Despite little supporting evidence, there appears to be an implicit assumption that the wakes of two-dimensional bluff bodies undergo transition to three-dimensional flow and eventually turbulence, through the same sequence of transitions as observed for a circular cylinder wake. Previous studies of a square cylinder wake support this assumption. In this paper, the transition to three-dimensional wake flow is examined for an elongated cylinder with an aerodynamic leading edge and square trailing edge. The three-dimensional instability modes are determined as a function of aspect ratio ($\hbox{\it AR}\,{=}\,$length to width). Floquet analysis reveals that three distinct instabilities occur. These are referred to as Modes A, B$^\prime$ and S$^\prime$ through analogy with the modes for circular and square cylinders. For aspect ratios less than approximately 7.5, Mode A is the most unstable mode. For aspect ratios greater than this, the most unstable mode switches to Mode B$^\prime$. This has the same spatio-temporal symmetry as Mode B for a circular cylinder, but a spanwise wavelength and near-wake features more in common with Mode S for a square cylinder. The dominant wavelength for this mode is approximately two cylinder thicknesses, much longer than for Mode B for a circular cylinder. It is found that the critical Reynolds number for the onset of the Mode A instability varies approximately with the square root of the aspect ratio. On the other hand, the critical Reynolds number for Mode B$^\prime$ is almost independent of aspect ratio. For large aspect ratios, the separation in Reynolds number between the critical Reynolds numbers is substantial; for instance, for $\hbox{\it AR}\,{=}\,17.5$, these values are approximately 450 and 700. In fact, for this aspect ratio, the third instability mode, Mode S$^\prime$, is more unstable than Mode A. These results suggest that the transition scenario for elongated bluff bodies may be distinctly different to short bodies such as circular or square cylinders. At the very least, the dominant spanwise wavelength in the turbulent wake is likely to be much longer than that for a circular cylinder wake. In addition, the reversal of the ordering of occurrence of the two modes with the different spatial symmetries is likely to affect the development of spatio-temporal chaos as a precursor to fully turbulent flow.

In conjunction with prior work, the current results indicate that nearly all three-dimensional instabilities of the vortex street can be identified as one of only a handful of transition modes.

The wake states from a circular cylinder undergoing controlled sinusoidal oscillation transverse to the free stream are examined. As the frequency of oscillation passes through the natural Kármán frequency there is a transition between two distinctly different wake states: the low- and high-frequency states. The transition corresponds to a change in the structure of the near wake and is also characterized by a jump in the phase and amplitude of both the total and vortex lift. Over the range of flow and oscillation parameters studied the wake states exhibit a number of universal features. The phases of the vortex lift and drag forces have characteristic values for the low- and high-frequency states, which appear to be directly related to the phase of vortex shedding. A split force concept is employed, whereby instantaneous force traces and images allow discrimination between the actual loading and the physics, and their conventional time-averaged representations. The wake states for the forced oscillations show some remarkable similarities to the response branches of elastically mounted cylinders. The equivalence between forced and self-excited oscillations is addressed in detail using concepts of energy transfer.

We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence.

Starting flow through a nozzle or orifice typically results in the transient formation of a leading vortex ring and trailing jet. Experiments are conducted to investigate the dynamics of this process in the case of a temporally variable nozzle exit diameter, with the aim of understanding these flows as they occur in Nature and emerging technologies. By kinematically decoupling the source flow from the nozzle motion, comparison across several classes of exit diameter temporal variation is facilitated. Kinematic models of the starting flows are used to accurately predict the fluid circulation produced by the vortex generators, and to emphasize the special role of the nozzle boundary layer in dictating the nature of the global flow patterns. A dimensionless temporal parameter is derived in order to track the vortex formation process for the various classes of nozzle motion. Dynamics of vortex ring disconnection from the source flow are studied in this new dimensionless framework. We show that temporally increasing the nozzle exit diameter as the starting flow develops results in higher-energy vortex ring structures with peak vorticity located further from the axis of symmetry relative to a static nozzle case. In addition, the normalized energy supplied by the vortex generator is increased in this process. We do not observe a delay in the onset of vortex ring disconnection from the trailing jet, as predicted by previous numerical simulations. In contrast, growth of the leading vortex ring is substantially augmented by temporally decreasing the nozzle exit diameter during fluid ejection, as noted in a previous experiment. Normalized vortex ring circulation is increased 35% in these cases, and the normalized energy of the generated vortex rings is equivalent to that of Hill's spherical vortex. These observed effects are explained by considering the measured vorticity distribution and energy of the starting flows. Strategies are suggested to exploit the discovered dynamics for various pulsed-jet applications.

