In order to study the memory of the larger eddies in turbulent shear flow, experiments have been conducted on plane turbulent wakes undergoing transition from an initial (carefully prepared) equilibrium state to a different final one, as a result of a nearly impulsive pressure gradient. It is shown that under the conditions of the experiments the equations of motion possess self-preserving solutions in the sense of Townsend (1956), but the observed behaviour of the wake is appreciably different when the pressure gradient is not very small, as the flow goes through a slow relaxation process before reaching final equilibrium. Measurements of the Reynolds stresse show that the approach to a new equilibrium state is exponential, with a relaxation length of the order of 103 momentum thicknesses. It is suggested that a flow satisfying the conditions required by a self-preservation analysis will exhibit equilibrium only if the relaxation length is small compared with a characteristic streamwise length scale of the flow.

We consider here the detailed application of a model Reynolds stress equation (Narasimha 1969) to plane turbulent wakes subjected to pressure gradients. The model, which is a transport equation for the stress exhibiting relaxation and diffusion, is found to be consistent with the observed response of a wake to a nearly impulsive pressure gradient (Narasimha & Prabhu 1971). It implies in particular that a wake can be in equilibrium only if the longitudinal strain rate is appreciably less than the wake shear.

We then describe a further series of experiments, undertaken to investigate the range of validity of the model. It is found that, with an appropriate convergence correction when necessary, the model provides excellent predictions of wake development under favourable, adverse and mixed pressure gradients. Furthermore, the behaviour of constant-pressure distorted wakes, as reported by Keffer (1965, 1967), is also explained very well by the model when account is taken of the effective flow convergence produced by the distortion. In all these calculations, only a simple version of the model is used, involving two non-dimensional constants both of which have been estimated from a single relaxation experiment.

Hot-film measurements in a fully developed channel flow have been made in an attempt to gain more insight into the process of Reynolds stress production. The background for this effort is the observation of a certain sequence of events (deceleration, ejection and sweep) in the wall region of turbulent flows by Corino (1965) and Corino & Brodkey (1969). The instantaneous product signal uv was classified according to the sign of its components u and v, and these classified portions were then averaged to obtain their contributions to the Reynolds stress $-\rho\overline{uv} $. The signal was classified into four categories; the two main ones were that with u negative and v positive, which can be associated with the ejection-type motion of Corino & Brodkey (1969), and that with u positive and v negative, associated with the sweep-type motion. It was found that over the wall region investigated, 3·5 [les ] y [les ] 100, these two types of motion give rise to a stress considerably greater than the total Reynolds stress. Two other types of motion, (i) u negative, v negative, corresponding to low-speed fluid deflected towards the wall, and (ii) u positive, v positive, corresponding to high-speed fluid reflected outwards from the wall, were found to account for the ‘excess’ stress produced by the first two categories, which give contributions of opposite sign.

The autocorrelations of the classified portions of uv were obtained to determine the relative time scales of these four types of motion. The positive stress producing motions (u < 0, v > 0 and u > 0, v < 0) were found to have significantly larger time scales than the negative stress producing motions (u < 0, v < 0 and u > 0, v > 0). It was further surmised that turbulent energy dissipation is associated with the Reynolds stress producing motions, since these result in localized shear regions in which the dissipation is several orders of magnitude greater than the average dissipation at the wall.

A circular cylinder in an infinite fluid rotates rigidly about a fixed axis which is parallel to, but does not coincide with, its geometric axis. It is found that, depending on the relative magnitude of the Reynolds number R and eccentricity ε, the flow may have two, one or no boundary layers. General solutions for R [Lt ] are obtained. It is found that owing to eccentricity there exist both a flow periodic in the circumferential direction and a non-periodic flow which is a function only of the radial distance from the centre of the cylinder. The non-periodic flow is caused by the nonlinear Reynolds stress and contributes to the torque experienced by the cylinder. The high Reynolds number case, \[ 1 \Lt R \Lt\epsilon^{-\frac{3}{2}}, \] is solved by matched asymptotic expansions. The stream function can be represented by Hankel functions of order 1/3 and a slight decrease in torque is found. In the low Reynolds number case, R [Lt ] 1, the torque is increased owing to eccentricity when R < 0·145 and decreased when R > 0·145. A physical explanation is presented.

