A study of the closed wakes behind large gas bubbles rising between two parallel plates reveals that they are elliptical in shape, rather than circular. Fitting an are of an ellipse, rather than a circle, to the curved upper surface of such a bubble eliminates a discrepancy between theory and experiment noted by Collins (1965a). Collins’ analysis of the effect of finite channel width is extended to the elliptical case, and the stability of the closed wake is examined.

The slow motion of a body in a viscous shearing field is examined. Variational principles are used to derive inequalities which approximate the elements of the shearing matrix M of a body of arbitrary shape, where M is the matrix relating the force, torque and stresslet exerted by the body on the fluid to the relative translational and rotational velocities of the body and the rate of deformation of the undisturbed linear field. An upper bound for the elements of M is obtained by showing that the quadratic form of M increases monotonically with B, the region occupied by the body, while a lower bound for this form is given in terms of the electrostatic properties of a conductor and a dielectric of the same shape as B. Particular attention is paid to bodies of revolution, for which certain more definitive results are obtained: for example, their resistance to a rotation with axial symmetry is always less than twice their resistance to a rotation perpendicular to their axis.

The rate of entrainment through the end of a plume or jet which impinges on a density interface has been determined in the laboratory, where the interface was produced by layers of fresh and salt water and the plume by a salt-water source. Observations of the impingement area indicated that entrainment was confined to a region about the size of the plume cross-section. It was thus concluded that the entrainment flux into the plume must be a function of the local width, velocity and buoyancy difference, and these can be combined into a single para- meter, the Froude number. Measurements of the volume flux showed it to be proportional to the cube of the Froude number. The flux of buoyancy from the salt-to fresh-water volumes is consequently proportional to the Froude number. In the second part of the study the density distribution in the initially fresh layer was derived and is verified by the experiments. This distribution has direct applications in the analysis of convective motions in the atmosphere and the ocean.

The stability of perturbed normal shock waves is considered. Shock perturbations depend directly upon the disturbances in the flow adjacent to the shock. In the present paper an initially stationary shock is assumed to be perturbed by acoustic waves reaching it from the downstream side. This case corresponds to the situation occurring in shock diffraction or reflexion. Two-dimensional problems of this type have been investigated previously, both analytically and experimentally. These previous analytic results have, in all cases, indicated that the perturbations of the shock decay with time as $t^{\frac{3}{2}}$, while experimentally both t½ and $t^{\frac{3}{2}}$ decays have been observed. It is demonstrated in the present investigation that, when waves are continuously generated at a point or points behind the shock, a t½ decay of the shock perturbations will occur, corresponding to the decay of the incident waves. However, when the source of waves is located only at the shock, as in a diffraction problem, $t^{-\frac{3}{2}}$ decay occurs owing to the cancellation, to lowest order, of the incident wave by its reflexion from the shock. These results explain the divergence between theory and experiment in this area, since the experiments giving the slower decay contained a source of waves behind the shock.

It is concluded that shock stability can only be considered in the context of the type of disturbances incident upon the shock.

A potential flow model of two-dimensional vortex shedding behind an inclined plate is developed. The free shear layers which emanate from the sides of the plate are represented by discrete vortices through the use of the appropriate complex-velocity potential, the Kutta condition and the Joukowsky transformation between a circle and the plate cross-section. The analysis is then applied to predict the kinematic and dynamic characteristics of the flow for various angles of attack. The results compare favourably with the available experimental data as far as the form of vortex shedding and the Strouhal number are concerned. The calculated normal-force coefficients are 20−25% yo larger than those measured by Fage & Johansen (1927).

Observations made in a well-developed, thermally stratified, horizontal, flat- plate boundary layer are used to study the effects of buoyancy on the mean flow and turbulence structure. These are represented in a similarity framework obtained from the concept of local equilibrium in a fully developed turbulent flow. Mean velocity and temperature profiles in both the inner and outer layers are strongly dependent on the thermal stratification, the former suggesting an increase in the thickness of the viscous sublayer with increasing stability. The coefficients of skin friction and heat transfer, on the other hand, decrease with increasing stability.

Normalized turbulent intensities, fluxes and their correlation coefficients also vary with buoyancy. In stable conditions, turbulence becomes rapidly suppressed with increasing stability as more and more energy has to be expended in over- coming buoyancy forces. The buoyancy effects are found to be more dominant in the stress budget than in the turbulent energy budget. The horizontal heat flux is much greater than the vertical heat flux and their ratio increases with stability. The ratio of the eddy diffusivities of heat and momentum, on the other hand, decreases with increasing stability. The spectra of velocity and temperature fluctuations indicate no buoyancy subrange, but the wavenumber corresponding to peak energy is found to increase with increasing stability.

