested that in a rapidly rotating fluid sphere, convection would be in the form of slowly drifting columnar rolls with small azimuthal scale (Roberts 1968; Busse 1970). The results in this paper show that there are two alternative convection modes which are preferred at small Prandtl numbers. The two new convection modes are, at leading order, essentially those inertial oscillation modes of the Poincaré equation with the simplest structure along the axis of rotation and equatorial symmetry: one propagates in the eastward direction and the other propagates in the westward direction; both are trapped in the equatorial region. Buoyancy forces appear at next order to drive the oscillation against the weak effects of viscous damping. On the basis of the perturbation of solutions of the Poincaré equation, and taking into account the effects of the Ekman boundary layer, complete analytical convection solutions are obtained for the first time in rotating spherical fluid systems. The condition of an inner sphere exerts an insignificant influence on equatorially trapped convection. Full numerical analysis of the problem demonstrates a quantitative agreement between the analytical and numerical analyses.
Gravitational and zero-drag motion of a spheroid adjacent to an inclined plane at low Reynolds number
Authors: Hsu Richard , Ganatos Peter
Published: April 2006

The first highly accurate solutions for the resistance tensor of an oblate or prolate spheroid moving near a planar wall obtained by Hsu & Ganatos (1989) are used to compute the translational and angular velocities and trajectories of a neutrally buoyant spheroid in shear flow and the gravitational settling motion of a non-neutrally buoyant spheroid adjacent to an inclined plane. The neutrally buoyant spheroid in shear flow undergoes a periodical motion toward and away from the wall as it continually tumbles forward. For some orientation angles it is found that the wall actually enhances the angular velocity of the particle. For certain inclinations a spheroid settling under gravity near an inclined plane reaches an equilibrium position, after which it translates parallel to the wall without rotation.
Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws
Authors: Samtaney Ravi , Zabusky Norman J.
Published: April 2006

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimenslons, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular. We present analytical expressions for Γ′, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, Γ′ scales as \[ \Gamma^\prime\propto (1-\eta^{-\frac{1}{2}})(\sin\alpha)(1+M^{-1}+2M^{-2})(M-1)(\gamma^{\frac{1}{2}}/\gamma + 1)\] where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, α is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For α ≤ 30°, the error in this approximation is less than 10% for 1.0 < M ≤ 1.32 for all η > 1, and 5.8 ≤ η ≤ 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code.

We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a ‘near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.
Modelling of rapid pressure—strain in Reynolds-stress closures
Authors: Johansson Arne V. , Hallbäck Magnus
Published: April 2006

The most general form for the rapid pressure—strain rate, within the context of classical Reynolds-stress transport (RST) closures for homogeneous flows, is derived, and truncated forms are obtained with the aid of rapid distortion theory. By a classical RST-closure we here denote a model with transport equations for the Reynolds stress tensor and the total dissipation rate. It is demonstrated that all earlier models for the rapid pressure—strain rate within the class of classical Reynolds-stress closures can be formulated as subsets of the general form derived here. Direct numerical simulations were used to show that the dependence on flow parameters, such as the turbulent Reynolds number, is small, allowing rapid distortion theory to be used for the determination of model parameters. It was shown that such a nonlinear description, of fourth order in the Reynolds-stress anisotropy tensor, is quite sufficient to very accurately model the rapid pressure—strain in all cases of irrotational mean flows, but also to get reasonable predictions in, for example, a rapid homogeneous shear flow. Also, the response of a sudden change in the orientation of the principal axes of a plane strain is investigated for the present model and models proposed in the literature. Inherent restrictions on the predictive capability of Reynolds-stress closures for rotational effects are identified.
Coherent structures in rotating three-dimensional turbulence
Authors: Bartello Peter , Métais Olivier , Lesieur Marcel
Published: April 2006

Numerical simulations investigating the formation and stability of quasi-two-dimensional coherent vortices in rotating homogeneous three-dimensional flow are described. In a numerical study of shear flows Lesieur, Yanase & Métais (1991) found that cyclones (respectively anticyclones) with |ω2D| ∼O(2Ω), where ω2Dis the vorticity and Ω is the rotation rate, are stabilized (respectively destabilized) by the rotation. A study of triply periodic pseudo-spectral simulations (643) was undertaken in order to investigate the vorticity asymmetry in homogeneous turbulence. Specifically, we examine (i) the possible three-dimensionalization of initially two-dimensional vortices and (ii) the emergence of quasi-two-dimensional structures in initially-isotropic three-dimensional turbulence. Direct numerical simulations of the Navier—Stokes equations are compared with large-eddy simulations employing a subgridscale model based on the second-order velocity structure function evaluated at the grid separation and with simulations employing hyperviscosity.

