We consider the flow of a slightly viscous homogeneous fluid over a small two-dimensional obstacle perpendicular to the vertical side walls in a channel rotating about a vertical axis. The flow in the channel is obtained from the solution of the quasi-geostrophic potential vorticity equation in the limit ε = D/L→ 0, where D is the obstacle width and L the channel width. The lowest order, interior flow is shown to be a combination of three effects: a rotational flow caused by vortex stretching and Ekman boundary-layer pumping; a significant irrotational flow induced by the magnitude of the former flow at the vertical boundaries; and the interior Ekman drift due to the basic current. The maximum streamline displacement is calculated and compares very well with recent experiments in the identical parameter range by Boyer (1971a, b). This theory explains how the side walls are responsible for the dependence of the maximum streamline displacement on Rossby number.

Numerical calculations are made of the torque required to sustain a wavy-vortex flow between rotating cylinders. The results are found to agree well with experimental work of Donnelly (1958), and give further confirmation of the validity of Davey, DiPrima & Stuart's (1968) analysis.

A deep surface-piercing wedge-ended hull model was towed through still water. Measurements of the surface wave pattern confirmed earlier findings for ship models, that the measured bow-wave cusp line often lies well forward of the position predicted by thin-ship theory, and that this shift increases with bow water-line angle and with decreasing model speed. Two possible explanations are considered here in terms of changes of wave phase speed with wave convection and steepness. Calculations based on a transformation method due to Guilloton predict more realistic wave profiles than linear theory, but account for less than half the observed shift. Some tentative conclusions are drawn.

The singularity in the Green's function double integral is removed by an improved method, which simplifies the numerical integration. The new integrand decays within one oscillation.

Vortex shedding from spheres has been studied in the Reynolds number range 400 < Re < 5 × 106. At low Reynolds numbers, i.e. up to Re = 3 × 103, the values of the Strouhal number as a function of Reynolds number measured by Möller (1938) have been confirmed using water flow. The lower critical Reynolds number, first reported by Cometta (1957), was found to be Re = 6 × 103. Here a discontinuity in the relationship between the Strouhal and Reynolds numbers is obvious. From Re = 6 × 103 to Re = 3 × 105 strong periodic fluctuations in the wake flow were observed. Beyond the upper critical Reynolds number (Re = 3.7 × 105) periodic vortex shedding could not be detected by the present measurement techniques.

The hot-wire measurements indicate that the signals recorded simultaneously at different positions on the 75° circle (normal to the flow) show a phase shift. Thus it appears that the vortex separation point rotates around the sphere. An attempt is made to interpret this experimental evidence.

