The hydrodynamic forces and couples that act on two spherical particles in slow motion through a quiescent fluid are determined as functions of the relative configuration of the particles from the solution of the Stokes equation for the motion of the fluid in the vicinity of the particles. General formulae that relate the translational and rotational velocities of the particles to the ratios of their radii a = a2/a1 and net densities I are obtained, as are asymptotic forms for the velocities in the limiting cases of very large and very small interparticle separation. Relative trajectories of the particles when they move solely under gravity and their own interaction are calculated for several values of I and a. A particularly interesting feature of the results is that, for certain ranges of values of I and a, trajectories of finite length and trajectories having the form of closed periodic orbits may occur.

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes: a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → 0, are derived.

When homogeneous turbulence acts upon a non-uniform magnetic field B0(x, t), a mean electromotive force [Escr ](x, t) is in general established owing to the correlation between the velocity field and the fluctuating magnetic field that is generated. The relation between [Escr ] and B0 is linear, and it is generally believed that, when the scale of inhomogeneity of B0 is sufficiently large, it can be expressed as a series involving successively higher spatial derivatives of B0: \[ {\cal E}_i = \alpha_{il} B_{0l}+\beta_{ilm}\partial B_{0l}/\partial x_m+\ldots, \] where the tensors αil, βilm,… are determined (in principle) by the statistical properties of the turbulence and the magnetic diffusivity Λ of the fluid. These tensors are of crucial importance in turbulent dynamo theory. In this paper the question of their asymptotic form in the limit Λ → 0 is considered. By putting Λ = 0 and assuming that the field is non-random at an initial instant t = 0, the developing form of the tensors αl(t) and βm(t) is determined. The expressions involve time integrals of Lagrangian correlation functions associated with the velocity field, which are comparable in structure with the eddy diffusion tensor in the analogous turbulent diffusion problem (Taylor 1921). Some doubts are expressed concerning the convergence of the time integrals as t → ∞, and it is concluded that a satisfactory treatment of the problem will require the inclusion of weak diffusion effects (as recognized by Parker 1955).

Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6·983, 10 ≤ Gr 2 × 104 and 0·05 [les ] A [les ] 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A [lsim ] 0·1 and Gr2A3Pr2 [lsim ] 105. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr → ∞.

Hot-film anemometer measurements have been carried out in a fully developed turbulent channel flow. An oil channel with a thick viscous sublayer was used, which permitted measurements very close to the wall. In the viscous sublayer between y+ ≃ 0·1 and y+ = 5, the streamwise velocity fluctuations decreased at a higher rate than the mean velocity; in the region y+ [lsim ] 0·1, these fluctuations vanished at the same rate as the mean velocity.

The streamwise velocity fluctuations u observed in the viscous sublayer and the fluctuations (∂u/∂y)0 of the gradient at the wall were almost identical in form, but the fluctuations of the gradient at the wall were found to lag behind the velocity fluctuations with a lag time proportional to the distance from the wall. Probability density distributions of the streamwise velocity fluctuations were measured. Furthermore, measurements of the skewness and flatness factors made by Kreplin (1973) in the same flow channel are discussed. Measurements of the normal velocity fluctuations v at the wall and of the instantaneous Reynolds stress −ρuv were also made. Periods of quiescence in the − ρuv signal were observed in the viscous sublayer as well as very active periods where ratios of peak to mean values as high as 30:1 occurred.

This paper deals with the motion of a heavy particle in a turbulent flow in an open channel with a smooth bottom. For the case when the particle stays in suspension in the main body of the flow almost all the time, (a) the probability density function of the projection on a cross-sectional plane of the particle position at any instant, and (b) the mean velocity and longitudinal dispersion coefficient of particles are determined analytically by employing the Eulerian formulation and applying the Aris moment transformations. It is found that the mean particle velocity decreases and the longitudinal dispersion coefficient of particles increases with the fall velocity.

It is shown that the surface wind drift in the ocean substantially reduces the maximum wave height ξx and wave orbital velocity that can be attained before breaking. If q is the magnitude of the surface drift at the point where the wave profile crosses the mean water level and c is the wave speed, then \[ \zeta_{\max} = \frac{c^2}{2g}\bigg(1-\frac{q}{c}\bigg)^2. \] Incipient breaking in a steady wave train is characterized by the occurrence of stagnation points at wave crests, but not necessarily by discontinuities in slope. After breaking, there is in the mean flow a stagnation point relative to the wave profile near the crest of the broken wave, on one side of which the water tumbles forward and behind which it recedes more smoothly to the rear. Some simple flow visualization studies indicate the general extent of the wake behind the breaking region.

