The buoyant motion of a bubble rising through a continuous liquid phase can be retarded by the adsorption onto the bubble surface of surfactant dissolved in the liquid phase. The reason for this retardation is that adsorbed surfactant is swept to the trailing pole of the bubble where it accumulates and lowers the surface tension relative to the front end. The difference in tension creates a Marangoni force which opposes the surface flow, rigidifies the interface and increases the drag coefficient. Surfactant molecules adsorb onto the bubble surface by diffusing from the bulk to the sublayer of liquid adjoining the surface, and kinetically adsorbing from the sublayer onto the surface. The surface surfactant distribution which defines the Marangoni force is determined by the rate of kinetic adsorption and bulk diffusion relative to the rate of surface convection. In the limit in which the rate of either kinetic or diffusive transport of surfactant to the bubble surface is slow relative to surface convection and surface diffusion is also slow, surfactant collects in a stagnant cap at the back end of the bubble while the front end is stress free and mobile. The size of the cap and correspondingly the drag coefficient increases with the bulk concentration of surfactant until the cap covers the entire surface and the drag coefficient is that of a bubble with a completely tangentially immobile surface. Previous theoretical research on the stagnant cap regime has not studied in detail the competing roles of bulk diffusion and kinetic adsorption in determining the size of the stagnant cap angle, and there have been only a few studies which have attempted to quantitatively correlate simulations with measurements.

This paper provides a more complete theoretical study of and a validating set of experiments on the stagnant cap regime. We solve numerically for the cap angle and drag coefficient as a function of the bulk concentration of surfactant for a spherical bubble rising steadily with inertia in a Newtonian fluid, including both bulk diffusion and kinetic adsorption. For the case of diffusion-limited transport (infinite adsorption kinetics), we show clearly that very small bulk concentrations can immobilize the entire surface, and we calculate the critical concentrations which immobilize the surface as a function of the surfactant parameters. We demonstrate that the effect of kinetics is to reduce the cap angle (hence reduce the drag coefficient) for a given bulk concentration of surfactant. We also present experimental results on the drag of a bubble rising in a glycerol–water mixture, as a function of the dissolved concentration of a polyethoxylated non-ionic surfactant whose bulk diffusion coefficient and a lower bound on the kinetic rate constants have been obtained separately by measuring the reduction in dynamic tension as surfactant adsorbs onto a clean interface. For low concentrations of surfactant, the experiments measure drag coefficients which are intermediate between the drag coefficient of a bubble whose surface is tangentially mobile and one whose surface is completely immobilized. Using the separately obtained value for the diffusion coefficient of the polyethoxylate, we undertake simulations which provide, upon comparison with the measured drag coefficients, a tighter bound on the kinetic rate constants than were otherwise obtained using dynamic surface tension measurement, and this suggests a new method for the measurement of kinetic rate constants.

The problem considered is that of a self-propagating pair of line vortices, of equal and opposite strengths, in a compressible inviscid fluid. An asymptotic solution is sought in the limit of small propagation speed compared with the sound speed in the medium. In most of the fluid region, the elementary incompressible flow solution limit is enhanced by a Rayleigh–Janzen approximation. This approximation fails at points that are either far from, or very close to, a vortex. For distant points, rescaled outer variables lead to an approximation that corresponds to the field induced by a moving dipole in compressible fluid. The main approximation also fails near each vortex line, where the analysis of Barsony-Nagy, Er-El & Yungster (J. Fluid Mech. vol. 178, 1987, p. 367) is used to express the local flow field in terms of a hypergeometric function. A particular feature of the problem is the propagation parameter $P$, which is proportional to $U' h'/K$, where $U'$ and $2h'$ denote the propagation speed and separation of the vortices and $K$ is the circulation. The parameter $P$ is a function of the Mach number $M$ and has the asymptotic value unity in the limit of incompressible flow. The analysis leads to the conclusion that $P=1+o(M^2)$ for small values of $M$; that is, the propagation number is unchanged to order $O(M^2)$. This differs from earlier work, which predicted the asymptotic development $P\sim 1-M^2/4$.

