The present study examines the steady, axisymmetric Stokes flow past a sphere coated with a thin, immiscible fluid layer. Inertial effects are neglected for both the outer fluid and the fluid film, and surface tension forces are assumed large compared with the viscous forces which deform the fluid film. Furthermore, the present analysis assumes that the mechanism driving the fluid circulation within the film is not too large. From force equilibrium on the film we find that a steady fluid film can only partially cover the sphere, i.e. the film must be held to the sphere by surface tension forces at the contact line. The extent of the sphere covered by the film is specified, in terms of the solid–fluid contact angle, by the condition of global force equilibrium on the fluid film.

Using a perturbation scheme based on the thinness of the fluid layer the solution to the flow field is obtained analytically, except for the fluid-film profile (i.e. the fluid–fluid interface) which requires numerical calculations. One of the principal results is an expression for the drag force on the fluid-coated particle. In particular, we find that the drag on a sphere is reduced by the presence of a fluid coating when the ratio of the film fluid viscosity to the surrounding fluid viscosity is less than ¼. Detailed numerical computations are conducted for a few typical cases. The calculations show that a film of prescribed areal extent, i.e. specified contact angle, is only possible when the magnitude of the driving force on the film is below some maximum value. A simple experiment was also performed, and photographs, which qualitatively illustrate the fundamental fluid-film configurations predicted by the theory, are presented.

Turbulent boundary-layer fluctuations in the incompressive domain are expressed in terms of fluctuating velocity-product 'sources’ in order to elucidate relative characteristics of fluctuating wall-shear stress and pressure in the subconvective range of streamwise wavenumbers. Appropriate viscous wall conditions are applied, and results are obtained to lowest order in this Strouhal-scaled wavenumber which serves as the expansion parameter. The spectral amplitudes of pressure and of the shear stress component directed along the wavevector both contain additive terms proportional to source integrals with exponential wall-distance weighting characteristic respectively of the irrotational and the rotational fields. At low wavenumbers, barring unexpected relative smallness of the pertinent boundary-layer source term, the rotational terms become dominant. There the wall pressure and shear-stress component have spectra that approach the same non-vanishing, wavevector-white but generally viscous-scale-dependent level and are totally coherent with phase difference ½π. The other, irrotational contributions to the shear-stress and pressure amplitudes likewise bear a simple and previously known, generally wavevector– and frequency-dependent, ratio to one another. In an inviscid limit this contribution to the pressure amplitude reduces to the one obtained previously from inviscid treatments. A representative class of models is introduced for the source spectrum, and the resulting rotational contribution to the spectral density of wall pressure and K-aligned shear stress at low (but incompressive) wavenumbers is estimated. It is suggested that this contribution may predominate and account for measured low-wavenumber levels of wall pressure.

A numerical investigation of the stability of an axisymmetric magnetic field is discussed. The magnetic field permeates a finitely conducting fluid contained within a rapidly rotating cylindrical annulus. The fluid is incompressible and viscid. The evolution of a non-axisymmetric perturbation to the axisymmetric magnetic field is governed by the momentum and induction equations which are integrated using a spectral timestep method. We follow the growth of the perturbation to finite amplitude and find that the character of the solution is dominated by the most unstable axially dependent mode found from the linear theory.

In the interior of an extended liquid a cyclic cavitation process can be set up by a focused acoustic wave field. A cluster of supercritical cavities is developed when the tensile strength of the liquid is exceeded. This occurs almost simultaneously in a confined region around the focal point where the sound speed rapidly vanishes when the cavities approach their critical size. As a consequence, the inner boundary condition for the acoustic field, carried by the single-phase liquid, changes from total reflection at the focal point without phase shift, as at a rigid wall, to reflection at the boundary of the cluster, which now forms a low-pressure two-phase kernel embedded in the single-phase liquid. The cluster is compliant to expansion waves but essentially rigid to compression waves. As soon as the cluster is formed its further development is determined jointly by the sound field and by the far field pressure of the liquid. The former first makes the cavities in the cluster grow and then contributes to its collapse, while the latter tends to bring about its collapse from the moment of cluster formation.

