Kinematics of nominally periodic surf zone waves have been measured in the laboratory using LDA (laser Doppler anemometry), above trough level as well as below, for weakly plunging breakers transforming into bores in shallower water. The aim was to determine, through phase- or ensemble-averaging, periodic flow structures in a two-dimensional vertical plane, from large-scale down to small-scale vortical structures. Coherent multiple vortical structures were evident at the initiation of breaking, becoming elongated along the surface during bore propagation. The initial region is likely to become more extensive as waves become more strongly plunging and could explain the difference in turbulence characteristics between plunging and spilling breakers observed elsewhere. Comparison of vorticity magnitudes with hydraulic-jump measurements showed some similarities during the initial stages of breaking, but these quickly grew less as breaking progressed into shallower water. Period-averaged kinematics and vorticity were also obtained showing shoreward mass transport above trough level and undertow below, with a thick layer of vorticity at trough level and a thin layer of vorticity of opposite rotation at the bed. There were also concentrated regions of mean vorticity near the end of the plunging region. Residual turbulence of relatively high frequency was presented as Reynolds stresses, showing marked anisotrophy. Dynamic pressure (pressure minus its hydrostatic component) was determined from the kinematics. The magnitudes of different effects were evaluated through the depth-integrated Reynolds-averaged Navier–Stokes (RANS) equations, which may be reduced to nine terms (the standard inviscid terms of the shallow-water equations conserving mass and momentum with hydrostatic pressure, and six additional terms), assuming that the complex, often aerated, free surface is treated as a simple interface. All terms were evaluated, assuming that a space/time transformation was justified with a slowly varying phase speed, and the net balance was always small in relation to the maxima of the larger terms. Terms due to dynamic pressure and vertical dispersion (due to the vertical variation of velocity) were as significant as the three terms in the inviscid shallow-water equations; terms involving residual turbulence were insignificant. The r.m.s. (root mean square) variation of each along the slope is highly irregular, with the inertia term due to (Eulerian) acceleration always greatest. This is consistent with complex, though repetitive, coherent structures. Modelling the flow with the shallow-water equations, using the surface elevation variation at the break point as input, nevertheless gave a good prediction of the wave height variation up the slope.

We consider the flow of a dilute suspension of equisized solid spheres in a viscous fluid. The viscosity of such a suspension is dependent on the volume fraction, c, of solid particles. If the particles are perfectly smooth, then solid spheres will not come into contact, because lubrication forces resist their approach. In this paper, however, we consider particles with microscopic surface asperities such that they are able to make contact. For straining motions we calculate the O(c2) coefficient of the resultant viscosity, due to pairwise interactions. For shearing motions (for which the viscosity is undetermined because of closed orbits on which the probability distribution is unknown) we calculate the c2 contribution to the normal stresses N1 and N2. The viscosity in strain is shown to be slightly lower than that for perfectly smooth spheres, though the increase in the O(c) term caused by the increased effective radius due to surface asperities will counteract this decrease. The viscosity decreases with increasing contact friction coefficient. The normal stresses N1 and N2 are zero if the surface roughness height is less than a critical value of 2.11 × 10−4 times the particle radius, and then become negative as the roughness height is increased above this value. N1 is larger in magnitude than N2.

The behaviour of internal gravity waves near a critical level is investigated by means of a transient two dimensional finite difference model. All the important non-linear, viscosity and thermal conduction terms are included, but the rotational terms are omitted and the perturbations are assumed to be incompressible. For Richardson numbers greater than 2·0 the interaction of the incident wave and the mean flow is largely as predicted by the linear theory–very little of the incident wave penetrates through the critical level and almost all of the wave's energy and momentum are absorbed by changes in the original wind. However, these changes in the wind are centred above the critical level, so that the change in the wind has only a small effect on the height of the critical level. For Richardson numbers less than 2·0 and greater than 0·25 a significant fraction of the incident wave is reflected, part of which could have been predicted by the linear theory. For these stable Richardson numbers a steady state is apparently reached where the maximum wind change continues to grow slowly, but the minimum Richardson number and wave magnitudes remain constant. This condition represents a balance between the diffusion outward of the added momentum and the rate at which it is absorbed. For Richardson numbers less than 0·25, over-reflexion, predicted from the linear theory, is observed, but because the system is dynamically unstable no over-reflecting steady state is ever reached.

