An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.

The results of Ackerberg (1970, 1971a), on the two-dimensional boundary-layer separation at a sharp trailing edge where a free streamline is attached, are extended to the axisymmetric case, with and without swirl. When the flow has swirl, the boundary-layer swirl velocity close to the wall may be opposite to that in the external flow; this may help explain the ‘bathtub vortex paradox’ observed by Sibulkin (1962). The solutions for the detached shear layers downstream of separation for two-dimensional and axisymmetric flows, with and without swirl, have been obtained. Some misprints in parts 1 and 2 are corrected in the appendix.

The dispersion equation for waves on an infinite uniform jet column of elliptic cross-section is derived, and approximated for large eccentricity with the aid of new asymptotics for the modified Mathieu functions. It is shown that the effect of appreciable eccentricity on lateral disturbances is greatly to reduce their growth rates below those for a circular jet, regardless of whether the disturbance grows spatially or temporally. For ‘vertical’ disturbances it is shown that the behaviour of waves of general length is qualitatively similar to that of long waves on a two-dimensional jet. Thus the mode symmetric about the major axis has small growth rate whether the mode grows temporally or spatially, while the mode antisymmetric about the major axis has small growth rate if temporally growing, but large growth rate if spatially growing. Comments are made as to the relevance of these results to the mode of action of jet silencers which squash a round jet into a flat ‘fish-tail’ shape.

Delta wings with conically subsonic cones-bodies mounted on their compressive side are analysed in the hypersonic small disturbance regime. The weakly three-dimensional conditions associated with slender parabolic Mach cones are used to validate a linearized rotational approximation of the flow field. A combined integral–series representation is obtained for the pressure distribution between the wing-body and shock wave for arbitrary body cross-sections, and is specialized to give analytical formulae for arbitrary-order polynomial transversal contours. Numerical results are presented for wedge, parabolic and higher order cross-sections illustrating the dominant character of the cross-flow stagnation singularity associated with sharp wing-body intersections having a finite slope discontinuity. It is shown that the pressure has a logarithmic infinity at this secondary leading edge, as in corresponding Prandtl–Glauert irrotational flows. The relation of this finding to Lighthill's theorem on cross-stream vorticity is discussed. Other features of the pressure field are considered with particular emphasis on their relationship to a recently derived area rule for such configurations, and possibilities for favourable interference.

An analysis is presented of laminar fully developed flow in a curved tube of circular cross-section under the influence of a pressure gradient oscillating sinusoidally in time. The governing equations are linearized by an expansion valid for small values of the parameter (a/R) [Ka/ων]2, where a is the radius of the tube cross-section, R is the radius of curvature, ν is the kinematic viscosity of the fluid and K and ω are the amplitude and frequency, respectively, of the pressure gradient. A solution involving numerical evaluation of finite Hankel transforms is obtained for arbitrary values of the parameter α = a(ω/ν)½. In addition, closed-form analytic solutions are derived for both small and large values of α. The secondary flow is found to consist of a steady component and a component oscillatory at a frequency 2ω, while the axial velocity perturbation oscillates at ω and 3ω. The small-α flow field is similar to the low Dean number steady flow configuration, whereas the large-α flow field is altered and includes secondary flow directed towards the centre of curvature.

It has been argued that the no-slip boundary condition, applicable when a viscous fluid flows over a solid surface, may be an inevitable consequence of the fact that all such surfaces are, in practice, rough on a microscopic scale: the energy lost through viscous dissipation as a fluid passes over and around these irregularities is sufficient to ensure that it is effectively brought to rest. The present paper analyses the flow over a particularly simple model of such a rough wall to support these physical ideas.

The method of solution of the Korteweg–de Vries equation outlined by Gardner et al. (1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

An experimental investigation into flow through shaped gauze screens in two-dimensional and axisymmetric situations has shown that there is disagreement between measured velocity profiles and those computed from the method developed by Elder (1958). This disagreement has been shown to be attributable to the retention of a term of second order in the basic linearization and comparisons between experimental and theoretical profiles omitting this term are in good agreement.

