A well-tested integral method has been used to calculate turbulent boundary-layer development for the distribution of external velocity given by U ∝ x−0.255. The results suggest that different values of the initial momentum thickness, so long as this is below some critical value, produce a range of equilibrium layers having widely different values of the form parameter G. For values of the initial momentum thickness greater than the critical value, layers are produced which proceed more or less rapidly to separation. These results provide a plausible explanation for conflicting experimental observations made in the past.

Additional calculations for the flows U ∝ x−0.15 and U ∝ x−0.35 suggest that, in the first case, a unique equilibrium condition is approached whatever the initial momentum thickness unless this exceeds some critical value; in the second case no equilibrium condition appears possible.

A realistic theoretical model of steady Langmuir circulations is constructed. Vorticity in the wind direction is generated by the Stokes drift of the gravity-wave field acting upon spanwise vorticity deriving from the wind-driven current. We believe that the steady Langmuir circulations represent a balance between this generating mechanism and turbulent dissipation.

Nonlinear equations governing the motion are derived under fairly general conditions. Analytical and numerical solutions are sought for the case of a directional wave spectrum consisting of a single pair of gravity waves propagating at equal and opposite angles to the wind direction. Also, a statistical analysis, based on linearized equations, is developed for more general directional wave spectra. This yields an estimate of the average spacing of windrows associated with Langmuir circulations. The latter analysis is applied to a particular example with simple properties, and produces an expected windrow spacing of rather more than twice the length of the dominant gravity waves.

The relevance of our model is assessed with reference to known observational features, and the evidence supporting its applicability is promising.

The drag on a cylindrical particle moving in a thin sheet of viscous fluid is calculated. It is supposed that the sheet is embedded in fluid of much lower viscosity. A finite steady drag is obtained, which depends logarithmically on the ratio of the viscosities. The Einstein relation is used to determine the diffusion coefficient for Brownian motion of the particle, with application to the movement of molecules in biological membranes. In addition, the Brownian motion is calculated using the Langevin equation, and a logarithmically time-dependent diffusivity is obtained for the case when the embedding fluid has zero viscosity.

The stability of individual inviscid barotropic planetary waves and zonal flow on a sphere to small disturbances is examined by means of numerical solution of the algebraic eigenvalue problem arising from the spectral form of the governing equations. It is shown that waves with total wavenumber n (the lower index of the Legendre function Pmn which describes the waves’ meridional structure) less than 3 are stable for all amplitudes, whereas those with n ≥ 3 are unstable if their amplitudes are sufficiently large. For travelling waves (m ≠ 0) with n = 3 and 4 and with disturbances comprised of 30 modes, the amplitudes required for instability are approximated by those obtained from triad interactions, and are smaller than those given by Hoskins (1973). For the zonal-flow modes (m = 0) the critical amplitudes are smaller than those predicted by triad interactions, and are close to those obtained from Rayleigh's classical criterion.

Long waves are generated in a laboratory-size rectangular basin, which is heated uniformly from below. Their subsequent decay is measured, and the decay component due to the action of convective turbulence isolated, using a combination of existing theories and interpretation techniques. An expression is proposed for the turbulent decay decrement as a function of the bulk Rayleigh number. The results agree as well as can be expected with a simple model based on a Reynolds-stress decay estimate obtained by superposing convective thermals on the oscillating flow associated with the long wave.

Calculated flow properties are compared with measurements obtained in two-dimensional isothermal wakes with and without recirculation. The equations of continuity and momentum were solved numerically together with equations which formed a turbulence model. Calculations were made using three turbulence models: the first comprised transport equations for turbulence kinetic energy and the rate of turbulence dissipation; the second and third comprised equations for the rate of turbulence dissipation and two forms of Reynolds-stress equations characterized by different redistribution terms. The results show that, for wakes without recirculation, the particular turbulence model is less important than the boundary condition assumed in the plane of the trailing edge of the body; though the Reynolds-stress models do, of course, provide a better representation of the individual normal stresses. In the case of wakes with recirculation, both the length of the recirculation region and the rate of spread of the downstream wake are underestimated. The second discrepancy is particularly evident and appears to stem from the form of the dissipation equation. A suggestion for improving the modelling of this equation is provided together with necessary justification.

A number of initial- and boundary-value problems for the Boussinesq equations are solved by a finite-difference technique, in an attempt to see how a stably-stratified horizontal shear layer rolls up into horizontally periodic billows of concentrated vorticity, such as are frequently observed in the atmosphere and oceans. This paper describes the methods, results and accuracy of the numerical simulations. The results are further analysed and approximately reproduced by a simple semi-analytic model in Corcos & Sherman (1976).

