A theoretical study is made of shearing flows bounded by a simple-harmonic wavy surface, the main object being to calculate the normal and tangential stresses on the boundary. The type of flow considered is approximately parallel in the absence of the waves, being exemplified by two-dimensional boundary layers over a plane. Account is taken of viscosity; but, as the Reynolds number is assumed to be large, its effects are seen to be confined within narrow ‘friction layers’, one of which adjoins the wave and another surrounds the ‘critical point’ where the velocity of flow equals the wave velocity. The boundary conditions are made as general as possible by including the three cases where respectively the boundary is rigid, flexible yet still solid, or completely mobile as if it were the interface with a second fluid.

The theory is developed on the model of stable laminar flow, although it is proposed that the same theory may usefully be applied also to examples of turbulent flow considered as ‘pseudo-laminar’ with velocity profiles corresponding to the mean-velocity distribution. Use is made of curvilinear co-ordinates which follow the contour of the wave-train. This admits a linearized form of the problem whose validity requires only that the wave amplitude be small in comparison with the wavelength, even when large velocity gradients exist close to the boundary. The analysis is made largely without restriction to particular forms of the velocity profile; but eventually consideration is given to the example of a linear profile and the example of a boundary-layer profile approximated by a quarter-period sinusoid. In § 7 some general methods are set out for the treatment of disturbed boundary-layer proses: these apply with greatest precision to thin boundary layers, but are also useful for the initially very steep but on the whole fairly diffuse profiles which occur in most practical instances of turbulent flow over waves.

The phase relationships found between the stresses and the wave elevation are discussed for several examples, and their interest in connexion with problems of wave generation by wind is pointed out. It is shown that in most circumstances the stresses are distributed in much the same way as if the leeward slopes of the waves were sheltered. For instance, the pressure distribution often has a substantial component in phase with the wave slope, just as if a wake were formed behind each wave crest—although of course actual separation effects are outside the scope of the present theory. In this aspect, the analysis amplifies the work of Miles (1957).

On the assumption that the rotational oscillations of a rigid plane are small, boundary-layer type solutions of the Navier-Stokes equations are attempted by an expansion of the velocity components in power series of the amplitude. First-and third-order approximations to the transverse velocity are obtained, from which a correction to the moment on a disk of finite radius is found.

The first non-vanishing approximation to the radial-axial flow (a second-order term) is seen to have a steady component and a component with frequency twice that of the plate. The former component appears to persist outside the boundary layer, and at large distances from the plate to have the character of an irrotational stagnation flow. A re-examination not involving the series approximation reveals that although steady radial flow does exist outside the boundary layer, it is a viscous flow and is confined within a secondary layer. The ratio of the thicknesses of the two layers is found to be inversely proportional to the amplitude of the oscillations. These results indicate that a second-order flow in a region where the first-order flow field vanishes should not be accepted without further discussion.

Previous observations of turbulent motion at large wave-numbers have revealed the existence of an uneven distribution of turbulent energy. The spotty distribution of the turbulent motion at high wave-numbers is here studied experimentally for the turbulent boundary layer. The high wave-number intermittency is observed at all locations through and along the boundary layer from near transition to near separation.

The flatness factors for the longitudinal turbulent component at different wave-numbers are measured to give a quantitative value for the intermittency at particular wave-numbers. Upstream of the separation region the flatness factors are found to depend on wave-number and longitudinal distance, but not on the distance from the wall. It appears that the intermittency develops in the transition region and does not diminish very rapidly with distance downstream. Near separation the flatness factors change radically in distribution near the wall, and are there no longer independent of distance from the wall.

An analytic solution for the velocity field of a vortex street generated in a viscous fluid is developed. A method is presented for the determination of the true transverse spacing of vortices. Experimental geometry and velocity data, obtained by hot-wire techniques, are presented.

The experimental results verified the validity of the analytic solution. The vortices of a real viscous vortex street were found to resemble very closely the exponential solution of the Navier-Stokes equations for an isolated axisymmetric rectilinear vortex. Three basic regions of vortex street behaviour were apparent at each Reynolds number investigated-a ‘formation region’ in which the vortex street is developed and large dissipation of vorticity occurs, a ‘stable region’ in which the vortices display a stable periodic laminar regularity, and an ‘unstable region’ in which the street disappears and turbulence develops. Geometry and velocities were determined.

To determine experimentally the mean value of a randomly fluctuating quantity, it may be necessary to measure the average value over a considerable interval of time. This problem arose in a recent study of the temperature fluctuations over a heated horizontal plate, and a system was used that depended on the counting of electrical pulses generated at a rate proportional to the quantity being measured. The advantage of this technique is that mean values may be measured over time intervals of almost unlimited length with little added difficulty for the experimenter. Circuits are described which measure: (a) the mean square of a fluctuating quantity and of its time-derivative, (b) the statistical distribution of the fluctuations, (c) the mean frequency of the fluctuation assuming a particular value, and (d) the mean product of two fluctuating quantities. Over the range of use, the stability and linearity of the calibrations is better than 1%, more than sufficient for work on natural convection. In its present form, the equipment responds uniformly to all frequencies below 100 c/s, but it would not be difficult to extend this range of response to higher frequencies.

First, from a volumetric formulation of the momentum theorem of linearized theory, a general analytic proof is presented of the invariance of the drag of an arbitrary spatial distribution of horseshoe vortices and sources under reversal of the undisturbed flow. By consideration of the interference drag of two such singularity distributions, a reverse-flow relation for steady subsonic or supersonic flow is then obtained. This relation, a generalization of the Ursell-Ward theorem, may be applied to configurations with bodies whose surfaces are not quasi-cylindrical and whose surface pressures are quadratically related to the perturbation velocity.

The relation is used to discuss several interfering two-body arrangements in supersonic flow. It is shown that, in certain cases, the drag and lift may be determined without knowledge of the interference flow field associated with the arbitrarily prescribed body geometry. The simplicity of the results permits the formulation of optimum problems. The invariance of the drag under flow reversal with unchanged geometry is also established.

During the last few years, many experimental devices have been built in which strong shock waves are generated in gases by electromagnetic forces on currentcarrying gas particles. The general theory of these devices is discussed, taking external circuit inductance into account. It is shown that a shock wave of constant speed is finally attained. This shock wave is travelling at 90% of its final speed when the circuit inductance has increased to 3·0 times its initial value.

For wings with supersonic edges and with arbitrary dihedral, twist, camber and thickness distribution, the pressure distribution on the wing exterior to and along the two Mach lines emanating from the vertex of the wing is equal to the corresponding pressure distribution for a planar wing. The problem is to find the pressure distribution inside the two Mach lines. In the present paper, the unknown pressure distribution is approximated by an elementary function of the two surface variables. The (as yet undetermined) constants in the function are then found by the conditions: (i) that the function takes on the corresponding planar values along the two Mach lines, (ii) that it fulfils certain generalized integral relationships (Ting 1959), and (iii) that it satisfies the averaging property of solutions of the wave equation to be developed in this paper. The generalized integral relationship relates the integral of the pressure distribution along the line of intersection of a Mach plane with the wing to the integral along the same line of the prescribed normal velocity. The averaging property relates the pressure distribution along the line of intersection of the surface of the dihedral wing to that on a planar wing.