bluuebunny/crossref_metadata_2025 huggingface dataset:
Local measurements of velocity fluctuations and diffusion coefficients for a granular material flow
Authors: Natarajan V. V. R. , Hunt M. L. , Taylor E. D.
Published: April 2006

Measurements were made of two components of the average and fluctuating velocities, and of the local self-diffusion coefficients in a flow of granular material. The experiments were performed in a 1 m-high vertical channel with roughened sidewalls and with polished glass plates at the front and the back to create a two-dimensional flow. The particles used were glass spheres with a nominal diameter of 3 mm. The flows were high density and were characterized by the presence of long-duration frictional contacts between particles. The velocity measurements indicated that the flows consisted of a central uniform regime and a shear regime close to the walls. The fluctuating velocities in the transverse direction increased in magnitude from the centre towards the walls. A similar variation was not observed for the streamwise fluctuations. The self-diffusion coefficients showed a significant dependence on the fluctuating velocities and the shear rate. The velocity fluctuations were highly anistropic with the streamwise components being 2 to 2.5 times the transverse components. The self-diffusion coefficients for the streamwise direction were an order-of-magnitude higher than those for the transverse direction. The surface roughness of the particles led to a decrease in the self-diffusion coefficients.
Compressible vortex reconnection
Authors: Virk D. , Hussain F. , Kerr R. M.
Published: April 2006

Reconnection of two antiparallel vortex tubes is studied as a prototypical coherent structure interaction to quantify compressibility effects in vorticity dynamics. Direct numerical simulations of the Navier-Stokes equations for a perfect gas are carried out with initially polytropically related pressure and density fields. For an initial Reynolds number (Re = Γ /v, circulation divided by the kinematic viscosity) of 1000, the pointwise initial maximum Mach number (M) is varied from 0.5 to 1.45. At M=0.5, not surprisingly, the dynamics are essentially incompressible. As M increases, the transfer of Γ starts earlier. For the highest M, we find that shocklet formation between the two vortex tubes enhances early Γ transfer due to viscous cross-diffusion as well as baroclinic vorticity generation. The reconnection at later times occurs primarily due to viscous cross-diffusion for all M. However, with increasing M, the higher early Γ transfer reduces the vortices’ curvature growth and hence the Γ transfer rate; i.e. for the Re case studied, the reconnection timescale increases with M. With increasing M, reduced vortex stretching by weaker ‘bridges’ decreases the peak vorticity at late times. Compressibility effects are significant in countering the stretching of the bridges even at late times. Our observations suggest significantly altered coherent structure dynamics in turbulent flows, when compressible.
Direct numerical simulations of two-dimensional chaotic natural convection in a differentially heated cavity of aspect ratio 4
Authors: Xin Shihe , Le Quéré Patrick
Published: April 2006

Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is investigated by direct numerical integration of the unsteady two-dimensional equations. Time integration is performed with a spectral algorithm using Chebyshev spatial approximations and a second-order finite-difference time-stepping scheme. Asymptotic solutions have been obtained for three values of the Rayleigh number based on cavity height up to 1010. The time-averaged flow fields show that the flow structure increasingly departs from the well-known laminar one. Large recirculating zones located on the outer edge of the boundary layers form and move upstream with increasing Rayleigh number. The time-dependent solution is made up of travelling waves which run downstream in the boundary layers. The amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At the largest Rayleigh number investigated they grow to such a point that they result in the formation of large unsteady eddies that totally disrupt the boundary layers. These eddies throw hot and cold fluid into the upper and lower parts of the core region, resulting in thermally more homogeneous top and bottom regions that squeeze a region of increased stratification near the mid-cavity height. It is also shown that these large unsteady eddies keep the internal waves in the stratified core region excited. These simulations also give access to the second-order statistics such as turbulent kinetic energy, thermal and viscous dissipation, Reynolds stresses and turbulent heat fluxes.
Merging flows in an arterial confluence: the vertebro-basilar junction
Authors: Ravensbergen J. , Krijger J. K. B. , Hillen B. , Hoogstraten H. W.
Published: April 2006

The basilar artery is one of the three vessels providing the blood supply to the human brain. It arises from the confluence of the two vertebral arteries. In fact, it is the only artery of this size in the human body arising from a confluence instead of a bifurcation. Earlier work, concerning flow computations in simplified models of the basilar artery, has demonstrated that a junction causes distinctive flow phenomena. This paper presents three-dimensional finite-element computations of steady viscous flow in a rigid symmetrical junction geometry representing the anatomical situation in a more realistic way. The geometry consists of two round tubes merging into a single round outlet tube. The Reynolds number for the basilar artery ranges from 200 to 600, and both symmetrical and asymmetrical inflow from the two inlet tubes has been considered.

