bluuebunny/crossref_metadata_2025 huggingface dataset:
Head-on collision of two coaxial vortex rings: experiment and computation
Authors: Chu Chin-Chou , Wang Chi-Tzung , Chang Chien-Cheng , Chang Ray-Yu , Chang Wen-Tyzh
Published: April 2006

Head-on collision of two coaxial vortex rings has been studied by joint experimental and numerical investigation. The Reynolds number, ReΓ, based on the initial circulation of the vortex rings, ranged from 400 to 2700. Besides numerical data, the vorticity field was also resolved by a non-intrusive visualization technique, LIPA, which enabled simultaneous measurement of velocities at multiple locations on a plane area. It was found that the enstrophy, rather than circulation, revealed three stages of evolution of the vortex rings prior to their breakdown. These include the free-travelling stage, stage of vortex stretching and the stage of viscous dissipation dominance. The results indicate that it would be incorrect to neglect the viscous effect, in particular, for the latter two stages of flow development. In fact, the rebound behaviour of the vortex rings for lower ReΓ is essentially a viscous phenomenon and is found to be closely related to the dissipation of enstrophy when the vortex rings are brought to interact actively with each other and is also related to the increase of the vorticity core diameter in the stage of dominance of viscous dissipation. Furthermore, an instant dimensionless group, Nt/ReΓ, based on the local vorticity distribution and the radius of a vortex ring, is found to be appropriate to characterize the onset of instability. Our investigation indicates that, in the range of observation, bulging instability will be observed during collision when Nt/ReΓ exceeds a critical value, (Nt/ReΓ)cr, which is a function of the initial core-size of the vortex ring. Comparisons showed that the numerical, measured, and visualization results were in consistent agreement; this not only enables us to assess the range of validity of the axisymmetry assumed for the numerical simulation, but also provides us with a rational basis for further analysis of azimuthal instability.
On turbulent spots in a laminar boundary layer subjected to a self-similar adverse pressure gradient
Authors: Seifert Avi , Wygnanski Israel J.
Published: April 2006

The characteristics of a turbulent spot propagating in a laminar boundary layer subjected to a self-similar adverse pressure gradient (defined by a Falkner–Skan parameter β = -0.1) were investigated experimentally. It was observed that some small differences in the normalized shape of the undisturbed velocity profile caused by the pressure gradient had a major influence on the spreading rate of the spot at comparable Reδ*. The rate of spread of the spot in the spanwise direction was affected most dramatically by the pressure gradient where the average angle at which the tips of the spots moved outward relative to the plane of symmetry was 21°. It was noted that the strength and duration of the disturbance initiating the spots had an effect on their spanwise rate of spread. For example, a strong impulsive disturbance and a disturbance caused by a stationary three-dimensional roughness generated spots which spread at a much smaller rate. The interaction of the spot with the wave packet existing beyond its tip was enhanced by the adverse pressure gradient because the Reynolds number of the surrounding boundary layer was everywhere supercritical. Thus, the maximum linear amplification rate in this case is approximately four times larger than in Blasius flow. Some features of the breakdown and their relationship to the shape and the perturbation velocities in the spot are discussed. The normalized length of the calmed region relative to the length of the spot is enhanced by the adverse pressure gradient and by an increase in the intensity of the disturbance.
The interaction of a moving fluid/fluid interface with a flat plate
Authors: Billingham J. , King A. C.
Published: April 2006

A well-known technique for metering a multiphase flow is to use small probes that utilize some measurement principle to detect the presence of different phases surrounding their tips. In almost all cases of relevance to the oil industry, the flow around such local probes is inviscid and driven by surface tension, with negligible gravitational effects. In order to study the features of the flow around a local probe when it meets a droplet, we analyse a model problem: the interaction of an infinite, initially straight, interface between two inviscid fluids, advected in an initially uniform flow towards a semi-infinite thin flat plate oriented at 90° to the interface. This has enabled us to gain some insight into the factors that control the motion of a contact line over a solid surface, for a range of physical parameter values.

The potential flows in the two fluids are coupled nonlinearly at the interface, where surface tension is balanced by a pressure difference. In addition, a dynamic contact angle boundary condition is imposed at the three-phase contact line, which moves along the plate. In order to determine how the interface deforms in such a flow, we consider the small- and large-time asymptotic limits of the solution. The small-time and linearized large-time problems are solved analytically, using Mellin transforms, whilst the general large-time problem is solved numerically, using a boundary integral method.