This is the second part of a series examining the behaviour of breaking-wave-induced macrovortices. The first part examined theoretically general behaviours during startup conditions, and gave computational results for macrovortex generation and their evolution on widely spaced breakwaters. In this paper, we extend and test qualitatively and quantitatively some of the basic results of Part 1, in particular the initial longshore vortex transport for a wide range of geometries ranging from narrow rip current topographies to isolated breakwaters. Accounting for the presence of a shoreline is found to be necessary for the representation of longshore vortex transport. Results show vortex motion ranging from strongly offshore in the case of a narrow rip current, to strongly longshore in the case of an isolated breakwater. Even with the significant approximations inherent in the analytical predictions, numerical computations using time-domain Boussinesq-type equations and laboratory experiments confirm the trends. Part 3 examines the horizontal mixing features of wave-induced flows over isolated (single-breakwater configuration) or multiple (rip-current configuration) submerged structures.

The series of papers on the flow dynamics due to wave-induced macrovortices is completed with a statistical analysis of the mixing of the shallow flows occurring around submerged structures used for coastal protection. This is investigated with specific focus on the role played by large-scale horizontal eddies shed in coastal areas by waves breaking corresponding to topographic features like submerged breakwaters. As in Part 2, conditions due to isolated or arrays of breakwaters are studied. Analysis of particle statistics is used to determine both the features of the induced quasi-two-dimensional flow and to derive general properties. In particular three distinct regimes are found to characterize the flow evolution. Asymptotic regimes for small and large times share in any of the features of typical ‘ballistic’ and ‘Brownian’ regimes. Focus is mainly placed on properties of the ‘intermediate time’ regime which are seen to depend on the chosen topographic configuration. In agreement with the deterministic results of Part 2, we find that, because of an intense longshore current, an isolated breakwater induces a larger dispersion than that due to an array of breakwaters, characterized by a rip current. Moreover, for the same topography, the diffusivity grows with the local wavelength. Comparison with field data suggests that results of scaled-down laboratory experiments reproduce well natural mixing conditions. A simple formulation of absolute diffusivity, to be used in practical applications related to environmental quality management, is, finally, proposed.

Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as $\mbox{\it Re} \to \infty$, the dissipation approaches zero no slower than $(\ln\mbox{\it Re})^{-1/2}$. The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation – but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

The von Kármán constant $k$ relates the flow speed profile in a wall-bounded shear flow to the stress at the surface. Recent laboratory studies in aerodynamically smooth flow report $k$ values that cluster around 0.42–0.43 and around 0.37–0.39. Recent data from the atmospheric boundary layer, where the flow is usually aerodynamically rough, are similarly ambiguous: $k$ is often reported to be significantly smaller than the canonical value 0.40, and two recent data sets suggest that $k$ decreases with increasing roughness Reynolds number $Re_{\ast}$. To this discussion, we bring two large atmospheric data sets that suggest $k$ is constant, 0.387$\,{\pm}\,$0.003, for $2\, {\le}\,\hbox{\it Re}_\ast \,{\le} \,100$.

The data come from our yearlong deployment on Arctic sea ice during SHEBA, the experiment to study the Surface Heat Budget of the Arctic Ocean, and from over 800 h of observations over Antarctic sea ice on Ice Station Weddell (ISW). These were superb sites for atmospheric boundary-layer research; they were horizontally homogeneous, uncomplicated by topography, and unobstructed and uniform for hundreds of kilometres in all directions.

During SHEBA, we instrumented a 20 m tower at five levels between 2 and 18 m with identical sonic anemometer/thermometers and, with these, measured hourly averaged values of the wind speed $U(z)$ and the stress $\tau (z)$ at each tower level $z$. On ISW, we measured the wind-speed profile with propeller anemometers at four heights between 0.5 and 4 m and measured $\tau $ with a sonic anemometer/thermometer at one height. On invoking strict quality controls, we gleaned 453 hourly $U(z)$ profiles from the SHEBA set and 100 from the ISW set. All of these profiles reflect near-neutral stratification, and each exhibits a logarithmic layer that extends over all sampling heights. By combining these profiles and our measurements of $\tau $, we made 553 independent determinations of $k$. This is, thus, the largest, most comprehensive atmospheric data set ever used to evaluate the von Kármán constant.

