The ability of small drops or bubbles to stick to non-horizontal solid surfaces is analysed. The principal results consist of identifying the critical value of the volume of a drop (or bubble) beyond which it will dislodge and move down (or up) the surface of the solid, and determining the speed at which it will move. In addition, the area of the solid wetted by the drop (or dried by the bubble) is calculated when its volume is at its critical value. All of the results are expressed in terms of experimentally measurable material properties. The most limiting restriction on the validity of the results is the assumption that the value of the contact angle hysteresis is small.

An experimental study was performed to further understanding of turbulent mixing in a two-layer fluid subjected to shear-free turbulence. At low Richardson numbers Ri (= ΔbD3*/K2, where Δb is the buoyancy jump, D* is the depth of a mixed layer and K is ‘action’) the entrainment seems to occur through the eroding effect of large eddies, whereas at high Ri the large eddies flatten at the density interface and the quasi-isotropic eddies near the interface are responsible for the entrainment. The buoyancy transfer can be well described by a gradient-transport model when the eddy diffusivity is properly defined. At or just above the entrainment interface, the buoyancy flux is of the same order as the dissipation, and the diffusive-flux Richardson number tends to a constant.

The thickness h of the interfacial layer was measured in three different ways and was found to grow linearly with D* in agreement with preliminary findings of an earlier investigation of Fernando & Long (1983). The buoyancy gradient in the interfacial layer was found to be constant, and the resulting buoyancy conservation law was experimentally verified. The frequency of the interfacial-layer waves appears to vary as Ri½. The present results, together with the results of the earlier work of Fernando & Long, show a good agreement with a theory of Long (1978b) for behaviour at high values of Ri. The closure assumptions of that theory were also verified by our measurements.

The effect of variable stratification on the linear bifurcations of a doubly diffusive plane parallel layer is examined numerically by expanding in a Fourier series. Because the motivation is analysis of solar-pond stability, a Prandtl number of 7 and ratio of diffusivities of $\frac{1}{80}$ is used in the study, with (large) solute Rayleigh numbers Rs ranging from 104 to 1012. Stratification of solute is cubic antisymmetric about midlayer; because temperature has a higher diffusivity, it is given a linear stratification. Exchange of stabilities results also solve the ‘fingering’ and thermal problems with cubic stratification. For the overstable case, the numerical results approach Walton's perturbation solution at large Rs, but differ significantly at smaller Rs (< 108). While both exchange of stabilities and overstable modes display an expected tendency to localize about the point of minimum solute gradient, the overstable modes behave in other, non-intuitive ways. Sublayers of reversed salinity gradient, if small enough, can be stable. Above Rs = 1012 computations become prohibitively expensive as a continuous spectrum is approached. A simple sublayer scaling rule defines an infinite family of Rs and stratification parameters on which the localized eigensolution is nearly invariant.

This paper presents a simple theory for evaluating the several measures used to characterize the intermittency of fine-scale turbulence, and corroborates the theoretical results from comparison with experimental data, some of which are new. The basic analytical tool is the envelope of the narrow-bandpass-filtered turbulent signal, defined via its Hilbert transform and the analytic signal. The contribution of this paper is twofold. First, it correctly identifies the roles played by the filter characteristics (such as the bandwidth) in determining the intermittency factor, the width of the active regions (pulses) in narrow-bandpass-filtered turbulent signals, and the pulse frequency; it also reveals that all dynamical characteristics of the signal enter indirectly through the peak pulse frequency and the threshold setting. Secondly, the theory suggests that, in the far-dissipation range, the most important feature of signals exhibiting internal intermittency is the stronger-than-algebraic roll-off of the spectral density in that region; it is argued that this feature of turbulence essentially determines the peak pulse frequency in that region. The theory is incomplete in that it does not show how the threshold setting depends on the signal dynamics, but here the discussion is supplemented by experimental data.

The trajectories of surface and subsurface particles of nonlinear gravity–capillary waves are calculated. Surface tension is found to have a small effect on the trajectories and drift velocity of pure gravity waves (down to 20 cm in length). On the other hand, pure capillary wave trajectories can be considerably altered even when the influence of gravity is small (waves of up to 2 cm in length). When the restoring forces are of comparable magnitude, some remarkable trajectories are possible, containing one or more subloops. Overall, the influence of increased surface tension is to increase the relative horizontal distance travelled by a particle, as well as the magnitude of the time-averaged drift velocity ratio at the surface and, as far as short waves are concerned, its penetration depth.

