A field of homogeneous turbulence generated at an initial instant by a distribution of random impulsive forces is considered. The statistical properties of the forces are assumed to be such that the integral moments of the cumulants of the force system all exist. The motion generated has the property that at the initial instant \[ E(\kappa) = C\kappa^2 + o(\kappa^2), \] where E(k) is the energy spectrum function, k is the wave-number magnitude, and C is a positive number which is not in general zero. The corresponding forms of the velocity covariance spectral tensor and correlation tensor are determined. It is found that the terms in the velocity covariance Rij(r) are O(r−3) for large values of the separation magnitude r.

An argument based on the conservation of momentum is used to show that C is a dynamical invariant and that the forms of the velocity covariance at large separation and the spectral tensor at small wave number are likewise invariant. For isotropic turbulence, the Loitsianski integral diverges but the integral \[ \int_0^{\infty} r^2R(r)dr = \frac{1}{2}\pi C \] exists and is invariant.

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

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.

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.

Various methods of establishing fluid media stably stratified in either temperature or the concentration of a foreign constituent are compared. A method used to construct a salt solution with precisely predetermined stratification is described. The Euler-Lagrange equations are solved exactly for a medium whose refractive index varies linearly in one dimension; the solution shows that a schlieren system may be used. For media with non-linear one-dimensional variation of refractive index, a modified shadowgraph technique may be employed. A device for the quantitative evaluation of a one-dimensional stratification is outlined and an example of its use is given.

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.

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.

Laboratory and numerical experiments on non-steady convective flows in a porous medium are reported. The main objective is to note the detailed comparison found between the time-dependent solutions and the time-like development of the iterative solutions of the steady equations originally pointed out by Garabedian (1956) and others.

Two flows are chosen for study. The first is the flow which develops when a blob of hot fluid is released at the base of a porous slab. The second is the flow which develops when a portion of the base of a porous slab is suddenly heated. The former flow is very simple and ideally suited for establishing the numerical scheme. The latter flow, however, produces several unexpected features. The gross features of the time development, when the motion is strongly non-linear, show an alternation between periods of slow gradual adjustment and periods of rapid change.

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.