An investigation of a selection of high-Reynolds-number bluff-body flows was conducted. Here in Part 2 phase-averaged velocity-field results will be presented for several far-wake flows generated by nominally two-dimensional and three-dimensional bodies. In these far-wake flows the shed vortices have approached a nearly constant convection velocity. Some mean velocity and phase-averaged and global Reynoldsstress measurements are also presented. The turbulent wake of a lift-producing three-dimensional body has been examined. Also included are the phase-averaged wake patterns behind a flapping flag and a windmill. The topological structure of these patterns is discussed and a preliminary classification of wake patterns is presented.

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate $x$. The bottom and top of the channel are at $z= 0, H$, and the cross-section is given by the quite general $- {f}_{1} (z)\leq y\leq {f}_{2} (z)$ for $0\leq z\leq H$. We develop a two-layer shallow-water (SW) formulation, and we implement it for the solution of the dam-break problem. The dependent variables are the position of the interface, $h(x, t)$, and the velocity (averaged over the area of the current), $u(x, t)$. The non-rectangular cross-section geometry enters the formulation via $f(h)$ and integrals of $f(z)$ and $zf(z)$, where $f(z)= {f}_{1} (z)+ {f}_{2} (z)$ is the width of the channel. For a given geometry $f(z)$, the free input parameters of the lock-release problem are: (i) the height ratio $H/ {h}_{0} $ of ambient to lock; and (ii) the density ratio of ‘light’ to ‘heavy’ fluids, $R$. We show that the dam-break problem is amenable to solution by the method of characteristics, but various complications (internal jumps, critical restrictions) appear when the return flow in the ambient is significant; these features are not captured by a one-layer shallow-water model. The role of the non-standard geometry in the detection and calculation of the internal and reflected jumps, and on the dam-break flow field, is elucidated. The methodology is illustrated for Boussinesq flow in typical geometries: power-law ($f(z)= b{z}^{\alpha } $, where $b, \alpha $ are positive constants), trapezoidal and circle (full or sector); we solved for full-depth ($H= 1$) and part-depth ($H\gt 1$) lock-release configurations. The approach seems to work well: in all the tested cases, the solution was obtained with simple mathematical and numerical tools (a Runge–Kutta integrator and a secant-method solver); the insights are compatible with our previous understanding of the problem; and the standard (rectangular) case is recovered in the appropriate limit (i.e. when the side-boundaries of the trapezium cross-section are vertical or far apart). A comparison of the speed of propagation with available experiments is also presented, and the agreement is satisfactory. The present solution is a significant generalization of the classical two-layer gravity–current dam-break problem. The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, $f(z)= $ const., in the wide domain of cross-sections covered by the new model.

We consider the flow of a slightly viscous homogeneous fluid over a small two-dimensional obstacle perpendicular to the vertical side walls in a channel rotating about a vertical axis. The flow in the channel is obtained from the solution of the quasi-geostrophic potential vorticity equation in the limit ε = D/L→ 0, where D is the obstacle width and L the channel width. The lowest order, interior flow is shown to be a combination of three effects: a rotational flow caused by vortex stretching and Ekman boundary-layer pumping; a significant irrotational flow induced by the magnitude of the former flow at the vertical boundaries; and the interior Ekman drift due to the basic current. The maximum streamline displacement is calculated and compares very well with recent experiments in the identical parameter range by Boyer (1971a, b). This theory explains how the side walls are responsible for the dependence of the maximum streamline displacement on Rossby number.

This paper describes one-dimensional (parallel) laminar natural convection in a viscous stably stratified fluid owing to temporally periodic variations in the surface temperature of infinite vertical plates and circular cylinders. Analytical solutions of the one-dimensional (parallel) Boussinesq equations of motion and thermodynamic energy are obtained for the periodic regime for arbitrary values of ambient stratification, Prandtl number and forcing frequency. The solutions for plates and cylinders are qualitatively similar and show that (i) the flows are composed of two waves that decay exponentially with distance from the surface: a fast long wave and a slow short wave; (ii) for forcing frequencies greater than the natural frequency of the corresponding inviscid system, these two waves propagate away from the surface; and (iii) for forcing frequencies less than this natural frequency, the short wave propagates away from the surface while the long wave propagates toward the surface. This latter case provides an example of a flow for which the conventional radiation condition is not appropriate. The analytical results are complemented, for the plate problem, with three-dimensional numerical simulations of flows that start from rest and are suddenly subjected to a periodic thermal forcing at the plate. The numerical results depict the transient (start-up) stage of the flow and the approach to a periodic regime. These results confirm that the analytical solutions provide the appropriate description of the periodic regime.