A companion paper (Part 1) employed a three-dimensional numerical simulation to examine the vorticity dynamics of the initial instabilities of a breaking internal gravity wave in a stratified, sheared, compressible fluid. The present paper describes the vorticity dynamics that drive this flow to smaller-scale, increasingly isotropic motions at later times. Following the initial formation of discrete and intertwined vortex loops, the most important interactions are the self-interactions of single vortex tubes and the mutual interactions of multiple vortex tubes in close proximity. The initial formation of vortex tubes from the roll-up of localized vortex sheets gives the vortex tubes axial variations with both axisymmetric and azimuthal-wavenumber-2 components. The axisymmetric variations excite axisymmetric twist waves or Kelvin vortex waves which propagate along the tubes, drive axial flows, deplete the tubes' cores, and fragment the tubes. The azimuthal-wavenumber-2 variations excite m=2 twist waves on the vortex tubes, which undergo strong amplification and unravel single vortex tubes into pairs of intertwined helical tubes; the vortex tubes then burst or fragment. Reconnection often occurs among the remnants of such vortex fragmentation. A common mutual interaction is that of orthogonal vortex tubes, which causes mutual stretching and leads to long-lived structures. Such an interaction also sometimes creates an m=1 twist wave having an approximately steady helical form as well as a preferred sense of helicity. Interactions among parallel vortex tubes are less common, but include vortex pairing. Finally, the intensification and roll-up of weaker vortex sheets into new tubes occurs throughout the evolution. All of these vortex interactions result in a rapid cascade of energy and enstrophy toward smaller scales of motion.

The propagation of gravity fronts of high density ratios has been studied experimentally (exchange flow) and by computer simulation. Non-Boussinesq fronts are known to occur in certain safety problems (chemical spills and fires), and we have investigated seven gas combinations giving density ratios from near unity to well over 20. The results are presented in terms of a density parameter ρ* which remains finite both in the weak (ρ* = 0) and the strong (ρ* = 1) limit. The front velocities, measured by means of hot wires, are found to fall on two distinct curves, one for the slower lightgas fronts and one for the faster heavy-gas fronts. Two fractional depths, Φ = ½ (lock exchange) and Φ = ⅙, have been investigated in detail and results for the interesting case Φ → 0 have been obtained by extrapolation. To aid in the extrapolation and for comparison, all experimental (and some intermediate) cases have been simulated by means of a general purpose CFD-code (PHOENICS). Good agreement is found for cases without convergence problems, i.e. for heavy-gas fronts of density ratio less than 5. Further information on frontal shape etc. has been obtained from visualization. The extrapolations to infinite depth indicate a limiting speed for both the heavy- and light-gas fronts close to the values predicted from shallow-layer theory for the analogous dam-break problem.

The far turbulent wake of a slightly heated cylinder was studied using our specially developed four-wire probe and digital processing of signals obtained. The processing method included a simultaneous solution of full nonlinear response equations for the four sensors yielding values of instantaneous velocity vector and temperature. The accuracy of results was improved substantially by the subtraction of free-stream velocity and temperature variations, first-order corrections for the d.c. drift of the signals, the elimination of 60 Hz related noise, and instantaneous correction for the streamwise displacement between the sensor wires. A low temperature threshold was used to discriminate between turbulent and potential flow.

The frequency of crossing of the turbulent-potential interface and statistics of turbulent and potential zone intervals are presented. Conditionally sampled fluctuating velocity and temperature fields are given for turbulent and potential zones as well as for a fluid at the downstream and upstream interface of the turbulent zones. Averaging was done with respect to conventional (Reynolds) as well as local zone or point means. The results on turbulence structure up to second-order correlations are presented in this paper.

It is hoped that the data will be useful for those interested in methods of prediction of turbulence, especially those based on the consideration of large structures. Researchers more interested in the physical phenomena involved might also find the data suitable for further analysis and interpretation.

We have obtained a closed-form solution for the sound radiation from multipole sources imbedded in an infinite cylindrical jet with an arbitrary velocity profile. It is valid in the limit where the wavelength is large compared with the jet radius. Simple formulae for the acoustic pressure field due to convected point sources are also obtained. The results show (in a simple way) how the mean flow affects the radiation pattern from the sources. For convected lateral quadrupoles it causes the exponent n of the Doppler factor (1 - M cosθ)−n multiplying the far-field pressure signal to be increased from the value of 3 used by Lighthill to 5.

