A verification of some of the predictions of the theory of Phillips (1955) is presented, and the relation between one-dimensional and two-dimensional wavenumber spectra is discussed. The convection velocity of the irrotational field deduced from measurements of the frequency spectrum alone appears to be about 0.9U1 in the frequency range carrying most of the energy. It follows that the pressure-fluctuation spectrum is proportional to the velocity-fluctuation spectrum and varies as ω2 at low frequency. The discrepancy between this result and measurements of wall-pressure spectra is plausibly attributed to extraneous sound.

In the flow-establishment region of an air jet issuing with an efflux velocity of about 35 ft./sec from a 1.0 ft. diameter nozzle into still air, measurements were made of mean axial and radial velocities, mean static pressure, turbulence intensities, turbulent shear, and pressure fluctuation. For the measurement of the latter a pressure probe using a ceramic piezo-electric tube was developed. Also included in the measurements were the temporal mean gradient and autocorrelation of the axial-velocity fluctuation and the intermittency factor. The fluctuating-pressure and turbulence-intensity fields were observed to be closely similar in form. Through use of the measured distributions of mean-flow and turbulence characteristics, all terms of the integral and differential forms of the momentum and mean-energy equations were evaluated throughout the region. The results are presented herein by curves of variation of each of the terms as they appear in the corresponding equations.

The Orr-Sommerfeld stability problem has been studied for velocity profiles appropriate to turbulent channel flow. The intent was to provide an evaluation of Malkus's theory that the flow assumes a state of maximum dissipation, subject to certain constraints, one of which is that the mean velocity profile is marginally stable. Dissipation rates and neutral stability curves were obtained for a representative two-parameter family of velocity profiles. Those in agreement with experimental profiles were found to be stable; the marginally stable profile of greatest dissipation was not in good agreement with experiments. An explanation for the apparent success of Malkus's theory is offered.

The wake behind a cone in incompressible flow has the form of a closed bubble. Measurements of velocity, turbulence and static pressure for various cone angles show that the wakes are all essentially similar. A wake Strouhal number may be defined, which is the same for all the models. A disk may be treated as a cone of 180° vertex angle.

A linear theory is developed for the motion of a viscous, incompressible fluid in a rotating cylinder with a sloping bottom.

An analysis of the normal modes of oscillation reveals that the presence of the bottom slope introduces a new set of low frequency inertial oscillations to replace the purely geostrophic modes which are not allowed in this geometry. The new waves possess mean circulation and are the mechanism by which the fluid adjusts to changes in the rotation rate of the container, a process discussed in detail.

The steady motion produced in the cylinder when the cylinder's upper surface rotates at a different rate than the bottom surface is studied. It is shown that the presence of the bottom slope inhibits the steady fluid motion in the body of the cylinder and introduces a non-symmetric, high velocity side wall boundary layer.

Experimental evidence, presented to validate the theory, reproduces certain important features of the oceanic circulation.

A 17 in. diameter shock tube was coupled to a 1 in. tube by a 10° half-angle conical convergence. Timing measurements showed that the shock waves emerged from the convergence at 2–3 times their entrance speeds, and then decelerated downstream. After the application of a viscous correction, the shock speeds at the convergence exit agreed to within 5% with equilibrium real-gas calculations of the model of Chester (1954), Chisnell (1957) and Whitham (1958). The downstream deceleration, due to viscosity and to higher-order interactions not included in the theory, was also briefly studied.

Theoretical solutions based on the expansion scheme for large x and large M∞, as proposed by Freeman (1962), are obtained for the asymptotic inviscid flow over plane bodies of the shape y/d = (x/d)m in the range $\frac{2}{3}/\gamma < m < \frac{2}{3}$ where blast wave theory applies as a first approximation. In particular, the second-order terms, which are necessary to satisfy the body boundary conditions for the normal velocity are computed. The magnitude of the second-order terms is found to increase from zero at $m=\frac{2}{3}/\gamma $ to infinity at $m = \frac{2}{3}.$

As a comparison with theory, experiments at M∞ = 8·2 were made with two plane power-law bodies in the range $\frac{2}{3}/\gamma < m < \frac{2}{3} $ and on a plane parabola with a tangent wedge nose. These consisted of the determination of shock-wave shapes, surface pressure distributions and detailed investigations of the distribution of pitot and static pressure across the shock layer.