The interaction of two pure helical (circularly polarized) velocity waves according to the incompressible Navier–Stokes equation produces modulation products of mixed helicity. In general, the interaction of waves of opposite helicity is stronger than that of waves with the same helicity. The inference is that strong net helicity depresses overall turbulent energy transfer. The conservation laws strongly inhibit energy transfer from higher to lower wavenumbers, when the helicity is large. The absolute equilibrium spectra of velocity and helicity for an inviscid flow system truncated at an upper wavenumber k2 are \[ U(k) = 2\alpha/(\alpha^2-\beta^2k^2),\quad Q(k) = 2\beta k^2/(\alpha^2-\beta^2k^2), \] where the velocity variance and helicity/unit volume are ∫U(k)d3k and ∫Q(k)d3k, respectively. The temperature parameters α and β are constrained by α > 0 and |βk2| < α. There are no analogues of the negative-temperature equilibrium states known for two-dimensional inviscid flow. It is argued that the inertial-range energy cascade in isotropic turbulence driven by helical input should not differ asymptotically from that of non-helical turbulence. The absolute equilibrium distributions suggest that, in contrast to the analogous two-dimensional situation, statistically steady helical input at middle wavenumbers should not produce a significant downward cascade of energy to lower wavenumbers.

We consider a drop of liquid hanging from a horizontal support and sandwiched between two vertical plates separated by a very narrow gap. Equilibrium profiles of such ‘two-dimensional’ drops were calculated by Neumann (1894) for the case when the angle of contact between the liquid and the horizontal support is zero. This paper gives the equilibrium profiles for other contact angles and the criterion for their stability. Neumann showed that, as the drop height increases, its cross-sectional area increases until a maximum is reached. Thereafter, as the height increases, the equilibrium area decreases. This behaviour is shown to be typical of all contact angles. When the maximum area is reached, the total energy is a minimum. It is shown that the drops are stable as long as the height and the area increase together.

This paper presents the analysis and numerical solution for the axisymmetric analogue of the problem considered by Proudman & Johnson (1962) and Robins & Howarth (1972).

The possible coexistence of laminar and turbulent flow in a triangular duct had been a point of controversy among several authors. Since the nature of the flow-laminar or turbulent-has a major influence on the head loss and heattransfer characteristics of the flow, resolution of this controversy is of practical as well as theoretical value.

It is shown that the multiplicative rule of the method of matched asymptotic expansions fails under certain conditions. The velocity distribution in front of an elliptic airfoil is given as an example. The reason for the breakdown is explained by inspecting the usual formal justification of multiplicative composition.

Experimental results are presented for the speed of travel of the spherical vortex at the front of a suddenly switched on, submerged, laminar jet in the Reynolds number range 80 < Re < 500. The results show that the speed of advance of the front is approximately one half of the speed of a fluid element on the axis of the steady jet. The experimental data are well correlated by an approximate model of the jet–vortex interaction in which the vortex is treated as a liquid drop with averaged properties. An auxiliary result is a new correlation for the axial variation of the velocity of a steady jet.

The method of multiple scales is used to derive equations governing the temporal and spatial variation of the amplitudes and phases of in viscid capillary–gravity travelling waves in the case of second-harmonic resonance (Wilton's ripples), but including the effects of: (i) near resonance, (ii) liquid depth, and (iii) pressure perturbations exerted by an external subsonic gas on the liquid–gas interface. The spatial form of the equations shows that, below a critical gas velocity, energy is transferred between the fundamental and its first harmonic in keeping with the energy conservation law. However, the amplitude of the first harmonic decreases with increasing gas velocity. Above the critical gas velocity, the displacement of the gas–liquid interface grows monotonically with distance. It is found that pure amplitude-modulated waves are possible only at perfect resonance. Pure phase-modulated, near-resonant waves are periodic, as the resonance forces a readjustment of the phases to produce perfect resonance. The effectiveness of the resonance in rippling the interface increases as the liquid depth decreases.

The problems of tsunami generation are treated by standard integral-transform and modified stationary-phase methods to yield asymptotic approximations to the surface disturbances. The effects of asymmetry are considered in a one-dimensional ocean. Series representations are used to produce sets of normal-mode oscillations in a two-dimensional ocean, and the magnitudes of the wave front and wave train are discussed in relation to the asymmetry of the generating region.