In this work we prove that the second-kind Fredholm's integral equations proposed by Rallison & Acrivos to solve the deformation and burst of a viscous drop in an extensional flow, with viscosity ratio λ, possess a unique continuous solution u(x) for any continuous datum F(x) when 0 < λ < ∞. In the original work they could only guarantee, analytically, the solvability of the integral equations in a small neighbourhood of λ = 1.

We study the statistics of a passive scalar $T({\bm{x},t)$ governed by the advection–diffusion equation with variations in the scalar produced by a steady source. Two important statistical properties of the scalar are the variance, $\sigma^2\equiv \langle T^2 \rangle$, and the entropy production, $\chi \equiv \kappa \langle|\bm{\nabla} T|^2\rangle$. Here $\langle\rangle$ denotes a space–time average and $\kappa$ is the molecular diffusivity of $T$. Using variational methods we show that the system must lie above a parabola in the $(\chi, \sigma^2)$-plane. The location of the bounding parabola depends on the structure of the velocity and the source. To test the bound, we consider a large-scale source and three two-dimensional model velocities: a uniform steady flow; a statistically homogeneous and isotropic flow characterized by an effective diffusivity; a time-periodic model of oscillating convection cells with chaotic Lagrangian trajectories. Analytic solution of the first example shows that the bound is sharp and realizable. Numerical simulation of the other examples shows that the statistics of $T({\bm{x},t)$ lie close to the parabolic frontier in the $(\chi, \sigma^2)$-plane. Moreover in the homogenization limit, in which the largest scale in the velocity field is much less than the scale of the source, the results of the simulation limit to the bounding parabola.

A three-dimensional two-layer, fully dispersive and strongly nonlinear interfacial wave model, including the interaction with a time-varying bottom topography, is developed. The method is based on a set of integral equations. The source and dipole terms are developed in series expansions in the vertical excursions of the interface and bottom topography, obtaining explicit inversion by Fourier transform. Calculations of strongly nonlinear interfacial waves with excursions comparable to the thinner layer depth show that the quadratic approximation of the method contains the essential dynamics, while the additional cubic terms always are small. Computations confirm the onset of wave train formation driven by topography, observed in experiments (Maxworthy, J. Geophys. Res., vol. 84(C1), 1979, pp. 338–346), depending on the Froude number and the topography height. Simulations of tidally driven three-dimensional internal wave formation show the formation of two wave trains per half tidal cycle for strong forcing and one wave train for weak forcing. Waves of both backward and forward curvature are calculated.

This paper aims to elucidate the structure of the turbulent mixing layers, especially, its dependence on initial disturbances. The mixing layers are produced by setting a woven-wire screen perpendicular to the freestream in the test section of a wind tunnel to obstruct part of the flow. Three kinds of model geometry are treated; these model screens produced mixing layers which may be regarded as the equivalents of the plane mixing layer and of two-dimensional and axisymmetric wakes issuing into ambient streams of higher velocity. The initial disturbances are imposed by installing thin rods of various sizes along the edge of the screen or at the origin of the mixing layer. Flow features are visualized by the smoke-wire method. Statistical quantities are measured by a laser-Doppler velocimeter. In all cases large-scale transverse vortices seem to persist, although comparatively small-scale vortices are superimposed on the flow field in the mixing layer. The mixing layers are in self-preserving state at least up to third-order moments, but the self-preserving state is different in each case. The growth rates of the mixing layer are shown to depend strongly on the initial disturbance imposed.

An experimental investigation has been carried out into the nature of the transport of airborne material in the near aerodynamic wakes of bluff bodies with simple shapes. The main attention was focused on the essential differences existing between axi- symmetric flows (as about disks) and two-dimensional flows (as about rectangular long thin flat plates). Measurements were made for such bodies of the near-wake residence time of injected small particles, along with other and more familiar near- wake properties such as the vortex-shedding frequency and base pressure. It was concluded for disks that the transport of material into and out of the near-wake region is dominated by turbulent diffusion, and is strongly influenced by free-stream turbulence, especially for free-stream turbulence whose length scale is substantially smaller than the disk diameter. For rectangular flat plates, transport is dominated by the periodic shedding of vortices, and to only a secondary extent by turbulent motions, and is not strongly influenced by free-stream turbulence.

