A two-dimensional, incompressible, irrotational, linearized flow model is employed in this analysis of two supercavitating, flat-plate hydrofoils in the presence of a free surface. The cavities are taken to have finite lengths, and gravity is neglected. The ensuing boundary-value problem is converted, by means of conformal mapping, to a mixed-boundary-value problem for the complex velocity in the upper half-plane. This altered problem is solved by use of the methods of thin-aerofoil theory and the solution involves digital-computer evaluation of a large number of incomplete elliptic integrals of the first and third kinds. Typical results are presented in graphs, and the results of the present work are compared with Yim's (1964) theory for a single supercavitating body near a free surface.

This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Bénard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.

The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).

The breakdown of the characteristics solution in the neighbourhood of the leading frozen characteristic is investigated for the flow induced by a piston advancing with finite acceleration into a relaxing gas and for the steady supersonic flow of a relaxing gas into a smooth compressive corner. It is found that the point of breakdown moves outwards along the leading characteristic as the relaxation time decreases and that there is no breakdown of the solution on the leading characteristic if the gas has a sufficiently small, but non-zero, relaxation time. A precise measure of this relaxation time is derived. The paper deals only with points of breakdown determined by initial derivatives of the piston path or wall shape. In the steady-flow case, the Mach number based on the frozen speed of sound must be greater than unity.

We discuss a one-parameter family of general solutions of the hydrostatic equations for liquid crystals. For certain values of the parameter, the energy may take on extremal values. We discuss special features of such solutions. Configurations similar to those considered naturally occur in liquid crystals of cholesteric type, suggesting that some such state has a relatively low energy.

The theory of stability of the flow of a viscous, electrically conducting fluid between rotating cylinders in the presence of an axial magnetic field is extended to the case where the cylinders are permeable and the primary flow includes a radial component. Numerical results pertaining to the stationary axially symmetric modes are presented, and the asymptotic stability behaviour for large values of the radial Reynolds number is derived.

Using a hot-wire system developed for this purpose, measurements of the velocity were made in a circular liquid-mercury jet issuing into a low-speed secondary flow and subject to a uniform axial magnetic field. The Reynolds number of the jet was about 10,000 while the magnetic interaction parameter varied from zero to slightly over one. The jet was strongly turbulent under all conditions investigated. The radial distribution of the mean and fluctuating component of the meridional velocity was measured at four axial stations located between 2 and 34 diameters from the nozzle exit. The results indicate that the rate of spreading of the jet is decreased, that the shape of the velocity profile changes and that the turbulent intensity decreases with increasing magnetic-field strength. The high-frequency components of the fluctuations in the second flow seem to be damped more strongly than fluctuations at low frequencies, while the reverse is observed within the core of the jet.

A power-law relation is derived between the Stanton number and the Reynolds number, expressing the law of heat transfer for a wall adjacent to a region of turbulent separated flow. The derivation is based on Prandtl's (1945) proposal for the laws of dissipation, diffusion and generation of turbulent kinetic energy. The constants appearing in these laws are determined by reference to experimental data for the hydrodynamic properties of the constant-stress and the linear-stress layers.

The agreement between the resulting predictions and the experimental data of other workers is sufficiently good to suggest that the actual mechanism of heat transfer from separated flows has much in common with that which is postulated. Closer agreement can be expected only after the present one-dimensional analysis has been superseded by a two-dimensional one.

Measurements of the low-speed flow up a step of height equal to 1·75 times the initial boundary-layer thickness show that the flow satisfies Stratford's (1959) condition for rapid separation, the extra stress gradients being confined to the first one-eighth of the boundary-layer thickness. The increase in turbulence intensity up to separation is small, and attributable to low-frequency fluctuations in separation position. Townsend's (1962) criterion predicts the separation point fairly accurately. A simple expression is found for the additional pressure rise that can be withstood by a boundary layer already fairly near separation, which gives tolerable results at any point in the flow up a step.

A mechanism is proposed for the manner in which the turbulent components support Reynolds stress in turbulent shear flow. This involves a generalization of Miles's mechanism in which each of the turbulent components interacts with the mean flow to produce an increment of Reynolds stress at the ‘matched layer’ of that particular component. The summation over all the turbulent components leads to an expression for the gradient of the Reynolds stress τ(z) in the turbulence \[ \frac{d\tau}{dz} = {\cal A}\Theta\overline{w^2}\frac{d^2U}{dz^2}, \]where ${\cal A}$ is a number, Θ the convected integral time scale of the w-velocity fluctuations and U(z) the mean velocity profile. This is consistent with a number of experimental results, and measurements on the mixing layer of a jet indicate that A = 0·24 in this case. In other flows, it would be expected to be of the same order, though its precise value may vary somewhat from one to another.

This paper extends the work of Wilson (1964) to include the effect of compressibility in the recurrence of dual solutions in the flow in the boundary layer on a semi-infinite, thermally insulated, flat plate placed at zero incidence to a uniform stream of electrically-conducting gas with an aligned magnetic field at large distances from the plate. Numerical integration of the boundary-layer equations has been performed for several values of the ratio, β, of the square of the Alfvén speed to the fluid speed in the undisturbed fluid, the conductivity parameter ε = 0·1 and ∞ and the square of the Mach number M2 = 0, 1/2, 1, 2, 2·5, 4 and 5. The effect of compressibility is to increase the value of β for which a solution can exist such that the skin friction at the plate is greater than zero. Dual solutions are seen to occur for non-zero Mach number and all values of ε but no attempt here has been made to explain this phenomenon. An analytic argument indicates that no solutions of the equations exist if the skin friction at the plate is greater than zero and if the vorticity and current decay exponentially, the condition for which is \[ M^2 < 1,\quad \beta > 1/(1-M^2). \] Nothing specific has been proved if this condition is not satisfied.

An attempt is made to understand the decay of a free vortex normal to a stationary, infinite boundary. For rapidly swirling flows in fluids of small viscosity, thin boundary layers develop along the rigid boundary and along the axis, the axial boundary layer being strongly influenced by the behaviour of the plate boundary. An over-all picture of the flow is sought, with only moderate success in the region far from the origin. Near the origin, the eruption of the plate boundary layer into the axial boundary layer is studied.

Detailed studies are reported of the free and forced oscillation of disks and a right-circular cylinder constrained to rotate about a fixed diametrical axis passing through the centre of the body and normal to the free-stream direction. When a disk is free to rotate, it oscillates at a definite frequency with slowly varying amplitude and phase. A right-circular cylinder also oscillates at a definite frequency but with rapidly increasing amplitude. When the amplitude becomes large, after a few cycles of oscillation, the cylinder rotates steadily in one direction.

Analogue computer elements, position sensors and a dynamic moment balance were used to study the static restoring moment, dynamic restoring moment, average damping moment, statistical properties of the disk motion and power spectrum of the turbulent moment. The behaviour of the disk and cylinder are explained using the measurements and the theory for random excitation of a linear system. The turbulent exciting moment is caused by the unsteady flow in the wake and can be changed by placing disks and splitter plates in the wake. A model is proposed for the unsteady flow field in the wake behind the disk. The model relates the turbulent moment to the vortex shedding process in the wake.