The reattachment of a supersonic jet with a turbulent separating boundary layer abruptly expanding into a two-dimensional parallel diffuser has been experimentally investigated using a surface-flow technique. The reattachment criterion proposed by Mukerjee & Martin (1969) for axisymmetric confined and unconfined flows is found to correlate equally well similar two-dimensional flow measurements in terms of the free-stream Mach number after separation.

An asymptotic solution of the boundary-layer equations, valid just upstream of a free streamline attached to the sharp trailing edge of a body, is compared with a numerical solution for the boundary-layer flow on a finite flat plate set perpendicular to a uniform stream. An arbitrary multiplicative constant in the asymptotic expansion, arising from an eigenfunction, is evaluated by requiring the skin friction to agree with a numerical value close to the free streamline. Using this value, the velocity profiles, computed from the asymptotic expansion, are in excellent agreement with the numerical solution.

An exact solution is obtained for the problem of fully-developed, radiating, laminar convective flow in a vertical heated channel. The effect of radiation is to decrease the temperature difference between the gas and the wall, thereby reducing the influence of natural convection. Thus, the reduction in velocity occurring in a heated upflow is less for a radiating gas. Graphs are presented for the dimensionless velocity and temperature profiles and for the volume and heat fluxes.

The basic method described by Bradshaw, Ferriss & Atwell (1967) is extended to compressible flow in two-dimensional boundary layers in arbitrary pressure gradient (excluding shock waves and expansion fans) by invoking Morkovin's hypothesis (Favre 1964) that the turbulence structure is unaffected by compressibility. Using the same empirical functions as in incompressible flow, skin friction in zero pressure gradient is predicted to within 3% of Spalding & Chi's (1964) correlation for free-stream Mach numbers less than 5. Comparisons with experiments in pressure gradient are restricted by the lack of data, but, since Morkovin's hypothesis does not depend on pressure gradient, methods which use it (of which the present method seems to be the first) can be checked fairly adequately by comparisons with data in zero pressure gradient.

No ‘compressibility transformations’ are needed, although the Crocco relation is used, provisionally, for the temperature: since the calculations take only about 20% longer than in incompressible flow, Morkovin's hypothesis does as much as any transformation could do. It is pointed out that, in supersonic flow, surface curvature which is large enough to induce a significant longitudinal pressure gradient is also large enough to have a very significant effect on the turbulence structure.

Sound waves of finite but small amplitude propagating into a quasi-steady, supersonic flow in a non-uniform duct are analyzed by means of a perturbation method. General properties of the flow and of the wave propagation are studied using a one-dimensional approximation. A shock propagation law in the unsteady flow is obtained. As an example, the formation and development of shock waves are discussed for a duct with a conical convergence. Comparisons of the theory with an experiment are also made; fairly good agreement is found.

A linear analysis of the perturbations associated with the jet flow, $\overline{u}(y) = {\rm sech}\,y$, of an inviscid compressible fluid is considered. Some subsonic stable solutions, not associated with stability boundaries, are first determined. Then a subsonic neutral solution is found and used as an aid in determining stability boundaries of the symmetric and antisymmetric disturbance modes. Numerical methods are also used to determine instability characteristics, including the Reynolds stress distributions. Comparisons are made with previous results obtained for the hyperbolic-tangent velocity profile and with unstable characteristics of the Bickley jet.

The optimum shape problems considered in this part are for those profiles of a two-dimensional flexible plate in time-harmonic motion that will minimize the energy loss under the condition of fixed thrust and possibly also under other isoperimetric constraints. First, the optimum movement of a rigid plate is completely determined; it is necessary first to reduce the original singular quadratic form representing the energy loss to a regular one of a lower order, which is then tractable by usual variational methods. A favourable range of the reduced frequency is found in which the thrust contribution coming from the leading-edge suction is as small as possible under the prescribed conditions, outside of which this contribution becomes so large as to be hard to realize in practice without stalling. This optimum solution is compared with the recent theory of Lighthill (1970); these independently arrived-at conclusions are found to be virtually in agreement.

