The motion of a heavy rigid ellipsoidal particle settling in an infinitely long circular tube filled with an incompressible Newtonian fluid has been studied numerically for three categories of problems, namely, when both fluid and particle inertia are negligible, when fluid inertia is negligible but particle inertia is present, and when both fluid and particle inertia are present. The governing equations for both the fluid and the solid particle have been solved using an arbitrary Lagrangian-Eulerian based finite-element method. Under Stokes flow conditions, an ellipsoid without inertia is observed to follow a perfectly periodic orbit in which the particle rotates and moves from side to side in the tube as it settles. The amplitude and the period of this oscillatory motion depend on the initial orientation and the aspect ratio of the ellipsoid. An ellipsoid with inertia is found to follow initially a similar oscillatory motion with increasing amplitude. Its orientation tends towards a flatter configuration, and the rate of change of its orientation is found to be a function of the particle Stokes number which characterizes the particle inertia. The ellipsoid eventually collides with the tube wall, and settles into a different periodic orbit. For cases with non-zero Reynolds numbers, an ellipsoid is seen to attain a steady-state configuration wherein it falls vertically. The location and configuration of this steady equilibrium varies with the Reynolds number.

A three-dimensional boundary-layer model of shallow-water flows assuming hydrostatic pressure with negligible numerical diffusion and wave damping has been extended to turbulent flow. A standard two-layer mixing-length model determines vertical length scales. The horizontal mixing length is made a multiple $\beta$ of the vertical value and $\beta$ is determined from comparison with experiment. Eddy viscosity is of a general three-dimensional form where, for example, the horizontal mixing length and associated strain rates determine the magnitude of eddy viscosity and hence vertical mixing (and vice versa). Direct comparison is made with previous experiments for subcritical flow around a conical island of small side slope which exhibits the transition from a vigorous vortex-shedding wake to a steady recirculating wake as the stability parameter, $St$, is increased. The value of $\beta$ influences wake structure, particularly for stability parameters close to the critical (the value at which the wake becomes steady or stable). The critical value in the experiments was 0.4 and this was reproduced in the model with $\beta \,{=}\,6$. Vortex shedding patterns with $\hbox{\it St} \,{=}\, 0.26$ and 0.36 were qualitatively reproduced. The flows were subcritical with an onset Froude number of about 0.2, with values approaching 0.6–0.7 in areas where depth-averaged vorticity magnitude was also greatest, at a small distance from the wet/dry intersection. At this intersection, depth-averaged vorticity approached zero while potential vorticity (depth-averaged vorticity/depth) was at a maximum, indicating the importance of the intersection as an origin for vorticity.

In this paper we study the essential role of Darrieus–Landau (DL), hydrodynamic, cellular flame-front instability in the propagation of expanding turbulent flames. First, we analyse and compare the characteristic time scales of flame wrinkling under the simultaneous actions of DL instability and turbulent eddies, based on which three turbulent flame propagation regimes are identified, namely, instability dominated, instability–turbulence interaction and turbulence dominated regimes. We then perform experiments over an extensive range of conditions, including high pressures, to promote and manipulate the DL instability. The results clearly demonstrate the increase in the acceleration exponent of the turbulent flame propagation as these three regimes are traversed from the weakest to the strongest, which are respectively similar to those of the laminar cellularly unstable flame and the turbulent flame without flame-front instability, and thus validating the scaling analysis. Finally, based on the scaling analysis and the experimental results, we propose a modification of the conventional turbulent flame regime diagram to account for the effects of DL instability.

A torsionally driven cavity, consisting of a fully enclosed cylinder with rotating bottom lid, is used to examine the confined swirling flow of low-viscosity Boger fluids for situations where inertia dominates the flow field. Flow visualization and the optical technique of particle image velocimetry (PIV) are used to examine the effect of small amounts of fluid elasticity on the phenomenon of vortex breakdown. Low-viscosity Boger fluids are used which consist of dilute concentrations of high molecular weight polyacrylamide or semi-dilute concentrations of xanthan gum in a Newtonian solvent. The introduction of elasticity results in a 20% and 40% increase in the minimum critical aspect ratio required for vortex breakdown to occur using polyacrylamide and xanthan gum, respectively, at concentrations of 45 p.p.m. When the concentrations of either polyacrylamide or xanthan gum are raised to 75 p.p.m., vortex breakdown is entirely suppressed for the cylinder aspect ratios examined. Radial and axial velocity measurements along the axial centreline show that the alteration in existence domain is linked to a decrease in the magnitude of the peak in axial velocity along the central axis. The minimum peak axial velocities along the central axis for the 75 p.p.m. polyacrylamide and 75 p.p.m. xanthan gum Boger fluids are 67% and 86% lower in magnitude, respectively, than for the Newtonian fluid at Reynolds number of Re ≈ 1500–1600. This decrease in axial velocity is associated with the interaction of elasticity in the governing boundary on the rotating base lid and/or the interaction of extensional viscosity in areas with high velocity gradients. The low-viscosity Boger fluids used in this study are rheologically characterized and the steady complex flow field has well-defined boundary conditions. Therefore, the results will allow validation of non-Newtonian constitutive models in a numerical model of a torsionally driven cavity flow.

