The problem of closing the Reynolds-stress and dissipation-rate equations at low Reynolds numbers is considered, specific forms being suggested for the direct effects of viscosity on the various transport processes. By noting that the correlation coefficient $\overline{uv^2}/\overline{u^2}\overline{v^2} $ is nearly constant over a considerable portion of the low-Reynolds-number region adjacent to a wall the closure is simplified to one requiring the solution of approximated transport equations for only the turbulent shear stress, the turbulent kinetic energy and the energy dissipation rate. Numerical solutions are presented for turbulent channel flow and sink flows at low Reynolds number as well as a case of a severely accelerated boundary layer in which the turbulent shear stress becomes negligible compared with the viscous stresses. Agreement with experiment is generally encouraging.

This study deals with the mass-transport velocity within the bottom boundary layer of cnoidal waves progressing over a smooth horizontal bed. Mass-transport velocity distributions through the boundary layer are derived and compared with that predicted by Longuet-Higgins (1953) for sinusoidal waves. The mass transport at the outer edge of the boundary layer is compared with various theoretical results for an inviscid fluid based on cnoidal wave theory and also with previous experimental results. The effect of the viscous boundary layer is to establish uniquely the bottom mass transport and this is appreciably greater than the somewhat arbitrary prediction for an inviscid fluid.

The classical theory of Brownian motion applies to suspensions which are so dilute that each particle is effectively alone in infinite fluid. We consider here the modifications to the theory that are needed when rigid spherical particles are close enough to interact hydrodynamically. It is first shown that Brownian motion is a diffusion process of the conventional kind provided that the particle configuration does not change significantly during a viscous relaxation time. The original argument due to Einstein, which invokes an equilibrium situation, is generalized to show that the particle flux in probability space due to Brownian motion is the same as that which would be produced by the application of a certain ‘thermodynamic’ force to each particle. We then use this prescription to deduce the Brownian diffusivities in two different types of situation. The first concerns a dilute homogeneous suspension which is being deformed, and the relative translational diffusivity of two rigid spherical particles with a given separation is calculated from the properties of the low-Reynolds-number flow due to two spheres moving under equal and opposite forces. The second concerns a suspension in which there is a gradient of concentration of particles. The thermodynamic force on each particle in this case is shown to be equal to the gradient of the chemical potential of the particles, which brings considerations of the multi-particle excluded volume into the problem. Determination of the particle flux due to the action of this force is equivalent to determination of the sedimentation velocity of particles falling through fluid under gravity, for which a theoretical result correct to the first order in volume fraction of the particles is available. The diffusivity of the particles is found to increase slowly as the concentration rises from zero. These results are generalized to the case of a (dilute) inhomogeneous suspension of several different species of spherical particle, and expressions are obtained for the diagonal and off-diagonal elements of the diffusivity matrix. Numerical values of all the relevant hydrodynamic functions are given for the case of spheres of uniform size.

The purpose of the present paper is to derive expressions for the pressure fields of various high frequency convected singularities immersed in a unidirectional sheared flow. These expressions include the simultaneous effects of fluid and source convection and refraction. These results are then combined to predict the far-field directivity of cold round jets. It is found that the agreement between experiment and the present theory is quite good at a source Strouhal number of unity but that this agreement deteriorates as the source frequency is increased. Our theoretical results show the explicit form of the ‘refraction integral’ and that convective amplification for the pressure of a quadrupole is increased by a factor of (1 — MJ cos Θ)−1 over the classical results, where MJ is the jet Mach number and Θ is the angle from the jet axis. Thus acoustic/mean-flow interaction not only implies refraction but also additional convective amplification due not to source convection but to fluid motion.