The present theory is further applied to predict the movement of a porpoise tail of large aspect-ratio and is found in satisfactory agreement with the experimental measurements. A qualitative discussion of the wing movement in flapping flight of birds is also given on the basis of optimum efficiency.

The optimum shape of a flexible plate is analysed for the most general case of infinite degrees of freedom. It is shown that the solution can be determined to a certain extent, but the exact shape is not always uniquely determinate.

Earlier papers have described a technique whereby a stratified shear flow may be produced under controlled conditions in the laboratory. A comparison between the experiments and theory is made here for small-amplitude unstable disturbances in an accelerating stratified free shear layer at the diffuse interface between layers of brine and water. In the early stages of the observed growth of the instability, which takes the form of growing waves, three measurable quantities can be compared with predicted values: the wavelength of the small amplitude waves, the time, which determines the flow conditions, and the growth rates of the waves. Some observations of the development of the disturbances to finite amplitude, the transition to turbulence and the resulting turbulence are reported.

Experiments on slightly tapered models of circular cross-section have shown that the vortex wake structure exists in a number of discrete cells having different shedding frequencies. Within each cell shedding is regular and periodic, the frequency being somewhat lower than that from a parallel cylinder of the same diameter. A similar type of wake behaviour has also been observed on a parallel model in a non-uniform mean flow. These results suggest that the discontinuities in the shedding law observed by Tritton could arise through non-uniformities in the flow.

The scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock. An incident plane wave is expanded in Bessel functions, and for each mode the problem is formulated in terms of the potential on the cylindrical surface containing the dock and extending to the bottom. The solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations, which are solved numerically by Galerkin's method. The convergence of the solution is discussed, and some numerical results are presented.

This problem has been investigated previously by Miles & Gilbert (1968) by a different method, but their work contained errors.

The influence of surface roughness on the cross-flow around a circular cylinder is the subject of the present experimental work. The investigations were carried out in a high-pressure wind tunnel, thus high Reynolds numbers up to Re = 3 × 106 could be obtained. Local pressure and skin friction distributions were measured. These quantities were evaluated to determine the total drag coefficient and the percentage of friction as functions of Reynolds number and roughness parameter. In addition the local skin friction distribution yields the angular position of boundary-layer transition from laminar to turbulent flow and the location of boundary-layer separation.

This paper seeks to evaluate the swimming flow around a typical slender fish whose transverse cross-section to the rear of its maximum span section is of a lenticular shape with pointed edges, such as those of spiny fins, so that these side edges are sharp trailing edges, from which an oscillating vortex sheet is shed to trail the body in swimming. The additional feature of shedding of vortex sheet makes this problem a moderate generalization of the paper on the swimming of slender fish treated by Lighthill (1960a). It is found here that the propulsive thrust depends not only on the virtual mass of the tail-end section, but also on an integral effect of variations of the virtual mass along the entire body segment containing the trailing side edges, and that this latter effect can greatly enhance the thrust-making.

The optimum shape problem considered here is to determine the transverse oscillatory movements a slender fish can make which will produce a prescribed thrust, so as to overcome the frictional drag, at the expense of the minimum work done in maintaining the motion. The solution is for the fish to send a wave down its body at a phase velocity c somewhat greater than the desired swimming speed U, with an amplitude nearly uniform from the maximum span section to the tail. Both the ratio U/c and the optimum efficiency are found to depend upon two parameters: the reduced wave frequency and a ‘proportionalloading parameter’, the latter being proportional to the thrust coefficient and to the inverse square of the wave amplitude. The basic mechanism of swimming is examined in the light of the principle of action and reaction by studying the vortex wake generated by the optimum movement.

A theory on the generation mechanism of directional acoustic radiation from a supersonic jet is proposed. The theory is based on the concept of instability of the shear layer at the boundary of the jet close to the nozzle. Theoretical prediction of the directional wave pattern is found to agree with shadowgraphic observation.

The most effective movements of swimming aquatic animals of almost all sizes appear to have the form of a transverse wave progressing along the body from head to tail. The main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation. The general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized in viscid flow theory. The large-time asymptotic behaviour of an initial-value harmonic motion shows the decay of the transient terms. For a flexible plate starting with a constant acceleration from at rest, the small-time solution is evaluated and the initial optimum shape is determined for the maximum thrust under conditions of fixed power and negligible body recoil.