Benilov, O'Brien & Sazonov (2003) and Benilov (2004) describe “a new type of instability” in a liquid film inside a rotating cylinder. Though their linear systems support only neutrally stable modes, they find explosive disturbances which become singular after a finite time. They suggest that this result casts doubt on the reliability of modal analysis for prediction of instability; and they claim that “Such cases have never been described in the literature, and they are probably extremely rare”. Here, other examples are given, some of which have been known (though not well-known) for many years. A common feature of all these singularities is a local phase synchronization of short-wave modes; but the configuration of Benilov et al. has the additional feature of eigenfunctions that exhibit very large changes in amplitude within the spatial domain. The relevance, or not, of such singularities to real physical systems is discussed.

Hot-film anemometry measurements are performed in a fully developed turbulent bubbly flow. For the bubble detection in the signal, both a threshold method and a new pattern recognition algorithm are employed. The measurements are carried out with gas fractions up to 3% and a mean water velocity of 0.20 m s$^{-1}$, corresponding to a Reynolds number of about $9\,{\times}\,10^4$. The typical bubble radius is 1–2 mm, corresponding to 10–20 Kolmogorov length scales. In this regime, a ‘bubblance’ parameter $b$ which compares the kinetic energy originating from the rising bubbles with that of the turbulence fluctuations is smaller than 1. Probability distribution functions, structure functions (with and without the extended self-similarity (ESS) method), and spectra of the water velocity time series are calculated. Both our results for the turbulent energy spectra and the second-order structure functions show qualitative agreement with numerical results by Massitelli, Lohse & Toschi (Phys. Fluids, vol. 15 (2003), p. L5), i.e. a more pronounced energy enhancement on small scales than on large scales owing to the presence of bubbles, leading to a less steep slope in the spectrum as compared to the Kolmogorov $-5/3$ law. These results are robust, i.e. do not depend on details of the bubble detection scheme.

In this paper, we consider nearshore rotational currents directly forced by unsteady multidirectional wave breaking. Scaling relationships, simplified analytical solutions, and asymptotic limits are developed for the maximum forced cross-shore and longshore velocities. In all cases, forced longshore velocities are considerably larger than cross-shore velocities. On longshore-uniform beaches, strong fluctuating velocities are found for (i) large waves; (ii) strong directional spreading; and (iii) short peak wave periods. When topographic inhomogeneities control longshore scales of wave breaking, overall scaling changes and the largest fluctuating velocities are found for ($a$) large waves; ($b$) long wave periods; and ($c$) topographic features that vary quickly in the longshore direction. The ratio of fluctuating rotational velocities to mean longshore current does not depend on the wave height or period, but instead on the bottom friction, slope, deep water wave angle, and details of the wave spectrum.

We consider the dynamics of a thin liquid film falling down a uniformly heated wall. The heating sets up surface tension gradients that induce thermocapillary stresses on the free surface, thus affecting the evolution of the film. We model this thermocapillary flow by using a gradient expansion combined with a Galerkin projection with polynomial test functions for both velocity and temperature fields. We obtain equations for the evolution of the velocity and temperature amplitudes at first- and second-order in the expansion parameter. These equations are fully compatible close to criticality with the Benney long-wave expansion. Models of reduced dimensionality for the evolution of the local film thickness, flow rate and interfacial temperature only, are proposed.

