Measurements are reported of the fluctuating lift acting on a sphere and the moment acting about the centre of a sphere at supercritical Reynolds numbers (R > 4 × 105). The lift and moment fluctuations are random functions of time which scale with the free-stream dynamic pressure and sphere dimensions. The power spectra of the lift and moment also scale with the above parameters and with the Strouhal number, nd/U. The spectra contain a maximum spectral density at very low frequencies (nd/U < 0·0003) and do not reveal appreciable effects of vortex shedding at discrete frequencies.

Hot wire anemometers were placed near the surface but outside the boundary layer along a great circle in the meridian plane in which the lift was measured. The fluctuating velocity component near the surface on the upstream hemisphere in this meridian plane is highly correlated with the fluctuating lift in the same meridian plane. The correlation betweeen the lift and tangential velocity near the surface suggests that the fluctuating lift is produced by the component of fluctuating bound vorticity about the sphere that is normal to the meridian plane in which the lift force is measured. The fluctuating moment measured about an axis passing through the centre of the sphere and perpendicular to the above meridian plane is almost perfectly correlated with the fluctuating lift (the measured correlation coefficients were 0·99 and 1·00). The fluctuating moment coefficient is very small ($\sqrt{C^2_m}\simeq 5\times 10^{-4}$) compared to the fluctuating lift coefficient ($\sqrt{C^2_L}\simeq 6\times 10^{-2}$). The exceptional correlation between the random lift and moment suggests that the unsteady moment about the sphere centre (which can be produced only by shear stress fluctuations) is caused by the fluctuations of bound vorticity (residing in the boundary layer and wake) that are responsible for the unsteady lift.

Internal gravity waves in a stratified atmosphere of unbounded inviscid incompressible fluid are considered. A class of non-linear waves is found in the Boussinesq approximation, whereby the inertial variation of density is neglected but the buoyancy is not, by reduction of the equations of motion to ordinary differential equations. To find similar non-linear waves when the atmosphere is slightly stratified, i.e. when the inertial variation of density is small but not entirely negligible, the equations of motion are first expressed in Lagrangian variables and derived from a variational principle. The Lagrangian variables are transformed to systematize the Boussinesq approximation. Finally, the properties of the non-linear waves in a slightly stratified atmosphere are found by Whitham's method of averaging. In terms of the transformed Lagrangian variables, strong non-linearity affects the linearized solution only by adding to the pressure a term proportional to the square of the wave amplitude. It follows that the amplitude of the waves is inversely proportional to the density of the atmosphere, even where the amplitude is not small and the linear approximation becomes invalid.

An experimental investigation has been carried out on the nature of disturbance amplification in the laminar natural convection boundary layer formed on a vertical flat surface with uniform heat flux input. A pair of calibrated hot-wire anemometers were used to obtain amplitude and phase profiles of disturbance velocities and to measure the distance amplification rate as a function of disturbance frequency. An interferometer was used to measure the amplitude and phase profiles of disturbances in the temperature field. The relative amplification of a temperature disturbance was measured as it was convected downstream. For this latter measurement a technique employing interferometric moiré patterns was used. All of these measurements are in good agreement with the theory. The results of this investigation further strengthen the conclusions drawn in an earlier theoretical investigation, that the frequencies which amplify fastest and lead to transition and turbulence occupy a band much higher than the frequencies which begin to amplify first. These results also support stability theory in the very close agreement between calculated and measured distributions of disturbance quantities.

A linearized stability analysis of the flow of water in a channel with a loose bed and straight banks is described. It is assumed that the wavelength of the perturbations, which develop into meanders or braids, is longer than the width of the channel. It is therefore long compared with the ripples or dunes which cover the bed of such a channel and whose wavelength is shorter than the width of the channel. The latter need be allowed for only as roughness elements creating resistance. The variation of resistance to flow and rate of transport of bed material with velocity are discussed briefly and taken into account. Instability is interpreted as leading to a meandering or braided channel and it is shown that all practicable channels are unstable. Wavelengths calculated for channels expecte to meander are compatible with those given by Inglis's empirical rule and wavelengths calculated for channels which become braided are approximately the same as those observed.

A conference on turbulent boundary-layer prediction was held at Stanford, California, from 18 to 23 August 1968. The meeting was sponsored by the Mechanics Division of the U.S. Air Force Office of Scientific Research and by the industrial sponsors of the Internal Flow Program of the Mechanical Engineering Department at Stanford University. Attendance was by invitation, and there were 75 participants. The following is a brief account of the organization and content of the meeting; the full proceedings are available in two volumes.These have been reproduced by direct photo offset process and may be obtained from the Mechanical Engineering Department, Stanford University, Stanford, California 94305, for $6.00 each, or $11.00 for the set of two, plus postage (U.S. 50c, Europe $1.00); hard covers $1.00 extra per volume.

Turbulent flow along the axis of a 90° corner is considered. Experimental measurements of velocities, wall shear stresses and $\overline{u^{\prime 2}}$, the turbulent normal stress in the streamwise direction, have been made. The data are considered both from the point of view of momentum integral and similarity analyses. A suitable momentum integral equation is easily derived but difficult to use due to the form of the terms which must be measured. Similarity techniques give a series of correlations which describe the flow over a narrow range of Reynolds number for which data are available. There is experimental evidence for secondary currents in this type of flow but their analysis presents considerable difficulty.

Experiments are described in which an oscillating horizontal cylinder was moved through a density stratified fluid at Reynolds numbers of between 1 and 100 based on the diameter and mean velocity of the cylinder. The phase configurations of the internal waves which were generated are shown to compare very well with Lighthill's theory for waves in dispersive systems.

