Radial flow takes place in a heterogeneous porous formation of random and stationary log-conductivity Y(x), characterized by the mean 〈Y〉, the variance σ2Y and the two- point autocorrelation ρY which in turn has finite and different horizontal and vertical integral scales, I and Iv, respectively. The steady flow is driven by a head difference between a fully penetrating well and an outer boundary, the mean velocity U being radial. A tracer is injected for a short time through the well envelope and the thin plume spreads due to advection by the random velocity field and to pore-scale dispersion. Transport is characterized by the mean front r=R(t) and by the second spatial moment of the plume Srr. Under ergodic conditions, i.e. for a well length much larger than the vertical integral scale, Srr is equal to the radial fluid trajectory variance Xrr.

The aim of the study is to determine Xrr(t) for a given heterogeneous structure and for given pore-scale dispersivities. The problem is more complex than the similar one for mean uniform flow. To simplify it, the well is replaced by a line source, the domain is assumed to be infinite and a first-order approximation in σ2Y is adopted. The solution is still difficult, being expressed with the aid of a few quadratures. It is found, however, that it can be derived quite accurately for a sufficiently small anisotropy ratio e=Iv/I by retaining only one term of the velocity two-point covariance. This major simplification leads to simple calculations and even to analytical solutions in the absence of pore-scale dispersion.

To compare the results with those prevailing in homogeneous media, apparent and equivalent macrodispersivities are defined for convenience.

The major difference between transport in radial and uniform flow is that the asymptotic, large-time, apparent macrodispersivity in the former is smaller by a factor of 3 than in the latter. For a three-dimensional point source the reduction is by a factor of 5. This effect is explained by the rapid change of the mean velocity during the period in which the velocities of two particles injected at the source become uncorrelated.

In contrast, the equivalent macrodispersivity tends to its value in uniform flow far from the well, where the flow is slowly varying in space.

High-accuracy three-dimensional numerical simulations of miscible displacements with gravity override in homogeneous porous media are carried out for the quarter five-spot configuration. Special emphasis is placed on describing the influence of viscous and gravitational effects on the overall displacement dynamics in terms of the vorticity variable. Even for neutrally buoyant displacements, three-dimensional effects are seen to change the character of the flow significantly, in contrast to earlier findings for rectilinear displacements. At least in part this can be attributed to the time dependence of the most dangerous vertical instability mode. Density differences influence the flow primarily by establishing a narrow gravity layer, in which the effective Péclet number is enhanced owing to the higher flow rate. However, buoyancy forces of a certain magnitude can lead to a pinch-off of the gravity layer, thereby slowing it down. Overall, an increase of the gravitational parameter is found to enhance mostly the vertical perturbations, while larger ${\hbox{\it Pe}}$ values act towards amplifying horizontal disturbances. The asymptotic rate of growth of the mixing length varies only with Péclet number. For large Péclet numbers, an asymptotic value of 0.7 is observed. A scaling law for the thickness of the gravity layer is obtained as well. In contrast to immiscible flow displacements, it is found to increase with the gravity parameter.

The spatial signatures of retrograde spanwise vortices in wall turbulence are assessed from particle-image velocimetry measurements in the streamwise–wall-normal plane of a zero-pressure-gradient turbulent boundary layer at Reτ≡u*δ/ν=2350. The present results suggest that a proportion of retrograde spanwise vortices have a well-defined spatial relationship with neighbouring prograde vortices. Two-point cross-correlations and conditionally averaged velocity fields given a retrograde vortex reveal that such structures are typically oriented either upstream of and below or downstream of and above a prograde core. While these pairings are consistent with the typical-eddy patterns reported by Falco and co-workers, we offer an alternative interpretation for a proportion of these retrograde/prograde pairs. In particular, the arrangement of a retrograde spanwise vortex upstream of and below a prograde core is also consistent with the spatial signature revealed if an omega-shaped hairpin structure were sliced through its shoulder region by a fixed streamwise–wall-normal measurement plane.