Isolated coherent two-dimensional vortices, obtained from a two-dimensional decay simulation, were superposed with a low-amplitude three-dimensional perturbation, and used to initialize the first set of simulations. With Ω = 0, a three-dimensionalization of all vortices was observed. This occurred first in the small scales in conjunction with the formation of longitudinal hairpin vortices with vorticity perpendicular to that of the initial quasi-two-dimensional flow. In agreement with centrifugal stability arguments, when 2Ω = [ω2D]rmsa rapid destabilization of anticyclones was observed to occur, whereas the initial two-dimensional cyclonic vortices persisted throughout the simulation. At larger Ω, both cyclones and anticyclones remained two-dimensional, consistent with the Taylor—Proudman theorem. A second set of simulations starting from isotropic three-dimensional fields was initialized by allowing a random velocity field to evolve (Ω = 0) until maximum energy dissipation. When the simulations were continued with 2Ω = [ω · Ω]rms/Ω, the three-dimensional flow was observed to organize into two-dimensional cyclonic vortices. At larger Ω, two-dimensional anticyclones also emerged from the initially-isotropic flow. The consequences for a variety of industrial and geophysical applications are clear. For quasi-two-dimensional eddies whose characteristic circulation times are of the order ofder of Ω−1, rotation induces a complete disruption of anticyclonic vortices, while stabilizing cyclonic ones.
Early-period dynamics of an incompressible mixing layer
Authors: Bun Y. , Criminale W. O.
Published: April 2006

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.
Slip over rough and coated surfaces
Authors: Miksis Michael J. , Davis Stephen H.
Published: April 2006

We study the effect of surface roughness and coatings on fluid flow over a solid surface. In the limit of small-amplitude roughness and thin lubricating films we are able to derive asymptotically an effective slip boundary condition to replace the no-slip condition over the surface. When the film is absent, the result is a Navier slip condition in which the slip coefficient equals the average amplitude of the roughness. When a layer of a second fluid covers the surface and acts as a lubricating film, the slip coefficient contains a term which is proportional to the viscosity ratio of the two fluids and which depends on the dynamic interaction between the film and the fluid. Limiting cases are identified in which the film dynamics can be decoupled from the outer flow.
The spatial structure of neutral atmospheric surface-layer turbulence
Authors: Mann Jakob
Published: April 2006

Modelling of the complete second-order structure of homogeneous, neutrally stratified atmospheric boundary-layer turbulence, including spectra of all velocity components and cross-spectra of any combination of velocity components at two arbitrarily chosen points, is attempted. Two models based on Rapid Distortion Theory (RDT) are investigated. Both models assume the velocity profile in the height interval of interest to be approximately linear. The linearized Navier–Stokes equation together with considerations of ‘eddy’ lifetimes are then used to modify the spatial second-order structure of the turbulence. The second model differs from the first by modelling the blocking by the surface in addition to the shear. The resulting models of the spectral velocity tensor contain only three adjustable parameters: a lengthscale describing the size of the largest energy-containing eddies, a non-dimensional number used in the parametrization of ‘eddy’ lifetime, and the third parameter is a measure of the energy dissipation.

Two atmospheric experiments, both designed to investigate the spatial structure of turbulence and both running for approximately one year, are used to test and calibrate the models. Even though the approximations leading to the models are very crude they are capable of predicting well the two-point second-order statistics such as cross-spectra, coherences and phases, on the basis of measurements carried out at one point. The two models give very similar predictions, the largest difference being in the coherences involving vertical velocity fluctuations, where the blocking by the surface seems to have a significant effect.
On mixing and transport at a sheared density interface
Authors: Sullivan Greg D. , List E. John
Published: April 2006

Mixing and transport of a stratifying scalar are investigated at a density interface imbedded in a turbulent shear flow. Steady-state interfacial shear flows are generated in a laboratory water channel for layer Richardson numbers, Ri, between about 1 and 10. The flow field is made optically homogeneous, enabling the use of laser-induced fluorescence with photodiode array imaging to measure the concentration field at high resolution. False-colour images of the concentration field provide valuable insight into interfacial dynamics: when the local mean shear Richardson number, Ris, is less than about 0.40–0.45, interfacial mixing appears to be dominated by Kelvin–Helmholtz (K–H) instabilities; when Ris is somewhat larger than this, interfacial mixing appears to be dominated by shear-driven wave breaking. In both cases, vertical transport of mixed fluid from the interfacial region into adjacent turbulent layers is accomplished by large-scale turbulent eddies which impinge on the interface and scour fluid from its outer edges.