An experimental investigation of the two-dimensional incompressible mixing layer was carried out. The measurements provide new information on the development of the mean and turbulent fields towards a self-preserving state and on the higher-order statistical characteristics of the turbulent field. The relevance of initial conditions to the development of the flow is discussed in the light of both present and previous data. Measurements of spectra, probability densities and moments to eighth order of all three velocity-component fluctuations at various transverse positions across the flow were carried out using an on-line digital data acquisition system. The probability density distributions of the derivative and the squared derivative of the longitudinal and lateral velocity fluctuations were also determined. Direct measurements of moments to eighth order of the velocity derivatives were attempted and are discussed in the light of the simultaneously measured histograms. The problems in obtaining higher-order statistical data are considered in some detail. Estimates of the integral time scale of many of the higher-order statistics are presented. The high wave-number structure was found to be locally anisotropic according to both spectral and turbulent velocity-gradient moment requirements. Higher-order spectra to fourth order of the longitudinal velocity fluctuations were measured and are discussed. Finally the lognormality of the squared longitudinal and lateral velocity-derivative fluctuations was investigated and the universal lognormal constant μ was evaluated.

A composite hydrodynamic-diffusion model of the arterial wall is presented to describe the vesicular transport of relatively inert macromolecules across the inner endothelial lining of the larger arteries of humans and animals and their subsequent diffusion in the underlying tissue of the intima and media. This model is motivated by the highly specialized ultrastructure of the arterial wall observed in electron microscopic studies and the recent experimental measurements of the time-dependent uptake of labelled macromolecules in animal arteries under carefully controlled in vitro conditions. The proposed dynamic model for the vesicular transport across the endothelial cell layer considers the constrained Brownian diffusion of 700 Å vesicles subject to long-range hydrodynamic and short-range London-van der Waals force interactions with the plasmalemma membranes of the endothelial cell. Approximate solutions are developed for the motion and the steady-state vesicle density distribution near the plasmalemma and in the interior of the cell using boundary-layer-like methods. The model for the vesicular transport just described appears as a novel boundary condition in the basic diffusion model for the underlying tissue. The latter is treated as a two-phase medium comprised of an interstitial fluid continuum with a uniformly dispersed smooth muscle phase as first proposed by Hills (1968). This model for the underlying tissue assumes that the smooth muscle cells contribute insignificantly to the macromolecule diffusion across the arterial wall but act as the principal storage reservoir for the macromolecules for large diffusion times because of their large volume fraction. The dimensionless parameters that arise in the theoretical model are determined by comparing the solutions for the time-dependent total wall uptake with Fry's (1973) experimental data for canine carotid artery.

The transition of homogeneous turbulence from an initially isotropic three-dimensional to a quasi-two-dimensional state is simulated numerically for a conducting, incompressible fluid under a uniform magnetic field B0. The magnetic Reynolds number is assumed to be small, so that the induced fluctuations of the magnetic field are small compared with the imposed magnetic field B0, and can be computed from a quasi-static approximation. If the imposed magnetic field is strong enough, all variations of the flow field in the direction of B0 are damped out. This effect is important e.g. in the design of liquid-metal cooling systems for fusion reactors, and the properties of the final state are relevant to atmospheric turbulence. An extended version of the code of Orszag & Patterson (1972) is used to integrate the Navier-Stokes equations for an incompressible fluid. The initial hydrodynamic Reynolds number is 60. The magnetic interaction number N is varied between zero and 50. Periodic boundary conditions are used. The resolution corresponds to 323 points in real space. The full nonlinear simulations are compared with otherwise identical linear simulations; the linear results agree with the nonlinear ones within 3% for about one-fifth of the large-scale turnover time. This departure is a consequence of the return-to-equilibrium tendencies caused mainly by energy transfer towards high wavenumbers. The angular energy transfer and the energy exchange between different components are smaller, and become virtually zero for large values of N. For N ≈ 50 we reach a quasi-two-dimensional state. Here, the energy transfer towards high wavenumbers is reduced for the velocity components perpendicular to B0 but relatively increased for the component parallel to B0. The overall behaviour is more similar to three-than to purely two-dimensional turbulence. This finding is of great importance for turbulence models of the atmosphere. The realization of a purely two-dimensional state does not seem to be possible for decaying turbulence. The magnetic field causes highly intensified pressure fluctuations, which contribute to the redistribution of the anisotropic Lorentz forcing.

