Calculations are presented of the development of the turbulent boundary layer on a rotating cylinder with its axis along the stream, and the results are compared with measurements made by Parr. With the choice of a suitable co-ordinate system, the boundary layer downstream of the nose of the cylinder approximates closely to a condition of two-dimensionality, and a simple integral method of solution can be applied. The only evidence of three-dimensionality lies in the destabilizing effect of rotation on the turbulence structure of the layer and an analysis of this instability has been made which relates changes in mixing length and entrainment to an instability parameter in the form of a Richardson number.

In spite of the changes in shear stress distribution and entrainment brought about by rotation, mean velocity profiles and skin friction values are found to be related to H and Rθ in the same way as for the stationary flat plate.

The two-dimensional sail is considered in a free-streamline model to complement the oft-considered airfoil model which is limited to small angles of attack. The shape of the sail, the lift and drag coefficients, and the moment are obtained for various angles of attack and states of tension.

This paper considers the surface response of a semi-infinite, uniformly rotating, constant depth, homogeneous ocean to a variable atmospheric force. For a general wind and pressure system it is shown that forced Kelvin-type waves can be generated and that only the longshore wind component and the pressure can generate them. In particular a semi-infinite wind and moving pressure pattern are shown to generate Kelvin waves that travel away from the force discontinuities at the speed of shallow-water waves. The waves in the latter case exhibit a frequency shift typical of non-dispersive waves from a moving source. Some numerical values for the amplitudes of the Kelvin waves are also given.

The stability of a sand bed in an alluvial channel is investigated by a two-dimensional mathematical model, based on the vorticity transport equation. The model takes account of the internal friction and describes the non-uniform distribution of the suspended sediment. It turns out that the inclusion of the friction and of a definite model of the sediment transport mechanism leads to results rather different from those obtained previously by potential-flow analysis.

This article describes experimental work on the mixed convection régime with flow normal to electrically heated cylinders. The forcing velocities used were in the range 0·0085–3 ft./sec (i.e. 10−2 < Ref < 45) and temperature differences in the range 30°C to 200°C (i.e. 10−3 < Ra < 10) were covered.

Correlations are proposed for the forced convection and natural convection conditions. A correlation is also developed for the combined forced and natural convection region by a vectorial addition of the flow parameters, which gives good agreement with the experiments except over a limited range in the contraflow régime.

In this paper the interchange flow between two reservoirs connected by a contraction and containing fluid of different densities is considered. The effect of the boundary layers on the floor and walls of the contraction on the depth of flow in the contraction is discussed for the case of single layer flowing from one reservoir to the other. Next the theory for a denser layer plunging under a stationary layer is developed. In this case there is a discontinuity at the point of intersection of the surfaces of the flowing and the stationary fluids and there are three possible flow régimes depending on whether this discontinuity occurs at, downstream of, or upstream of the contraction.

Finally, the case where there is an interchange flow with fluid flowing from each reservoir into the other is introduced. This latter theory parallels that developed by Wood (1968) for the case of two layers flowing from one reservoir through a contraction into another reservoir and as in this case there are two points of control, one at the position of minimum width and one (the virtual point of control) away from this position of minimum width.

Experiments are described for a single layer flowing through the contraction and the results of these are used to obtain an indication of the accuracy that could be expected from the experiments with the more complicated exchange flow. The experiments with the exchange flow verified the major elements of the theory.

The Hunt–Stewartson technique of estimating fluid velocity and magnetic flux profiles in rectangular ducts is generalized for the entire secondary boundary layer.

A theory is developed for calculating oscillations of harbours of constant depth and arbitrary shape. This theory is based on the solution of a singular integral equation. Numerical results have been calculated for rectangular harbours so as to check the accuracy of the method. Examples for wave amplification factor and velocity field for both rectangular and actual complex-shaped harbours are given.

The problem of a piston moving in a cylinder is studied experimentally using flow visualization techniques. A vortex motion is observed at the piston face and cylinder wall interface as the cylinder wall moves toward the piston. Non-dimensional scaling parameters for the vortex size and stability are determined and semi-empirical theories for the size of the vortex are presented.

Experiments are carried out to establish the relationship between the nature of the flow instability and the inclination angle of the plate. The angular dependence of the Rayleigh number characterizing the onset of instability is also determined. An electrochemical flow visualization technique is utilized to expose the patterns of fluid motion. It is found that for inclination angles of less than 14° (relative to the vertical), waves are the mode of instability. On the other hand, for inclination angles in excess of 17°, the instability is characterized by longitudinal vortices. The range between 14° and 17° is a zone of continuous transition, with the two modes of instability co-existing.

Experiments are reported in which the error of a sharp-edged, circular static pressure hole normal to the boundary of a moving fluid is deduced from measurements made with pressure transducers, the diaphragms of which were set flush with the boundary surface. An error curve is presented which covers the range of hole Reynolds number up to a value of 2000.

A technique is described for measuring the local phase speed in a turbulent flow. The technique has been used to measure the phase speed of the Fourier components of the longitudinal velocity fluctuations in grid turbulence. These measurements are unique in that the probe spacing is only twice the Kolmogoroff length scale. The velocity fluctuations were measured with linearized constant-temperature hot-wire anemometers, the outputs of which were digitally sampled and recorded in real time. Digital Fourier analysis techniques were then used to calculate the cross-spectral density of the two velocity measurements. From this, the phase, phase speed, and coherence were calculated. The coherence has been used to estimate the variance of these measurements.

The steady-state solution of the non-linear equation \[ h_t + hh_x + h_{xxx} = \delta h_{xx} \] with both damping and dispersion is examined in the phase plane. For small damping an averaging technique is used to obtain an oscillatory asymptotic solution. This solution becomes invalid as the period of the oscillation approaches infinity, and is matched to a straightforward expansion solution. The results obtained are compared with a numerical integration of the equation.