A simple derivation is given of the parabolic flow first described by John (1953) in semi-Lagrangian form. It is shown that the scale of the flow decreases like t−3, and the free surface contracts about a point which lies one-third of the way from the vertex of the parabola to the focus.

The flow is an exact limiting form of either a Dirichlet ellipse or hyperbola, as the time t tends to infinity.

Two other self-similar flows, in three dimensions, are derived. In one, the free surface is a paraboloid of revolution, which contracts like t−2 about a point lying one-quarter the distance from the vertex to the focus. In the other, the flow is non-axisymmetric, and the free surface contracts like t−5.

The parabolic flow is shown to be one of a general class of self-similar flows in the plane, described by rational functions of degree n. The parabola corresponds to n = 2. When n = 3 there are two new flows. In one of these the scale varies as t12/7 and the free surface has the appearance of a trough filling up. In the other, the free surface resembles flow round the end of a rigid wall; the scale varies as t−4·17.

A multi-layer model is used to describe a ‘two-dimensional’ continuously stratified fluid. We use a momentum theorem in each layer to derive an ordinary differential equation describing the vertical structure behind a jump. This equation is compared with the corresponding equation for continuous flow. As one would expect from the classical one-layer theory, they are identical up to second order for weak disturbances. The energy change across a jump is also derived. By requiring that energy be lost through a jump, we calculate when a weak jump is possible in general. Algorithms for computing jumps of arbitrary strength are given. To ensure that the flow after the jump is stable and also for a numerical reason to be stated in § 8, we require that the Richardson number after the jump be equal to or greater than S¼S. Numerical examples are given to show the range of parameters within which jumps are possible; the velocity profiles related to different kinds of jumps also appear. Since hydraulic jumps in a continuously stratified fluid have not yet been observed in any laboratory, it should be of interest to verify these calculations experimentally.

A theory is developed for the dependence of the Nusselt number on the Rayleigh number in turbulent thermal convection in horizontal fluid layers. The theory is based on a number of assumptions regarding the behaviour in the molecular boundary layers and on the assumption of a buoyancy-defect law in the interior analogous to the velocity-defect law in flow in pipes and channels. The theory involves an unknown constant exponent s and two unknown functions of the Prandtl number. For either s = ½ or s = 1/3, corresponding to two different theories of thermal convection, and for a given Prandtl number, constants can be chosen to give excellent agreement with existing data over nearly the whole explored range of Rayleigh numbers in the turbulent case. Unfortunately, comparisons with experiment do not permit a definite choice of s, but consistency with the chosen form of the buoyancy-defect law seems to suggest s = 1/3, corresponding to similarity theory.

Inviscid methods of treating secondary flows due to streamline curvature are well known in the literature. The development of secondary motions at high Reynolds number in a bifurcation characteristic of the lung airways or the cardiovascular system is considered. In particular, the secondary motions in the initial section of a bifurcation, which can be considered a tube of slowly varying ellipticity, are calculated. Providing viscous effects are confined to regions near the walls, the calculated velocity field agrees with experimental observation. Thus the structure of the secondary motions can be considered an effect of stretching of vortex lines by the tube boundary.

The detailed dynamics of an unstable free shear layer are examined for a gravitationally stable or neutral fluid. This first article focuses on the part of the evolution that precedes the first subharmonic interaction. This consists of the transformation of selectively amplified sinusoidal waves into periodically spaced regions of vorticity concentration (the cores) joined by thin layers (the braids), in which vorticity is also concentrated. The thin layers are the channels along which vorticity is advected into the cores, and the cores provide the strain which creates the braids. For moderately long waves an analysis is given of the braid structure as a function of time. For gravitationally stable shear layers at high Reynolds numbers, the local vorticity reaches such large values as to cause secondary shear instability on a small (length) and short (time) scale. A physical account of the primary instability and its self-limiting mechanism is used as a basis for a computation, which yields growth rates and maximum amplitude as a function of initial layer parameters. The computation supplies the wavelength of waves that grow to achieve the largest (absolute) amplitude. Finally, the model makes it clear that, in the absence of secondary instability, this initial phase of the nonlinear development of the layer contributes only a modicum of additional mixing, especially at high Reynolds numbers.