Motivated by the experimental findings, a model for interfacial mixing and entrainment is developed. A local equilibrium is assumed in which the rate of loss of interfacial fluid by eddy scouring is balanced by the rate of production (local mixing) by interfacial instabilities and molecular diffusion. When a single layer is turbulent and entraining, the model results are as follows: in the molecular-diffusion-dominated regime, δ/h ∼ Pe−1/2 and E ∼ Ri−1Pe−1/2; in the wave-breaking-dominated regime, δ/h ∼ Ri−1/2 and E ∼ Ri−3/2; and in the K–H-dominated regime, δ/h ∼ Ri−1 and E ∼ Ri−2, where δ is the interface thickness, h is the boundary-layer thickness, Pe is the Péclet number, and E is the normalized entrainment velocity. In all three regimes, the maximum concentration anomaly, [gcy ]m ∼ Ri−1. When both layers are turbulent and entraining, E and δ depend on combinations of parameters from both layers.
The effect of viscosity on the height of disks floating above an air table
Authors: Hinch E. J. , Lemaître J.
Published: April 2006

A similarity solution is used to analyse the flow in the thin gap between a floating disk and an air table, the similarity solution being that for an axisymmetric stagnation point but with an upper boundary condition within the boundary layer. Viscous effects increase the height of the floating disk by 20% even at a Reynolds number of 50. Theoretical predictions are compared with experimental observations. The effect of the pressure distribution under the disk on the air flux through the table is examined.
Oblique-mode breakdown and secondary instability in supersonic boundary layers
Authors: Chang Chau-Lyan , Malik Mujeeb R.
Published: April 2006

Laminar–turbulent transition mechanisms for a supersonic boundary layer are examined by numerically solving the governing partial differential equations. It is shown that the dominant mechanism for transition at low supersonic Mach numbers is associated with the breakdown of oblique first-mode waves. The first stage in this breakdown process involves nonlinear interaction of a pair of oblique waves with equal but opposite angles resulting in the evolution of a streamwise vortex. This stage can be described by a wave–vortex triad consisting of the oblique waves and a streamwise vortex whereby the oblique waves grow linearly while nonlinear forcing results in the rapid growth of the vortex mode. In the second stage, the mutual and self-interaction of the streamwise vortex and the oblique modes results in the rapid growth of other harmonic waves and transition soon follows. Our calculations are carried all the way into the transition region which is characterized by rapid spectrum broadening, localized (unsteady) flow separation and the emergence of small-scale streamwise structures. The r.m.s. amplitude of the streamwise velocity component is found to be on the order of 4–5 % at the transition onset location marked by the rise in mean wall shear. When the boundary-layer flow is initially forced with multiple (frequency) oblique modes, transition occurs earlier than for a single (frequency) pair of oblique modes. Depending upon the disturbance frequencies, the oblique mode breakdown can occur for very low initial disturbance amplitudes (on the order of 0.001% or even lower) near the lower branch. In contrast, the subharmonic secondary instability mechanism for a two-dimensional primary disturbance requires an initial amplitude on the order of about 0.5% for the primary wave. An in-depth discussion of the oblique-mode breakdown as well as the secondary instability mechanism (both subharmonic and fundamental) is given for a Mach 1.6 flat-plate boundary layer.
Pattern formation of granules on the bottom of a differentially rotating tank
Authors: Thomas P. J.
Published: April 2006

The formation of patterns of granules on the bottom of a differentially rotating tank, partially filled with a fluid, is described. The pattern formation is initiated when the tank is spun up from an initial constant rotation rate by a sufficiently large increment to another constant higher rotation rate. The granules are observed to be set into motion, slide across the bottom of the tank and organize themselves into two distinct, superposed geometrical patterns. The first pattern constitutes a system of tightly wound equi-angular spirals and is identified as being simply a visualization of the well-known stationary Class B waves of the unstable boundary layer. This pattern is not considered further here. The second pattern, which is considered here, is shown not to be a visualization of some other known stationary wave mode. It consists of a number of large-scale spiral arms extending outward towards the wall of the tank and originating at a patch of salt in the tank's centre. The presence of Class B waves is apparently a necessary condition for the spiral arm formation. The experiments show that the radius r0 of the inner salt patch is inversely proportional to the increment Δω of the rotation rate for the larger values of Δω. The angle ε(r) between the direction of the spiral arms and the tangential direction is observed to decrease with the distance r from the centre as rb where b = −0.64 ± 0.1. The relationship between the number n of spiral arms and the associated Reynolds number Re and Rossby number Ro is found from the experimental observations and dimensional considerations to be \[n^2 = Re \, Ro. \] Various aspects related to the mechanism involved in the pattern formation are discussed. However, no conclusive model for the spiral arm formation can at present be suggested.
Dynamics of vortical structures in a homogeneous shear flow
Authors: Kida Shigeo , Tanaka Mitsuru
Published: April 2006