We analyse the regularized reduced model derived in Part 1 (Ruyer-Quil et al. 2005). Our investigation is two-fold: (i) we demonstrate that the linear stability properties of the model are in good agreement with the Orr–Sommerfeld analysis of the linearized Navier–Stokes/energy equations; (ii) we show the existence of nonlinear solutions, namely single-hump solitary pulses, for the widest possible range of parameters. We also scrutinize the influence of Reynolds, Prandtl and Marangoni numbers on the shape, speed, flow patterns and temperature distributions for the solitary waves obtained from the regularized model. The hydrodynamic and Marangoni instabilities are seen to reinforce each other in a non-trivial manner. The transport of heat by the flow has a stabilizing effect for small-amplitude waves but promotes the instability for large-amplitude waves when a recirculating zone is present. Nevertheless, in this last case, by increasing the shear in the bulk and thus the viscous dissipation, increasing the Prandtl number decreases the amplitude and speed of the waves.

Downslope flows into density-stratified environments have been observed to have the character of detraining gravity currents on small slopes, and of entraining plumes on steep slopes. In this paper, observations of flows on slopes of intermediate (20$^{\circ}$–30$^{\circ}$) steepness are described, and their mixing properties quantified. Both gravity-current-like and plume-like flows are observed, and an observational boundary between these two types is identified. Theoretical models for the bulk properties of these flows are presented, and their predictions are compared with the observations. A theoretical criterion is derived for the limit of applicability of the gravity-current model in terms of the Buoyancy number, the bottom slope and the bottom drag coefficient. This provides a criterion for the boundary between the plume-like and gravity-current-like flows, which is consistent with the observations. These results have implications for the modelling of downslope flows in nature, and indicate where the appropriate dynamical model may change from one type to the other.

This paper focuses on the effects of an opening placed along a tube on the propagation of a pressure wavefront. Such a configuration has been chosen for its relevance to many of the countermeasures envisaged in reducing strong pressure transients in tunnels due to the entry of high-speed trains (this installation includes a perforated entrance hood and ventilation shaft). We will start by establishing that when a compression wavefront passes through an opening, the front is split into an infinite number of smaller pressure steps, with their amplitude expressed as the terms of a mathematical series. The main parameter of the series is a transmission coefficient of the opening. The shape of each of the smaller pressure steps is driven by the transmission–reflection process that takes place at the opening. Both experimental and numerical studies have been carried out to carefully estimate both the transmission coefficient and the shape of the transmitted and reflected pressure waves. Three major parameters are identified: the relative surface area of the opening to the tube cross-section, the ratio of the incident front length to the longitudinal opening length, and the incident front amplitude.

It will be shown that the transmission coefficient decreases exponentially with the relative surface area of the opening and is significantly influenced by the incident front amplitude. Both the length and shape of the transmitted front are similar to those of the incident front. The reflected front length, however, increases linearly with the incident front length as well as with the longitudinal opening length. The shape of the reflected front is greatly influenced by the incident front length. A linear analysis has been conducted and shows that the transmission coefficient can be predicted in a straightforward manner. These results are deemed to be of help not only in the design of countermeasures for the train/tunnel entry problem, but also for technological applications involving transient pressure pulses in branched pipe flows (e.g. pulsed flow in exhaust pipes).

Owing to the inertial effect of the flow, an unsteady hydrodynamic force will act on a particle of arbitrary shape undergoing a steady rigid-body motion with small but finite Reynolds number if the axis of rotation of the particle is not its axis of rotational symmetry. Unsteady flow field is generated owing to such rotation of the body and as a result the particle experiences a time-dependent translational resistance. In this paper, we analyse this time-dependent hydrodynamic force and obtain its higher-order correction by systematically expanding the Navier–Stokes equation in small Reynolds number.

