bluuebunny/crossref_metadata_2025 huggingface dataset:
Finite-amplitude convection in the presence of an unsteady shear flow
Authors: Hall Philip
Published: April 2006

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.
Low Reynolds number motion of bubbles, drops and rigid spheres through fluid–fluid interfaces
Authors: Manga Michael , Stone H. A.
Published: April 2006

The low Reynolds number buoyancy-driven translation of a deformable drop towards and through a fluid–fluid interface is studied using boundary integral calculations and laboratory experiments. The Bond numbers characteristic of both the drop and the initially flat fluid–fluid interface are sufficiently large that the drop and interface become highly deformed, substantial volumes of fluid may be entrained across the interface, and breakup of both interfaces may occur. Specifically, drops passing from a higher- to lower-viscosity fluid are extended vertically as they pass through the interface. For sufficiently large drop Bond numbers, the drop may deform continuously, developing either an elongating tail or enlarging cavity at the back of the drop, analogous to the deformation characteristic of a single deformable drop in an unbounded fluid. The film of fluid between the drop and interface thins most rapidly for those cases that the drop enters a more viscous fluid or has a viscosity lower than the surrounding fluids. In the laboratory experiments, bubbles entering a less viscous fluid are extended vertically and may break into smaller bubbles. The column of fluid entrained by particles passing through the interface may also break into drops. Further experiments with many rigid particles indicate that the spatial distribution of particles may change as the particles pass through interfaces: particles tend to form clusters.
Regeneration mechanisms of near-wall turbulence structures
Authors: Hamilton James M. , Kim John , Waleffe Fabian
Published: April 2006

Direct numerical simulations of a highly constrained plane Couette flow are employed to study the dynamics of the structures found in the near-wall region of turbulent flows. Starting from a fully developed turbulent flow, the dimensions of the computational domain are reduced to near the minimum values which will sustain turbulence. A remarkably well-defined, quasi-cyclic and spatially organized process of regeneration of near-wall structures is observed. This process is composed of three distinct phases: formation of streaks by streamwise vortices, breakdown of the streaks, and regeneration of the streamwise vortices. Each phase sets the stage for the next, and these processes are analysed in detail. The most novel results concern vortex regeneration, which is found to be a direct result of the breakdown of streaks that were originally formed by the vortices, and particular emphasis is placed on this process. The spanwise width of the computational domain corresponds closely to the typically observed spanwise spacing of near-wall streaks. When the width of the domain is further reduced, turbulence is no longer sustained. It is suggested that the observed spacing arises because the time scales of streak formation, breakdown and vortex regeneration become mismatched when the streak spacing is too small, and the regeneration cycle at that scale is broken.
Investigation of the effects of a compliant surface on boundary-layer stability
Authors: Lee T. , Fisher M. , Schwarz W. H.
Published: April 2006

Some of the effects of a passive, single-layer, viscoelastic compliant surface on the stability of a Blasius boundary layer were investigated in a low-turbulence wind tunnel. Measurements of the wavelength and growth rates of vibrating-ribbon-excited harmonic waves were made by hot-wire anemometry. The data for three compliant surfaces with different shear moduli, material damping coefficients, and thicknesses were compared to rigid-surface data. The flow-induced surface displacements were measured using an electro-optic displacement transducer. The results show that the growth rates of unstable Tollmien–Schlichting waves, and the extent of the unstable region in the (F, Rδ*)-plane are reduced over the compliant surfaces relative to those over a rigid surface with the absence of flow-induced surface instabilities. The suppression of the Tollmien–Schlichting waves is accompanied by a surface motion driven by the flow field at the excitation frequency. The experimental results suggest that a delay of the onset to turbulence is possible in air by using appropriately tuned surface characteristics. Further experiments are needed to study the three-dimensional disturbance mode, the flow-induced surface instabilities and the breakdown process.
Fingering instabilities in vertical miscible displacement flows in porous media
Authors: Manickam O. , Homsy G. M.
Published: April 2006

The fingering instabilities in vertical miscible displacement flows in porous media driven by both viscosity and density contrasts are studied using linear stability analysis and direct numerical simulations. The conditions under which vertical flows are different from horizontal flows are derived. A linear stability analysis of a sharp interface gives an expression for the critical velocity that determines the stability of the flow. It is shown that the critical velocity does not remain constant but changes as the two fluids disperse into each other. In a diffused profile, the flow can develop a potentially stable region followed downstream by a potentially unstable region or vice versa depending on the flow velocity, viscosity and density profiles, leading to the potential for ‘reverse’ fingering. As the flow evolves into the nonlinear regime, the strength and location of the stable region changes, which adds to the complexity and richness of finger propagation. The flow is numerically simulated using a Hartley-transform-based spectral method to study the nonlinear evolution of the instabilities. The simulations are validated by comparing to experiments. Miscible displacements with linear density and exponential viscosity dependencies on concentration are simulated to study the effects of stable zones on finger propagation. The growth rates of the mixing zone are parametrically obtained for various injection velocities and viscosity ratios.
Near-contact electrophoretic particle motion
Authors: Loewenberg Michael , Davis Robert H.
Published: April 2006

