A theory is developed to describe the properties of waves on the free surface of a liquid in turbulent motion. The distinction between the interacting wave and turbulent motions is achieved by separating the velocity field uniquely into a surface-induced contribution characteristic of the wave motion and a vorticity-induced contribution associated with the turbulence. The theory is applied to the scattering of gravity waves passing over the surface of deep water which has a turbulent motion of sufficiently low mean square vorticity. Expressions are derived for the directional distribution of the scattered wave (equation 4.19) and for the logarithmic decrement resulting from the scattering (4.20). It is shown that, under typical conditions in open sea, the attenuation from scattering will be greater than that from direct viscous dissipation for wavelengths greater than about 3 m.

Expressions are derived for the velocity of two spheres, moving slowly under external forces through a viscous fluid, as a function of their separation and radii. They compare favourably with the available experimental data. A discussion of the interactions of three particles and some general comments on the settling of a swarm of spheres are also included.

The previous work of Thomas (1956) on the turbulent convection over a single, heated, horizontal surface has been extended using improved methods of analysis of the temperature fluctuations, and it has been possible to measure the distributions of mean temperature, the mean squares of the temperature fluctuation, the temperature gradient and the rate-of-change of temperature, and the statistical distributions of these quantities. These measurements were made for three different values of the convective heat flux, and the results are consistent with the dimensional consequences of assuming that the convection near the surface is independent of the distant boundaries and determined by the heat flux and the viscosity and conductivity of the fluid.

The most striking feature of the observations is that the fluctuations of temperature, temperature gradient, and rate-of-change of temperature, all show periods of activity, characterized by large fluctuations, alternating with periods of quiescence with comparatively small ones. Both the proportion and frequency of occurrence of the active periods decrease with increasing distance from the surface and they probably occur when rising columns of hot air pass through the point of observation. The quiescent periods occur when the point of observation lies outside the columns, and analysis of the statistical distributions of the various fluctuations shows that, during these periods, they are nearly independent of height. It is concluded that the quiescent fluctuations are typical of the turbulent convection far from the surface while the active fluctuations are the manifesta- tion of the convective processes arising near the rigid boundary. These processes may be described as the detachment of columns of hot air from the edge of the conduction layer and the erosion of these rising columns by contact with the surrounding air which is in vigorous turbulent motion. Since the variations of the intensities with height is dominated by the contributions of the active periods, it is not surprising that no agreement is found with the predictions of the similarity theory which assumes the convection to be independent of the conduction layer at a sufficient distance from it. The Malkus theory of turbulence, which empha- sizes dependence on the conduction layer, is in qualitative agreement with this inferred mechanism of the convection and is in quantitative agreement with the observed distribution of mean temperature. A brief discussion is given of the effect of a horizontal shearing motion on the convection and of the relation of these measurements to measurements of the temperature distribution in the earth's boundary layer with upward flux of total heat.

A method has been developed for calculating the viscous effects of the hypersonic flow on the fore part of various bodies. This method takes into account the finite leading-edge thickness of the body and the detached shock wave. The calculated pressure distributions agree with all experimental results for both air and helium over a wide range of Mach numbers and Reynolds numbers. The calculation predicts skin frictions of the order of twice those predicted by ordinary boundary-layer theory.

Steady, plane flow of incompressible fluid past thin cylindrical obstacles is treated with two different orientations of the undisturbed, uniform magnetic field; namely, parallel and perpendicular, respectively, to the undisturbed, uniform stream. In the first case, the flow of an infinitely conducting fluid is shown to be irrotational and current-free except for surface curents at the walls of the obstacles. With large but finite conductivity the surface currents are replaced by thin boundary layers of large current density.

In the second case, for infinite conductivity the flow field is made up of an irrotational current-free part and a system of waves involving currents and vorticity extending out from the body. For large, finite conductivity these waves attenuate exponentially with distance from the body.

In both cases the forces on sinusoidal walls and on airfoils are calculated. In the second case positive drag occurs.

The behaviour of weak homogeneous turbulence subjected to a uniform distortion is studied by means of linearized equations that include the effect of inertial interaction between the mean flow and the turbulent fluctuations and of viscous dissipation. Three particular cases of distortion are investigated in varying degrees of detail; those of uniform rotation, uniform shear, and uniform irrotational distortion. For the first two, the total energy associated with the turbulence is found to decay, but for the third it is found in general to increase without limit. A general solution in terms of the spectrum functions for the restricted case of irrotational distortions is given, particular consideration being paid to the asymptotic limits approached as the distortions (not necessarily the rates of distortion) become large. These show that for an arbitrary initial turbulent spectrum the asymptotic rates of growth are functions of the precise geometrical nature of the distortion, of an integral parameter of the initial spectrum function, of a Reynolds number based on the rate of distortion and the length scale of the initial turbulence, and of the contraction. A particular example, that of initially isotropic turbulence in its final period of decay subjected to an axisymmetric distortion, is worked out in detail and te solution (described in terms of mean turbulent intensities) displayed graphically for several values of the Reynolds number.

As a primary step towards understanding the flow of a partially ionized gas in a magnetic field, we have studied both theoretically and experimentally the problem in which the gas flow is one-dimensional. This simplification permits a detailed calculation of the flow field and a quantitative comparison of the theory with observations made in a shock tube.

An ionized gas is composed of three species: electrons, ions and neutral particles. To take complete account of all the phenomena occurring when the high-velocity gas interacts with the magnetic field, the motion of all three species must be considered. When this is done, it is found that the electrical conductivity of the gas is a tensor dependent on both the magnitude and geometry of the magnetic field. However, when the collision frequency for the electrons is greater than their cyclotron frequency in a magnetic field, the gas may be treated as a continuum with a scalar conductivity. For such gas states, all of the observed effects for two experimental geometries in which the gas current forms closed loops in the magnetic field can be explained with a simple theory. The interaction produces a flow completely analogous to pipe flow with friction and no heat transfer, where the wall friction force is replaced by the magnetic body force which can choke the flow if the body forces are larger than a certain minimum value.

Using the same experimental geometries, the gas state is then adjusted so that the electrical conductivity is a tensor. Outstanding among the observed effects are ion slip, where the ions and neutrals travel through the field at different velocities, and Hall currents, generated by the drift of the charged particles across magnetic field lines. The observed effects again agree with the predicted values.

The equation of motion is derived for the steady flow under gravity of a liquid in a uniform channel, including the effects of the friction of the bed and of the eddy viscosity for lateral mixing. The equation is solved to find the distribution of mean velocity across the channel when the cross-section is rectangular, triangular or trapezoidal. Numerical values are also given to indicate the extent to which eddy viscosity may affect the lateral distribution of mean velocity.

Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.