When a particle is placed in a fluid in which there is a non-uniform concentration of solute, it will move toward higher or lower concentration depending on whether the solute is attracted to or repelled from the particle surface. A quantitative understanding of this phenomenon requires that the equations representing conservation of mass and momentum within the fluid in the vicinity of the particle are solved. This is accomplished using a method of matched asymptotic expansions in a small parameter L/a, where a is the particle radius and L is the length scale characteristic of the physical interaction between solute and particle surface. This analysis yields an expression for particle velocity, valid in the limit L/a → 0, that agrees with the expression obtained by previous researchers. The result is cast into a more useful algebraic form by relating various integrals involving the solute/particle interaction energy to a measurable thermodynamic property, the Gibbs surface excess of solute Γ. An important result is that the correction for finite L/a is actually O(Γ/C∞a), where C∞ is the bulk concentration of solute, and could be O(1) even when L/a is orders of magnitude smaller.

Fluid injection into the deep subsurface, such as injection of carbon dioxide (CO$_2$) into deep saline aquifers, often involves two-fluid flow in confined geological formations. Similarity solutions may be derived for these problems by assuming that a sharp interface separates the two fluids, by imposing a suitable no-flow condition along both the top and bottom boundaries, and by including an explicit solution for the pressure distribution in both fluids. When the injected fluid is less dense and less viscous than the resident fluid, as is the case for CO$_{2}$ injection into a resident brine, gravity override produces a fluid flow system that is captured well by the similarity solutions. The similarity solutions may be extended to include slight miscibility between the two fluids, as well as compressibility in both of the fluid phases. The solutions provide the location of the interface between the two fluids, as well as drying fronts that develop within the injected fluid. Applications to cases of supercritical CO$_{2}$ injection into deep saline aquifers demonstrate the utility of the solutions, and comparisons to solutions from full numerical simulations show the ability to predict the system behaviour.

The lift and drag forces were measured on both a single circular cylinder and tandem circular cylinders in uniform flow at Reynolds numbers from 40 to 104, to investigate the stability of an oscillating cylinder. A cylinder (the downstream one in the tandem case) was made to oscillate in either the transverse or longitudinal direction (perpendicular or parallel to the stream). In the case of a single cylinder, its oscillation causes the so-called synchronization in a frequency range around the Strouhal frequency (transverse mode) or double the Strouhal frequency (longitudinal mode). The aerodynamic damping for transverse oscillation becomes negative in the synchronization range. In the case of tandem cylinders, at low Reynolds numbers in the pure Kármán range synchronization was observed to occur only when the downstream cylinder oscillated inside the vortex-formation region of the upstream one, and at high (low subcritical) Reynolds numbers synchronization occurred irrespective of the cylinder spacing in either oscillating mode. In the tandem case, too, the transverse oscillation of the downstream cylinder becomes unstable in the range of synchronization.

The lateral migration of a neutrally buoyant rigid sphere suspended in a second-order fluid is studied theoretically for unidirectional two-dimensional flows. The results demonstrate the existence of migration induced by normal stresses whenever there is a lateral variation of the shear rate in the undisturbed flow. The migration occurs in the direction of decreasing absolute shear rate, which is towards the centre-line for a plane Poiseuille flow and towards the outer cylinder wall for Couette flow. The direction of migration agrees with existing experimental data for a viscoelastic suspending fluid, and qualitative agreement is found between the theoretically predicted and experimentally measured sphere trajectories.

The behaviour of the velocity and pressure fluctuations in the logarithmic and outer layers of turbulent flows is analysed using spectral information and probability density functions from channel simulations at Reτ≤2000. Comparisons are made with experimental data at higher Reynolds numbers. It is found, in agreement with previous investigations, that the intensity profiles of the streamwise and spanwise velocity components have logarithmic ranges that are traced to the widening spectral range of scales as the wall is approached. The same is true for the pressure, both theoretically and observationally, but not for the normal velocity or for the tangential stress cospectrum, although even those two quantities have structures with lengths of the order of several hundred times the wall distance. Because the logarithmic range grows longer as the Reynolds number increases, variables which are ‘attached’ in this sense scale in the buffer layer in mixed units. These results give strong support to the attached-eddy scenario proposed by Townsend (1976), but they are not linked to any particular eddy model. The scaling of the outer modes is also examined. The intensity of the streamwise velocity at fixed y/h increases with the Reynolds number. This is traced to the large-scale modes, and to an increased intensity of the ejections but not of the sweeps. Several differences are found between the outer structures of different flows. The outer modes of the spanwise and wall-normal velocities in boundary layers are stronger than in internal flows, and their streamwise velocities penetrate closer to the wall. As a consequence, their logarithmic layers are thinner, and some of their logarithmic slopes are different. The channel statistics are available electronically at http://torroja.dmt.upm.es/ftp/channels/.

