The viscous and steady flow about two distinct parallel infinite rotating circular cylinders is theoretically investigated. Because any inner steady Stokes flow is not quiescent far from the cylinders, a strictly steady analysis requires matching an inner steady Stokes approximation with an outer solution of the steady Navier–Stokes equations. However, except for the case of identical cylinders of equal angular velocities, it is impossible to determine this outer solution. In the same spirit as Nakanishi et al. (1997) and Ueda et al. (2001), the present work therefore first addresses the unsteady viscous flow induced by cylinders impulsively set into both steady rotation and translation ${\bm W}.$ Using integral representations of the stream function and the vorticity, the resulting long-time flow is approximated in the limit of large viscosity. Letting time tend to infinity for ${\bm W}$ non-zero extends Lee & Leal (1986) and agrees with Watson (1996), whereas the required steady flow is obtained by making ${\bm W}$ vanish before letting time go to infinity. At the obtained leading-order approximation, the ‘lift’ and ‘drag’ forces on each cylinder (parallel and normal to the line of centres) are respectively zero and independent of the Reynolds number. The drag experienced by each body is plotted versus the gap between the cylinders for several values of the rotation, both for identical and non-identical cylinders.

Chandrasekhar's (1961) solution to the eigenvalue equation arising from the Kelvin-Helmholtz stability problem for a rotating fluid is shown to be incorrect. The unstable modes are correctly enumerated with the aid of Cauchy's principle of the argument. Various previously published solutions using Chandrasekhar's analysis are corrected and extended.

A study has been made of the effects of non-axisymmetric bottom topography in a differentially heated, rotating fluid annulus. In the absence of free baroclinic waves the disturbance produced by topography of vertical dimensions typically much less than the depth of the fluid and horizontal dimensions comparable to the gap width is steady and confined to the lower part of the fluid. The maximum amplitude of this disturbance depends on the angular speed of rotation of the apparatus and varies with height, being very small above the level where the basic azimuthal flow reverses. The flow pattern shows a closed circulation displaced relative to the topography in the downstream direction. In the presence of free baroclinic waves the topographic wave extends throughout the whole depth of the fluid. Determinations of the amplitude and phase of the various azimuthal Fourier components present show that the topographically forced components exchange energy with the free components as they drift relative to the topography.

Over the past decades a large amount of work has been done on inviscid, steady progressive, gravity waves on a free surface. However, waves which occur in nature are never, in fact, free-surface waves, since they are always beneath a fluid of finite density, if only air. In spite of this, very little work has been done on finite-amplitude interfacial waves. Tsuji & Nagata (1973) carried out a perturbation expansion in wave amplitude to fifth order, for interfacial waves between two stationary fluids. Holyer (1979) extended the calculation using the computer and resorted to Pade approximants to sum the resulting series. Meiron & Saffman (1983) investigated numerically the limiting highest wave and demonstrated the existence of over-hanging gravity waves of permanent form.

This paper describes a comparison of experimental data and theoretical results for the motions in waves of large flat-bottomed barges having zero forward speed. The experimental data are derived from model tests at two scales (1:36 and 1:108). These consist of measurements of motions in surge, sway, heave, roll, pitch and yaw to long-crested seas approaching the barge models at two orientation angles (head and beam seas). The experimental data are compared with results computed from linearized potential-flow theory, which accounts for the diffraction and radiation of gravity waves around the barge. The boundary-integral scheme employed to solve this potential-flow problem is briefly reviewed.

The experimental data at two scales and the theoretical results show that potential-flow theory is in reasonable agreement with experimental data for all motions except roll near resonance. These roll-motion discrepancies, due to vortex shedding from bilge-keel edges, are discussed. Some of the observable effects of scale in the model tests are highlighted.

The stability of three-dimensional salt fingers to three-dimensional perturbations is considered. It is found that when square fingers are subjected to long-wavelength perturbations, then the same types of instability exist as for two-dimensional salt fingers. There are quantitative changes to the growth rates and the positions of marginal stability.

The Kraichnan-Phillips theorem asserts that, for incompressible, homogeneous, turbulent flow over a plane, rigid wall, the wall-pressure wavenumber-frequency spectrum P(k, ω) → 0 as the planar wavenumber k → 0 provided the frequency ω ≠ 0. A proof of this theorem is given by use of a general formula that expresses the normal force on an arbitrary rigid body in terms of volume and surface integrals involving the vorticity. Implications for the theory of flow-induced surface vibrations are briefly discussed.

The statistics of rotational motion of small, inertialess triaxial ellipsoids are computed along Lagrangian trajectories extracted from direct numerical simulations of homogeneous isotropic turbulence. The total particle angular velocity and its components along the three principal axes of the particle are considered, expanding on the results of Chevillard & Meneveau (J. Fluid Mech., vol. 737, 2013, pp. 571–596) who showed results of the rotation rate of the particle’s principal axes. The variance of the particle angular velocity, referred to as the particle enstrophy, is found to increase as particles become elongated, regardless of whether they are axisymmetric. This trend is explained by considering the contributions of vorticity and strain rate to particle rotation. It is found that the majority of particle enstrophy is due to fluid vorticity. Strain-rate-induced rotations, which are sensitive to shape, are mostly cancelled by strain–vorticity interactions. The remainder of the strain-rate-induced rotations are responsible for weak variations in particle enstrophy. For particles of all shapes, the majority of the enstrophy is in rotations about the longest axis, which is due to alignment between the longest axis and fluid vorticity. The integral time scale for particle angular velocities about different axes reveals that rotations are most persistent about the longest axis, but that a full revolution is rare.

