The interaction of a simple wave, in steady supersonic flow, with a two-dimension mixing region is treated by applying Fourier analysis to the linearized equations of motion. From asymptotic forms for the Fourier transforms of physical quantities, for large wave-number, the dominant features of the resulting flow pattern are predicted; in particular it is found that a shock wave, incident on the mixing region, is reflected as a logarithmically infinite ridge of pressure. For two particular Mach-number distributions in the undisturbed flow, numerical solutions are obtained, showing greater detail than the results predicted by the asymptotic approach. A method is given whereby the linear theory may be improved to take into account some non-linear effects; and the reflected wave, for an incident shock wave, is then seen to consist of a shock wave, gradually diminishing in strength, followed by the main expansion wave.

When a hollow circular cylinder with its axis horizontal is partially filled with water and rotated rapidly about its axis, an almost rigid-body motion results with an interior free surface. The emotion is analysed assuming small perturbations to a rigid rotation, and a criterion is found for the stability of the motion. This is confirmed experimentally under varying conditions of water depth and angular velocity of the cylinder. The modes of oscillation (centrifugal waves) of the free surface are examined and a frequency equation deduced. Two particular modes are considered in detail, and satisfactory agreement is found with the frequencies observed.

This paper is intended to give some indication of the impact forces of a water wave on a wall. The effect of gravity forces in the small time interval of impact considered will be small and is neglected. The shape of the wave before impact is considered to be a two-dimensional wedge which is assumed to strike a wall at right angles to its path. The wedge is assumed to be infinite in extent and to have uniform translational velocity V before impact. The choice of a wedge shape enables the problem to be formulated in terms of similarity variables x/Vty/Vt, where the origin of the x, y plane is at the initial point of contact of the vertex of the wedge with the wall. The solution presented here can be easily adapted to the problem of an axi-symmetric cone of water striking a wall, but this is not pursued in the present paper.

A systematic analysis is made of published measurements of the magnitude of temperature fluctuations in the atmospheric boundary layer. These cover a wide range of height, wind speed, and thermal stratification. Within appropriate ranges of the variables, there is evidence for the existence of a dominantly forced convection régime, and also one wherein the predictions of the similarity theory of free convection are fairly closely approached. Subject to the limitations set by the recording systems used, regression relations are obtained for the magnitude of the fluctuations in terms of height and vertical temperature gradient or heat flux.

An experimental investigation of the hydrodynamic stability of the laminar hypersonic boundary layer was carried out with the aid of a hot-wire anemometer. The case investigated was that of a flat surface at zero angle of attack and no heat transfer.

The streamwise amplitude variation of both natural disturbances and of disturbances artifically excited with a siren mechanism was studied. In both cases it was found that such small fluctuations amplify for certain ranges of frequency and Reynolds number Rθ, and damp for others. The demarcation boundaries for the amplification (instability) zone were found to resemble the corresponding limits of boundary-layer instability at lower speeds. A ‘line of maximum amplification’ of disturbances was also found. The amplification rates and hence the degree of selectivity of the hypersonic layer were found, however, to be considerably lower than those at the lower speeds. The disturbances selected by the layer for maximum amplifications have a wavelength which was estimated to be about twenty times the boundary-layer thickness δ.

Townsend (1954) has shown that turbulent vorticity may rotate and strain a diffusion wake, thereby increasing the contribution of molecular diffusion to the total mean dispersion over short diffusion times. To test whether any such effect occurs at longer diffusion times, the lateral dispersion of both helium and of carbon dioxide in air were measured downstream from a continuous point source in the turbulence produced by a grid in a wind tunnel. The data show that, for long diffusion times, accelerated molecular diffusion is negligible, so that molecular diffusion makes only an independent contribution to the total dispersion.

An empirical relationship for the torque transmitted by fluid friction to an outer cylinder as a function of the angular velocity of the inner cylinder has been obtained by analysis of experimental data published by Wendt, Taylor and Donnelly. With one exception it is found that the torque G has the functional form $G = a \Omega^{-1}_1 + b\Omega ^{1 \cdot 36}_1,$ where Ω1 is the angular velocity of the inner cylinder and a and b are constants determined from the data. The formula applies to a range of values of Ω1 above the onset of instability extending to about 10 times the critical angular velocity. The experiments also show that the finite amplitude analysis recently advanced by Stuart gives the correct variation of torque over a short range above the critical speed. At speeds well beyond critical it is found that G varies approximately as Ω1.51, and that the variation of torque with gap width can be expressed as a simple power law with exponent about 0.31. In an appendix Dr G. K. Batchelor shows that these latter relations are consistent with the supposition that the flow is steady and consists of inviscid cores surrounded by boundary layers.