Just downstream of the confluence a ‘double hump’ axial velocity profile is found. In the transition zone the flow pattern appears to have a complicated structure. In the symmetrical case the axial velocity profile shows a sharp central ridge, whereas in the asymmetrical case the highest ‘hump’ crosses the centreline of the tube. The flow has a highly three-dimensional character with secondary velocities easily exceeding 25% of the mean axial flow velocity. The secondary flow pattern consists of four vortices. Under all simulated flow conditions, the inlet length turns out to be much larger than the average length of the human basilar artery.

To validate the computational model, a comparison is made between numerical and experimental results for a junction geometry consisting of tubes with a rectangular cross-section. The experiments have been performed in a Perspex model with laser Doppler velocimetry and dye injection techniques. Good agreement between experimental and computational results is found. Moreover, all essential flow phenomena turn out to be quite similar to those obtained for the circular tube geometry.
Directional solidification into static stability
Authors: Brattkus Kirk
Published: April 2006

Consider the directional solidification of a binary alloy rejecting a heavy solute as it solidifies upward. If the solidification front is planar, the fluid melt ahead of the front is stably stratified and convection is not expected. In this paper we analyse the linear stability of planar solidification asymptotically in the limit of large solutal Rayleigh number, R. Three distinct linear modes are found which correspond to internal waves, buoyancy edge waves, or morphological modes. Of these three modes, only the morphological modes are subject to an instability. We find that for large Rayleigh number this instability first occurs at long wavelengths with wavenumbers that scale on R−1/14. The scalings derived from the linear analysis are used to construct a nonlinear theory for the morphological instability in the large Rayleigh number limit. Similarity solutions are found which describe steadily convecting, non-planar growth reminiscent of an observed phenomenon known as steepling.
The interaction of waves with horizontal cylinders in two-layer fluids
Authors: Linton C. M. , McIver M.
Published: April 2006

We consider two-dimensional problems based on linear water wave theory concerning the interaction of waves with horizontal cylinders in a fluid consisting of a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. In a single-layer fluid there are a number of reciprocity relations that exist connecting the various hydrodynamic quantities that arise. These relations are systematically extended to the two-fluid case. It is shown that for symmetric bodies the solutions to scattering problems where the incident wave has wavenumber K and those where it has wavenumber k are related so that the solution to both can be found by just solving one of them. The particular problems of wave scattering by a horizontal circular cylinder in either the upper or lower layer are then solved using multipole expansions.
On hydromagnetic instabilities driven by the Hartmann boundary layer in a rapidly rotating sphere
Authors: Zhang Keke , Busse F. H.
Published: April 2006

The instability of an electrically conducting fluid of magnetic diffusivity λ and viscosity v in a rapidly rotating spherical container of magnetic diffusivity$\hat{\lambda}$in the presence of a toroidal magnetic field is investigated. Attention is focused on the case of a toroidal magnetic field induced by a uniform current density parallel to the axis of rotation, which was first studied by Malkus (1967). We show that the internal ohmic dissipation does not affect the stability of the hydromagnetic solutions obtained by Malkus (1967) in the limit of small λ. It is solely the effect of the magnetic Hartmann boundary layer that causes instabilities of the otherwise stable solutions. When the container is a perfect conductor,$\hat{\lambda}$= 0, the hydromagnetic instabilities grow at a rate proportional to the magnetic Ekman number of the fluidEλ; when the container is a nearly perfect insulator,$\lambda/\hat{\lambda}\ll 1$, the hydromagnetic instabilities grow at a rate proportional toE1/2λ; when the container is a nearly perfect conductor, λ 1, the growth rates are proportional to λ, where λ is the magnetic Ekman number based on the diffusivity λ of the container. The main characteristics of the instabilities are not affected by varying magnetic properties of the container. In light of the destabilizing role played by the Hartmann boundary layer, we also examine the corresponding magnetoconvection in a rapidly rotating fluid sphere with the perfectly conducting container and stress-free velocity boundary conditions. Analytical magnetoconvection solutions in closed form are obtained and implications are discussed.
A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid
Authors: Liu Y. J. , Liao T. Y. , Joseph D. D.
Published: April 2006