Microorganisms must approach other suspended organisms or particles in order to interact with them during a host of life processes including feeding and mating. Microorganisms live at low Reynolds number where viscosity dominates and strongly affects the hydrodynamics of swimmer and nearby cells and objects. Viscous hydrodynamics makes it difficult for two surfaces to approach closely at low Reynolds numbers. Nonetheless, it is observed that microorganisms in fluid are still able to approach closely enough to interact with each other or suspended particles. Here, we study how the physical constraints provided by viscous hydrodynamics affects the feasibility of direct approach of flagellated and ciliated microorganisms to targets of different sizes. We find that it is feasible for singly flagellated swimmers to approach targets that are the same size or bigger. On the other hand, for squirmers, the feasibility of approach depends on near-field flows that can be controlled by the details of their swimming strokes.

The fundamental challenge to characterize and quantify thermal transport in the strongly nonlinear regime of Rayleigh–Bénard convection – the buoyancy-driven flow of a horizontal layer of fluid heated from below – has perplexed the fluid dynamics community for decades. Rayleigh proposed controlling the temperature of thermally conducting boundaries in order to study the onset of convection, in which case vertical heat transport gauges the system response. Conflicting experimental results for ostensibly similar set-ups have confounded efforts to discriminate between two competing theories for how boundary layers and interior flows interact to determine transport through the convecting layer asymptotically far beyond onset. In a conceptually new approach, Bouillaut, Lepot, Aumaître and Gallet (J. Fluid Mech., vol. 861, 2019, R5) devised a procedure to radiatively heat a portion of the fluid domain bypassing rigid conductive boundaries and allowing for dissociation of thermal and viscous boundary layers. Their experiments reveal a new level of complexity in the problem suggesting that heat transport scaling predictions of both theories may be realized depending on details of the thermal forcing.

The flow of liquids through paper is challenging to model due to the complexity and disordered layout of the fibre matrix. The expanding use and capability of microfluidic paper-based analytical devices ($\unicode[STIX]{x1D707}$PADs) and their requirement for precision has increased the need to accurately predict the flow of liquids through paper. Many studies have developed models and revealed some of the physical mechanisms responsible for the flow behaviour, but we still lack a complete understanding, particularly in relation to how the fluid fills the various voids with a wide range of shapes and sizes in the fibre matrix of paper. In the featured article, Chang et al. (J. Fluid Mech., vol. 845, 2018, 36–50) used a combined experimental and theoretical approach to uncover the importance of the liquid filling the intra-fibre pores, showed that this results in deviation from the flow behaviour predicted by the Lucas–Washburn equation and developed a model which accounts for this effect.

A linear and irrotational model is constructed to represent the formation of water waves by ground motions of a sloping bed. A survey of the constant depth case, given first, helps in understanding the mechanism of formation, and, in this oversimplified case, wave propagation away from a source, which is usually very asymmetric. The importance of asymmetry, which may produce trapped waves, is illustrated by an estimate of the propagation in a three-dimensional case. The formation of waves by a ground motion on a slope is then studied in detail. The problem is reduced to linear integral equations of the first kind. Using an inversion technique one constructs a source–response pair in which the source is ‘δ-like’ and the response is close to that which would be found if the depth was constant around the source. A general approximate solution is then derived, in both the two-dimensional and three-dimensional cases. Results for the sloping-bottom case are given for small times. They give initial values of surface displacement. They also enable one to determine the important physical parameters in the ground motion and to evaluate the efficiency of wave production.

This study makes a specific comparison between two different two-dimensional free shear layers: the T-layer which develops in time from an initial tangential velocity discontinuity separating the two half-spaces; and the S-layer which develops downstream of the origin where two uniform streams of unequal velocity are brought into tangential contact. The method of comparison is to assume that the vorticity of the S-layer is given parabolically by a Galilean mapping of that of the T-layer; to satisfy the appropriate boundary conditions in the S-layer and to compute the velocity induced at any point in the S-layer by its vorticity field; and to compare this velocity to that which can be derived from the velocity of the T-layer at corresponding points by a Galilean transformation of the velocity itself. The purpose of this calculation is to assess approximately how far the flow in the S-layer is from parabolic and, in particular, to what extent the perturbations induced upstream by large concentrations of vorticity found downstream are instrumental in hastening or retarding the subharmonic instability that leads to the formation of these large structures. The calculations suggest that this elliptic influence, or the feedback, in a mixing layer is relatively small, at least for small velocity ratios.

