A solution is presented for the flow field in and around a single spherical droplet or bubble moving axially at an arbitrary radial location, within a long circular tube. In the tube there is viscous fluid flowing with a constant Poiseuillian velocity distribution far from the droplet.

The settling velocity of the droplet or bubble is \begin{eqnarray*} U = \frac{2(\rho_i-\rho_e)ga^2}{9\mu_e}\frac{1+\alpha}{\frac{2}{3}+\alpha}\left[1-\frac{2+3\alpha}{3(1+\alpha)}\left(\frac{a}{R_0}\right)f\left(\frac{b}{R_0}\right)\right]+U_0\left[1-\left(\frac{b}{R_0}\right)^2\right.\\ \left. - \frac{2\alpha}{2+3\alpha}\left(\frac{a}{R_0}\right)^2\right] + O\left(\frac{a}{R_0}\right)^3. \end{eqnarray*} This is a general equation and it contains as special cases the familiar solutions of Stokes, Hadamard-Rybczynski, Brenner & Happel, Greenstein & Happel and Haberman & Sayre.

The function describing the deviation of the interface from sphericity is solved and an iterative procedure is suggested for obtaining higher order solutions.

A flat plate x = 0, |y| < L is initially at rest in an electrically conducting, inviscid, incompressible fluid permeated by a uniform magnetic field (B0, 0, 0). The plate is impulsively accelerated to a small velocity (— U, 0, 0) which is then kept constant. It is assumed that LV/λ 1 and U/V [Lt ] 1, where V is the Alfven velocity, and λ is the magnetic diffusivity.

Four stages in the development of the flow are distinguished, the last three being:

1. (ii) L [Lt ] Vt [Lt ] (λ)½. During this stage the initial potential flow is being disturbed by propagation of electric current and vorticity from the plate. The initial discontinuity on the plate has only propagated a small distance away compared to L, but a large distance compared to the length scale of diffusion (λt)½. Exact solutions to the flow are found in the neighbourhood of y = — L and y = 0 (x = 0).

2. (iii) Vt [Lt ] L [Lt ] (λt)½. The asymptotic behaviour of the electric current and vorticity on the plate are determined showing that a column of fluid of length Vt moves with the plate, aligned to the magnetic field. The transverse diffusion of the current sheets bounding the column accelerates the fluid in layers of thickness O(λt)½.

3. (iv) (λt)½ [Lt ] L. Unlike stages (ii) and (iii), where the motion is dominated by the propagation of vorticity and electric current as Alfven waves from the plate, diffusive mechanisms completely dominate the motion. ‘Slug’ flow is maintained. The nature of the flow, including the structure of the layers bounding the column are determined for |x| [Lt ] (λt)½.

This and the subsequent part of the Journal of Fluid Mechanics will be devoted to the publication of papers on turbulence. Each of these papers was the subject of an oral report at a Symposium on Turbulence sponsored by the Boeing Company and organized by the Flight Sciences Laboratory of the Boeing Scientific Research Laboratories, a t Seattle, Washington, U.S.A., from 23 to 27 June 1969. The Organizing Committee consisted of Dr A. Goldburg, Prof. L. S. G. Kovasznay (Chairman), andDr Y.-H. Pao (Secretary). At this Symposium invited participants listened both to review lectures and to contributed papers, and tried by discussion to assess the present position in the study of turbulence.

Some preliminary results on the behaviour of controlled wave disturbances introduced artificially into turbulent channel flow are reported. Weak plane-wave disturbances are introduced by vibrating ribbons near each wall. The amplitude and relative phase of the streamwise component of the induced wave is educed from a hot wire signal, allowing the wave speed and attenuation characteristics and the wave shape to be traced downstream. The normal component and wave Reynolds stress have been inferred from these data. It appears that Orr–Sommerfeld theories attempted to date are inadequate for description of these waves.

The phenomenon of salt fingers has been investigated optically to determine the geometry of the cells as seen from above. When the fingers are short, the flow appears to be highly turbulent, though a dominant scale is evident. When the fingers are longer, a cellular structure is clear. This structure changes only slowly, apparently in response to disturbances in the convecting layers which bound the fingers above and below, and becomes more nearly stationary as the fingers grow. Cell boundaries show a strong tendency to intersect at right angles, which favours the emergence of cells with a square horizontal section. As the fingers get longer the cell width increases, but more slowly than the length.

