We present experimental results for the wake structure of spheres moving in homogeneous and stratified fluid. In homogeneous fluid, the results of Kim & Durbin (1988) are confirmed and it is shown that the two characteristic frequencies of the wake correspond to two instability modes, the Kelvin–Helmholtz instability and a spiral instability. For the stratified wake four general regimes have been identified, depending principally on the Froude number F. For F > 4.5 the near wake is similar to the homogeneous case, and for F < 0.8 it corresponds to a triple-layer flow with two lee waves, of amplitude linear in F, surrounding a layer dominated by quasi-two-dimensional motion. Froude numbers close to one (F∈]0.8, 1.5[) give rise to a saturated lee wave of amplitude equal to half the sphere radius, which suppresses the separation region or splits it into two. Between F = 1.5 and 4.5 a more complex regime exists where the wake recovers progressively its behaviour in homogeneous fluid: the axisymmetry of the recirculating zone, the Kelvin–Helmholtz instability and, finally, the spiral instability.

Space–time correlations and frequency spectra of wall-pressure fluctuations, obtained from direct numerical simulation, are examined to reveal the effects of pressure gradient and separation on the characteristics of wall-pressure fluctuations. In the attached boundary layer subjected to adverse pressure gradient, contours of constant two-point spatial correlation of wall-pressure fluctuations are more elongated in the spanwise direction. Convection velocities of wall-pressure fluctuations as a function of spatial and temporal separations are reduced by the adverse pressure gradient. In the separated turbulent boundary layer, wall-pressure fluctuations are reduced inside the separation bubble, and enhanced downstream of the reattachment region where maximum Reynolds stresses occur. Inside the separation bubble, the frequency spectra of wall-pressure fluctuations normalized by the local maximum Reynolds shear stress correlate well compared to those normalized by free-stream dynamic pressure, indicating that local Reynolds shear stress has more direct influence on the wall-pressure spectra. Contour plots of two-point correlation of wall-pressure fluctuations are highly elongated in the spanwise direction inside the separation bubble, implying the presence of large two-dimensional roller-type structures. The convection velocity determined from the space–time correlation of wall-pressure fluctuations is as low as 0.33U0 (U0 is the maximum inlet velocity) in the separated zone, and increases downstream of reattachment.

The acoustic and entropy transfer functions of quasi-one-dimensional nozzles are studied analytically for both subsonic and choked flows with and without shock waves. The present analytical study extends both the compact nozzle solution obtained by Marble & Candel (J. Sound Vib., vol. 55, 1977, pp. 225–243) and the effective nozzle length proposed by Stow, Dowling & Hynes (J. Fluid Mech., vol. 467, 2002, pp. 215–239) and by Goh & Morgans (J. Sound Vib., vol. 330, 2011, pp. 5184–5198) to non-zero frequencies for both modulus and phase through an asymptotic expansion of the linearized Euler equations. It also extends the piecewise-linear approximation of the velocity profile in the nozzle proposed by Moase, Brear & Manzie (J. Fluid Mech., vol. 585, 2007, pp. 281–304) to any arbitrary profile or equivalently any nozzle geometry. The equations are written as a function of three variables, namely the dimensionless mass, total temperature and entropy fluctuations, yielding a first-order linear system of differential equations with varying coefficients, which is solved using the Magnus expansion. The solution shows that both the modulus and the phase of the transfer functions of the nozzle have a strong dependence on the frequency. This holds for both choked flows and subsonic converging–diverging nozzles. The method is used to compare two different nozzle geometries with the same inlet and outlet Mach numbers, showing that, even if the compact solution predicts no differences between the transfer functions of the two nozzles, significant differences are found at non-zero frequencies. A parametric study is finally performed to calculate the indirect to direct noise ratio for a model combustor, showing that this ratio decreases at higher frequencies.

