In steady, fully developed and unventilated cavity flows occurring in practice, air (originally dissolved in the water) and heat are diffused through the fluid towards the interface providing a continuous supply of air and vapour to the cavity. This must be balanced by the rate of entrainment of volume of air and vapour away from the cavity in the wake. These equilibria which determine respectively the partial pressure of air within the cavity and the temperature differences involved in the flow are studied in this paper. The particular case of the cavitating flow past a spherical headform has been investigated in detail. Measurements indicate a near-linear relation between the partial pressure of air in the cavity and the total air content of the water. From a second set of experiments, designed to estimate the volume rates of entrainment under various conditions by employing artificial ventilation, it appears that this is a function only of tunnel speed and cavity size within the range of the experiments. A simplified theoretical approach involving the turbulent boundary layer on the surface of the cavity is then used to estimate the rates of diffusion into the cavity. The resulting air balance yields a partial pressure of air/air content relation compatible with experiment. The water vapour or heat balance suggests that the temperature differences involved are likely to be virtually undetectable experimentally.

A smoke visualization technique was used to study the laminar wake behind a single cylinder. Observations of the wake revealed the nature of the process of formation of a Kármán vortex street: instability of the shear layers was coupled with the rolling up process induced by the concentration of vorticity.

An unexpected mass transfer was observed through the Kármán vortex street which cause mixing of the fluid surrounding the wake.

The thermohaline stability problem previously treated by Stern, Walin and Veronis is examined in greater detail. An error in an earlier paper is corrected and some new calculations made. It is shown, for instance, that direct convection can occur for thermal Rayleigh number R much less than 100 Rs when Rs [gsim ] 0·1, where Rs is the salinity Rayleigh number. A graphical presentation is devised to show the relative importance of the different terms in the equations of motion as a function of R and Rs. The most unstable mode over all wave-numbers for each R, Rs is found and it is shown that where both unstable direct and oscillating modes are present, the most unstable mode is direct in most cases.

This investigation demonstrates the measurement of low turbulence intensities in a flow of mercury using the constant-temperature hot-film technique. A simple equation for the determination of turbulence intensity is derived on the basis of observed calibration data for short, cylindrical, quartz-insulated, platinum hot-film sensors. This equation has the advantage of being independent of the changeable effects at the sensor-fluid interface and of the usually troublesome drift in anemometer output signal. The flow speeds considered are low, so that the Péclet number based on the outside diameter of the insulated sensor is less than unity. Various problems associated with the application of hot-film anemometry to mercury flows are discussed.

Calculations have been made of the development of the turbulent boundary layer on a disk rotating in free air, using circumferential and radial momentum-integral equations and an auxiliary equation of entrainment. In the calculations, circumferential velocity profiles are represented by Thompson's (1965) two-parameter family, while radial profiles are given by Mager's (1952) quadratic expression. The circumferential component of skin friction follows from the use of Thompson's profile family for the circumferential velocity. The entrainment, in dimensionless form, is assumed to be determined uniquely by the circumferential velocity profile in the same way as was proposed by Head (1958) for a two-dimensional turbulent boundary layer.

Detailed measurements have been made of the development of the turbulent boundary layer on the rotating disk, and the calculations are found to be in excellent agreement with the results when a suitable adjustment is made to Head's two-dimensional entrainment curve.

The oscillations of a drop moving in another fluid medium have been studied at low values of Reynolds number and Weber number by taking into consideration the shape of the drop and the viscosities of the two phases in addition to the interfacial tension. The deformation of the drop modifies the Lamb's expression for frequency by including a correction term while the viscous effects split the frequency into a pair of frequencies—one lower and the other higher than Lamb's. The lower frequency mode has ample experimental support while the higher frequency mode has also been observed. The two modes almost merge with Lamb's frequency for the asymptotic cases of a drop in free space or a bubble in a dense viscous fluid but the splitting becomes large when the two fluids have similar properties. Instead of oscillations, aperiodic damping modes are found to occur in drops with sizes smaller than a critical size ($\sim\hat{\rho}\hat{\nu}^2/T $). With the help of these calculations, many of the available experimental results are analyzed and discussed.

A differential approximation for the equations of radiative transfer in a grey gas is applied in a study of the effects of thermal radiation upon the classical problem of the compressive action of a plane, cylindrical or spherical piston. The ambient gas ahead of the precursor shock wave is supposed cool and the shock wave transparent, whilst the piston is taken to be neither an emittor nor reflector of radiative energy. It is shown that self-similar flow patterns may arise if the ambient density and piston speed are both non-uniform with variations linked to the absorption coefficient which is assumed to be density and temperature dependent. Detailed flow patterns are obtained in the case of general opacity and also in the transparent limit from which it is deduced that under certain conditions the approximation provided by the latter may be rather dubious.

