A number of initial- and boundary-value problems for the Boussinesq equations are solved by a finite-difference technique, in an attempt to see how a stably-stratified horizontal shear layer rolls up into horizontally periodic billows of concentrated vorticity, such as are frequently observed in the atmosphere and oceans. This paper describes the methods, results and accuracy of the numerical simulations. The results are further analysed and approximately reproduced by a simple semi-analytic model in Corcos & Sherman (1976).

The flow in a pipe of circular cross-section which is coiled in a circle is studied, the pressure gradient along the pipe varying sinusoidally in time with frequency . The radius of the pipe a is assumed small in relation to the radius of curvature of its axis R. Of special interest is the secondary flow generated by centrifugal effects in the plane of the cross-section of the pipe, and an asymptotic theory is developed for small values of the parameter = (2/a2), where is the kinematic viscosity of the fluid. The secondary flow is found to be governed by a Reynolds number , where is a typical velocity along the axis of the pipe, and asymptotic theories are developed for both small and large values of this parameter. For sufficiently small values of it is found that the secondary flow in the interior of the pipe is in the opposite sense to that predicted for a steady pressure gradient, and this is verified qualitatively by an experiment described at the end of the paper.

The permeability of a random array of fixed spheres has been calculated over the range of volume fractions from dilute to almost closest packing, by assuming pairwise-additive (low-Reynolds-number) hydrodynamic interactions within an effective medium. Non-convergent pair interactions arising from the long-range decay of the Stokeslet were removed by renormalizing the Stokes equation to determine the permeability of the effective medium, i.e. to include the mean screening effect of the other spheres. Pair interactions in this Brinkman medium were calculated by the method of reflections in the far field and boundary collocation in the near field.

The permeability predicted by the theory asymptotes correctly to established results for dilute arrays, and compares favourably (within 15%) with the Carman correlation for volume fractions between 0.3 and 0.5. The magnitude also falls within the range of exact results for periodic arrays at the higher concentrations, but our model does not reproduce the dependence on structure.

Use of the Brinkman equation with an effective viscosity leads to an apparent slip velocity at the boundary of a porous medium. Our calculation of the bulk stress via volume averaging determines the effective viscosity and hence the slip coefficient unambiguously for dilute porous media. However, at concentrations corresponding to the available experimental results the lengthscale characterizing pressure or velocity gradients becomes comparable to the interparticle spacing, and the averaging technique fails. Indeed the Brinkman equation itself is no longer valid.

A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.

Experiments on ‘head-on’ collisions between two solitary waves show that the waves reach a maximum amplitude greater than twice the initial wave amplitude and that they suffer a time delay during their interaction. These results are compared with available theories and found to be in qualitative but not quantitative agreement.

The flow over a backward-facing step with laminar separation was investigated experimentally under controlled perturbation for a Reynolds number of 11000, based on a step height h and a free-stream velocity UO. The reattaching shear layer was found to have two distinct modes of instability: the ‘shear layer mode’ of instability at Stθ ≈ 0.012 (Stθ ≡ fθ/UO, θ being the momentum thickness at separation and f the natural roll-up frequency of the shear layer); and the ‘step mode’ of instability at Sth ≈ 0.185 (Sth ≡ fh/U0). The shear layer instability frequency reduced to the step mode one via one or more stages of a vortex merging process. The perturbation increased the shear layer growth rate and the turbulence intensity and decreased the reattachment length compared to the unperturbed flow. Cross-stream measurements of the amplitudes of the perturbed frequency and its harmonics suggested the splitting of the shear layer. Flow visualization confirmed the shear layer splitting and showed the existence of a low-frequency flapping of the shear layer.

We present the first closed-loop separation control experiment using a novel, model-free strategy based on genetic programming, which we call ‘machine learning control’. The goal is to reduce the recirculation zone of backward-facing step flow at $\mathit{Re}_{h}=1350$ manipulated by a slotted jet and optically sensed by online particle image velocimetry. The feedback control law is optimized with respect to a cost functional based on the recirculation area and a penalization of the actuation. This optimization is performed employing genetic programming. After 12 generations comprised of 500 individuals, the algorithm converges to a feedback law which reduces the recirculation zone by 80 %. This machine learning control is benchmarked against the best periodic forcing which excites Kelvin–Helmholtz vortices. The machine learning control yields a new actuation mechanism resonating with the low-frequency flapping mode instability. This feedback control performs similarly to periodic forcing at the design condition but outperforms periodic forcing when the Reynolds number is varied by a factor two. The current study indicates that machine learning control can effectively explore and optimize new feedback actuation mechanisms in numerous experimental applications.