When an air bubble rises in a viscoelastic fluid there is a critical capillary number for cusping and jump in velocity: when the capillary number is below critical, which is about 1 in our data, there is no cusp at the tail of a (smooth) air bubble. For larger volumes, a two-dimensional cusp, sharp in one view and broad in the orthogonal view, is in evidence. Measurements suggest that the cusp tip is in the generic form y = ax2/3 satisfied by analytic cusps. The intervals of volumes for which dramatic changes in air bubble shape take place is very small and the two to ten fold increase in the rise velocity which accompanies the small change of volume could be modelled as a discontinuity. A second drag transition and an orientational transition occurred when U/c > 1 where U is the rise velocity of an air bubble and c is the shear wave speed. For U/c < 1, U is proportional to d2, where d is the equivalent diameter for a sphere of diameter d having the same volume, and when U/c > 1 then U is proportional to d and the Deborah number does not change with U. Moreover the bubble shapes when U/c < 1 are overall prolate (with or without a cusped tail) with the long side parallel to gravity, in contrast to the oblate shapes which are always observed in Newtonian fluids and in viscoelastic fluids with U/c > 1 when inertia is dominant. The formation of cusps occurs in all kinds of columns of different sizes and shapes. Cusping is generic but the orientation of the broad edge with respect to the sidewalls is an issue. There is no preferred orientation in columns with round cross-sections, or in the case of walls far away from the rising bubble. In columns with rectangular cross-sections, three relatively stable configurations can be observed: the cusp can be observed in the wide window and the broad edge in the narrow window; the cusp can be observed in the narrow window and the broad edge in the wide window or, less frequently, the broad edge lies along a diagonal. These orientational and drag alternatives are directly analogous to those which are observed in the settling of long or broad solid bodies (Liu & Joseph 1993).
Unsteady separation and heat transfer upstream of obstacles
Authors: Puhak R. I. , Degani A. T. , Walker J. D. A.
Published: April 2006

The development of a laminar boundary layer upstream of both two- and threedimensional obstacles mounted on a plane wall is considered. The motion is impulsively started from rest, and it is shown that the boundary layer upstream of the obstacle initially develops independently from that on the obstacle itself. Numerical solutions for the unsteady boundary-layer flow on the plane wall are obtained in both Eulerian and Lagrangian coordinates. It is demonstrated that in both situations the flow focuses into a narrow-band eruption characteristic of separation phenomena at high Reynolds number. For the three-dimensional problem, results are obtained on a symmetry plane upstream of the obstacle which indicate the evolution, and subsequent sharp compression, of a spiral vortex in the near-wall flow in a manner consistent with recent experimental studies. The eruptive response of the two-dimensional boundary layer is found to be considerably stronger than the corresponding event in three dimensions. Calculated results for the temperature distribution are obtained for the situation where the wall temperature is constant but different from that of the mainstream. It is shown that a concentrated response develops in the surface heat transfer rate as the boundary layer starts to separate from the surface.
Two-dimensional vortex motion in the cross-flow of a wing-body configuration
Authors: De Laat T. W. G. , Coene R.
Published: April 2006

For a two-dimensional potential flow, Föppl obtained the equilibrium positions for a symmetric vortex pair behind a circular cylinder in a uniform oncoming flow. In this article it is shown that such an equilibrium is in general possible for a vortex in a stagnation flow (e. g. in a corner). Furthermore it is found that a vortex near such an equilibrium position will rotate with a definite frequency around this equilibrium. Expressions are derived for the frequencies associated with the closed orbits of the vortices in the case of equilibrium of a vortex in a stagnation flow and for the equilibrium of the symmetric vortex pair behind a circular cylinder in oncoming flow. For the large-amplitude case the vortex trajectories are claculated using a fifth-order Runge-Kutta integration method. The analysis is then extended to the case of a simple wing-body combination in a cross-flow such as arises for a slender aircraft at an angle of attack with vortices generated by strakes or at the front part of the body. At the wint-body junctions the motions of the vortices may be periodic, quasi-periodic or the vortices may be swept away, depending on the initial conditions.
Liquid-metal flow in an insulated rectangular expansion with a strong transverse magnetic field
Authors: Walker John S. , Picologlou Basil F.
Published: April 2006

This paper concerns a steady liquid-metal flow through an expansion or contraction with electrically insulated walls, with rectangular cross-sections and with a uniform, transverse, externally applied magnetic field. One pair of duct walls is parallel to the applied magnetic field, and the other pair diverges or converges symmetrically about a plane which is perpendicular to the field. The magnetic field is assumed to be sufficiently strong that inertial effects can be neglected and that the well-known Hartmann-layer solution is valid for the boundary layers on the walls which are not parallel to the magnetic field. A general treatment of three-dimensional flows in constant-area ducts is presented. An error in the solution of Walker et al. (1972) is corrected. A smooth expansion between two different constant-area ducts is treated. In the expansion the flow is concentrated inside the boundary layers on the sides which are parallel to the magnetic field, while the flow at the centre of the duct is very small and may be negative for a large expansion slope. In each constant-area duct, the flow evolves from a concentration near the sides at the junction with the expansion to the appropriate fully developed flow far upstream or downstream of the expansion. The pressure drop associated with the three-dimensional flow increases as the slope. increases.
A numerical study of turbulent supersonic isothermal-wall channel flow
Authors: Coleman G. N. , Kim J. , Moser R. D.
Published: April 2006