An experiment has been carried out to verify the existence of the resonant interaction between trains of gravity waves, predicted by Phillips (1960). As suggested by Longuet-Higgins (1962), two trains of waves in mutually perpendicular directions were generated in a rectangular wave tank. The ratio σ1/σ2 of the wave frequencies was varied (1·4 < σ1/σ2 < 2·1). When σ1/σ2 [eDot ] 1·7357 it was expected that a resonant interaction would take place, generating a wave of frequency (2σ1−σ2). The amplitude of the third wave was expected to increase almost linearly in the direction of wave propagation. The shape of the response curve as a function of σ1/σ2 was also predicted.

In the present experiments rather large wave amplitudes had to be used, and the theoretical shape of the response curve was distorted by non-linear detuning. Nevertheless the peak amplitude of the resonant wave was found to increase with distance in very nearly the manner predicted.

These experiments were carried out in 1961 but publication was deferred pending a similar but more accurate investigation by McGoldrick, Phillips, Huang & Hodgson (1966). Much of the theoretical discussion given in the present paper is relevant to their work.

Direct numerical simulations are employed to investigate the effects of discrete suction orifices at the wall on the disturbance evolution in laminar three-dimensional boundary-layer flows with favourable pressure gradient. Suction panels with many suction orifices can excite unstable crossflow (CF) modes even if the orifice spacing is smaller than the chordwise/spanwise wavelengths of unstable modes, caused by imperfections in the orifice order or suction strength. It has been found that the most unstable steady vortex mode leads to strong CF vortices that invoke turbulence by secondary instability even on the active suction panel. The deliberate excitation and support of stabilizing vortices that have less than two-thirds of the spanwise wavelength of the most amplified ones – known from the upstream flow deformation or micrometre-sized roughness elements technique – by the orifice order on the panel can secure the desired stabilizing effect of suction and lower the necessary suction amount significantly.

The response, evolution, and modelling of subgrid-scale (SGS) stresses during rapid straining of turbulence is studied experimentally. Nearly isotropic turbulence with low mean velocity and Rλ˜290 is generated in a water tank by means of spinning grids. Rapid straining (axisymmetric expansion) is achieved with two disks pushed towards each other at rates that for a while generate a constant strain rate. Time-resolved, two-dimensional velocity measurements are performed using cinematic PIV. The SGS stress is subdivided to a stress due to the mean distortion, a cross-term (the interaction between the mean and turbulence), and the turbulent SGS stress τ(T)ij. Analysis of the time evolution of τ(T)ij at various filter scales shows that all scales are more isotropic than the prediction of rapid distortion theory, with increasing isotropy as scales decrease. A priori tests show that rapid straining does not affect the high correlation and low square-error exhibited by the similarity model. Analysis of the evolution of total SGS energy dissipation reveals, surprisingly, that the Smagorinsky model with a constant coefficient (determined from isotropic turbulence data) underpredicts the dissipation during rapid straining. While the partial dissipation −〈τ(T)ijS˜ij〉 (due only to the turbulent part of the stress) is overpredicted by the Smagorinsky model, addition of the cross-terms reverses the trend. The similarity model with a constant coefficient appropriate for isotropic turbulence, on the other hand, overpredicts SGS dissipation. Owing to these opposite trends a linear combination of both models (mixed model) provides better prediction of SGS dissipation during rapid straining. However, the mixed model with coefficients determined from dissipation balance underpredicts the SGS stress.

The motion of a spherical particle near the interface between two immiscible viscous fluids undergoing simple shear flow is considered in the limit of small Reynolds and capillary numbers where the interface exhibits negligible deformation. Taking advantage of the rotational symmetry of the boundaries of the flow with respect to the axis that is normal to the interface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown components of the traction and velocity along the particle and interface contours. The results document the particle translational and angular velocities, and reveal that the particle slips while rolling over the interface under the influence of a simple shear flow, for any viscosity ratio. In the second part of the investigation, the motion of an axisymmetric particle straddling a planar interface is considered. The results confirm a simple exact solution when a particle with top-down symmetry is immersed half-way in each fluid and translates parallel to the interface, reveal a similar simple solution for a particle that is held stationary in simple shear flow, and document the force and torque exerted on a spherical particle for more general arrangements. The onset of a non-integrable singularity of the traction at the contact line prohibits the computation of the translational and angular velocities of a freely suspended particle convected under the action of a shear flow.