Direct numerical simulation is used to study the turbulent flow over a smooth wavy wall undergoing transverse motion in the form of a streamwise travelling wave. The Reynolds number based on the mean velocity $U$ of the external flow and wall motion wavelength $\lambda$ is 10 170; the wave steepness is $2\pi a/\lambda=0.25$ where $a$ is the travelling wave amplitude. A key parameter for this problem is the ratio of the wall motion phase speed $c$ to $U$, and results are obtained for $c/U$ in the range of $-1.0$ to $2.0$ at $0.2$ intervals. For negative $c/U$, we find that flow separation is enhanced and a large drag force is produced. For positive $c/U$, the results show that as $c/U$ increases from zero, the separation bubble moves further upstream and away from the wall, and is reduced in strength. Above a threshold value of $c/U\approx 1$, separation is eliminated; and, relative to small- $c/U$ cases, turbulence intensity and turbulent shear stress are reduced significantly. The drag force decreases monotonically as $c/U$ increases while the power required for the transverse motion generally increases for large $c/U$; the net power input is found to reach a minimum at $c/U\approx 1.2$ (for fixed $U$). The results obtained in this study provide physical insight into the study of fish-like swimming mechanisms in terms of drag reduction and optimal propulsive efficiency.

The plane steady decelerated flow of a dust-gas mixture is analysed in an approximate manner. The problem, which has a five-parameter family of solutions, is reduced to a form such that the analysis can be completed by the integration of a first-order non-linear differential equation and a quadrature. A few integral curves are given and the characterizing features of the flow field are discussed.

This paper presents laser-Doppler measurements of the mean velocity and statistical moments of turbulent velocity fluctuations in the near-wall region of a fully developed pipe flow at low Reynolds numbers. A refractive-index-matched fluid was used in a Duran-glass test section to permit access to the near-wall region without distortion of the laser beams. All measurements were corrected for the influence of the finite size of measuring control volume. Measurements of long-time statistical averages of all three fluctuating velocity components in the near-wall region are presented. It is shown that the turbulence intensities in the wall region do not scale with inner variables. However, the limiting behaviour of the intensity components very close to the wall show only small variations with the Reynolds number. Measurements of higher-order statistical moments, the skewness and flatness factors, of axial and tangential velocity components confirm the limiting behaviour of these quantities obtained from direct numerical simulations of turbulent channel flow. The comparison of measured data with those obtained from direct numerical simulations reveals that noticeable discrepancies exist between them only with regard to the flatness factor of the radial velocity component near the wall. The measured v’ flatness factor does not show the steep rise close to the wall indicated by numerical simulations. Analysis of the measured data in the near-wall region reveals significant discrepancies between the present LDA measurements and experimental results obtained using the hot-wire anemometry.

This investigation presents a unique and elaborate set of experiments relating the generation of noise to the evolution of large-scale turbulence structures within an ideally expanded, Mach 1.28, high-Reynolds-number $(1.03\,{\times}\,10^{6})$ jet. The results appear to indicate many similarities between the noise generation processes of high-speed low-Reynolds-number and high-speed high-Reynolds-number jets. Similar to the rapid changes observed in theregion of noise generation in low-Reynolds-number jets in previous experimental and computational work, a series of robust flow features formed approximately one convective time scale before noise emission and then rapidly disintegrated shortly before the estimated moment of noise emission. Coincident with the disintegration, a positive image intensity fluctuation formed at the jet centreline in a region that is immediately past the endof the potential core. This indicates mixed fluid had reached the jet core. These results are consistent with the formation of large-scale structures within the shear layer, which entrain ambient air into the jet, and their eventual interaction and disintegration apparently result in noise generation. These results are quite different from the evolution of the jet during prolonged periods that lacked significant sound emission. The observations presented in this work were made through the use of well-established technique that were brought together in an unconventional fashion. The sources of large-amplitude sound waves were estimated in time and three-dimensional space using a novel microphone array/beamforming algorithm while the noise-generation region of the mixing layer was simultaneously visualized on two orthogonal planes (one of which was temporally resolved). The flow images were conditionally sampled based on whether or not a sound wave wascreated within the region of the flow while it was being imaged and a series of images was compiled that was roughly phase-locked onto the moment of sound emission. Another set of images was gathered based on a lack of sound waves reaching the microphone array over several convective time scales. Proper orthogonal decomposition (POD) was then used tocreate a basis for the flow images and this basis was used to reconstruct the evolution of the jet.