Flow around a wall-mounted cube is an example of a turbulent flow around a three-dimensional bluff body attached to a surface. The main experimentally observed feature of this type of flow is the appearance of complex vortical structures, e.g. a horseshoe vortex originating in front of the body and enveloping it. The current paper is a follow-up to Yakhot, Liu & Nikitin (2006) in which we presented results of direct numerical simulation (DNS) of turbulent flow around a cube. Here, it is shown that unsteadiness of the considered flow is caused by inviscid–viscous interaction between the horseshoe vortex and the narrow band of positive vorticity attached to the surface in front of the cube. Details of the spatio-temporal evolution of large-scale vortical structures, including samples of long-term visualization and turbulence statistics, are presented. For the normal-to-the-wall velocity, in the vicinity of the cube's front face, the results reveal an anomalous probability distribution, namely, a bimodal distribution and one with high kurtosis.

In this paper a reservoir connected through a horizontal contraction to a channel is considered. Both the reservoir and the channel are considered to contain a stable multi-layered system of fluids. The conditions under which there is a flow in only one layer, and the depth in this flowing layer decreases continuously from its depth in the reservoir to its depth in the channel, give the maximum discharge that can be obtained with a flow only from this single layer. For this case the volume discharge calculations are carried out at a single section (the section of minimum width). Where there are velocities in only two layers and the depth in each of these layers decreases continuously from their depths in the reservoir to their depths in the channel, the theory involves computations at two sections in the flow. These are the section of minimum width and a section upstream of the position of minimum width (the virtual point of control). For this flow it is shown that the solution is the one in which the velocity and density distributions are self similar and that the depths of the layers at the point of maximum contraction are two-thirds of those far upstream. It is then shown that for any stable continuous or discrete density stratification in the reservoir a self similar solution will satisfy the conditions for the depths of the flowing layers to decrease smoothly from the reservoir to downstream of the contraction. Again the ratio of the depth at the contraction to that far upstream is two-thirds.

When there is a very large density difference between the fluid in the lower dead water and that in the lowest flowing streamline then this streamline becomes horizontal and may be considered as a frictionless bed. The flow when the bed is not horizontal but where there is a small rise in the channel at the position of maximum contraction is considered for the case where two discrete layers flow under a volume of dead water. In this case the velocity and density profiles are not self similar.

Experiments have been carried out with a contraction in a flume for the withdrawal of two discrete layers from a three layer system and the withdrawal from a fluid with a linear density gradient. In both cases the reservoir and channel bed and hence the lowest streamline was effectively horizontal. These experiments confirmed the theoretical predictions.

We study direct numerical simulations (DNS) of a jet in cross-flow at low values of the jet-to-cross-flow velocity ratio . We observe that, as the ratio increases, the flow evolves from simple periodic vortex shedding (a limit cycle) to more complicated quasi-periodic behaviour, before finally becoming turbulent, as seen in the simulation of Bagheri et al. (J. Fluid. Mech., vol. 624, 2009b, pp. 33–44). The value of at which the first bifurcation occurs for our numerical set-up is found, and shedding of hairpin vortices characteristic of a shear layer instability is observed. We focus on this first bifurcation, and find that a global linear stability analysis predicts well the frequency and initial growth rate of the nonlinear DNS at the critical value of and that good qualitative predictions about the dynamics can still be made at slightly higher values of where multiple unstable eigenmodes are present. In addition, we compute the adjoint global eigenmodes, and find that the overlap of the direct and the adjoint eigenmode, also known as a ‘wavemaker’, provides evidence that the source of the first instability lies in the shear layer just downstream of the jet.