The flow normal to an infinite circular cylinder which is uniformly accelerated from rest in a viscous fluid is considered. The flow is assumed to remain symmetrical about the direction of motion of the cylinder. Two types of solution are presented. In the first an expansion in powers of the time from the start of the motion is given which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds numbers. Physical properties of the flow for small times and finite but large Reynolds numbers are calculated from this expansion. In the second method of solution the Navier-Stokes equations are integrated by an accurate procedure which is a logical extension of the solution in powers of the time. Results are obtained for R2 = 97·5, 5850, 122 × 103 and ∞, where R is the Reynolds number. This is defined as R = 2a(ab)½/v, where a is the radius of the cylinder, b the uniform acceleration and v the kinematic viscosity of the fluid. The methods are in good agreement for small times.

The numerical method of integration has been carried to moderate times and various flow properties have been calculated. The growth of the length of the separated wake behind the cylinder for R2 = 97·5, 5850 and 122 × 103 is compared with the results of recent experimental measurements. The agreement is only moderate for R2 = 97·5 but it improves greatly as R increases. The numerical integrations were continued in each case until the implicit method of integration failed to converge, which terminated the procedure. A secondary vortex appeared on the surface of the cylinder for the case R2 = 122 × 103.

In two earlier papers (Gallagher & Mercer 1962, 1964) the results for the first four eigenvalues of the problem of the stability of plane Couette flow were given. The first twelve eigenvalues have been calculated by the same method and the results show that the manner in which the eigenvalues join to form complex pairs depends on α, the wavenumber of the disturbance. They also would appear to indicate that mode-crossing always occurs.

The steady motion of water in an enclosed rectangular cavity with differentially heated vertical end walls was studied experimentally, and the results are compared with the findings of parts 1 and 2. The depth-to-length ratios of the cavities were 102 and 1·9 × 102, and the Rayleigh number was allowed to vary sufficiently to enable a study to be made of the transition from a flow driven by the vertical wall boundary layers to one sustained by a longitudinal temperature gradient in the central sections of the cavity.

An investigation is made of selective withdrawal of a linear stratified fluid from a line sink in a channel of depth d. The flow is characterized by a densimetric Froude number F = Q/Nd2, where Q is the discharge per unit width and N is the Väisälä frequency. The dynamics of establishment of flow are investigated theoretically. Analytic results are obtained from a linearized theory based on a systematic perturbation scheme for small values of F. These results lead to a proper identification of the successive arrival of ‘columnar disturbance modes’ as the mechanism responsible for the development of flow concentration in the withdrawal region. Viscous and diffusive effects are also examined. For larger times and higher discharges (higher values of F), nonlinear effects become important, and the full Navier-Stokes equations are now solved numerically by a finite-difference procedure in a ‘stretched’ co-ordinate system. The solutions indicate that the establishment of the steady flow field is due to the successive arrival and interaction of ‘columnar disturbance modes’. Steady-state solutions are also presented, and a similarity profile is obtained. Comparison of the theoretical findings with experimental results are presented in part 2. Agreement with experimental measurements is found to be excellent.

The laminar boundary layer on a rotating thin blade of an axial turbomachine is discussed. A method of deriving the basic equations of the flow for a generalized blade configuration is described, treating a typical blade in practical use as an example. A perturbation technique is applied to obtain a series solution composed of two groups of similar functions. The results give insight into the properties of the boundary layer on helical non-loaded blades in a uniform stream, indicating the influence purely of three-dimensionality of the blade configuration.

Experiments were conducted to examine the development and establishment of the withdrawal current in the selective withdrawal of a stratified fluid from a line sink. The experiments were performed in a Plexiglas channel 30·5 ft long 14 in. wide, and filled with a linearly stratified salt solution to a depth of 18 in. The line sink was located at mid-depth. The flow was symmetric about the middepth. Velocity measurements and flow visualization were obtained by neutrally buoyant liquid droplets and vertical dye lines. Density measurements were made by a salinity probe. The development of the upstream velocity field was found by measurement to be brought about by the successive arrival of ‘columnar disturbance modes’, discussed in part 1. Agreement with the theoretical predictions is excellent. A similarity profile of the steady-state horizontal velocity is obtained, and the Froude number based on the thickness of the flowing zone is found to approach a constant value in the essentially inviscid region. The results are compared with field data from the Tennessee Valley Authority (TVA) reservoirs.