When an electric field is applied to a drop suspended in another liquid the drop deforms. The relation between the applied field and the mode and magnitude of the deformation have been studied extensively. Nevertheless, Torza, Cox & Mason (1971) found that quantitative agreement between the leaky dielectric theory (Taylor 1966) and experiment is quite poor. Here we describe results from a new series of experiments. Drops suspended in weakly conducting liquids were deformed into spheroids with both steady and oscillatory fields. Drop deformation, interfacial tension, and the electrical properties of the fluids were measured for each system to provide a definitive test of the theory. The agreement between the leaky dielectric model and our results for drop deformations in steady fields is much improved over previous results, although discrepancies remain for some systems. Drop deformations in oscillatory fields consist of steady and oscillatory parts because of the quadratic dependence on the field strength. Measurements of the steady part at 60 Hz, where the oscillatory deformation is negligible, are in excellent agreement with the theory. The effects of frequency on the steady deformation were studied by measuring oblate deformations at a series of frequencies and field strengths; the agreement with theory is good. Finally, the time-dependent total deformation was measured under conditions where both parts of the deformation are commensurate. Good agreement was found between the measured and predicted maximum and minimum deformations. Nevertheless, only a small range of fluid properties could be studied owing to the need to avoid droplet sedimentation.

Gravity waves on the surface of an inviscid incompressible fluid of finite depth are considered. The waves are assumed to be periodic in time and in the horizontal direction. The surface profile, potential function, pressure and frequency of the motion are determined (to third order) as series in powers of the amplitude divided by the wavelength. It is found that the frequency increases with amplitude for depths less than a certain multiple of the wavelength and decreases with increasing amplitude for greater depths. Graphs of the surface profile and of the pressure as a function of depth are included.

Diffusiophoresis, the motion of a particle in response to an externally imposed concentration gradient of a solute species, is analysed from both the traditional coarse-grained macroscopic (i.e. continuum) perspective and from a fine-grained micromechanical level in which the particle and the solute are treated on the same footing as Brownian particles dispersed in a solvent. It is shown that although the two approaches agree when the solute is much smaller in size than the phoretic particle and is present at very dilute concentrations, the micromechanical colloidal perspective relaxes these restrictions and applies to any size ratio and any concentration of solute. The different descriptions also provide different mechanical analyses of phoretic motion. At the continuum level the macroscopic hydrodynamic stress and interactive force with the solute sum to give zero total force, a condition for phoretic motion. At the colloidal level, the particle's motion is shown to have two contributions: (i) a ‘back-flow’ contribution composed of the motion of the particle due to the solute chemical potential gradient force acting on it and a compensating fluid motion driven by the long-range hydrodynamic velocity disturbance caused by the chemical potential gradient force acting on all the solute particles and (ii) an indirect contribution arising from the mutual interparticle and Brownian forces on the solute and phoretic particle, that contribution being non-zero because the distribution of solute about the phoretic particle is driven out of equilibrium by the chemical potential gradient of the solute. At the colloidal level the forces acting on the phoretic particle – both the statistical or ‘thermodynamic’ chemical potential gradient and Brownian forces and the interparticle force – are balanced by the Stokes drag of the solvent to give the net phoretic velocity.

For a particle undergoing self-phoresis or autonomous motion, as can result from chemical reactions occurring asymmetrically on a particle surface, e.g. catalytic nanomotors, there is no imposed chemical potential gradient and the back-flow contribution is absent. Only the indirect Brownian and interparticle forces contribution is responsible for the motion. The velocity of the particle resulting from this contribution can be written in terms of a mobility times the integral of the local ‘solute pressure’ – the solute concentration times the thermal energy – over the surface of contact between the particle and the solute. This was the approach taken by Córdova-Figueroa & Brady (Phys. Rev. Lett., vol. 100, 2008, 158303) in their analysis of self-propulsion. It is shown that full hydrodynamic interactions can be incorporated into their analysis by a simple scale factor.

The Ranque–Hilsch effect, observed in swirling flow within a single tube, is a spontaneous separation of total temperature, with the colder stream near the tube centreline and the hotter air near its periphery. Despite its simplicity, the mechanism of the Ranque–Hilsch effect has been a matter of long-standing dispute. Here we demonstrate, through analysis and experiment, that the acoustic streaming, induced by orderly disturbances within the swirling flow is, to a substantial degree, a cause of the Ranque–Hilsch effect. The analysis predicts that the streaming induced by the pure tone, a spinning wave corresponding to the first tangential mode, deforms the base Rankine vortex into a forced vortex, resulting in total temperature separation in the radial direction. This is confirmed by experiments, where, in the Ranque–Hilsch tube of uniflow arrangement, we install acoustic suppressors of organ-pipe type, tuned to the discrete frequency of the first tangential mode, attenuate its amplitude, and show that this does indeed reduce the total temperature separation.