This paper presents a numerical method for solving linearized water-wave problems with oscillatory time dependence. Specifically it considers the diffraction problem for oblique plane waves incident upon an infinitely long fixed cylinder on the free surface. The numerical method is based on a variational principle equivalent to the linearized boundary-value problem. Finite-element techniques are used to represent the velocity potential; and the variational principle is used to determine the unknown coefficients in the solution throughout the fluid domain. To illustrate this method, reflexion and transmission coefficients and the diffraction forces and moment are computed for oblique waves incident upon a vertical flat plate, a horizontal flat plate and rectangular cylinders, where the comparison is made with the existing results by others. Also considered is the associated sinuous forced-motion problem, where comparison is made with the results for a circular cylinder obtained by Bolton & Ursell (1973).

The stability of a baroclinic zonal current to symmetric perturbations on a meridionally unbounded f-plane is considered. The lower boundary is at rest but the upper one moves with a constant velocity in keeping with the velocity of the zonal current. Following Stone (1966) a horizontal length scale O(Ro) is taken, where Ro is the Rossby number, with the Richardson number Ri = O(1). Instability sets in when the wavelength is O(E1/3), where E is the Ekman number based on the distance between the rigid horizontal boundaries, which corresponds to Stone's inviscid value zero, and to McIntyre's (1970) value infinity on a length scale O(E½).

A nonlinear analysis about the point of onset of instability yields the result that for the monotonic mode zonal momentum is convected polewards. The possible implications of this result for the dynamics of Jupiter's atmosphere are discussed.

This paper presents a summary of the results of an extensive experimental investigation of the problem considered by Tatsumi (1952a, b) and more recently by Huang & Chen (1974a, b). The results, like the analyses, show that the linear instability is confined to the non-similar inlet region of the pipe.

The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate ε. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows.

Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.

Euromech 52, on drag reduction in turbulent flows due to additives, was held from 27-30 August 1974in Stockholm, Sweden, under the chairmanship of M. T. Landahl, and was intended to bring together specialists in the various aspects of the drag-reduction phenomenon to discuss current experimental and theoretical work, and to identify promising directions for future research. There were forty-seven participants from eleven countries and twenty-two papers were presented, with ample time for discussion. These papers, which are briefly summarized in this report, could be grouped roughly under the headings of §§ 2-6 below.

Some of the consequences of the conservation of magnetic helicity $\int \rm{a.b}\it{d}^{\rm{3}}\rm{r\qquad (a\; =\; vector\; potential\; of\; magnetic\; field\; b)}$ for incompressible three-dimensional turbulent MHD flows are investigated. Absolute equilibrium spectra for inviscid infinitely conducting flows truncated at lower and upper wavenumbers kmin and kmax are obtained. When the total magnetic helicity approaches an upper limit given by the total energy (kinetic plus magnetic) divided by kmin, the spectra of magnetic energy and helicity are strongly peaked near kmin; in addition, when the cross-correlations between the velocity and magnetic fields are small, the magnetic energy density near kmin greatly exceeds the kinetic energy density. Several arguments are presented in favour of the existence of inverse cascades of magnetic helicity towards small wavenumbers leading to the generation of large-scale magnetic energy.

Published results on transition in a Stokes layer indicate a wide range of transition Reynolds numbers. As thermal effects in a resonance tube (Merkli & Thomann 1975) depend on the state of the boundary layer, the transition Reynolds number was determined, and a critical Reynolds number Ac ≈ 400 was found. The observations were made with hot wires and with flow visualization by means of smoke, and provide new details on turbulence in a Stokes layer. With this knowledge an explanation of the large discrepancies between some stability theories and the experiments is suggested. The main point is that turbulence occurs in the form of periodic bursts which are followed by relaminarimtion in the same cycle and do not lead to turbulent flow during the whole cycle.

A further, unexpected result of the present investigation is the discovery of vortex patterns superimposed on the normal laminar acoustic motion.

Flow visualization by the schlieren technique in the neighbourhood of a fully developed cavity on two axisymmetric headforms has shown the existence of laminar boundary-layer separation upstream of cavitation separation, and the distance between the two separations to be strongly dependent on Reynolds number. Based on present results, a semi-empirical method is developed to predict the position of cavitation separation on a smooth body. The method applies only in the Reynolds-number range when the cavitating body possesses laminar boundary-layer separation under non-cavitating conditions. Calculated positions of cavitation separation on a sphere by the method show good agreement with experimentally observed positions.