The mechanism of generation, development and interaction of vortical structures, extracted as concentrated-vorticity regions, in homogeneous shear turbulence is investigated by the use of the results of a direct numerical simulation of the Navier-Stokes equation with 1283 grid points. Among others, a few of typical vortical structures are identified as important dynamical elements, namely longitudinal and lateral vortex tubes and vortex layers. They interact strongly with each other. Longitudinal vortex tubes are generated from a random fluctuating vorticity field through stretching of fluid elements caused by the mean linear shear. They are inclined toward the streamwise direction by rotational motion due to the mean shear. There is a small (about 10°) deviation in direction between the longitudinal vortex tubes and vorticity vectors therein, which makes the vorticity vectors turn toward the spanwise direction (against the mean vorticity) until the spanwise components of the fluctuating vorticity become comparable in magnitude with the mean vorticity. These longitudinal vortex tubes induce straining flows perpendicular to themselves which generate vortex layers with spanwise vorticity in planes spanned by the tubes and the spanwise axis. These vortex layers are unstable, and roll up into lateral vortex tubes with concentrated spanwise vorticity through the Kelvin-Helmholtz instability. All of these vortical structures, through strong mutual interactions, break down into a complicated smallscale random vorticity field. Throughout the simulated period an oblique stripe structure dominates the whole flow field: initially it is inclined at about 45° to the downstream and, as the flow develops, the inclination angle decreases but eventually stays at around 10°–20°.
The hydraulic jump in a viscous laminar flow
Authors: Higuera F. J.
Published: April 2006

The hydraulic jump appearing in the viscous laminar flow of a thin liquid layer over a finite horizontal plate is studied using the boundary-layer approximation for the flow in and around the jump. The position and structure of the jump are determined by numerically solving the resulting problem with a boundary condition at the edge of the plate that expresses the matching of the layer with the shorter region where the liquid turns around and falls under the action of gravity. When the Froude number of the flow ahead of the jump is very large, the jump is much shorter than the horizontal extent of the layer, though still much longer than its depth. An asymptotic description of the inner structure of such a jump is given, building upon the analysis of Bowles & Smith for the short interaction region at the leading end of the jump. This structure consists of a fast moving separated flow in the upper part of the layer that progressively slows down by ingesting new fluid across its lower boundary, until the hydrostatically generated adverse pressure gradient makes it recirculate in the lower part of the layer. The effects of the surface tension and the cross-stream pressure variation owing to the curvature of the streamlines are taken into account in the jump and in the flow approaching the edge of the plate, showing that they can lead to quantitative and also qualitative changes of the jump structure, including a local breakdown of the boundary-layer approximation.
The effect of spanwise system rotation on Dean vortices
Authors: Matsson O. John E. , Alfredsson P. Henrik
Published: April 2006

An experimental study is reported of the flow in a high-aspect-ratio curved air channel with spanwise system rotation, utilizing hot-wire measurements and smoke visualization. The experiments were made at two different Dean numbers (De), approximately 2 and 4.5 times the critical De for which the flow becomes unstable and develops streamwise vortices. For the lower De without system rotation the primary Dean instability appeared as steady longitudinal vortices. It was shown that negative spanwise system rotation, i.e. the Coriolis force counteracts the centrifugal force, could cancel the primary Dean instability and that for high rotation rates it could give rise to vortices on the inner convex channel wall. For positive spanwise system rotation, i.e. when the Coriolis force enhanced the centrifugal force, splitting and merging of vortex pairs were observed. At the higher De secondary instabilities occurred in the form of travelling waves. The effect of spanwise system rotation on the secondary instability was studied and was found to reduce the amplitude of the twisting and undulating motions for low negative rotation. For low positive rotation the amplitude of the secondary instabilities was unaffected for most regions in parameter space.
The asymptotic study of dissipation and breakdown of a wing-tip vortex
Authors: Trigub V. N. , Blokhin A. B. , Simakin I. N.
Published: April 2006