The form of the dynamic contact angle as a function of contact line velocity is the most important factor in determining how an interface deforms as it meets and moves over the plate. Depending on this, the three-phase contact line may, at one extreme, hang up on the leading edge of the plate or, at the other extreme, move rapidly along the surface of the plate. At large times, the solution asymptotes to an interface configuration where the contact line moves at the far-field velocity.
A three-dimensional computation of the force and torque on an ellipsoid settling slowly through a viscoelastic fluid
Authors: Feng J. Feng , Joseph D. D. , Glowinski R. , Pan T. W.
Published: April 2006

The orientation of an ellipsoid falling in a viscoelastic fluid is studied by methods of perturbation theory. For small fall velocity, the fluid's rheology is described by a second-order fluid model. The solution of the problem can be expressed by a dual expansion in two small parameters: the Reynolds number representing the inertial effect and the Weissenberg number representing the effect of the non-Newtonian stress. Then the original problem is split into three canonical problems: the zeroth-order Stokes problem for a translating ellipsoid and two first-order problems, one for inertia and one for second-order rheology. A Stokes operator is inverted in each of the three cases. The problems are solved numerically on a three-dimensional domain by a finite element method with fictitious domains, and the force and torque on the body are evaluated. The results show that the signs of the perturbation pressure and velocity around the particle for inertia are reversed by viscoelasticity. The torques are also of opposite sign: inertia turns the major axis of the ellipsoid perpendicular to the fall direction; normal stresses turn the major axis parallel to the fall. The competition of these two effects gives rise to an equilibrium tilt angle between 0° and 90° which the settling ellipsoid would eventually assume. The equilibrium tilt angle is a function of the elasticity number, which is the ratio of the Weissenberg number and the Reynolds number. Since this ratio is independent of the fall velocity, the perturbation results do not explain the sudden turning of a long body which occurs when a critical fall velocity is exceeded. This is not surprising because the theory is valid only for slow sedimentation. However, the results do seem to agree qualitatively with ‘shape tilting’ observed at low fall velocities.
Nonlinear instability of liquid jets with thermocapillarity
Authors: Mashayek F. , Ashgriz N.
Published: April 2006

The breakup mechanism of a capillary jet with thermocapillarity is investigated. Effects of the heat transfer from the liquid to the surrounding ambient, the liquid thermal conductivity, and the temperature-dependent surface tension coefficient on the jet instability and the formation of satellite drops are considered. Two different disturbances are imposed on the jet. In the first case, the jet is exposed to a spatially periodic ambient temperature. In addition to the thermal boundary condition, an initial surface disturbance with the same wavenumber as the thermal disturbance is also imposed on the jet. Both in-phase and out-of-phase thermal disturbances with respect to surface disturbances are considered. For the in-phase thermal disturbances, a parameter set is obtained at which capillary and thermocapillary effects can cancel each other and the jet attains a stable configuration. No such parameter set can be obtained when the thermocapillary flows are in the same direction as the capillary flows, as in the out-of-phase thermal disturbances. In the second case, only an initial thermal disturbance is imposed on the surface of the liquid while the ambient temperature is kept spatially and temporally uniform.
On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 1. Fundamental principles
Authors: Vladimirov V. A. , Moffatt H. K.
Published: April 2006

A new frozen-in field w (generalizing vorticity) is constructed for ideal magnetohydrodynamic flow. In conjunction with the frozen-in magnetic field h, this is used to obtain a generalized Weber transformation of the MHD equations, expressing the velocity as a bilinear form in generalized Weber variables. This expression is also obtained from Hamilton's principle of least action, and the canonically conjugate Hamiltonian variables for MHD flow are identified. Two alternative energy-type variational principles for three-dimensional steady MHD flow are established. Both involve a functional R which is the sum of the total energy and another conserved functional, the volume integral of a function Φ of Lagrangian coordinates. It is shown that the first variation δ1R vanishes if Φ is suitably chosen (as minus a generalized Bernoulli integral). Expressions for the second variation δ2R are presented.
Multi-mode models of flow and of solute dispersion in shallow water. Part 1. General derivation
Authors: Smith Ronald
Published: April 2006

Instead of considering just the vertically averaged current and the vertically averaged concentration, a multi-mode model is derived in which more of the vertical structure can be computed directly rather than being lumped into a dispersion coefficient. Test cases, of laminar flows, are used to quantify the accuracy of the lowest non-trivial truncation (two modes) in replicating both the flow and the dispersion process.
Three-dimensional calculations of the simple shear flow around a single particle between two moving walls
Authors: Nirschl H. , Dwyer H. A. , Denk V.
Published: April 2006