This paper describes an experiment performed with a fluid-filled sphere whose rotation speed about a fixed axis was forcibly varied in a slight but sinusoidal manner about a non-zero mean value. The object of this experiment was both to excite axisymmetric inertial eigen-oscillations within the relatively low viscosity fluid through the mild pumping action of the oscillatory Ekman boundary layer near the wall, and to measure and compare with theory some of the properties of such modes.

Seven distinct fluid resonances were detected via pressure measurements made along the axis for various ratios of the excitation to the mean rotation frequency. For the three most pronounced of those modes, the observed frequency ratios agree within ½ of 1% with the corresponding ratios predicted from linear, small viscosity theory. The response amplitudes at the various resonances and the rates of decay upon switching off the excitation also compare favourably with theory, although the observed amplitudes are systematically lower and the decays more rapid by a few per cent to several tens of per cent.

The theory referred to above is largely that of Greenspan (1964, 1968). It is in part rederived here from energy considerations.

An experimental investigation of the non-equilibrium behaviour of the shock stand-off distance ahead of spheres at low supersonic Mach number is reported. An intermittent wind tunnel operating with a reacting gas mixture with one non-equilibrium mode is described. A non-equilibrium parameter after Damköhler is determined for the flow and the experiments cover a range of this variable from near-frozen to near-equilibrium states. The shock stand-off distance is measured and found to vary as expected between these two bounds with a low value at equilibrium. With all other variables governing shock standoff distance held constant, such measurements can also be used to determine the relaxation time of the non-equilibrium mode.

Noise intensity measurements per octave band have been carried out in the far field of a small circular nozzle, at angles 0-150° from the jet axis, with and without vortex generators. The results, interpreted in the light of recent theoretical conclusions, confirm Lighthill's original suggestion that the dominant noise radiator is the pressure-shear x-r quadrupole at all but the lowest frequencies. At very low frequencies Reynolds stress-shear x-x and x-r quadrupoles contribute comparable amounts of radiation. When the product of Strouhal number (based on nozzle diameter and jet velocity) and jet Mach number exceeds unity the geometrical acoustics approximation becomes valid, but even at such high frequencies the shear noise contribution dominates. Vortex generators, while reducing noise intensity, do not appear to modify drastically the above composition of noise radiators.

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.

Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.

As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

An analytical description for the initial phases of collapse of a spherical or cylindrical shock wave in a perfect gas is given in the present paper. The shock wave is initiated by the instantaneous and uniform deposition of a finite quantity of energy per unit surface area at a finite radius R0. For the initial shock motion where xs = (Rs - R0)/R0 is small, analytical solutions are obtained by a power-series expansion of the dependent variables in xs. The classical self-similar solution for a strong planar blast wave is recovered as the present zero-order solution. Non-similar effects arising from both finite shock strengths and the presence of a characteristic length R0 are accounted for simultaneously in the present perturbation scheme. The analysis is carried out up to third order in xs. For very large values of the initiation energy where the shock wave remains strong throughout its collapse, it is found that the present perturbation solution can adequately describe a significant portion of the collapse processes. The solution indicates that the shock decays rather rapidly initially and later begins to accelerate as a result of the additional adiabatic compression of the shocked states due to flow-area convergence. However, for weak initiation where the energy released is small, the present perturbation solution is an asymptotic series and diverges very rapidly as the shock propagates away from the wall. The range of validity then is limited to very small values of xs.

Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basic N-shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all times t after a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals $(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$, where Δp is the pressure jump across the shock and tc is a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer, tc is calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.

The flow of an incompressible, viscous fluid past a sphere is considered for small values of the Reynolds number. In particular the drag is found to be given by \[ D = D_s\{1+{\textstyle\frac{3}{8}}R+{\textstyle\frac{9}{40}}R^2(\log R+\gamma + {\textstyle\frac{5}{3}}\log 2 - {\textstyle\frac{323}{360}})+{\textstyle\frac{27}{80}}R^3\log R+O(R^3)\}, \] where Ds is the Stokes drag, R is the Reynolds number and γ is Euler's constant.

A viscous fluid is confined between two parallel, infinite planes which perform torsional oscillations of small amplitude about a common axis. The resulting flow is studied for the case of high-frequency oscillations, when boundary layers form adjacent to moving surfaces. Particular analysis is made of the second-order, steady, radial-axial streaming. It is shown that in certain circumstances viscosity may be effective throughout the domain of flow, while in others there is a region in which viscosity is negligible.

This paper is part of a study of quasigeostrophic waves, which depend on the topography of the ocean floor and the curvature of the earth.