In this paper it is shown how earlier results for buoyant vortex rings may be extended to describe the corresponding two-dimensional case, which arises in the theory of bent-over plumes. It is again assumed that in uniform surroundings the circulation remains constant while the buoyancy acts to increase the momentum of the pair. The behaviour in two dimensions is quite different from that in three, however; a buoyant vortex ring spreads linearly with height, whereas a buoyant pair spreads exponentially with height, or linearly with time (and therefore, in a bent-over plume, linearly with distance downwind).

The theory has been extended to describe the rise of buoyant rings and pairs through stably stratified surroundings having a linear density gradient. The behaviour near the maximum height reached is found to depend critically in both cases on the relative rates at which the circulation and the momentum fall to zero. If these reach zero together, the rings or pairs will steadily increase in size and come to rest at a finite height and with a finite radius. If the circulation is non-zero when the momentum vanishes, the radius begins to decrease soon after the buoyancy becomes zero, and the vortices will therefore tend to break up suddenly and mix into their surroundings. There is a considerable increase in the final height which should be attained by vortex rings or bentover plumes if the initial circulation is increased; it is suggested that releasing smoke intermittently, rather than continuously, at high velocity might be a means of increasing the effective height of chimneys in calm conditions. When the circulation reaches zero before the momentum does, the solutions indicate that the radius becomes very large near the level of zero buoyancy.

Hydrodynamic stability of free boundary-layer flows is treated in general. It is found that the situations at low Reynolds numbers are universal for all velocity profiles of free boundary-layer type. Curves of constant amplification are calculated as far as O(R3). In particular, the asymptotic form of the neutral curves for R [eDot ] 0 is found to be α = R/(4√3), so that the critical Reynolds numbers of these flows are identically zero. The phase velocity of the disturbance is also found to be zero, for all disturbances, up to the second approximation.

A method of normalizing the velocity profiles is suggested, and existing results for the stability of various profiles at large Reynolds numbers are discussed from a new point of view.

The hot-wire anemometer, used for recording speed variations in turbulent flow, involves in its working principle the unsteady heat transfer from a hot fixed surface to a fluctuating air stream moving past the surface. If the wire is maintained at a constant (high) temperature, the rate of loss of heat from the wire changes with the velocity of the incident stream, and the compensating rate of gain of heat, produced by the Joule heating effect of the electric current, changes, correspondingly. The accompanying change of current can be measured, and used to calculate the varying velocity of the air stream. The hot wire may have a diameter as low as 10−4 in. and the Reynolds number of the flow is then of the order of 0.05 for each ft. per sec of velocity. With low velocities, of the order of 10 or 20 ft./sec, the flow past the wire is in the range of small Reynolds number, and the exact equations of flow may be approximated by simpler equations in the manner of Oseen's theory (Lamb 1932). The approximate equations are not easy to solve when the flow is compressible, as it will be in the presence of the large temperature differences imposed by the heat of the wire. If, however, the temperature differences are assumed to be small, the approximate energy equation is no longer linked with the equations of continuity and momentum, and it may be solved without knowledge of the velocity field. The purpose of this note is to give the solution for the temperature field when a warm circular cylinder or a warm sphere is held at rest in a fluctuating stream.

A two-dimensional, small-perturbation theory for the steady motion of thin lifting airfoils in an incompressible conducting fluid, with the uniform applied magnetic field perpendicular to (and in the plane of) the undisturbed, uniform flow field, is described. The conductivity of the fluid is assumed to be such that the magnetic Reynolds number, Rm, of the flow is large but finite. Within this assumption, a theory based on superposition of sinusoidal modes is constructed and applied to some simple thin airfoil problems.

It is shown that with this particular field geometry the Alfvén wave mechanism is important in making possible very deep penetration into the flow field of currents and their associated vorticity. It is also shown that the current penetration for an airfoil is much larger than for a wavy wall of wavelength equal to the airfoil chord.

A value of Rm = 5 is found to be a good approximation to infinity in this study; in fact, use of the present technique for values of Rm of the order of unity is permissible. These results provide an indication of what is meant by ‘large’ magnetic Reynolds number in two-dimensional magneto-aerodynamics.

The model proposed by Phillips (1957) for the generation of water waves by the random fluctluations of normal pressure already present in a turbulent wind is generalized to include energy transfer associated with the interaction between surface wave and mean air flow (Miles 1957). It is found that this energy transfer may increase by an order of magnitude the surface displacements produced by a given distribution of pressure fluctuations in the principal stage of development.