The steady axisymmetrical wing-tip vortex is studied in this paper by means of asymptotic methods within the limit of high Reynolds numbers. The smooth regrouping of the vortex under the action of viscous forces is described by a quasicylindrical approximation. The solutions of the quasi-cylindrical approximation are thoroughly analysed numerically and it is shown that a saddle-point bifurcation appears at certain critical values of circulation. At these values the solution may be continued in two ways: as a supercritical branch which approaches the Batchelor limit far downstream; and a subcritical one, which passes the second, nodal-point bifurcation. The parabolic quasi-cylindrical equations past this point allow the downstream disturbances to propagate upstream, like for example, boundary-layer equations in the regime of strong hypersonic interaction. The flow past the second bifurcation point was studied numerically and it was shown that solutions of the quasicylindrical approximation with large reversed-flow regions exist. An asymptotic expansion of such solutions far downstream was constructed, and it turned out that the reversed-flow region expands exponentially. This process is halted by elliptical effects in the external flow. An asymptotic theory of large reversed-flow regions is suggested including viscosity and elliptical effects. Numerical solutions for unbounded vortex breakdown parabolically expanding far downstream are presented. Then the general asymptotic problem statement which describes the flow near the bifurcation points is used to study the asymptotic solutions near the first bifurcation point. The problem is investigated numerically and two kinds of solution, which may be treated as transcritical jumps and marginal vortex breakdown, are found and discussed.
Spin-up of a rapidly rotating heavy gas in a thermally insulated annulus
Authors: Lindblad Ingemar A. A. , Bark Fritz H. , Zahrai Said
Published: April 2006

The linear spin-up problem for a rapidly rotating viscous diffusive ideal gas is considered in the limit of vanishing Ekman number E. Particular attention is given to gases having a large molecular weight. The gas is enclosed in a cylindrical annulus, with flat top and bottom walls, which is rotating around its axis of symmetry with rotation rate Ω. The walls of the container are adiabatic. In a rotating gas (of any molecular weight), the Ekman layers on adiabatic walls are weak, which implies that there is no distinct non-diffusive response of the gas outside the Ekman and Stewartson boundary layers on the timescale E−1/2Ω−1 for spin-up of a homogeneous fluid. For the case of adiabatic walls, it is shown that the spin-up mechanisms due to viscous diffusion and Ekman suction are, from a formal point of view, equally strong. Therefore, the gas will adjust to the increased rotation rate of the container on the diffusive timescale E−1Ω−1. However, if E1/3 [Lt ] γ – 1 [Lt ] 1 and M [siml ] 1, which characterizes rapidly rotating heavy gases (where γ is the ratio of specific heats of the gas and M the Mach number), it is shown that the gas spins up mainly by Ekman suction on the shorter timescale (γ–1)2E−1Ω−1. In such cases, the interior motion splits up into a non-diffusive part of geostrophic character and diffusive boundary layers of thickness (γ – 1) outside the Ekman and Stewartson layers. The motion approaches the steady state of rigid rotation algebraically instead of exponentially as is usually the case for spin-up.
Experiments on a round turbulent buoyant plume
Authors: Shabbir Aamir , George William K.
Published: April 2006

This paper reports a comprehensive set of hot-wire measurements of a round buoyant plume which was generated by forcing a jet of hot air vertically up into a quiescent environment. The boundary conditions of the experiment were measured, and are documented in the present paper in an attempt to sort out the contradictory mean flow results from the earlier studies. The ambient temperature was monitored to ensure that the facility was not stratified and that the experiment was conducted in a neutral environment. The axisymmetry of the flow was checked by using a planar array of sixteen thermocouples and the mean temperature measurements from these are used to supplement the hot-wire measurements. The source flow conditions were measured to ascertain the rate at which the buoyancy was added to the flow. The measurements conserve buoyancy within 10%. The results are used to determine balances of the mean energy and momentum differential equations. In the mean energy equation it is found that the vertical advection of energy is primarily balanced by the radial turbulent transport. In the mean momentum equation the vertical advection of momentum and the buoyancy force balance the radial turbulent transport. The buoyancy force is the second largest term in this balance and is responsible for the wider (and higher) velocity profiles in plumes as compared to jets. Budgets of the temperature variance and turbulent kinetic energy are also determined in which thermal and mechanical dissipation rates are obtained as the closing terms. Similarities and differences between the two balances are discussed. It is found that even though the direct effect of buoyancy in turbulence, as evidenced by the buoyancy production term, is substantial, most of the turbulence is produced by shear. This is in contrast to the mean velocity field where the effect of the buoyancy force is quite strong. Therefore, it is concluded that in a buoyant plume the primary effect of buoyancy on turbulence is indirect, and enters through the mean velocity field (giving larger shear production).
Spontaneous acoustic emission from strong ionizing shocks
Authors: Mond M. , Rutkevich I. M.
Published: April 2006