Three-dimensional solutions have been obtained for the steady simple shear flow over a spherical particle in the intermediate Reynolds number range 0.1 [les ] Re [les ] 100. The shear flow was generated by two walls which move at the same speed but in opposite directions, and the particle was located in the middle of the gap between the walls. The particle-wall interaction is treated by introducing a fully three-dimensional Chimera or overset grid scheme. The Chimera grid scheme allows each component of a flow to be accurately and efficiently treated. For low Reynolds numbers and without any wall influence we have verified the solution of Taylor (1932) for the shear around a rigid sphere. With increasing Reynolds numbers the angular velocity for zero moment for the sphere decreases with increasing Reynolds number. The influence of the wall has been quantified with the global particle surface characteristics such as net torque and Nusselt number. A detailed analysis of the influence of the wall distance and Reynolds number on the surface distributions of pressure, shear stress and heat transfer has also been carried out.
Nonlinear Taylor–Couette flow of helium II
Authors: Henderson Karen L. , Barenghi Carlo F. , Jones Chris A.
Published: April 2006

We solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations of motion of helium II for the first time and investigate the configuration of quantized vortex lines in Taylor–Couette flow. The results are interpreted in terms of quantities which can be observed by measuring the attenuation of second sound. Comparison is made with existing experimental results.
Inception of channelization and drainage basin formation: upstream-driven theory
Authors: Izumi Norihiro , Parker Gary
Published: April 2006

The ubiquitous presence of river drainage basins in the terrestrial environment suggests that distributed overland flow generated by rainfall tends to spontaneously organize itself into dendritic systems of discrete channels. Several recent numerical models describe the evolution of complete drainage basins from the initial condition of rainfall on a flat, tilted plateau, the surface of which has been provided with random elevation perturbations. These analyses model overland flow via the assumption of a perfect balance between gravitational and frictional terms, i.e. in terms of normal flow.

Linear stability analysis applied to the normal flow model has been shown, however, to fail to select a wavelength corresponding to a finite distance of separation between incipient basins. This suggests that the normal flow model may not be a sufficient basis for studying drainage basin development, especially at the finest scales of morphologic significance.

Here the concept of a threshold condition for bed erosion is combined with an analysis of the full equations of shallow overland flow in order to study wavelength selection. Classical linear stability analysis is shown to be inadequate to analyse the problem at the level of inception. An alternative linear analysis of bed perturbations based on the threshold condition is developed, and shown to lead to the selection of finite wavelength of the correct order of magnitude.

The analysis here is driven from the upstream direction in that bed erosion is first caused only when sufficient flow has gathered from upstream due to rainfall. A downstream-driven theory of incipient channelization that is not necessarily dependent upon rainfall is presented in Izumi (1993), and is presently in preparation for publication.
Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory
Authors: Kushner Paul J. , Shepherd Theodore G.
Published: April 2006

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.

In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms.

The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent.

This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii).

In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.
Critical microjets in collapsing cavities
Authors: Longuet-Higgins Michael S. , Oguz Hasan
Published: April 2006

Inward microjets are commonly observed in collapsing cavities, but here we show that jets with exceptionally high velocities and accelerations occur in certain critical flows dividing jet formation from bubble pinch-off. An example of the phenomenon occurs in the family of flows which evolve from a certain class of initial conditions: the initial flow field is that due to a moving point sink within the cavity.

A numerical study of the critical flow shows that in the neighbourhood of microjet formation the flow is self-similar. The local accelerations, velocities and distances scale as tβ-2, tβ-1 and tβ respectively, where β = 0.575. The velocity potential is approximately a spherical harmonic of degree ¼.
The flow induced by the torsional oscillations of an elliptic cylinder
Authors: Riley N. , Wybrow M. F.
Published: April 2006

We consider the fluid motion induced when an elliptic cylinder performs small-amplitude torsional oscillations about an axis parallel to a generator which passes through either the centre or a point on the major or minor axis of the ellipse. In common with other fluid flows dominated by oscillatory motion, a time-independent, or steady streaming flow develops. This steady streaming exhibits several unusual and unexpected features, which are confirmed by experiment.
Modulation of three resonating gravity–capillary waves by a long gravity wave
Authors: Trulsen Karsten , Mei Chiang C.
Published: April 2006

We consider a resonant triad of gravity–capillary waves, riding on top of a much longer gravity wave. The long-wave phase is assumed to vary on the same scale as the slow modulation of the short waves. Envelope equations are first deduced in the Lagrangian description. By perturbation analysis for a weak long wave, we then find that the long wave can resonate the natural modulation oscillations of the triad envelope, giving rise to various bifurcations in the Poincaré map. Numerical integration for a stronger long wave reveals that chaos can emerge from these bifurcations. The bifurcation criterion of Chen & Saffman (1979) for collinear Wilton's ripples is generalized to arbitrary non-collinear triads, and is found to play an important role as a criterion for the onset of chaotic behaviour.
Quasi-periodic cylinder wakes and the Ginzburg–Landau model
Authors: Albarède Pierre , Provansal Michel
Published: April 2006

