We present the results of an experimental and numerical study of the spin-up from rest to solid-body rotation of a fluid-filled torus. In separate experiments, the rotation rate of the container is suddenly increased to a fixed value and the final rotation rate is used to define a non-dimensionalized control parameter, C. At low values of C, the observed flows during the transient phase are axisymmetric and spin-up is achieved through viscous diffusion. This in turn is followed by significant secondary flow and the appearance of ‘fronts’ as C is increased. During the transient phase the fluid motion near the inner wall of the container is dynamically unstable according to Rayleigh's criterion. Thus at higher values of C wave-like structures break the axisymmetry, non-uniqueness in the details of the process is found and finally, an innear wall instability is observed directly. A plot of the spin-up time versus C shows breaks in the slope at transition points between each of the above dynamical regimes but the overall trend is found to be insensitive to the details of the fluid motion. Further elucidation of the dynamical processes is provided by a novel variant of the now standard phase-space reconstruction techniques. The results show a systematic splitting of the phase paths as C is increased.

Finally, in the complementary numerical study, the time-dependent Navier–Stokes equations are solved for axisymmetric flows. Here, the flow is computed using a velocity–streamfunction–vorticity formulation in a two-dimensional plane with a velocity component normal to this plane. The quantitative and qualitative agreement between the numerical and experimental results is excellent for moderate values of the dynamical control parameter C.

In this paper an attempt is made to explain the period-doubling of wind-generated gravity-capillary waves as observed in the experiment of Choi (1977). It is conjectured that period-doubling is closely related to the phenomenon of second-harmonic resonance. In order to obtain a simple dynamical model, results of McGoldrick (1970) and Simmons (1969) are extended to include the effect of wind input and shear in the current. For pure gravity–capillary waves (no wind, no current) the condition for energy transfer from the second harmonic to the fundamental wave of Chen & Saffman (1979) is recovered. We also discuss the effect of wind and we find that wind input gives rise to a very sudden period-doubling. Qualitative agreement with experiment is obtained.

Longitudinal and transverse components of the velocity were simultaneously measured for various vertical locations at 20 off-centreline positions in turbulent spots artificially generated in a zero-pressure-gradient laminar boundary layer. Global ensemble-averaged velocity diagrams and vertical vorticity contours were computed in similarity coordinates ξ = (x − x0)/(U∞(t−t0)) and ζ = z/(U∞(t−t0)) for different heights η = y/δ* above the plate. This global averaging technique inadequately describes the spot, which is not a single large vortex structure. A discriminative averaging technique was developed to construct a ‘statistically most-probable’ spot with sufficient resolution to include some of the largest substructures detected in visual studies. Four eddies were identified in vertical slices of the central region of the spot, while several rows appeared in the plan view. These features of the velocity and vorticity contours of the statistically most-probable spot exhibit similarities with structures observed in flow visualizations.

Laminar flows in annular ducts of rectangular cross-section subjected to a constant axial magnetic field B0 are considered. The equations of flow are treated by a perturbation method involving infinite series expansions in ascending powers of the ratio Λ = a/R0 (where a and R0 are respectively the height and the mean radius of the annular duct).

The leading terms of the series are calculated in the range of high values of the Hartmann number M by means of a boundary-layer technique. When M is large, the secondary flow pattern exhibits two profoundly distinct features. Firstly, in the Hartmann and interior regions, secondary flows have a one-dimensional structure and involve no inertial effects if the curvature of the duct is small enough; secondly, in thin layers near the walls parallel to the magnetic field, the secondary flow pattern is dramatically influenced by the conductivities of the walls: varying these conductivities gives rise to either one or several counter-rotating eddies. When the Reynolds number of the flow increases, inertial effects emanating from these layers penetrate the core of the duct by convective transport. Order-of-magnitude arguments show that the mean velocity is affected by secondary flow effects when $KM^{-\frac{5}{4}}$ becomes large, where K is the Dean number of the flow.

The theory of rotating open-channel flow between two basins is presented and experimentally verified for three cases. The first case considered is the submerged weir-flow, i.e. the height of the channel floor is below the free-surface height of the fluid in the lower basin. Flow for this case is found to depend on the fluid level in the downstream basin when the fluid level there is high and is independent of the fluid level downstream as soon as the level falls below a critical height; at that point the flow in the channel becomes critical (controlled). The dependence of the flow on upstream potential vorticity and on rotation rate is also examined. It is shown that transport of the flow decreases whenever rotation rate or upstream potential vorticity is increased. The second case studied is controlled flow through a channel of irregular cross section (the truncated channel). Transport of this flow is shown to increase at some moderate rotations if the channel floor has a cross-channel slope similar to that of the surface of the current. Otherwise, the transport always decreases with rotation rate. The final case is concerned with supercritical separation of the current downstream of the control section, i.e. the current separates from the left wall of the channel (looking downstream). Visual evidence and measurements of formation of such a boundary current are obtained. Comparisons between the theory and the experiment for the three cases are generally reasonable except when flow separates in the control section; the theory is found inapplicable to such separation.

The laminar, boundary-layer, natural convection flow adjacent to a vertical, heated or cooled, flat surface submerged in quiescent cold water was studied. The results demonstrate for the first time the existence of multiple steady-state solutions in a natural convection flow. The calculated velocity profiles, over a range of the parameters, are compared with recent corresponding velocity measurements of Wilson & Vyas (1979) and of Carey & Gebhart (1981). The newly found additional steady-state solutions are of considerable practical interest because the heat-transfer rates for a pair of solutions (with determining physical parameters and boundary conditions otherwise identical) are sometimes vastly different. An important consequence of this study is the possible relationship of multiple steady-state solutions to the recently observed unsteadiness in some such flows.