The transition mechanism in a spatially developing two-dimensional wake is studied by means of direct numerical simulations. Five different types of forcing of the inlet are investigated: fundamental mode, fundamental and one or two subharmonics, fundamental mode and random noise, and random noise only. The effects of the amplitude levels of the perturbations on the development of the layer are also investigated. Statistical analyses are performed and some numerical results are compared with experimental measurements. When only a fundamental mode is forced, the energy spectra show amplification of the fundamental frequency and its higher harmonics, and the development of a stable vortex street. When the inlet flow is forced by a fundamental mode and two subharmonics, a vortex street also appears downstream, but the shape of the vortices is distorted. The amplitude of the subharmonic grows only after the saturation of the fundamental. Amplification of modes close to the fundamental mode is observed when random noise of large amplitude is added to the fundamental mode. The phase jitter around the fundamental frequency plays a critical role in generating vortices of random shape and spacing. Large- and small-scale distortions of the flow structure are observed. Pairing of vortices of the same sign is observed, as well as coupling of vortices of opposite sign. When the inlet profile is forced by random noise of amplitude 10−5 times the free-stream velocity, one frequency close to the most unstable one is amplified more than the others. The energy spectrum is otherwise full. When the same low amplitude (10−5) is used to force the fundamental mode and its two subharmonics, bands of energy develop around the forced modes and their harmonics. Finally, we find that large-deficit wakes are globally unstable when the size of the absolutely unstable region is greater than about three times the half-width of the wake

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniformly heated surface. The effect of a non-zero contact angle is imposed through a disjoining–conjoining pressure term. The numerical results for two different molecular interactions are compared: on the one hand, London–van der Waals and ionic–electrostatics molecular interactions that account for polar liquids; on the other hand, long- and short-range molecular forces that model molecular interactions of non-polar fluids. In addition, the effect of gravity on the velocity of the drop is analysed. We find that for small contact angles, the long-time dynamics is independent of the molecular potential, and the footprint of the droplet increases with the square root of time. For intermediate contact angles we observe that polar droplets are more likely to break up into smaller volumes than non-polar ones. A linear stability analysis allows us to predict the number of droplets after breakup occurs. In this regime, the effect of gravity is stabilizing: it reduces the growth rates of the unstable modes and increases the shortest unstable wavelength. When breakup is not observed, the droplet moves steadily with a profile that consists in a capillary ridge followed by a film of constant thickness, for which we find power law dependencies with the cross-sectional area of the droplet, the contact angle and the temperature gradients. For large contact angles, non-polar liquids move faster than polar ones, and the velocity is proportional to the Marangoni stress. We find power law dependencies for the velocity for the different regimes of flow. The numerical results allow us to shed light on experimental facts such as the origin of the elongation of droplets and the existence of saturation velocity.

A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of $\langle ({\it\delta}q)^{2}\rangle$ along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of $\langle q^{2}\rangle$ and the decay rate constants of $\langle u^{2}\rangle$ and $\langle v^{2}\rangle$ are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity $U$ and the streamwise distance $x-x_{0}$ from the virtual origin (i.e. $U\propto (x-x_{0})^{-1}$), is sufficient to predict a power-law decay for the turbulence kinetic energy ($\langle q^{2}\rangle \propto (x-x_{0})^{m}$), rather than requiring a power-law decay ($m=-2$) as an additional ad hoc assumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate, $\langle {\it\epsilon}\rangle$, varies as $(x-x_{0})^{m-2}$. These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for $11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$ over the range $10\leqslant x/D\leqslant 90$. For the present experiments, $\langle q^{2}\rangle$ exhibits power-law behaviour with $m=-1.83$. The validity of this solution is confirmed using other experimental data where $\langle q^{2}\rangle$ follows a power law with $-1.89\leqslant m\leqslant -1.78$. The present similarity form of the transport equation for $\langle ({\it\delta}q)^{2}\rangle$ is also shown to be closely satisfied by the experimental data.