The time-periodic phenomena occurring at low Reynolds numbers (Re [lsim ] 180) in the wake of a circular cylinder (finite-length section) are well modelled by a Ginzburg–Landau (GL) equation with zero boundary conditions (Albarède & Monkewitz 1992). According to the GL model, the wake is mainly governed by a rescaled length, based on the aspect ratio and the Reynolds number. However, the determination of coefficients is not complete: we correct a former evaluation of the nonlinear Landau coefficient, we show difficulties in obtaining a consistent set of coefficients for different Reynolds numbers or end configurations, and we propose the use of an ‘influential’ length. New two-point velocimetry results are presented: phase measurements show that a subtle property is shared by the three-dimensional wake and the GL model.

Two time-quasi-periodic phenomena – the second mode observed for smaller aspect ratios, and the dislocated chevron observed for larger aspect ratios – are presented and precisely related to the GL model. Only the linear characteristics of the second mode are readily explained; its existence depends on the end conditions. Moreover, through a quasi-static variation of the length, the second mode evolves continuously to end cells (and vice versa). Observations of the dislocated chevron are recalled. A very similar instability is found on the chevron solution of the GL equation, when the model parameters (c1, c2) move towards the phase diffusion unstable region. The early stages of this instability are qualitatively similar to the observed patterns.
A slender ship moving at a near-critical speed in a shallow channel
Authors: Chen Xue-Nong , Sharma Som Deo
Published: April 2006

The problem solved concerns a slender ship moving at a near-critical steady speed in a shallow channel, not necessarily in symmetric configuration, involving the special phenomenon of generation of solitary waves. By using the technique of matched asymptotic expansions along with nonlinear shallow-water wave theory, the problem is reduced to a Kadomtsev–Petviashvili equation in the far field, matched with a nearfield solution obtained by an improved slender-body theory, taking the local wave elevation and longitudinal disturbance velocity into account. The ship can be either fixed or free to squat. Besides wave pattern and wave resistance, the hydrodynamic lift force and trim moment are calculated by pressure integration in the fixed-hull case; running sinkage and trim, by condition of hydrodynamic equilibrium in the free-hull case. The numerical procedure for solving the KP equation consists of a finite-difference method, namely, fractional step algorithm with Crank–Nicolson-like schemes in each half step. Calculated results are compared with several published shipmodel experiments and other theoretical predictions; satisfactory agreement is demonstrated.
Air entrapment by a falling water mass
Authors: Oguz Hasan N. , Prosperetti Andrea , Kolaini Ali R.
Published: April 2006

The impact of a nearly cylindrical water mass on a water surface is studied both experimentally and theoretically. The experiments consist of the rapid release of water from the bottom of a cylindrical container suspended above a large water tank and of the recording of the free-surface shape of the resulting crater with a high-speed camera. A bubble with a diameter of about twice that of the initial cylinder remains entrapped at the bottom of the crater when the aspect ratio and the energy of the falling water mass are sufficiently large. Many of the salient features of the phenomenon are explained on the basis of simple physical arguments. Boundary-integral potential-flow simulations of the process are also described. These numerical results are in fair to good agreement with the observations.
Unicellular natural circulation in a shallow horizontal porous layer heated from below by a constant flux
Authors: Kimura S. , Vynnycky M. , Alavyoon F.
Published: April 2006

Natural convection in a saturated horizontal porous layer heated from below and cooled at the top with a constant flux is studied both analytically and numerically. Linear stability analysis indicates that unicellular recirculation remains a stable mode of flow as the aspect ratio (A) of the layer is increased, in contrast to the situation for an isothermally heated and cooled layer. An analytical solution is presented for fully developed counterflow in the infinite-aspect-ratio limit; this flow is found to be linearly stable to transverse disturbances for Rayleigh number (Ra) as high as 506, at which point a Hopf bifurcation sets in; however, further analysis indicates that an exchange of stability due to longitudinal disturbances will occur much sooner at Ra ≈ 311.53. The velocity and temperature profiles of the counterflow solution, whilst not strictly speaking valid in the extreme end regions of the layer, otherwise agree very well with full numerical computations conducted for the ranges 25 [les ] Ra [les ] 1050, 2 [les ] A [les ] 10. However, for sufficiently high Rayleigh number (Ra between 630 and 650 for A = 8 and Ra between 730 and 750 for A = 4, for example), the computations indicate transition from steady unicellular to oscillatory flow, in line with the Hopf bifurcation predicted by the linear stability analysis for infinite aspect ratio.
Instability of a liquid film flow over a vibrating inclined plane
Authors: Woods David R. , Lin S. P.
Published: April 2006