In this paper we study the development of large-scale wavelike eddies, or instability waves, in a turbulent free shear flow. The model is based on splitting the flow into three components: the mean flow, the instability wave and the fine-scale turbulence. The wave is considered to be sufficiently weak so that it is developing in a pre-existing, known turbulent mean shear flow. The basis for the wave development is its time-averaged kinetic energy flux equation in integral form and the wave description is obtained through a shape assumption: the amplitude is determined by the energy equation; the shape function and local characteristics are determined by the local linear stability theory. The wave energy changes as it is convected into a different streamwise position where its instability properties change. The energy balancing mechanisms are production, work done by the wave pressure gradients and the energy transfer between the wave and the fine-scale turbulence via the wave-induced Reynolds stresses. The latter is taken to be dissipative via an eddy-viscosity model, inertia—elastic effects not being considered. According to forceful evidence from observations in turbulent free shear flows, the wave development is taken as being upstream controlled and begins from a distinct origin rather than being the result of local forcing by variations of the fine-scale turbulent Reynolds stresses. The wave energy flux initially grows via energy supplied by the inflexional mean flow when the shear layer is relatively thin but eventually decays through action of the finescale turbulence, directly via the dissipative energy transfer and indirectly via the turbulence-diffused, rapidly thickened mean shear flow, which renders the production mechanism less available. Numerical calculations are carried out for a turbulent mean shear flow, with speed Tie on one side and zero in the ambient region, its distribution being approximated by a sine profile in the Howarth-Dorodnitsyn co-ordinate. The flow develops from an initial boundary layer of finite thickness δ0 to a similar free-mixing layer far downstream. The wave is characterized by a dimensionless frequency parameter β0 formed from the wave frequency β*, ue, and δ0. Convection speeds, in general, increase in the downstream direction. They are subsonic initially for Mach numbers Me < 2 and remain subsonic for Me < 1.5. For Me > 2 peaking in the local intensity levels occurs when convection speeds are supersonic and this may explain the observed supersonic far-field radiation a t the higher jet speeds. Induced wave patterns in the ambient region are determined by the complex instability-wave speed rather than the real convection speed alone, consequently ambient wave patterns exist even a t subsonic convection speeds, but are more heavily damped near the origin and fan out laterally downstream for a given β0. According to the present model, if the waves are given an upstream excitation level about 10−3–10−2 times that of Ue, resembling, for instance, the levels of the upstream wall turbulent boundary-layer fluctuations over a wide, low-frequency, spectrum or other possible disturbances a t the nozzle exit, the development of the calculated near noise field as a function of downstream distance bears striking resemblances to the observed near jet noise field and is thus fully sufficient to explain such observations. This comparison leads to the suggestion that the essential energetics of the large-scale wavelike eddies are that they are formed at the origin, amplified and subsequently decay in a developing mean turbulent free shear flow. Therefore this leads also to a most important indication of shear-layer instabilities and noise control. If their historical evolution can be controlled so can the noise from the damaging wavelike eddies. Methods of control on the basis of this model are discussed.

Solutions are obtained for the baroclinic instability problem for situations in which the static stability and mean shear vary geminately with height. The simple solution given by Eady is shown to be a special limiting case of a class of exact solutions for flows whose basic states have a vanishing interior potential vorticity gradient. The generalized solutions show that the temperature amplitude distribution is particularly sensitive to vertical variations in static stability but that phases and other amplitudes are only slightly influenced by such variations. When the static stability and shear increase (decrease) with height an enhanced temperature maximum occurs at the upper (lower) surface in comparison with the standard Eady solution.

The generalized solutions also help to explain the character of annulus waves and predict a short-wave cut-off that is the same as that given by Eady's theory provided that it is based on the vertically averaged gravitational frequency.

The results of an experimental investigation of a turbulent vortex street in the range 103 [lsim ] Re [lsim ] 2 × 104 are presented. The vortex street was created by the motion of a circular cylinder in a motionless fluid (mercury). Photographs obtained showed that the turbulent street, created by the vortex shedding behind the cylinder, persisted at longer downstream distances and higher Reynolds numbers than previously reported in the literature. A theory was developed to account for the experimental measurements pertaining to the change of the geometrical characteristics, h (distance between the two rows of vortices) and α (longitudinal distance between two consecutive vortices on the same row), of the street in the downstream direction. The implications of the structure of the vortex street on the entrainment mechanism of the turbulent wake are discussed. Some observations of the flow in the formation region of the vortices are discussed in relation to existing work.

The turbulent boundary layer over a flat plate was observed visually using two dye injectors and a normal bubble wire. One dye injector was a standard wall slot; the other one was a small Pitot probe which could be placed anywhere in the flow. The purpose of these experiments was to improve understanding of the relationships between the bursting of low-speed streaks near the wall and the flow field farther away from the wall.

On the basis of these observations, it seems that (i) each lift-up is associated with a disturbance which originates in the logarithmic region and is characterized by a mean motion towards the wall, and that (ii) such disturbances are generated by the interaction of an earlier burst from further upstream with the fluid motion in the logarithmic region.