A fully nonlinear numerical model is used to investigate spontaneous wave generation during two-dimensional frontogenesis forced by a horizontal strain field. The model uses the idealised configuration of an infinitely long straight front and uniform potential vorticity, with a uniform imposed convergent strain across the front. Shakespeare & Taylor (J. Fluid Mech., vol. 757, 2014, pp. 817–853) formulated a generalised analytical model (ST14) for this system that extends the classical Hoskins & Bretherton (J. Atmos. Sci., vol. 29, 1972, pp. 11–37) model (HB) to large strain rates (${\it\alpha}\sim f$). Here, we use a numerical model to simulate the fully nonlinear problem and compare the results with the predictions of the analytical model for a variety of strain rates. Even for weak strains (${\it\alpha}=0.2f$), the confinement of the secondary circulation and the spontaneous generation of waves, predicted by ST14, are shown to be important corrections to the HB solution. These inviscid predictions are also robust for an equilibrated front where strain-forced frontogenesis is balanced by diffusion. For strong strains the wavefield becomes of leading-order importance to the solution. In this case the frontal circulation is tightly confined, and the vertical velocity is an order of magnitude larger than in the HB model. The addition of a strain field that weakens with time allows the release and propagation of the spontaneously generated waves. We also consider fronts with both large vorticity and strain rate, beyond the validity of the ST14 model.

Two-dimensional, incompressible flows are discussed by a generalization of the line-vortex model. A large number of structures are randomly distributed initially. Each individual structure is convected by the superposed flow field of all the others. The statistical properties of the resulting space–time varying random flow are studied. Analytical expressions for both Eulerian and Lagrangian correlation functions are obtained for the limit where the density of structures is large. The analytical results compare favourably with numerical simulations. The study serves as a special test on proposed relations between Eulerian and Lagrangian averages which can be generally valid, i.e. also for three-dimensional, turbulent flows.

Nonlinear mechanisms leading to the ejection of dipolar vortices from a laminar wall jet are being investigated using highly accurate Navier–Stokes simulations. With a set of well-defined numerical experiments for a forced Glauert wall jet, the nonlinear resonant interaction between the large-amplitude harmonic disturbance and a small-amplitude wave packet is systematically explored using two-dimensional simulations. Generated by a small-amplitude pulse, the wave packet experiences rapid resonant growth in the subharmonic part of its spectrum resulting in vortex mergings and, ultimately, the ejection of a pair of counter-rotating vortices from the wall jet. This two-dimensional subharmonic instability, if not mitigated by competing three-dimensional instabilities, can lead to the detachment of the entire wall jet from the surface. As shown using three-dimensional direct numerical simulations, vortex ejection still occurs in a forced transitional wall jet if the two-dimensional wave packet can reach a large amplitude level upstream of the region of three-dimensional turbulent breakdown. Movies are available with the online version of the paper.

We consider the temporal evolution of a viscous incompressible fluid in a torus of finite curvature; a problem first investigated by Madden & Mullin (J. Fluid Mech., vol. 265, 1994, pp. 265–217). The system is initially in a state of rigid-body rotation (about the axis of rotational symmetry) and the container’s rotation rate is then changed impulsively. We describe the transient flow that is induced at small values of the Ekman number, over a time scale that is comparable to one complete rotation of the container. We show that (rotationally symmetric) eruptive singularities (of the boundary layer) occur at the inner or outer bend of the pipe for a decrease or an increase in rotation rate respectively. Moreover, on allowing for a change in direction of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which eruptive singularities can occur at both the inner and outer bend simultaneously. We also demonstrate that the flow is susceptible to a combination of axisymmetric centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability arises as a consequence of the developing eruption and is shown to be in qualitative agreement with the experimental observations of Madden & Mullin (1994). Throughout our work, detailed quantitative comparisons are made between asymptotic predictions and finite- (but small-) Ekman-number Navier–Stokes computations using a finite-element method. We find that the boundary-layer results correctly capture the (finite-Ekman-number) rotationally symmetric flow and its global stability to linearised perturbations.

A theory and simulation code are developed to study non-steady sources as means to control sonic booms of supersonic aircraft. A key result is that the source of sonic boom pressure is not confined to the length of the aircraft but occupies an extensive segment of the flight path. An aircraft in non-steady flight functions as a synthetic aperture antenna, generating complex acoustic waves with no simple relation to instantaneous volume or lift distributions.

The theory applies linear acoustics to slender non-steady sources but requires no far-field approximation. The solution for pressure contains a term not seen in Whitham's theory for sonic booms of distant supersonic aircraft. The term describes a pressure field that decays algebraically behind the Mach cone and, in the case of steady flight, integrates to a ground load equal to the weight of the aircraft. The algebraic term is separate from those that describe the sonic boom.

Two non-steady source phenomena are evaluated: periodic velocity changes (surge), and periodic longitudinal lift redistribution (slosh). Surge can attenuate a sonic boom and covert it into prolonged weak reverberation, but accelerations needed to produce the phenomenon seem too large for practical use. Slosh may be practical and can alter sonic booms but does not, on average, result in boom attenuation. The conclusion is that active sonic boom abatement is possible in theory but maybe not practical.