The problem of the onset of instability in a liquid layer flowing down a vibrating inclined plane is formulated. For the solution of the problem, the Fourier components of the disturbance are expanded in Chebychev polynomials with time-dependent coefficients. The reduced system of ordinary differential equations is analysed with the aid of Floquet theory. The interaction of the long gravity waves, the relatively short shear waves and the parametrically resonated Faraday waves occurring in the film flow is studied. Numerical results show that the long gravity waves can be significantly suppressed, but cannot be completely eliminated by use of the externally imposed oscillation on the incline. At small angles of inclination, the short shear waves may be exploited to enhance the Faraday waves. For a given set of relevant flow parameters, there exists a critical amplitude of the plane vibration below which the Faraday wave cannot be generated. At a given amplitude above this critical one, there also exists a cutoff wavenumber above which the Faraday wave cannot be excited. In general the critical amplitude increases, but the cutoff wavenumber decreases, with increasing viscosity. The cutoff wavenumber also decreases with increasing surface tension. The application of the theory to a novel method of film atomization is discussed.
Supercritical withdrawal from a two-layer fluid through a line sink
Authors: Hocking G. C.
Published: April 2006

Accurate numerical solutions to the problem of finding the location of the interface between two unconfined regions of fluid of different density during the withdrawal process are presented. Supercritical flows are considered, in which the interface is drawn directly into the sink. As the flow rate is reduced, the interface enters the sink more steeply, until the solution method breaks down just before the interface enters the sink vertically from above, and becomes flow from the lower layer only. This lower bound on supercritical flow is compared with the upper bound on single-layer (free surface) flow with good agreement.
A vertically oscillating plate disturbing the development of a boundary layer
Authors: Miau J. J. , Chen C. R. , Chou J. H.
Published: April 2006

A vertically oscillating plate in a boundary layer regulates the vorticity flux rate with respect to time and displaces the vorticity away from the wall. These phenomena are discussed for non-dimensional frequencies of the oscillating plate K = 0, 0.006, 0.01 and 0.02. The velocity data obtained by a split-fibre probe near the wall in the region immediately downstream of the oscillating plate lead to a discussion on the behaviour of the flow structures with respect to the non-dimensional frequency. The physical understanding deduced is complementary to the findings of a smoke-wire flow visualization conducted in this study. An integral analysis of the momentum equation indicates that the mean vorticity flux rate of the present flow is composed of contributions from both the parallel shear layer and the curving streamline. This analysis further suggests that the mean vorticity flux rate can be obtained through a combination of pressure measurements at the wall and in the irrotational region of the flow.
A new mechanism of small-scale transition in a plane mixing layer: core dynamics of spanwise vortices
Authors: Schoppa W. , Hussain F. , Metcalfe R. W.
Published: April 2006

We present a new mechanism of small-scale transition via core dynamics instability (CDI) in an incompressible plane mixing layer, a transition which is not reliant on the presence of longitudinal vortices (‘ribs’) and which can originate much earlier than ribinduced transition. Both linear stability analysis and direct numerical simulation are used to describe CDI growth and subsequent transition in terms of vortex dynamics and vortex line topology. CDI is characterized by amplifying oscillations of core size non-uniformity and meridional flow within spanwise vortices (‘rolls’), produced by a coupling of roll swirl and meridional flow that is manifested by helical twisting and untwisting of roll vortex lines. We find that energetic CDI is excited by subharmonic oblique modes of shear layer instability after roll pairing, when adjacent rolls with out-of-phase undulations merge. Starting from moderate initial disturbance amplitudes, twisting of roll vortex lines generates within the paired roll opposing spanwise flows which even exceed the free-stream velocity. These flows collide to form a nearly irrotational bubble surrounded by a thin vorticity sheath of a large diameter, accompanied by folding and reconnection of roll vortex lines and local transition. We find that accelerated energy transfer to high wavenumbers precedes the development of roll internal intermittency; this transfer, inferred from increased energy at high wavenumbers and an intensification of roll vorticity, occurs prior to the development of strong opposite-signed (to the mean) spanwise vorticity and granularity of the roll vorticity distribution. We demonstrate that these core dynamics are not reliant upon special symmetries and also occur in the presence of moderate-strength ribs, despite entrapment of ribs within pairing rolls. In fact, the roll vorticity dynamics are dominated by CDI if ribs are not sufficiently strong to first initiate transition; thus CDI may govern small-scale transition for moderate initial 3D disturbances, typical of practical situations. Results suggest that CDI constitutes a new generic mechanism for transition to turbulence in shear flows.
Modelling and analysis of meniscus roll coating
Authors: Gaskell P. H. , Savage M. D. , Summers J. L. , Thompson H. M.
Published: April 2006

Three mathematical models are developed for meniscus roll coating in which there is steady flow of a Newtonian fluid in the narrow gap, or nip, between two contrarotating rolls in the absence of body forces.