These results can have implications for steep waves where parasitic capillaries are generated and for observations of the wind-drift current.

Experiments in which a long vertical, heated wire is surrounded by concentric annuli of a melt and its crystalline solid show that the convection state changes from a stable unicell surrounded by a stationary cylindrical solid–liquid interface, to a complex time-dependent flow surrounded by a rotating, helical solid–liquid interface. This transition occurs at a Grashof number of approximately 150, which is an order of magnitude less than the critical Grashof number calculated for a liquid annulus surrounded by rigid walls. A linear stability analysis has been carried out for an infinitely tall vertical annulus. When the deformable nature of the crystal–melt interface is taken into account in the boundary conditions, two new modes of instability arise. The most dangerous mode is asymmetrical and corresponds to helical waves travelling vertically upwards. The critical Grashof number and the scaling properties of the eigenstate agree with experiments. The results clearly demonstrate the coupling of convection with crystal–melt interfacial instabilities.

The observation of large-amplitude ‘kink’ waves on the vortex cores produced by an oscillating grid in a rotating fluid (Hopfinger, Browand & Gagne 1982) has motivated the study of such waves under more controlled circumstances. We have experimentally observed the properties of helical waves, rotating, plane standing waves and evolving, isolated kink-waves. Their characteristics have been related to theories based on the localized induction equation of Arms & Hama (1965), the ‘cut-off’ theory of Crow (1970) as extended by Moore & Saffman (1972), and an extension of Pocklington's (1895) dispersion relationship for ‘hollow-core’ vortices. It is shown that the latter dispersion relation and the Moore & Saffman theory are good approximations to our experimental results. Using these, we present new results on solitary kink-wave properties of concentrated vortex flows, and in particular show that envelope solitons are possible only for a restricted range of carrier wavenumbers. A second class of waves was also observed: the axisymmetric solitary waves of Benjamin (1967). These were found to become unstable to spiral disturbances when their amplitude exceeded a certain magnitude, as has been found in the study of the related vortex-breakdown phenomenon. All of these observations are used to interpret the experiments presented by HBG and to discuss qualitatively the dynamics of rotating turbulence. In the Appendix we propose a possible mechanism by which concentrated vortices can be formed in a rotating turbulent fluid.

Detailed understanding of flowing granular materials is severely hampered by the deficiencies of present experimental methods. To help increase the information base, a computer simulation has been developed to describe two-dimensional unidirectional flows of inelastic fully rough particles. This paper presents the results of a Couette shear-flow simulation. The results include distributions of velocity, density and granular temperature (a measure of the kinetic energy contained in the random particle motions). The effects of density and shear rate on the granular temperature are explored. Shear and normal forces on the solid walls are compared with experimental and theoretical results. The behaviour of the particles in the simulated flow is examined and assessments are made of the collision angle and velocity distributions. The development of a distinct, ‘layered’ microstructure is observed in high-density granular flows.

We reexamine the related problems of instability of parallel shear flows and over-reflection of internal waves at a critical level, concentrating on the stratified case. Our primary aim is to delineate the specific aspects of a flow that permit overreflection and instability. A related and partly realized aim is to develop a mechanistic ‘picture’ of how over-reflection and instability work. In the course of this study we have also uncovered some new results concerning the instability of stratified shear flows – showing how regions of enhanced static stability and enhanced damping can destabilize otherwise stable flows.

For the scattering of steady plane waves, we show that, of the conditions found by Lindzen & Tung (1978), in the unstratified case, only the existence of wave-propagation regions above and below the critical level is always necessary for over-reflection (at least in the absence of damping), although a trapping region around the critical level and a reflecting surface bounding the upper wave region may play crucial roles in some cases. Our results suggest that the role of the upper wave region may be to allow a wave flux through the critical level. Moreover, we show numerically that the effect of an upper wave region can be mimicked by a region of localized damping which leads to over-reflection as well.

We also consider an initial-value problem, using numerical methods. When a wave is incident on the incident level, the reflection and transmission coefficients grow smoothly to their final values. The rate of growth depends on the flow parameters, but there is some evidence to suggest there is a characteristic timescale involved that depends only on the shear (and not on wave travel time). This fits a mechanistic picture of over-reflection and instability that we describe, in which the essential part is a kinematic interaction between wave and mean flow at the critical level, depending only on shear.