The region of finite vorticity near the edge of a diffracting wedge is investigated. Dimensional analysis gives the dependence of the circulation and the velocity of the vortex region on the pulse strength. A close estimate of the magnitude of the magnitude of these quantities is obtained by replacing the vortex region by a single concentrated vortex. The theoretical conditions at the sharp edge are discussed and compared with observations of real fluid behaviour. A short account of the theory of the core of the spiral vortex sheet in a perfect fluid is appended.

The research reported herein was supported by the United States Air Force, through the Office of Scientific Research of the Air Research and Development Command.

The physical mechanisms by which the Reynolds shear stress is produced from dynamically evolving vortical structures in the wall region of a direct numerical simulation of turbulent channel flow are explored. The complete set of quasistreamwise vortices are systematically located and tracked through the flow by the locus of the points of intersection of their centres of rotation with the (y, z) numerical grid planes. This approach assures positive identification of vortices of widely differing strengths, including those whose amplitude changes significantly in time. The process of vortex regeneration, and the means by which vortices grow, distort and interact over time are noted. Ensembles of particle paths arriving on fixed planes in the flow are used to represent the physical processes of displacement and acceleration transport (Bernard & Handler 1990a) from which the Reynolds stress is produced. By interweaving the most dynamically significant of the particle paths with the evolving vortical structures, the dynamical role of the vortices in producing Reynolds stress is exposed. This is found to include ejections of low-speed fluid particles by convecting structures and the acceleration and deceleration of fluid particles in the cores of vortices. Sweep dominated Reynolds stress close to the wall appears to be a manifestation of the regeneration process by which new vortices are created in the flow.

The characteristics of a turbulent boundary layer overlying a complex roughness topography were explored with stereo particle-image velocimetry measurements in the wall-normal–spanwise $(y\mbox{--}z)$ plane. The roughness under consideration was replicated from a turbine blade damaged by deposition of foreign materials containing a broad range of topographical scales arranged in a highly irregular manner. The single-point turbulence statistics displayed strong spanwise heterogeneity, in particular spanwise-alternating low- and high-momentum flow pathways in the mean flow marked by enhanced Reynolds stresses and turbulent kinetic energy. The spanwise regions between high- and low-momentum flow pathways were occupied by swirling motions, suggesting the generation and sustainment of turbulent secondary flows due to the spanwise heterogeneity of the complex roughness under consideration. Similar observations were recently reported for more ordered spanwise roughness transitions by Nugroho, Hutchins & Monty (Intl J. Heat Fluid Flow vol. 41, 2013, pp. 90–102) and Willingham et al. (Phys. Fluids vol. 26, 2014, 025111).

Numerical simulations are used to investigate the flow associated with a bluegill sunfish (Lepomis macrochirus) pectoral fin during steady forward motion. The simulations are intended to match the experiments of Lauder et al. (Bioinsp. Biomim., vol. 1, 2006, p. S25), and the results obtained from the simulations complement the experimental analysis. The focus of the current paper is on the quantitative characterization of the propulsive performance of the pectoral fin, which undergoes significant deformation during its stroke. This includes a detailed analysis of the thrust production mechanisms as well as their connection to the vortex dynamics and other flow features. The simulations indicate that the fish fin produces high propulsive performance by employing a complex fin gait driven by active and passive fin deformation. By connecting the vortex dynamics and fin kinematics with the surface distribution of the force on the fin, it is found that during abduction, the fin moves such that the tip of the fin undergoes a complex, three-dimensional flapping motion that produces a strong and long-lasting, attached tip vortex. This tip vortex is associated with most of the thrust production during the abduction phase of the stroke. During the adduction phase, the fin motion is similar to a ‘paddling’ stroke. Comparisons are made with rigid flapping foils to provide insights into the remarkable performance of the fish fin and to interpret the force production from the viewpoint of functional morphology.