The near-contact axisymmetric electrophoretic motion of a pair of spherical particles with thin electric double layers and differing surface zeta-potentials is analysed for low Reynolds numbers and moderate surface potentials. Near-contact electrophoretic motion of a spherical particle normal to a planar conducting boundary is analysed under the same assumptions. Pairwise motion is computed by considering touching particles in point contact; relative motion is described by a perturbation about the touching state using lubrication theory. Analytical formulae are derived for two particles of disparate sizes, and for the motion of a single particle towards a boundary; numerical calculations are performed for all size ratios. The results have a universal form with respect to the particle zeta-potentials. All results indicate that the electrophoresis is a much more efficient mechanism of near-contact motion than is buoyancy. An explanation for this finding is given in terms of the electro-osmotic slip velocity on the particle surfaces that facilitates fluid removal from between approaching surfaces.
Baroclinic instability of quasi-geostrophic flows localized in a thin layer
Authors: Benilov E. S.
Published: April 2006

This paper examines the baroclinic instability of a quasi-geostrophic flow with vertical shear in a continuously stratified fluid. The flow and density stratification are both localized in a thin upper layer. (i) Disturbances whose wavelength is much smaller than the deformation radius (based on the depth of the upper layer) are demonstrated to satisfy an ‘equivalent two-layer model’ with properly chosen parameters. (ii) For disturbances whose wavelength is of the order of, or greater than, the deformation radius we derive a sufficient stability criterion. The above analysis is applied to the subtropical and subarctic frontal currents in the Northern Pacific. The effective time of growth of disturbances (i) is found to be 16–22 days, the characteristic spatial scale is 130–150 km.
The flow behind rings: bluff body wakes without end effects
Authors: Leweke T. , Provansal M.
Published: April 2006

Recent studies have demonstrated the strong influence of end effects on low-Reynoldsnumber bluff body wakes, and a number of questions remain concerning the intrinsic nature of three-dimensional phenomena in two-dimensional configurations. Some of them are answered by the present study which investigates the wake of bluff rings (i.e. bodies without ends) both experimentally and by application of the phenomenological Ginzburg–Landau model. The model turns out to be very accurate in describing qualitative and quantitative observations in a large Reynolds number interval. The experimental study of the periodic vortex shedding regime shows the existence of discrete shedding modes, in which the wake takes the form of parallel vortex rings or ‘oblique’ helical vortices, depending on initial conditions. The Strouhal number is found to decrease with growing body curvature, and a global expression for the Strouhal–Reynolds number relation, including curvature and shedding angle, is proposed, which is consistent with previous straight cylinder results. A secondary instability of the helical modes at low Reynolds numbers is discovered, and a detailed comparison with the Ginzburg–Landau model identifies it as the Eckhaus modulational instability of the spanwise structure of the near-wake formation region. It is independent of curvature and its clear observation in straight cylinder wakes is inhibited by end effects.

The dynamical model is extended to higher Reynolds numbers by introducing variable parameters. In this way the instability of periodic vortex shedding which marks the beginning of the transition range is characterized as the Benjamin–Feir instability of the coupled oscillation of the near wake. It is independent of the shear layer transition to turbulence, which is known to occur at higher Reynolds numbers. The unusual shape of the Strouhal curve in this flow regime, including the discontinuity at the transition point, is qualitatively reproduced by the Ginzburg–Landau model. End effects in finite cylinder wakes are found to cause important changes in the transition behaviour also: they create a second Strouhal discontinuity, which is not observed in the present ring wake experiments.
Instability of a high-speed submerged elastic jet
Authors: Rallison J. M. , Hinch E. J.
Published: April 2006