Experiments are conducted to generate progressive wave fields in deep water with two-dimensional surface patterns for which two parameters are systematically varied: (i) the aspect ratio of the cells comprising the surface patterns and (ii) a measure of nonlinearity of the input wave field. The goal of these experiments is to determine whether these patterns persist, what their main features are, whether standard models of waves describe these features, and whether there are parameter regimes in which the patterns are stable. We find that in some parameter regimes, surface patterns in deep water do persist with little change of form during the time of the experiment. In other parameter regimes, particularly for large-amplitude experiments, the patterns evolve more significantly. We characterize the patterns and their evolutions with a list of observed features. To describe the patterns and features, we consider two models: ($a$) the standard ($2+1$) nonlinear Schrödinger equation and ($b$) coupled nonlinear Schrödinger equations for two interacting wavetrains. Exact solutions of these models provide qualitative explanations for many of the observed features.

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a continuous flexible substrate. Of the two rolls that make a forward or reverse roll coating gap, one is often covered by a layer of more-or-less deformable elastomer. Liquid carried into the converging side of the nip can develop high enough pressure to deform the resilient cover, which changes the nip profile and thus alters the velocity and pressure field. This elastohydrodynamic coupled action is not yet well understood. Theoretical modelling has to take into account the viscous flow, the roll deformation and the free-surface effects in order to predict the flow behaviour.

In this work the flow between a rigid and a deformable counter-rotating roll that shares the same angular speed is described by the lubrication approximation together with a viscocapillary model, based in the film-flow equation, for the film-split region. The deformation of the elastomer layer is modelled by Hookean springs oriented radially, which constitute a one-dimensional model. The stability of the system to transverse perturbation is analysed by examining the time-dependent response to infinitesimal disturbances in order to identify those that grow fastest.

The corresponding system of equations is solved by Newton's method with first-order continuation. The relationship between coating thickness, operational parameters (loading force, gap setting, roll velocities, etc.), liquid properties and the properties of the cover is reported, as well as the critical capillary number for onset of ribbing and wavelengths of the ribbing pattern predicted by the mathematical model. The results indicate how a deformable cover may be used in order to delay the onset of ribbing for a desired coating thickness.

In order to validate the theoretical predictions of the viscocapillary/Hookean spring model, the symmetric film-split flows between a pair of rigid rolls and a pair consisting of a deformable roll and a rigid one were also analysed experimentally.

A direct-numerical-simulation study of spatially evolving compressible zero-pressure-gradient turbulent boundary layers is presented for a fine-meshed range of Mach numbers from 0.3 to 2.5. The use of an identical set-up for all subsonic and supersonic cases warrants proper comparability and allows a highly reliable quantitative evaluation of compressible mean-flow scaling laws and the settlement on a commonly accepted compressible mean-flow velocity profile in the considered Mach and Reynolds number range. All data are compared to the literature data-base where significant data scattering can be observed. The skin-friction distribution was found in excellent agreement with the prediction by the van Driest-II transformation. Contrary to the prevailing appraisal, the wake region of the mean-velocity profile is observed to scale much better with the momentum-thickness Reynolds number calculated with the far-field-viscosity than with the wall-viscosity. The time-averaged velocity fluctuations, density-scaled according to Morkovin’s hypothesis, are found to be noticeably influenced by compressibility effects in the inner layer as well as in the wake region. Allowing wall-temperature fluctuations affects neither the density nor velocity fluctuations.

Two-dimensional time-dependent buoyancy-induced flows above a horizontal line heat source inside rectangular vessels, with adiabatic sidewalls and top and bottom walls maintained at uniform temperature, are studied numerically. Transitions to unsteady flows are performed by direct simulations for various depths of immersion of the source in the central vertical plane of air-filled vessels. For a square vessel and a line source near the bottom wall, the numerical solutions exhibit a sequence of instabilities, called natural swaying motion of confined plumes, beginning with a periodic regime having a high fundamental frequency followed by a two-frequency locked regime. Then, broadband components appearing in the spectra indicate chaotic behaviour and a weakly turbulent motion arises via an intermittent route to chaos. For rectangular vessels of aspect ratio greater than 2 and depths of immersion greater than the width, the flow undergoes a pitchfork bifurcation. This symmetry breaking is driven by the destabilization of an upper unstable layer of stagnant fluid above the plume. Then a subcritical Hopf bifurcation occurs. On the other hand, if the depth of immersion is lower than the width of the vessel, a stable layer of fluid is at rest below the line source. Then penetrative convection sets the whole air-filled vessel in motion and an oscillatory motion of very low frequency arises through supercritical Hopf bifurcation followed by a two-frequency locked state.