The free-convective flow induced when a semi-infinite horizontal fuel surface burns in a quiescent oxidizing atmosphere is considered. For high values of the Grashof number the dominant feature of the flow is a boundary layer close to the surface within which there is a flame or intense reaction zone. An outer flow or fire-wind is induced by entrainment into this boundary layer. A simple experiment supports this overall picture of the flow.

The frequencies of vortex shedding from circular cylinders forced to oscillate transversely in low-turbulence uniform and shear flows were investigated. The stream velocity in the shear flow varied linearly with spanwise distance.

In both flows the vortex shedding frequency locked on to the cylinder frequency and to submultiples of the cylinder frequency. In uniform flow the range of cylinder frequencies for locking-on was dependent on the amplitude of oscillation and Reynolds number. At the boundaries of locking-on at the cylinder frequency locked-on shedding was intermittent with unforced shedding and locking-on was accompanied by a change in wake width. At a particular cylinder frequency near mid-range it is conjectured that the wake width jumped from being greater to being less than that for the stationary cylinder. In shear flow the spanwise extent of locking-on at the cylinder frequency was explained by considering the uniform flow results and the inclination of shed vortices in shear flow. At the spanwise boundaries of this locking-on, locked-on cells were shed intermittently with unforced cells which were more stable in frequency than the corresponding cells for the stationary cylinder.

Space-time correlations have been measured in the turbulent supersonic wake behind a slender axisymmetric body. The data were obtained with two independently-positioned hot-wire anemometers, and interpreted with the aid of an extension of compressible anemometry theory. Comparison with earlier single-point Eulerian spectra showed a breakdown of Taylor's hypothesis near the wake edge, and thus also the expected differences in the scale lengths. The eddy convection speed appears equal to that of the mean flow, and the eddy shape is inclined to the wake axis, although it becomes nearly parallel to it at the wake edge. As a consequence, radial and longitudinal space correlation functions appear nearly equal. An upstream-downstream asymmetry was noted in the latter.

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

This work is concerned with the dispersion of a buoyant solute in a straight horizontal pipe of circular cross-section where dispersion is affected by molecular diffusion, the laminar flow along the pipe and density currents. Erdogan & Chatwin (1967) have derived an equation for the mean concentration $\overline{C}$ of a buoyant solute using a relatively simple asymptotic model, and have predicted that the dispersion induced by buoyancy effects depends on the Péclet number of the flow. In this part of this study an approximate expression is derived for $\overline{C}$ from Erdogan & Chatwin's equation, and an asymptotic form is obtained for the second moment of distributions of buoyant solutes. The examination of the second moment leads to a simple, but important, result: the dispersion induced by density currents at large times is small compared with the dispersion induced by density currents at times when transient effects are significant.

Using simple Fourier transform techniques and extensions to stationary-phase methods, the behaviour of surface gravity waves is determined near triply coalescing roots of the dispersion relation. It is shown that the amplitude of the surface wave is proportional to (∂U/∂x)−¼ at the location of the triple root. Far from the triple root it satisfies conservation of action. The internal wave is modelled simply by its surface current U.

Asymptotic orders of magnitude are also given for the case ∂U/∂x = 0 at the triple root.

Wavenumber frequency spectra have been measured in a two-dimensional incompressible mixing layer, using linearized hot-wire anemometers. Spectra of two dimensions (frequency and wavenumber) have been measured for lateral and longitudinal turbulent velocities, and used to construct three-dimensional spectra. The validity of the separation assumption used to construct these spectra was tested. Spectra of the velocity product responsible for the mean shear stress and the lateral gradient of this spectrum have been determined, as has a structure constant for the lateral velocity fluctuations that Phillips (1967) suggested is relevant to the maintenance of the shear stress gradient. Phillips’ (1967) stress model fails the tests proposed in this study.

Free-streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored, vortices in an inviscid incompressible fluid. If each vortex has area A and the separation is L, there are two possible shapes if A½/L is less than a critical value 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable while the less deformed shape is stable.