The effect of an incoming wake on the flow around and heat transfer from the stagnation region of a circular cylinder was studied using direct numerical simulations (DNSs). Four simulations were carried out at a Reynolds number (based on free-stream velocity and cylinder diameter D) of ReD = 13200: one two-dimensional (baseline) simulation and three three-dimensional simulations. The three-dimensional simulations comprised a baseline simulation with a uniform incoming velocity field, a simulation in which realistic wake data – generated in a separate precursor DNS – were introduced at the inflow plane and, finally, a simulation in which the turbulent fluctuations were removed from the incoming wake in order to study the effect of the mean velocity deficit on the heat transfer in the stagnation region. In the simulation with realistic wake data, the incoming wake still exhibited the characteristic meandering behaviour of a near-wake. When approaching the regions immediately above and below the stagnation line of the cylinder, the vortical structures from the wake were found to be significantly stretched by the strongly accelerating wall-parallel (circumferential) flow into elongated vortex tubes that became increasingly aligned with the direction of flow. As the elongated streamwise vortical structures impinge on the stagnation region, on one side they transport cool fluid towards the heated cylinder, while on the other side hot fluid is transported away from the cylinder towards the free stream, thereby increasing the heat transfer. The DNS results are compared with various semi-empirical correlations for predicting the augmentation of heat transfer due to free-stream turbulence.

As a model of an almost fully extended macromolecule, a small, flexible, inextensible, nearly straight thread in a shearing flow with weak Brownian motions is considered. The hydrodynamic resistance to motion is included using the slenderbody theory for Stokes flow. The variation of the small transverse displacements along the thread is expressed as a truncated sum of Fourier components, with appropriately chosen modal functions. A diffusion equation is derived in the Fourier space and solved. The expected deformation of the thread is then given for axisymmetric and two-dimensional straining flows. The transverse displacement of the ends and the small shortening of the projected length of the thread are both found to be sensitive to the truncation of the Fourier representation, although it becomes clear on physical grounds that the ratio of the shortening to the typical transverse distortion should increase with the number of degrees of freedom. In simple shear flow the deformation increases as the thread aligns with the flow, until the analysis breaks down when the entire thread is no longer in the extensional quadrants. The influence of the 2:1 ratio of the resistance coefficients from the slender-body theory is found to be a small numerical factor.

Steady seepage is determined by a head drop upstream/downstream of a water-retaining wall. Due to its erratic variations, hydraulic log-conductivity $Y=\ln K$ is modelled as a stationary random space function (RSF). We deal with a highly anisotropic porous formation, i.e. an axisymmetric medium where the horizontal correlation integral scale of $Y$ is much larger than the vertical one. The goal of computing the resulting flow field within a stochastic framework is complicated by non-uniformity of the mean flow. Simple (closed-form) expressions for the correlation functions of the flow variables as well as the mean head are derived. We use these results to quantify the impact of spatial variability of $Y$ upon the probability that the exit volumetric flow rate downstream of the wall is greater than that obtained by regarding the formation as homogeneous (with constant hydraulic conductivity). In particular, we show that the spatial variability of $Y$ may lead to predictions (and consequently to design choices) which significantly differ from those achieved by regarding the porous formation as homogeneous.

The transition from diffusion-dominated to inertia-dominated behaviour in the transport of condensable heavy molecules carried in a continuum subsonic He jet that impinges on a solid surface is studied experimentally. The Stokes number S, or ratio between the heavy-molecule relaxation time and the fluid-dynamic time, is varied in the interval 0 [lsim ] S [lsim ] 1 by changing the jet Mach number at a constant value of the Reynolds number. Although the heavy species departs considerably from equilibrium at all but the smallest values of S, the helium jet is always near equilibrium conditions. At values of S of order unity the observed rate of deposition at the stagnation point asymptotes to a value some six times greater than in the diffusion region (where S → 0), implying that the process is governed by the large inertia of the heavy species, very much like in aerosol impactors. As a result, it is argued that the concept of pressure diffusion is unsuitable to explain the observed behaviour. An approximate theoretical description of the transport process is given for the region S [Lt ] 1 where the kinetic problem is amenable to a hydrodynamic treatment. Finally, the analogy with the inertia-dominated behaviour of aerosols is used to assess the relative merits of various aerodynamics schemes aiming at separating isotopes.

