A singularity is encountered in the flow field about two-dimensional and axisymmetric bodies characterized by a sharp corner, where the fluid velocity becomes sonic. Investigation shows that the problem in question belongs, as do many other discontinuity problems, to the family of asymptotic or ‘boundary-layer’ phenomena of mathematical physics. The solution of a first approximation to the flow equations is given by a series in powers of a variable measuring the distance from the corner, with coefficients depending on an appropriate similarity variable. The leading coefficient of the series is independent of three-dimensional and rotationality effects, in complete analogy to the well-known solution of the corner problem in supersonic flow. Detailed results are presented for the leading singularity and for the first two corrections due to rotationality and axial symmetry of the flow.

The electrical conductivity of shock-ionized argon has been measured at temperatures between 6000 °K and 13,000 °K and electron number densities in the range 1016 to 1018 cm−3. The results are compared with theoretically computed values and show sufficient agreement to be plotted on a linear scale. The experimental technique is based on the measurement of voltages generated in a search coil by the flux from currents induced in a plasma moving rapidly through a magnetic field.

An experiment has been performed to determine the cavity pressure and cavity length for the flow under a curved two-dimensional turbulent jet discharging parallel to a flat plate at some distance from the surface. The measurements agree quite closely with the predictions of a simple analysis of the flow when a value is assigned to the spread parameter for the curved jet which is somewhat different from that for the plane jet.

Detailed pressure and velocity measurements show that the rate of growth of the curved jet is in fact closely similar to that of the plane jet. This suggests that the adjustment of the spread parameter should be regarded simply as a convenient device for taking into account the quantitative errors of the assumptions made in the analysis.

The stability and transition Reynolds numbers for the flow due to a disk rotating in still air were increased by low rates of suction through either a woven wire-cloth or a slitted surface. Observations on the slitted disk at rotational speeds between 550 and 1250 r.p.m. showed that the critical value of the Reynolds number r2ω/ν for instability increased from about 135,000 without suction to nearly 250,000 for a value of the suction parameter a of 0·4. The corresponding values for transition increased from about 275,000 to about 400,000. A given increase in stability Reynolds number required about 75% more suction than that theoretically predicted for uniform distributed suction, a satisfactory result in view of the limitations of the apparatus.

At higher rates of suction (0·4 < a < 1·6), the reduction in secondary flow allowed transverse turbulent contamination to spread inwards from the rim. Consequently the vortices associated with secondary-flow instability were not found, though disturbances of larger wavelength appeared. Intermittent turbulent flow was spread over a much larger region of the disk and no laminar flow could be obtained above a Reynolds number of 400,000, Owing to this feature of the flow, it was not possible to extend laminar flow to values of the unit Reynolds number (ratio of stream velocity to kinematic viscosity) corresponding to flight conditions on a swept-back wing. It is concluded that the rotating disk is not a satisfactory tool for the investigation of the effects of suction on secondary-flow instabilities such as arise in the case of a swept-back wing, or for the testing of suction surfaces.

The method of images is used to determine the flow perturbation caused by a small point source of fluid in a stream containing a vortex sheet, a number of vortex sheets, or a continous two-dimensional distribution of vorticity. Calculations are carried out for a variety of such shear flows, to illustrate the range of applicability of the method.

The image source distribution provides an alternative approach to that given by Lighthill for calculating small disturbances to a parallel shear flow, and suggests a simple physical picture by which to interpret the complicated effects associated with the distortion and stretching of vortex lines.

An experimental investigation of the explosions of 2 in. diameter glass spheres under high internal pressure has been made. The spheres were initially filled with air or helium at 400 and 326 p.s.i., respectively, and were exploded in air at atmospheric pressure. Experiments on the simulation of high-altitude explosions are also described. Schlieren and spark shadowgraph records of explosion phenomena, and pressure records of the reflexion of the spherical shock wave at various radii, are presented.

An account of some initial experiments on the implosion of 5 in. diameter glass spheres is given. The results were not very satisfactory because of the failure of the spheres to shatter in a desirable manner while under an external pressure of 65 p.s.i.

Numerical solutions to the air and helium sphere explosions are described and the experimental wave phenomena are shown to be in good quantitative agreement with the theoretical predictions, in that they exhibit all the main features that were predicted and are modified only by the physical limitations of the glass diaphragm. A formation process is associated with the spherical shock waves in practice, resulting in initial shock velocities which are lower than the theoretical values.

The theory of hydrodynamic stability and the impact on it of recent work with turbulent spots is discussed. Emmons's (1951) assumptions about the growth and interaction of turbulent spots are found experimentally to be substantially correct. In particular it is shown that the region of turbulent flow on a flat plate is simply the sum of the areas that would be obtained if all spots grew independently.