Wave propagation across the continental shelf is studied by analogy with transmission-line theory. Fourier transformation along the contours of constant depth, which are assumed parallel to a straight coastline, yields a Sturm-Liouville equation for a prescribed depth profile h(x). The modal spectrum of the profile, which comprises a finite, discrete spectrum of trapped modes and a continuous spectrum of radiated modes, is established. The Green's function for a point source on the coastline is constructed by Fourier superposition over this spectrum. Detailed results are calculated for a two-step model (level shelf separated from level abyss by vertical cliff) and for a gradually sloping shelf that merges smoothly into a level abyss. The radiation impedance of a harbour is calculated, and the effects of the continental shelf on the resonant response of the harbour to a tsunami are discussed.

Some numerical solutions of a variable-coefficient Korteweg-de Vries equation are presented. This particular equation was derived by the author recently (Johnson 1972) in an attempt to describe the development of a single solitary wave moving onto a shelf. Soliton production on the shelf was predicted and this is confirmed here. Results for two and three solitons are reproduced and two intermediate shelf depths are also considered. In these latter two cases both solitons and an oscillatory wave occur. One of the profiles corresponds to the integrations performed by Madsen & Mei (1969) and a comparison is made.

The stability of Poiseuille flow in a pipe of circular cross-section to azimuthally varying as well as axisymmetric disturbances has been studied. The perturbation velocity and pressure were expanded in a complete set of orthonormal functions which satisfy the boundary conditions. Truncating the expansion yielded a matrix differential equation for the time dependence of the expansion coefficients. The stability characteristics were determined from the eigenvalues of the matrix, which were calculated numerically. Calculations were carried out for the azimuthal wavenumbers n = 0,…, 5, axial wavenumbers α between 0·1 and 10·0 and αR [les ] 50000, R being the Reynolds number. Our results show that pipe flow is stable to infinitesimal disturbances for all values of α, R and n in these ranges.

A theoretical study of the spatial stability of Poiseuille flow in a rigid pipe to infinitesimal disturbances is presented. Both axisymmetric and non-axisymmetric disturbances are considered. The coupled, linear, ordinary differential equations governing the propagation of a disturbance that has a constant frequency and is imposed at a specified location in the fluid are solved numerically for the complex eigenvalues, or wavenumbers, each of which defines a mode of propagation. A series solution for small values of the pipe radius is derived and step-by-step integration to the pipe wall is then performed. In order to ascertain the number of eigenvalues within a closed region, an eigenvalue search technique is used. Results are obtained for Reynolds numbers up to 10000. For these Reynolds numbers it is found that the pipe Poiseuille flow is spatially stable to all infinitesimal disturbances.

The present work is an analytical study of the stability of interfaces between fluids in motion, special attention being given to the role of surface tension without consideration of viscous effects. A variational approach based upon the principle of minimum free energy, which was first formulated for stagnant fluids, is applied to fluids in motion. This generalization is possible if viscous and inertia effects are unimportant as far as stability is concerned. One stability problem is studied in detail: a gas jet impinging on a free liquid. The analytical results obtained by this variational technique lie within the range of accuracy (15%) of the experimental results for this gas-jet problem. The method is very general and therefore can be applied to quite a number of interface stability problems.

The formation of convective cells in a fluid between two horizontal rigid boundaries with time-periodic temperature distribution is studied by the use of the Floquet theory. Numerical results for the critical Rayleigh number are given for a Prandtl number of 0·73 (air) and for various values of the frequency and magnitude of the primary temperature oscillation. Some numerical results for a Prandtl number of 7·0 (water) are also given. The most striking feature of the results is that the disturbances (or convection cells) oscillate either synchronously or with half frequency.

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.

The motion induced in a layer of Boussinesq fluid by moving periodic thermal waves is obtained by numerically solving the complete nonlinear two-dimensional momentum and temperature equations. Three sets of boundary conditions are treated: rigid upper and lower boundaries with symmetrical heating; free upper boundary and rigid lower boundary with heating only at the top; free upper and lower boundaries with symmetrical heating. The nonlinear streamline patterns show that, when the velocity fluctuations are larger than the phase speed of the thermal wave and the mean flow, the convection cells have shapes governed by fluctuating nonlinear interactions. Significant mean velocities can be created even without the characteristic tilt in the convection cells expected on the basis of linear theory. Nonlinear interactions can lead to a mean shear even in the absence of motion of the thermal source. When the viscous diffusion time across the fluid layer is less than or of the same order as the period of the thermal wave, the order of magnitude of the induced mean velocity does not exceed that of the phase speed of the wave, even for intense thermal forcing.