The linearized inertial instability of the parallel shear flow of a viscoelastic liquid is considered. An elastic Rayleigh equation is derived, for high Reynolds numbers and high Weissenberg numbers, and for a viscoelastic liquid whose first normal stress dominates other stresses. The equation is used to investigate the stability of a submerged jet, that may be planar or axisymmetric, having a parabolic velocity profile. The sinuous mode is found to be fully stabilized by sufficiently large elasticity. The varicose mode in the planar case is partially stabilized, being unstable only at longer wavelengths and with a reduced growth rate. An axisymmetric jet, which is stable to varicose perturbations at zero elasticity, is found to be unstable to short-wave disturbances for small non-zero elasticity. This novel instability involves elastic waves in the shear. It is also present in other modes but does not have the fastest growth rate.
Modified Boussinesq equations and associated parabolic models for water wave propagation
Authors: Chen Yongze , Liu Philip L.-F.
Published: April 2006

The modified Boussinesq equations given by Nwogu (1993a) are rederived in terms of a velocity potential on an arbitrary elevation and the free surface displacement. The optimal elevation where the velocity potential should be evaluated is determined by comparing the dispersion and shoaling properties of the linearized modified Boussinesq equations with those given by the linear Stokes theory over a range of depths from zero to one half of the equivalent deep-water wavelength. For regular waves consisting of a finite number of harmonics and propagating over a slowly varying topography, the governing equations for velocity potentials of each harmonic are a set of weakly nonlinear coupled fourth-order elliptic equations with variable coefficients. The parabolic approximation is applied to these coupled fourth-order elliptic equations for the first time. A small-angle parabolic model is developed for waves propagating primarily in a dominant direction. The pseudospectral Fourier method is employed to derive an angular-spectrum parabolic model for multi-directional wave propagation. The small-angle model is examined by comparing numerical results with Whalin's (1971) experimental data. The angular-spectrum model is tested by comparing numerical results with the refraction theory of cnoidal waves (Skovgaard & Petersen 1977) and is used to study the effect of the directed wave angle on the oblique interaction of two identical cnoidal wavetrains in shallow water.
The unified Kadomtsev–Petviashvili equation for interfacial waves
Authors: Chen Yongze , Liu Philip L.-F.
Published: April 2006

In this paper, the propagation of interfacial waves in a two-layered fluid system is investigated. The interfacial waves are weakly nonlinear and dispersive and propagate in a slowly rotating channel with varying topography and sidewalls, and a weak steady background current field. An evolution equation for the interfacial displacement is derived for waves propagating predominantly in the longitudinal direction of the channel. This new evolution equation is called the unified Kadomtsev–Petviashvili (uKP) equation because most of the KP-type equations existing in the literature for both surface water waves and interfacial waves are special cases of the new evolution equation. The Painlevé PDE test is used to find the conditions under which the uKP equation can be solved by the inverse scattering transform. When these conditions are satisfied, elementary transformations are found to reduce the uKP equation to one of the completely integrable equations: the KP, the Korteweg–de Vries (KdV) or the cylindrical KdV equations. The integral invariants associated with the uKP equation for waves propagating in a varying channel are obtained and their relations with the conservation of mass and energy are discussed.
On the alignment and axisymmetrization of a vertically tilted geostrophic vortex
Authors: Viera F.
Published: April 2006

We investigate the evolution of a vertically tilted geostrophic vortex of cylindrical shape and circular horizontal cross-section using the recently developed method of boundary surface dynamics. The vortex consists of a finite volume of constant potential vorticity immersed in a spatially unbounded fluid of uniform density stratification. The fully nonlinear three-dimensional problem is then reduced to the calculation of the Lagrangian evolution of the boundary surface of the vortex region, thus decreasing the dimensionality by one. In the numerical simulations presented here, the vortex shows a general tendency to attain vertical alignment and a horizontal axisymmetrical shape by wobbling about its centre and going through three basic stages of evolution: (a) the circular horizontal cross-sections of the upper and lower parts of the vortex distort and become elongated; (b) the upper and lower sections then become vertically aligned by reducing their horizontal intercentroid distances; and (c) the distorted horizontal cross-sections relax towards axisymmetry, often through the process of filamentation. For a given vortex height, if the horizontal scale of the flow is close to the internal radius of deformation, or equivalently, the density stratification is not too strong, the processes of filamentation and vertical alignment are enhanced. However, for stronger stratifications, both filamentation and vertical alignment are found to be greatly inhibited. For relatively small initial inclination angles, filamentation only occurs in the upper and lower sections of the vortex. Increasing the angle of tilt also increases the tendency of the surface to steepen and filament in the middle sections of the vortex. For a fixed value of the ratio of horizontal scale of the flow to the deformation radius, taller vortices have an increased tendency to align and axisymmetrize than shorter vortices of equal inclination angle.
Capillary–gravity and capillary waves generated in a wind wave tank: observations and theories
Authors: Zhang Xin
Published: April 2006

