The fingering instability of an interface between two immiscible fluids in a Hele Shaw cell is simulated numerically. The algorithm used is based on a transcription of the equations of motion for the interface in which it formally becomes a generalized vortex sheet. The evolution of this sheet is computed using a variant of the vortex-in-cell method. The resulting scheme and code make it possible to follow the collective behaviour of many competing and interacting fingers well into the nonlinear, large-amplitude regime. It is shown that in this regime the evolution is controlled essentially by just one dimensionless parameter, the ratio of fluid viscosities. The effects of varying this parameter are studied and the results compared with experimental investigations. Scaling properties of the average density profile across the evolving mixed layer between the two homogeneous fluid phases are investigated. Many phenomena are observed that must be characterized as collective interactions and thus cannot be understood in terms of flows with just a single finger.

The effects of ‘crossing trajectories’ and inertia on the dispersion of particles suspended in a field of grid-generated turbulence were investigated experimentally. The effect of particle trajectories crossing the trajectories of fluid elements, under the influence of a potential field (usually gravity), is to force the particles from one region of highly correlated flow to another. In this manner, particles lose velocity correlation more rapidly than the corresponding fluid points and as a result disperse less.

A homogeneous decaying turbulent field was created behind a square biplanar grid in a wind tunnel. Particles were charged by a corona discharge then passed into the test section through a small plastic tube. A uniform electric field within the test section was used to simulate the effect of gravity, forcing the charged particles out of regions of correlated fluid at a higher than normal rate, therefore inducing the effects of crossing trajectories. Two sizes of glass beads (5 μm and 57 μm diameter) were employed in order to observe inertial effects. Laser-Doppler anemometry was used to measure particle mean-square displacement, autocorrelation coefficient, and mean-square velocity, from which dispersion coefficients were calculated.

For the two particle sizes used in the tests, it was found that the particle diffusion coefficient, after a suitably long time from their release, was influenced primarily by the effect of crossing trajectories. Only in the particle mean-square velocity was the particle inertia seen to have any effect. The ratio of the particle relaxation time to the Kolmogoroff timescale was found to be a good indicator for the effects of particle inertia.

Finite-amplitude solitary waves in water of arbitrary uniform depth are considered. A numerical scheme based on series truncation is presented to calculate the highest solitary wave. It is found that the ratio of the amplitude of the wave versus the depth is 0.83322. This value is compared with the values obtained by previous investigators. In addition, another numerical scheme based on an integral-equation formulation is derived to compute solitary waves of arbitrary amplitude. These calculations confirm and extend the calculations of Byatt-Smith & Longuet-Higgins (1976) for very steep waves.

Air flows past a fixed thin body of a general planform, at a non-uniform small clearance from a plane ground surface. The flow beneath the body is described by a linear two-dimensional partial differential equation, in which the clearance appears as an input non-constant coefficient. Solutions are required subject to separate leading-edge and (nonlinear) trailing-edge boundary conditions, at the edge contour of the planform. The transition points between leading and trailing edge are not necessarily at the lateral extremities of this contour, and are to be determined as part of the solution. As an illustration, a solution is obtained for a circular planform with an exponentially varying clearance. The general problem is relevant to vehicle aerodynamics, especially for racing cars, and some qualitative discussion of the nature of the negative-lift ground-effect problem for such vehicles, and of the effect of ‘skirts’, is presented here.

Experiments were conducted with three delta wings and two rectangular wings to investigate the evolution of trailing vortices in stratified and unstratified water. The vortex trajectories were determined as a function of the normalized time V0t/b0, stratification parameter Nb0/V0 and an effective vortex-core size re/b0. The results have shown that the vortices rise only to a finite height as they decay gradually at first and rapidly thereafter under the influence of turbulence, sinusoidal instability, and core bursting. The effect of stratification is to reduce the lifespan of vortices and the maximum height attained by them.

Wind and tracer-concentration fluctuations, and hence the budgets for tracer variance, vertical flux and streamwise flux, have been measured in the dispersing plume from an elevated lateral line source in an equilibrium turbulent surface layer, using heat as a passive tracer. The results are analysed by testing closure assumptions for models of turbulent dispersion at first and second order. Except close to the source, a first-order (gradient-diffusion) model satisfactorily predicts both the vertical and streamwise tracer fluxes.

The tracer-variance budget is essentially a balance between advection and dissipation, with production becoming significant as fetch increases. The vertical and streamwise heat-flux budgets have advection and turbulent-transport terms which are in balance (almost exactly for the vertical flux, only approximately for the streamwise flux), leaving balances between local production and pressure-gradient interaction. The turbulence-interaction component of the pressure term cannot be modelled as $-\overline{u^{\prime}_{i}\theta^{\prime}}/\tau, \overline{u^{\prime}_{i}\theta^{\prime}}$ being the flux vector and τ a scalar timescale.