Gravity-driven free-surface flow of a suspension of neutrally buoyant particles down an inclined plane channel of constant width has been studied experimentally and by flow modelling. A uniform suspension of spheres, sieved to radii of $a \,{=}\, 53--125 $\umu$m or 125–150$\,\umu$m, was introduced to create films of initial depth $h_o$. The flow was always at small film-depth-based Reynolds numbers. The film depth and the mixture flow profile were measured at the initial and two locations at least $200 h_o$ and $400 h_o$ downstream. The bulk particle volume fraction, $\phi_{\rm B}$, was varied in the range $0.01\,{\le}\, \phi_{\rm B} \,{\le}\, 0.5$; $h_o\,{\approx}\, 1.8$–3.2 mm and the inclination angle relative to horizontal, $0.1^{\circ} \,{<}\, \alpha\,{\le}\, 90^{\circ}$, were also varied. Analysis of the particle velocity was performed by stereoscopic imaging to determine particle location followed by particle correlation velocimetry. A two-layer Newtonian viscosity model was applied to the velocimetry results in order to infer particle concentration information. Measured velocity profiles and film depth show that film thickness decreases from $h_o$, while the velocity gradient at the wall and the mean velocity increase, as the mixture flows down the plane. The free surface, examined using direct imaging, becomes progressively more deformed as $\alpha$ and $\phi_{\rm B}$ increase, with the onset of observable deformation found at a particle-scale capillary number of $\hbox{\it Ca}_{p} \,{\sim}\, {\rho g_x a^2}/{\sigma}\,{=}\, O(10^{-4})$; $\rho$ is the mixture density, $g_x\,{=}\, g \sin \alpha$ is the axial component of gravitational acceleration and $\sigma$ is the surface tension of the suspending liquid. An existing model for suspension flow which describes phase migration as driven by normal stresses caused by the suspended particles is used to predict the flow, with satisfactory agreement for film depth and development distance for the non-uniform local solid volume fraction, $\phi$. The agreement with the detailed $\phi$ profile is less good, as the model fails to predict the observed $\phi \,{\approx}\, 0$ near the solid boundary while $\phi$ is overpredicted adjacent to the free surface.

Analytical solutions are developed for non-Boussinesq turbulent plumes rising from horizontal area sources in unconfined quiescent environments of uniform density. The approach adopted follows and extends an earlier approach for Boussinesq plumes and replaces the non-Boussinesq area source of interest and located at $z\,{=}\,0$ with an idealized point source located at a virtual origin $z\,{=}\,z_v$ such that the flow above the idealized source approximates that from the actual source. Asymptotic analytical expressions are developed for the location of the virtual source that are valid for large vertical distances above the non-Boussinesq source. The non-Boussinesq source is characterized by a non-dimensional parameter $\Gamma_{\hbox{\scriptsize{\it nb}}}$ which is a measure of the relative strengths of the mass, momentum and density deficit fluxes at, or at a specified height above, the source. The vertical distance between the actual and virtual sources scales on the length scale $\ell$ that characterizes the height over which the flow is non-Boussinesq and expressions for $z_v/\ell$ are developed for lazy ($\Gamma_{\hbox{\scriptsize{\it nb}}}\,{>}\,1$) and forced plume ($\Gamma_{\hbox{\scriptsize{\it nb}}}\,{<}\,1$) sources. For pure-plume source conditions $\Gamma_{\hbox{\scriptsize{\it nb}}}\,{=}\,1$, and the virtual source provides an exact representation of the actual plume above $z\,{=}\,0$. The limiting cases of a nearly pure lazy plume and of a highly lazy plume are also explored analytically. For fire plumes, $\Gamma_{\hbox{\scriptsize{\it nb}}}$ is determined from the balance of fluxes immediately above the combustion region and a procedure for estimating these fluxes is given. Solutions expressing the dependence of the mass flux with height are also developed for the near-field flow regions and thereafter an approximation for the mass and momentum fluxes valid for all heights and for source conditions yielding $0\,{<}\,\Gamma_{\hbox{\scriptsize{\it nb}}}\,{<}\,\infty$ is deduced. Applications of the model may include plumes above fires and forced releases of highly buoyant gas into the atmosphere, for example, following the rupturing of a pressurized container vessel.

