The 51st Euromech Colloquium, on mixing in stratified fluids, was held at the University of Cambridge from 24-27 June 1974, with one of the present authors (J.S.T.) acting as Chairman of the Organizing Committee. There were fifty-two participants from twelve countries, and thirty-two contributions were presented and discussed during the four days of the meeting.

Free convection has been studied in gaseous helium at low temperatures in a cylindrical vessel of aspect ratio (diameterlheight) 2·5. Compared with measurements in fluids at room temperature the present arrangement has the advantages of small size, a short time constant and improved accuracy. As the Rayleigh number was varied from 60 to 2 × 109, the Nusselt number rose from 1 to 69, obeying the relation Nu = 0·173Ra0·2800±0·0005 over the upper four decades of Ra. The critical Rayleigh number was 1630, but the conditions of the experiment did not allow reliable measurements at such low values of Ra. The very high sensitivity within a given experiment showed the presence of several ‘discrete transitions’, which were often step like and not merely a change of gradient as reported by other workers. The largest of these, at Ra = 3 · 105, involved a drop in heat flux of some 6% and was somewhat hysteretic. The temperature fluctuations increased markedly as the step was crossed.

First, the linear stability of the flow between two concentric cylinders when the outer one is a t rest and the inner has angular velocity Ω{1+ εcosωt} is considered. In the limit in which ε and ω tend to zero it is found that the critical Taylor number a t which instability first occurs is decreased by an amount of order ε2 from its unmodulated value, the stabilizing effect a t order ε2ω2 being slight. The limit in which ω tends to infinity with ε arbitrary is then studied. In this case it is found that the critical Taylor number is decreased by an amount of order ε2ω−3 from its unmodulated value.

Second, the effect of taking nonlinear terms into account is investigated. It is found that equilibrium perturbations of small but finite amplitude can exist under slightly supercritical conditions in both the high and low frequency limits. Some comparisons with experimental results are made, but these indicate that further theoretical work is needed for a broad band of values of ω. In appendix B it is shown how this can be done by an alternative formulation of the problem.

Stability characteristics of a single-phase free convection loop are presented. In the experiments, water was placed inside a toroidal glass loop oriented in a vertical plane. The lower half of the loop was heated and the upper half was cooled. At low heat-transfer rates and also a t high heat-transfer rates the free convection flow was observed to be steady. For the intermediate range, however, the flow was found to be highly oscillatory. Stability predictions are also developed. The comparison between theory and experiment yields favourable agreement.

Observations of unstable behaviour have been reported previously for single-phase fluids in the vicinity of the thermodynamic critical point. In these situations it has been assumed that the unusual behaviour of the fluid properties in the near-critical region necessarily constitutes the underlying cause of such instabilities. In contrast t o this view, analyses by Keller (1966) and Welander (1967) indicate that instabilities can occur for ordinary fluids as well. Results of the present study confirm this contention, since instabilities were clearly observed for water at atmospheric pressure and moderate temperatures.

This paper is concerned with the nonlinear stability of the flow between two long eccentric rotating cylinders. The problem, which is of interest in lubrication technology, is an extension both of the authors’ earlier work on the linear eccentric case and of still earlier work by Davey and others on the nonlinear concentric analysis of Taylor-vortex development. There are four parameters which are assumed small in the analysis; they are the mean clearance ratio, the eccentricity, the amount by which the Taylor number exceeds its critical value; and the Taylor-vortex amplitude. Following the earlier work mentioned above, relation-ships are specified between these parameters in order to develop a satisfactory perturbation scheme. Thus a non-local solution is obtained to the nonlinear stability problem, in which the whole flow field is taken into account.

Of some importance in the analysis is the fact that it is necessary to allow for the development of a pressure field substantially bigger than that associated with Taylor vortices in the concentric case, owing to the Reynolds lubrication effect in a viscous fluid moving through a converging passage. I n order to achieve this mathematically, it is necessary to solve the continuity equation to a higher order than is necessary for the momentum equations.

It is found that the angular position for maximum vortex activity, which is 90° downstream of the maximum gap in the linear case, can taken on any value between 0 and 90°, depending on the value of the supercritical Taylor number. For a particular experiment of Vohr (1968) acceptable agreement is obtained for this angle (50°), though the ‘small’ parameters are somewhat outside the expected range of perturbation theory. Formulae are obtained for the torque and forces acting on the inner cylinder.

The evolution of a small disturbance in a three-dimensional steady free-surface flow is investigated. The radius of curvature of the free surface and the length scale characterizing the non-uniformity of the velocity are assumed to be of the same order of magnitude. It is shown that the local rate of growth of the amplitude of the disturbance depends on both the normal pressure gradient (as in the case of Taylor instability) and the rate of strain on the free surface. Application of the theory to rising gaseous bubbles and gravity water waves is discussed.

Measurements have been made concerning the fine structure of the turbulence in the part adjacent to the wall of the wall region of a plane turbulent boundary layer. The objective was to gain further information concerning the larger-scale disturbance mechanism which is mainly responsible for the generation of turbulence. Hot-wire anemomet.ry was used and information on the fine structure was obtained by differentiating and filtering the hot-wire signal.