A study of compressible supersonic turbulent flow in a plane channel with isothermal walls has been performed using direct numerical simulation. Mach numbers, based on the bulk velocity and sound speed at the walls, of 1.5 and 3 are considered; Reynolds numbers, defined in terms of the centreline velocity and channel half-width, are of the order of 3000. Because of the relatively low Reynolds number, all of the relevant scales of motion can be captured, and no subgrid-scale or turbulence model is needed. The isothermal boundary conditions give rise to a flow that is strongly influenced by wall-normal gradients of mean density and temperature. These gradients are found to cause an enhanced streamwise coherence of the near-wall streaks, but not to seriously invalidate Morkovin's hypothesis : the magnitude of fluctuations of total temperature and especially pressure are much less than their mean values, and consequently the dominant compressibility effect is that due to mean property variations. The Van Driest transformation is found to be very successful at both Mach numbers, and when properly scaled, statistics are found to agree well with data from incompressible channel flow results.
Thermo-viscous fingering of flow in a thin gap: a model of magma flow in dikes and fissures
Authors: Helfrich Karl R.
Published: April 2006

Flow of a fluid with a strongly temperature-dependent viscosity in a finite-length slot is analysed as a model of magma flow in dikes. The slot walls are held at a fixed temperature, thus cooling and increasing the viscosity of the fluid as it moves along the gap. Poiseuille flow and temperature advection, averaged across the slot, are used to study the stability of this basic one-dimensional flow to lateral perturbations. A linear stability analysis shows that for sufficiently strong cooling and viscosity increase with decreasing temperature, the flow is unstable to a fingering instability. Warm fluid is focused into relatively fast flowing zones and suffers only modest cooling, while cold, slow flowing regions experience more cooling and an increase in viscosity, which acts to locally clog the slot. The necessary condition for instability is the presence of multiple solutions for velocity (fast, intermediate and slow branches) in the basic one-dimensional flow. The intermediate branch, where the thermal adjustment lengthscale is comparable to the slot length, is unstable and the analysis indicates that the instability continues onto the slow branch. The parametric regions of instability and the growth rates are dependent on the choice of boundary conditions at the slot entrance (i. e. the magma source): either uniform flux, or uniform pressure. The latter case is the more geophysically realistic and has the larger unstable region and growth rates. Numerical solutions of the nonlinear equations show that at finite-amplitude the hot, low-viscosity, fast-flowing fingers continue to speed up, while the slow, cold regions continue to cool and slow down. At the slot exit fluid issues from the gap in isolated hot, low-viscosity spouts separated by zones of cold, nearly still fluid. Application of the model to geophysical settings indicates that the instability is expected for realistic parameter values. The model may help explain the observed focusing of fissure eruptions.
Inviscid separated flow over a non-slender delta wing
Authors: Moore D. W. , Pullin D. I.
Published: April 2006

We consider inviscid incompressible flow about an infinite non-slender flat delta wing with leading-edge separation modeled by symmetrical conical vortex sheets. A similarity solution for the three dimensional steady velocity potential Φ is sought with boundary conditions to be satisfied on the line which is the intersection of the wing sheet surface with the surface of the unit sphere. A numerical approach is developed based on the construction of a special boundary element or ‘winglet’ which is effectively a Green function for the projection of ∇2Φ = 0 onto the spherical surface under the similarity ansatz. When the wing semi-apex angle γo is fixed satisfaction of the boundary conditions of zero normal velocity on the wing and zero normal velocity and pressure continuity across the vortex sheet then leads to a nonlinear eigenvalue problem. A method of ensuring a condition of zero lateral force on a lumped model of the inner part of the rolled-up vortex sheet gives a closed set of a equations which is solved numerically by Newton's method. We present and discuss the properties of solutions for γ0 in the range 1.30 < γ <89.50. The dependencies of these solutions on γ0 differs qualitatively from predictions of slender-body theory. In particular the velocity field is in general not conical and the similarity exponent must be calculated as part of the global eigenvalue problem. This exponent, together with the detailed flow field including the position and structure of the separated vortx sheet, depend only on γ0. In the limit of small γ0, a comparison with slender-body theory is made on the basis of an effective angle of incidence.
Deep-water internal solitaty waves
Authors: Stamp Andrew P. , Jacka Marcus
Published: April 2006

An experimental investigation of mode-2 (’lump-Like’) Solitary waves propagaling on a thin interface between two deep layers of different densities is presented. Small-and large-amplitude waves behaved differently: small waves carried energy and momentum, whereas sufficiently large waves also carried mass. Weakly nonlinear theory anticipated the result for amplitudes a/h [les ] 0.5 but did not provide even a qualitative description of the large-amplitude waves. In particular, the prediction that for waves to maintain permanent form their wavelength must decrease with increasing amplitude failed; instead the wavelength of large waves was observed to increase with increasing amplitude. Furthermore, whilst the waves were expected to emerge from interactions along their precollision trajectories, the large waves actually suffered a backward shift.
Analogy between turbulent flows in curved pipes and orthogonally rotating pipes
Authors: Ishigaki Hiroshi
Published: April 2006