Apart from the well known pictorial studies of eddies in a stream by Leonardo da Vinci, it seems to me that other engineers were the first to make any progress in the study of turbulence. Except for Osborne Reynolds (1883) they limited their work to empirical formulae giving the results of hydraulic experiments in which they measured mean flow distributions resulting from turbulence. Osborne Reynolds seems to have been the first to show that with given boundaries the flow can be steady up to a certain speed and then suddenly become turbulent. This was a great advance and contributed to the urge by Orr (1907), Rayleigh (1915) and many others to study mathematically the stability of steady flow reacute;gimes. Meanwhile the meteorologists were attracted to the field. Arthur Schuster, a wealthy man who was professor of physics at Manchester University, wanted to encourage a mathematician to study meteorology and try to introduce mathematical reasoning into what seemed almost pure empiricism. He set aside provision for a readership in dynamical meteorology. I was appointed in 1911.

An experimental study of the interaction of a turbulent boundary layer with a wavy wall was conducted in a wind tunnel. A smooth neoprene rubber sheet comprising a portion of the floor of the tunnel was mechanically deformed into 12 sinusoidal waves which progressed upwind or down at controlled speed. The turbulent layer thickness was a little less than the wavelength. The mean velocity profile was linear on a semi-log plot over a substantial range of vertical height.

The wall pressure was observed to be asymmetrical about the wave profile, resulting in a pressure drag. Flow separation was not the cause of the drag. The drag was found to be larger than that predicted by the inviscid wave generation theory. The measurements indicate that the waves strongly modulate the turbulent structure. The phase of the turbulent stresses with respect to the waves varies with wave speed, indicating that the dynamical reaction time of the turbulence is not negligible in comparison with the wave period.

The conditions under which stationary waves may exist on a stream of water of finite depth are investigated theoretically for the case of a current which is uniform except for a constant defect in velocity in a region near the free surface. The analysis is extended to provide a two-dimensional theory for the surface profile induced by a simplified model of a hovering craft. The relevance of this work to the use of high speed flumes is discussed, and an example demonstrates the importance of the velocity distribution near the free surface.

Results for three interrelated problems are obtained by making use of solutions of boundary-value problems obtained in a different context. The first one concerns a thin rigid circular disk rotating in a slow stream of viscous fluid, both when the fluid is conducting and when it is non-conducting. For the case of a conducting fluid formulae are given for both small and large Hartmann numbers. The second problem concerns a disk performing simple harmonic rotary oscillations about its axis of symmetry in a non-conducting viscous fluid which is at rest at infinity. The last problem is that of an arbitrary axisymmetric solid oscillating about its axis of symmetry in a bounded viscous fluid, and the solution is illustrated by the case of an oscillating disk.

Although the entrainment of non-turbulent fluid into a turbulent flow occurs across sharply defined boundaries, its rate is not determined solely by the turbulent motion adjacent to the interface but depends on overall properties of the flow, in particular, on those that control the energy balance. In the first place, attention is directed to the many observations which show that the motion in many turbulent shear flows has a structure closely resembling that produced by a rapid, finite, plane shearing of initially isotropic turbulence. The basic reasons for the similarity are the stability and permanence of turbulent eddies and the finite distortions undergone by fluid parcels in free turbulent flows. Next, the existence of eddy similarity and the condition of overall balance of energy are used to account for the variation of entrainment rates within groups of broadly similar flows, in particular mixing layers between streams of different velocities and wall jets on curved surfaces. For some flows which satisfy the ordinary conditions for self-preserving development, no entrainment rate is consistent with the energy balance and self-preserving development is not possible. Examples are the axisymmetric, small-deficit wake and the distorted wake. Finally, the implications of an entrainment rate controlled by the general motion are discussed. It is concluded that the relatively rapid entrainment in a plane wake depends on an active instability of the interface, not present in a constant-pressure boundary layer whose slow rate of entrainment is from ‘passive’ distortion of the bounding surface by eddies of the main turbulent motion. Available observations tend to support this conclusion.

