Microscale breaking wind waves cover much of the surface of open waters exposed to moderate wind forcing. Recent studies indicate that understanding the nature and key features of the surface skin flows associated with these small waves is fundamental to explaining the dramatic enhancement of constituent exchange that occurs in their presence. We describe a laboratory study in which velocity measurements were made within a few hundred micrometres of the surface of microscale breaking wind waves without bubble entrainment, using flow visualization and particle image velocimetry (PIV) techniques for a range of wind speed and fetch conditions. Our measurements show that for each experiment, the mean surface drift directly induced by the wind on the upwind faces and crests of these waves is ($0.23\,{\pm}\,0.02$)${u}^a_\ast$ in the trough increasing to ($0.33\,{\pm}\,0.07$)${u}^a_\ast$ at the crest, where ${u}^a_\ast$ is the wind friction velocity. About these mean values, there is substantial variability in the instantaneous surface velocity up to approximately ${\pm}\,0.17{u}^a_\ast$ in the trough and ${\pm}\,0.37{u}^a_\ast$ at the crest. This variability can be attributed primarily to the modulation of the wave field, with additional contributions arising from fluctuations in wind forcing and near-surface turbulence generated by shear in the drift layer or by the influence of transient microscale breaking.

Strongly nonlinear heat transport across a porous layer is studied using Howard's (1963) variational method. The analysis explores a bifurcation property of Busse's (1969) multi-a solution of this variational problem and complements the 1972 study of Busse & Joseph by further restricting the fields which are allowed to compete for the maximum heat transported a t a given temperature difference. The restriction arises, as in the case of infinite Prandtl number convection studied by Chan (1971), from letting a parameter tend to infinity from the outset; here, however, the parameter which is assumed infinitely large (the Prandtl-Darcy number) is actually seldom smaller than O(107).

The theoretical bounding heat-transport curve is computed numerically. The maximizing Nusselt number (Nu) curve is given a t first by a functional of the single-a solution; then this solution bifurcates and the Nusselt number functional is maximized for an interval of Rayleigh numbers (R) by the two-a solution. The agreement between the numerical analysis and recent experiments is striking. The theoretical heat-transport curve is found to be continuously differentiable but has piecewise discontinuous second derivatives.

The results of an asymptotic (R → ∞) analysis following Chan (1971) are in qualitative agreement with the results of numerical analysis and give the asymptotic law Nu = 0.016R. This law is consistent with the result of the porous version of the well-known dimensional argument leading to the one-third power law for regular convection. The asymptotic results, however, do not appear to be in good quantitabive agreement with the numerical results.

We perform a direct numerical simulation (DNS) investigation of the incompressible temporally developing turbulent boundary layer. The approach is inspired by temporal simulations of flows which are generally thought of as developing in space, such as wakes and mixing layers. Compressible boundary layers have previously been studied in this manner yet the temporal approach appears to be under-exploited in the literature concerning incompressible boundary layers. The flow is the turbulent counterpart to the laminar Rayleigh problem or Stokes’ first problem, in which a fluid at rest is set into motion by a wall moving at constant velocity. An initial profile that models the effect of a wall-mounted trip wire is implemented and allows the characterisation of initial conditions by a trip Reynolds number. For the current set-up, a trip Reynolds number of 500 based on the trip-wire diameter successfully triggers transition yet only mildly perturbs the flow so it assumes a natural development at the lowest possible Reynolds number based on momentum thickness. A systematic trip study reveals that as the ratio of momentum thickness to trip-wire diameter approaches unity, our flow approaches a state free from the effects of its starting trip Reynolds number. The transport of a passive scalar by this flow is also simulated. The role played by domain size is investigated with two boxes, sized to accommodate two chosen final Reynolds numbers. Comparisons of the skin friction coefficient, velocity and scalar statistics demonstrate that the temporally developing boundary layer is a good model for the spatially developing boundary layer once initial conditions can be neglected. Analysis of similarity solutions suggests such a rapprochement of the spatial and temporal boundary layers may be expected at high Reynolds numbers given that the only terms that asymptotically persist are those common to both cases. If one seeks statistics for the turbulent boundary layer, the temporal boundary layer is therefore a viable method if modest convergence is sufficient. We suggest that such a temporal set-up could prove useful in the study of turbulence dynamics.

In a stably stratified shear layer, thin vorticity layers (‘baroclinic layers’) are produced by buoyancy effects and strain in between the Kelvin–Helmholtz vortices. A two-dimensional numerical study is conducted, in order to investigate the stability of these layers. Besides the secondary Kelvin–Helmholtz instability, expected but never observed previously in two-dimensional numerical simulations, a new instability is also found.