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, $\mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p$, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-$\mbox{\textit{St}}$ expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in $\mbox{\textit{St}}$. The equations for relative trajectories are then solved, to $O(\mbox{\textit{St}})$, in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-$\mbox{\textit{St}}$ open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains $O(\mbox{\textit{St}})$, while the scaling of the gradient displacement increases from $O(\mbox{\textit{St}})$ for far-field open trajectories, to $O(\mbox{\textit{St}}^{{1}/{2}})$ for open trajectories with $O(\mbox{\textit{St}}^{{1}/{2}})$ upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is $O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2)$, while that in the gradient direction scales as $O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2)$ and $O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2)$ in the limits $\phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}}$ and $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi \,{\ll}\, 1$, respectively. Further, the region of zero-Stokes closed trajectories is destroyed, and there exists a new attracting limit cycle whose location in the shearing plane is, at leading order, independent of $\mbox{\textit{St}}$. The extension of the present analysis to include a generic linear flow, and the implications of the finite-$\mbox{\textit{St}}$ trajectory modifications for coagulating systems are discussed.

We study the temporal stability of the Orr–Sommerfeld and Squire equations in channels with turbulent mean velocity profiles and turbulent eddy viscosities. Friction Reynolds numbers up to $Re_\tau \,{=}\, 2\,{\times}\, 10^4$ are considered. All the eigensolutions of the problem are damped, but initial perturbations with wavelengths $\lambda_x \,{>}\, \lambda_z$ can grow temporarily before decaying. The most amplified solutions reproduce the organization of turbulent structures in actual channels, including their self-similar spreading in the logarithmic region. The typical widths of the near-wall streaks and of the large-scale structures of the outer layer, $\lambda_z^+ \,{=}\, 100$ and $\lambda_z/h \,{=}\, 3$, are predicted well. The dynamics of the most amplified solutions is roughly the same regardless of the wavelength of the perturbations and of the Reynolds number. They start with a wall-normal $v$ event which does not grow but which forces streamwise velocity fluctuations by stirring the mean shear ($uv\,{<}\,0$). The resulting $u$ fluctuations grow significantly and last longer than the $v$ ones, and contain nearly all the kinetic energy at the instant of maximum amplification.

The connections between the random elements of a discrete random flow field and the uncertainty in the hierarchical set of its spatio-temporal scales, obtained by the symmetric version of the proper orthogonal decomposition (POD) method, are investigated. It is shown that the relevant statistics for the energy levels, the temporal modes and the spatial modes can be expressed in an explicit form as power series of the flow field standard deviation. Such expansions characterize accurately interesting phenomena of mixing between different flow scales. The basis of the present work is the assumption that the randomness is characterized by a Gaussian uncorrelated random field. Two applications of the theory developed are proposed: to the incompressible flow past a cylinder at Reynolds number $\mbox{\textit{Re}}\,{=}\,100$ and to the natural convective flow over an isothermal horizontal plate at Rayleigh number $\mbox{\textit{Ra}}\,{=}\,4.75\,{\times}\,10^6$. The theoretical predictions are confirmed well by Monte Carlo simulations and interesting relations between the random flows and the relevant statistics of their POD spatio-temporal scales are determined and discussed.

The interaction of an oblique shock wave impinging on a turbulent boundary layer at Mach number 2.3 is experimentally investigated for a wide range of shock intensities. Characteristic time and length scales of the unsteady reflected shock and inside the downstream interaction region are obtained and compared with already existing results obtained in compression ramp experiments as well as in subsonic detached flows. Dimensionless characteristic frequencies are highlighted to characterize low-frequency shock unsteadiness as well as the different large scales which develop inside the initial part of the interaction. The possibility of describing the spatial development of the large scales inside the interaction zone using a mixing-layer scheme including compressibility effects is tested for a wide range of Mach numbers, shock intensities and geometrical configurations. Moreover, strong evidence of a statistical link between low-frequency shock movements and the downstream interaction is given. Finally, the downstream evolution of the structures shed into the boundary layer is characterized and shows features specific of our configuration.