The experimental results are in good agreement with the theory in the case m =1/2, where the second-order effects are small. At m = 5/8 the region of validity of the theory is limited to much larger distances from the nose of the body and larger Mach numbers. Accordingly, the prediction for the deviation from firstorder theory, although being correct in sign, is too small. Shock-wave shapes on bodies of the same power but of different size are correlated by the similarity theory when scaled with respect to the dimension d.

The experimental results obtained with the wedge-parabola are in very good agreement with a characteristics solution by C. H. Lewis (1965, unpublished).

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

The wakes formed behind sharp-edged, bluff, elliptical bodies have elliptical cross-sections, but the major axis of the wake is aligned with the minor axis of the body. This effect was observed in both mean velocity and turbulent intensity data in wakes throughout the range of the experiment, from several body diameters to distances of 250 diameters downstream of the body. The turbulent energy in the wake flow near the body displayed a periodicity which was correlated using a Strouhal number. Over the Reynolds-number range from 8 × 103 to 7 × 104, the Strouhal number depended only on the body eccentricity.

The boundary conditions at the surface of a small bubble rising in a liquid are examined theoretically, and it is shown by order-of-magnitude arguments, which are confirmed by detailed calculation in a special case, that although surfacetension gradients must always exist around the bubble, they are too small to affect the motion appreciably unless surface-active substances are present. This is because gradients of surface tension imply gradients of temperature in a pure liquid, and these turn out to be always small near the bubble if they tend to zero at large distances from it. The same is true for drops of one fluid in another.

Using simple flow-visualization techniques, it has been shown that as a large volume of air rises through a quiescent liquid it produces a well-defined wake and that the drag on the bubble appears as a momentum defect within this wake.

The second European Mechanics Colloquium, on the subject of three-dimensional boundary layers, was held at Liverpool University from 4 to 7 January 1966 and was attended by thirty-eight people closely associated with current work in this field.

The meeting was opened by an introductory review by J. C. Cooke, who successfully sought to provoke discussion by emphasizing areas of apparent agreement and disagreement. The discussions which followed were based on a series of contributions by participants in the Colloquium, and covered the following topics:

1. Laminar boundary layers.

2. Three-dimensional perturbations of two-dimensional turbulent boundary layers.

3. Corner and secondary flows.

4. Boundary layers associated with flow past obstacles.

5. Flow over delta wings.

6. Separation.

7. Flow over rotating surfaces.

8. Heat and mass transfer.

9. Miscellaneous topics.

The results of an experimental investigation into the development of a turbulent plane jet issuing into a parallel moving airstream are described. On the basis of a simple dimensional argument, it is shown that the results for the spread of jets with different ratios of jet nozzle to free-stream velocity can be collapsed into a single universal curve provided the effective origins of the various sets of data can be shifted. Evidence is found of a change in structure of the jet from a self-preserving plane jet flow near the origin of the flow towards a selfpreserving wake type of flow far downstream from the origin. This change of structure is compared with a prediction based on a simple application of Town-send's large-eddy hypothesis. It is shown that the hypothesis does not account for the way in which the jet structure changes and possible reasons for this are briefly discussed. Finally, some comments are made on the usefulness of the various theories of jet spreading.

The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with the more usual expression formed from kinetic minus potential energy.

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whether kh0 is less than or greater than 1.36, where k is the wave-number per 2π and h0 is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable if kh0 > 1·36, The instability of deep-water waves, kh0 > 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.