By specializing to the case of unit Lewis number and Prandtl number equal to ¾, a number of general results for the structure of a plane steady compressible flow field, within which chemical energy is being liberated by a simple Arrhenius type of combustion reaction, can be acquired by the use of elementary arguments. The field is of the semi-infinite variety, with a ‘flameholder’ presumed to exist at the origin of coordinates. In these circumstances it is necessary to specify the velocity gradient at inlet to the system or, equivalently, the pressure difference across the field. These quantities cannot be selected arbitrarily, and the nature and extent of the restrictions upon them is fully explored. Since the Mach number of the stream is hypothesized to be a quantity of order unity, local Damköhler numbers are always small. Therefore the field is shown to consist of relatively long regions within which the combustion activity takes place, with embedded thin domains of rapid, almost chemically inert, gasdynamical adjustment, whose dimension is typically that of the conventional shock wave. When the inlet Mach number is less than unity the gasdynamical adjustment domains are always adjacent to the origin, and this is also true under most supersonic inlet conditions.

However, there are some special circumstances for which the shock is detached from the flameholder and is established in the middle of the combustion activity. A specific example is provided by a shock within the induction domain.

These special circumstances are shown to be ultrasensitive to pressure difference across the whole domain. It is also shown that wholly supersonic combustion does exist, but only under similar conditions of extreme sensitivity to pressure difference.

The general arguments are supported and illuminated by asymptotic analysis based on the large activation energy of the Arrhenius reaction. Space precludes a full asymptotic treatment of the combustion activity but a companion paper that analyses these parts of the general field is being prepared in collaboration with D. R. Kassoy. Analysis of the shock within the induction domain, together with results from the case of subsonic inlet Mach numbers, shows that gasdynamical effects can prevent ignition by channelling combustion energy into kinetic energy of the flow at the expense of thermal energy.

When two layers of different densities flow in opposite directions over a sill, it is possible for a stagnation point to occur at the interface. Experiments show that the flow towards such a point causes mixing and the continuous formation of a layer of intermediate density. Flow of this type may occur at specific locations in a fjord where the entrance is constricted by a sill.

We consider a porous solid, partially filled with a non-wetting fluid, just above the injection threshold: the injected regions have the topology of the infinite cluster in a percolation problem, and they split into a ‘backbone’ part plus ‘dead ends’. Under a steady Darcy flow, a dye molecule moves by convection on the backbone, and by molecular diffusion on the dead ends. The process has some similarity with solute transport in chromatographic columns. However, because the ‘dead ends’ have a broad distribution of sizes, special singularities may occur, which are reminiscent of non-Gaussian transport for charge carriers in amorphous semiconductors.

We ultimately predict the existence of a well-defined diffusion coefficient D|| for motion parallel to the average flow, in the limit of slow molecular diffusion. We find $D_{\|}\approx (\overline{U}\xi)^2/D_{\rm a}$, where ξ is the percolation correlation length, $\overline{U}$ the macroscopic flow velocity and Da the diffusion coefficient of an ‘ant’ on the infinite cluster (i.e. the macroscopic diffusion constant measured in the absence of flow).

Interfacial conditions for a cylindrical vortex sheet or a cylindrical fluid layer with radius-dependent density, velocity and magnetic fields are derived for isentropic compressible swirling flows subjected to arbitrary disturbances. Surface tension is included for possible immiscible fluids. These conditions are valid for both spatially and temporally growing waves and for flow profiles with or without discontinuities. The deformation of the sheet or the layer affects the flow in two ways: perturbing the total pressure field and disturbing the centrifugal force field created by the azimuthal components of the velocity and the magnetic flux. The latter seems to be straightforward, but is easily overlooked as in some of the previous analyses. We will show that failure to consider such a perturbation to a stable centrifugal force field will lead to the improper destabilization of certain modes with smaller axial and azimuthal wavenumbers.

The interfacial conditions and their corresponding stability characteristics are further examined for a general class of incompressible flows subject to temporal perturbations. Unlike the single role of destabilization played by the velocity in two-dimensional stratified flows or axisymmetric jet flows, the rotating velocity in vortex motions plays a dual role in flow stability: the angular-velocity gradient generates tangential shear, and the angular velocity itself creates a centrifugal force field. While the former always destabilizes the flow, the latter can either stabilize or destabilize the flow depending on whether the resultant force is centrifugally stable or unstable. These characteristics are demonstrated by examining three general types of perturbations.

