Nano-bubbles have recently been observed experimentally on smooth hydrophobic surfaces; cracks on a surface can likewise be the site of bubbles when partially wetting fluids are used. Because these bubbles may provide a zero shear stress boundary condition and modify considerably the friction generated by the solid boundary, it is of interest to quantify their influence on pressure-driven flow, with particular attention given to small geometries. We investigate two simple configurations of steady pressure-driven Stokes flow in a circular pipe whose surface contains periodically distributed regions of zero surface shear stress. In the spirit of experimental studies probing slip at solid surfaces, the effective slip length of the resulting flow is evaluated as a function of the degrees of freedom describing the surface heterogeneities, namely the relative width of the no-slip and no-shear stress regions and their distribution along the pipe. Comparison of the model with experimental studies of pressure-driven flow in capillaries and microchannels reporting slip is made and a possible interpretation of the experimental results is offered which is consistent with a large number of distributed slip domains such as nano-size and micron-size nearly flat bubbles coating the solid surface. Further, the possibility is suggested of a shear-dependent effective slip length, and an explanation is proposed for the seemingly paradoxical behaviour of the measured slip length increasing with system size, which is consistent with experimental results to date.

The dynamic characteristics of the pressure and velocity fields of the unsteady incompressible laminar wake behind a circular cylinder are investigated numerically and analysed physically. The governing equations, written in a velocity—pressure formulation and in conservative form, are solved by a predictor—corrector pressure method, a finite-volume second-order-accurate scheme and an alternating-direction-implicit (ADI) procedure. The initiation mechanism for vortex shedding and the evaluation of the unsteady body forces are presented for Reynolds-number values of 100, 200 and 1000.

The vortex shedding is generated by a physical perturbation imposed numerically for a short time. The flow transition becomes periodic after a transient time interval. The frequency of the drag and lift oscillations agree well with the experimental data.

The study of the interactions of the unsteady pressure and velocity fields shows the phase relations between the pressure and velocity, and the influence of different factors: the strongly rotational viscous region, the convection of the eddies and the almost inviscid flow.

The interactions among the different scales of structures in the near wake are also studied, and in particular the time-dependent evolution of the secondary eddies in relation to the fully developed primary ones is analysed.

The objective of this study is to investigate for turbulent flow the fluid motions very near a solid boundary, and to create a physical picture which relates these motions to turbulence generation and transport processes. An experimental technique was developed which permitted detailed observations of the regions very near a pipe wall, including the viscous sublayer, without requiring the introduction of any injection or measuring device into the flow. This technique involved suspending solid particles of colloidal size in a liquid, and photographing their motions with a high-speed motion picture camera moving with the flow. To provide greater detail, the field of view was magnified.

Fluid motions were observed to change in character with distance from the wall. The sublayer was continuously disturbed by small-scale velocity fluctuations of low magnitude and periodically disturbed by fluid elements which penetrated into the region from positions further removed from the wall. From a thin region adjacent to the sublayer, fluid elements were periodically ejected outward toward the centreline. Often there was associated with these events a zone of high shear at the interface between the mean flow and the decelerated region that gave rise to the ejected element. When the ejected element entered this shear zone, it interacted with the mean flow and created intense, chaotic velocity fluctuations. These ejections and resulting fluctuations were the most important feature of the wall region, and are believed to be a factor in the generation and maintenance of turbulence.

An experimental and theoretical-numerical investigation has been carried out to extend existing knowledge of velocity and temperature distributions and local heat-transfer coefficients for naturel convection within a horizontal annulus. A Mach—Zehnder interferometer was used to determine temperature distributions and local heat-transfer coefficients experimentally. Results were obtained using water and air at atmospheric pressure with a ratio of gap width to inner-cylinder diameter of 0·8. The Rayleigh number based on the gap width varied from 2·11 × 104to 9·76 × 105. A finite-difference method was used to solve the governing constant-property equations numerically. The Rayleigh number was changed from 102 to 105 with the influence of Prandtl number and diameter ratio obtained near a Rayleigh number of 104. Comparisons between the present experimental and numerical results under similar conditions show good agreement.