Short water surface waves generated by wind in a water tunnel have been measured by an optical technique that provides a synoptic picture of the water surface gradient over an area of water surface (Zhang & Cox 1994). These images of the surface gradient can be integrated to recover the shape of the water surface and find the two-dimensional wavenumber spectrum. Waveforms and two-dimensional structures of short wind waves have many interesting features: short and steep waves featuring sharp troughs and flat crests are very commonly seen and most of the short waves are far less steep than the limiting wave forms; waveforms that resemble capillary–gravity solitons are observed with a close match to the form theoretically predicted for potential flows (Longuet-Higgins 1989); capillaries are mainly found as parasitics on the downwind faces of gravity waves, and the longest wavelengths of those parasitic capillaries found are less than 1 cm; the phenomenon of capillary blockage (Phillips 1981) on dispersive freely travelling short waves is also observed. The spectra of short waves generated by low winds show a characteristic dip at the transition wavenumber between the gravity and capillary regimes, and the dip becomes filled in as the wind increases. The spectral cut-off at high wavenumbers shows a power law behaviour with an exponent of about minus four. The wavenumber of the transition from the dip to the cut-off is not sensitive to the change of wind speed. The minus fourth power law of the extreme capillary wind wave spectrum can be explained through a model of energy balances. The concept of an equilibrium spectrum is still useful. It is shown that the dip in the spectrum of capillary–gravity waves is a result of blockage of both capillary–gravity wind waves and parasitic capillary waves.
Available potential energy and mixing in density-stratified fluids
Authors: Winters Kraig B. , Lombard Peter N. , Riley James J. , D'Asaro Eric A.
Published: April 2006

A conceptual framework for analysing the energetics of density-stratified Boussinesq fluid flows is discussed. The concept of gravitational available potential energy is used to formulate an energy budget in which the evolution of the background potential energy, i.e. the minimum potential energy attainable through adiabatic motions, can be explicitly examined. For closed systems, the background potential energy can change only due to diabatic processes. The rate of change of background potential energy is proportional to the molecular diffusivity. Changes in the background potential energy provide a direct measure of the potential energy changes due to irreversible diapycnal mixing. For open systems, background potential energy can also change due to boundary fluxes, which can be explicitly measured. The analysis is particularly appropriate for evaluation of diabatic mixing rates in numerical simulations of turbulent flows. The energetics of a shear driven mixing layer is used to illustrate the analysis.
An asymptotic model for the hydrodynamics of the thermal bar
Authors: Farrow D. E.
Published: April 2006

An asymptotic solution is found for the temperature and circulation structure for an unsteady two-dimensional model of the thermal bar phenomenon. The non-rotating case considered here is relevant to laboratory models of the thermal bar. The main implication of the asymptotic results is that the thermal bar might propagate out from the shore more slowly than predicted by a purely heat-balance-based estimate. The solution is discussed in the context of available experimental results.
A simple representation of a developing contaminant concentration field
Authors: Sawford B. L. , Sullivan P. J.
Published: April 2006

Chatwin & Sullivan (1990) have demonstrated that, for a wide range of self-similar scalar fields, the moments of the probability density function of concentration have a very simple form. Here, an extension to this simple form which takes account of the source distribution is developed. This extension has two effects. Firstly it modifies the values of the two parameters appearing in the original theory and in particular explains the observed behaviour of these parameters very near to a line source of heat in grid tubulence. Secondly, it introduces an additional parameter in the description of each moment beyond the second. It is shown that these additional parameters are necessary in order to describe measurements of the first four central moments throughout the concentration field from a continuous line source of heat in grid-generated turbulence.
Nonlinear wave loads on a slender vertical cylinder
Authors: Faltinsen O. M. , Newman J. N. , Vinje T.
Published: April 2006

The diffraction of water waves by a vertical circular cylinder is considered in the regime where the wave amplitude A and cylinder radius a are of the same order, and both are small compared to the wavelength. The wave slope is small, and a conventional linear analysis applies in the outer domain far from the cylinder. Significant nonlinear effects exist in the complementary inner domain close to the cylinder, associated with the free-surface boundary condition. Using inner coordinates scaled with respect to a, it is shown that the leading-order nonlinear contribution to the velocity potential includes terms proportional to both A2a and A3. The wave load which acts on the cylinder near the free surface includes second- and third-harmonic components which are proportional respectively to A2a2 and A3a. In a conventional perturbation analysis, where A [Lt ] a, these components would be ordered in magnitude corresponding to the different powers of A, but here they are of the same order. The second- and third-order components of the total force are of comparable magnitude for practical values of the wave slope.
Airfoil leading-edge suction and energy conservation for compressible flow
Authors: Amiet R. K.
Published: April 2006

