In this paper slender-body theory is applied to flexible bodies. The bodies are assumed to have a constant forward velocity normal to the crests of a regular train of two-dimensional waves and to move at a nearly constant depth. A flexible recoil mode is defined which is the dynamic counterpart of the stretched straight fish position defined by Lighthill (1960) for uniform flow conditions. Expressions are derived for the mean thrust and for the mean rate of working, and permit the evaluation of conditions for efficient propulsion. By properly adapting the motions of the body to the oncoming waves, energy can be extracted from the waves in such a way that it can be used for propulsion. This phenomenon may help to explain the high speeds that cetacea are observed to sustain over long periods of time.

A flow of an electrolyte in a seventy-diameter length of round pipe is subjected to a two-dimensional electric current and magnetic field which give a controllable streamwise electromagnetic body force. As the body-force distribution is axisymmetric and the effects of induced currents are negligible the fully developed pipe flow is axisymmetric and longitudinally homogeneous, but it can have severely distorted mean velocity and turbulence profiles. Measurements of the mean velocity and turbulence intensity are presented for different levels of distortion and the results are discussed with reference to classical turbulence theories. The inadequacy of these theories is thus demonstrated. The extra degree of freedom provided by the body force combines with the relative simplicity of the fully developed flow to give a useful tool for investigating the nature of shear-flow turbulence and for studying the assumptions involved in analytical approaches. The technique also produces distorted laminar pipe flows with inflexion-point velocity profiles, which are of interest in stability studies.

Measurements of higher-order spectra of turbulent velocity fluctuations in the atmospheric boundary layer over the open ocean and land produce the interesting result that, in the wavenumber range designated originally by Kolmogorov as an inertial subrange, the functional dependence of the spectra on wavenumber is practically independent of the order of the spectrum. These results confirm the observation of Dutton & Deaven that their extension by a dimensional similarity argument of the original Kolmogorov theory to higher-order spectra was not valid. In the present work, we derive an alternative generalization of the Kolmogorov ideas for spectra of arbitrary order. The results of this generalization describe the dependence upon wavenumber of the available data quite well. We also present theoretical calculations based on a Gaussian model for the fluctuating velocity field which furnish quantitative predictions for spectra of arbitrary order that are also in good agreement with the measurements, both in functional form and in absolute value.

Comparison of results based on the Gaussian model with laboratory measurements obtained in a free shear layer shows that the Gaussian theory predicts accurately all the available normalized higher-order spectra for all frequencies. When the corresponding measured higher-order moments are close to those expected for a Gaussian process, the Gaussian theory also correctly predicts the absolute magnitudes of the higher-order spectra.

The secondary and tertiary motions which were observed by Elder (1965) during his experimental study of natural convection in a vertical cavity have been obtained in a numerical solution of the Boussinesq equations. Aspect ratios up to 20, Rayleigh numbers up to 3000000, a Prandtl number of 1000, and up to almost 11000 effective mesh points were used.

Brownian motion of particles suspended in a fluid is studied, and expressions derived for the particle diffusivity and velocity autocorrelation function. The theory of thermal noise in a general linear system is applied to both the particles and the fluid in a novel formulation. This enables the recent modification of the Langevin equation to include the effect of fluid inertia to be seen as just a necessary but simple reinterpretation of the original analysis, without introducing the theory of non-Markovian processes.

The paper is the second part of a study of the failure of the insulation of a layer of dielectric fluid of arbitrary volume, occupying a hole in a solid dielectric sheet, when stressed by an applied electric field. In part 1 symmetric and asymmetric equilibria were found for the two-dimensional problem, using an approximation given by Taylor (1968) for the electric field, which is valid for large holes. In this paper axisymmetric equilibria are given for a circular hole, under the same conditions. In addition the points of bifurcation of asymmetric solutions have been found, and provide sufficient information to give the stability characteristics. It is found that when the volume-excess fraction δ exceeds a value of approximately −0·3 instability occurs in an asymmetric form reported earlier for large holes by Michael, O'Neill & Zuercher (1971) in the case δ = 0. For δ < −0·3 the nature of the instability changes to an axisymmetric form of failure associated with a maximum of the loading parameter.

The analysis given shows that axisymmetric displacements of ‘sausage’ mode type, that is, symmetric about a centre-plane, are associated with small changes in the static pressure in the dielectric layer. Such modes have not previously been examined in this context, and in an appendix to this paper Michael & O'Neill give an analysis of them when δ = 0, valid for all hole sizes, by extending the small perturbation analysis of Michael, O'Neill & Zuercher. These modes however do not provide the most unstable displacements for any configuration, and do not therefore affect the stability from a physical point of view.