The exact solution of the problem of river morphodynamics derived in Part 1 is employed to formulate and solve the problem of planimetric evolution of river meanders. A nonlinear integrodifferential evolution equation in intrinsic coordinates is derived. An exact periodic solution of such an equation is then obtained in terms of a modified Fourier series expansion such that the wavenumbers of the various Fourier modes are time dependent. The amplitudes of the Fourier modes and their wavenumbers satisfy a nonlinear system of coupled ordinary differential equations of the Landau type. Solutions of this system display the occurrence of two possible scenarios. In the sub-resonant regime, i.e. when the aspect ratio of the channel is smaller than the resonant value, meandering evolves according to the classical picture: a periodic train of small-amplitude sine-generated meanders migrating downstream evolve into the classical, upstream skewed, train of meanders of Kinoshita type. Evolution displays all the experimentally observed features: the meander growth rate increases up to a maximum and then decreases, while the migration speed decreases monotonically. No equilibrium solutions are found. In the super-resonant regime the picture is essentially reversed: downstream skewing develops while meanders migrate upstream.

Numerical solutions of the planimetric evolution equation are obtained for the case when the initial channel pattern exhibits random small perturbations of the straight configuration. Under these conditions, the evolution displays the typical features of solutions of the Ginzburg–Landau equation, in particular, the occurrence of spatial modulations of the meandering pattern which organizes itself in the form of wavegroups. Furthermore, multiple loops develop in the advanced stage of meander growth.

Structural properties of regions of uniform streamwise momentum in turbulent boundary layers are examined using experimental databases obtained from particle image velocimetry. This investigation employs a large range of Reynolds numbers, spanning more than an order of magnitude ($Re_{{\it\tau}}=10^{3}{-}10^{4}$), enabling us to provide a detailed description of uniform momentum zones as a function of Reynolds number. Our analysis starts by examining the identification criterion used by Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) to report the presence of uniform momentum zones in turbulent boundary layers. This criterion is then applied to show that a zonal-like structural arrangement is prevalent in all datasets examined, emphasising its importance in the structural organisation. Streamwise velocity fluctuations within the zones are observed to be small but they are bounded by distinct step changes in streamwise momentum which indicate that shear layers of intense vorticity separate each zone. A log-linear increase in the number of these zones with increasing Reynolds number is revealed, together with an increase in the thicknesses of zones with increasing distance from the wall. These results support a hierarchical length-scale distribution of coherent structures, which generate zonal-like organisation within turbulent boundary layers. Interpretation of these findings is aided by employing synthetic velocity fields generated using a model based on the attached eddy hypothesis, which is described in the work of Perry and co-workers. Comparisons between the model and experimental results show that a hierarchy of self-similar structures leads to population densities and length-scale distributions of uniform momentum zones that closely adhere to those observed experimentally in this study.

In this paper a theoretical investigation of convection currents in anisotropic porous media is performed. The porous layer is homogeneous and bounded by two infinite perfectly heat-conducting horizontal planes. The criterion for the onset of convection is derived. The supercritical steady two-dimensional motion, the heat transport and the stability of the motion are investigated. Some of the results may be applied in insulation techniques.

In high-Reynolds-number two-dimensional turbulence with a −5/3 power-law energy spectrum, the clustering of inertial particles reflects the clustering of acceleration stagnation points for all particle relaxation times smaller than the integral time scale $T$ of the turbulence. Acceleration stagnation points and small inertial particles on these points are swept together by large-scale motions. In synthetic turbulence where there is no sweeping and acceleration stagnation points do not cluster, inertial particles do nevertheless cluster as a result of the repelling action of persistent velocity stagnation-point clusters. This repelling action has a negligible effect on the clustering of inertial particles in the presence of acceleration stagnation points clustering.

Two concentric spheres are supposed to rotate about the same axis with almost the same angular velocity, so that the viscous stresses over the surfaces of the spheres induce a flow which may be represented by a small perturbation superimposed upon a rigid body rotation of the fluid as a whole. The governing equations are therefore linearized in the magnitude of the perturbation, and it appears that the validity of this linearization is independent of the Reynolds number of the primary rotation. Attention is then restricted to the case in which the Reynolds number is large, the principal object of the note being to exemplify some of the properties of rotating systems at large Reynolds numbers in terms of a particularly simple mathematical model.