By the ‘Poincaré problem’ is meant the determination of the free oscillations of a contained rotating fluid, its velocity being linearized around a state of solid rotation. Compressibility requires one to introduce a basic thermodynamic profile as well as a basic velocity distribution. Here the temperature gradient has been supposed proportional to the adiabatic gradient, by introduction of a proportionality constant α (α = 0 in the isothermal case; α = 1 in the adiabatic case). In this formulation the system is reducible to a single second-order ordinary differential equation and its boundary condition.

It is proved that if α = 1 the oscillation frequencies in the rotating system cannot equal plus or minus twice the rotation frequency. The negative case is pathological in the sense that there are solutions arbitrarily near the forbidden solution, and a solution curve of frequency as a function of rotation rate crosses the forbidden frequency.

The basic system is expanded in terms of a power series in γ − 1, where γ is the ratio of specific heats. The zeroth-order set of equations is solved in terms of confluent hypergeometric functions, and a solvability condition on the first-order set gives frequency shifts as functions of α. Several zeroth-order frequencies have been calculated, together with four first-order frequency shifts.

The theory describing the swimming mechanism of an elongated body is extended to cover the cases of locomotion through unsteady streams in pipes. Such an extension is essential for the artificial fish ‘Pod’, a medical device which swims in the patient's blood vessel. Two approaches are considered. First, potential theory is considered, and the results achieved show that the main influence of the pipe is on evaluation of the proper virtual mass. Next the flow is assumed to be viscous. The consideration of viscosity is obviously necessary for flows in pipes. In that case the virtual mass is replaced by another equivalent mass depending on the viscosity and on the angular frequency of the lateral motion and in addition new terms appear in the local lift expressions. These are recognized as the viscous damping force.

Plate tectonics provides a remarkably accurate kinematic description of the motion of the earth's crust but a fully dynamical theory requires an understanding of convection in the mantle. Thus the properties of plates and of the mantle must be related to a systematic study of convection. This paper reviews both the geophysical information and the fluid dynamics of convection in a Boussinesq fluid of infinite Prandtl number. Numerical experiments have been carried out on several simple two-dimensional models, in which convection is driven by imposed horizontal temperature gradients or else by heating either internally or from below. The results are presented and analysed in terms of simple physical models. Although the computations are highly idealized and omit variation of viscosity and other major features of mantle convection, they can be related to geophysical measurements. In particular, the external gravity field depends on changes in surface elevation; this suggests an observational means of investigating convection in the upper mantle.

The 41st EUROMECH Colloquium on flows with concentrated vorticity was held in Norwich from 17 to 21 September 1973. There were sixty-five participants from nine countries and the author was the chairman of the organizing committee.

This paper deals with local flow characteristics of subsonic turbulent jets in the presence of a cross-flow. For the various types of jet considered (a cylindrical jet and coaxial jets) the experimental results concern the axes and the velocity profiles in the plane of symmetry of the flow. In the case of the cylindrical jet, the shape of the universal axial velocity profile is defined, as are the law of velocity decay along the axis and the laws of variation of the thicknesses of the jet. Finally, the existence of a link between the axis equation and the law of axial velocity decay in the zone of similarity of the velocity profiles is established.

By measuring the heat dispersion behind a heated wire stretched across a wind tunnel (Taylor 1921, 1935), the Lagrangian velocity autocorrelation was determined in an approximately isotropic, grid-generated turbulent flow. The techniques were similar to previous ones, but the scatter is less. Assuming self-preservation of the Lagrangian velocity statistics in a form consistent with recent measurements of decay in this flow (Comte-Bellot & Corrsin 1966, 1971), a stationary and an approximately self-preserving form for the dispersion were derived and approximately verified over the range of the experiment.

