A recent paper by Needham, Riley & Smith (1988) considers the flow of a jet of inviscid, incompressible fluid emerging from a circular pipe into a weak crossflow. In this paper we extend their results. In particular we show how the Kutta condition at the pipe lip may be satisfied, and we present details of the flow field obtained from both the formal solution and a field calculation.

A study was undertaken to examine the flat plate relaxation behaviour of a turbulent boundary layer recovering from 90° of strong convex curvature (δ0/R = 0.08), for a length of ≈ 90δ0 after the end of curvature, where δ0 is the boundary layer thickness at the start of the curvature. The results show that the relaxation behaviour of the mean flow and the turbulence are quite different. The mean velocity profile and skin friction coefficient asymptotically approach the unperturbed state and at the last measuring station appear to be fully recovered. The turbulence relaxation, however, occurs in several stages over a much longer distance. In the first stage, a stress ‘bore’ (a region of elevated stress) is generated near the wall, and the bore thickens with distance downstream. Eventually it fills the whole boundary layer, but the stress levels continue to rise beyond their self-preserving values. Finally the stresses begin a gradual decline, but at the last measuring station they are still well above the unperturbed levels, and the ratios of the Reynolds stresses are distorted. These results imply a reorganization of the large-scale structure into a new quasi-stable state. The long-lasting effects of curvature highlight the sensitivity of a boundary layer to its condition of formation.

In Part 1 (Kim & Yue 1989), we considered the second-order diffraction of a plane monochromatic incident wave by an axisymmetric body. A ring-source integral equation method in conjunction with a novel analytic free-surface integration in the entire local-wave-free domain was developed. To generalize the second-order theory to irregular waves, say described by a continuous spectrum, we consider in this paper the general second-order wave–body interactions in the presence of bichromatic incident waves and the resulting sum- and difference-frequency problems. For completeness, we also include the radiation problem and second-order motions of freely floating or elastically moored bodies. As in Part 1, the second-order sum- and difference-frequency potentials are obtained explicitly, revealing a number of interesting local behaviours of the second-order pressure. For illustration, the quadratic transfer functions (QTF's) for the sum- and difference-frequency wave excitation and body response obtained from the present complete theory are compared to those of existing approximation methods for a number of simple geometries. It is found that contributions from the second-order potentials, typically neglected, can dominate the total load in many cases.

The origin of steady asymmetric flows in a symmetric sudden expansion is studied using experimental and numerical techniques. We show that the asymmetry arises at a symmetry-breaking bifurcation and good agreement between the experiments and numerical calculations is obtained. At higher Reynolds numbers the flow becomes time-dependent and there is experimental evidence that this is associated with three-dimensional effects.

Corrected relationships are presented between integral properties of periodic gravity surface waves in the case of a non-zero mean Eulerian velocity. Also considered is the special case of a reference frame in which the mean wave momentum vanishes.

The intermediate long-wave (ILW) equation is a weakly nonlinear integrodifferential equation which governs the evolution of long internal waves in a stratified fluid of finite depth. It reduces to the Korteweg–de Vries (KdV) and to the Benjamin–Ono (BO) equations for shallow and large depths respectively. Solitary wave solutions of the ILW equation are well known, however analytic expressions for periodic solutions of the same equation do not seem to exist. Such expressions are derived in this paper and a remarkable property discovered for these periodic waves is that they can be represented as an infinite sum of spatially repeated solitons. Thus, nonlinear periodic solutions of the ILW equation are obtained by linear superposition of solitons.

Knotted closed-curve solutions of the equation of self-induced vortex motion are studied. It is shown that there are invariant torus knots which translate and rotate as rigid bodies. The general motion of ‘small-amplitude’ torus knots and iterated (cabled) torus knots is described and found to be almost periodic in time, and for some, but not all, initial data, the topology of the knot is shown to be invariant.

Two impinging two-dimensional incompressible inviscid fluid jets of known widths and velocities produce two outgoing jets. The speeds of the outgoing jets are readily determined from the Bernoulli equation. Their two widths and two directions (four quantities) are related by conservation of mass and conservation of two components of momentum (three relations). Because these three conservation relations do not suffice to determine the four unknowns, Milne-Thomson (1968) states on p. 302 that ‘a unique solution is, in general, not possible’. He incorrectly attributes this indeterminateness to disregard of ‘the initial conditions from which this steady motion is supposed to arise’.