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.

An aerodynamic investigation was conducted to determine the laminar-flow drag coefficient of spheres of various sizes in a subsonic wind tunnel. The tests were conducted using the M.I.T.-N.A.S.A. prototype magnetic-balance system. By measuring the drag of different sized spheres without model support interference the tunnel wall effect can be deduced. The present results indicate that the classical wind tunnel correction does not completely account for the effects of model size and wall interference. That is, the corrected drag coefficient data for the different sphere sizes differ among themselves in the region of Reynolds number overlap.

A comparison of the present sphere drag results with those of numerous other investigations including free-flight and ballistic-range data is given. The drag coefficients presented here are slightly lower than those of other workers for Reynolds numbers ranging from 20 000 to 150 000, but fall between the limits of experimental scatter for Reynolds numbers from 150 000 to 260 000.

An analysis of the estimated error in the present data indicates the primary source to be measurement of the wind tunnel parameters rather than errors resulting from the balance system.

Nonlinear internal gravity waves in an inviscid incompressible fluid are discussed for the case when the properties of the medium vary slowly on a scale determined by the local wave structure. A two-time-scale technique is used to obtain transport equations which describe the slowly varying modulations of the waves. Various solutions of these transport equations are discussed.

A solution has been found for the transient behaviour of resonant growing standing waves by using a perturbation expansion. Comparison with laboratory experiments as well as a numerical nonlinear solution of the same problem leads to the conclusion that: (i) the transient behaviour and the nonlinear tendency of the standing waves are described well by the analytic expression; (ii) the numerical results describe the solution very well until the wave starts to break; (iii) from the laboratory experiments and the numerical results, the standing internal gravity waves break owing to local gravitational instability at a critical amplitude which is similar to the one predicted by the expansion theory; (iv) the critical amplitude seems to be the maximum amplitude that a wave can reach; (v) when the generation of turbulence is violent, the small eddies begin forcing a secondary flow characterized by layers of strong jets separated by patches of turbulence.

In this paper the linear stability of the flow between two long eccentric rotating circular cylinders is considered. The problem, which is of interest in lubrication technology, is an extension of the classical Taylor problem for concentric cylinders. The basic flow has components in the radial and azimuthal directions and depends on both of these co-ordinates. As a consequence the linearized stability equations are partial differential equations rather than ordinary differential equations. Thus standard methods of stability theory are not immediately useful. However, there are two small parameters in the problem, namely δ, the clearance ratio, and ε, the eccentricity. By letting these parameters tend to zero in such a way that δ½ is proportional to ε, a global solution to the stability problem is obtained without recourse to the concept of ‘local instability’, or ‘parallel-flow’ approximation, so commonly used in boundary-layer stability theory. It is found that the predictions of the present theory are at variance with what is given by a ‘local’ theory. First, the Taylor-vortex amplitude is found to be largest at about 90° downstream of the region of ‘maximum local instability’. This result is given support by some experimental observations of Vohr (1968) with δ = 0·1 and ε = 0·475, which yield a corresponding angle of about 50°. Second, the critical Taylor number rises with ε, rather than initially decreasing with ε as predicted by local stability theory using the criteria of maximum local instability. The present prediction of the critical Taylor number agrees well with Vohr's experiments for ε up to about 0·5 when δ = 0·01 and for ε up to about 0·2 when δ = 0.1.

Convective motions in rectangular boxes with one side horizontal have been studied. The critical Rayleigh numbers were determined. In most cases cell patterns with rolls parallel to the shorter side wall of the rectangular box were observed. The results have been compared with the known theoretical results of Davis (1967). In general, good agreement has been found.

Alfvén-gravitational waves are found to propagate in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform transverse magnetic field in which the mean horizontal velocity U is independent of vertical height z. The governing wave equation is a fourth-order ordinary differential equation with constant coefficients and is not singular when the Doppler-shifted frequency Ωd = 0, but is singular when the Alfvén frequency ΩA = 0.If Ω2d < Ω2A the waves are attenuated by a factor exp − [2ΩA(N2−Ω2d)½−Ω2d + Ω2A]z, which tends to zero as z → ∞. This attenuation is similar to the viscous attenuation of waves discussed by Hughes & Young (1966). The interpretation of upward and downward propagation of waves is given.

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.