This paper analyses the effect of an alternating magnetic field of low frequency ω on a cylindrical tank of liquid metal. Previous work with higher-frequency fields has focused attention on the mean recirculating motion, but in the low-frequency limit periodic motion and surface waves become important. We show that a system of forced standing axisymmetric waves of frequency 2ω is established, and that the growth of non-axisymmetric modes is governed by a coupled system of Mathieu-type equations. The stability regions associated with this system are discussed and it is shown that the most easily excited transition to a non-axisymmetric mode is subharmonic, with frequency ω. Comparison with experiment shows that the theory gives qualitatively correct predictions.

The influence of different boundary conditions applied in the contact line region on the outer meniscus shape is analysed by means of a finite-element numerical simulation of the steady movement of a liquid-gas meniscus in a capillary tube. The free-surface steady shape is obtained by solving the unsteady creeping-flow approximation of the Navier–Stokes equations starting from some initial shape. Comparisons of the outer solutions obtained using two different inner models, together with that published by Lowndes (1980), indicate the relative insensitivity of the outer solution to the type of model utilized in the contact line region.

Consider two infinitely long cylinders of different radii with one inside the other but off-centred. The gap between the two cylinders is partially filled with a viscous fluid. As the cylinders rotate with independent velocities U1 and U2, a thin liquid film coats each of their surfaces all the way around except in the region where the viscous fluid completely fills the gap. Interface conditions that connect solutions of averaged equations in the viscous fluid region with solutions in the thin film region are derived. For the two-interface problem analysed here, two types of instabilities occur depending on the amount of viscous fluid between the cylinders. For large fluid volume, the primary supercritical instability occurs when the front interface becomes unstable as the cylinder velocities are increased. For small fluid volume, the back interface passes through the region where the gap width is a minimum to the same side as the front interface. Steady state solutions with straight interface edges exhibit a turning point with respect to the cylinder velocities. The back interface becomes unstable at the turning point; this inverse instability is subcritical.

A literature survey (Berger, Talbot & Yao 1983) indicates that laminar viscous flow in curved pipes has been extensively investigated. Most of the existing analytical results deal with the case of circular cross-section. The important studies dealing with elliptical cross-sections are mainly due to Thomas & Walters (1965) and Srivastava (1980). The analysis of Thomas & Walters is based on Dean's (1927, 1928) approach in which the simplified forms of the momentum and continuity equations have been used. The analysis of Srivastava is essentially a seminumerical approach, in which no explicit expressions have been presented.

In this paper, using elliptic coordinates and following the unsimplified formulation of Topakoglu (1967), the flow in a curved pipe of elliptical cross-section is analysed. Two different geometries have been considered: (i) with the major axis of the ellipse placed in the direction of the radius of curvature; and (ii) with the minor axis of the ellipse placed in the direction of the radius of curvature. For both cases explicit expressions for the first term of the expansion of the secondary-flow stream function as a function of the ellipticity ratio of the elliptic section have been obtained. After selecting a typical numerical value for the ellipticity ratio, the secondary-flow streamlines are plotted. The results are compared with that of Thomas & Walters. The remaining terms of the expansion of the flow field are not included, but they will be analysed in a future paper.

If an electric current of uniform density j0 is passed axially through a stationary fluid between concentric cylinders of radii r1 and r2 (> r1), the fluid is stable to axisymmetric disturbances only if the damping provided by viscosity and electrical resistivity is sufficiently large. It is shown herein that the fluid may also be stabilized by passing a line current J along the axis, sufficient conditions for stability being $J \le -\pi j _0(r^2_2 - r^2_1)$, or $\ge J\; \pi j _0r^2_1$

The values of J needed to stabilize the fluid when the fluid has non-zero viscosity and finite conductivity are calculated for the case r2-r1 [Lt ] r1. In this latter case, the ring vortices which exist under conditions of neutral stability are exactly the same as those for flow between rotating cylinders if J and j0 have the same sign, and if J is not very small compared with πj0r2.