The viscous gravity current that results when fluid flows along a rigid horizontal surface below fluid of lesser density is analysed using a lubrication-theory approximation. It is shown that the effect on the gravity current of the motion in the upper fluid can be expressed as a condition of zero shear on the unknown upper surface of the gravity current. With the supposition that the volume of heavy fluid increases with time like tα, where α is a constant, a similarity solution to the governing nonlinear partial differential equations is obtained, which describes the shape and rate of propagation of the current. The viscous theory is shown to be valid for t [Gt ] t1, when α < αc and for t [Lt ] t1 when α > αc, where t1, is the transition time at which the inertial and viscous forces are equal, with $\alpha_{\rm c} = \frac{7}{4}$ for a two-dimensional current and αc = 3 for an axisymmetric current. The solutions confirm the functional forms for the spreading relationships determined for α = 1 in the preceding paper by Didden & Maxworthy (1982), as well as evaluating the multiplicative factors appearing in the relationships. The relationships compare very well with experimental measurements of the axisymmetric spreading of silicone oils into air for α = 0 and 1. There is also very good agreement between the theoretical predictions and the measurements of the axisymmetric spreading of salt water into fresh water reported by Didden & Maxworthy and by Britter (1979). The predicted multiplicative constant is within 10% of that measured by Didden & Maxworthy for the spreading of salt water into fresh water in a channel.

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

Measurements of the amount of fluid left behind when a viscous liquid is blown from an open-ended tube are described.

It is shown that all parallel inviscid shear flows of constant density are unstable to a wide class of initial infinitesimal three-dimensional disturbances in the sense that, according to linear theory, the kinetic energy of the disturbance will grow at least as fast as linearly in time. This can occur even when the disturbance velocities are bounded, because the streamwise length of the disturbed region grows linearly with time. This finding may have implications for the observed tendency of turbulent shear flows to develop a longitudinal streaky structure.

Cavitation bubbles were produced by focusing giant pulses of a Q-switched ruby laser into distilled water. The dynamics of the bubbles in the neighbourhood of a solid boundary were studied by means of high-speed photography using a rotating-mirror camera with framing rates of up to 300000 frame/s. Bubble motion was evaluated from the frames with the aid of a digital computer using a graphical input device. Smoothed distance-time curves of different portions of the bubble wall were obtained also, allowing a reliable calculation of bubble-wall velocities (except at the actual instant of collapse). One of the numerical examples of the collapse of a spherical bubble near a plane solid boundary obtained by Plesset & Chapman could be realized experimentally. A comparison of the bubble shapes shows good agreement.

An experimental investigation of the binary droplet collision dynamics was conducted, with emphasis on the transition between different collision outcomes. A series of time-resolved photographic images which map all the collision regimes in terms of the collision Weber number and the impact parameter were used to identify the controlling factors for different outcomes. The effects of liquid and gas properties were studied by conducting experiments with both water and hydrocarbon droplets in environments of different gases (air, nitrogen, helium and ethylene) and pressures, the latter ranging from 0.6 to 12 atm. It is shown that, by varying the density of the gas through its pressure and molecular weight, water and hydrocarbon droplets both exhibit five distinct regimes of collision outcomes, namely (I) coalescence after minor deformation, (II) bouncing, (III) coalescence after substantial deformation, (IV) coalescence followed by separation for near head-on collisions, and (V) coalescence followed by separation for off-centre collisions. The present result therefore extends and unifies previous experimental observations, obtained at one atmosphere air, that regimes II and II do not exist for water droplets. Furthermore, it was found that coalescence of the hydrocarbon droplets is promoted in the presence of gaseous hydrocarbons in the environment, suggesting that coalescence is facilitated when the environment contains vapour of the liquid mass. Collision at high-impact inertia was also studied, and the mechanisms for separation of the coalescence are discussed based on time-resolved collision images. A coalescence/separation criterion defining the transition between regimes III and IV for the head-on collisions was derived and found to agree well with the experimental data.