We consider a disturbance that evolves from a strictly linear finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatial-evolution, nonlinear-convection, and viscous-diffusion terms are of the same order of magnitude in the interactive critical-layer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern vis-a-vis the inviscid case, and the instability-wave growth ultimately becomes algebraic. This leads to a new dominant balance between linear- and nonlinear-convection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat's-eye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.

A consistent slender-body approximation is developed for the flow past a fish- like body with arbitrary combinations of body thickness and low-aspect-ratio fin appendages, but with the fins confined to the plane of symmetry of the body. Attention is focused on the interaction of the fin lifting surfaces with the body thickness, and especially on the dynamics of the vortex sheets shed from the fin trailing edges. This vorticity is convected by the (non-lifting) flow past the stretched-straight body, and departs significantly from the purely longitudinal orientation of conventional lifting-surface theory. Explicit results are given for axisymmetric bodies having fins with abrupt trailing edges, and calculations of the total lift force are presented for bodies with symmetric and asymmetric fin configurations, moving with a constant angle of attack.

The novel idea is to use a disk pump to drive each fluid layer independently round a closed-return density-stratified water channel. A small-scale model was built and tested to demonstrate the feasibility of the idea. A hydrogen-bubble technique was used for both flow visualization and velocity measurement. Density measurements were made with an electrical conductivity probe developed by the authors. The tests have demonstrated the usefulness of such a facility, especially for long steady-state experiments in density-stratified flows.

The near-wall flow structure of a zero-pressure-gradient flat-plate turbulent boundary layer with a single-layer viscoelastic compliant surface was visualized using the hydrogen-bubble technique. The compliant materials were made by mixing silicone elastomer with silicone oil. The flow-visualization experiments indicated low-speed wall streaks with increased spanwise spacing and elongated spatial coherence compared to those obtained on a rigid surface. More interestingly, an intermittent relaminarization-like phenomenon was observed at low Reynolds numbers for the particular compliant surface investigated. Apparently, the observed changes in the near-wall flow structure over the compliant surface are caused by the stable interaction between the compliant surface and the turbulent flow-field. Optical holographic interferometry and laser Doppler velocimetry were also employed to obtain the basic parameters of the turbulent boundary layers and the flow-induced compliant-surface displacements to better understand the physics of the interaction between a turbulent boundary layer and a passive compliant surface.

Interactions between waves in a stably stratified fluid with a free or fixed upper surface may occur at second order when the horizontal wave-numbers, ki, and frequencies, σi, of the three interacting waves satisfy the relations \[ {\bf k}_1 \pm{\bf k}_2 \pm {\bf k}_3 = 0,\quad \sigma_1 \pm \sigma_2 \pm \sigma_3 = 0. \] These relations may be satisfied in the case when two free surface waves interact with an internal wave, or in the case when all three waves are internal, provided that they do not all belong to the same mode. The theory is applied to situations which might be realized in the laboratory.

Receptivity of compressible mixing layers to general source distributions is examined by a combined theoretical/computational approach. The properties of solutions to the adjoint Navier–Stokes equations are exploited to derive expressions for receptivity in terms of the local value of the adjoint solution. The result is a description of receptivity for arbitrary small-amplitude mass, momentum, and heat sources in the vicinity of a mixing-layer flow, including the edge-scattering effects due to the presence of a splitter plate of finite width. The adjoint solutions are examined in detail for a Mach 1.2 mixing-layer flow. The near field of the adjoint solution reveals regions of relatively high receptivity to direct forcing within the mixing layer, with receptivity to nearby acoustic sources depending on the source type and position. Receptivity ‘nodes’ are present at certain locations near the splitter plate edge where the flow is not sensitive to forcing. The presence of the nodes is explained by interpretation of the adjoint solution as the superposition of incident and scattered fields. The adjoint solution within the boundary layer upstream of the splitter-plate trailing edge reveals a mechanism for transfer of energy from boundary-layer stability modes to Kelvin–Helmholtz modes. Extension of the adjoint solution to the far field using a Kirchhoff surface gives the receptivity of the mixing layer to incident sound from distant sources.