A quantitative analogy between fully developed turbulent flows in curved pipes and orthogonally rotating pipes will be described through similarity arguments, the use of experimental data and computational results. A pair of similarity parameters will be derived for each turbulent flow, so that they have the same dynamical meaning as those of laminar flows. When the second parameter for each flow is large enough, it will be shown that friction factors, as well as heat transfer rates, of the two flows coincide for equal values of the fundamental parameters. Computed contours of velocity and temperature will also reveal strong similarities between the two flows.
Global characteristics of jet impact
Authors: Korobkin Alexander
Published: April 2006

This paper is concerned with the impact of a jet of arbitrary cross-section onto a rigid plane. At the initial stage of the impact the liquid motion is described within the framework of the acoustic approximation. Behind the shock front which is generated under the impact, the pressure distribution is calculated analytically for an arbitrary cross-section of the jet. It is shown that a quarter of the total impact energy transfers into the internal energy of the compressed liquid. The focusing of the compression and relief waves in the axisymmetrical case is discussed.
A new oscillatory instability in a mushy layer during the solidification of binary alloys
Authors: Anderson D. M. , Worster M. Grae
Published: April 2006

We consider the solidification of a binary alloy in a mushy layer and analyse the linear stability of a quiescent state with specific interest in identifying an oscillatory convective instability. We employ a near-eutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to examine the dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. We consider also the limit of large Stefan number, which incorporates a key balance necessary for the existence of the oscillatory instability. The model we consider here contains no double-diffusive effects and no region in which a statically stable density gradient exists. The mechanism underlying the oscillatory instability we discover is instead associated with a complex interaction between heat transfer, convection and solidification.
Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid
Authors: Spedding G. R. , Browand F. K. , Fincham A. M.
Published: April 2006

Late wakes (Nt> 20) of towed spheres in a stably stratified fluid were analysed in a plane using a reliable, customized DPIV technique that provides sufficient spatial and temporal resolution to cover all important scales of motion in this freely decaying geophysical flow. Systematic experiments were conducted with independent variation ofRe∈ [103, 104] andF∈ [1, 10] (F ≡ 2U/NDis an internal Froude number based on the buoyancy frequency,N, and the sphere radius,D/2), and for selected {Re, F} pairs above this range.

The normalized wake width grows at approximately the same rate as in a three-dimensional unstratified wake, but it becomes narrower, not wider, with decreasingF(i.e. as stratification effects become more important). The centreline defect velocity, on the other hand, reaches values an order of magnitude above those measured for three-dimensional unstratified wakes at equivalent downstream locations. Both observations are argued to be consequences of the very high degree of order and coherence that emerge in the late-wake vortex structures.

Streamwise-averaged turbulence quantities, such as the velocity fluctuation magnitude, and mean-square enstrophy, show similar power law behaviour for allRe≤ 5 × 103, with exponents equal to those expected in three-dimensional axisym-metric turbulent wakes. There is no obvious physical reason why three-dimensional arguments are so successful in such a flow, and at such long evolution times. The scaling collapses none of the cases forRebelow 4 – 5 × 103, appearing to establish a minimumRefor a class of self-similar stratified wake flows that evolve from fully turbulent initial conditions.

Individual vortex cross-sections appear to be well approximated by Gaussian distributions at allRe, F andNtstudied here. The scaling behaviour of individual vortices mimics that of the statistical, wake-averaged quantities, and differs measurably from a simple two-dimensional viscous diffusion model. The importance of formulating a realistic three-dimensional model is discussed, and some limited steps in this direction point to future useful experiments and modelling efforts.
First- and higher-order effects of curvature and torsion on the flow in a helical rectangular duct
Authors: Bolinder C. Jonas
Published: April 2006

A series expansion method is employed to determine the first-order terms in curvature ε and torsion η of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers,De=Reε1/2, up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germano numbers,Gn= ηRe, up to at least 50 whereReis the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in ε and η neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently smallReis dominated by torsion effects; and that for increasingRe, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall. For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.
Two-dimensional vortex dynamics in a stratified barotropic fluid
Authors: Arendt Steve C.
Published: April 2006