A simple model is suggested to describe flow in a Hele-Shaw cell when the Hele-Shaw parameter Λ is not necessarily small. The averaged flow is potential with a conservative body force proportional to the local velocity. The elementary ramifications of this are deduced and comparisons made with experiment. In particular no separation is predicted if Λ is less than O(1), in agreement with experiment. Furthermore, the separation cavities occurring for large Λ are completely stagnant. The theory predicts attached viscous shear layers in the wake of a lifting body, reminiscent of certain MHD problems. These tails were not observed experimentally.

The outer intermittent region of a fully developed turbulent boundary layer with zero pressure gradient was extensively explored in the hope of shedding some light on the shape and motion of the interface separating the turbulent and non-turbulent regions as well as on the nature of the related large-scale eddies within the turbulent regime. Novel measuring techniques were devised, such as conditional sampling and conditional averaging, and others were turned to new uses, such as reorganizing in map form the space-time auto- and cross-correlation data involving both the U and V velocity components as well as I, the intermittency function. On the basis of the new experimental results, a conceptual model for the development of the interface and for the entrainment of new fluid is proposed.

The two-dimensional incompressible mixing layer was investigated by using constant-temperature, linearized hot wire anemometers. The measurements were divided into three categories: (1) the conventional average measurements; (2) time-average measurements in the turbulent and the non-turbulent zones; (3) ensemble average measurements conditioned to a specific location of the interface. The turbulent energy balance was constructed twice, once using the conventional results and again using the turbulent zone results. Some differences emerged between the two sets of results. It appears that the mixing region can be divided into two regions, one on the high velocity side which resembles the outer part of a wake and the other on the low velocity side which resembles a jet. The binding turbulent–non-turbulent interfaces seem to move independently of each other. There is a strong connexion between the instantaneous location of the interface and the axial velocity profile. Indeed the well known exponential mean velocity profile never actually exists at any given instant. In spite of the complexity of the flow the simple concepts of eddy viscosity and eddy diffusivity appear to be valid within the turbulent zone.

Measurements of the turbulent pressure field on the outer surface of a 3 in. diameter cylinder aligned with the flow were made at a point approximately 24 ft. downstream of the origin of the turbulent boundary layer in an air stream of 145 ft./sec. The boundary-layer thickness was 2·78 in. and the Reynolds number based on momentum thickness was 2·62 × 104. The wall-pressure measurements were made with pressure transducers constructed from 0·06 in. diameter lead–zirconate–titanate disks mounted flush with the wall. The measurements including root-mean-square, power spectrum, and correlations of the wall pressure are compared with the existing experimental results for the turbulent pressure field beneath a plane boundary layer. The streamwise convection speed deduced from longitudinal space-time correlation measurements was almost identical to that obtained in the plane boundary layer. The rate of decay of the maxima of the space-time correlation of the pressure produced by the convected eddies was double that in a plane boundary layer. The longitudinal and transverse scales of the pressure correlation were approximately equal (in a plane boundary layer the transverse scale is larger than longitudinal scale) and were one-half or less than the longitudinal scale in the plane boundary layer. It is concluded that the effect of the transverse curvature of the wall is an overall reduction in size of pressure-producing eddies. The reduction in transverse scale of the larger eddies is greater than that of the smaller eddies. In general, the smaller eddies decay more rapidly and produce greater spectral densities at high frequencies owing to the unchanged convection speed.

This paper presents an experimental investigation of the conditions of instability and post-critical flow in an electrically driven shear layer which has been previously studied both theoretically and experimentally. The shear layer forms as a primarily azimuthal flow in an electrically conducting fluid between the edges of two parallel and coaxial circular electrodes, mounted in insulating planes, when a current is passed between them in the presence of a strong, axially applied magnetic field. The aximuthal velocity at any radial position in the layer is maximum near the electrode planes and decreases to zero at the centreplane between the electrodes. Radial and axial secondary velocity components apparently exist, though smaller in magnitude than the azimuthal velocity throughout most of the layer.