Possibly the most important aspect of this experiment is that data were available in the same flow on the simplest Eulerian velocity autocorrelation in time, the correlation at a fixed spatial point translating with the mean flow (Comte-Bellot & Corrsin 1971). Thus, the Lagrangian velocity autocorrelation coefficient function calculated from the dispersion data could be compared with this corresponding Eulerian function. It was found that the Lagrangian Taylor micro-scale is very much larger than the analogous Eulerian microscale (76 ms compared with 6.2ms), contrary to an estimate of Corrsin (1963). The Lagrangian integral time scale is roughly equal to the Eulerian one, being larger by about 25 %.

A calculation procedure for three-dimensional parabolic flows is applied to predict the velocity and temperature fields in helically coiled pipes. The curvature produces a secondary flow and causes departures from the symmetric velocity profile of Poiseuille flow. Predictions are presented of flow and heat transfer in the developing and fully developed regions. Comparisons of the developing and fully developed velocity profiles with experimental data exhibit good agreement. The development of the wall temperature for the case of axially uniform heat flux with an isothermal periphery has been compared with experimental data and the agreement is good. Predictions for fully developed temperature profiles and heat-transfer coefficients also exhibit good agreement with experimental data. Effects of the Dean number on the friction factor and heat transfer are presented.

The effects of both viscous and pressure stress work are considered in laminar natural convection on flat surfaces. Whereas in previous investigations only viscous stress work effects were studied, it is established here that pressure work effects are generally rather more important both for gases and liquids. Variations in the properties of the fluid outside the natural convection boundary layer are shown to occur inevitably in this problem and are found to lead to effects comparable with those produced by pressure work. Property variations in the boundary layer itself are also found to be of importance. Such variations, both inside and outside the boundary layer, have been ignored in previous studies. Examples of flows are discussed in which the fluids are either perfect gases or fluids undergoing small changes in temperature and pressure. For both of these fluids, the effects of conditions of either constant temperature or constant entropy in the fluid outside the boundary layer are examined.

In this paper we consider the flow field induced by a periodic magnetic field in and about a conducting liquid drop immersed in an incompressible insulating fluid. It is assumed that at infinity the magnetic field varies with time t as cos ωt, where ω is a constant. This magnetic field is associated with a periodic electric field which produces a net electric stress, dependent on the spatial variables, normal to the drop surface. This stress sets up a flow field, in and about the liquid drop, that creates an appropriate viscous stress so that there is stress balance at the drop surface. The flow field is periodic with angular frequency 2ω. For small drop deformations the drop shape at any instant is a spheroid. It is shown that for large ω the amplitude of the velocity field for a conducting drop is approximately independent of ω and for a non-conducting drop it is proportional to ω. The larger velocity amplitude for a non-conducting drop is probably due to the fact that in this case there is no dissipation of electromagnetic energy. The electric stress over the drop surface increases with ω and it is suggested that the drop will burst at large ω unless the amplitude of the applied magnetic field is suitably decreased.

Stokes’ infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes’ expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Padé approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth. An alternative expansion procedure proposed by Havelock for the computation of waves short of the highest has been reconsidered and found to be defective.

Experiments were conducted in two curved rectangular ducts of different radii and the results compared with flow in a straight duct. The turbulent boundary layers developing on the curved walls were examined for mean flow properties. The law of the wall takes a modified form and the usual logarithmic portion applies over a very small range of the profiles on the inner walls and a slightly larger range on the outer walls. Similarity based on a defect law did not exist for any part of the flow examined.

Two-dimensional incompressible motion is generated by a steady external body force varying sinusoidally with a transverse co-ordinate. Such flow is found to be unstable for Reynolds numbers greater than 2½, and under these conditions evolves towards a new steady state. This ‘steady-eddy’ state is itself unstable in a sense, and its breakdown suggests the catastrophic onset of a cascade of turbulence. The mechanics of this cascade can be represented by a kind of recursion system in which the turbulence dynamics of one scale is repeated in the next, and a law of turbulent stress results. The spectrum of kinetic energy generated by a steady input of momentum at a discrete wavelength shows a rapid decrease (as k−5) towards shorter wavelengths but a much slower decrease (as k) towards longer wavelengths.