Alfvén-gravitational waves propagating in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform aligned magnetic field in which the mean horizontal velocity U(z) depends on height z only are considered. The governing wave equation has three singularities, at the Doppler-shifted frequencies Ωd = 0, ± ΩA, where ΩA is the Alfvén frequency. Hence the effect of the Lorentz force is to introduce two more critical levels, called hydromagnetic critical levels, in addition to the hydrodynamic critical level. To study the influence of magnetic field on the attenuation of waves two situations, one concerning waves far away from the critical levels (i.e. Ωd [Gt ] ΩA) and the other waves at moderate distances from the critical levels (i.e. Ωd > ΩA), are investigated. In the former case, if the hydrodynamic Richardson number JH exceeds one quarter the waves are attenuated by a factor exp{−2π(JH −¼)½} as they pass through the hydromagnetic critical levels, at which Ωd = ± ΩA, and momentum is transferred to the mean flow there. Whereas in the case of waves at moderate distances from the critical levels the ratio of momentum fluxes on either side of the hydromagnetic critical levels differ by a factor exp {−2π(J −¼)½}, where J (> ¼) is the algebraic sum of hydrodynamic and hydromagnetic Richardson numbers. Thus the solutions to the hydromagnetic system approach asymptotically those of the hydrodynamic system sufficiently far on either side of the magnetic critical layers, though their behaviour in the vicinity of such levels is quite dissimilar. There is no attenuation and momentum transfer to the mean flow across the hydrodynamic critical level, at which Ωd = 0. The general theory is applied to a particular problem of flow over a sinusoidal corrugation. This is significant in considering the propagation of Alfvén-gravity waves, in the presence of a geomagnetic field, from troposphere to ionosphere.

The existence of a new time scale t = σ-¼(L2/kN)½, where σ, k, L and N are the Prandtl number, the thermometric conductivity, a typical length and the Brunt-Väisälä frequency, respectively, is clarified for a temperature adjustment process of a Boussinesq fluid in a circular cylinder.

Data for the fluctuating mass-transfer coefficient to point electrodes on the surface of a rotating disk are presented for transition and turbulent flow, along with analysis of the energy spectrum. The results corroborate previous evidence of standing vortices on the surface in the transition region. It is shown that the electrochemical method gives a more sensitive probe of flow instabilities than has been possible previously; the transition region is found to be wider than has been previously thought to be and lies between roughly Re = 1·7 × 105 and 3·5 × 105, where Re = r2ω/v is the Reynolds number, ω being the angular velocity of the disk (in rad/s) and v the kinematic viscosity. As the Reynolds number is increased, the stationary vortices in the transition region propagate primarily with a frequency matching that of the disk speed. After a peak energy has been reached at Re = 2·6 × 105, the vortices break down into a fully developed turbulent flow.

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.

Salt fingering across a density interface produced by sugar and salt is investigated experimentally for varying values of initial concentrations and for times sufficiently long to observe the régimes of growth, equilibrium and final decay. Simultaneous measurements of sugar and salt concentrations and interface geometry are made. Calculations based on Stern's collective instability model allow explicit evaluation of the coefficients of the $\frac{4}{3}$-law for the sugar flux in the form suggested by Stern & Turner (1969) for the equilibrium régime. The results are verified by the experimental data. Predictions of the finger wavelengths from this model are in general agreement with the results of previous authors, while the flux-law coefficient is an order of magnitude smaller. It is concluded that the collective instability mechanism is an adequate physical description of the salt-finger mechanism.

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.

Results on the behaviour of controlled wave disturbances introduced artificially into turbulent channel flow are reported. Weak plane-wave disturbances are introduced by vibrating ribbons near each wall. The amplitude and relative phase of the streamwise component of the induced wave is educed from a hot-wire signal, allowing the wave speed, the attenuation characteristics and the wave shape to be traced downstream. These results form a basis for evaluation of closure models for the dynamical equations describing wave components in shear-flow turbulence.

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.

Measurements of static pressure and wave height are used to describe the waveinduced pressure field above generating sea waves. A large hump in the pressure spectra is observed at the wave frequencies. The amplitude of this hump increases and the rate of its vertical decay decreases as the mean wind speed increases. The phase difference between the pressure and the waves during active generation is about 135°, pressure lagging the waves, and does not change vertically for measurements at heights greater than the wave crests. In the present data, active wave generation appears to occur only when the wind at a height of 5 metres is greater than or about equal to twice the phase speed of the waves.

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.

The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

The dynamical equations governing small amplitude wave disturbances in turbulent shear flows are derived. These equations require additional equations or assumptions about the wave-induced fluctuations in the turbulence Reynolds stresses before a closed system can be obtained. Some simple closure models are proposed, and the results of calculations using these models are presented. When the predictions are compared with our data for channel flow, we find it essential that these oscillations in the Reynolds stresses be included in the model. A simple eddy-viscosity representation serves surprisingly well in this respect.