The distributions of the Kolmogorov microscale and of the flatness and skewness factors of the axial fluctuating velocity u and its first and second derivative determined at two Reynolds numbers suggest the existence of Reynolds number similarity. In the region y+ < 15 the flatness and skewness factors of u increase with decreasing y+. At approximately y+ = 15 the flatness factor shows a minimum value, while the skewness factor becomes zero. This location agrees with that where the turbulence intensity u′ has a maximum value. In the outer part of the wall region (y+ > 100) the flatness and skewness factors approach values obtained in shear-free turbulence at the same turbulence Reynolds number.

The fine structure of the turbulence is strongly associated with and dominated by the random, larger-scale, intermittent inrush-ejection cycle. In the viscous sublayer both the fine structure, and the large-scale mechanism of the turbulence are influenced mainly by the inrush phase, while further out in the wall region (y+ > 40) they are influenced by both inrush and ejection. As a result, in the viscous sublayer the average burst periods of the high frequency turbulence components and their flatness factors (of ∂u/∂t and of ∂2u/∂t2) attain values twice those in the outer part.

The change in the mechanism of the fine structure with distance from the wall is clearly demonstrated by the spectra of non-negative variables, i.e. (∂u/∂t)2 and (∂2u/∂t2)2. The spectra agree with each other and decrease with increasing frequency, following a power law as predicted by Gurvich & Yaglom (1967). The power law applies to almost the whole frequency range, when the highest, viscous, frequency range is excluded. However, the exponent is different for the viscous sublayer and the outer part of the wall region. In the buffer layer the spectra have two distinct power-law regions. In the lower frequency range the exponent is the same as that for the viscous sublayer, while in the higher frequency range it is the same as that for the outer part of the wall region.

A simple approximation is proposed for the problem of the dispersion of marked particles in an incompressible fluid in random motion when the probability distribution of the velocity field is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. Approximations for the Lagrangian velocity correlation function and the dispersion are given and compared with exact computer calculations of Kraichnan. Agreement is found to be good except for time-independent velocity fields and singular wavenumber spectral functions.

The equilibrium statistical mechanics of inviscid two-dimensional flow are re-examined both for a continuum truncated at a top wavenumber and for a system of discrete vortices. In both cases, there are negative-temperature equilibria for finite flows. But for spatially infinite flows, there are only positive-temperature equilibria, and both the continuum and discrete system exhibit proper, extensive, thermodynamic limits a t all realizable values of the energy and enstrophy density. The negative-temperature behaviours of the continuum and discrete system are semi-quantitatively the same, except for a supercondensation phenomenon in the discrete case a t the smallest realizable values of negative temperature. The supercondensed states have very large energy and in them all vortex cores of the same sign are clumped within an area small eompared with the mean area per vortex. The approach of the continuum system to absolute equilibrium by enstrophy cascade to high wavenumbers and energy cascade to low wavenumbers is examined. It is argued that the enstrophy cascade is closely analogous to distortion of a passive scalar field by straining of large spatial scale. This implies that high intermittency of spatial derivatives of the vorticity field can develop but that there is no associated change in the previously proposed log-corrected k−1 enstrophy spectrum law. On the other hand, intermittency build-up in the downward energy cascade can result in a change of the exponent in the energy spectrum law to a negative value of smaller magnitude than 5/3. Intermittency effects in the non-equilibrium energy cascade seem a more plausible explanation for vortex clumping observed in recent computer experiments than do the spatially smooth condensation phenomena associated with the negative-temperature absolute equilibria.

A secondary flow is set up when a fluid flows through a stationary curved pipe. The fluid in the middle of the pipe moves outwards and that near the wall inwards. Dean showed that the dynamical similarity of this fully developed flow depends on a non-dimensional parameter $D = 2(a/R)^{\frac{1}{2}}(a\overline{W}/\nu) $, where $\overline{W} $ is the mean velocity along the pipe, v is the coefficient of kinematic viscosity and a is the radius of the pipe, which is bent into a coil of radius R. Dean's analysis was limited to small values of D. Later, Barua developed an asymptotic boundary-layer theory for large values of D and showed for these values of D that the resistance coefficient γc is much larger than that for the corresponding straight pipe. The present work deals with the flow in a curved pipe as it develops from a uniformly distributed velocity at the entrance to a fully developed profile. Barua's results for the fully developed flow are adopted as downstream conditions in the present work. The ratio of the entry lengths of the curved ipe and the corresponding straight one is shown to be proportional to D−1/2 when D is large. Thus, the entry length for a curved pipe is much shorter than that for the corresponding straight pipe.

A small perturbation analysis is made of the flow inside a spherical explosion created in a gas absorbing energy from an array of lasers with a common focus. The influence of a small counterpressure outside the leading shock wave is investigated for power deposition constant in time. The departure from self-similarity is described by a power series in the variable y, the inverse square of a Mach number based on the wave-front velocity. To first order, the flow equations are solved numerically, the shock velocity is obtained and the radial profiles of velocity, density and pressure are presented graphically. The flow singularity at the radiation focus is discussed.