The influence of the Reynolds number (Re) upon the dynamics of the baroclinic layers is first studied. The layers reach an equilibrium state, whose features have been described theoretically by Corcos & Sherman (1976). An excellent agreement between those predictions and the results of the numerical simulations is obtained. The baroclinic layers are found to remain stable almost up to the time the equilibrium state is reached, though the local Richardson number can reach values as low as 0.05 at the stagnation point. On the basis of the work of Dritschel et al. (1991), we show that the stability of the layer at this location is controlled by the outer strain field induced by the large-scale Kelvin–Helmholtz vortices. Numerical values of the strain rate as small as 3% of the maximum vorticity of the layer are shown to stabilize the stagnation point region.

When non-pairing flows are considered, we find that only for Re ≤ 2000 does a secondary instability eventually amplify in the layer. (Re is based upon half the initial vorticity thickness and half the velocity difference at the horizontally oriented boundaries.) This secondary instability is not of the Kelvin–Helmholtz type. It develops in the neighbourhood of convectively unstable regions of the primary Kelvin–Helmholtz vortex, apparently once a strong jet has formed there, and moves along the baroclinic layer while amplifying. It next perturbs the layer around the stagnation point and a secondary instability, now of the Kelvin–Helmholtz type, is found to develop there.

We next examine the influence of a pairing upon the flow behaviour. We show that this event promotes the occurrence of a secondary Kelvin–Helmholtz instability, which occurs for Re ≥ 400. Moreover, at high Reynolds number (≥ 2000), secondary Kelvin–Helmholtz instabilities develop successively in the baroclinic layer, at smaller and smaller scales, thereby transferring energy towards dissipative scales through a mechanism eventually leading to turbulence. Because the vorticity of such a two-dimensional stratified flow is no longer conserved following a fluid particle, an analogy with three-dimensional turbulence can be drawn.

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or with each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point-singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head–tail reversal of the swimmer, in the near field, microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of approximately two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

The present work deals with the buckling phenomenon characteristic of highly viscous liquid jets slowly impinging upon a plate. The quasi-one-dimensional equations of the dynamics of thin liquid jets are used as the basis for the theoretical analysis of buckling. With the problem linearized, the characteristic equation is obtained. Its solutions show that instability (buckling) sets in only in the presence of axial compression in the jet, and when the distance between the nozzle exit and the plate exceeds some critical value. The latter is calculated. It is shown that buckling instability corresponds to the rectilinear jet/folding jet bifurcation point. The value of the folding frequency is calculated at the onset of buckling. The theoretical results are compared with Cruickshank & Munson's (1981) and Cruickshank's (1988) experimental data and the agreement is fairly good.

Experiments on bifurcation of rotating liquid drops into two-lobed shapes were conducted during a Space shuttle flight. The drops were levitated in air and spinned using acoustic fields in the low-gravity environment. These experiments have successfully resolved the discrepancies existing between the previous experimental results and the theoretical predictions. In the simplest case of a rotating drop that is free from deformation by external forces, the results agree well with the existing theoretical predictions. In the case of a rotating drop subjected to flattening by the acoustic radiation stress, deliberately or otherwise, the experiments suggest the existence of a family of curves, with the free drop as the limiting case.

An initial value problem with relevance to jet noise is investigated. A plane parallel jet flow is subjected to a spatially localized initial disturbance and is then left to evolve according to the two-dimensional compressible Navier–Stokes equations. The hydrodynamic response is in the form of a convecting vortex packet. The Ffowcs Williams–Hawkings approach is formulated in the time domain and used to extrapolate from the simulated near field to the acoustic far field. The predominant downstream sound radiation comes from an early stage of nonlinear development of the vortex packet. Two simplified models to account for the radiation are introduced, based on nonlinear mode interactions on a prescribed base flow. The first uses two sets of linearized Euler equations, coupled via the inviscid Lilley–Goldstein acoustic analogy. This formulation separates the linear sound field from the sound field driven by nonlinear interactions; qualitative agreement of the latter with the Navier–Stokes computations demonstrates the importance of nonlinear interactions. The second model uses combinations of linear inviscid eigenmodes to drive the sound field, which allows extraction of the dominant mode interactions responsible for the observed radiation pattern. The results indicate that a difference-wavenumber nonlinear interaction mechanism dominates sound radiation from subsonic instability modes in shear flows.