Low-Prandtl-number convection is investigated in vertical axisymmetric turbulent buoyant sodium jets discharging into a slowly moving ambient. Measurements of mean velocity, mean temperature and temperature fluctuations are performed simultaneously using a miniature permanent-magnet flowmeter probe. By varying the ratio of momentum to buoyancy flux, or the densimetric Froude number, different intensities of buoyancy are obtained giving a range of conditions encompassing forced-convection jets, buoyant jets and plumes. In line with the classical properties of jets the radial velocity and temperature profiles can be described by the Gaussian function, independent of the flow regime, at all axial measuring positions. The axial decay of the centreline mean velocity for sodium is the same as for fluids of higher Prandtl number, governed by power laws with indices of −1 for forced convection, −2/3 for the transitional buoyant region and −1/3 for plume flow. In contrast, the centreline mean temperatures for sodium plumes decrease with a power of −1 compared with the −5/3 decay for fluids of higher Prandtl number. The different behaviour in sodium is due to the dominance of molecular diffusion in heat transport, while momentum transport is dictated by turbulent diffusion, which gives a similarity solution for forced-convection jets but not for buoyant jets or plumes. The radial profiles of the temperature r.m.s. values can be described by an axisymmetric curve with two maxima, independent of the flow regime, at all axial measuring positions and the two maxima are more pronounced than in conventional fluids. The temperature fluctuations are analysed to give statistical parameters such as minimum and maximum values, skewness, flatness, probability density functions and spectral distribution. The spectral distributions display both a convective subrange and the conductive subrange predicted for fluids of low Prandtl number. Integral length scales of the temperature fluctuations are evaluated and found to be significantly smaller than turbulent velocity scales.

This paper presents a study of the steady, axisymmetric, creeping translation of a fluid sphere in a tube for the case in which surfactant is adsorbed onto the fluid sphere interface. Marangoni stresses caused by the convective redistribution of surfactant are computed perturbatively in the limit of sorption-controlled uniform retardation, and fully converged numerical solutions of the creeping-flow equations including the Marangoni stress are obtained by a collocation technique.

The results indicate that when the fluid sphere moves in a liquid which is at rest at infinity, the Marangoni stress retards the particle velocity. This retardation generally increases with the sphere to tube diameter ratio up to a value of approximately 0.6, whereupon the retardation begins to level off or even become reduced. When the sphere is suspended in a Poiseuille flow, stagnation rings develop on the sphere surface, and the Marangoni stresses that derive from this surface convection pattern can accelerate the fluid particle when the particle velocity is small with respect to the Poiseuille centreline velocity, but in the same direction as that velocity.

We investigate cryogenic flows of liquid 4He between two grids oscillating in phase, at temperatures ranging from approximately 1.3 to 2.5 K, resulting in suitably defined Reynolds numbers up to $10^{5}$. We specifically study the flow-induced motions of small particles suspended in the fluid by using the particle tracking velocimetry technique. We focus on turbulent flows of superfluid 4He that occur below approximately 2.2 K and are known to display, in certain conditions, features different from those observed in flows of classical viscous fluids, such as water. We find that, at large enough length scales, larger than the mean distance between quantized vortices, representing the quantum length scale of the flow, the shapes of the velocity and velocity increment statistical distributions are very similar to those obtained in turbulent flows of viscous fluids. The experimental outcome strongly supports the view that, in the range of investigated parameters, particles probing flows of superfluid 4He behave as if they were tracking classical flows.

A series expansion is derived for the three-dimensional boundary-layer flow over a flat plate, arising from a general main-stream flow over the plate. The series involved are calculated as far as terms of order ξ2, where ξ is a non-dimensional parameter defining distance measured from the leading edge of the plate. The results are applied to an example in which the main stream arises from the disturbance of a uniform stream by a circular cylinder mounted downstream from the leading edge of the plate, the axis of the cylinder being normal to the plate. Calculations are made for shear stress components on the plate, and for the deviation of direction of the limiting streamlines from those in the main stream.

The geometrically different planforms of near-wall plume structure in turbulent natural convection, visualized by driving the convection using concentration differences across a membrane, are shown to have a common multifractal spectrum of singularities for Rayleigh numbers in the range $10^{10}$–$10^{11}$ at Schmidt number of 602. The scaling is seen for a length scale range of $2^5$ and is independent of the Rayleigh number, the flux, the strength and nature of the large-scale flow, and the aspect ratio. Similar scaling is observed for the plume structures obtained in the presence of a weak flow across the membrane. This common non-trivial spatial scaling is proposed to be due to the same underlying generating process for the near-wall plume structures.