When a flat-plate airfoil at zero angle of attack encounters a vertical gust in an otherwise uniform flow, it experiences a force along the chord. This leading-edge suction force is examined for compressible flow with a time-dependent gust. A simple derivation of the thrust force is based on the fact that the leading edge is a singular point so that the flow here is dominated by the leading-edge dipole strength. From the viewpoint of a fluid-fixed observer the fluid does work on the airfoil, and this energy must come from the incident gust. Demonstrating energy conservation is not surprising, but it gives a better understanding of the relationship between the individual energy terms. The derivation shows that the acoustic energy can be calculated using compact assumptions at low frequency, but that it must be calculated non-compactly at high frequency. For a general gust the work done on the airfoil is shown to equal the energy taken from the fluid, the energy transfer occurring at the leading edge. For a sinusoidal gust the energy contained in the incident gust is shown to equal the sum of the energy remaining in the wake, the work done on the airfoil and the acoustic energy radiated away. The relative proportions of the incident energy going to these three energy types depends on the gust frequency, the acoustic radiation becoming more efficient as the frequency increases. For a fixed gust frequency, the thrust force goes to zero at a Mach number of one, and for an incident gust consisting of vorticity on the airfoil axis, the entire energy of the gust is radiated as acoustic energy at this Mach number.
Sedimentation and entrainment in dense layers of suspended particles stirred by an oscillating grid
Authors: Huppert Herbert E. , Turner J. Stewart , Hallworth Mark A.
Published: April 2006

Many flows, including those containing suspended particles, are kept turbulent by the action of the bottom stress, and this turbulence is also responsible for maintaining sedimenting particles in suspension and in some cases entraining more particles from the bed. A convenient one-dimensional analogue of these processes is provided by laboratory experiments conducted in a mixing box, where a characterizable turbulence is generated by the vertical oscillation of a horizontal grid. In the present paper we report the results of a series of experiments with a grid located close to the bottom boundary to simulate the action of stresses acting at a rough boundary, and compare the results with those obtained using the more extensively studied geometry in which a similar grid is located in the interior of a stirred fluid layer. Experiments have been conducted both with dense, particle-free fluid layers and with layers containing sufficiently high concentrations of dense particles to have a significant effect on the bulk density. In the fluid case, the interface at the top of the stirred dense layer continues to rise as lighter fluid is entrained across the interface. Sediment layers are distinctly different, because the particles responsible for the density difference between the layers can fall out of the suspension as it changes in thickness. The work done in keeping particles in suspension and the effect of this on the turbulence above the grid must be taken into account. The mechanism of resuspension of particles depends on the level of turbulence near the bottom boundary, below the grid. As the stirring rate, and thus the intensity of turbulence, are increased three possible equilibrium states can be attained sequentially: the particles eventually all precipitate; or some particles precipitate while the remainder are held indefinitely in suspension; or all the particles are suspended. In the last two cases a stable, self-limited suspension layer is produced, separated from the overlying fluid by a sharp density interface at a fixed height. Theoretical arguments are presented which provide a satisfactory scaling of the experimental data. These are compared with previous theories and numerical experiments aimed at modelling both the one-dimensional problem and the corresponding processes in turbulent gravity currents. Comparisons are also made with sediment-laden channel flows and convecting layers containing sedimenting particles. Similar results will hold for light, positively buoyant particles or non-coalescing bubbles.
Dissolution or growth of soluble spherical oscillating bubbles: the effect of surfactants
Authors: Fyrillas Marios M. , Szeri Andrew J.
Published: April 2006