The physical mechanisms underlying the relaxation process leading to thermal equilibrium behind ionizing shock waves in argon have been studied through use of optical techniques. The non-equilibrium condition in the relaxation region was investigated experimentally by measuring the shift in the fringes due to a change in the refractive index of the medium with a Mach–Zehnder interferometer. Both electron- and mass-density profiles from the shock front to the equilibrium region were determined. The experimental work has been supplemented by a theoretical analysis of the ionization mechanism to explain the measured profiles and relaxation times.

This paper investigates the three-dimensional laminar boundary layer over a blunt body (a prolate spheroid) at low incidence and with reversed flow. Results reflecting the general characteristics of such a problem are presented. More significant are the features relating to the circumferential flow reversal. Some of these features confirm our early hypotheses concerning the existence of a reversed region ahead of separation and the role of the zero-cfθ line in the general context of separation in three dimensions. Other features are unexpected, including the distribution of cfμ and the shape of the separation line. Here cfθ and cfμ denote, respectively, the circumferential and meridional components of the skin friction.

Direct numerical solutions of the incompressible Navier–Stokes equations have been obtained under the Boussinesq approximation for the temporal evolution of a turbulent jet-like flow subjected to off-source volumetric heating, of the kind that occurs in a cloud due to latent heat release on condensation of water vapour. The results show good qualitative agreement with available experimental data on spatially growing jets. Thus, heating accelerates the flow and arrests jet growth; and turbulence velocities increase with heating but not as rapidly as mean velocities, so normalized intensities drop. It is shown that the baroclinic torque resulting from the heating enhances the vorticity dramatically in all three directions, with a preferential amplification at the higher wavenumbers that results in a rich fine structure at later times in the evolution of the jet. Streamwise vortex pairs, rendered stronger by mean flow acceleration, appear to be responsible for large expulsive motions at certain transverse cross-sections in the ambient fluid near the heated flow; together with the disruption of the toroidal component of the coherent vorticity achieved by heating, this results in an entraining velocity field that is qualitatively different from that around unheated turbulent jets. This mechanism may provide a plausible explanation for the experimentally observed drop in entrainment with off-source heating.

A floating horizontal cylinder of infinite length is performing simple harmonic oscillations of small amplitude in the free surface of a uniform inviscid fluid under gravity. The cylinder intersects the mean free surface at right angles, and the fluid is bounded below by a fixed horizontal plane. Let the corresponding two-dimensional velocity potential be expressed as a distribution of simple wave sources over the boundary of the body. Then it is known that the source density satisfies a Fredholm integral equation of the second kind which has a unique solution except at those frequencies (the irregular frequencies) at which the Fredholm determinant vanishes. The present work is concerned with the irregular frequencies. Let the simple wave sources be replaced by a fundamental solution which consists of a simple wave source together with additional wave singularities inside the cylinder. It is shown how irregular frequencies can be eliminated by an appropriate choice of these singularities.

We present an experimental investigation of entrainment and the dynamics near the turbulent/non-turbulent interface in a dense gravity current. The main goal of the study is to investigate changes in the interfacial physics due to the presence of stratification and to examine their impact on the entrainment rate. To this end, three-dimensional data sets of the density and the velocity fields are obtained through a combined scanning particle tracking velocimetry/laser-induced fluorescence approach for two different stratification levels with inflow Richardson numbers of $\mathit{Ri}_{0}=0.23$ and $\mathit{Ri}_{0}=0.46$, respectively, at a Reynolds number around $\mathit{Re}_{0}=3700$. An analysis conditioned on the instantaneous position of the turbulent/non-turbulent interface as defined by a threshold on enstrophy reveals an interfacial region that is in many aspects independent of the initial level of stratification. This is reflected most prominently in matching peaks of the gradient Richardson number $\mathit{Ri}_{g}\approx 0.1$ located approximately $10{\it\eta}$ from the position of the interface inside the turbulent region, where ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ is the Kolmogorov scale, and ${\it\nu}$ and ${\it\epsilon}$ denote the kinematic viscosity and the rate of turbulent dissipation, respectively. A possible explanation for this finding is offered in terms of a cyclic evolution in the interaction of stratification and shear involving the buildup of density and velocity gradients through inviscid amplification and their subsequent depletion through molecular effects and pressure. In accordance with the close agreement of the interfacial properties for the two cases, no significant differences were found for the local entrainment velocity, $v_{n}$ (defined as the propagation velocity of an enstrophy isosurface relative to the fluid), at different initial stratification levels. Moreover, we find that the baroclinic torque does not contribute significantly to the local entrainment velocity. Comparing results for the surface area of the convoluted interface to estimates from fractal scaling theory, we identify differences in the interface geometry as the major factor in the reduction of the entrainment rate due to density stratification.