Internal gravity waves excited by the turbulent motions in a bottom Ekman layer are examined using large-eddy simulation. The outer flow is steady and uniformly stratified while the density gradient is set to zero at the flat lower wall. After initializing with a linear density profile, a mixed layer forms near the wall separated from the ambient stratification by a pycnocline. Two types of internal wave are observed. Waves with frequencies larger than the free-stream buoyancy frequency are seen in the pycnocline, and vertically propagating internal waves are observed in the outer layer with characteristic frequency and wavenumber spectra. Since a signature of the pycnocline waves is observed in the frequency spectrum of the mixed layer, these waves may affect the boundary-layer turbulence. The dominant outer-layer waves have a group velocity directed 35-60° from the vertical axis, which is consistent with previous laboratory studies. The energy flux associated with the radiated waves is small compared to the integrated dissipation in the boundary layer, but is of the same order as the integrated buoyancy flux. A linear model is proposed to estimate the decay in wave amplitude owing to viscous effects. Starting from the observed wave amplitudes at the bottom of the pycnocline, the model prediction for the spectral distribution of the outer layer wave amplitude compares favourably with the simulation results.

Experimental studies of a forced transitional jet at moderate Reynolds numbers reveal interesting transitions between low-dimensional states, namely tangent bifurcations, intermittency and hysteresis. The experiments were carried out in an axisymmetric air jet with a laminar top-hat exit profile in the low-noise ambient of a large anechoic chamber, using forcing amplitude and frequency as control parameters. Tangent bifurcations are seen to occur in two different transitions from periodicity to chaos: (i) from stable pairing to nearly periodic modulations of pairing and (ii) from stable double pairing to a quarter-harmonic chaotic attractor. In case (i), an empirically derived mapping closely reproduces the temporal dynamics of one chaotic attractor at a point just after the tangent bifurcation. In case (ii), the intermittency was characterized by estimating the scaling exponent of the p.d.f. of the periodic durations, which was found to be close to the characteristic value for type-II intermittency. Hysteresis is seen at higher Strouhal numbers in the transitions between aperiodic modulations and the periodic/chaotic double pairing states. Based on simultaneous flow visualization and velocimetry, the hysteresis appears to be associated with intermittent tilting of (otherwise) axisymmetric vortices. These transitions are explained in terms of feedback-driven dynamics from vortex roll-ups and pairings, which can be phase-locked (periodic) or unlocked (nearly quasi-periodic and chaotic). The observed transitions connect large regions of deterministic behaviour in the phase diagram, confirming the existence of a low-dimensional dynamical system in transitional jets-an open flow of technological relevance.

Two-point coherence measurements indicate that spatial coupling (and, hence, coherent motion from pairing dynamics) extends for five to eight diameters from the exit, well beyond the locations of pairing and double pairing. This justifies the use of single-point measurements and confirms our hypothesis that the dynamics in this convectively unstable flow are primarily temporal rather than spatio-temporal.

A new theoretical formulation is developed for the effects of surfactants on mass transport across the dynamic interface of a bubble which undergoes spherically symmetric volume oscillations. Owing to the presence of surfactants, the Henry's law boundary condition is no longer applicable; it is replaced by a flux boundary condition that features an interfacial resistance that depends on the concentration of surfactant molecules on the interface. The driving force is the disequilibrium partitioning of the gas between free and dissolved states across the interface. As in the clean surface problem analysed recently (Fyrillas & Szeri 1994), the transport problem is split into two parts: the smooth problem and the oscillatory problem. The smooth problem is treated using the method of multiple scales. An asymptotic solution to the oscillatory problem, valid in the limit of large Péclet number, is developed using the method of matched asymptotic expansions. By requiring that the outer limit of the inner approximation match zero, the splitting into smooth and oscillatory problems is determined unambiguously in successive powers of [weierp ]−1/2, where [weierp ] is the Péclet number. To leading order, the clean surface solution is recovered. Continuing to higher order it is shown that the concentration field depends on RI[weierp ]−1/2, where RI is the (dimensionless) interfacial resistance associated with the presence of surfactants. Although the influence of surfactants appears at higher order in the small parameter, surfactants are shown to have a very significant effect on bubble growth rates owing to the fact that the magnitude of RI is approximately the same as the magnitude of [weierp ]1/2 at conditions of practical interest. Hence the higher-order ‘corrections’ happen numerically to be of the same magnitude as the leading-order, clean surface problem. This is the fundamental reason for major increases in the bubble growth rates associated with the addition of surfactants. This is in contrast to the case of a still, surfactant-covered bubble, in which the first-order correction to the growth rate is of order RI[weierp ]−1 and presents a [weierp ]−1/2 correction. Finally, although existing experimental results have shown only enhancement of bubble growth in the presence of a surfactant the present theory suggests that it is possible for a surfactant, characterized by weak dependence of interfacial resistance on surface concentration, to inhibit rather than enhance the growth of bubbles by rectified diffusion.

