The potential problem for the flow at high Reynolds numbers R outside the boundary layer and wake of a thin flat plate at small incidence with allowance for displacement thickness is not fully denned unless the position of the wake is known in advance. The Kutta–Joukowski hypothesis does not provide a satisfactory first approximation to this because of the singularity in curvature of the streamline springing from the trailing edge in inviscid flow, which implies that the initial curvature of the wake in the real flow will be large enough to cause a modification to the potential flow. The net vorticity per unit length in a curved wake is approximately U∞ δ2∞dθ0/ds, where U∞, δ2∞ and dθ0/ds are respectively the undisturbed stream velocity, momentum thickness at infinity and curvature. The outer potential problem is set up with a vortex sheet of this strength to represent the wake, leading to a singular integro-differential equation for θ0(s). From the particular solution we obtain a proportionate correction – (CD/4π) (log4/CD) to the Kutta–Joukowski circulation, where CD is the drag coefficient. For laminar flow this is of order R−½ log (1/R). However, the solution also contains an arbitrary constant which cannot be settled without an examination of the near wake. The recent work of Brown & Stewartson (1970) suggests that this may lead to a term of the lower order $R^{-\frac{3}{8}}$, but depends on the assumption, not supported by the present analysis, that the pressure rise across the wake is o(R−¼).

Turbulent boundary-layer wall-pressure measurements were made with ‘pinhole’ microphones three times smaller (relative to a boundary-layer displacement thickness) than microphones used in earlier work. The improved high-frequency resolution permitted examination of the influence of high-frequency eddies on smooth-wall pressure statistics. It was found that the space-time decay rate is considerably higher than previously reported. Measurements of cross-spectral density made with 5 Hz bandwidth filters disclosed low phase speeds at low frequency and small separation. Measurements were repeated on rough walls and parallels were drawn from knowledge of a smooth-wall boundary-layer structure to propose a structure for a rough-wall boundary layer. The effect of independently varying roughness height and separation on the large and small-scale turbulence structure was deduced from the measurements. It was found that roughness separation affected the very large-scale structure, whereas the roughness height influenced the medium and very small-scale turbulence.

Photographs and simultaneous temperature measurements show that the thermal oscillations which occur in air between horizontal plates with unstable stratification are associated with lateral oscillations of wavy convection rolls. The phenomenon appears to consist of an oscillating motion nearly independent of depth superimposed upon the convective motions, and is best viewed from above. For large Prandtl number the photographic evidence suggests that the first unsteady motions to occur consist of a very similar oscillation, except that the features which oscillate are updraft and downdraft lines of short length. The periods of the oscillations obtained visually agree well with those reported previously from fixed temperature measurements, and a weak oscillation of doubled frequency is explained.

A numerical study has been made of steady laminar convection in an infinite horizontal layer of fluid, bounded above by a rigid plate held at constant temperature, and below by a thermal insulator. The fluid is heated by a uniform distribution of heat sources. Recent methods for the solution of the Navier-Stokes equations by finite differences are discussed, and the results of integrations in two and three space dimensions and time are presented in an attempt to determine the planform and other flow properties as functions of the Rayleigh and Prandtl numbers.

These results are compared with a preliminary study by Roberts (1967) using a mean-field equation approach and with experimental observations by Tritton & Zarraga (1967) and Hooper (private communication).

We obtain exact solutions of the linearized perturbation equations for the convective motions in a Boussinesq fluid contained in an infinite rectangular channel and heated from below. The top and bottom are assumed to be perfect heat conductors. The sides can either be conducting or insulating. We assume that the sidewalls are rigid, but allow the top and bottom to be free so that we can separate variables.

We find that the preferred modes of convection closely resemble transverse ‘finite rolls’ [as predicted by Davis (1967) for convection in a box] for channels with height to width ratios outside the range 0·1 to 1. Inside this range they show noticeable departures from roll form. We prove, however, that, except when the sidewalls are relaxed to infinity, ‘finite rolls’ are never exact solutions of the linearized equations, even though in most cases they are good approximations.

We also find that bringing the sidewalls closer together inhibits convection and, generally, produces thinner cells at the onset of convection.

It is shown that a random superposition of inertial waves in a rotating conducting fluid can act as a dynamo, i.e. can systematically transfer energy to a magnetic field which has no source other than electric currents within the fluid. Dynamo action occurs provided the statistical properties of the velocity field lack reflexional symmetry, and this occurs when conditions are such that there is a net energy flux (positive or negative) in the direction of the rotation vector Ω.

If the magnetic field grows from an infinitesimal level, then the mode of maximum growth rate dominates before the back-reaction associated with the Lorentz force becomes significant. This mode is first determined, and then the back-reaction associated with it alone is analysed. It is shown that the magnetic energy grows exponentially during the stage when the Lorentz forces are negligible, then reaches a maximum depending on the values of the parameters \[ R_m = u_0 l/\lambda,\quad Q = \Omega l^2/\lambda, \] (u0 = initial r.m.s. velocity, l = length scale characteristic of the velocity field, λ = magnetic diffusivity) and ultimately decays as t−1 (equation (5.15)). This decay is coupled with a decay of the velocity field due to ohmic dissipation, and it occurs because there is no external source of energy for the fluid motion.