The electron-beam fluorescence technique has been employed in measuring simultaneous density and temperature fluctuations in a hypersonic (M ≃ 16), adiabatic wall boundary layer. The paper discusses this technique, as it is applied to the conditions of relatively high density associated with turbulent flows. It presents general considerations concerning the attainable frequency response and spatial resolution of the technique. It describes results from initial measurements in a boundary layer on a wind-tunnel wall. These results show that the r.m.s. of the density, temperature and pressure fluctuations are large, much larger than observed in supersonic boundary layers.

Kraichnan's (1967) predictions concerning a simultaneous direct enstrophy cascade and inverse energy cascade for high Reynolds number two-dimensional turbulence are tested numerically using a variant of the eddy-damped quasi-normal approximation. For the initial-value problem, an analytic study using this theory shows that, in the zero-viscosity limit, energy and enstrophy are conserved for arbitrarily long times, contrary to the three-dimensional case, where the energy is conserved for only a finite time, after which it is dissipated. Non-local effects in the enstrophy inertial range, which are difficult to treat by conventional numerical schemes (Leith 1971; Leith & Kraichnan 1972), are shown to be representable by an additional diffusion term in the spectral equation. The resulting equation, including non-local effects, is integrated numerically. When enstrophy and energy are continuously injected at a fixed wavenumber, it is shown numerically that a quasi-steady regime is obtained where enstrophy cascades to large wavenumbers across a k−3 inertial range with zero energy transfer while energy flows indefinitely to small wavenumbers across a $k^{-\frac{5}{3}}$ inertial range with zero enstrophy transfer.

In a conventional hydrostatic thrust bearing, the lubricant is supplied from the centre and flows radially outwards. It has been found that the load capacity of such a bearing decreases with increasing angular speed of the rotor. The bearing fails when a critical rotor speed is reached at which the load capacity becomes zero. In this paper a modified feeding system is suggested in which the lubricant is supplied from a ring-shaped groove situated between the exits at the centre and the edge of the bearing. Analysis shows that, by reversing the flow direction near the centre, the load capacity of a bearing of proper geometry can be made to decrease at a much slower rate than that of a conventional bearing as the rotor speed increases. The non-central feeding system might be considered for use in high-speed thrust bearings.

A series of experiments designed to measure the wave resistance of an accelerating two-dimensional air-cushion vehicle is described. A model was towed at a constant acceleration from rest over water of various depths. In addition, the effects of different levels of acceleration and different cushion pressures were examined. The results are compared with linearized potential-flow theory and show particularly encouraging agreement. The predicted humps and hollows in the curve of wave resistance vs. time are verified, but with a small shift with respect to time. Also, the theoretical region of negative wave resistance in water of finite depth is demonstrated.

A baroclinic-type instability in a gas centrifuge heated from above is discussed. The instability is shown to be of overstable type, and estimates of the growth time scale and oscillation period of the unstable mode are given.

We examine two-dimensional motion of a stably stratified fluid containing two solutes with different molecular diffusivities in an inclined slot. The two solutes have continuous opposing gradients with the slower-diffusing one more dense at the bottom. It is found that, in the steady state, there exists a slow upward flow along the slope driven by the slight buoyancy difference near the wall, not unlike the solution found by Phillips (1970) for a single solute. The magnitude of the flow is less than that in Phillips’ solution by a factor of approximately (1-λ)/(1-λτ), where λ is the ratio of the density gradient and τ−1 is the ratio of the diffusivity of the faster-diffusing solute to that of the slower-diffusing one. For the time-dependent flow resulting from switching on the diffusivities at t = 0, there may be a dramatic reversal of the flow near the walls depending on the relative magnitude of λ and τ. If λ is somewhat greater than τ, the initial flow is downward, along the slope, reaching a maximum magnitude about one order of magnitude greater than the steady-state value. Then the ‘reverse’ flow gradually diminishes and approaches the steady state rather slowly. For λ [gsim ] τ, the approach to the steady state is monotonic; there is no ‘reverse’ flow near the wall. The existence of the downward flow, which was observed by Turner & Chen (1974), may lead to double-diffusive instabilities which eventually result in horizontal convecting layers.