This paper describes the development of a numerical model for studying the evolution of a wave train, shoaling and breaking in the surf zone. The model solves the Reynolds equations for the mean (ensemble average) flow field and the k–ε equations for the turbulent kinetic energy, k, and the turbulence dissipation rate, ε. A nonlinear Reynolds stress model (Shih, Zhu & Lumley 1996) is employed to relate the Reynolds stresses and the strain rates of the mean flow. To track free-surface movements, the volume of fluid (VOF) method is employed. To ensure the accuracy of each component of the numerical model, several steps have been taken to verify numerical solutions with either analytical solutions or experimental data. For non-breaking waves, very accurate results are obtained for a solitary wave propagating over a long distance in a constant depth. Good agreement between numerical results and experimental data has also been observed for shoaling and breaking cnoidal waves on a sloping beach in terms of free-surface profiles, mean velocities, and turbulent kinetic energy. Based on the numerical results, turbulence transport mechanisms under breaking waves are discussed.

A fluid-fluid interface that joins a solid surface forms a common line. If the common line moves along the solid, a mutual displacement process is involved and is studied here. Some simple experiments motivate the formulation of the basic assumption of the analysis. The basic assumption is a formalization of the idea that the fluid-fluid interface rolls on or unrolls off the solid. This forms an axiom for the mostly kinematical analysis that follows. The predictions are tested through a series of qualitative experiments. The role of the no-slip boundary condition at the solid surface is discussed.

The effect of vertical boundaries on convection in a shallow layer of fluid heated from below is considered. By means of a multiple-scale perturbation analysis, results for horizontally unbounded layers are modified so that they ‘fit in’ to a rectangular region. The critical Rayleigh number and critical wave-number are determined. Motion is predicted to have the form of finite ‘rolls’ whose axes are parallel to the shorter sides of the dish. Aspects of the non-linear development and stability of this motion are studied. The general question of convective pattern selection in a bounded layer is discussed in the light of available theoretical and experimental results.

Meso-scale structures that take the form of clusters and streamers are commonly observed in dilute gas–particle flows, such as those encountered in risers. Continuum equations for gas–particle flows, coupled with constitutive equations for particle-phase stress deduced from kinetic theory of granular materials, can capture the formation of such meso-scale structures. These structures arise as a result of an inertial instability associated with the relative motion between the gas and particle phases, and an instability due to damping of the fluctuating motion of particles by the interstitial fluid and inelastic collisions between particles. It is demonstrated that the meso-scale structures are too small, and hence too expensive, to be resolved completely in simulation of gas–particle flows in large process vessels. At the same time, failure to resolve completely the meso-scale structures in a simulation leads to grossly inaccurate estimates of inter-phase drag, production/dissipation of pseudo-thermal energy associated with particle fluctuations, the effective particle-phase pressure and the effective viscosities. It is established that coarse-grid simulation of gas–particle flows must include sub-grid models, to account for the effects of the unresolved meso-scale structures. An approach to developing a plausible sub-grid model is proposed.

In this paper we investigate the relationship between the large- and small-scale energy-containing motions in wall turbulence. Recent studies in a high-Reynolds-number turbulent boundary layer (Hutchins & Marusic, Phil. Trans. R. Soc. Lond. A, vol. 365, 2007a, pp. 647–664) have revealed a possible influence of the large-scale boundary-layer motions on the small-scale near-wall cycle, akin to a pure amplitude modulation. In the present study we build upon these observations, using the Hilbert transformation applied to the spectrally filtered small-scale component of fluctuating velocity signals, in order to quantify the interaction. In addition to the large-scale log-region structures superimposing a footprint (or mean shift) on the near-wall fluctuations (Townsend, The Structure of Turbulent Shear Flow, 2nd edn., 1976, Cambridge University Press; Metzger & Klewicki, Phys. Fluids, vol. 13, 2001, pp. 692–701.), we find strong supporting evidence that the small-scale structures are subject to a high degree of amplitude modulation seemingly originating from the much larger scales that inhabit the log region. An analysis of the Reynolds number dependence reveals that the amplitude modulation effect becomes progressively stronger as the Reynolds number increases. This is demonstrated through three orders of magnitude in Reynolds number, from laboratory experiments at Reτ ~ 103–104 to atmospheric surface layer measurements at Reτ ~ 106.

Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300. The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 188.5±1.0. The critical spanwise wavelength is 3.96 ± 0.02 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability. At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0.822 diameters at onset. Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300.

An investigation is made into the moving contact line dynamics of a Cahn–Hilliard–van der Waals (CHW) diffuse mean-field interface. The interface separates two incompressible viscous fluids and can evolve either through convection or through diffusion driven by chemical potential gradients. The purpose of this paper is to show how the CHW moving contact line compares to the classical sharp interface contact line. It therefore discusses the asymptotics of the CHW contact line velocity and chemical potential fields as the interface thickness ε and the mobility κ both go to zero. The CHW and classical velocity fields have the same outer behaviour but can have very different inner behaviours and physics. In the CHW model, wall–liquid bonds are broken by chemical potential gradients instead of by shear and change of material at the wall is accomplished by diffusion rather than convection. The result is, mathematically at least, that the CHW moving contact line can exist even with no-slip conditions for the velocity. The relevance and realism or lack thereof of this is considered through the course of the paper.

The two contacting fluids are assumed to be Newtonian and, to a first approximation, to obey the no-slip condition. The analysis is linear. For simplicity most of the analysis and results are for a 90° contact angle and for the fluids having equal dynamic viscosity μ and mobility κ. There are two regions of flow. To leading order the outer-region velocity field is the same as for sharp interfaces (flow field independent of r) while the chemical potential behaves like r−ξ, ξ = π/2/max{θeq, π − θeq}, θeq being the equilibrium contact angle. An exception to this occurs for θeq = 90°, when the chemical potential behaves like ln r/r. The diffusive and viscous contact line singularities implied by these outer solutions are resolved in the inner region through chemical diffusion. The length scale of the inner region is about 10√μκ – typically about 0.5–5 nm. Diffusive fluxes in this region are O(1). These counterbalance the effects of the velocity, which, because of the assumed no-slip boundary condition, fluxes material through the interface in a narrow boundary layer next to the wall.

The asymptotic analysis is supplemented by both linearized and nonlinear finite difference calculations. These are made at two scales, experimental and nanoscale. The first set is done to show CHW interface behaviour and to test the qualitative applicability of the CHW model and its asymptotic theory to practical computations of experimental scale, nonlinear, low capillary number flows. The nanoscale calculations are carried out with realistic interface thicknesses and diffusivities and with various assumed levels of shear-induced slip. These are discussed in an attempt to evaluate the physical relevance of the CHW diffusive model. The various asymptotic and numerical results together indicate a potential usefullness for the CHW model for calculating and modelling wetting and dewetting flows.

The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 2563 modes) and by power-series analysis in time.

The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance $\hat{\delta}(t)$ of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that $\hat{\delta}(t)$ decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity ($\hat{\delta}(t) = 0$) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t44 to order t80. Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time tmax at which the maximum energy dissipation is achieved. Early-time, high-R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near tmax the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near tmax when R = 3000 and Rλ = 110:

1. a k−n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times);

2. th energy dissipation has considerable spatial intermittency; its spectrum has a k−1+μ inertial range with the codimension μ ≈ 0.3−0.7.

Skewness and flatness results are also presented.

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)

Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, ε, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, ε → 0, δ → 0, independently. Retaining terms O(ε), O(δ2) and O(εδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.