When a box of fluid-saturated porous material is heated from below, it is known that either a two- or three-dimensional convection pattern can occur depending on the initial configuration. By means of an analytic eigenfunction-expansion technique and a study of the phase-space dynamics of finite-amplitude disturbances we obtain (i) the regions within the space of initial conditions which lead to one or other of these competing states, and thereby (ii) the probability that a certain pattern will be realized, as well as (iii) the explicit form of the heat transferred by the patterns as it depends on aspect ratios. Cubic and nearly cubic boxes are considered, and the analysis applies for values of Rayleigh parameter from convection onset to 1.5 times critical. Our results correct several details appearing in the literature and explain observations made in previous numerical studies.

This paper examines an often overlooked point in the theory of dispersion of passive contaminants in parallel flows – the behaviour of a cloud of solute which has been injected into the flow over a period of time. The problem is linear if the solute has neutral buoyancy and the diffusivity is independent of concentration, and so it can be treated by a Fourier transform in time. For such a Fourier-transform method to be successful, a non-standard eigenvalue problem has to be solved to determine the concentration pattern, the speed at which it is convected downstream and its decay distance downstream. The eigenvalue problem is not self-adjoint, the eigenvalue enters it nonlinearly if longitudinal diffusion is included, and it involves both regular and singular perturbation aspects. The eigenvalue problem is examined generally and the conclusions of Chatwin (1973a) are re-assessed. The eigenvalue problem is then solved numerically for three cases of interest. Two of these cases (dispersion from a harmonically varying source in Poiseuille and plane Couette flow) reveal most unusual eigenvalue structure, whilst the third (dispersion from a harmonically varying source in turbulent channel flow) is not exceptional. The effect of weak longitudinal diffusion is examined theoretically (for general applications) and numerically in one instance (the application to Poiseuille flow). Longitudinal diffusion, even if weak, has a marked effect on the eigenvalue structure. The paper concludes with a suggestion for an alternative attack on the original problem of dispersion from time-dependent sources.

The forcing of convection due to time-dependent heating is calculated in terms of an inhomogeneous boundary condition on the complex envelope function at the sidewalls of the container. Conditions for eliminating the forcing are derived. Application is made to recent experiments on the timescale for the onset of convection.

The primary objective of this paper is to obtain the detailed description of the flow field near an elliptic cylinder that is placed perpendicularly in a uniform stream at low Reynolds number. Attention is paid to the shape effects due to the flattening of the cylinder and to the inertial effects of the fluid. The analysis resorts to the method of matched asymptotic expansions. The main part of the inner expansion describes the near flow field as a Stokes flow, which is characterized by the singularities arranged at the two foci of the ellipse. The first three terms $O({\mathbb R}) ({\mathbb R}$ = Reynolds number) in the inner expansion are developed, and the flow aspects under the influence of the fluid inertia are investigated. The streamline patterns with one or two vortices round a finite flat plate of zero thickness, which is a special case of the elliptic cylinder, are presented.

A circular shear zone is created in a thin layer of fluid. The Kelvin-Helmholtz instability induces regular, steady patterns of m vortices. The experimental conditions are such that neither the centrifugal nor the Coriolis forces play a role in the motion. The state of the flow is defined by a Reynolds number, the value of which is controlled by the imposed velocities. The pattern of vortices can be characterized by its wavevector k or by m, the order of its symmetry. As k is quantized, its evolution, due to an increase or a decrease of the controlled stress, leads to transitions between patterns of different m. The transitory states between different symmetries are investigated. The experiments are performed with a soap film which provides a new type of visualization of an air flow.

This paper describes an experimental investigation of transport processes in the near wake of a circular cylinder at a Reynolds number of 140000. The flow in the first eight diameters of the wake was measured using X-array hot-wire probes mounted on a pair of whirling arms. This flying-hot-wire technique increases the relative velocity component along the probe axis and thus decreases the relative flow angle to usable values in regions where fluctuations in flow velocity and direction are large. One valuable fringe benefit of the technique is that rotation of the arms in a uniform flow applies a wide range of relative flow angles to the X-arrays, making them inherently self-calibrating in pitch. An analog circuit was used to generate an intermittency signal, and a fast surface-pressure sensor was used to generate a phase signal synchronized with the vortex-shedding process. The phase signal allowed sorting of the velocity data into 16 populations, each having essentially constant phase. An ensemble average for each population yielded a sequence of pictures of the instantaneous mean flow field, with the vortices frozen as they would be in a photograph. In addition to globally averaged data for velocity and stress, the measurements yield non-steady mean data (in the sense of an average a t constant phase) for velocity, intermittency, vorticity, stress and turbulent-energy production as a function of phase for the first eight diameters of the near wake. The stresses were resolved into a contribution from the periodic motion and a contribution from the random motion at constant phase. The two contributions are found to have comparable amplitudes but quite different geometries, and the time average of their sum (the conventional global Reynolds stress) therefore has a quite-complex structure. The non-steady mean-vorticity field is obtained with good resolution as the curl of the non-steady mean-velocity field. Less than half of the shed circulation appears in the vortices, and there is a slow decay of this circulation for each shed vortex as it moves downstream. In the discussion, considerable emphasis is put on the topology of the non-steady mean flow, which emerges as a pattern of centres and saddles in a frame of reference moving with the eddies. The kinematics of the vortex-formation process are described in terms of the formation and evolution of saddle points between vortices in the first few diameters of the near wake. One important conclusion is that a substantial part of the turbulence production is concentrated near the saddles and that the mechanism of turbulence production is probably vortex stretching at intermediate scales. Entrainment is also found to be closely associated with saddles and to be concentrated near the upstream-facing interface of each vortex.