The zero flux model predicts a constant pressure gradient within the central core and two eddies, each with an inner structure, in qualitative agreement with observation. The small flux model takes account of a small inlet flux and employs the lubrication approximation to represent fluid velocity as a combination of Couette and Poiseuille flows. Results show that the meniscus coating regime is characterized by small flow rates (λ [Lt ] 1) and a sub-ambient pressure field generated by capillary action at the upstream meniscus. Such flows are found to exist for small modified capillary number, Ca(R/H0)1/2 [lsim ] 0.15, where Ca and R/H0 represent capillary number and the radius to semi-gap ratio, respectively.

A third model incorporates the full effects of curved menisci and nonlinear free surface boundary conditions. The presence of a dynamic contact line, adjacent to the web on the upper roll, requires the imposition of an apparent contact angle and slip length. Numerical solutions for the velocity and pressure fields over the entire domain are obtained using the finite element method. Results are in accord with experimental observations that the flow domain consists of two large eddies and fluid transfer jets or ‘snakes’. Furthermore, the numerical results show that the sub-structure of each large eddy consists of a separatrix with one saddle point, two sub-eddies with centres, and an outer recirculation.

Finally finite element solutions in tandem with lubrication analysis establish the existence of three critical flow rates corresponding to a transformation of the pressure field, the emergence of a ‘secondary snake’ (another fluid transfer jet) and the disappearance of a primary snake.
A spectral boundary element approach to three-dimensional Stokes flow
Authors: Muldowney G. P. , Higdon J. J. L.
Published: April 2006

A novel method is introduced for solving the three-dimensional Stokes equations via a spectral element approach to the boundary integral method. The accuracy and convergence of the method are illustrated through applications involving rigid particles, deformable droplets and interacting particles. New physical results are obtained for two applications in low Reynolds number flow: the permeability of periodic models of a porous membrane and the instability of a toroidal droplet subject to non-axisymmetric perturbations. Further applications are described in the companion paper (Higdon & Muldowney 1995).
Peristaltic transport of two immiscible viscous fluids in a circular tube
Authors: Rao Adabala Ramachandra , Usha Srinivasan
Published: April 2006

Peristaltic motion of two immiscible viscous incompressible fluids in a circular tube is studied in pumping and copumping ranges under long-wavelength and low-Reynoldsnumber assumptions. The effect of the peripheral-layer viscosity on the time-averaged flux and the mechanical efficiency is studied. The formation and growth of the trapping zone in the core and the peripheral layer are explained. It is observed that the bolus volume in the peripheral layer increases with an increase in the viscosity ratio. The limits of the time-averaged flux $\bar{Q}$ for trapping in the core are obtained. The trapping observed in the peripheral layer decreases in size with an increase in $\bar{Q}$ but never disappears. The development of the complete trapping of the core fluid by the peripheral-layer fluid with an increase in the time-averaged flux is demonstrated. The effect of peripheral-layer viscosity on the reflux layer is investigated. It is also observed that the reflux occurs in the entire pumping range for all viscosity ratios and it is absent in the entire range of copumping.
On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers
Authors: Kerswell R. R.
Published: April 2006

The Ekman boundary layer scalings associated with small oscillatory flows in a rapidly rotating system are well known to break down at the critical latitudes. The effect of these anomalous boundary regions on the interior flow has long been an issue of some concern. We argue, using analytical results derived in a model problem based on Stewartson's (1957) split-disk configuration, that these boundary regions spawn internal shear layers. These shear layers have their natural length scale of E1/3 and carry velocities only O(E1/6) smaller than the Rossby number – where E is the Ekman number – if the boundary is concave away from the fluid at the critical latitude. Otherwise, only a weaker shear layer is produced of length scale E1/5 containing velocities of O(E3/10) smaller than the Rossby number rather than the usual O(E1/5) value for viscous flows in the interior. We show how these layers can carry angular momentum by themselves or more significantly in combination with the interior inviscid flow, so as to influence the mean flow at leading order. These conclusions are then discussed in the context of a precessing oblate spheroid of fluid for which detailed experimental observations are available.
Magnetic damping of jets and vortices
Authors: Davidson P. A.
Published: April 2006