An investigation of the conditions required for breakdown to turbulence near a wall, that is, to initiate a turbulent spot, suggests that regardless of how disturbances are generated in a laminar boundary layer and independent of both the Reynolds number and the spatial extent of the disturbances, breakdown to turbulence occurs by the initiation of a turbulent spot at all points at which the velocity fluctuation exceeds a critical intensity. Over most of the layer this intensity is about 0·2 times the free-stream velocity. The Reynolds number is important merely in respect of the growth of disturbances prior to breakdown.

The formula for transfer of $\overline {\theta^2}$-stuff across the θ-spectrum, which was obtained by Batchelor, Howells & Townsend (1959) for wave-numbers at which molecular conduction is important, is extended here to smaller wave-numbers by means of a simplified general picture of the mechanism involved. The interaction of velocity and temperature fields is represented by a combination of eddy conductivity due to the smaller eddies, and a straining action due to the larger eddies, and this leads to an approximate equation for the θ-spectrum, for a fluid of arbitrary Prandtl number, over at least the equilibrium range of wave-numbers.

In this paper some simple theoretical considerations concerning the movement of ships in restricted waterways are discussed, and it is shown that for a ship towed from the bank, or by any external force, there are three distinct speed ranges: subcritical, critical and supercritical. In the subcritical and supercritical ranges, Bernoulli's equation and the continuity equation are satisfied everywhere by a state of steady motion relative to the ship, but in the critical range these laws require that a quantity of fluid is piled up continuously ahead of the ship in the form of a bore. Experimental confirmation is given by means of photographs of model tests.

A similarity solution is obtained up to the first order in an effective Rayleigh number for the behaviour of very weak thermal vortex rings produced by the rapid release of heat at one point of a large region of fluid.

The laminar pattern of flow is similar to that in ordinary vortex rings, but the temperature decreases outwards from the centre in all directions with some asymmetry about the horizontal plane through the centre, and there is no accumulation of heat into the vortex ring. The vortex propagates slowly in relation to its rate of growth, and the process is dominated by viscous and thermal diffusion.

The expansion method introduced by Friedrichs (1948) for the systematic development of shallow-water theory for water waves of large wavelength was used by Keller (1948) to obtain the first approximation for the finite-amplitude solitary wave of Boussinesq (1872) and Rayleigh (1876), as well as for periodic waves of permanent type, corresponding to the cnoidal waves of Korteweg & de Vries (1895).

The present investigation extends Friedrich's method so as to include terms up to the fourth order from shallow-water theory for a flat horizontal bottom, and thereby obtains the complete second approximations to both cnoidal and solitary waves. These second approximations show that, unlike the first approximation, the vertical motions cannot be considered as negligible, and that the pressure variation is no longer hydrostatic.

The Rayleigh instability of an incompressible, infinitely conducting, inviscid fluid of variable density is investigated under the influence of an horizontal magnetic field and coriolis forces. After establishing the equations of the problem when both the density and the magnetic field vary with distance in the upward direction, two special cases of density distribution are studied in detail. Both the magnetic field and coriolis forces are found to have a stabilizing influence on the configuration and further it is concluded that they may bring about stability in the configuration when it is thoroughly unstable without them.

An approximate solution is presented to the problem of the propagation of a temperature wave through a stratified medium which both diffuses and radiates heat. The solution is a combination of two waves whose relative amplitudes vary with the distance from the lower boundary. Apparent diffusivities computed from phase lag and attenuation coefficients can differ greatly from each other and can vary with height even if the actual diffusivity does not.

An example, using parameters simulating the earth's atmosphere, suggests that analysis of the propagation of the diurnal wave upward from the earth's surface is likely to be inadequate if radiative effects are not considered.

Self-excited oscillations have been discovered experimentally in a supersonic laminar boundary layer along a flat plate. By the use of appropriate measuring techniques, the damping and amplification of the oscillations are studied and the stability limits determined at free-stream Mach numbers 1·6 and 2·2. The wave-like nature of the oscillations is demonstrated and their wave velocities are measured using a specially designed ‘disturbance generator’. It is shown empirically that the stability limits expressed in terms of the boundary-layer-thickness Reynolds number are independent of the Mach number and dependent only on the oscillation frequency. The main effect of compressibility is an increase in wave velocity with Mach number. This has the consequence that the disturbances, although possessing the same dimensionless amplification coefficient as in the incompressible case, have less time (per unit distance) to grow in amplitude. Thus, the adiabatic compressible boundary layer is shown to be more stable than the incompressible one. In general, the experiments confirm the basic assumptions and predictions of the existing stability theory and also suggest the desirability of improvement in the theory in certain phases of the problem. Finally, on the basis of these results a rough estimate of the transition Reynolds number is made in the compressible flow range.