An experiment has been performed, using pulsed dye injection on an aerofoil in a Hele-Shaw cell. The purpose was to observe the form of the trailing-edge flow when the Reynolds number was high enough to permit separation and the initiation of a Kutta condition. The experiment provides a successful confirmation of the existence of a ‘viscous tail’ as predicted by Buckmaster (1970) although there is an unexplained quantitative discrepancy.

Complementary variational formulations are developed for the scattering of a gravity wave by a circular dock. These formulations, which are based on assumed distributions of the radial velocity and the potential, respectively, on the projection of the cylindrical boundary, yield lower and upper bounds to an impedance parameter that determines the difference between the scattered wave for the dock and the corresponding wave for a circular cylinder. Numerical results, using trial functions based on the incident wave, are compared with the results implied by a Galerkin solution (Garrett 1971). The maximum errors in the variational approximations to the total scattering cross-section are found to be of the order of 2% for a typical depth/radius ratio, draft/depth ratios of 0, ½ and 1, and all wavelengths. The axisymmetric component of the scattering cross-section is found to be very close to the value for scattering by a circular cylinder (dock extending to bottom). The intensity of the scattered wave on the forward axis for long wavelengths and a certain range of the geometric parameters is significantly less than that for a circular cylinder, and may vanish for critical combinations of these parameters.

According to the Lighthill acoustic analogy, the sound induced by a region of turbulence is the same as that due to an equivalent distribution of quadrupole sources within the fluid. It is known that the presence of scattering bodies situated near such multipoles can convert some of their intense near field energy into the form of sound waves whose amplitude is far greater than that of the incident field. Calculations are here presented to determine the extent of this conversion, for hard and soft bodies of various shapes, making use of the reciprocal theorem to recast the problem into one of finding the field, near the obstacle, induced by an incident plane wave. If the obstacle is small compared with a wavelength, then its presence is equivalent to an additional dipole (or source) whose greater efficiency as a sound radiator implies that the familiar intensity law I ∝ U8, for far field intensity I against typical turbulence velocity U for an unbounded flow, is replaced by I ∝ U6 (or I ∝ U4) for a hard (or soft) body. For the situation where the scatterer is large compared with wavelength, the prototype problem of a wedge of exterior angle (p/q)π is shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces. This result is shown to hold for the more general ‘wedge-like’ surfaces, whose dimensions are large scale and whose edges may be smoothed out on a small scale, compared with wavelength. The method used involves the matching of an incompressible flow, on the fine scales typical of the edge geometry, to an outer flow determined by the large scale features of the surface. Favourable comparisons are made with previous results pertaining to the two-dimensional semi-infinite duct and to the half-plate of finite thickness.

The waves generated by the steady motion of an obstacle along the axis of a uniformly rotating, electrically conducting homogeneous fluid have been studied by Lighthill's technique. The wave-number surface consists of a sphere and four coincident planes. The waves corresponding to points on the sphere travel ahead or trail behind the obstacle according as a1 the Alfvén velocity, is greater or less than U, the velocity of the obstacle. By drawing the appropriate normals to the four planes, it is seen that the formation of a Taylor column ahead of the obstacle is possible even at large Rossby numbers when U < a1 in contrast with the non-magnetic case and the case with U > a1.

The motion of a horizontal plate moving in its own plane in a rotating, stratified fluid is studied to establish the parameter conditions specifying the onset of boundary-layer blocking for the entire range of Rossby and Russell numbers. Régimes in Rossby–Russell number space defining the range of applicability of the inertia–viscous, buoyancy–viscous, and Coriolis–viscous boundary-layer balances are presented, and similarity solutions valid over a limited region of that space are derived. The plate drag and heat transfer are computed from the similarity solutions.

Schwinger's variational formulation is applied to the radiation of surface waves due to small oscillation of bodies. By means of Haskind's theorem the wave forces on a stationary body due to a plane incident wave are found using only far-field properties. Results for horizontal rectangular and vertical circular cylinders are presented.