This paper presents some new velocity correlation measurements in a two-dimensional zero-pressure-gradient turbulent boundary layer; these extend the work of Grant (1958). The most detailed of the new measurements concern: correlations between velocity fluctuations in the mean flow direction and fluctuations perpendicular to the wall at a different point, usually but not always in the same plane parallel to the wall; correlations between velocity fluctuations in the mean flow direction and fluctuations perpendicular to this but parallel to the wall, usually but not always close to the wall with the separation in the same direction as the latter velocity fluctuation; and various correlations with a fixed separation normal to the wall and a simultaneous variable separation in the mean flow direction.

The results are not consistent with any of the various models of the large eddy structure that have been proposed. The features of the present work most relevant to the formulation of a substitute are brought together in §9. Included are suggestions that the similarities between the wall and outer regions are more marked than the differences, and that the description of the large eddies in the wall region as a coherent eruption from the viscous sublayer is unsatisfactory.

The new experiments are considered in connexion with three questions posed by the previous work. These concern the large eddy contribution to the Reynolds stress; the asymmetry between upstream and downstream separations when there is also a separation normal to the wall; and the similarity between boundary layers and channel flow. A complicated situation surrounds the first point; there are arguments suggesting that the large eddy contribution might in some places be of opposite sign to the total Reynolds stress, but this no longer seems so likely. The upstream-downstream asymmetry is also more complicated than had been previously supposed, but the results can be systematically interpreted if the peaks of the correlation curves are considered in connexion with continuity and the tails of the curves are considered in connexion with the shearing action of the mean flow. Regarding the third point, the present experiments give no indication of any difference in the large eddy structure in the wall regions of boundary layer turbulence and channel flow turbulence.

Section 5 gives a brief account of some experiments with a vibrating ribbon in the turbulent boundary layer.

We consider the free boundary problem for the evolution of a nearly straight slender fibre of viscous fluid. The motion is driven by prescribing the velocity of the ends of the fibre, and the free surface evolves under the action of surface tension, inertia and gravity. The three-dimensional Navier–Stokes equations and free-surface boundary conditions are analysed asymptotically, using the fact that the inverse aspect ratio, defined to be the ratio between a typical fibre radius and the initial fibre length, is small. This first part of the paper follows earlier work on the stretching of a slender viscous fibre with negligible surface tension effects. The inclusion of surface tension seriously complicates the problem for the evolution of the shape of the cross-section. We adapt ideas applied previously to two-dimensional Stokes flow to show that the shape of the cross-section can be described by means of a conformal map which depends on time and distance along the fibre axis. We give some examples of suitable relevant conformal maps and present numerical solutions of the resulting equations. We also use analytic methods to examine the coupling between stretching and the evolution of the cross-section shape.

This paper presents an approximate solution of the problem of the onset of convection between plane-parallel plates heated from below when the fluid between them absorbs and emits thermal radiation. A complete solution to this problem would be extremely difficult, and the equation of radiative transfer is therefore developed in two approximate forms, one appropriate to an opaque medium, the other to a transparent medium. This equation is then combined with the dynamical equations of the problem.

The initial static state is investigated by use of the Milne-Eddington approximation, and it is shown that there can be very large variations of temperature near to the boundaries.

The conditions for marginal stability are investigated both for motions which are restricted to the temperature boundary layer, and for motions which take place in the body of the fluid. In the former case it is found that a complete solution is provided by the approximate form for a transparent medium, and in the latter case a reasonable interpolation has to be made between results for the two approximate forms in order to complete the solution.

The effect of radiative transfer both on the initial static state and on the dynamical equations is such that the fluid is stabilized. This stabilization could probably be detected in the laboratory under certain conditions. In the earth's atmosphere the critical Rayleigh number for large scale motions may be increased by a factor 105, while at the surface of the solar photosphere the factor may be as large as 1012.

