The second-order slender-body solution is derived for an unyawed elliptic cone in supersonic flow. The result is used as the basis for a critique of various approximations in compressible flow theory: slender-body, linearized, first- and second-order thin-wing theories; edge corrections; and the method of linearized characteristics.

This paper proposes a theory of collisions between small drops in a turbulent fluid which takes into account collisions between equal drops. The drops considered are much smaller than the small eddies of the turbulence and so the collision rates depend only on the dimensions of the drops, the rate of energy dissipation ε and the kinematic viscosity v. Reasons are given for believing that the collision rate due to the spatial variations of turbulent velocity is shown to be $N = 1\cdot 30(r_1+r_2)^2n_1 n_2(\epsilon|v)^{\frac {1}{2}}$, valid for $r_1|r_2$ between one and two. A numerical integration has been performed using this expression to show how an initially uniform distribution will change because of collisions. An approximate calculation is then made to take account also of collisions which occur between drops of different inertia because of the action of gravity and the turbulent accelerations.

The results are applied to the case of small drops in atmospheric clouds to test the importance of turbulence in initiating rainfall. Estimates of ε are made for typical conditions and these are used to calculate the initial rates of collision, the change in mean properties and the rate of production of large drops. It is concluded that the effects of turbulence in clouds of the layer type should be small, but that moderate amounts of turbulence in cumulus clouds could be effective in broadening the drop size distribution in nearly uniform clouds where only the spatial variations of velocity are important. In heterogeneous clouds the collision rates are increased, and the effects due to the inertia of the drop soon become predominant. The effect of turbulence in causing collisions between unequal drops becomes comparable with that of gravity when ε is about 2000 cm2 sec−3.

Sir Charles Darwin has advocated a study of the ‘drift’ of material surfaces in the classically investigated irrotational flows past bodies. This suggestion is followed up and given further support in the present paper. In particular, it is shown how secondary flows can be evaluated by use of the ‘drift function’ t for the primary flow. This is a function (§1) such that material surfaces initially at right angles to the stream drift into shapes expressible by equations t=constant.

The analysis leads to a simple expression (§4) for the secondary velocity field in the flow past an infinite cylinder of any cross-section, with the upstream velocity normal to its axis and increasing linearly with distance along the axis-a problem in which only the secondary vorticity field was previously known. The drift past a sphere is computed and illustrated (§6), and the secondary vorticity field in shear flow past a sphere is tabulated (§7). There is also a detailed study (§3) of the asymptotic form of the secondary velocity field in flow past any body, based on a result of Darwin concerning ‘hydrodynamic mass’.

Consider a wing-jet combination placed in a wind tunnel; the measured thrust on the wing due to a high speed jet emerging from it will require to be corrected to give the infinite stream value. This paper provides a theory of these wind tunnel corrections, and incidently establishes that the ideal thrust is almost independent of the jet exit angle.

A converging channel of area ratio 34 : 1 has been developed to produce strong shock waves in a tube. Shock waves of speeds M1=3−7·5 in oxygen and M1 = 5−10 in nitrogen have now been studied with an interferometer, and values of the relaxation time τ for the approach to vibrational equilibrium behind the shocks have been measured. The value of τ (atmospheric) varies from 54μsec at 800° K to 1·3 μsec at 3000° K for oxygen and from 19 μsec at 3500° K to 5 μsec at 5500° K for nitrogen. For oxygen, the graph of log τ against T−1/3 is not quite the straight line predicted by the Landau-Teller theory. The density ratios across the shocks were measured and compared with values calculated by the Bethe-Teller method for variable specific heats. Agreement between the measured and calculated values is satisfactory. Experiments were also performed on oxygen-nitrogen mixtures to determine the effect of nitrogen on the approach to equilibrium of the oxygen. It was found that O2 and N2 collisions at approximately 2000° K are 40% as effective in transferring energy to the oxygen as O2 and O2 collisions. A device that detects a shock with a time lag of less than 1μsec, consisting of an evaporated gold film which changes its resistance when heated by the shock, was also developed.

In this paper the problem is considered of long gravity waves approaching a semi-infinite barrier which extends parallel to the wave crests, the whole system being in rotation. It is well known that, when the rotation is zero, there is a ‘shadow region’ behind the barrier in which the disturbance diminishes rapidly with distance from the edge. However, it is shown that the rotation gives rise to an additional wave in the shadow region. The crests of this wave are at right-angles to the incident wave, and it travels along the barrier without attenuation in that direction. The amplitude falls off exponentially with distance from the barrier, as in a Kelvin wave. The amplitude at the barrier may exceed that of the incident waves.

The problem arises in connexion with the propagation of tides and storm surges in the ocean.

This paper considers the two-dimensional laminar boundary layer on an infinite flat plate normal to an oncoming stream, the plate making transverse oscillations in its own plane. The exact solution is shown to depend on a single ordinary differential equation, containing the frequency of oscillation as a parameter. Series methods are employed to evaluate the solution for small and large values of the frequency, and enough terms are calculated to give the solution with satisfactory accuracy over the whole frequency range. It is observed that the solution satisfies the full Navier-Stokes equations.

For a cylinder of arbitrary section making transverse or rotational oscillations, it is shown how the results for a flat plate can be used to describe the boundary layer in the neighbourhood of the front stagnation point. Difficulties which arise in extending the solution to cover the remainder of the cylinder are discussed, and an estimate is made of the fluctuating torque on a circular cylinder making transverse oscillations.

The region of finite vorticity near the edge of a diffracting wedge is investigated. Dimensional analysis gives the dependence of the circulation and the velocity of the vortex region on the pulse strength. A close estimate of the magnitude of the magnitude of these quantities is obtained by replacing the vortex region by a single concentrated vortex. The theoretical conditions at the sharp edge are discussed and compared with observations of real fluid behaviour. A short account of the theory of the core of the spiral vortex sheet in a perfect fluid is appended.

The research reported herein was supported by the United States Air Force, through the Office of Scientific Research of the Air Research and Development Command.