Several aspects of the oscillations of a gas bubble in a slightly compressible liquid are discussed by means of a simplified model based on the assumption of a spatially uniform internal pressure. The first topic considered is the linear initial-value problem for which memory effects and the approach to steady state are analysed. Large-amplitude oscillations are studied next in the limit of large and small thermal diffusion lengths obtaining, in the first case, an explicit expression for the internal pressure, and, in the second one, an integral equation of the Volterra type. The validity of the assumption of uniform pressure is then studied analytically and numerically. Finally, the single-bubble model is combined with a simple averaged-equation model of a bubbly liquid and the propagation of linear and weakly nonlinear pressure waves in such a medium is considered.

Detailed measurements of turbulent concentration parameters in a round free methane jet up to 70 diameters downstream from the jet source are described. The concentration changes were obtained using a Raman spectrometer designed for the measurement of a methane vibrational transition function, and processed using a photon correlator. Mean and fluctuating concentration levels are given, together with the concentration probability density distribution.

A theory for interpreting the statistical aspects of the signal is presented and results are compared with other data from the literature on jets of uniform and non-uniform density. It is shown that the intensity of concentration fluctuations has an asymptotic value of 28·5% in the far field region of the jet. One interesting feature of the data is the deviation from Gaussian statistics along the jet centre-line.

Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods. In this paper, the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored. Accuracy of simulation is tested empirically using a simple scalar-convection test problem and the Taylor–Green vortex-decay problem. It is demonstrated empirically that the Galerkin (Fourier) equations involving Np degrees of freedom, where p is the number of space dimensions, give simulations at least as accurate as finite-difference simulations involving (2N)p degrees of freedom. The theoretical basis for the improved accuracy of the Galerkin (Fourier) method is explained. In particular, the nature of aliasing errors is examined in detail. It is shown that ‘aliasing’ errors need not be errors at all, but that aliasing should be avoided in flow simulations. An eigenvalue analysis of schemes for simulation of passive scalar convection supplies the mathematical basis for the improved accuracy of the Galerkin (Fourier) method. A comparison is made of the computational efficiency of Galerkin and finite-difference simulations, and a survey is given of those problems where Galerkin methods are likely to be applied most usefully. We conclude that numerical simulation of many of the flows of current interest is done most efficiently and accurately using the spectral methods advocated here.

A viscous-liquid drop spreads on a smooth horizontal surface, which is uniformly heated or cooled. Lubrication theory is used to study thin drops subject to capillary, thermocapillary and gravity forces, and a variety of contact-angle-versus-speed conditions. It is found for isothermal drops that gravity is very important at large times and determines the power law for unlimited spreading. Predictions compare well with the experimental data on isothermal spreading for both two-dimensional and axisymmetric configurations. It is found that heating (cooling) retards (augments) the spreading process by creating flows that counteract (reinforce) those associated with isothermal spreading. For zero advancing contact angle, heating will prevent the drop from spreading to infinity. Thus, the heat transfer serves as a sensitive control on the spreading.

A theoretical study has been made of an experiment by Tritton & Zarraga (1967) in which eonvective motions were generated in a horizontal layer of water (cooled from above) by the application of uniform heating. The marginal stability problem for such a layer is solved, and a critical Rayleigh number of 2772 is obtained, at which patterns of wave-number 2·63 times the reciprocal depth of the layer are marginally stable.

The remainder of the paper is devoted to the finite amplitude convection which ensues when the Rayleigh number, R, exceeds 2772. The theory is approximate, the basic simplification being that, to an adequate approximation, Fourier decompositions of the convective motions in the horizontal (x, y) directions can be represented by their dominant (planform) terms alone. A discussion is given of this hypothesis, with illustrations drawn from the (better studied) Bénard situation of convection in a layer heated below, cooled from above, and containing no heat sources. The hypothesis is then used to obtain ‘mean-field equations’ for the convection. These admit solutions of at least three distinct forms: rolls, hexagons with upward flow at their centres, and hexagons with downward flow at their centres. Using the hypothesis again, the stability of these three solutions is examined. It is shown that, for all R, a (neutrally) stable form of convection exists in the form of rolls. The wave-number of this pattern increases gradually with R. This solution is, in all respects, independent of Prandtl number. It is found, numerically, that the hexagons with upward motions in their centres are unstable, but that the hexagons with downward motions at their centres are completely stable, provided R exceeds a critical value (which depends on Prandtl number, P, and which for water is about 3Rc), and provided the wave-number of the pattern lies within certain limits dependent on R and P.