A new theoretical formulation is developed for the effects of surfactants on mass transport across the dynamic interface of a bubble which undergoes spherically symmetric volume oscillations. Owing to the presence of surfactants, the Henry's law boundary condition is no longer applicable; it is replaced by a flux boundary condition that features an interfacial resistance that depends on the concentration of surfactant molecules on the interface. The driving force is the disequilibrium partitioning of the gas between free and dissolved states across the interface. As in the clean surface problem analysed recently (Fyrillas & Szeri 1994), the transport problem is split into two parts: the smooth problem and the oscillatory problem. The smooth problem is treated using the method of multiple scales. An asymptotic solution to the oscillatory problem, valid in the limit of large Péclet number, is developed using the method of matched asymptotic expansions. By requiring that the outer limit of the inner approximation match zero, the splitting into smooth and oscillatory problems is determined unambiguously in successive powers of [weierp ]−1/2, where [weierp ] is the Péclet number. To leading order, the clean surface solution is recovered. Continuing to higher order it is shown that the concentration field depends on RI[weierp ]−1/2, where RI is the (dimensionless) interfacial resistance associated with the presence of surfactants. Although the influence of surfactants appears at higher order in the small parameter, surfactants are shown to have a very significant effect on bubble growth rates owing to the fact that the magnitude of RI is approximately the same as the magnitude of [weierp ]1/2 at conditions of practical interest. Hence the higher-order ‘corrections’ happen numerically to be of the same magnitude as the leading-order, clean surface problem. This is the fundamental reason for major increases in the bubble growth rates associated with the addition of surfactants. This is in contrast to the case of a still, surfactant-covered bubble, in which the first-order correction to the growth rate is of order RI[weierp ]−1 and presents a [weierp ]−1/2 correction. Finally, although existing experimental results have shown only enhancement of bubble growth in the presence of a surfactant the present theory suggests that it is possible for a surfactant, characterized by weak dependence of interfacial resistance on surface concentration, to inhibit rather than enhance the growth of bubbles by rectified diffusion.
Effects of soluble and insoluble surfactant on laminar interactions of vortical flows with a free surface
Authors: Tsai Wu-Ting , Yue Dick K. P.
Published: April 2006

We study the two-dimensional, laminar interactions between a contaminated free surface and a vortical flow below. Two canonical vortical flows are considered: a pair of vortex tubes impinging onto the free surface; and an unstable shear wake behind a body operating on the surface. A quantitative model for free-surface viscous flows in the presence of soluble or insoluble surfactants is developed. For the low to moderate Froude numbers considered here, for which weakly nonlinear free-surface boundary conditions are valid, the surface boundary layer and vorticity production are weak for clean water and the vortical flow evolution does not differ qualitatively from that under a free-slip boundary. When even a small amount of contamination is present, the flow can be dramatically affected. The vortical flow creates gradients in the surfactant surface concentration which leads to Marangoni stresses, strong surface vorticity generation, boundary layers, and even separation. These significantly influence the underlying flow which itself affects surfactant transport in a closed-loop interaction. The resulting flow features are intermediate between but qualitatively distinct from those under either a free- or no-slip boundary. Surfactant effects are most prominent for insoluble surface contamination with likely development of surfactant shocks and associated surface features such as Reynolds ridges. For soluble surfactant with initially uniform bulk concentration, surface concentration variations are moderated by sorption kinetics between the surface and bulk phases, and the overall effects are generally diminished. For initially stratified bulk concentrations, however, the evolution dynamics becomes more varied and surfactant effects may be amplified relative to the insoluble case. The dependence of these results on the properties of the contamination is studied.
Passive scalar dispersion and mixing in a turbulent jet
Authors: Tong Chenning , Warhaft Z.
Published: April 2006

The dispersion and mixing of passive scalar (temperature) fluctuations is studied in a turbulent jet. The temperature fluctuations were produced by heated fine wire rings placed axisymmetrically in the flow. Typically the ring diameters were of the same order as the jet,Dj, and they were placed in the self-similar region. However, other initial conditions were studied, including a very small diameter ring used to approximate a point source. Using a single ring to study dispersion (which is analogous to placing a line source in a planar flow such as grid turbulence), it was found that the intense local thermal field close to the ring disperses and fills the whole jet in approximately 1.5 eddy turnover times. Thereafter the thermal field evolves in the same way as for the traditional heated jet experiments. Two heated rings were used to study the mixing of two independently introduced scalar fields. Here an inference method (invoking the principle of superposition) was used to determine the evolution of the cross-correlation coefficient, ρ, and the segregation parameter, α, as well as the coherence and co-spectrum. While initially strongly dependent on ring locations and spacing, ρ and α reached asymptotic values of 1 and 0.04, respectively, also in about 1.5 eddy turnover times. These results are contrasted with mixing and dispersion in grid turbulence where the evolution is slower. Measurements in the far field of the jet (where ρ = 1) of the square of the scalar derivative conditioned on the scalar fluctuation itself, as well as other conditional statistics, showed strong dependence on the measurement location, as well as the direction in which the derivative was determined. The cross-correlation between the square of the scalar derivative and the signal showed a clear Reynolds-number trend, decreasing as the jet Reynolds number was varied from 2800 to 18 000. The far-field measurements, using the heated rings, were corroborated by new heated jet experiments.
The motion of long bubbles in polygonal capillaries. Part 1. Thin films
Authors: Wong Harris , Radke C. J. , Morris S.
Published: April 2006

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.

Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 (Ca ≡ μU/σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 [Lt ] x [Lt ] Ca−1, the film profile is frozen with a thickness of order Ca2/3 at the centre and order Ca at the sides. For x ∼ Ca−1, surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca4/3. For x [Gt ] Ca−1, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca−5/3, the height of the parabola is order Ca2/3. Finally, for x [Gt ] Ca−5/3, the height decreases as Ca1/4x−1/4.
A rapid-pressure covariance representation consistent with the Taylor—Proudman theorem materially frame indifferent in the two-dimensional limit
Authors: Ristorcelli J. R. , Lumley J. L. , Abid R.
Published: April 2006

A nonlinear variable-coefficient representation for the rapid-pressure covariance appearing in the Reynolds stress and heat-flux equations, consistent with the Taylor–Proudman theorem, is presented. The representation ensures that the modelled second-order equations are frame indifferent with respect to rotation in a number of different flows for which such an invariance is required. The model coefficients are functions of the state of the turbulence; they are valid for all states of a mechanical turbulence, attaining their limiting values only when the limit state is achieved. This is accomplished by a special ansatz that is used to obtain – analytically – the coefficients valid away from the realizability limit. Unlike other rapid-pressure representations in which extreme states are used to set model constants, here the coefficients are variable functions asymptotically consisted with – not fixed by – the limit states of the turbulence field. The mathematical principles invoked do not specify all the coefficients in the model; undetermined coefficients appear as free parameters which are used to ensure that the representation is asymptotically consistent with an experimentally determined equilibrium state of homogeneous sheared turbulence. This is done by ensuring that the modelled evolution equations have the same fixed points as those obtained from numerical and laboratory experiments for the homogeneous shear. Results of computations of homogeneous shear, with rotation and with curvature, are shown. Results are better, in a wide class of planar flows for which the model has not been calibrated, than those of other nonlinear models.
Adjoint systems and their role in the receptivity problem for boundary layers
Authors: Hill D. C.
Published: April 2006

The effectiveness with which various sources excite convective instabilities in a boundary layer is found by a simple method. Chosen field values of the adjoint to the Tollmien–Schlichting eigensolution, normalized appropriately, indicate the amplitude of the unstable disturbance which will result for direct time-harmonic forcing by sources of momentum, mass and vorticity, as well as by boundary motions. For the Blasius boundary layer, forcing in the vicinity of the critical layer induces the largest response. At this position, the response to forcing in the wall-normal direction is typically 5% of that resulting from streamwise forcing of the same magnitude. At the wall, normal motions elicit a much stronger response than streamwise motions. Forcing close to the lower branch of the neutral stability curve leads to the largest response. The adjoint field values are equivalent to the residues of Fourier-inversion integrals. This equivalence is discussed for two problems; the vibrating ribbon problem and excitation of an inviscid free shear layer by a vorticity source. The efficiency factor is calculated for the scattering of ‘acoustic’ waves into Tollmien–Schlichting waves in the presence of small surface roughness, at a finite Reynolds number, based on the Orr–Sommerfeld operator. This is achieved by using the solution of an inhomogeneous adjoint problem. The results are compared with the asymptotic solutions obtained from triple-deck theory, and agree with previous finite-Reynolds-number calculations.
Effect of islands upon dispersion in rivers
Authors: Smith Ronald
Published: April 2006

A general formulation is given for the dispersion of conservative tracers in steady flow in multi-connected channels. A multi-index is used to distinguish the different routes for tracer between the source and the observation position. For each route exact formulae are obtained for the time integral, time centroid, and cross-channel average of the temporal variance. The total concentration is the superposition of the contributions from the different routes.
Influence of incidence angle on sound generation by airfoils interacting with high-frequency gusts
Authors: Myers Matthew R. , Kerschen E. J.
Published: April 2006

A theoretical model is developed for the sound generated when a convected vortical or entropic gust encounters an airfoil at non-zero angle of attack. The theory is based on a linearization of the Euler equations about the steady subsonic flow past the airfoil. High-frequency gusts, whose wavelengths are short compared to the airfoil chord, but long compared to the displacement of the mean-flow stagnation point from the leading edge, are considered. The analysis utilizes singular-perturbation techniques and involves four asymptotic regions. Local regions, which scale on the gust wavelength, are present at the airfoil leading and trailing edges. Behind the airfoil a ‘transition’ region, which is similar to the transition zone between illuminated and shadow zones in optical problems, is present. In the outer region, far away from the airfoil edges and wake, the solution has a geometric-acoustics form. The primary sound generation is found to be concentrated in the local leading-edge region. The trailing edge plays a secondary role as a scatterer of the sound generated in the leading-edge region. Parametric calculations are presented which illustrate that moderate levels of airfoil steady loading can significantly affect the sound field produced by airfoil–gust interactions.
Pressure-driven flow of a thin viscous sheet
Authors: Van De Fliert B. W. , Howell P. D. , Ockenden J. R.
Published: April 2006