A discussion of the applicability of an effective-viscosity approach to turbulent flow suggests that there are flow situations where the approach is valid and yet present hypotheses fail. The general form of an effective-viscosity formulation is shown to be a finite tensor polynomial. For two-dimensional flows, the coefficients of this polynomial are evaluated from the modelled Reynolds-stress equations of Launder, Reece & Rodi (1975). The advantage of the proposed effective-viscosity formulation, equation (4.3), over isotropie-viscosity hypotheses is that the whole velocity-gradient tensor affects the predicted Reynolds stresses. Two notable consequences of this are that (i) the complete Reynolds-stress tensor is realistically modelled and (ii) the influence of streamline curvature on the Reynolds stresses is incorporated.

Stability experiments were made on plane Poiseuille flow generated in a long channel of a rectangular cross-section with a width-to-depth ratio of 27·4. By reducing the background turbulence down to a level of 0·05 %, we succeeded in maintaining the flow laminar at Reynolds numbers up to 8000, which is much larger than the critical Reynolds number of the linear theory, about 6000. The downstream development of the sinusoidal disturbance introduced by the vibrating ribbon technique was studied in detail at various frequencies in the range of Reynolds number from 3000 to 7500. This paper presents the experimental results and clarifies the linear stability, the nonlinear subcritical instability and the breakdown leading to the transition.

The characteristic function defining the eigenvalues of the Orr-Sommerfeld equation is discussed and it is shown how the expected analytic properties of this function can be exploited to generate series expansions defining eigenvalues within the circle of convergence. This technique is applied to the modes arising in the Blasius flat-plate boundary layer (treated as a parallel flow), for which the complex wavenumber α can be expanded as a convergent power series in the complex frequency parameter β in various regions of the β plane. Such power series are effectively equivalent to Fourier expansions and the properties of the latter are used to find the coefficients.

A square-root singularity in the relationship between α and β is found and it is shown how α can, nevertheless, be expressed in terms of β as the sum of one regular series and the square root of a second regular series. The loci of the real and imaginary parts of α have been computed from these series and show the behaviour in the neighbourhood of the branch point.

The series description provides a particularly simple and rapid method of evalauting eigenvalues and their derivatives within any given region.

Normal and tangential resistance coefficients are calculated for a rigid slender body close to a planar no-slip boundary or midway between and close to two such boundaries. The important length scale is found to be the separation distance from the boundaries, and the forces per unit length acting on the slender body are approximately constant along most of its length. Owing to the presence of the walls, the ratio of the normal and tangential resistance coefficients can be greater than 2, its maximum limiting value in the infinite-fluid case. Applications to the movements of flagellated micro-organisms are discussed.

A similarity solution is obtained for a model of the turbulent starting plume comprising a steady plume feeding mass, momentum and buoyancy into a vortex ring. Bulk equations representing the time rate of increase of ring momentum and ring buoyancy, together with equations (dependent on broad features of the ring structure) representing the velocity of propagation and time rate of circulation increase are used to determine the motion of the vortex ring. The similarity solution is found to exist only for diffuse distributions of vorticity and buoyancy within the ring. Further, the ratio of ring velocity to plume velocity, which is assumed to be constant, is found to take a value which agrees with that obtained from experimental observations.

The phase velocities of axial disturbances to a free jet have been investigated experimentally. For a certain range of low Strouhal numbers previous theories predicted the existence of phase velocities exceeding the mean velocity of the jet. These ‘ultra-fast’ phase velocities were found in the present experiments too.

When a mutually perpendicular horizontal magnetic field and electric current are imposed upon an electrolytic liquid in a tank covered by a lighter nonconducting fluid, the interface can become unstable. In the experiment described in this paper the tank floor is stepped so that only part of the interface becomes unstable. The result is a distinctive arch-like configuration of the electrolyte, which finally disintegrates to break the current path, but which will continue to repeat itself.

A description is given of the high Reynolds number (R [Gt ] 1) laminar fluid motion in the neighbourhood of the trailing edge of a flat plate undergoing small amplitude sinusoidal oscillations in a uniform supersonic stream. It is shown that for oscillations of frequency ω* = O(R¼) and amplitude h* = O(R−½) a rational description of the flow at the trailing edge is based on a ‘triple-deck’ structure, which is a familiar feature of steady trailing-edge flows. The theory may be extended in a straightforward manner to include slow oscillations of the plate, and it is shown in general that the occurrence of separation at the trailing edge is dependent upon the magnitude of the product of the frequency and amplitude of oscillation, and that if ω* [les ] O(R½) then the flow is maintained right up to the trailing edge provided that h*ω* [Lt ] R−¼. The precise condition for the occurrence of separation is found for frequencies in the range ω* R−¼.