The two-dimensional diffraction of a long surface wave by a deformation of the bottom is calculated through a conformal-mapping algorithm (Kreisel 1949). The result is applied to obtain the complex reflection coefficient for a rectangular trench. The corresponding reflection coefficient for oblique incidence is obtained through a variational formulation.

This work involves the study of the development of Type 1 stationary and travelling cross-flow modes in the three-dimensional boundary layer over a rotating disk. In order to control the characteristics of the stationary modes, we utilized organized patterns of roughness which were applied to the disk surface. These consisted of ink dots which were equally spaced in the azimuthal direction at a fixed radius in order to enhance particular azimuthal wavenumbers. Logarithmic spiral patterns of dots were also used to enhance azimuthal wave angles. Velocity fluctuation time series were decomposed into the components corresponding to the stationary and travelling modes using the instantaneous disk position as a reference. Their development was documented through the linear and nonlinear stages leading to turbulence. The linear stage agreed well with linear stability predictions for both modes. In the nonlinear stage we documented a triad coupling between pairs of travelling modes and a stationary mode. The strongest of these was a difference interaction which led to the growth of a low-azimuthal-number, stationary mode. This mode had the largest amplitude and appeared to dominate transition. In retrospect, we can observe the signs of this mechanism in past flow visualization (Kobayashi, Kohama & Takamadate 1980), and it can account for the ‘jagged’ front normally associated with cross-flow-dominated transition on swept wings.

The stability of a viscous fluid between concentric cylinders is analysed, for the case in which the basic velocity distribution is the sum of a velocity distribution due to the rotation of the cylinders (Taylor 1923), and a ‘pumping’ velocity distribution due to a pressure gradient acting round the cylinders (Dean 1928). The critical Taylor number is computed for a wide range of values of the ratio of average velocity of pumping to average velocity of rotation for the case in which the outer cylinder is stationary. It is assumed that the spacing between the cylinders is small.

Thermocapillary convection in three-dimensional rectangular finite containers with rigid lateral walls is studied. The upper surface of the fluid layer is assumed to be flat and non-deformable but is submitted to a temperature-dependent surface tension. The realistic ‘no-slip’ condition at the sidewalls makes the method of separation of variables inapplicable for the linear problem. A spectral Tau method is used to determine the critical Marangoni number and the convective pattern at the threshold as functions of the aspect ratios of the container. The influence on the critical parameters of a non-vanishing gravity and a non-zero Biot number at the upper surface is also examined. The nonlinear regime for pure Marangoni convection (Ra = 0) and for Pr = 104, Bi = 0 is studied by reducing the dynamics of the system to the dynamics of the most unstable modes of convection. Owing to the presence of rigid walls, it is shown that the convective pattern above the threshold may be quite different from that predicted by the linear approach. The theoretical predictions of the present study are in very good agreement with the experiments of Koschmieder & Prahl (1990) and agree also with most of Dijkstra's (1995a, b) numerical results. Important differences with the analysis of Rosenblat, Homsy & Davis (1982b) on slippery walls containers are emphasized.

We consider the flow of a viscous liquid film on the surface of a cylinder that is heated or cooled. Lubrication theory is used to study a thin film under the influence of gravity, capillary, thermocapillary, and intermolecular forces. We derive evolution equations for the interface shapes as a function of the azimuthal angle about the cylinder that govern the behaviour of the film subject to the above coupled forces. We use both analytical and numerical techniques to elucidate the dynamics and steady states of the thin layer over a wide range of thermal conditions and material properties. Finally, we extend our derivation to the case of three-dimensional dynamics and explore the stability of the film to small axial disturbances.