An estimate is obtained of the heat transfer from a constant-temperature hot wire in a two-dimensional low Reynolds number flow (R [Lt ] 1). The flow has a small sinusoidally fluctuating velocity superimposed on the mean velocity. For the fluctuating components it is shown that at low frequencies the heat transfer is in phase with the velocity, the magnitude of the heat transfer being given by the result for steady heat transfer, but at high frequencies the heat transfer lags behind the velocity, the relative magnitude of the heat transfer decreasing as (frequency)−1.

As part of a general investigation of complex turbulent flows, extensive one-point measurements have been made of the turbulence structure of the mixing layer bounding a normally impinging plane jet with an irrotational core. The ratio of shear-layer thickness to streamline radius of curvature reaches a maximum of about 0.2, the sense of the curvature being stabilizing. Downstream of the impingement region the shear layer returns asymptotically to being a classical plane mixing layer. The most striking feature of the results is that the return is not monotonic: after decreasing in the region of stabilizing curvature, the Reynolds stresses, triple products, energy dissipation rate and other turbulence quantities overshoot the plane-layer values before finally decreasing. Some conclusions are drawn about the nature of the turbulent transport of Reynolds stress, and about the representation of this and other processes in calculation methods for complex turbulent flows. An incidental result of the work is a comprehensive set of measurements in a plane mixing layer.

Experiments have been performed in a low-speed wind tunnel to determine the detention time of airborne smoke particles that become trapped in the wake vortex (or bubble) region behind flat disks placed perpendicular to the flow. Using a laser transmissometer to detect the smoke, its detention time was obtained from the time-dependent decay of the smoke in the disk bubble during the time immediately following the removal of the source of smoke. The dimensionless group H, the product of the detention time and the mainstream air velocity divided by the disk diameter, is seen to be a constant equal to 7.44 ± 0.52 for Reynolds numbers in the range 2000–40000. This result is compatible with a simple fluid-mechanical model which describes the transport of fluid-borne scalar entities across the bubble boundary by turbulent diffusion. The investigation suggests that H should be unique for the flow about a disk over a wide range of conditions, and further suggests the possibility that similar unique values for H can exist for flow about other obstacles. The number H has potential applications in a number of physical and engineering research areas.

In this paper we consider the flow field induced in an incompressible viscous conducting fluid in a hemispherical bowl by a symmetric discharge of electric current from a point source at the centre of the plane end of the hemisphere. This plane end is a free surface. We construct an analytic solution for the slow viscous flow and a numeriacl solution for the nonlinear problem. The streamlines in an axial cross-section form two sets of closed loops, one on either side of the axis. Our computations indicate that, for a given fluid, when the discharged current reaches a certain magnitude the velocity field breaks down. This breakdown probably originates at the vertex of the hemispherical container.

This paper describes part of a detailed study of the initial region of coaxial jets of three different mean-velocity ratios. Similarity of the mean-velocity and turbulent-intensity profiles within the two mixing regions inside the initial merging zone, and within the mixing region inside the fully-merged zone, has been observed. Similarity with single-jet results has been obtained. Overall pressure measurements and hot-wire and microphone spectra inside coaxial jets yield two pronounced peaks, suggesting the existence of two types of vortices at different frequencies. The higher-frequency or primary vortices are found to be generated in the primary or inner mixing region, and the lower-frequency or secondary ones in the secondary or outer mixing region. It seems that the former are generated further upstream than the latter. Both types of vortices are found in the initial merging zone. However, their growth or decay and their dominance depend on the mean-velocity ratio. Low mean-velocity ratios indicate the dominance of the high-frequency vortices. At high mean-velocity ratios, the low-frequency vortices are dominant. The Strouhal number of the vortices plays an important part in their growth and decay. The complicated flow structure of coaxial jets can be very simply related to, and described by, the much simpler structure of single jets.

A numerical exploration of multiple-cell solutions for the flow between a rotating and a stationary disk is carried out systematically by the continuation method. The paper confirms and extends the work of Mellor, Chapple & Stokes. The results include the discovery of one-, two-, three- and five-cell solution regions which have not been reported before.

Results of hot-wire measurements in a plane incompressible jet are reported. The flow was found to be self-preserving beyond x/d > 40 and measurements were made up to x/d = 120. The quantities measured include mean velocities, turbulence intensities and third- and fourth-order terms, as well as two-point correlations and the intermittency factor. Conditional sampling techniques were used to obtain exclusively data within the turbulent zone of the jet. The results are compared with previous investigations.

This is the third paper in a sequence providing data on turbulent free shear flows.

This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, arêtes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.