It is found that the cylindrical surface that touches the inner sphere (the axis being the axis of rotation) is a singular surface in which velocity gradients are very large. Everywhere outside this cylinder, the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder, the velocity distribution in the central (inviscid) core of the motion is shown to be determined by the velocity distribution in the boundary layers over the spheres, and explicit solutions are obtained for all these velocity distributions. The mechanics of the cylindrical shear layer itself is also discussed, though no explicit solution is obtained in this case.

Results are given for experiments on water-lubricated pipelining of 6.01 P cylinder oil in a vertical apparatus in up- and downflow in regimes of modest flow rates, less than 3 ft/s. Measured values of the flow rates, holdup ratios, pressure gradients and flow types are presented and compared with theoretical predictions based on ideal laminar flow and on the predictions of the linear theory of stability. New flow types, not achieved in horizontal flows, are observed: bamboo waves in upflow and corkscrew waves in downflow. Nearly perfect core–annular flows are observed in downflows and these are nearly optimally efficient with values close to the ideal. The holdup ratio in upflow and fast downflow is a constant independent of the value and the ratio of values of the flow rates of oil and water. A vanishing holdup ratio can be achieved by fluidizing a long lubricated column of oil in the downflow of water. The results of experiments are compared with computations from ideal theory for perfect core–annular flow and from the linear theory of stability. Satisfactory agreements are achieved for the celerity and diagnosis of flow type. The wave is shown to be nearly stationary, convected with the oil core in this oil and all oils of relatively high viscosity. These results are robust with respect to moderate changes in the values of viscosity and surface tension. The computed wavelengths are somewhat smaller than the average length of the bamboo waves which are observed. This is explained by stretching effects of buoyancy and lubrication forces induced by the wave. Other points of agreement and disagreement are reviewed.

Caflisch & Luke (1985) showed that, owing to the long range of the hydrodynamie interactions, the variance of the sedimentation velocity in a random suspension with uniform probability for the positions of the particles is divergent in the sense that it grows without bound as the macroscopic linear dimension of the settling vessel is increased. It is shown here, however, that a Debye-like screening of a particle's velocity disturbance, leading to a finite variance, will occur if the pair probability reflects a net deficit of one particle in the vicinity of each particle. The three-particle interactions, which determine the structure of a dilute, monodisperse suspension of spheres, lead to a deficit of neighbouring particles. The magnitude and range of this deficit are shown to be sufficient to lead to a Debye-like screening of the velocity disturbance at a radial distance of order aϕ−1, where a is the particle radius and ϕ their volume fraction. A self-consistent approximation to the screened conditional average velocity field and pair distribution is presented. The screening leads to a variance of the particle velocity and a particle tracer diffusion coefficient that are finite and of order Us2 and Usaϕ−1, respectively, where Vs is the Stokes settling velocity of the particles in unbounded fluid.

The coupling of Brownian displacements and shear-induced convection of spherical colloidal particles in dilute suspensions is examined using solutions of appropriate convective diffusion equations for the time-dependent probability density and also by calculation of relevant statistical quantities for an ensemble of diffusing particles from Langevin equations. Based on a fundamental solution for convective diffusion from a point in a general linear field, analytical expressions for the probability density fα(r; t) are given for the case of an arbitrary, two-dimensional linear flow field. The parameter α, which characterizes the flow, may range from − 1 (pure rotation), through zero (simple shear), to + 1 (pure elongation). The Langevin approach offers interesting insights into the physical mechanism of diffusive–convective coupling, and may also be used to obtain rigorous expressions for moments of the probability density appropriate to a particle diffusing in an unbounded quadratic (Poiseuille) flow. Preliminary experiments are described which qualitatively verify the theoretical predictions for Poiseuille flow, and which suggest a simple, direct method for measuring particle diffusivities. Finally the effect of bounding walls on convective diffusion is considered by means of Monte Carlo calculations. Results show that particle-wall interactions significantly affect the average behaviour of particles located initially within distances of a few particle radii of the wall, since the frictional force is no longer isotropic.