The morning glory is a meteorological phenomenon which occurs in northern Australia and takes the form of a series of roll clouds. The morning glory is generated by the interaction of nocturnal seabreezes over Cape York Peninsula and propagates in a south-westerly direction over the Gulf of Carpentaria. In the present work, it is shown that the morning glory can be modelled by the resonant flow of a two-layer fluid over topography, the topography being the mountains of Cape York Peninsula. In the limit of a deep upper layer, the equations of motion reduce to a forced Benjamin–Ono equation. In this context, resonant means that the underlying flow velocity of the seabreezes is near a linear long-wave velocity for one of the long-wave modes. The morning glory is then modelled by the undular bore (simple wave) solution of the modulation equations for the Benjamin–Ono equation. This modulation solution is compared with full numerical solutions of the forced Benjamin–Ono equation and good agreement is found when the wave amplitudes are not too large. The reason for the difference between the numerical and modulation solutions for large wave amplitude is also discussed. Finally, the predictions of the modulation solution are compared with observational data on the morning glory and good agreement is found for the pressure jump due to the lead wave of the morning glory, but not for the speed and half-width of this lead wave. The reasons for this are discussed.

We perform particle image velocimetry (PIV) measurements of various terms of the non-homogeneous Kármán–Howarth–Monin equation in the most inhomogeneous and anisotropic region of grid-generated turbulence, the production region which lies between the grid and the peak of turbulence intensity. We use a well-documented fractal grid which is known to magnify the streamwise extent of the production region and abate its turbulence activity. On the centreline around the centre of that region the two-point advection and transport terms are dominant and the production is significant too. It is therefore impossible to apply usual Kolmogorov arguments based on the Kármán–Howarth–Monin equation and resulting dimensional considerations to deduce interscale flux and spectral properties. The interscale energy transfers at this location turn out to be highly anisotropic and consist of a combined forward and inverse cascade in different directions which, when averaged over directions, gives an interscale energy flux that is negative (hence forward cascade on average) and not too far from linear in $r$, the modulus of the separation vector $\boldsymbol{r}$ between two points. The energy spectrum of the streamwise fluctuating component exhibits a well-defined $-5/3$ power law over one decade, even though the streamwise direction is at a small angle to the inverse cascading direction.

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either raacroscopically quiescent or in linear flow at arbitrary Peclet number $Pe = \dot{\gamma}a^2/2D$, where $\dot{\gamma}$ is the shear rate, a is the particle radius, and D = kBT/6πa is the diffusion coefficient of an isolated particle. Here, kB is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by kBT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction $\phi = \frac{4}{3}\pi a^3n$ and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, D∞s, is given by the sum of D0s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D0s and D∞ are anisotropic, in general, with the anisotropy of D0s due solely to that of the steady microstructure. The influence of flow upon D∞s is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations.

The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(øPe3/2) for Pe ≤ 1 and ø ≤ 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D0s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with $\dot{\gamma}$. Regardless of whether or not the particles interact hydrodynamically, flow influences D∞s at O(øPe) and O(øPe3/2). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of O(øPe3/2), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe3/2DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor.

At high ø a scaling theory based on the approach of Brady (1994) is used to approximate D∞s. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of $\bar{P}e = \dot{\gamma}a^2/2D^s_0(\phi)$. At small $\bar{P}e$ the dependence on $\bar{P}e$ is the same as at low ø.

A study is made of intrusions of fixed volumes of fluid of constant density into a uniformly stratified and uniformly rotating environment. In particular, the eventual steady-state configuration is sought for the ideal fluid case. Exact two-dimensional solutions can be found using a co-ordinate transformation which converts the equations satisfied outside the intrusion into the Cauchy-Riemann equations. The same technique does not, however, yield linear equations in the axisymmetric case.

Effects of friction are also considered for cases where the quasi-geostrophic approximation can be made. Vertical advection by the viscosity-induced motion is shown to have the same effect on the temperature field as a greatly enhanced lateral diffusion, and therefore tends to spread the intrusion out laterally.

Nonlinear, two-dimensional magnetoconvection has been investigated numerically for a fixed Rayleigh number of 104, with the ratio ζ of the magnetic to the thermal diffusivity in the range 0·4 ≥ ζ ≥ 0·05. As the Chandrasekhar number Q is decreased, convection first sets in as overstable oscillations, which are succeeded by steady convection with dynamically active flux sheets and, eventually, with kinematically concentrated fields. In the dynamical regime spatially asymmetrical convection, with most of the flux on one side of the cell, is preferred. As Q increases, these asymmetrical solutions become time-dependent, with oscillations about the steady state which develop into large-scale oscillations with reversals of the flow. Although linear theory predicts that narrow cells should be most unstable, the nonlinear results show that steady convection occurs most easily in cells that are roughly twice as wide as they are deep.