A surface distribution of stress is imposed on an ocean enclosed by two continental boundaries; the resulting transport circulation is studied between two latitudes of zero surface wind-stress curl, within which the curl reaches a single maximum. Under the assumption that turbulent transfer of relative vorticity has a minimum effect on the mean circulation, inviscid flow patterns are deduced in the limit of small transport Rossby number. Inertial currents, or naturally scaled regions of high relative vorticity, occur on both the eastern and the western continental coasts. Limits on the relative transports of the currents are obtained and found to depend on the direction of variation of the wind-stress curl with latitude, relative to that of the Coriolis accelerations. The most striking feature of the inviscid flow is a narrow inertial current the axis of which lies along the latitude of maximum wind-stress curl. All eastward flow occurs in this midlatitude jet.

A feature of the flow which cannot remain essentially free of turbulent processes is the integrated vorticity relationship, since the imposed wind-stress distribution acts as a net source of vorticity for the ocean. Heuristic arguments are used together with this integral constraint to deduce the presence and strength of the turbulent diffusion which must occur in the region of the mid-latitude jet. It is further inferred that the turbulent meanders of the jet must effect a net meridional transport of relative vorticity.

Numerical solutions are presented for steady two-dimensional incompressible flow past an infinite row of cylinders (of unit radii, with distances W between their centres). The calculations cover R [les ] 700 for 5 [les ] W [les ] ∞ and also R = 800 for 5 [les ] W [les ] 100 (where R denotes the Reynolds number based on the cylinder diameters). The recirculation regions (wake bubbles) are found to grow in length approximately linearly with R in all cases. For high values of R, a major change occurs in their character when W is increased past Wcrit ≈ 16. While they have remained slender up to this point (essentially only stretching in length in proportion to R), their centres of circulation have moved towards their ends. As W is further increased, the wake bubbles widen rapidly, beginning from the rear of the wakes. In the limit of W→∞, the present results agree with the previous ones for a single cylinder as reported by Fornberg (1985).

Ensemble averaging combined with multiple scattering ideas is applied to the Stokes flow over a stochastic rough surface. The surface roughness is modelled by compact protrusions on an underlying smooth surface. It is established that the effect of the roughness on the flow far from the boundary may be represented by replacing the no-slip condition on the exact boundary by a partial slip condition on the smooth surface. An approximate analysis is presented for a sparse distribution of arbitrarily shaped protrusions and explicit numerical results are given for hemispheres. Analogous conclusions for the two-dimensional case are obtained. It is shown that in certain cases a traction force develops on the surface at an angle with the direction of the flow.

This paper examines the problem of the survival of Salter's duck wave-energy device in extreme waves, both experimentally in a narrow tank at Edinburgh University and theoretically, using programs originally designed to study nonlinear ship motions (Vinje & Brevig 1981). Both approaches are essentially two-dimensional, corresponding to waves normally incident upon the duck string, and comparison of the two sets of results for a duck on a fixed axis shows good agreement.

Linear perturbation analysis is applied to the problem of the onset of convection in a horizontal layer of fluid heated uniformly from below, when the fluid is bounded below by a rigid plate of inlinite conductivity and above by a solid layer of finite conductivity and finite thickness. The critical Rayleigh number and wave-number are found for various thickness ratios and thermal conductivity ratios. Both numbers are reduced by the presence of a boundary of finite (rather than infinite) conductivity in qualitative agreement with the observation of Koschmieder (1966).

The analysis of the sound field generated by the passage of isotropic turbulence through a shock of finite strength (Ribner 1953, 1954) has been extended to provide the flux of acoustic energy emanating from unit area on the downstream side of the shock. This is motivated by the problem of estimating the sound power emerging from a supersonic jet containing shock waves. The energy flux is found to vary almost linearly with shock density ratio, reaching a maximum at infinite Mach number of 0[sdot ]062 of the flux of turbulence kinetic energy convected into unit area of the shock.

Direct comparison with a result obtained by Lighthill (1953) is misleading. His energy relations, reckoned relative to a frame moving with the fluid, must be converted to the shock-fixed frame used herein. The converted results of his theory (weak shocks) and the results of our theory (arbitrary shocks) appear to show a similar asymptotic behaviour for vanishing shock strength; they diverge with increasing shock strength.