A cylinder containing liquid with a free surface is subjected to a vertical oscillation of amplitude εg/ω2 and frequency 2ω, where ω is within O(εω) of the natural frequency of a particular (primary) mode in the surface-wave spectrum and 0 < ε 1. A Lagrangian formulation, which includes terms of second and fourth order in the primary mode and second order in the secondary modes (which are excited by the primary mode), together with the method of averaging, leads to a Hamiltonian system for the slowly varying amplitudes of the primary mode. The fixed points (which correspond to harmonic motions) and phase-plane trajectories and their perturbations due to linear damping are determined. It is shown that ε > δ, where δ is the damping ratio (actual/critical) of the primary mode, is a necessary condition for subharmonic response of that mode. Explicit results are given for the dominant axisymmetric and antisymmetric modes in a circular cylinder. Internal resonance, in which a pair of modes have frequencies that approximate ω and 2ω, is discussed separately, and the fixed points and their stability for the special case ω2 = 2ω1 are determined. Internal resonance for ω2 = ω1 is discussed in an appendix.

In 1964, Yeh & Cummins demonstrated that coherent light sources could be used for the measurement of steady fluid velocities by observing the Doppler shift in the frequency of light scattered from small particles moving with the fluid. Since 1964 many investigators have attempted to extend this technique to the measurement of turbulent velocity fluctuations.

A fundamental limitation on this type of velocimeter is the Doppler ambiguity introduced by the finite transit time of particles through the scattering volume, turbulent velocity fluctuations across the scattering volume, mean velocity gradients and electronic noise. A unified account of the effect of the Doppler ambiguity on the measurement of the instantaneous velocities is presented and results are interpreted using the power spectrum. The influence of the ambiguity on the measurement of other statistical quantities is also examined.

Limitations on the spatial and temporal resolution imposed by the finite sampling volume are examined using the power spectrum and criteria for optimization of the response are proposed.

An operational laser-Doppler velocimeter is described and measurements of spectra in both laminar and turbulent flow are presented. The experimental results are seen to be in excellent agreement with theoretical predictions.

The rheological properties of a cell suspension may play an important role in the flow field generated by populations of swimming micro-organisms (e.g. in bioconvection). In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 identical squirmers in a simple shear flow field, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. The restriction to pairwise additivity of forces is expected to be justified if the suspension is semi-dilute. The results for non-bottom-heavy squirmers show that the squirming does not have a direct influence on the apparent viscosity. However, it does change the probability density in configuration space, and thereby causes a slight decrease in the apparent viscosity at O(c2), where c is the volume fraction of spheres. In the case of bottom-heavy squirmers, on the other hand, the stresslet generated by the squirming motion directly contributes to the bulk stress at O(c), and the suspension shows strong non-Newtonian properties. When the background simple shear flow is directed vertically, the apparent viscosity of the semi-dilute suspension of bottom-heavy squirmers becomes smaller than that of inert spheres. When the shear flow is horizontal and varies with the vertical coordinate, on the other hand, the apparent viscosity becomes larger than that of inert spheres. In addition, significant normal stress differences appear for all relative orientations of gravity and the shear flow, in the case of bottom-heavy squirmers.

The release from a point source of relatively heavy fluid into a saturated porous medium above an impermeable boundary is considered. A theoretical relationship is compared with experimental data for the rate of propagation of the front of the resulting gravity current and its shape. A motivation of the study, the problem of carbon dioxide sequestration, is briefly discussed.

Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented.

A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem.

A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion.

Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.