The characteristics of the flow around a smooth circular cylinder in the critical Reynolds-number range have been investigated experimentally on the basis of instantaneous mean-pressure-distribution measurements on the cylinder and of hot-wire velocity-fluctuation measurements in the cylinder wake. Two subregions have been identified in the critical or lower transition; the first characterized by symmetric pressure distributions, intense vortex shedding, and gradual and significant variations in characteristic parameters as the Reynolds number increases, and the second by intense flow oscillations associated with formation and bursting of laminar-separation bubbles on one or both sides of the cylinder, without preference for side. The spectra of the velocity fluctuations in the second, unsteady subrange appear in general with broad band peaks. The spectral peak in the twin-bubble regime which follows this second subrange is sharp but has little energy compared to subcritical peaks.

The present study examines the motion of a slender body in the presence of a plane fluid–fluid interface with an arbitrary viscosity ratio. The fluids are assumed to be at rest at infinity, and the particle is assumed to have an arbitrary orientation relative to the interface. The method of analysis is slender-body theory for Stokes flow using the fundamental solutions for singularities (i.e. Stokeslets and potential doublets) near a flat interface. We consider translation and rotation, each in three mutually orthogonal directions, thus determining the components of the hydrodynamic resistance tensors which relate the total hydrodynamic force and torque on the particle to its translational and angular velocities for a completely arbitrary translational and angular motion. To illustrate the application of these basic results, we calculate trajectories for a freely rotating particle under the action of an applied force either normal or parallel to a flat interface, which are relevant to particle sedimentation near a flat interface or to the processes of particle capture via drop or bubble flotation.

The second-order transport terms in a monatomic gas, originally derived by Burnett from Boltzmann's equation via the Chapman–Enskog iteration, have the curious property of being dependent on the observer's reference frame. However, frame-indifference has long been accepted as an essential property of constitutive equations in continuum mechanics. Various attempts have been made to find errors in the kinetic theory, but these have been countered by physical accounts of the framedependent terms that are independent of the Chapman–Enskog theory.

It is shown in this paper that both the Chapman–Enskog theory and the physical models are based on an inappropriate definition of the peculiar velocity. The Chapman–Enskog theory is easily corrected, and the result is a kinetic theory in harmony with the principle of material frame-indifference. A brief survey of the debate on this topic is presented, and corrected expressions are given for the Burnett terms.

Nonlinear effects on the refraction of water waves are discussed. The existence of conjugate solutions for wave fields, one set of which corresponds to the ‘anomalous’ refraction solutions of Peregrine & Ryrie (1983) provides the stimulus for consideration of jumps in wave-field properties. Just such a jump is described by Yue & Mei (1980) for a case where near-linear waves are reflected with small deflection by a rigid wall.

Wave jumps between conjugate solutions appear to be possible for finite-amplitude wavetrains. These are examined and the structure of wave jumps in the near-linear, small-deflection case is elucidated. They have a structure directly analogous to that of an undular bore on shallow water with surface tension (‘hydraulic analogy’). Numerical results of Yue & Mei (1980) provide valuable guidance and confirmation.

Linear waves are reflected at a caustic, and can be described with Airy functions. Although equivalent weakly nonlinear solutions exist, the results from reflection by a wall and from use of the hydraulic analogy show that, unlike the caustics of linear theory, nonlinear caustics should not be considered in isolation. Caustic cusps, or wave focusing, must be considered, unless bed topography has discontinuities. A qualitative discussion of focusing based on the behaviour of unsteady waves in the hydraulic analogy shows that wave jumps can be expected. The relationship to linear theory is also put in perspective. Nonlinearity causes the linear rays to split into two sets of characteristics. The splitting of a ray focus leads to two wave jumps.

Consideration of the case of a semi-infinite beach shows that anomalous refraction is most unlikely to occur because there is an offshore influence of the beach on the wave field which changes the incident wave conditions to prevent anomalous refraction.