Chatwin (1970) has described the approach to normality of a cloud of solute which is injected into a pipe containing solvent in steady laminar flow. This paper is concerned with the modification of Chatwin's theory when there is a small density difference between the solvent and the dissolved solute. Asymptotic series are derived for the induced density currents and for the distribution of solute in the case when the molecular diffusivity is constant throughout the pipe's cross-section. These series lead to the asymptotic forms of the moments of the distribution, thereby describing some additional deviations from normality caused by buoyancy effects at large times. The theory predicts that the additional dispersion due to buoyancy effects is proportional to the square of the Rayleigh number and depends on the Péclet number of the flow. There is excellent agreement between the results and those previously obtained by the author (1976) from Erdogan & Chatwin's (1967) model of dispersing buoyant solutes. The results confirm that Erdogan & Chatwin's intuitive theory correctly models the significant features of the situation for large Schmidt numbers.

The paper deals with steady laminar film flow which is set up at the cylindrical surface of an idealized horizontal ‘road’ when homogeneous ‘rain’ is falling onto the road in a vertical downward direction. It is shown that a particular solution of the Navier-Stokes equations is possible for which the depth of the liquid film is constant. In that case the Navier-Stokes equations reduce to the equations governing plane stagnation-point flow. However, the boundary conditions differ from those for the classical stagnation-point problem. Solutions for nearly inviscid flow and predominantly viscous flow are derived analytically. In particular, simple formulae for the depth of the film are found in both cases. Finally, the importance of the particular solution as a member of a whole class of solutions is discussed on the basis of a momentum integral approximation.

The oxygen vibrational and dissociation relaxation behind regular reflected shocks has been calculated and measured. Numerical calculations using published rate coefficients supplied the relaxation-zone data needed to estimate the range of most useful experimental conditions. Then photographs of the shock reflexion were taken using a complementary double-exposure interferometer. The density profiles in the relaxation zones behind the reflected shocks were measured by means of a multibeam laser-differential interferometer. The results of these experiments confirmed the theoretical model adopted for the calculations within a certain range of experimental conditions, but clearly revealed the need for revising the rate coefficients. New calculations with different vibrational relaxation times and dissociation rate coefficients then had the result that the best fit of calculated to measured profiles was obtained when the following values were inserted.

Vibration\begin{eqnarray*} & p\tau_v = A_v\exp(B_vT^{-\frac{1}{3}}),\\ & A_v = (2.1\pm 0.2)\times 10^{-5}\,{\rm kg/ms},\quad B_v = 129\,{}^{\circ}{\rm K}^{\frac{1}{3}}. \end{eqnarray*}

Dissociation: O2 + O_2[rlarr ] 2O + O_2 \begin{eqnarray*} & {\mathop {k_1}\limits^{\rightharpoonup}} = A_1T^{-2.5}\exp (-\theta_D/T),\\ & A_1 = (6.2 \pm 0.5)\times 10^{18}\,{\rm m}^3\,{}^{\circ}{\rm K}^{2.5}/{\rm mol}\,{\rm s},\quad\theta_D = 59\,136\,{}^{\circ}{\rm K}. \end{eqnarray*}

Dissociation: O2 + O[rlarr ]3O \begin{eqnarray*} & {\mathop {k_1}\limits^{\rightharpoonup}} = A_2T^{-1.0}\exp (-\theta_D/T),\\ & A_2 = (4.0 \mp 0.5)\times 10^{-13}\,{\rm m}^3\,{}^{\circ}{\rm K}/{\rm mol}\,{\rm s}. \end{eqnarray*}

Possibilities of high shoreline amplification and run-up are investigated. A shoreline amplification of magnitude 5·38 and a tsunamigenic (deep water) amplification of magnitude 5·71 are obtained from single waves without analytic or computational difficulties. It is not claimed that these are a maximum, but rather it is conjectured that arbitrarily high run-up and amplification can be obtained provided that the correct initial wave trains are chosen.