Dimensional analysis is applied to the velocity profile U(y) of turbulent boundary layers subjected to adverse pressure gradients. It is assumed that the boundary layer is in moving or local equilibrium in the sense that the free-stream velocity U∞ and kinematic pressure gradient α = ρ−1dP/dx vary only slowly with the co-ordinate x. This assumption implies a rather complicated general equation for the velocity gradient dU/dy which may be considerably simplified for several specific regions of the flow. A general family of velocity profiles is derived from the simplified equations supplemented by some experimental information. This family agrees well with almost all existing data on velocity profiles in adverse-pressure-gradient turbulent boundary layers. It may be used for the derivation of a skin-friction law which predicts satisfactorily the values of the wall shear stress at any non-negative value of the pressure gradient. The variation of the boundary-layer thickness with x is also predicted by dimensional considerations.

This paper describes in detail the overall pressure and spectral results in the outer mixing region of three annular jets. The outer mixing region can be considered as the result of the shearing of a single jet by the ambient air. The coherent structures found consist of an array of toroidal jet vortices convecting downstream in the outer mixing region. The structures tend to agree with the ones for a single jet.

The phenomenon of buoyancy-induced convection in an infinite porous medium with a concentrated heat source is studied analytically. The transient and steady-state temperature distribution and flow pattern around the source are determined using a perturbation analysis in the Rayleigh number based on the heat generation rate at the source. The first-order transient solution derived in the paper is valid for Rayleigh numbers less than 10. The transient flow pattern consists of an expanding vortex ring situated in the horizontal plane containing the source. The steady-state solution, valid for Rayleigh numbers of the order of 20 or less, reveals an upward flow pattern which becomes very intense near the source. The upward flow extends throughout the medium. Both solutions show that as the Rayleigh number increases the region situated above the source is effectively heated by natural convection in addition to direct heat conduction from the source.

The transmission of sound through a turbulent shear layer is reviewed both with respect to the mathematical theory of the distortion of signals during propagation and also as the basis of the analysis in part 2 of the spectral broadening evinced by experimental results relevant to the study of aircraft noise. The reflexion and transmission coefficients, which involve both amplitude and phase changes, are obtained for scattering by an irregular and unsteady interface convected between two media; diffraction of high frequency sound by small-scale turbulence in a shear layer is accounted for by means of a phase shift and conservation laws for energy, wavenumber and amplitude. These results may be used to construct the sound field transmitted through a turbulent shear layer from a source in the interior of a jet; multiple internal reflexions are accounted for in the case of transmission through two parallel shear layers, i.e. a jet of finite width shielding a noise source. A statistical description is given of the process of attenuation of the transmitted wave as energy is diffracted by the turbulence, and of the partial compensation by interference between correlated components of the refracted wave; the reference case is a monochromatic point source, which, when placed behind a system of shear layers, has its energy redistributed directionally and spread over a range of frequencies. The expressions obtained for the energy flux include as particular cases the results of Howe for the plane (1975) and impedance (1976) layers, the effects of turbulence being shown to be consistent with the experiments of Schmidt & Tilmann (1970) and Ho & Kovasznay (1976a, b).

It has been observed experimentally by Candel, Julienne & Julliand (1975) that a monochromatic test tone generated by a source inside a jet is received outside as a broad frequency band of definite shape. This phenomenon of spectral broadening occurs during transmission through the shear layer, which generally has a randomly irregular and unsteady shape, contains in addition distributed turbulence, and separates the jet and the ambient medium. We show in the first place that, in the audible range of frequencies, neither the approximation which treats the shear layer as a scattering interface with a convected undulating shape nor the opposite, high frequency limit obtained by means of asymptotic estimation of integrals derived for the diffraction of rays in turbulence is sufficient to provide a satisfactory theory of the observations. The refraction integrals obtained in part 1 have to be evaluated exactly in order to account for the phenomenon of spectral broadening, the methods used possibly being of interest in other branches of wave theory. The formation of the transmitted spectrum from an incident tone can be illustrated by representing a simple shear layer as an array of elements each re-radiating energy received from the source with its own characteristic attenuation and frequency shift. A computer program is used to obtain spectra under conditions corresponding to the experiments of Candel, Guédel & Julienne (1975) and gives encouraging agreement with their measurements, which were made with high frequency sources immersed in low speed jets. The theory can also be applied to the prediction of spectra received at various angles to the axis of high subsonic jets, but depends on extrapolation when supersonic exhausts are considered. We conclude with an example of the possible relevance of spectral broadening as a means of reducing the noise disturbance from current jet-powered aircraft, such as Concorde.