We show that two-dimensional ‘point’ vortex dynamics in both a polytropic fluid of γ = 3/2 and an isothermal fluid stratified by a constant gravitational field can be written in Hamiltonian form. We find that the formulation admits only one constant of the motion in addition to the Hamiltonian, so that two vortices are the most for which the motion is generally integrable. We study in detail the two-vortex problem and find a rich collection of behaviour: closed trajectories analogous to the circular orbits of the uniform-fluid two-vortex problem, open trajectories for which the self-propelled vortices scatter off each other, and both unstable and stable steadily translating pairs of vortices. Comparison is made to the case of two vortices in a uniform-density fluid bounded by a wall.
Axisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer
Authors: Damiano E. R. , Duling B. R. , Ley K. , Skalak T. C.
Published: April 2006

A closed-form analytic solution for the motion of axisymmetric rigid pellets suspended in a Newtonian fluid and driven under a pressure gradient through a rigid impermeable cylindrical tube lined with a porous deformable biphasic wall layer is derived using mixture and lubrication theories. The analysis details the velocity distributions in the lubrication and wall layers as well as the solid-phase displacement field in the wall layer. Expressions for the shear stress and pressure gradient are obtained throughout the lubrication and wall layers. Results are presented in terms of resistance, volume flow, and driving pressure relative to smooth-walled tubes for cases both with and without rigid spheres flowing in the free lumen. The analysis is motivated by its possible relevance to the rheology of blood in the microcirculation wherein the endothelial-cell glycocalyx – a carbohydrate-rich coat of macromolecules consisting of proteoglycans and glycoproteins expressed on the luminal surface of the capillary wall – might exhibit similar behaviour to the wall layer modelled here. Estimates of the permeability of the glycocalyx are taken from experimental data for fibrinogen gels formed in vitro. In a tube without pellets lined with a porous wall layer having a thickness which is 15% of the tube radius and having a permeability in the range of fibrinogen gels, approximately a 70% greater pressure drop is required to achieve the same volume flow as would occur in an equivalent smooth-walled tube without a wall layer. If, in the presence of this same wall layer, a rigid spherical pellet is introduced which is 99.5% of the free-lumen radius, the apparent viscosity increases by as much as a factor of four with a concomitant reduction in tube hematocrit of about 10% relative to the corresponding values in an equivalent smooth-walled tube having the same sphere-to-tube diameter ratio without a wall layer.
A numerical simulation of unsteady flow in a two-dimensional collapsible channel
Authors: Luo X. Y. , Pedley T. J.
Published: April 2006

The collapse of a compressed elastic tube conveying a flow occurs in several physiological applications and has become a problem of considerable interest. Laboratory experiments on a finite length of collapsible tube reveal a rich variety of self-excited oscillations, indicating that the system is a complex, nonlinear dynamical system. Following our previous study on steady flow in a two-dimensional model of the collapsible tube problem (Luo & Pedley 1995), we here investigate the instability of the steady solution, and details of the resulting oscillations when it is unstable, by studying the time-dependent problem. For this purpose, we have developed a time-dependent simulation of the coupled flow – membrane problem, using the Spine method to treat the moving boundary and a second-order time integration scheme with variable time increments.

It is found that the steady solutions become unstable as tension falls below a certain value, say Tu, which decreases as the Reynolds number increases. As a consequence, steady flow gives way to self-excited oscillations, which become increasingly complicated as tension is decreased from Tu. A sequence of bifurcations going through regular oscillations to irregular oscillations is found, showing some interesting dynamic features similar to those observed in experiments. In addition, vorticity waves are found downstream of the elastic section, with associated recirculating eddies which sometimes split into two. These are similar to the vorticity waves found previously for flow past prescribed, time-dependent indentations. It is speculated that the mechanism of the oscillation is crucially dependent on the details of energy dissipation and flow separation at the indentation.

As tension is reduced even further, the membrane is sucked underneath the downstream rigid wall and, although this causes the numerical scheme to break down, it in fact agrees with another experimental observation for flow in thin tubes.
Classification of self-organized vortices in two-dimensional turbulence: the case of a bounded domain
Authors: Chavanis P. H. , Sommeria J.
Published: April 2006

We calculate steady solutions of the Euler equations for any given value of energy and circulation (and angular momentum in the case of a circular domain). A linear relationship between vorticity and stream function is assumed. These solutions correspond to the predicted self-organization into a maximum-entropy state, in the limit of strong mixing. Vorticity mixing is then only weakly restricted by the constraint of energy conservation. While maximum-entropy solutions depend in general on the whole probability distribution of vorticity levels, these linearized results depend only on a single control parameter, yet keeping much of the general structure of the problem. A convenient classification of the maximum-entropy states is thus provided. We show furthermore how to extend these linearized results as an expansion in energy, involving successive moments of the vorticity probability distribution. They are applied to a rectangular domain and compared with existing numerical and laboratory results. We predict that the flow organizes into a single vortex in the square domain, but into a two-vortex dipolar state in a rectangle with aspect ratio greater than 1.12. The case of a circular domain is also explicitly solved, taking into account the conservation of the angular momentum.
Formation of cells in natural convection in a vertical slot at large Prandtl number
Authors: Wakitani Shunichi
Published: April 2006