We study numerically the behaviour of and interaction between Burgers vortices – a known equilibrium solution to the Navier–Stokes equations which incorporates a balance between viscous diffusion and strain intensification of vorticity. A hybrid spectral/finite-difference method is employed to solve the Navier–Stokes equations in vorticity–stream function form for a unidirectional vorticity field on an infinite domain in the presence of a uniform three-dimensional strain field, one principal axis of which is parallel to the vorticity. Merging of two strained vortices is studied over a range of Reynolds numbers Re = Γ/2πν = 10–1280, and the results are used to calculate an energy spectrum for three-dimensional, homogeneous turbulence. The cancellation of two strained vortices with opposing circulation is investigated for Reynolds numbers Re = Γ0/2πν = 0.1–160 (τ0 is the circulation about one vortex), over a range of strain rates in the direction parallel to the line joining the vortex centres. A solution of the Navier–Stokes equations describing vorticity cancellation in the strain-induced collision of vortex layers (Kambe 1984) is used to estimate the asymptotic, timewise decay of circulation for each vortex. Good agreement with the present numerical results is obtained. Vortex core pressures calculated during the cancellation event are compared to a simple analytical model based on Moore & Saffman (1971).

The previously established criterion for the stability of a vortex sheet with respect to two-dimensional disturbances in a compressible fluid is extended to three-dimensional disturbances, and it is shown that disturbances travelling at sufficiently oblique angles with respect to the undisturbed flow are unstable. Slip-line instability downstream of a Mach reflexion is discussed briefly.

An experimental study of a turbulent boundary layer at Rθ≈1070 and Rτ≈543 was conducted. Detailed measurements of the velocity vector and the velocity gradient tensor within the near-wall region were performed at various distances from the wall, ranging from approximately y+=14 to y+=89. The measured mean statistical properties of the fluctuating velocity and vorticity components agree well with previous experimental and numerically simulated data. These boundary layer measurements were used in a joint probability density analysis of the various component vorticity and vorticity–velocity gradient products that appear in the instantaneous vorticity and enstrophy transport equations. The vorticity filaments that contribute most to the vorticity covariance Ω[bar]xΩ [bar]y in this region were found to be oriented downstream with angles of inclination to the wall, when projected on the streamwise (x, y)-plane, that decrease with distance moving from the buffer to the logarithmic layer. When projected on the planview (x, z)- and cross-stream (y, z)-planes, the vorticity filaments that most contribute to the vorticity covariances Ω [bar]xΩ [bar]z and Ω [bar]yΩ [bar]z have angles of inclination to the z-ordinate axis that increase with distance from it. All the elements of the ΩiΩj ∂Ui/∂xj term in the enstrophy transport equation, i.e. the term that describes the rate of increase or decrease of the enstrophy by vorticity filament stretching or compression by the strain-rate field, have been examined. On balance, the average stretching of the vorticity filaments is greater than compression at all y+ locations examined here. However, some individual velocity gradient components compress the vorticity filaments, on average, more than they stretch them.

In this paper, an experimental study of laminar magnetohydrodynamic (MHD) buoyancy-driven flow in a cylindrical cell with axis horizontal is described. A steady uniform magnetic field is applied vertically to the mercury-filled cell, which is also subjected to a horizontal temperature gradient. The main features of this internal MHD thermogravitational flow are made experimentally evident from temperature and electric potential measurements. Whatever the level of convection, raising the Hartmann number Ha to a value of the order of 10 is sufficient to stabilize an initially turbulent flow. At much higher values of the Hartmann number (Ha∼100) the MHD effects cause a change of regime from boundary-layer driven to core driven. In this latter regime an inviscid inertialess MHD core flow is bounded by a Hartmann layer on the horizontal cylindrical wall and viscous layers on the endwalls. Since the Hartmann layer is found to stay electrically inactive along the cell, the relevant asymptotic (Ha[Gt ]1) laws for velocity and heat transfer are found from the balance between the curl of buoyancy and Lorentz forces in the core, together with the condition that the flow of electric current between core and Hartmann layer is negligible. A modified Rayleigh number RaG/Ha2, which is a measure of the ratio of thermal convection to diffusion when there is a balance between buoyancy and Lorentz forces, is the determining parameter for the flow.

An asymptotic expansion is found for the low Reynolds number flow induced in an axisymmetric tube by long peristaltic waves of arbitrary shape. Expressions are determined for the relationship between the mean pressure gradient and the volume flux, for the mean rate of working by the wall of the tube and for the shear stress a t the wall. A necessary and sufficient condition for the occurrence of trapping (that is, regions of separated flow near the axis of the tube in a reference frame moving at the wave speed) is obtained. It is shown that reflux (that is, a mean flux in the negative axial direction in a layer of fluid adjacent to the wall when the net mean flux is positive) occurs whenever there is an adverse mean pressure gradient, independently of the shape of the wave. A n estimate of the amount of reflux is derived.