The Boussinesq equations for Rayleigh–Bénard convection are simulated for a cylindrical container with an aspect ratio near 1.5. The transition from an axisymmetric stationary flow to time-dependent flows is studied using nonlinear simulations, linear stability analysis and bifurcation theory. At a Rayleigh number near 25000, the axisymmetric flow becomes unstable to standing or travelling azimuthal waves. The standing waves are slightly unstable to travelling waves. This scenario is identified as a Hopf bifurcation in a system with $O(2)$ symmetry.

A body of salt-stratified fluid in a vertical slot can undergo double-diffusive instabilities when laterally heated. A previous study has indicated the possibility that vibrations could induce instabilities in the cases of a strong salinity gradient in regimes that are not only linearly stable, but also nonlinearly stable. We investigate this limit using the method of averaging and confirm that any level of high-frequency vibrations will lead to a reduction in the heating required for instability for a sufficiently strong salinity gradient, but that this is probably not of great importance to terrestrial experimenters.

This paper describes the effect of gravity on the dynamics of a turbulent boundary layer laden with microbubbles. A poly-dispersed distribution of air bubbles with diameters in the 5–1000 $\mu$m range was injected at the leading edge of a surface-piercing vertical flat plate. Reynolds numbers, based on the momentum thickness, up to 2000 and bubble void fractions up to 5 % were studied. The streamwise velocity profile was found to satisfy the logarithmic law characteristic of single-phase turbulent boundary layers, for the range of bubble void fractions studied. It was found that the effect of the bubbles could be modelled as an offset from the wall which increases as the bubble void fraction is increased. Bubbles were observed to segregate by size within the boundary layer, giving rise to strong inhomogeneities in the local void fraction. Buoyancy acting on this region of high bubble void fraction leads to a strongly sheared vertical velocity profile, which divides the boundary layer into three subregions: an inner region closest to the wall populated by very small bubbles with low vertical velocity; an intermediate region where large bubbles accumulate, and which also had the largest number density, resulting in a large vertical velocity induced by the buoyancy; and an outer region with significantly fewer bubbles than the two previously described and of smaller diameters. These three regions correlated well with the presence of a secondary flow normal to the wall in which the wall-normal velocity was positive in the inner region (fluid moved away from the wall), went through zero in the intermediate region (where the vertical velocity reached its maximum) and was negative in the outer region (fluid moved towards the wall). This secondary flow was found to depend strongly on the bubble void fraction and did not scale well with the viscous scales of the boundary layer.

Experiments with cylinders undergoing forced oscillations show a coupling between the fluctuating lift and drag forces as a result of nonlinear wave interactions. The fluctuating lift and drag components are computed from the instantaneous pressure around the azimuth of a circular cylinder measured with an array of miniature microphones. Quadratic nonlinear interaction between the von Kármán vortex-shedding modes and the forcing field produces sum and difference modes, which appear in the spectrum of the surface pressure signals when the forcing frequency is different from the von Kármán vortex-shedding frequency, ($f_{o}$). The spatial symmetry of the interacting modes determines whether the combination modes appear in the lift spectrum or the drag spectrum. Furthermore, the spatial symmetry of the modes can be predicted from the symmetries of the two interacting parent modes. Crossflow excitation of the cylinder produces combination modes that appear in the drag spectrum. Consequently, attempts to enhance the fluctuating lift by crossflow excitation will necessarily affect the fluctuating drag through nonlinear interaction.