The thermo-fluid-dynamic field that arises when an infinite thick plate is impulsively accelerated to a constant speed in a laminar regime is studied, taking into account the coupling of the convection and conduction in the fluid with the conduction in the solid. Two significant cases are discussed depending on the boundary condition imposed on the unwetted side of the plate: constant temperature or adiabatic wall. The work is particularly focused on analysing the singularities arising in the field at the initial time. For this purpose an exact analytical solution of the problem governed by the Navier–Stokes equations with constant properties and by the energy equations in the fluid and in the solid is proposed and discussed. The non-dimensional parameter governing the conjugated effects is shown to be the ratio between the thermal effusivities in the fluid and in the solid. The results have also been extended to the analysis of compressible flows by the Stewartson–Dorodnitsin transformation.

The couple on a ball rotating relative to an otherwise quiescent suspension of comparably-sized, neutrally buoyant spheres is studied both experimentally and numerically. Apparent ‘slip’ relative to the analytical solution for a sphere spinning in a Newtonian fluid (based upon the viscosity of the suspension) is determined in suspensions with volume fractions $c$ ranging from 0.03 to 0.50. This apparent slip results in a decrease of the measured torque on the spinning ball when the radius of the ball becomes comparable with that of the suspended spheres. Over the range of our data, the slip becomes more pronounced as the concentration $c$ increases. At $c\,{=}\,$0.25, three-dimensional boundary-element simulations agree well with the experimental data. Moreover, at $c\,{=}\,$0.03, good agreement exists between such calculations and theoretical predictions of rotary slip in dilute suspensions.

An experimental investigation is conducted into the collapse of granular columns inside rectangular channels. The final shape is documented for slumps inside relatively wide channels, and for collapses inside much narrower slots. In both cases, the collapse is initiated by withdrawing a swinging gate or sliding door, and the flow remains fairly two-dimensional. Four different granular media are used; the properties of the materials vary significantly, notably in their angles of friction for basal sliding and internal deformation. If $H$ is the initial height of the column, $h_{\infty}$ the maximum final height of the column and $a$ the initial aspect ratio, then the data suggest that $H/h_{\infty} \,{\sim} a^{0.6}$ in wide channels and $H/h_{\infty} \,{\sim}\, a^{0.5}$ for narrow slots. For the runout, we find that $(l_{\infty}\,{-}\,L)/L \,{\sim}\, a^{0.9\pm 0.1}$ for wide channels, and $(l_{\infty}\,{-}\,L)/L \,{\sim}\, a^{0.65\pm0.05}$ or $l_\infty/L \,{\sim}\, a^{0.55\pm0.05}$ for narrow slots, where $l_{\infty}$ is the maximum runout of the material and $L$ the initial length of the column along the channel ($a\,{:=}\,H/L$). In all cases, the numerical constant of proportionality in these scaling relations shows clear material dependence. In wide slots, there is no obvious universal scaling behaviour of the final profile, but such a behaviour is evident in narrow slots. The experimental results are compared with theoretical results based on a shallow granular-flow model. The qualitative behaviour of the slump in the wide slot is reproduced by the theoretical model. However, there is qualitative disagreement between theory and the experiments in the narrow slot because of the occurrence of secondary surface avalanching.

The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range $31 \,{\times}\, 10^3 \,{\le}\,\hbox{\it Re}_D \,{\le}\, 35 \,{\times}\, 10^6$. The constants of Prandtl's ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Kármán constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.