The effect of surface roughness on the flow past spheres has been investigated over the Reynolds number range 5 × 104 < Re < 6 × 106. The drag coefficient has been determined as a function of the Reynolds number for five surface roughnesses. With increasing roughness parameter the critical Reynolds number decreases. At the same time the transcritical drag coefficient rises, having a maximum value of 0·4.

The vortex shedding frequency has been measured under subcritical flow conditions. It was found that the Strouhal number for each of the various roughness conditions was equal to its value for a smooth sphere. Beyond the critical Reynolds number no prevailing shedding frequency could be detected by the measurement techniques employed.

The drag coefficient of a sphere under the blockage conditions 0·5 < ds/dt < 0·92 has been determined over the Reynolds number range 3 × 104 < Re < 2 × 106. Increasing blockage causes an increase in both the drag coefficient and the critical Reynolds number. The characteristic quantities were referred to the flow conditions in the smallest cross-section between sphere and tube. In addition the effect of the turbulence level on the flow past a sphere under various blockage conditions was studied.

A recent assessment of available direct numerical simulation (DNS) data from turbulent boundary layer flows (Schlatter & Örlü, J. Fluid Mech., vol. 659, 2010, pp. 116–126) showed surprisingly large differences not only in the skin friction coefficient or shape factor, but also in their predictions of mean and fluctuation profiles far into the sublayer. While such differences are expected at very low Reynolds numbers and/or the immediate vicinity of the inflow or tripping region, it remains unclear whether inflow and tripping effects explain the differences observed even at moderate Reynolds numbers. This question is systematically addressed by re-simulating the DNS of a zero-pressure-gradient turbulent boundary layer flow by Schlatter et al. (Phys. Fluids, vol. 21, 2009, art. 051702). The previous DNS serves as the baseline simulation, and the new DNS with a range of physically different inflow conditions and tripping effects are carefully compared. The downstream evolution of integral quantities as well as mean and fluctuation profiles is analysed, and the results show that different inflow conditions and tripping effects do indeed explain most of the differences observed when comparing available DNS at low Reynolds number. It is further found that, if transition is initiated inside the boundary layer at a low enough Reynolds number (based on the momentum-loss thickness) , all quantities agree well for both inner and outer layer for . This result gives a lower limit for meaningful comparisons between numerical and/or wind tunnel experiments, assuming that the flow was not severely over- or understimulated. It is further shown that even profiles of the wall-normal velocity fluctuations and Reynolds shear stress collapse for higher irrespective of the upstream conditions. In addition, the overshoot in the total shear stress within the sublayer observed in the DNS of Wu & Moin (Phys. Fluids, vol. 22, 2010, art. 085105) has been identified as a feature of transitional boundary layers.

The light-scatter technique has been used to study the nozzle-fluid concentration field in an isothermal, turbulent, axisymmetric air/air free jet with the nozzle air marked by an oil smoke. The data on the mean concentration field appear to be the most accurate yet obtained, due to the peculiar advantages of the technique. The turbulent concentration fluctuations have been characterized as to intensity, spectral distribution, and two-point correlation. The intermittency factor has been measured and the properties of the turbulent fluid computed. Comparison with the results of other investigators who used heat to mark the nozzle fluid indicates a close similarity between the concentration and temperature fields.

Cylinders of various cross-section were subjected to controlled oscillations in a direction transverse to the incident flow. Excitation was at frequency fe, relative to the formation frequency f*0 of large-scale vortices from the corresponding stationary cylinder, and at Reynolds numbers in the range 584 [les ] Re [les ] 1300. Modifications of the near wake were characterized by visualization of the instantaneous flow structure in conjunction with body displacement-flow velocity correlations.

At fe/f*0 = ½, corresponding to subharmonic excitation, as well as at fe/f*0 = 1, the near wake structure is phase-locked (synchronized) to the cylinder motion. However, the synchronization mechanism is distinctly different in these two regimes. Near or at fe/f*0 = 1, the phase of the shed vortex with respect to the cylinder displacement switches by approximately π. Characteristics of this phase switch are related to cylinder geometry. It does not occur if the cylinder has significant afterbody.

Over a wide range of fe/f*0, the perturbed near wake rapidly recovers to a largescale antisymmetrical mode similar in form to the well-known Kármán vortex street. The mechanisms of small-scale (fe) vortex interaction leading to recovery of the large-scale (f0) vortices are highly ordered and repeatable, though distinctly different, for superharmonic excitation (fe/f*0 = n = 2, 3, 4) and non-harmonic excitation (non-integer values of fe/f*0).