Dyakov (1954) and Kontorovich (1957) formulated the conditions for corrugation instability of shock waves as well as for spontaneous emission of sound and entropy-vortex waves from them. For the first time since then, it is shown here that physical circumstances do exist under which shocks in gases spontaneously emit sound waves. Such circumstances are provided by strong ionizing shocks. In order to see that, the coefficient of reflection of an acoustic wave from a shock is derived as a function of the wave's frequency and the ionization degree. Spontaneous emission of sound occurs when the reflection coefficient becomes infinitely large. It is shown that the relevant frequency range for the occurrence of spontaneous emission is that for which the electrons are not in local thermodynamic equilibrium with the heavy particles. The special properties of acoustic perturbations behind the ionizing shock are considered for this frequency range and the sound velocity in a partially ionized gas is derived. In addition, the condition for spontaneous emission of sound is modified in order to take into account the difference between the electrons and heavy-particle perturbed temperatures. It is shown, by numerical calculations, that the criterion for spontaneous emission is satisfied behind ionizing shocks in argon. In particular, for an initial pressure of 5 Torr, the threshold for the occurrence of the spontaneous emission is found to be M1 = 15. This critical value of the shock Mach number, as well as other calculated physical features, agree very well with those obtained experimentally by Glass & Liu (1978) who observed the occurrence of instability behind shocks in argon.
A formula for ‘wave damping’ in the drift of a floating body
Authors: Aranha J. A. P.
Published: April 2006

A simple formula for ‘wave damping’ is derived, exact within the context of the proposed theory, namely: potential flow correct to second order in the wave amplitude and to leading order in U/c, where U is the drift velocity and c the wave celerity. The analysis is restricted to a two-dimensional problem although the extension to three dimensions seems possible.
Motion of a rigid particle in a rotating viscous flow: an integral equation approach
Authors: Tanzosh John P. , Stone H. A.
Published: April 2006

A boundary integral method is presented for analysing particle motion in a rotating fluid for flows where the Taylor number${\cal T}$is arbitrary and the Reynolds number is small. The method determines the surface traction and drag on a particle, and also the velocity field at any location in the fluid.

Numerical results show that the dimensionless drag on a spherical particle translating along the rotation axis of an unbounded fluid is determined by the empirical formula$D/6\pi = 1+(4/7){\cal T}^{1/2}+(8/9\pi){\cal T}$, which incorporates known results for the low and high Taylor number limits. Streamline portraits show that a critical Taylor number${\cal T}_c\ap 50$exists at which the character of the flow changes. For${\cal T} < {\cal T}_c$the flow field appears as a perturbation of a Stokes flow with a superimposed swirling motion. For${\cal T} > {\cal T}_c$the flow field develops two detached recirculating regions of trapped fluid located fore and aft of the particle. The recirculating regions grow in size and move farther from the particle with increasing Taylor number. This recirculation functions to deflect fluid away from the translating particle, thereby generating a columnar flow structure. The flow between the recirculating regions and the particle has a plug-like velocity profile, moving slightly slower than the particle and undergoing a uniform swirling motion. The flow in this region is matched to the particle velocity in a thin Ekman layer adjacent to the particle surface.

A further study examines the translation of spheroidal particles. For large Taylor numbers, the drag is determined by the equatorial radius; details of the body shape are less important.
Surface-wave damping in a circular cylinder with a fixed contact line
Authors: Henderson D. M. , Miles J. W.
Published: April 2006

The natural frequencies and damping ratios for surface waves in a circular cylinder are calculated on the assumptions of a fixed contact line, Stokes boundary layers, and either a clean or a fully contaminated surface. These theoretical predictions are compared with the measurements for the first six modes in a brimfull, sharp-edged cylinder of radius 2.77 cm and depth 3.80 cm. The differences between the predicted and observed frequencies were less than 0.5% for all but the fundamental axisymmetric mode with a clean surface. The difference between the predicted and observed damping ratio for the dominant mode with a clean surface was 20%; this difference was significantly larger for the higher modes with a clean surface and for all of the modes with a contaminated surface.