In a homogeneous, β-plane ocean, motion of the fluid across contours of constant f/h releases relative vorticity (f is the Coriolis parameter and h the depth). This well-known effect provides a restoring tendency for either Rossby waves (with h constant) or topographic waves over a slope. The long waves in general obey an elliptic partial differential equation in two space variables. Because the equation has been integrated in the vertical direction, the exact inviscid bottom boundary condition appears in variable coefficients.

When the depth varies in only one direction the equation is separable at the lowest order in ω, the frequency upon f. With a simple slope, |[xdtri ]h/h| = constant, the transition from Rossby to topographic waves occurs at |[xdtri ]h| ∼ h/Re, where Re is the radius of the earth. Isolated topographic features are considered in §2. It is found that a step of fractional height δ on an otherwise flat ocean floor reflects the majority of incident Rossby waves when δ > 2ω. In the ocean ω is usually small, due to continental barriers, so even slight depth variations are important. A narrow ridge does not act as a great obstruction but calculations show, for example, that the Mid-Atlantic Ridge is broad enough to reflect all but the lowest mode Rossby waves in the North Atlantic.

Besides isolating oceanic plains from one another, steps and ridges support trapped topographic waves of greatest frequency ∼ δ/2, analogous to the potential well solutions in quantum mechanics. These waves cannot carry energy along abrupt topography, but they disperse more rapidly over broader slopes; the phase and group speeds may be hundreds of cm/sec. The continentalshelf waves found by Robinson are an example of the latter case. There are many such wave guides, where the f/h contours are crowded, in the deep ocean.

The theory suggests that measurement of Rossby waves will rarely be possible at the coast of a continent.

Some recent work on the existence of vibrational de-excitation shocks (δ-shocks) in expanding non-equilibrium nozzle flows is extended to include situations in which an adiabatic shock (δ-shocks) may be embedded within the de-excitation shock. A discussion of some further properties of the shock solution is given and some examples are worked out. Numerical solutions of the full equations are also presented. These solutions confirm the existence of the δ-shocks but bring to light certain anomalies in the simple approximate solution. The modifications necessary to remove these discrepancies are outlined, and the implications of the numerical results are briefly discussed. Finally, some comments on the nature of the asymptotic solution for an arbitrary rate process are made.

On the assumptions of incompressibility, and negligible thermal conduction, salinity diffusion and viscosity, simple expressions are derived for the conservation equations of mass, momentum and energy when internal waves encounter an unsteady non-uniform current. These expressions of conservation equations are valid for all kinds of internal waves without regard to their different characteristics. From the dynamical conservation equations, we find that a stress-like term, the ‘excess momentum flux tensor’, plays an important role in the interaction between internal waves and an unsteady non-uniform current. Furthermore, it is deduced from the energy balance equation that, in the encounter of interfacial waves with a steady non-uniform current in a two-liquid system, the waves are amplified in an adverse current but suppressed in an advancing current as a result of interaction of the waves with the current. This conclusion may explain the large amplitudes sometimes observed in internal waves near the confluence of currents and near fronts at the thermocline, the region in the ocean where the density gradient is a maximum.

Trajectories are calculated for small particles introduced upstream into a fluid flowing past a fixed sphere. Unseparated potential flow is taken as the velocity profile for the fluid, and the effect of gravity is included in the formulation when it acts along the axis of symmetry. Using a numerical procedure, particle trajectories which graze the sphere, and the corresponding collision efficiencies, are calculated for values of the Stokes number σ. When gravity is neglected, an analytic solution is obtained for large values of σ which is in good agreement with the numerical results for σ as low as 5. These results are compared with those of Sell (1931) and Langmuir & Blodgett (1946). When gravity is included, a critical value of the Stokes number σc is calculated for which no collisions occur until σ > σc.

We consider slow oscillations trapped about axisymmetric islands and seamounts. ω is in the range [lsim ] δ/2 (ω is the frequency divided by f, the Coriolis parameter, and δ the fractional change in depth). The periods, for example, are [gsim ] 2·4 days for an island with a sloping ‘skirt’, h [vprop ] r½, where h is the depth and (r, θ) are polar co-ordinates in the plane tangent to the mean sea surface. Energy leaks slowly away from the topography in Rossby waves. In the limiting case of a cylindrical island with vertical walls there are no such trapped motions, but incident Rossby waves are scattered anisotropically. If γ, the ratio of the island radius, a, to the Rossby wavelength, is small, the scattering cross-section ∼ γ3a. The free oscillations at seamounts and islands with skirts allow much stronger scattering (with cross-section ∼ a/γ, ∼ a wavelength), when one of their frequencies is near that of the incident wave.

The theory suggests that measurements of Rossby waves will be possible at small islands, but that the many local oscillations in the same frequency range will add some confusion.