European Mechanics Colloquium number 17 was held at Cambridge from 1 to 3 July 1970, when the subject of bluff bodies and vortex shedding was discussed. The following report summarizes some of the papers presented. No formal proceedings of the meeting will be published.

Drag reduction caused by dilute, distilled water solutions of four polyethylene-oxides and one polyacrylamide, molecular weights respectively 0·1 × 106 to 8 × 106 and 13 × 106, was studied experimentally in one smooth and three sand-roughened pipes, relative roughnesses (R/k) 15, 23, 35, all of about 0·34, in inside diameter. The onset of drag reduction in the rough pipes occurred at the same wall shear stress as in smooth; the onset wall shear stress was essentially independent of polymer concentration, varied inversely as the square of polymer radius of gyration and was unaffected by the flow régime, hydraulically smooth, transitional or fully rough, during which onset occurred. Following onset a flow régime was observed wherein the fractional slip, i.e. fractional increase in mean velocity relative to solvent at a given friction velocity, obtained with a given polymer solution in a rough pipe was the same as the fractional slip in the smooth pipe despite marked differences in the respective rough and smooth friction factors. This ‘effectively smooth’ régime prevailed for values of non-dimensional roughness k+* < k+ < k+es from onset, k+*, to an upper limit given by k+es ∼ 50 for all of the present experiments. For k+ < k+es, the fractional slip in the rough pipes was always less than that corresponding to smooth and was a function of relative roughness as well as flow and polymeric parameters. The maximum drag reduction possible in the rough pipes was limited by an asymptote which was independent of polymeric parameters. Under asymptotic conditions, friction factors in all the rough pipes identically obeyed the smooth pipe friction factor relation for k+ < 12; the onset of roughness at k+ ∼ 12 indicated that the maximum viscous sublayer thickness attained during drag reduction was approximately 2½ times Newtonian.

An analytical solution is developed to describe the steady, closed streamline velocity field within a cylindrical cavity with a uniformly translating wall at low Reynolds numbers. The solution has application for the case of two-phase flow in a tube where regions of fluid are segmented by a moving train of bubbles or plugs, such as in the pulmonary and peripheral capillaries of the body where segments of plasma are trapped between red blood cells. The mathematical approach presented in this study can in principle be useful in the analysis of a wide class of closed-streamline creeping-flow problems.

An analytical solution is developed to describe the unsteady-state heat transfer to a cylindrical cavity with circulating flow induced by a moving wall. The previously derived velocity field at low Reynolds numbers is incorporated into the energy equation, and the case of heat transfer to a fluid segmented by highly conducting plugs flowing in a tube with constant wall temperature is considered. Calculations of temperature distributions, average temperatures, and heat transfer coefficients as functions of time and Péclet number are presented for a specific cavity geometry, and the degree of enhancement in heat transfer caused by the recirculating flow is determined.

The methods developed in this study may be useful in obtaining analytical solutions to a variety of closed-streamline heat and mass transfer problems with known velocity fields.

In this, the first part of a two-part experimental study of the behaviour of impinging jets, the mean properties of the flow field are established. Velocity profiles are given for three types of jet flow issuing from a circular convergent nozzle. Measured distributions of surface pressure are given which result when the jets impinge both normally and obliquely at various distances on several surface shapes. The pressure distributions are used to compute the radial velocity gradient at the impingement stagnation point. It is found that for normal impingement this gradient correlates with the free jet centreline velocity and half-radius at the same axial location. A fall-off in the correlated value is noted as the impingement is made oblique. Measurements of the azimuthal distribution of momentum flux in the resulting wall jet are also given. The general behaviour of all three types of jet is found to be similar at locations downstream of any local effects due to the shock waves present in the under-expanded types. A special study of the close-range impingement of an under-expanded jet containing a normal shock disk reveals a region of separated flow surrounding the stagnation point. This condition results in a negative value of the radial velocity gradient at the centreline.

Two problems representative of the gravity flow of granular materials are considered in the context of a theory presented by Goodman (1970). The problems consist of steady fully-developed flow of a granular material down an inclined plane and between vertical parallel plates. It is shown that the dynamical behaviour of these materials is quite different from that of a viscous fluid. For the inclined flow problem, the normal stresses are not only unequal but vary non-linearly with depth. Also the maximum value of the mass flux distribution does not necessarily occur at the upper surface. For the vertical channel-flow problem, the material behaves somewhat like a Bingham fluid in that a plug region exists in the central part of the channel. The interesting feature of this problem is that the concentration of material volume in the shearing region outside the plug may either increase or decrease from the plug to the channel wall, depending on the boundary conditions. Experimental evidence for these phenomena in real granular materials is cited.