Rivers and canals with perimeters composed of non-cohesive sand and silt have self-formed active beds and banks. They thus provide a most interesting fluid flow problem, for which one must determine the container as well as the flow. If bed load alone occurs across the perimeter of a wide channel, gravity will pull particles down the lateral slope of the banks; bank erosion is accomplished and the channel widens. In order to maintain equilibrium, this export of material from the banks must be countered by an import of sediment from the channel centre.

The mechanism postulated for this import is lateral diffusion of suspended sediment, which overloads the flow near the banks and causes deposition. The model is formulated analytically with the aid of a series of approximate but reasonable assumptions. Singular perturbation techniques are used to define the channel geometry and obtain rational regime relations for straight channels. A comparison with data lends credence to the model.

It is hoped that a first step has been made towards a more general treatment, which would include various complicating factors that are important features of natural rivers but are not essential to the maintenance of channel width. Among these factors are meandering, sediment sorting and seepage.

The propagation of disturbances through an atmosphere that is, in its undisturbed condition, undergoing a spatially uniform chemical explosion is analysed on the assumption that the disturbances are of small amplitude. When the latter are arbitrarily small, and therefore classifiable as acoustic, the progress of the ambient explosion is undisturbed to first order and a rather complete history of the acoustic waves (including weak shock waves) can be constructed. The generally amplifying effect of the explosion on the disturbances, which has previously been identified at wave heads, is found to occur throughout the disturbed domains.

When the disturbance amplitude becomes comparable to the ratio of the thermal energy of the gas to the combustion-reaction activation energy, the ambient explosion becomes involved in the disturbance to first order. For large activation energies a small disturbance theory can be constructed to account for the disturbance behaviour; at present it is limited to time intervals from initiation that are shorter than the ‘no-depletion’ homogeneous ignition time.

A viscous-inviscid interaction is produced when a compressible laminar boundary layer encounters a corner. The correct mathematical structure for such interactions at large Reynolds number is given by the asymptotic triple-deck theory. In the present work the triple-deck equations for supersonic and hypersonic flows are solved for both compression and expansion corners. Results are presented for a range of corner angles, including separated cases, and are compared with experimental data and with finite Reynolds number calculations based on an interacting boundary-layer model.

The excitation of instability waves in a plane compressible shear layer by sound waves is studied. The problem is formulated mathematically as an inhomogeneous boundary-value problem. A general solution for arbitrary incident sound wave is found by first constructing the Green's function of the problem. Numerical values of the coupling constants between incident sound waves and excited instability waves for a range of flow Mach numbers are calculated. The effect of the angle of incidence in the case of a beam of acoustic waves is analysed. It is found that for moderate subsonic Mach numbers a narrow beam aiming at an angle between 50 to 80° to the flow direction is most effective in exciting instability waves.

Rivers are capable of transporting their own bed material without altering their width. However, a naive extension of the threshold theory of canals in coarse alluvium to straight reaches of gravel rivers leads to the stable-channel paradox: transport of bed material is incompatible with a stable width. In this paper singular perturbation techniques are used to obtain a bed stress distribution which allows a mobile bed but immobile banks at bankfull or dominant discharge. This result is used to obtain regime relations for straight rivers with bed and banks composed of coarse gravel.

The analysis, although dependent on a series of approximate assumptions for Reynolds-stress closure and sediment transport, provides reasonable agreement with data.

The stability of two-dimensional convection rolls has been studied as a function of the Rayleigh number, wavenumber and variation in viscosity. The experiments used controlled initial conditions for the wavenumber, Rayleigh numbers up to 25 000 and variations in viscosity up to a factor of about 20. The parameter range of stable rolls is bounded by a hexagonal-cell regime at small Rayleigh numbers and large variations in viscosity. Otherwise, the rolls are subject to the same transitions as have already been studied in fluids of uniform viscosity. The bimodal instability leading to a stable three-dimensional pattern occurs at smaller values of the average Rayleigh number as the variations in viscosity increase. This appears to be a consequence of the low viscosity of the warm thermal boundary layer associated with the original rolls.

The propagation of internal Alfvén-inertio-acoustic gravity waves in a perfectly electrically conducting, stratified, inviscid, non-isothermal, rotating atmosphere permeated by a non-uniform magnetic field is investigated. These waves exhibit singular properties at the critical levels at which the magnetic field and the sound velocity are such that \[ (\omega^2 - S^2)\{(c^2+V^2)\omega^2-c^2S^2\}-(c^2+V^2)\overline{R}^2=0, \] where ω is the frequency of the waves, $S = kV_x + lV_y,\overline{R} = 2\Omega_z\omega$, Vx and Vy are the x and y components of the Alfvén velocity, k and l are the corresponding wavenumbers and c is the sonic velocity. These levels act like valves which permit waves to penetrate them from one side only and absorb them when they propagate from the other side. In contrast to the incompressible results of Acheson (1972), we show that the valve effect in compressible flow no longer requires the presence of non-zero components of rotation in the plane normal to the direction in which the medium varies. We find that the compressibility increases the probability of a valve effect and so increases the capacity of a hydromagnetic wave to propagate across a field line, rather than being absorbed at some critical level.