It is well known that the imposition of a static magnetic field tends to suppress motion in an electrically conducting liquid. Here we look at the magnetic damping of liquid-mental flows where the Reynolds number is large and the magnetic Reynolds number is small. The magnetic field is taken as uniform and the fluid is either infinite in extent or else bounded by an electrically insulating surface S. Under these conditions, we find that three general principles govern the flow. First, the Lorentz force destroys kinetic energy but does not alter the net linear momentum of the fluid, nor does it change the component of angular momentum parallel to B. In certain flows, this implies that momentum, linear or angular, is conserved. Second, the Lorentz force guides the flow in such a way that the global Joule dissipation, D, decreases, and this decline in D is even more rapid than the corresponding fall in global kinetic energy, E. (Note that both D and E are quadratic in u). Third, this decline in relative dissipation, D / E, is essential to conserving momentum, and is achieved by propagating linear or angular momentum out along the magnetic field lines. In fact, this spreading of momentum along the B-lines is a diffusive process, familiar in the context of MHD turbulence. We illustrate these three principles with the aid of a number of specific examples. In increasing order of complexity we look at a spatially uniform jet evolving in time, a three-dimensional jet evolving in space, and an axisymmetric vortex evolving in both space and time. We start with a spatially uniform jet which is dissipated by the sudden application of a transverse magnetic field. This simple (perhaps even trivial) example provides a clear illustration of our three general principles. It also provides a useful stepping-stone to our second example of a steady three-dimensional jet evolving in space. Unlike the two-dimensional jets studied by previous investigators, a three-dimensional jet cannot be annihilated by magnetic braking. Rather, its cross-section deforms in such a way that the momentum flux of the jet is conserved, despite a continual decline in its energy flux. We conclude with a discussion of magnetic damping of axisymmetric vortices. As with the jet flows, the Lorentz force cannot destroy the motion, but rather rearranges the angular momentum of the flow so as to reduce the global kinetic energy. This process ceases, and the flow reaches a steady state, only when the angular momentum is uniform in the direction of the field lines. This is closely related to the tendency of magnetic fields to promote two-dimensional turbulence.
Drift velocity of spatially decaying waves in a two-layer viscous system
Authors: Piedra-Cueva Ismael
Published: April 2006

This paper analyses the mass transport velocity in a two-layer system induced by the action of progressive waves. First the movement inside the two layers is obtained. Next the mass transport of spatially decaying waves is calculated by solving the momentum and mass conservation equations in the Lagrangian coordinate system. Two different physical situations are analysed: the first is waves in a closed channel and the second is waves in an unbounded domain, where the steady-state mass flux may be non-zero. The influence of the viscous properties of the lower layer on the mass transport in both layers is studied. Comparison with the experiments of Sakakiyama & Bijker (1989) in a water-mud system shows good agreement. The results show that the mass transport velocity can be quite different from the velocity given by the rigid bed theory, depending on the physical properties of the lower layer.
Vortex intensification and collapse of the Lissajous-elliptic ring: single- and multi-filament Biot-Savart simulations and visiometrics
Authors: Fernandez Victor M. , Zabusky Norman J. , Gryanik Vladimir M.
Published: April 2006

The collapsing ‘Lissajous-elliptic’ (LE) vortex ring is examined via quantifications of Single- and multi-filament Biot-Savart numerical simulations. In the single-filament simulations, parametric studies show simple relationships between the collapse boundary and the impulse and energy invariants. Collapse becomes non-monotonic in time, for a sufficiently small initial core ‘radius’. Self-similar, singular-like behaviour of the off-filament strain-rate growth has been observed in a small interval, just prior to core overlapping. The computation of the strain-rate eigenvalues and vortex stretching in a diagnostics box surrounding the collapse region yields patterns observed previously in continuum simulations. New diagnostics are presented, including line densities of the energy and the linear and angular momentum, all of which approach zero in the collapse region of the ring. These diagnostics may provide critical parameters for initiating surgery in a topology-changing algorithm. Our multi-filament simulations exhibit layer-like vortex regions and a ‘torus’-shaped vortex stretching pattern observed previously in continuum periodic-domain simulations of vortex reconnection. Quantifications in a cross-section of the collapse region indicate that the circulation tends to concentrate in the head or frontside of the convecting dipolar structure. This is also the location of the incipient ‘bridge’ which is evolving from the weak filaments that have been convected from the initially outer-vortex regions. The formation of this smaller scale vortex structure exhibits the largest vorticity amplification in the variable-core model simulations.
Dynamical behaviour of a premixed turbulent open V-flame
Authors: Rhee C. W. , Talbot L. , Sethian J. A.
Published: April 2006