This paper presents several features at supercritical conditions of natural convection of large Prandtl number fluids in a vertical slot. Both experimental and numerical results show that even at slightly supercritical conditions the multicellular flow is in an unsteady state which is attributed to the moving and merging of cells. The effect of variable viscosity on the unsteadiness is investigated numerically. Successive merging and appearance of cells is characteristic of this convection so that the number of secondary cells changes with time. The bifurcation phenomena characterized by several modes of unsteady cellular flow are examined.
An inverse energy cascade in two-dimensional low Reynolds number bubbly flows
Authors: Esmaeeli Asghar , Tryggvason Gréatar
Published: April 2006

Two direct numerical simulations of several buoyant bubbles in a two-dimensional periodic domain are presented. The average rise Reynolds number of the bubbles is close to 2, and surface tension is high, resulting in small bubble deformation. The void fraction is relatively high, and the bubbles interact strongly. Simulations of the motion of both 144 and 324 bubbles show the formation of flow structures much larger than the bubble size, and a continuous increase in the energy of the low-wavenumber velocity modes. Plots of the energy spectrum show a range of wavenumbers with an approximately -5/3 slope. This suggests that a part of the work done by the buoyant bubbles is not dissipated, but instead increases the energy of flow structures much larger than the bubbles. This phenomenon, which is also seen in numerical simulation of forced two-dimensional turbulence, prevents the appearance of a statistically steady-state motion that is independent of the size of the computational domain.
Thermoacoustic and buoyancy-driven transport in a square side-heated cavity filled with a near-critical fluid
Authors: Zappoli Bernard , Amiroudine Sakir , Carles Pierre , Ouazzani Jalil
Published: April 2006

The mechanisms of heat and mass transport in a side-heated square cavity filled with a near-critical fluid are explored, with special emphasis on the interplay between buoyancy-driven convection and the Piston Effect. The Navier–Stokes equations for a near-critical van der Waals gas are solved numerically by means of an acoustically filtered, finite-volume method. The results have revealed some striking behaviour compared with that obtained for normally compressible gases: (i) heat equilibration is still achieved rapidly, as under zero-g conditions, by the Piston Effect before convection has time to enhance heat transport; (ii) mass equilibration is achieved on a much longer time scale by quasi-isothermal buoyant convection; (iii) due to the very high compressibility, a stagnation-point effect similar to that encountered in high-speed flows provokes an overheating of the upper wall; and (iv) a significant difference to the convective single-roll pattern generated under the same conditions in normal CO2 is found, in the form of a double-roll convective structure.
Measurement of the lift force on a particle fixed to the wall in the viscous sublayer of a fully developed turbulent boundary layer
Authors: Mollinger A. M. , Nieuwstadt F. T. M.
Published: April 2006

We have investigated the lift force on a small isolated particle which is attached to a flat smooth surface and embedded within the viscous sublayer of the turbulent boundary layer over this surface. We have developed a novel experimental technique with which it is possible to measure both the mean and fluctuating lift force by gluing the particle on top of a silicium cantilever. The deflection of this cantilever is measured with a focused laser beam. The sensitivity of the focus detection system allows us to measure a lift force with an average value around 10−8N and with a standard deviation of approximately 5% of the mean. This means that our device is at least a factor of 100 more sensitive than previous devices and at the same time able to measure the lift forces on smaller particles. Data for the mean lift force (FL+) as a function of the particle radius (a+), where both parameters have been non-dimensionalized with the kinematic viscosityvand the friction velocityu*, are obtained in the range 0.3 <a+< 2. The data support the relationship:FL+= (56.9 ± 1.1) (a+)1.87±0.04. Also results on the fluctuating lift force have been obtained. We find that the ratio of the r.m.s. to the mean lift force is approximately 2.8.
On finite-amplitude patterns of convection in a rectangular-planform container
Authors: Daniels P. G. , Weinstein M.
Published: April 2006

This paper considers the development of finite-amplitude patterns of convection in rectangular-planform containers. The horizontal dimensions of the container are assumed to be large compared with the critical wavelength of the motion. An interaction between rolls parallel and perpendicular to the lateral boundaries is modelled by a coupled pair of nonlinear amplitude equations together with appropriate conditions on the four lateral boundaries. At Rayleigh numbers above a critical value a steady-state solution is established with rolls parallel to the shorter lateral sides, consistent with the predictions of linear theory. At a second critical value this solution becomes unstable to cross-rolls near the shorter sides and a new steady state evolves. This consists of the primary roll pattern together with regions near the shorter sides where there is a combination of rolls parallel and perpendicular to the boundary.