When a drop impacts on a liquid surface it entraps a small amount of air under its centre as the two liquid surfaces meet. The contact occurs along a ring enclosing a thin disk of air. We use the next-generation ultra-high-speed video camera, capable of 1 million f.p.s. (Etoh et al. 2002), to study the dynamics of this air sheet as it contracts due to surface tension, to form a bubble or, more frequently, splits into two bubbles. During the contraction of the air disk an azimuthal undulation, resembling a pearl necklace, develops along its edge. The contraction speed of the sheet is accurately described by a balance between inertia and surface tension. The average initial thickness of the air sheet decreases with higher impact Reynolds numbers, becoming less than one micron. The total volume of air entrapped depends strongly on the bottom curvature of the drop at impact. A sheet of micro-bubbles is often observed along the original interface. Oguz–Prosperetti bubble rings are also observed. For low Weber numbers (We<20) a variety of other entrapment phenomena appear.

Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity field induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100[ratio ]1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13–0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7–8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg–de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin–Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We find that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

A direct numerical simulation was carried out of plane turbulent Couette flow at a Reynolds number of 750, based on half the velocity difference between the walls and half the channel width. Particular attention was paid to choosing a computational box that is large enough to accommodate even the largest scales of the turbulence. In the central region of the channel very large elongated structures were observed, in accordance with earlier findings. The study is focused on the properties of these structures, but is also aimed at obtaining accurate turbulence statistics. Terms in the energy budget were evaluated and discussed. Also, the limiting values of various quantities were determined and their relevance in high Reynolds number flows discussed. The large structures were shown to be very sensitive to an imposed system rotation. They could be essentially eliminated with a stabilizing system rotation (around the spanwise axis) small enough for only minor damping of the rest of the scales. Despite the fact that the large structures dominate the appearance of the flow field their energy content was shown to be relatively small, on the order of 10% of the total turbulent kinetic energy.

A non-orthogonal helical co-ordinate system is introduced to study the effect of curvature and torsion on the flow in a helical pipe. It is found that both curvature and torsion induce non-negligible effects when the Reynolds number is less than about 40. When the Reynolds number is of order unity, torsion induces a secondary flow consisting of one single recirculating cell while curvature causes an increased flow rate. These effects are quite different from the two recirculating cells and decreased flow rate at high Reynolds numbers.

The correct asymptotic theory for the linear onset of instability of a Boussinesq fluid rotating rapidly in a self-gravitating sphere containing a uniform distribution of heat sources was given recently by Jones et al. (2000). Their analysis confirmed the established picture that instability at small Ekman number $E$ is characterized by quasi-geostrophic thermal Rossby waves, which vary slowly in the axial direction on the scale of the sphere radius $r_o$ and have short azimuthal length scale $O(E^{1/3}r_o)$. They also confirmed the localization of the convection about some cylinder radius $s\,{=}\,s_M$ roughly $r_o/2$. Their novel contribution concerned the implementation of global stability conditions to determine, for the first time, the correct Rayleigh number, frequency and azimuthal wavenumber. Their analysis also predicted the value of the finite tilt angle of the radially elongated convective rolls to the meridional planes. In this paper, we study small-Ekman-number convection in a spherical shell. When the inner sphere radius $r_i$ is small (certainly less than $s_M$), the Jones et al. (2000) asymptotic theory continues to apply, as we illustrate with the thick shell $r_i\,{=}\,0.35\,r_o$. For a large inner core, convection is localized adjacent to, but outside, its tangent cylinder, as proposed by Busse & Cuong (1977). We develop the asymptotic theory for the radial structure in that convective layer on its relatively long length scale $O(E^{2/9}r_o)$. The leading-order asymptotic results and first-order corrections for the case of stress-free boundaries are obtained for a relatively thin shell $r_i\,{=}\,0.65\,r_o$ and compared with numerical results for the solution of the complete PDEs that govern the full problem at Ekman numbers as small as $10^{-7}$. We undertook the corresponding asymptotic analysis and numerical simulation for the case in which there are no internal heat sources, but instead a temperature difference is maintained between the inner and outer boundaries. Since the temperature gradient increases sharply with decreasing radius, the onset of instability always occurs on the tangent cylinder irrespective of the size of the inner core radius. We investigate the case $r_i\,{=}\,0.35\,r_o$. In every case mentioned, we also apply rigid boundary conditions and determine the $O(E^{1/6})$ corrections due to Ekman suction at the outer boundary. All analytic predictions for both stress-free and rigid (no-slip) boundaries compare favourably with our full numerics (always with Prandtl number unity), despite the fact that very small Ekman numbers are needed to reach a true asymptotic regime.