The development of laminar flow in a 180° section of a curved square duct (R/d = 6.45) was studied by laser-Doppler anemometry (LDA). The streamwise flow-velocity component Vϕ and the secondary flow component Vr were measured as a function of Dean number and of the azimuthal angle ϕ. The development of the streamwise flow component was found to be connected with a strong momentum transfer towards the outer wall between ϕ = 0° and ϕ = 60°, with a partial back-transfer of the momentum towards the duct centre (ϕ = 45°–108°), and with little further change of the momentum between ϕ = 108° and 180° near the outer wall. The measurements of the Vr component showed at least one vortex pair in the secondary flow. A second vortex pair with opposing sense of circulation was found to develop near the outer wall only for Dean numbers between 150 and 300, in agreement with numerical calculations. This second vortex pair was found in the region between ϕ = 108 and 171°. Between ϕ = 60 and 108° it was not possible to identify the second vortex pair in the developing flow. However, developing laminar-flow numerical calculations by Humphrey (1982) show that it also arises for K = 485 in a 180° square duct with R/d = 3.35 and Re = 890, and it is a function of inlet flow conditions. From the measured stream-function maximum of the second vortex pair it may be deduced that a curved duct with 180° (20 hydraulic diameters) bend angle is not sufficient to reach fully developed flow conditions at d/R = 1/6.45 and K = 226.

The existence of surface waves trapped above a submerged horizontal cylinder was shown by Ursell to depend upon the vanishing of a certain infinite determinant. Here, the determinant is evaluated numerically and the dispersion curves found. It is shown that the mild slope equation may be used to determine the dispersion relation and surface profiles with good accuracy and with less computational effort than the full linear theory.

Radiation and diffraction of free-surface waves due to a submerged body in a uniform current is considered. The fluid layer is infinitely deep and the motion is two-dimensional. Applying the method of integral equations, the radiation problem and the diffraction problem for a submerged circular cylinder are examined. For small speed U of the current a forced motion of a given frequency will give rise to four waves. It is shown, however, that, for a circular cylinder, an incoming harmonic wave gives rise to two waves only. Depending on the frequency, the new generated wave may be considered as a transmitted or a reflected wave. The mean second-order force is computed. For the radiation problem the first-order damping force is also obtained. It is shown that, for some values of the parameters, the damping force is negative. This result is closely related to the fact that a harmonic wave travelling upstream with a phase velocity less than U conveys negative energy downstream. The forces remain finite as Uσ/g (σ ≡ the frequency, g ≡ the acceleration due to gravity) approaches ¼.

Because of the inherent nonlinearity in the boundary conditions on the free surface, water waves with frequencies from neighbouring parts of the sea spectrum interact and force low-frequency oscillations at the second order. Since the physical phenomenon involves vastly different timescales, the perturbation method of multiple scales is applied here to a rectangular cylinder in beam seas. It will be shown that at the second order there are two kinds of long waves; one is locked to the envelopes of the incident, reflected and transmitted short waves, while the other propagates away from the body at the long-wave velocity (gh)½. The latter contributes to the damping of slow-drift oscillations of the body. Analytical results for the displacement amplitudes of the slow sway and for the radiated long waves are derived. The transient evolution due to incident envelopes of finite and semi-infinite duration is also predicted.

An exact derivation is given of the dynamo equation in the presence of an arbitrary incompressible mean flow v0. A mixed representation is used specifying v0 in terms of a Lagrangian displacement, while Eulerian coordinates are employed for the turbulent velocity v superimposed on v0.

When the first-order-smoothing approximation is made (FOSA; valid when the turbulent velocity v has a short correlation time τc) the usual dynamo equation is recovered, except that the turbulent velocity v in the tensors αis and βisk is replaced by $\overline{\boldmath v}$. The bar represents the effect of advection and is expressed solely in terms of the Lagrangian coordinate specifying the mean flow v0. Thus the intuitive idea is confirmed that dynamo action depends only on velocity correlation functions measured at a point comoving with the mean flow. The result admits easy evaluation in actual model situations. This is illustrated with an example tailored to the solar dynamo. A shear in v0 causes a (kinematic) anisotropy in the tensors αis and βisk. This can be a large effect, which comes on top of the intrinsic (dynamical) anisotropy in the velocity correlation functions. Subsequently, the analysis is extended beyond FOSA up to arbitrary order, relevant for long correlation times τc on the basis of the work of Van Kampen (1974). It is shown that the same formalism is also applicable to the problem of turbulent transport of a scalar.