Analytical and numerical methods are used to describe both the evolution and steady-state structure of the solution, and a comparison is made with the results of full numerical simulations and experiments.
High-Reynolds-number weakly stratified flow past an obstacle
Authors: Chernyshenko S. I. , Castro Ian P.
Published: April 2006

Stably stratified steady flow past a bluff body in a channel is considered for cases in which the stratification is not sufficiently strong to give solutions containing wave motions. The physical mechanisms by which stratification influences the flow are revealed. In particular, the drag reduction under weak stratification, observed in experiments, is explained. This is achieved by constructing an asymptotic laminar solution for high Reynolds number (Re) and large channel width, which explicitly gives the mechanisms, and using comparisons with numerical results for medium Re and experiments for turbulent flows to argue that these mechanisms are expected to be common in all cases. The results demonstrate the possibility, subject to certain restrictions, of using steady high-Re theory as a tool for studying qualitative features of real flows.
Diascalar flux and the rate of fluid mixing
Authors: Winters Kraig B. , D'Asaro Eric A.
Published: April 2006

We define the rate at which a scalar θ mixes in a fluid flow in terms of the flux of θ across isoscalar surfaces. This flux θd is purely diffusive and is, in principle, exactly known at all times given the scalar field and the coefficient of molecular diffusivity. In general, the complex geometry of isoscalar surfaces would appear to make the calculation of this flux very difficult. In this paper, we derive an exact expression relating the instantaneous diascalar flux to the average squared scalar gradient on an isoscalar surface which does not require knowledge of the spatial structure of the surface itself. To obtain this result, a time-dependent reference state θ(t,z*.) is defined. When the scalar is taken to be density, this reference state is that of minimum potential energy. The rate of change of the reference state due to diffusion is shown to equal the divergence of the diffusive flux, i.e. (∂/∂z*)θd.

This result provides a mathematical framework that exactly separates diffusive and advective scalar transport in incompressible fluid flows. The relationship between diffusive and advective transport is discussed in relation to the scalar variance equation and the Osborn–Cox model. Estimation of water mass transformation from oceanic microstructure profiles and determination of the time-dependent mixing rate in numerically simulated flows are discussed.
Secondary flow in weakly rotating turbulent plane Couette flow
Authors: Bech Knut H. , Andersson Helge I.
Published: April 2006

As in the laminar case, the turbulent plane Couette flow is unstable (stable) with respect to roll cell instabilities when the weak background angular velocity Ωk is antiparallel (parallel) to the spanwise mean flow vorticity (-dU/dy)k. The critical value of the rotation number Ro, based on 2Ω and dU/dy of the corresponding laminar flow, was estimated as 0.0002 at a low Reynolds number with fully developed turbulence. Direct numerical simulations were performed for Ro = ±0.01 and compared with earlier results for non-rotating Couette flow. At the low rotation rates considered, both senses of rotation damped the turbulence and the number of near-wall turbulence-generating events was reduced. The destabilized flow was more energetic, but less three-dimensional, than the non-rotating flow. In the destabilized case, the two-dimensional roll cells extracted a comparable amount of kinetic energy from the mean flow as did the turbulence, thereby decreasing the turbulent kinetic energy. The turbulence anisotropy was practically unaffected by weak spanwise rotation, while the secondary flow was highly anisotropic due to its inability to contract and expand in the streamwise direction.
Mixing in turbulent jets: scalar measures and isosurface geometry
Authors: Catrakis Haris J. , Dimotakis Paul E.
Published: April 2006

Experiments have been conducted to investigate mixing and the geometry of scalar isosurfaces in turbulent jets. Specifically, we have obtained high-resolution, high-signal-to-noise-ratio images of the jet-fluid concentration in the far field of round, liquid-phase, turbulent jets, in the Reynolds number range 4.5 × 103≤Re≤ 18 × 103, using laser-induced-fluorescence imaging techniques. Analysis of these data indicates that this Reynolds-number range spans a mixing transition in the far field of turbulent jets. This is manifested in the probability-density function of the scalar field, as well as in measures of the scalar isosurfaces. Classical as well as fractal measures of these isosurfaces have been computed, from small to large spatial scales, and are found to be functions of both scalar threshold and Reynolds number. The coverage of level sets of jet-fluid concentration in the two-dimensional images is found to possess a scale-dependent-fractal dimension that increases continuously with increasing scale, from near unity, at the smallest scales, to 2, at the largest scales. The geometry of the scalar isosurfaces is, therefore, more complex than power-law fractal, exhibiting an increasing complexity with increasing scale. This behaviour necessitates a scale-dependent generalization of power-law-fractal geometry. A connection between scale-dependent-fractal geometry and the distribution of scales is established and used to compute the distribution of spatial scales in the flow.