Systematic asymptotic expansions are used to find the leading-order equations for the pressure-driven flow of a thin sheet of viscous fluid. Assuming the fluid geometry to be slender with non-negligible curvatures, the Navier–Stokes equations with appropriate free-surface conditions are simplified to give a ‘shell-theory’ model. The fluid geometry is not known in advance and a time-dependent coordinate frame has to be employed. The effects of surface tension, gravity and inertia can also be incorporated in the model.
Viscous effects on fully coupled resonant-triad interactions: an analytical approach
Authors: Wu Xuesong
Published: April 2006

This paper is concerned with viscous effects on the development of a fully coupled resonant triad consisting of Rayleigh waves. Complementary to the numerical study of Lee (1995), we attack this problem analytically. The fully coupled amplitude equations are derived with all the kernels involved being expressed in closed forms. The amplitude equations are then solved numerically. It is found that viscosity reduces the growth of the disturbance in the parametric-resonance stage and delays the final occurrence of the finite-time singularity. But viscosity does not appear to be able to eliminate the singularity. While the analysis is performed for the temporally evolving instability waves, we demonstrate its broad application by showing that it can be slightly modified to obtain the amplitude equations for the spatially growing Rayleigh waves, and the equations which describe the development of the resonant-triad of Tollmien–Schlichting waves in the fully interactive stage.
On vapour flow in a hot porous layer
Authors: Fitzgerald Shaun D. , Woods Andrew W.
Published: April 2006

The motion of isothermal vapour in a permeable rock is governed by a nonlinear diffusion equation for the vapour pressure. We analyse vapour flow described by this equation in both bounded and unbounded domains. We then apply these solutions to describe the controls on the rate of vaporization of liquid invading a hot permeable rock. In an unbounded domain, we determine asymptotic similarity solutions describing the motion of vapour when it is either supplied to or removed from the reservoir. Owing to the compressibility, these solutions have the property that vapour surfaces migrate towards the isobar on which the vapour has the maximum speed.

In contrast, if vapour is supplied to or removed from a closed bounded system sufficiently slowly then the vapour density and pressure rapidly become approximately uniform. As more vapour is added, the mean pressure gradually increases and vapour surfaces become compressed. If liquid slowly invades a hot bounded porous layer and vaporizes, the vapour pressure becomes nearly uniform. As more liquid is added, the reservoir gradually becomes vapour saturated and the vaporization ceases.

In an open bounded system, with a constant rate of vapour injection, the flux of vapour across the reservoir becomes uniform. If liquid is injected slowly and vaporizes then again the vapour flux becomes spatially uniform. However, the vapour flux now increases slowly as the liquid invades further into the rock, as a result of the decreased resistance to vapour flow from the interface to the far boundary.
A simple theory of capillary–gravity wave turbulence
Authors: Glazman Roman E.
Published: April 2006

Employing a recently proposed ‘multi-wave interaction’ theory (Glazman 1992), inertial spectra of capillary-gravity waves are derived. This case is characterized by a rather high degree of nonlinearity and a complicated dispersion law. The absence of scale invariance makes this and some other problems of wave turbulence (e.g. nonlinear inertia-gravity waves) intractable by small-perturbation techniques, even in the weak-turbulence limit. The analytical solution obtained in the present work for an arbitrary degree of nonlinearity is shown to be in reasonable agreement with experimental data. The theory explains the dependence of the wave spectrum on wind input and describes the accelerated roll-off of the spectral density function in the narrow sub-range separating scale-invariant regimes of purely gravity and capillary waves, while the appropriate (long- and short-wave) limits yield power laws corresponding to the Zakharov-Filonenko and Phillips spectra.
Marangoni instabilities in liquid mixtures with Soret effects
Authors: Joo S. W.
Published: April 2006

The stability of a binary liquid mixture heated from above is analysed. The heat transfer is driven by the imposed temperature difference between the horizontal bottom plate and the ambient gas. The mass flux in the layer is induced by the Soret effect. The gravitational effects are ignored, and the instability is driven by solutocapillarity and retarded by thermocapillarity. The interface is allowed to deform, and both the small-wavenumber and the Pearson-type instabilities are studied. Oscillatory instability can exist when the thermocapillary is destabilizing and the solutocapillarity is stabilizing.