The related questions concerning the transmission of electromagnetic waves are considered:

1. The reflexion and transmission of plane waves at a perfectly conducting layer of gas in an otherwise non-conducting atmosphere, when there is a uniform external magnetic field perpendicular to the layer. Here the main result is that a layer of finite depth h is an almost perfect filter, being transparent to waves of frequency nπA0/h (A0 = Alfvén velocity, n an integer).

2. The existence of plane surface waves for such a finite layer. There is always one such wave and, for certain ranges of frequency, two. The first becomes ‘choked’ at the filter frequencies, its velocity first tending to zero and then jumping to a finite value. The second chokes at the frequencies nπA0a0/h √(a20 + A20 (a0 = acoustic velocity).

The purpose of this note is to describe a particular class of steady fluid flows, for which the techniques of classical hydrodynamics and boundary-layer theory determine uniquely the asymptotic flow for large Reynolds number for each of a continuously varied set of boundary conditions. The flows involve viscous layers in the interior of the flow domain, as well as boundary layers, and the investigation is unusual in that the position and structure of all the viscous layers are determined uniquely. The note is intended to be an illustration of the principles that lead to this determination, not a source of information of practical value.

The flows take place in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. When fluid is extracted through both walls there are boundary layers on both walls and the flow outside these layers is irrotational. When fluid is extracted through one wall and injected through the other, there is a boundary layer only on the former wall and the inviscid rotational flow outside this layer satisfies the no-slip condition on the other wall. When fluid is injected through both walls there are no boundary layers, but there is a viscous layer in the interior of the channel, across which the second derivative of the tangential velocity is discontinous, and the position of this layer is determined by the requirement that the inviscid rotational flows on either side of it must satisfy the no-slip conditions on the walls.

An exact solution is presented of the equations and boundary conditions governing the steady longitudinal motion of a semi-infinite non-conducting plate in an oblique magnetic field. The discussion covers the distortion of the boundary layer, the structure of the induced electric and magnetic fields, the current-density distribution, and the behaviour of the fields near the edge of the plate.

An experimental study of the incompressible flow near the side edge of a finite flat plate at zero incidence is reported for the Reynolds number range 104 to 106, but comparison with data already published shows that the conclusions are quantitatively valid for Reynolds numbers up to 109. The laminar velocity field is everywhere convex and does not contain any secondary flow other than that of the normal diffusive growth of the layer, but has a logarithmic singularity at the edge where the stress is controlled by the local radius of curvature. The excess skin friction due to the edge is considerably greater than that given by calculations based on the Rayleigh approximation but agrees with a recent Pohlhausen calculation by Varley (1958). Near the edge the high stress and ease of momentum diffusion makes the flow very unstable and turbulent spots occur well upstream of the normal transition zone in the middle of the plate. The spots originate from a nearly point-like region at the edge of the plate and grow linearly in time at the same rate as ordinary turbulent spots to sweep out a narrow tongue of turbulent fluid near the edge until they merge with the normal transition zone. Within this tongue a weak secondary flow of Prandtl's second kind, driven by the anisotropic Reynolds stresses, begins to develop. In fully turbulent flow when the secondary flow is largely localized to within a few boundary-layer thicknesses of the edge the secondary flow velocities are everywhere less than 0.04 of the free-stream velocity. Nevertheless, the secondary flows from each of the side edges interact, regardless of the width of the plate, to increase the total drag coefficient by an amount 0·0004 which is independent both of the Reynolds number and the width of the plate except when it is very narrow. This simple result allows apparent discrepancies between various formulations of the drag coefficient of a finite plate to be reduced to less than ± 1% S.D. Of these formulations Schoenherr's (1932) empirical relation agrees best with the present data.

Some results of an experimental investigation of forces associated with the subsonic flow of air around a circular cylinder in a wind tunnel are presented. The oscillating forces due to the downstream vortex street are studied for Reynolds numbers in the ‘critical’ range 4 × 104 to 6 × 105. Of particular interest is the observation, at the onset of transition to turbulence, of a spanwise wave or cell pattern near the cylinder surface, which is stabilized in a striking manner by the use of the fine threads as a visualization technique.

A transformation is introduced which, for a class of outer pressure distributions, reduces the unsteady incompressible laminar boundary-layer equations in two dimensions to an equation in which the time does not appear explicitly. A formally exact solution of the resulting equation is then presented in the form of a series and it is shown that the solution can be expressed in terms of universal functions.