The motion generated in an initially quiescent, incompressible, stratified, and/or rotating fluid of infinite extent when a spherical source begins to breathe fluid in and out periodically is considered. The properties of the resulting flow may be understood in terms of the inertial-internal waves which may propagate energy in the fluid. At all points located a finite distance from the source, except those points falling on certain conical surfaces which are tangent to the source and which contain the group velocity vector for waves at the source frequency, the flow is ultimately a steady oscillation at the source frequency. The manner in which the flow depends on source frequency is discussed in detail.

Measurements of the fluctuating turbulent stresses (uiuj)’ in the mixing layer of a two-dimensional jet are presented. These include the intensity, spatial correlation and the wave-number frequency spectra of the fluctuating stresses.

The acoustic spectrum perpendicular to the axis of a jet can be approximated with the aid of assumptions about jet similarity and a knowledge of the four-dimensional wave-number frequency spectrum at zero wave-number. In the present paper the four-dimensional spectrum was inferred from measured two-dimensional spectra and the self-noise spectrum perpendicular to a round jet estimated. Good agreement was obtained with measured acoustic spectra.

The gain and loss integrals in the Boltzmann equation for a rigid sphere gas are evaluated in closed form for a distribution which can be expressed as a linear combination of Maxwellians. Application to the Mott-Smith bimodal distribution shows that the gain is also bimodal, but the two modes in the gain are less pronounced than in the distribution. Implications of these results for simple collision models in non-equilibrium flow are discussed.

‘Best’ solutions for the shock-structure problem are obtained by solving the Boltzmann equation for a rigid sphere gas by applying minimum error criteria on the Mott-Smith ansatz. The use of two such criteria minimizing respectively the local and total errors, as well as independent computations of the remaining error, establish the high accuracy of the solutions, although it is shown that the Mott-Smith distribution is not an exact solution of the Boltzmann equation even at infinite Mach number. The minimum local error method is found to be particularly simple and efficient. Adopting the present solutions as the standard of comparison, it is found that the widely used v2x-moment solutions can be as much as a third in error, but that results based on Rosen's method provide good approximations. Finally, it is shown that if the Maxwell mean free path on the hot side of the shock is chosen as the scaling length, the value of the density-slope shock thickness is relatively insensitive to the intermolecular potential. A comparison is made on this basis of present results with experiment, and very satisfactory quantitative agreement is obtained.

The direct-simulation Monte Carlo procedure for the modelling of rarefied gas flows on a digital computer has been used to generate representations of the simulated flows on a computer display unit. Photographs prepared from the displays illustrate the structure of the flows at the molecular level. The method has been applied to the steady hypersonic flow past circular cylinders, spheres, flat plates and blunt cones. These flows are represented either by several hundred typical molecular paths or by several thousand small arrows, each representing the position and velocity of a sample molecule at a particular instant. Colour was used to differentiate between undisturbed free-stream molecules, molecules that had struck the surface of the body and molecules that had been indirectly affected by the presence of the body. The method was also applied to the flow of a binary gas mixture past a circular cylinder, with colour used to differentiate between the components of the mixture.

The moment method proposed by Lees (1959) is applied to the problem of vapour condensation on and evaporation from spherical liquid droplets when the droplet is not in equilibrium with its surrounding. Using a four-moment solution, an analytical expression is derived for the mass flux to or from the droplet surface when the droplet is surrounded by a pure vapour. By neglecting changes in temperature, an analytical solution is also obtained for the mass flux when the droplet is immersed in a vapour-gas mixture. The results of both of these analyses are applicable in the range from λ/R → 0 to λ/R ∞, where λ is the mean free path and R the droplet radius, and it is shown that in the limits the results reduce to the appropriate free molecule and continuum expressions.

Sound generation by turbulent two-phase flow is considered by the methods of Lighthill's theory of aerodynamic noise. An inhomogeneous wave equation is derived, in which the effects of one phase on the other are represented by monopole, dipole and quadrupole distributions. The resulting power outputs are obtained for the case of a distribution of small air bubbles in water. The monopole radiation resulting from volumetric response of the bubbles to the turbulent pressure field overwhelms that from the quadrupoles equivalent to the turbulent flow, the increase in acoustic power output being about 70 dB for a volume concentration of 10%. The monopole radiation occurs through the forced response of the bubbles at the turbulence frequency; resonant response is shown to be impossible when the excitation is due to turbulence alone. Surface radiation arises from the edge of a cloud of bubbles. This radiation is important when the region containing bubbles is in the form of a sheet with thickness smaller than the length scale of the turbulent motion. Dipole radiation is also considered, and found to be negligible whenever monopole sources are present. In the case of a dusty gas, only dipole and quadrupole sources are present, and here it is shown that the dipole radiation is equivalent to an increase in the usual quadrupole radiation. The increase depends upon the mass concentration of dust, and is significant for mass concentrations in excess of unity.

The two-dimensional adiabatic transverse normal modes of an inviscid compressible fluid, having solid body rotation about the axis of its cylindrical container, are considered. Relative to the rotating fluid there are two trains of harmonic waves, propagating in opposite directions. The first five modes of the first-order harmonic wave and the first mode of the harmonic waves of order two, five, ten, fifteen and twenty have been considered. The period and amplitude of the waves is considerably modified by the rotation. Relative to a fixed coordinate system the angular velocity of the waves initially propagating in the same direction as the rotating fluid, is larger than that of the waves propagating in the opposite direction, as might be expected. However, in the case of the first mode, relative to the rotating fluid, the waves propagating in the opposite direction are the faster.

The stability of a horizontal layer of Maxwellian fluid heated from below is considered. Critical Rayleigh numbers, wave-numbers, and frequencies for overstability are determined for both free and rigid boundaries. Elasticity is found to destabilize the fluid, and the presence of rigid boundaries is found to be slightly stabilizing.