The level-set approach of Osher & Sethian to tracking interfaces is successfully adapted to the simulation of a premixed turbulent open V-flame including the effects of exothermicity and baroclinicity. In accord with experimental observations this algorithm, along with a flame anchoring scheme, predicts flame cusping for a case in which a strong vortex pair interacts with the flame front. The computed velocity and scalar statistics obtained for the turbulent V-flame compare reasonably well with experimental results by Cheng & Shepherd, and demonstrate the importance of flame-generated vorticity in the determination of flame dynamics and product velocity characteristics.
Extensions of the mild-slope equation
Authors: Porter D. , Staziker D. J.
Published: April 2006

The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction-refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory.
Reconsideration of oblique shock wave reflections in steady flows. Part 1. Experimental investigation
Authors: Chpoun A. , Passerel D. , Li H. , Ben-Dor G.
Published: April 2006

The reflection of shock waves over straight reflecting surfaces in steady flows was investigated experimentally using the supersonic wind tunnel of Laboratoire d'Aerothermique du CNRS, Meudon, France. The results for a flow Mach number M0 = 4.96 contradict the state of the art regarding the regular [harr ] Mach reflection transition in steady flows. Not only was a hysteresis found to exist in this transition, but, unlike previous reports, regular reflection configurations were found to be stable in the dual-solution domain in which theoretically both regular and Mach reflection are possible.
Numerical investigation of Richtmyer–Meshkov instability using front tracking
Authors: Holmes Richard L. , Grove John W. , Sharp David H.
Published: April 2006

Front tracking simulations of the Richtmyer-Meshkov instability produce significantly better agreement with experimentally measured growth rates than obtained in nontracking computations. Careful analysis of the early stages of the shock acceleration process show that nonlinearity and compressibility play a critical role in the behaviour of the shocked interface and are responsible for the deviations from the linear theories. The late-time behaviour of the interface growth rate is compared to a nonlinear potential flow model of Hecht et al.
Analytical description of the breakup of liquid jets
Authors: Papageorgiou Demetrios T.
Published: April 2006

A viscous or inviscid cylindrical jet with surface tension in a surrounding medium of negligible density tends to pinch owing to the mechanism of capillary instability. We construct similarity solutions which describe this phenomenon as a critical time is encountered, for three distinct cases: (i) inviscid jets governed by the Euler equations, (ii) highly viscous jets governed by the Stokes equations, and (iii) viscous jets governed by the Navier-Stokes equations. We look for singular solutions of the governing equations directly rather than by analysis of simplified models arising from slender-jet theories. For Stokes jets implicitly defined closed-form solutions are constructed which allow the scaling exponents to be fixed. Navier-Stokes pinching solutions follow rationally from the Stokes ones by bringing unsteady and nonlinear terms into the momentum equations to leading order. This balance fixes a set of universal scaling functions for the phenomenon. Finally we show how the pinching solutions can be used to provide an analytical description of the dynamics beyond breakup.
A numerical study of pulsating flow behind a constriction
Authors: Rosenfeld Moshe
Published: April 2006

The flow field behind a constricted channel is studied numerically. A pulsating incoming flow with a non-vanishing mean is imposed at the entrance and the flow field is investigated for a wide range of Reynolds and Strouhal numbers (1500 > Re > 45, 12 > St > 0.01). In most cases (except at the two ends of the Strouhal number regime or for Re < 90), propagating vortices are found downstream of the constriction with a wavy core flow between them. The size and number of coexisting vortices depend on St but less on Re. The strength and structure of the vortical regions depend on both Re and St. The formation of the vortices is discussed for the various St regimes and the characteristics of the vortical flow are described.
Evolution of near-inertial waves
Authors: Kloosterziel R. C. , Müller P.
Published: April 2006

The three-dimensional evolution of near-inertial internal gravity waves is investigated for the case of a laterally unbounded fluid layer of constant finite depth. A general Green's function formulation is derived which can be used to solve initial value problems or study the effect of forcing. The Green's function is expanded in vertical normal modes, and is very singular. Convolutions with finite-sized initial conditions lead however to well-behaved solutions. Expansions in similarity solutions of the diffusion equation are shown to be an alternative for finding exact solutions to initial value problems, with respect to one normal mode. For the case of constant buoyancy frequency normal modes expansions are shown to be equivalent to expansions in an alternative series of which the first term is the response on the infinite domain, all the others being corrections to account for the no-flux boundary condition on the upper and lower boundaries.