The main difficulty in second-order diffraction analysis stems from the contribution of the second-order potential, which obeys an inhomogeneous free-surface boundary condition. In some applications it is sufficient to know the second-order hydro-dynamic force, which can be calculated without explicitly evaluating this second-order potential. This technique cannot however be used for calculating other quantities such as the hydrodynamic pressure at any point, the sectional force and bending moment in the cylinder or the free-surface elevation due to the second-order effects. This paper provides a detailed analysis of the second-order diffraction problem of a uniform vertical circular cylinder in regular waves. This furnishes results not only on the cylinder surface, but also on the free surface, and in principle in the fluid domain surrounding the body. The analysis may help to throw some light on the physical interpretation of the second-order theory and its mathematical description. Moreover, this information is intended to complement the development of general numerical methods for arbitrary bodies.

Mixing in a turbulent patch, that was generated within a linearly stratified fluid of buoyancy frequency No, was studied experimentally. The turbulence within the patch was induced by sustained oscillations of a mono-planar grid. Grid oscillations produced a local turbulent region, which initially grows rapidly as in a non-stratified fluid and then assumes a quasi-stationary thickness (Lp)c when the stratification inhibits its vertical growth at a time Not ≈ 4. The growth beyond (Lp)c occurs slowly via breaking of interfacial waves at the entrainment interface. As mixing proceeds, the buoyancy frequency within the patch N decreases. The time evolution of N, the buoyancy lengthscale, the Thorpe lengthscale, the maximum Thorpe displacement, the overturning lengthscale and the available potential energy fluctuations were measured and their relationships were investigated. Various aspects of the wave radiation from the patch to the outer stratified layer, the trapping of interfacial waves at the entrainment interface, and the effects of grid parameters on the evolution of the patch were also studied.

A potential-flow modelling of flow past an oscillating circular cylinder with separated wake is developed here based on Parkinson & Jandali's (1970) wake-source model for steady flow. The phase-averaged pressure distributions, the in-line force coefficients, as well as the drag and added-mass coefficients, for an in-line oscillating circular cylinder in a steady free-stream flow are computed using the present ‘unsteady wake-source model’. The results show that Morison's equation is in some cases a satisfactory model in the study of unsteady bluff-body aerodynamics.

The two-dimensional incompressible potential-flow model simulates the effect of flow separation in unsteady flow by placing surface sources, with time-dependent strength and angular positions on the rear wetted surface of the body, and downstream sinks to form a closed wake model in the transformed plane. The unsteady Bernoulli equation is used to obtain the time-dependent pressure distributions over the front wetted surface, from which the in-line force coefficients are obtained through integration.

The in-line force equation obtained from the present model is shown to be comprised of an uncoupled drag term and inertia terms. The corresponding hydrodynamie coefficients obtained for the case of oscillatory flow are also more realistic than those obtained in a potential-flow calculation without flow separation which gives a drag coefficient of zero and a constant inertia coefficient of 2.0. The in-line force equation is reduced to the familiar Morison's equation with some simplifications and thus provides some support to the much criticized Morison's equation in the study of unsteady separated flow.

Another interesting feature of the present model is that it enables the calculation of instantaneous drag and inertia coefficients which have not been successfully obtained previously. In the cases considered here, the variations of drag and inertia coefficients over a cycle are shown to be small and thus the Morison's equation using mean coefficients is shown to predict the in-line forces rather precisely.

The present model was compared with experimental measurements obtained by oscillating a 0.1 m diameter circular cylinder along the direction of free-stream flow. The pressure distributions and in-line force coefficients agree well with the experimental measurements for velocity ratio rω/U∞ up to 0.25, reduced velocity U∞T/d down to 50 and Keulegan—Carpenter number 2πr/d up to 17, where r, ω, T, U∞ and d are the amplitude of oscillation, angular frequency, period of oscillation, free-stream velocity and diameter of the cylinder respectively. The computed drag and inertia coefficients also agree well with those obtained experimentally by previous investigators.

We present a calculation of the hydrodynamic self-diffusion coefficient of a tagged particle in a dilute mono-dispersed suspension of small neutrally buoyant spheres undergoing a steady simple shearing motion. The displacement of the tagged particle parallel to the longitudinal or streamwise direction resulting from a ‘collision’ with one other particle is calculated on the assumption that inertia and Brownian motion effects are negligible. Summing over different pairs leads to a logarithmically divergent integral for the diffusivity which is rendered finite by allowing for the cutoff due to the occasional presence of another pair of particles. The longitudinal shearinduced self-diffusion coefficient is thus found to be 0.267a2γ{cln c−1 + O(c)], where γ denotes the applied shear rate, a is the radius of the spheres and c their volume concentration.

At any free surface at which the tangential stress τns vanishes there must be a surface vorticity ω = −2κq, where κ is the curvature and q the tangential velocity. In a surface wave on water, this condition produces a (Stokes) boundary layer with thickness of order δ = (2ν/σ)½, where ν is the kinemetic viscosity and σ the radian frequency of the wave. To first order in the wave steepness parameter ak, the vorticity remains within the boundary layer, but at second order some escapes through the Stokes layer. The mean vorticity $\overline{\omega}$ at the outer edge of the Stokes layer is of order 2(ak)2σ, twice the mean vorticity generated at the free surface.

These results are applied to steep capillary waves, particularly the parasitic capillaries often seen on the forward face of short gravity waves. Because of the high value of σ for the capillaries, the vorticity they generate is much larger than that generated by the gravity wave itself. Hence the capillaries contribute significantly to the vortex (roller) often found at the crest of short gravity waves, when capillaries are present. It is argued that the crest roller and the capillaries form a cooperative system, a ‘capillary roller’ in which each supports the other, with the aid of surface tension and viscosity. Energy is supplied by the gravity wave.

A capillary roller is one instance of a more general phenomenon: a ‘capillary bore’, which is a noticeable feature of many disturbed water surfaces.

Evidence from direct numerical simulations and from a recent weakly nonlinear theory is presented which shows that the weakly nonlinear results of Daudpota, Hall & Zang (1988) make incorrect predictions for the influence of the Tollmien–Schlichting wave on the Dean vortex.