Assuming information about the mean velocity and vertical turbulent velocity, it is possible to calculate the flow direction wavenumber spectrum of pressure fluctuations ϕ(k1 δ)/τ02δ. The law of the wall plus Cole's wake function represented the mean velocity profiles. A scale-anisotropic model of R22 was used and the component intensity û2 was assumed to vary across the boundary layer in constant proportionality to the Reynolds stress. Calculated zero-pressure-gradient spectra rise as k11.5 at low wavenumbers. Curves for various Reynolds numbers are closely similar, and diverge only slightly around the peak in the spectrum. A high wavenumber spectrum ϕk1v/u*. u*/τ02v is independent of Reynolds number. The calculations reveal an overlap region in which ϕ ∼ k1−1. Imposing an equilibrium pressure gradient increases the spectrum at the low and mid wavenumbers, but has no effect in the overlap region. The spectrum peak for II = 6 is a factor 102 higher than for the zero-pressure-gradient layer. It is proposed that the convective velocity Uc(k1) has an overlap region. The overlap law is found to be \[ \frac{U_c}{u_{*}} = -\frac{1}{\kappa}\ln k_1\delta +\frac{1}{\kappa}\ln\frac{u_{*}\delta}{\nu}+A, \] where K and A are the same constants as in the mean velocity expression. Comparison with experiments shows very good agreement. A rough convective ‘wake’ function is formulated for the low-wavenumber range. Wavenumber spectra are converted to frequency spectra, and compared with experiments. Data from a zero pressure gradient and an adverse pressure gradient II = 3 show reasonable agreement with the calculations.

Attention is drawn to a principal difference between the transfer of a horizontal magnetic field by turbulence and by three-dimensional cell convection.

If the motion of the conducting medium in the cells is such that the heated material ascends at the centre while descending along the sides of the cells, then the magnetic tubes of force will be carried downwards by peripheral flows. Discrete ascending flows separated from one another by descending material carry only closed magnetic field loops. Such loops do not transfer net magnetic flux. As a result, the magnetic flux becomes blocked at the base of the convective layer.

Numerical solutions for laminar incompressible fluid flows past an abruptly started elliptic cylinder at 45° incidence are presented. Various finite-difference schemes for the stream-function/vorticity formulation are used and their merits briefly discussed. Almost steady-state solutions are obtained for Re = 15 and 30, whereas for Re = 200 a Kármán vortex street develops. The transient period from the start to the steady or quasi-steady state is investigated in terms of patterns of streamlines and lines of constant vorticity and drag, lift and moment coefficients.

The structure of density interfaces upon which turbulent motions have been imposed is investigated in the laboratory. Particular attention is paid to the profiles of density across such interfaces, and the temporal response of the interface to the imposed turbulence. Interfaces over a range of Péclet numbers and Richardson numbers are examined. It is found that the interface thickness h/l (non-dimensionalized with respect to the length scale l of the turbulence) is a function of the Péclet number but independent of the Richardson number. At low Péclet numbers (Pe [lsim ] 200) molecular diffusion is important in the determination of the interface structure and a diffusive core (across which all transport occurs by molecular diffusion) is formed in the centre of the interface. At higher Péclet numbers the interface structure appears to be determined by non-diffusive processes and h/l becomes approximately constant at a value of about 1·5. Some information concerning the intermittent nature of the mixing process at high Péclet number is obtained from records of salinity fluctuations measured at a fixed point. Finally, the implications of these data concerning the interpretations of the measurements of entrainment across an interface made by Turner (1968) are discussed.

The sublayer of a turbulent boundary-layer flow with and without polymer additives was measured by a simple optical technique having a high degree of spatial resolution. The reduction of skin friction with polymer additives was found to be due to a decrease in the effective viscosity of water at the interface between the mucus (glycoprotein) or polymer (polyethylene oxide) and the water and a thickening of the laminar sublayer. The rate at which mucus diffuses away from a medium-sized fish is almost constant for all swimming speeds.

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.