The frequency f0 of the recovered vortex street downstream of the body shows substantial departure from the shedding frequency f*0 from the corresponding stationary body. It locks-on to resonant modes corresponding to f0/fe = 1/n. This wake response involves strictly hydrodynamic phenomena. It shows, however, a resonant behaviour analogous to that of coupled flow-acoustic systems where the shear layer is convectively unstable

Analytical formulae or the effect of interaction between pairs of rigid spherical particles on the mean velocity of each species in a statistically homogeneous dilute polydisperse system were given in Part 1 (Batchelor 1982), and are here evaluated numerically. We have calculated the pair-distribution function and the associated value of the sedimentation coefficient for a wide variety of conditions of the two interacting species, including different values of the ratio of the radii of the spheres (λ), different values of the ratio of their (reduced) densities (γ), small and large values of the Péclet number of the interaction, and different forms of the potential of the mutual force exerted directly between the two spheres. Values of λ and γ such that some of the trajectories of one sphere centre moving under gravity alone relative to another are of finite length lie outside the scope of the calculations at large Péclet number, and the change of behaviour across the boundary of this excluded set of values leads to a complicated dependence of the sedimentation coefficient on λ and γ. At small Péclet number the behaviour is simpler, and a formula which represents the calculated values of the sedimentation coefficient over the whole range of values of λ and γ (on which the dependence is known to be linear) with fair accuracy in the absence of interparticle forces is devised. Our calculations of the effect of an interparticle force were based on the assumption of a high Coulomb barrier at a certain sphere separation which could be varied, and a van der Waals attractive force at larger separations. It appears that the direct contribution to the sedimentation coefficient made by gravity is always appreciably larger than that made either by relative Brownian diffusion of the two interacting spheres or by the interparticle force. However, all three of these (effective) forces normally have a significant influence on the pair-distribution function and thereby also affect the sedimentation coefficient indirectly. Some published observations of the mean particle velocity in monodisperse systems are interpreted in the light of the present calculations of the effect of interparticle forces.

Round jets originating from a pipe nozzle are computed by large-eddy simulations (LES) to investigate the effects of the nozzle-exit conditions on the flow and sound fields of initially laminar jets. The jets are at Mach number 0.9 and Reynolds number 105, and exhibit exit boundary layers characterized by Blasius velocity profiles, maximum root-mean-square (r.m.s.) axial velocity fluctuations between 0.2 and 1.9% of the jet velocity, and momentum thicknesses varying from 0.003 to 0.023 times the jet radius. The far-field noise is determined from the LES data on a cylindrical surface by solving the acoustic equations. Jets with a thinner boundary layer develop earlier but at a slower rate, yielding longer potential cores and lower centreline turbulent intensities. Adding random pressure disturbances of low magnitude in the nozzle also increases the potential core length and reduces peak r.m.s. radial velocity fluctuations in the shear layer. In all the jets, the shear-layer transition is dominated by vortex rolling-ups and pairings, which generate strong additional acoustic components, but also amplify the downstream-dominant low-frequency noise component when the exit boundary layer is thick. The introduction of inlet noise however results in weaker pairings, thus spectacularly reducing their contributions to the sound field. This high sensitivity to the initial conditions is in good agreement with experimental observations.

An experimental adaptation of the well-known laser-Doppler anemometry technique is developed for measuring the velocity and concentration profiles in concentrated suspension flows. To circumvent the problem of optical turbidity, the refractive indices of the solid and liquid phases are closely matched. The residual turbidity, owing to small mismatches of the refractive indices, as well as impurities in the particles, allows a Doppler signal to be detected when a particle passes through the scattering volume. By counting the number of Doppler signals in a period of time, the local volume fraction is also measured.

This new technique is utilized to study concentrated suspension flows in a rectangular channel. The general behavior of the suspension is that the velocity profile is blunted while the concentration profile has a maximum near the centre. Comparisons are made with theoretical predictions based on the shear-induced particle migration theory.

The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.

Turbulent boundary-layer wall-pressure measurements were made with ‘pinhole’ microphones three times smaller (relative to a boundary-layer displacement thickness) than microphones used in earlier work. The improved high-frequency resolution permitted examination of the influence of high-frequency eddies on smooth-wall pressure statistics. It was found that the space-time decay rate is considerably higher than previously reported. Measurements of cross-spectral density made with 5 Hz bandwidth filters disclosed low phase speeds at low frequency and small separation. Measurements were repeated on rough walls and parallels were drawn from knowledge of a smooth-wall boundary-layer structure to propose a structure for a rough-wall boundary layer. The effect of independently varying roughness height and separation on the large and small-scale turbulence structure was deduced from the measurements. It was found that roughness separation affected the very large-scale structure, whereas the roughness height influenced the medium and very small-scale turbulence.