An experimental investigation is described in which principal emphasis is given to revealing the nature of the motions in the non-linear range of boundary-layer instability and the onset of turbulence. It has as its central purpose the evaluation of existing theoretical considerations and the provision of a sound physical model which can be taken as a basis for a theoretical approach. The experimental method consisted of introducing, in a two-dimensional boundary layer on a flat plate at ‘incompressible’ speeds, three-dimensional disturbances under controlled conditions using the vibrating-ribbon technique, and studying their growth and evolution using hot-wire methods. It has been definitely established that longitudinal vortices are associated with the non-linear three-dimensional wave motions. Sufficient data were obtained for an evaluation of existing theoretical approaches. Those which have been considered are the generation of higher harmonics, the interaction of the mean flow and the Reynold stress, the concave streamline curvature associated with the wave motion, the vortex loop and the non-linear effect of a three-dimensional perturbation. It appears that except for the latter they do not adequately describe the observed phenomena. It is not that they are incorrect or may not play a role in some aspect of the local behaviour, but from the over-all point of view the results suggest that it is the non-linear effect of a three-dimensional perturbation which dominates the behaviour. A principal conclusion to be drawn is that a new perspective, one that takes three-dimensionality into account, is required in connexion with boundary-layer instability. It is demonstrated that the actual breakdown of the wave motion into turbulence is a consequence of a new instability which arises in the aforementioned three-dimensional wave motion. This instability involves the generation of ‘hairpin’ eddies and is remarkably similar in behaviour to ‘inflexional’ instability. It is also shown that the results obtained from the study of controlled disturbances are equally applicable to ‘natural’ transition.

A general theory is put forward for the effect of wall proximity on the Stokes resistance of an arbitrary particle. The theory is developed completely for the case where the motion of the particle is parallel to a principal axis of resistance. In this case, the wall-effect correction can be calculated entirely from a knowledge of the force experienced by the particle in an unbounded fluid, providing (i) that the wall correction is already known for a spherical particle and (ii) that the particle is small in comparison to its distance from the boundary. Experimental data are cited which confirm the theory. The theory is extended to the wall effect on a particle rotating near a boundary.

A surface distribution of stress is imposed on an ocean enclosed by two continental boundaries; the resulting transport circulation is studied between two latitudes of zero surface wind-stress curl, within which the curl reaches a single maximum. Under the assumption that turbulent transfer of relative vorticity has a minimum effect on the mean circulation, inviscid flow patterns are deduced in the limit of small transport Rossby number. Inertial currents, or naturally scaled regions of high relative vorticity, occur on both the eastern and the western continental coasts. Limits on the relative transports of the currents are obtained and found to depend on the direction of variation of the wind-stress curl with latitude, relative to that of the Coriolis accelerations. The most striking feature of the inviscid flow is a narrow inertial current the axis of which lies along the latitude of maximum wind-stress curl. All eastward flow occurs in this midlatitude jet.

A feature of the flow which cannot remain essentially free of turbulent processes is the integrated vorticity relationship, since the imposed wind-stress distribution acts as a net source of vorticity for the ocean. Heuristic arguments are used together with this integral constraint to deduce the presence and strength of the turbulent diffusion which must occur in the region of the mid-latitude jet. It is further inferred that the turbulent meanders of the jet must effect a net meridional transport of relative vorticity.

A one-dimensional steady solution of the equations of motion of a cold plasma in a magnetic field is obtained. The plasma is of semi-infinite extent, bounded by a plane interface which separates it from a vacuum or medium at rest. The particles approach from infinity, are reflected at the front, and return to infinity in the opposite direction. At infinity, the magnetic field is parallel and anti-parallel to the plasma streams, and is inclined at an angle to the normal to the interface. The front is a current sheet across which the lines of force are bent, with the component of the magnetic field in the plane of the front changing direction. The inertia of the electrons is neglected, and the characteristic frequency associated with the front is the ion gyro-frequency.

The motions of single disks suspended in a liquid undergoing Couette flow have been studied in detail. The angular rotation and the axial spin of the disks were found to be in good agreement with the theory of Jeffery, provided the equivalent axis ratio re was used instead of the measured axis ratio r. It was found that non-interacting disks move in constant orbits without a tendency to drift into orbits corresponding to a minimum dissipation of energy. Discontinuous changes in orbit were observed to occur following two-body collisions.

The theoretical work reported herein makes a departure from the many previous analyses of the solitary wave which have treated the wave as an example of irrotational fluid motion. The present analysis is of more general scope in that it covers the whole category of examples in which the wave may propagate in either direction on a horizontal stream whose primary velocity distribution U(y) is an arbitrary function (i.e. there is no restriction on the extent of the variations of U(y)). An approximate form of the wave profile is found in general to be a sech2 {(x − ct)/b}, as it is according to previous theories applicable to the wave on a uniform stream, but the relationships amongst the wave amplitude a, the length scale b, and the two propagation velocities c (positive downstream and negative upstream) depend in complicated fashion on the form of U(y).

It is shown, with a high degree of certainty, that a general compressible laminar boundary layer can develop a singularity at a point of zero skin-friction only if the heat transfer at that point is zero.

An almost rigid rotation of a viscous fluid is produced by dividing the containing cylinder into two sections and rotating them at slightly different speeds. The fluid velocity can be separated into two parts, a swirl about the axis and a streaming motion in the axial planes. When the difference in the speeds of rotation of the two sections is small, the equations of motion can be linearized. The solution is found for large Reynolds numbers and provides an illustration of the way in which the conditions of geostrophic flow (no velocity variation in the axial direction and an inability to insist on undistrubed flow at infinity) are approached as the Reynolds number tends to infinity.

This paper gives a direct derivation of some results obtained by Longuet-Higgins & Stewart (1960, 1961) on the amplitude variation of waves propagating on a non-uniform stream. The derivation raises a number of subtle points in the use of the equations of conservation of mass, momentum and energy. These points are of general interest and are discussed in detail with other applications, since they do not seem to have been pointed out previously.

The problem considered is that of the two-dimensional motion of the fluid in a cylinder of finite radius after the outer portion of the fluid is given an initial uniform velocity. The primary purpose of the investigation is the study of the changes in the energy distribution in the fluid as the initial motion decays. The appropriate flow equations are developed and then approximated by finite-difference equations. Numerical solutions of these equations are presented, and the energy-transfer processes are discussed in some detail. During the early stages of the flow, it is found that the spatial distribution of energy depends strongly on the Prandtl number. During the later stages, however, there is a net outward flow of energy for the case of a liquid and a net inward flow for a gas.