This paper investigates the three-dimensional stability of a pair of co-rotating vertical vortices in a rotating strongly stratified fluid. In a companion paper (Otheguy, Chomaz & Billant 2006), we have shown that such a basic flow in a strongly stratified fluid is affected by a zigzag instability which bends the two vortices symmetrically. In the non-rotating flow, the most unstable wavelength of this instability scales as the buoyancy length and its growth rate scales as the external strain that each vortex induces on the other one. Here, we show that the zigzag instability remains active whatever the magnitude of the planetary rotation and is therefore connected to the tall-column instability in quasi-geostrophic fluids. Its growth rate is almost independent of the Rossby number. The most amplified wavelength follows the universal scaling $\lambda\,{=}\, 2 \pi F_h b \sqrt{{\gamma_1}/{\hbox{\it Ro}^2} +{\gamma_2}/{\hbox{\it Ro}} +\gamma_3 }$, where $b$ is the separation distance between the two vortices, ($\gamma_1$, $\gamma_2$, $\gamma_3$) are constants, $F_h$ is the horizontal Froude number and $\hbox{\it Ro}$ the Rossby number ($F_h=\Gamma / \pi a^2 N$, $\hbox{\it Ro}= \Gamma / \pi a^2 f$, where $\Gamma$ is the circulation of each vortex, $a$ the vortex radius, $N$ the Brunt–Väisälä frequency and $f$ the Coriolis parameter). When $\hbox{\it Ro}\,{=}\,\infty$, the scaling $\lambda \,{\propto}\, F_h b$ found in the companion paper Otheguy et al. (2006) is recovered. When $\hbox{\it Ro} \,{\rightarrow}\, 0$, $\lambda \,{\propto}\,b f/N$ in agreement with the quasi-geostrophic theory. In contrast to previous results, the wavelength is found to depend on the separation distance between the two vortices $b$, and not on the vortex radius $a$.

Fluid material line growth in turbulent flow has been measured by tagging lines with small hydrogen bubbles in the nearly isotropic turbulence behind a regular grid in a water tunnel. The average three-dimensional line lengths were inferred by an intersection-counting method carried out on one-plane photographs. The measurements cover ‘small’ time intervals only.

In this study the flow through and around a parallel-walled channel with an obstruction (flat plate) placed at the channel inlet is investigated. Depending on the position of the obstruction, the flow inside the channel is in a direction opposite to that outside, stagnant or in the same direction as outside but with reduced magnitude. Flow visualization in water is used to examine the fluid motion, although some wind tunnel measurements have been made and are also reported. The parameters that have been varied are the gap between the obstruction and the entry to the channel, the length of the channel and the Reynolds number. The maximum value of the reverse flow velocity is found to be about 20% of that of the flow outside. The maximum forward velocity inside the channel (when it occurs) is only about 65% of the outside velocity even for very large gaps between the obstruction and the channel entrance. A tentative explanation is offered for the observed features.

This paper presents the stochastic properties of orbital velocities of random water waves in intermediate water depth. Both the emergence effect and weak nonlinear effects are studied; the theoretical predictions are compared with measured kinematics and the deviations from linear theory are quantified.

This study includes new ideas in fluid dynamics. An analytic formula for probability distribution for velocities modified by the emergence effect as well as by nonlinearities of the wave motion in intermediate water depth is developed. This probability function gives us the first statistical moment, the second statistical moment for modified velocities in an analytical form, and by numerical integration the third statistical moment for modified velocities.

The theoretical formulae for the statistical moments for surface elevation and for velocities up to third order, with nonlinearities of the motion taken into account, for the case when the emergence effect can be neglected, i.e. below the surface layer, have been developed. This includes a generalized formula for free-surface elevation setdown and calculation of the asymmetry of the horizontal velocity, which is found to be negative in agreement with measurements of Anastasiou et al. (1982b).

From the first statistical moment of the modified horizontal velocity, the mean flux between any two levels in the wave flume may be calculated. When the integration is carried out from the bottom up to + ∞, it leads in approximation to the formula for total mean flux found by Phillips (1960). This agreement with Phillips’ formula encourages one to interpret the positive mean value of horizontal velocities as a ‘real current’. This interpretation also provides a new understanding of the fluid dynamic implications of results presented by Tung (1975).

Theoretical prediction of the measured kinematics has allowed a better estimation of the return flow in the wave flume, and in the vicinity of the mean water level currents in two different directions are noted. Firstly, the emergence effect gives rise to a current at the mean water level in the direction of the wave advance. Secondly, a flow in the opposite direction, interpreted as a return current in the wave flume, is noticed just below that level.