Experimental measurements of the frequencies of discrete tones induced by flow over rectangular cavities were carried out over a range of low subsonic Mach numbers to provide a reliable data base for (aircraft wheel well) cavity noise consideration. A mathematical model of the cavity tones and pressure oscillation phenomenon based on the coupling between shear layer instabilities and acoustic feedback is developed to help in understanding the tone generation mechanism. Good agreement is found between discrete tone frequencies predicted by the model and experimental measurements over a wide range of Mach numbers. Evidence of tones generated by the cavity normal mode resonance mechanism at very low subsonic Mach numbers is also presented.

Turbulence measurements under the influence of a transverse magnetic field have been made at Purdue University's Magneto-Fluid-Mechanic Laboratory in a high aspect ratio channel. The Reynolds number range covered was 25000 ≤ Re 282000; the geometry and experimental conditions were such that the experiment approximated turbulent Hartmann flow. The aspect ratio of the channel was 5·8:1, its walls were electrically insulated and the working fluid was mercury. Measurements in the presence of a magnetic field were made of the skin friction coefficient, the mean velocity profiles, the turbulence intensity profiles (both u’ and v’) and the Reynolds stress profiles.

A sudden change in the damping of the Reynolds stresses was manifested by a ‘hump’ in the curves of Cf versus M/Re taken with the Reynolds number held constant. This ‘hump’ occurs as a gentle rise and sudden drop to the Hartmann laminar line of the Cf data. Close examination of the $\overline{u^{\prime}v^{\prime}}$ data near the wall confirms this behaviour, indicating that the turbulent contribution to the shear stress is the controlling factor in this behaviour of Cf. The Reynolds stresses were completely suppressed to zero at high values of the magnetic field, though the turbulence intensities of u’ and v’ were not. The Reynolds stress data are fundamental in revealing the mechanisms which are at work during the suppression of turbulence by a magnetic field.

It was also found that at high magnetic fields, when most of the turbulence was damped, the skin friction coefficient fell below the values predicted by Hartmann's (1937) laminar solution for high values of M/Re. This result was linked to the presence of ‘M-shaped’ velocity profiles in the direction perpendicular to both the magnetic field and the mean velocity vector. The presence of ‘M-shaped’ profiles has not previously been linked to a reduction in Cf.

An experimental programme has been carried out to examine the spread of heat as a passive scalar contaminant in a turbulent shear flow. The situation involves a slightly heated two-dimensional jet expanding into a quiescent medium on one side and a uniform stream with velocity equal to that of the warm jet on the other. Thus the developed flow is a typical mixing layer with an asymmetric mean temperature profile superimposed on it. Measurements of the mean and fluctuating velocity and temperature fields show the existence of a region where the production of temperature fluctuations is negative. Spectral analysis in this zone indicates a separation of large and small wavenumber components of the cospectrum into two regimes. The sign of the high-frequency portion is consistent with gradient-transport concepts while the low-frequency component is of opposite sign. From this it is inferred that the large eddies are mainly responsible for the negative production. A mathematical model has been developed to describe the transport within this region.

A theoretical and experimental study is presented of the longitudinal propagation of disturbances in an unstably stratified fluid layer, especially those generated by periodic temperature fluctuations at a side boundary. The most important characteristics of the waves at both supercritical Rayleigh numbers, in the presence of steady roll-like cells, and subcritical Rayleigh numbers are determined.

The persistence of the large vortices formed at the origin of wakes and mixing layers constitutes a kind of memory of initial conditions by the turbulence. In order to study the fading of this turbulence memory, and its effect on the rate of approach to the fully developed state, two wakes with different initial conditions have been examined experimentally. The wake of a sphere was compared with the wake of a porous disk which had the same drag, but did not exhibit vortex shedding. Measurements were made of the mean and fluctuating velocities, the anisotropy of the turbulence, and the intermittency. It was found that the wake of the sphere developed self-preserving behaviour more rapidly than the wake of the disk, and that even after both wakes became self-preserving there were differences between them in the structure of the turbulence and the scale of the mean flow. From this it is concluded that the behaviour of self-preserving wakes does not depend on the drag alone, but also on the structure of the dominant eddies. Generalizing these results, it is suggested that reported differences in the value of the entrainment constant of jets, wakes, and mixing layers are due to differences in the structure of the dominant eddies, rather than differences in the type of flow.