The experiments employ a miniature hot-film sensor as an instantaneous velocity indicator in mercury. The following experimental results are reported. \begin{eqnarray*} & R_c = 2420,\quad 150 \leqslant M \leqslant 350,\\ & R_c = 345\,M^{\frac{1}{3}},\quad 350\leqslant M \leqslant 650, \end{eqnarray*} where Rc is the critical Reynolds number of the shear layer, based upon the maximum velocity in the layer and the layer thickness, and M is the Hartmann number based upon the distance from the electrodes to the centreplane. For values of M below about 300, oscillograms of the hot-film signal show that the layer goes unstable in a chaotic fashion. For M ≥ 300, however, the instabilities grow to produce a wave-like post-critical flow of steady amplitude. The waves are apparently parallel to the applied magnetic field and drift in the azimuthal direction at a constant proportion (10–15%) of the critical velocity. The maximum intensity of the peak to peak fluctuations from the measured mean azimuthal velocity appears to be about 0·05. These fluctuations decrease to zero at the centreplane. By an approximate measurement of wavelength it is determined that the integral number of waves in the post-critical flow increases with M.

The physical mechanism governing the onset of instability has not been established. The manner in which hydrodynamic and electromagnetic forces interact with the destabilizing disturbances cannot be fully understood without further theoretical analysis.

In rotating flow moving axially through a straight tube, a helical vortex will be generated if the angular momentum flux is sufficiently large relative to the flux of linear momentum. This paper describes an experimental study of the occurrence, frequency and peak-to-peak amplitude of the wall pressure generated by this vortex. The experimental results are displayed in dimensionless form in terms of a Reynolds number, a momentum parameter and tube geometry.

A three-dimensional boundary layer developing along a semi-infinite swept stagnation line from a starting edge and evolving into that associated with such a line of infinite extent is calculated. A series solution useful for assessing the counteracting effects of cross-flow and mass transfer near the starting edge and for providing initial data for a subsequent streamwise, numerical solution is developed. The asymptotic behaviour far from the starting edge is examined and shown to involve only eigenfunction contributions associated with the far upstream flow. However, it is not presently possible to determine the relevant eigenvalues and eigenfunctions. Numerical solutions based on a difference-differential analysis yield the entire development of the boundary layer and indicate the streamwise length required for the case of the boundary layer at an infinite stagnation line to be obtained.

With a transverse array of channels of equal widths but differing resistances, we have generated an improved approximation to spatially homogeneous turbulent shear flow. The scales continue to grow with downstream distance, even in a region where the mean velocity gradient and one-point turbulence moments (component energies and shear stress) have attained essentially constant values. This implies asymptotic non-stationarity in the basic Eulerian frame convected with the mean flow, behaviour which seems to be inherent to homogeneous turbulent shear flow.

Two-point velocity correlations with space separation and with space-time separation yield characteristic departures from isotropy, including clear ‘upstream–downstream’ unsymmetries which cannot be classified simply as axis tilting of ellipse-like iso-correlation contours.

The high wave-number structure is roughly locally isotropic although the turbulence Reynolds number based on Taylor ‘microscale’ and r.m.s. turbulent velocity is only 130. Departures from isotropy in the turbulent velocity gradient moments are measurable.

The approximation to homogeneity permits direct estimation of all components of the turbulent pressure/velocity-gradient tensor, which accounts for inter-component energy transfer and helps to regulate the turbulent shear stress. It is found that its principal axes are aligned with those of the Reynolds stress tensor. Finally, the Rotta (1951, 1962) linear hypothesis for intercomponent energy transfer rate is roughly confirmed.

The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.

In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

This paper surveys the current state of analytical attempts at a theory of turbulence. The formulation of the problem in terms of moments is discussed. The difficulty posed by the closure problem is examined in detail using the quasinormal approximation as an example. The notion of dynamical relaxation by non-linear scrambling leads to the introduction of eddy relaxation times and the direct-interaction approximation. The properties of the direct-interaction approximation are indicated. Finally, a comparison is made between numerical solution of the equations of turbulence their and direct numerical simulation of the Navier–Stokes equations.