Conditions for applicability of the work are (1) very large magnetic Reynolds number, (2) incompressible flows v and v0, (3) stationary mean flow v0, and (4) correlation time τc [Lt ] period of the dynamo.

A partially submerged propeller theory was developed by employing a singularity distribution method. Unsteady pressure doublets and pressure sources represented the blade camber and blade-and-cavity thickness respectively. The induced velocities were derived by reducing the formula to a lifting-line configuration. The free-surface effect was considered by use of the image method. The induced velocities contained the singular integrals of 5th order, which are usually numerical unstable. An effort was made to derive numerically stable formulae from these singular equations by applying a method similar to the induction-factor method of Morgan & Wrench (1965) and Lerbs (1952) used for steady-state fully wetted propeller problems. These new formulae are not only applicable to the present partially submerged ventilated propeller problem, but also to general unsteady subcavitating and cavitating propeller problems such as propeller starting-up and non-periodic loading problems. By combining the two-dimensional water entry-and-exit theory of Wang (1979), the thrust and torque coefficients were calculated for representative partially submerged propellers and favourably compared with the experimental data.

We present observations of the development of nonlinear surface gravity-wave groups. We calculate the amplitude and phase modulations using Hilbert-transform techniques. With increasing propagation distance and wave steepness, the phase modulation develops local phase reversals whose locations correspond to amplitude minima or nodes. The concomitant frequency modulation develops jumps or discontinuities. The observations are compared with recent similar results for wavetrains. The observations are modelled numerically using the cubic nonlinear Schrödinger equation. The motivation is twofold: to examine quantitatively the evolution of phase as well as amplitude modulation and to test the inviscid predictions for the asymptotic behaviour of groups versus long-time observations. Although dissipation rules out recurrence, there is a long-time coherence of the groups. The phase modulation is found to distinguish between dispersive and soliton behaviour.

The present work is an investigation into the scale effects on cavitation inception in submerged water jets. Three scale effects were studied: (i) jet size, (ii) jet velocity and (iii) the nuclei population as a function of dissolved air content in the jet. Holography and schlieren photography were utilized to observe the flow. The results obtained in the present work were then used to illustrate the importance of previously overlooked flow variables in scaling cavitation inception data. An interesting finding of the present investigation was that the cavities at inception did not occur in the cores of the large vortex structures within the shear layer of the jets, as is commonly believed. A new parameter called the ‘probable cavitation occurrence’ parameter is also introduced in this paper. This parameter incorporates the effects of the nuclei number-density function and peak pressure fluctuations and it shows great promise in scaling cavitation-inception data.

The cubic Schrödinger equation, which governs the complex amplitude of a cross-wave that is subharmonically excited by a symmetric wavemaker in a deep wave tank (Jones 1984), is solved for prescribed boundary conditions at the wavemaker and a null condition at infinity. This solution (which is a limiting case of a one-parameter family of cnoidal waves) typically describes a trapped wave that decays exponentially in a semi-infinite tank, although it may be fitted to a tank of finite length for a particular combination of parameters. It is closely related to the solution for trapped waves excited through vertical oscillation of a long channel (Wu, Keolian & Rudnick 1984).

We have observed steady and oscillatory flow through a two-dimensional channel expansion. The experimental results are supported by numerical solutions of the unsteady Navier–Stokes equations. This work was prompted by the recent discovery of vortex waves during steady flow past a moving indentation in a channel wall. Our work deals with both asymmetric channels, in which we show that vortex waves are observed during oscillatory flow with rigid walls, and with symmetric channels, in which a vortex street is observed. We believe that the vortex street is not a vortex wave, but the result of a shear-layer instability.

A wind-wave flume which allows mechanically generated water waves to propagate in opposition to a boundary-layer air flow has been used to measure the air-flow structure in an opposing wind–wave situation. Measurements of the wave-induced pressure closely follow the predictions of potential flow theory, with the pressure in antiphase with the water surface. Hence, in contrast to the following wind situation, there is no appreciable air–water energy flux due to normal stresses. The vertical and horizontal wave-induced velocities deviate slightly in magnitude from the potential flow result while still following it qualitatively. Based on these velocity measurements it is determined that the Reynolds stress $-\rho_{\rm a}\overline{\tilde{u}\tilde{u}}$ is the dominant term in causing the decay of waves in opposing winds. The predicted rate of decay has a squared dependence on the wave slope and the ratio of wind speed to wave phase speed.