The Navier-Stokes vorticity equation is solved numerically for the circulation induced in a vertical plane, by a constant stress acting on a liquid, enclosed in a basin of uniform depth and vertical sides.

Solutions of the linearized vorticity equation are obtained for all Reynolds numbers (τsD2/4ρν2 where νs is the surface stress, ρ is the density, ν is the kinematic viscosity, and D is the depth of the liquid) and solutions of the complete vorticity equation for Reynolds numbers 0–400.

The notable feature of the solutions is the totally different end circulations. At the upwind end the flow becomes very slack, and the vorticity equation has a boundary-layer limit, while at the downwind end a damped wave occurs and the equation has an inviscid limit.

At Reynolds numbers between 400 and 600, the streamlines at the downwind end lead to a condition of hydrodynamic instability, in approximate agreement with some experimental observations by G. H. Keulegan.

A method is proposed for obtaining regular solutions of the boundary-layer equations near a point of vanishing skin friction for steady incompressible laminar two-dimensional flow. The boundary-layer equations are integrated numerically in the usual way until the separation point is approached; then the displacement thickness is prescribed as a regular function of the distance along the surface, leaving the pressure gradient to be calculated from the consequent solution. Numerical solutions are obtained with reversed flow and shallow separation bubbles inside the boundary layer without the occurrence of a singularity at the separation point.

Experimental evaluation of the radial momentum equation near the surface for an axisymmetrie turbulent wall jet is reported. The equation for flow contains six terms of magnitudes which cannot be neglected. The pressure gradient across the flow as well as along the flow is found to be of major importance. Thus, for a wall jet flow it is impossible to treat the radial and vertical momentum equations as independent of one another.

The dynamics of rotating fluids was, in the main, developed by methods special to the field, using the equationsof motion of a fluid inarotatingframe of reference. It is, nevertheless, possible to derive all the leading results from the classical principles of fluid dynamics in non-rotating frames; specifically, from the rules governing rate of change of vorticity. Although writers on the subject have adopted this approach increasingly often in recent years, the author believes that a broad survey of the field, deriving results from those classical rules concerning vorticity, has not previously been given and may be of some interest to fluid dynamicists in general.

The present survey was read to the IUTAM Symposium on Rotating Fluid Systems at La Jolla, California, on 28 March 1966. It states briefly (§2) the rules governing rate of change of vorticity, and then applies them, first, to problems of steady relative motion of rotating fluids; in particular, of the atmosphere (§3), of rotating fluids in the laboratory (§4), and of the oceans (§5). Waves and wavy movements are then studied, first (§6) for systems with constant Coriolis parameter, including inertial waves, surface waves, ‘long waves’ and internal waves, and, secondly (§7)) for systems with variable Coriolis parameter, including Rossby-Haurwitz waves with and without the influence of tidal effects, as well as problems of barotropic and baroclinic instability. Vorticity principles are used as the sole theoretical tool throughout the survey.

Periodic oscillations are found in a one-dimensional model of thermal convection. The model consists of a fluid-filled tube bent into rectangular shape and standing in a vertical plane. The fluid is heated at the centre of the lower horizontal segment and cooled at the centre of the upper horizontal segment. When a certain parameter exceeds unity, a periodic motion of the fluid is found in which the flow is always in the same direction but in which the speed varies. Inertia is unimportant for this oscillation, which depends upon the interplay between frictional and buoyancy forces.

Experimental evidence suggests that it may be profitable to separate the discussion of the high-frequency components of jet noise from low-frequency components. Theory then leads one to the recognition that the physical mechanism of noise generation is slightly different for the two components and for this reason one may speak of high-frequency ‘self noise’ and low-frequency ‘shear noise’ (Lilley 1958; Ribner 1964).

Both theory and experiment indicate that mean-flow refraction effects on self noise are appreciable. Using geometrical acoustics a description of the far-field radiation pattern of the high-frequency end of the jet-noise spectrum is obtained, in good qualitative agreement with observations. A conservation law of acoustic energy, applying between the frame of reference in which the sound sources move, and the fixed frame in which the observations are carried out, results in the absence of the convection amplification effect (over and above the U8 law) for total sound power at high frequencies, a result which helps explain the uniform validity of the U8 law.

An analysis of the shear-noise source term suggests that this part of the sound radiation is due to a combination of quadrupoles which have at least one axis parallel to the jet. This then explains the observed concentration of low-frequency noise around the jet axis.

The flow field in the nose region of a blunt body in hypersonic flow is studied by considering the transport of vorticity and enthalpy. The entire region between body and shock is considered to be viscous, not necessarily thin in comparison with the nose radius of the body and to be of slowly varying density. The (given) post-shock vorticity need not be small and the density ratio ρ∞/ρs may either be small or near unity, the analysis being valid asymptotically at both limits.

It is found that the vorticity equation may be uncoupled from the total enthalpy equation if μ√ρ is constant. While the equations are not expected to be necessarily restricted to the immediate vicinity of the stagnation line, only there can the solution be written down explicitly; elsewhere, numerical integration is required.

The paper deals with the refraction of a plane shock wave by an interface between two different gases. It is shown that the equations of motion can be reduced to a single polynomial equation of degree 12. Detailed numerical results are presented for the air-CH4 and the air-CO2 interfaces, which are respectively ‘slowfast’ and ‘fast–slow’ combinations. When the results are compared with experiment good agreement is obtained. The numerical data are multi-valued, but it is found that it is always the weakest solution that agrees with the experimental data. The multiple roots of the equation are often found to be associated with the appearance of degenerate and irregular wave systems and some attempt is made to analyse and discuss these systems.

A technique has been developed to measure the instantaneous shear stress at the boundary over which a liquid is flowing. It is being used to study turbulence in the immediate vicinity of a pipe wall. A reaction is conducted on an electrode mounted flush with a solid wall at high enough voltages to reduce the concentration of the reacting species to zero at the surface. Under these conditions, the rate of reaction is controlled by the rate of mass transfer. The electrode is analogous to a constant-temperature hot-wire anemometer in that the surface concentration is kept constant and the current flowing in the circuit is related to the surface shear stress. For fully developed turbulence the limiting velocity intensity, based on the local average velocity, is 0·32. Some of the velocity fluctuations are as large as the local average velocity and their distribution is nearly symmetric about the average. The ratio of the longitudinal to circumferential scale is about 30 : 1, because the circumferential scale is very small. There is some indication that close to a wall the velocity fluctuations in the circumferential direction are much smaller than the longitudinal fluctuations.