The stability of two-dimensional convection rolls has been studied as a function of the Rayleigh number, wavenumber and variation in viscosity. The experiments used controlled initial conditions for the wavenumber, Rayleigh numbers up to 25 000 and variations in viscosity up to a factor of about 20. The parameter range of stable rolls is bounded by a hexagonal-cell regime at small Rayleigh numbers and large variations in viscosity. Otherwise, the rolls are subject to the same transitions as have already been studied in fluids of uniform viscosity. The bimodal instability leading to a stable three-dimensional pattern occurs at smaller values of the average Rayleigh number as the variations in viscosity increase. This appears to be a consequence of the low viscosity of the warm thermal boundary layer associated with the original rolls.

A uniform-mean-gradient shear flow was produced using a ten-layer closed-loop water channel, providing long enough dimensionless flow development times (τ = (x/Ū) (∂ Ū/∂z)) for the turbulence to grow. The rate of growth of the turbulence compares well with similar measurements in wind-tunnel-generated uniform shear flows for which the mean shears and centreline velocities are larger by an order of magnitude. Preliminary investigations were undertaken to study the growth of the turbulent intensity as functions of the mean shear, centreline velocity, and initial disturbance lengthscales. Initial disturbance lengthscales were varied by using grids of different mesh sizes.

Turbulent intensities were found to increase nearly linearly with τ. Differences in grid mesh size produce different offsets in the turbulent intensity level, with a larger grid mesh producing a higher positive offset. This offset persists throughout the growth of the turbulent intensity. These observations provide valuable insight in interpreting previous wind-tunnel measurements, in particular the high-shear experiments of Karnik & Tavoularis (1983). Comparison with the theoretical predictions of Tavoularis (1985) allows for an improved universal characterization of evolving turbulence in a uniform mean shear.

The incompressible fluid flow in a rectangular container driven by two facing sidewalls which move steadily in anti-parallel directions is investigated experimentally for Reynolds numbers up to 1200. The moving sidewalls are realized by two rotating cylinders of large radii tightly closing the cavity. The distance between the moving walls relative to the height of the cavity (aspect ratio) is Γ = 1.96. Laser-Doppler and hot-film techniques are employed to measure steady and time-dependent vortex flows. Beyond a first threshold robust, steady, three-dimensional cells bifurcate supercritically out of the basic flow state. Through a further instability the cellular flow becomes unstable to oscillations in the form of standing waves with the same wavelength as the underlying cellular flow. If both sidewalls move with the same velocity (symmetrical driving), the oscillatory instability is found to be tricritical. The dependence on two sidewall Reynolds numbers of the ranges of existence of steady and oscillatory cellular flows is explored. Flow symmetries and quantitative velocity measurements are presented for representative cases.

Direct numerical simulations (DNS) are employed to investigate laminar boundary layer separation and its control by pulsed vortex generator jets (VGJs), i.e. by injecting fluid into the flow through a spanwise array of small holes. Particular focus is directed towards identifying the relevant physical mechanisms associated with VGJ control of low-Reynolds-number separation, as encountered in low-pressure turbine applications. Pulsed VGJs are shown to be much more effective than steady VGJs when the same momentum coefficient is used for the actuation. From our investigations we have found that the increased control effectiveness of pulsed VGJs can be explained by the fact that linear hydrodynamic instability mechanisms are exploited. When pulsing with frequencies to which the separated shear layer is naturally unstable, instability modes are shown to develop into large-scale, spanwise coherent structures. These structures provide the necessary entrainment of high-momentum fluid to successfully reattach the flow.

The trapped modes which can occur in a long narrow wave channel containing any number of different-sized bottom-mounted circular cylinders arbitrarily spaced along the centreline of the channel are considered. The modes, all of which are antisymmetric with respect to the centreplane of the channel are of two types: Neumann modes, in which the fluid has normal velocity zero on the channel walls corresponding to a localized sloshing near the cylinders, and Dirichlet modes, in which the dynamic pressure vanishes on the channel walls. These latter modes have no physical meaning in the water-wave context but have been observed in a related acoustic context where the same governing equations and boundary conditions apply.

It is shown that in general there are [les ]N trapped modes for any configuration of N cylinders, the precise number depending critically on the geometry of the configuration. Both types are of importance in predicting the exciting forces on individual cylinders within a large but finite periodic arrangement of cylinders.