Detailed experimental data are presented on the transition between regular and Mach reflexion. Data have been obtained for steady, pseudo-steady and un-steady flows, and include a study of the continuous and discontinuous transitions predicted by previous researchers. It is found that the criterion often used to calculate the transition condition is wrong in every case that we have investigated. In its place we propose an alternative criterion which has the property that the system remains always in mechanical equilibrium during transition.

Three-dimensional convection in a Boussinesq fluid confined between horizontal rigid boundaries is studied in a series of numerical experiments. Convection in air, whose Prandtl number Pr = 0·71, is systematically investigated, together with another model for Pr = 1. Convection with a steadily changing mean temperature is also considered. Two-dimensional rolls over the Rayleigh number range 4500 [les ] Ra [les ] 24000 and three-dimensional flow patterns over the range 26000 [les ] Ra ≤ 32000 are shown to be stable in air when the mean temperature of the layer is constant ($\partial \overline{T}/\partial t = \eta = 0$). Discrete changes in the slope of the heat-flux curve are shown to exist in the ranges \[ 7000\leqslant Ra\leqslant 8000,\quad 12000\leqslant Ra\leqslant 14000\quad{\rm and}\quad 24000\leqslant Ra\leqslant 26000 \] in air. Only the last discrete transition in the heat flux is asSociated with a significant transition in the flow pattern. Two-dimensional rolls with a horizontally asymmetric distribution of upward and downward motions over the range 4500 [les ] Ra [les ] 8000, and three-dimensional flow patterns over the range 10 000 [les ] Ra [les ] 20 000 are shown to be stable when the mean temperature varies with time. The circulation in a three-dimensional cell depends on the sign of the mean temperature change: downward motions occupy the centre of the cell when $\partial\overline{T}/\partial t > 0$, and upward motions when $\partial\overline{T}/\partial t < 0 $. Motions start to be time dependent for Ra > 20000. Transitions in the planform are asSociated with discrete changes in the slope of the heat-flux curve. Transitions in both the heat flux and flow pattern depend quantitatively on the Prandtl number.

The problem of obtaining a numerical solution for the steady flow between two coaxial infinite disks, one fixed and porous, the other rotating, is reduced by von Kámán's hypothesis to solution of a system of nonlinear equations. A Newton-type iteration results in several solutions to these equations, as a number of authors have already indicated. Nevertheless, an interval in which only one solution is found exists for small values of the Reynolds number based on the angular velocity of the rotating disk, the distance between the disks and the kinematic viscosity of the fluid. At large values of this Reynolds number, two solutions appear and have been the subject of intense controversy.

In this paper, both physical and numerical arguments are presented which support a Batchelor-type solution for the flow between infinite disks, in which part of the fluid rotates as a solid body. The other solution, following Stewartson, assumes that the velocity of the fluid outside the boundary layers is entirely axial. This only seems to be verified experimentally when the distance between the disks is large compared with the (finite) radius of the disks.

Measurements of the streamwise development of a variety of velocity correlations in a uniform shear flow are given and discussed. It is concluded that, in the region in which the Reynolds stresses are approximately constant, the turbulence structure is practically independent of all the initial conditions except the initial length scale. It appears that the preferential amplification of some eddy structures by the mean shear effectively destroys information about the initial conditions, apart from the initial length scale, after a total strain of about 1.5.

In this paper we apply the techniques of higher-order boundary-layer theory to study the steady streaming induced in the neighbourhood of a cylinder which vibrates harmonically, perpendicular to its generators, in an unbounded fluid. The theoretical predictions are compared with the results of experiments performed at high streaming Reynolds numbers. Improved agreement between theory and experiment is achieved although unresolved discrepancies remain.

A linear stability analysis is used to investigate the stability of rotational Couette flow of sbratified fluids. The linearized time-dependent perturbation equations are solved using explicit finite-difference approximations. Small random axisymmetric perturbations of a given wavelength are initially distributed in the flow field, and their development in time is obtained by numerical integration. It is found that the kinetic energy of the perturbations oscillates in time owing to the periodic transformation of the disturbance flow field from a one-vortex system to a two-vortex system and vice versa. The neutral condition is defined as the state in which the maxima of the perturbation kinetic energy curve no longer change in time. A neutral-stability curve is obtained using the experimentally observed critical wavelengths. It is in general agreement with the experimental data, and it confirms the experimental result that stable density stratification enhances stability.