Mean streamwise vorticity in turbulent flow is shown to arise both from mean flow skewing and from the inhomogeneity of anisotropic wall turbulence. The structure of the Reynolds stress tensor is examined in several flows where the latter mechanism predominates. On the basis of a simple model for the anisotropy, the direction of the secondary currents is deduced for the corner boundary layer, the salient edge flow, and in the non-uniform nominally two-dimensional boundary layer.

As an extension of our previous work (Sato & Kuriki 1961), an experimental study was made on the laminar-turbulent transition of a two-dimensional wake, with emphasis on the non-linear interaction of velocity fluctuations. In the natural transition, a sinusoidal velocity fluctuation first appears as a result of the selective amplification of irregular disturbances. When the amplitude of the fluctuation becomes large, a non-linear interaction takes place: higher harmonics of the sinusoidal fluctuation and a slow, irregular fluctuation are generated. The transition into turbulence is gradual, and there are no abrupt ‘breakdowns’.

When a sound with two frequencies (f1 and f2) is introduced into the wake, two velocity fluctuations are induced. They are amplified independently when amplitudes are small. If amplitudes exceed certain values, a mutual interaction takes place, and the growth of each component is suppressed by the presence of the other. At the same time, fluctuations with frequencies of f1 − f2, 2(f1 − f2) … and f1 + f2, 2(f1 + f2) … are produced. The behaviour of the slow, irregular fluctuation in the natural transition closely resembles that of the f1 − f2 component. They seem to be produced by the same mechanism.

In the transition region of the wake, a small variation in the amplitude of velocity fluctuation is enhanced by the suppression effect, and a small irregularity in the frequency is amplified by the generation of the f1 − f2 component. These are two elementary processes of randomization of regular fluctuations, which lead to the development of turbulence.

Experiments to explore the effect of initial disturbance length-scale on turbulence developed in the presence of a uniform mean shear are the subject of this paper.

Flows with nearly the same mean shear (8·6 sec−1) and initially different turbulent scales are generated in a wind tunnel test-section by placing grids just downstream of a honeycomb of uniform cell diameter (¼ in.) and non-uniform cell length. Both round-rod grids of uniform square mesh and parallel-rod construction with roughly equal solidity (0.34) are used. Grid mesh sizes range from $\frac{1}{24}$ in. to 2 in.

From the results it is concluded that for a given value of mean shear the imposed length scale fixes the energy level of the resulting turbulence, provided the scale is sufficiently large. When it is reduced below some minimum value the turbulence decays. Also, it is found that two-dimensional flow-generator geometries are more effective than three-dimensional geometries in producing a roughly homogeneous turbulent field with a higher fluctuation level in a shorter distance.

The results of a numerical evaluation of the Navier-Stokes equations of motion for the case of a viscous fluid streaming past a sphere are presented in terms of the length of the standing eddy behind the sphere and in terms of the angle of flow separation at the sphere. Emphasis was placed on calculating these quantities at Reynolds numbers between 20 and 40 where no reliable theoretical or experimental values are available. In support of these calculations, it is shown that the values for the drag on a sphere previously calculated by us from the Navier-Stokes equations of motion by the same numerical technique as that used for calculating the eddy length and angle of flow separation agree well with our recent, extensive drag measurements for a wide Reynolds number interval. Our results are used to make a comparison between drag and flow field as predicted by analytical solutions and numerical solutions to the Navier-Stokes equations of motion. Some limitations of the analytical solutions to predict correct values for the drag, and to describe the correct nature of the flow field, are pointed out. It is shown further that a plot of [(D/Ds) – 1] versus log NRe, where D is the actual drag on a sphere, Ds is the Stokes drag, and NRe is the Reynolds number, reveals that the variation of the drag on a sphere with Reynolds number follows well defined régimes, which correlate well with the régimes of the flow field around a sphere. A similar relationship between ‘drag-régime’ and flow field pattern is discussed for the case of viscous flow past a cylinder.

A solid long slender body is considered placed in a fluid undergoing a given undisturbed flow. Under conditions in which fluid inertia is negligible, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio of the cross-sectional radius to body length. Specific examples are given for the resistance to translation of long slender bodies for cases in which the body centre-line is curved as well as for those for which the centre-line is straight.

The meniscus profiles determined numerically by White & Tallmadge (1965) have been found to be in error. Accordingly, the reported values of meniscus height and curvature at the top of the meniscus are also in error. These errors were initially suspected by observing that top curvatures calculated from the reported values of height fell below the minimum possible (flat plate) value at large radii.

A uniformly valid asymptotic solution for large Reynolds number is constructed for plane steady laminar flow of a liquid into the channel between two semi-infinite parallel plates. The entry condition is taken as either that for a cascade of plates in a uniform oncoming stream, or uniform flow directly at the inlet. A paradox in the standard solution of Schlichting—that near the inlet the flow due to displacement would not be the accelerated uniform core on which his expansion is based—is resolved by showing that his series for small as well as large distance actually applies only to conditions far downstream, and matches with another expansion valid near the inlet. Good agreement is found with three independent numerical solutions of the full Navier-Stokes equations, except for a discrepancy in one solution for uniform entry that is traced to erroneous neglect of inlet vorticity.