A solution satisfying the usual radiation conditions is found to the problem of an internal wave propagating towards a corner. It is found that, far from the corner, and the characteristic emanating from the corner, the solution is asymptotically equivalent to the solution found by plane wave reflexions from an infinite wall. The present solution shows that, by imposing the radiation condition, a singularity predicted by the ray theory along the corner characteristic is absent. A further singularity in the present solution along the same characteristic is shown to be due to an inability of the usual linear internal wave equations to fully describe the motion. The solution is for restricted corner angles.

An oscillating current such as a tidal stream or an inertial oscillation may have a horizontal scale of the order of many times the local depth of water. Thus an island projecting from an otherwise uniform sea bed will give rise to a local, periodic disturbance near the island. It is shown that this disturbance may be resolved into two waves travelling in opposite senses round the island. If the particle orbits at large distances are circular, then only one of these waves has non-zero amplitude.

In addition to the oscillatory motion, however, there is a steady d.c. streaming, or mass-transport velocity, whose magnitude is of order u2/σa where u denotes the magnitude of the oscillatory velocity at large distances, σ denotes the radian frequency, and a is the radius of the island. In this paper the profile of the streaming velocity is calculated for circular islands, with or without shoaling regions offshore. It is shown that resonance with the free modes trapped by the shoaling regions may greatly increase the streaming velocity. Viscosity (or horizontal mixing) also tends to increase the streaming velocity close to the shoreline.

The conclusions are supported by some simple model experiments. It is suggested that such streaming may partly account for the observed pattern of currents near Bermuda.

Finite-difference solutions of the equations of motion for steady incompressible flow around a circular cylinder have been obtained for a range of Reynolds numbers from R = 5 to R = 100. The object is to extend the Reynolds number range for reliable data on the steady flow, particularly with regard to the growth of the wake. The wake length is found to increase approximately linearly with R over the whole range from the value, just below R = 7, at which it first appears. Calculated values of the drag coefficient, the angle of separation, and the pressure and vorticity distributions over the cylinder surface are presented. The development of these properties with Reynolds number is consistent, but it does not seem possible to predict with any certainty their tendency as R → ∞. The first attempt to obtain the present results was made by integrating the time-dependent equations, but the approach to steady flow was so slow at higher Reynolds numbers that the method was abandoned.

This paper describes some experiments in rotating flows in which solitary waves were observed.

In one set of experiments the waves were generated on a swirling flow whose circumferential velocity distribution resembled that of the Rankine combined vortex. This flow was established by stirring the liquid in a large cylindrical container, in much the same way as one stirs a cup of tea, and it was often found at the cessation of the stirring that a wave had been generated. This wave propagated along the vortex core and was reflected at the bottom of the container and at the free surface of the liquid and displayed the remarkable permanence characteristic of solitary waves. It appears that, to a first approximation, the speed of the waves may be calculated simply from the depression of the free surface of the liquid at the centre of the vortex. These waves are the rotating-fluid counterpart to the solitary waves in fluids of great depth recently discussed by Benjamin (1967b) and by Davis & Acrivos (1967).

In a second set of experiments, solitary waves were generated in a long cylindrical tube and are analogous to the familiar solitary wave of open-channel flows. The theory indicates that these waves are possible in any swirling flow in which the angular velocity is distributed non-uniformly. Thus, a long liquid-filled tube was started rotating about its axis with a uniform angular velocity, and waves were generated before the fluid had reached a state of uniform rotation. Using the known velocity distribution for a tube of infinite length, comparisons have been made between the observed wave forms and the theoretical calculations of Benjamin (1967a). There is good agreement between the observed wave forms and the theoretical predictions.

Two hypotheses concerning the turbulent flow over an infinitesimal-amplitude travelling wave are investigated. One hypothesis, originally made by Miles, is that the wave does not affect the turbulence and therefore the turbulent Reynolds stresses are dependent only on height above the mean wave surface. Alternatively, the proposal that turbulent stresses are primarily dependent on height above the instantaneous wave surface is examined. Numerical solutions of the appropriate equations are compared with Stewart's recent experimental results and with the approximate solutions employed by Miles and others. No definite conclusion can be reached from comparison with experimental results since the predicted flows are quite sensitive to details of the mean velocity profile near the wave surface where no data was taken. It is found that the asymptotic results do not apply for the conditions investigated.

The diagonal pattern of bed form which appears in sand-bed channels when the channel width to depth ratio is great and the flow is nearly critical is discussed from the theoretical viewpoint by using the method of characteristics. Some photographs illustrate the phenomenon.

This paper describes the measurements of heat transfer and drag for the flow of dilute polymer solutions around very small cylinders. The thermal experiments were carried out at Reynolds numbers less than 50, and the results establish the dependence of the heat transfer on fluid velocity, cylinder diameter, solution concentration, and polymer molecular weight. The drag measurements were conducted with the same type of solutions and in the same Reynolds-number range. To complement the heat-transfer and drag measurements, the flows around a cylinder and through an orifice were examined visually. These flow-visualization studies showed that the streamline pattern with dilute polymer solutions can be significantly different from that with Newtonian fluids because of viscoelastic effects.

An analysis of Rouse's theory of macromolecules shows that for low accelerations a dilute polymer solution behaves mechanically like a Maxwell model. The analysis thereby produces a relaxation time, a single parameter representing the elasticity of the fluid, which can be related to the properties of the solute and solvent. This relaxation time is contained in a new dimensionless group which governs dynamic similarity when induced elastic stresses dominate viscous stresses in the flow around a circular cylinder. The dimensionless group is shown to correlate the thermal data when the heat transfer does not depend on the free stream velocity.