A computational study is reported of the instability and growth of fingers for liquid films driven over heterogeneous surfaces. Computations are performed using a variation of the precursor-film model, in which a disjoining pressure term is used to introduce variation in the static contact angle, which in turn models surface heterogeneity. The formulation is shown to yield results consistent with the Tanner–Hoffman–Voinov dynamic contact angle formula for sufficiently small values of the precursor film thickness. A modification of the disjoining pressure coefficient is introduced which yields correct variation of dynamic contact angle for finite values of the precursor film thickness. The fingering instability is examined both for cases with ordered strips of different static contact angle and for cases with random variation in static contact angle. Surface heterogeneity is characterized by strip width and amplitude of static contact angle variation for the case with streamwise strips and by correlation length and variance of the static contact angle variation from its mean value for the random distribution case.

This paper describes particle transport in Stokes flow in a two-dimensional corner whose walls oscillate, which is a simple model for particle transport in the pulmonary alveoli. Formally speaking, the wall motion produces a perturbation to the well-known Moffatt corner eddies. However, this ‘perturbation’ is dominant as the corner is approached. The motion of particles is regular near to the corner. Far from the corner, chaotic motion within the main part of the flow is restricted to very small regions. We deduce that there is competition between the far-field motion that generates eddies and the wall motion. The relative strengths of these two motions determines whether a given particle moves regularly or chaotically. Consequently, there is an intermediate region in which chaotic transport is maximized.

A numerical study of quasi-steady doubly periodic monochromatic short-crested wave patterns in deep water is conducted using a high-order Boussinesq-type model. Simulations using linear wavemaker conditions in the nonlinear model are initially used to approximate conditions from recent laboratory experiments. The computed patterns share many features with those observed in wavetanks, including bending (both frontwards and backwards) of the wave crests, dipping at the crest centrelines, and a pronounced long modulation in the direction of propagation. A new and simple explanation for these features is provided, involving the release of spurious free first harmonics, due to the neglect of steady third-order components in the three-dimensional wave generation. A comparison with the experimentally observed beat length and amplitude matches the theoretical/numerical predictions well. Additionally, direct inclusion of steady third-order components in the wave generation is shown to reduce significantly the modulations (and other unsteady features), further confirming the explanation. This numerical work makes apparent some previously unknown difficulties associated with the physical generation of even the simplest three-dimensional waves, adding significant insight into the interpretation of recent experimental observations.

Recent experiments by Hammack et al. (J. Fluid Mech., vol. 532, 2005, p. 1) on deep-water waves with two-dimensional, periodic surface patterns showed several features, some of which were unsteady. Fuhrman & Madsen (J. Fluid Mech., vol. 559, 2006, p. 391) explained three of these as being the consequence of sinusoidal forcing by the wavemaker array that did not include forced harmonics, either in time or in the direction parallel to the wavemaker. They predicted that neglected third-harmonic terms cause more serious consequences than neglected second-harmonic terms when the generated wavefields have two-dimensional surface patterns. This paper presents experiments that provide strong evidence that their explanation is correct: including the third-harmonic terms in the wavemaker forcing results in wave patterns that propagate with nearly permanent form.

In this work we study the classical Landau–Levich problem of dip-coating. While in the clean interface case and in the limit of low capillary numbers it admits an asymptotic solution, its full study has not been conducted. With the help of an efficient numerical algorithm, based on a boundary-integral formulation and the appropriate set of interfacial and inflow boundary conditions, we first study the film thickness behaviour for a clean interface problem. Next, the same algorithm allows us to investigate the response of this system to the presence of soluble surface active matter, which leads to clarification of its role in the flow dynamics. The main conclusion is that pure hydrodynamical modelling of surfactant effects predicts film thinning and therefore is not sufficient to explain the film thickening observed in many experiments.

The instability behaviour of non-Newtonian liquid jets moving in an inviscid gaseous environment is investigated theoretically for three-dimensional disturbances. The corresponding dispersion relation between the wave growth rate and the wavenumber is derived. Results for axisymmetrical non-Newtonian jets, the Newtonian jets, and the inviscid jets are recovered, and it is shown that two-dimensional disturbances are the most dangerous for the considered set of parameters.