The results of this investigation suggest that the gravity flow of granular materials is essentially governed by two factors-a material characteristic length, which is possibly related to the grain size, and the externally imposed constraints such as the gravity field or the pressure exerted upon the granular material from the confining plates.

An experiment with the pressure-driven flow down a long rotating channel is described. For zero rotation the flow is quasi-parabolic, laminar, and one-dimensional up the channel. With slight rotation Ω there is a weak double-vortex secondary circulation aligned with the channel. At intermediate Ω there exists an instability in the form of logitudinal rolls of non-dimensional wavenumber 5. The instability disappears at high rotation rates.

The general stability problem for a rotating zonal flow $\overline{U}(y)$ is considered theoretically. For perturbations independent of the co-ordinate in the direction of the flow, the problem is exactly analogous to the stability problem of a temperature-stratified fluid where the stratification $\overline{T}_z(z)$ corresponds to the quantity \[ (\partial\overline{U}/\partial y)(1/2\Omega)-1. \] This analogy extends to much more general mean fields (e.g. non-linear or time dependent) than does the oft-quoted analogy between thermal convection and cylindrical Couette flow. The instability theory is in qualitative agreement with the experiment.

The asymptotic stability of a rapidly rotating, horizontally bounded fluid, heated from below, is treated using boundary-layer methods. It is shown that the rotational constraint is so strong as to preclude instabilities, if the interior regions of the fluid are considered to be inviscid. The correct formulation allows this constraint to be broken by introducing horizontal diffusive effects into the interior, while vertical diffusion is confined to Ekman layers on the horizontal surfaces; no vertical layers exist. Moreover, the mechanism of instability is (to the lowest order) associated with energy conversions entirely within the interior region. The present formulation elucidates the role of the Ekman layers in producing high-order corrections to the limiting critical Rayleigh number, and the asymptotic results are extended to include higher-order terms. The effect of rigid side walls on the critical Rayleigh number, and on the azimuthal wavenumber, is considered. Except for very tall cylinders, the critical Rayleigh number is unaffected by the presence of side walls; the results for different azimuthal modes of convection are inconclusive, but indicate that no great error occurs if disturbances are assumed axisymmetric.

Wave-induced oscillations in harbours of constant depth but arbitrary shape in the horizontal plane connected to the open-sea are investigated both theoretically and experimentally. A theory termed the ‘arbitrary-shape harbour’ theory is developed. The solution of the Helmholtz equation is formulated as an integral equation which is then approximated by a matrix equation. The final solution is obtained by equating, at the harbour entrance, the wave amplitudes and their normal derivatives obtained from the solutions for the regions outside and inside the harbour. Special solutions using the method of separation of variables for the region inside circular and rectangular harbours are also obtained. Experiments were conducted to verify the theories. Four specific harbours were investigated: two circular harbours with 10° and 60° openings respectively, a rectangular harbour, and a model of the East and West Basins of Long Beach Harbour, California. In each case, the theoretical results agreed well with the experimental data.

Two-dimensional isotropic turbulence is investigated in its development from an arbitrarily specified initial flow through its transformation into a statistically self-preserving decaying flow. Numerical simulation is the principal method of investigation. The early development is characterized by rapid growth of the mean squared vorticity gradient, and this growth is found to be predicted satisfactorily by the quasi-normal hypothesis. During the later states of decay the numerical results are found to be generally consistent with the predictions by Kraichnan, Leith and Batchelor of a k−3 inertial range spectrum. The dimensionless constant of the spectrum is found to be near 2, about half the value found earlier for turbulence maintained by a constant forcing amplitude. The results are also consistent with Batchelor's predictions of the time-dependent behaviour of certain quadratic moments: An inconsistency in those predictions is pointed out, however, which can be resolved by altering the inertial range spectrum by a logarithmic term, as suggested by Kraichnan. The most important two-point Eulerian correlation functions are exhibited. An investigation is made of the Gaussianity of the flow with results indicating a strong tendency toward intermittency in the enstrophy dissipation.