Surface waves in a channel can be produced by the horizontal motion of a plane wavemaker at one end of the channel. The amplitude and the frequency of the waves depend on both surface tension and gravity, as well as on the condition imposed at the contact line between the free surface and the wavemaker. Some of the previous work on the generation of capillary–gravity waves has been based on the unjustified assumption that the slope of the free surface at the contact line can be prescribed. A more acceptable condition is one that relates the slope to the motion of the contact line relative to the wavemaker; in this way the dynamic properties of the contact angle can be incorporated. The waves generated by a plane wavemaker in fluid of infinite depth and in fluid of a depth equal to that of the wavemaker are determined. An important reason for including surface tension is that in its absence the transient motion initiated by an impulsive start is singular; when surface tension is included this singularity is removed.

We have conducted further numerical experiments on two-dimensional fully compressible convection in an imposed vertical magnetic field and interpreted the results by reference to appropriate low-order models. Here we focus on streaming instabilities, involving horizontal shear flows, which form an important mechanism for the breakdown of steady convection in relatively weak fields for boxes of sufficiently small aspect ratio. While these shearing modes can arise even in the absence of a field, they will typically lead only to travelling and modulated waves unless there is a field to provide a restoring force. For magnetoconvection a new and dramatic form of pulsating wave appears after a complex sequence of secondary bifurcations.

Steady, one-dimensional flow of a compressible fluid through a collapsible tube is analysed. A general model is employed, incorporating axial variations in the parameters of the conducting system, such as the tube unstressed cross-section area and wall stiffness, the external pressure and energy exchange with the environment. The flow variables are described in differential form as functions of the conduit system parameters. A coupled set of equations for the dependent flow variables is summarized in a table of influence coefficients, which provides a clear and simple description of the effects produced by the system parameters. Examples of the effects of fluid compressibility in the respiratory system are presented for forced expiration manoeuvres. The effects are found to be generally small, but are most accentuated when breathing heavy gases and when the airways are pathologically stiffened.

When a fluid in a tube is occluded, one finds a static configuration in which the occluding free surface of the fluid is an equilibrium capillary surface spanning the tube. We extend known criteria for existence and non-existence of such a surface, leading to an explicit mathematically rigorous occlusion criterion for cylindrical tubes in a transverse body force field, depending on the force magnitude and contact angle. For any contact angle γ ≠ π/2, we provide further an explicit design of a tube section, which will not occlude in a downward gravity field, regardless of the field strength. In addition, we derive a precise analytic occlusion criterion for liquid partially filling a circular vessel spinning about its axis.

We describe experiments of the entrainment and mixing of water (sub-Kolmogorov scale) droplets across a turbulent–non-turbulent interface (TNI) as well a turbulent–turbulent interface (TTI) in shearless grid turbulence, over a time scale in which evaporation is insignificant. The flow is produced by means of a splitter plate with an active grid and water sprays on one side and screens or an active grid on the other side. The Taylor microscale Reλ on the turbulent side is 275 and the average dissipation scale Stokes number, Stη ≈ 0.2, and based on the integral scale, Stl ≈ 0.003. By changing the orientation of the grid system, gravitational effects may be excluded or included. We show that in the absence of gravity, for the Stokes number range studied (0.06 ≤ Stη ≤ 1.33), the droplet distribution does not change across the interface. With gravity, the larger drops are selectively mixed and this is more pronounced for the TNI than for the TTI. The particle concentration distribution is an error function for the TTI but departs significantly for the TNI due to the intermittency in the flow. In terms of particle concentration, the entrainment is most efficient for the TTI with gravity. The results are related to droplet entrainment in clouds.

The evolution of artificially generated localized disturbances in the shape of hairpin vortices, in laminar axisymmetric rotating shear flows, is investigated experimentally. The results are compared with the predictions of a theoretical model (Levinski & Cohen 1995) with respect to the growth of such disturbances. Hairpin vortices were generated at the surface of the inner cylinder of an axisymmetric Couette apparatus, employing an injection–suction technique. The flow field was analysed from flow visualization using top and side views and by measurements of the mean and instantaneous velocity fields, carried out using laser Doppler anemometry and particle image velocimetry. An instability domain, within the range of base flow parameters where the flow is known to be linearly stable, was found. The marginal ratio between the angular velocities of the inner and outer cylinders beyond which the flow is stable to finite-amplitude localized disturbances agrees with the theoretical prediction based on the measured mean flow in the region of the disturbance. The dependence of the hairpin's inclination angle on the ratio between the two angular velocities is fairly well predicted by the theoretical model.

The three-dimensional particle paths due to a helical beat pattern of the flagellum of a sessile choanoflagellate, Salpingoeca Amphoridium (SA), are modelled and compared to the experimental observations of Pettitt (2001). The organism’s main components are a flagellum and a cell body which are situated above a substrate such that the interaction between these entities is crucial in determining the fluid flow around the choanoflagellate. This flow of fluid can be characterized as Stokes flow and a flow field analogous to one created by the flagellum is generated by a distribution of stokeslets and dipoles along a helical curve.

The model describing the flow considers interactions between a slender flagellum, an infinite flat plane (modelling the substrate) and a sphere (modelling the cell body). The use of image systems appropriate to Green’s functions for a sphere and plane boundary are described following the method of Higdon (1979a). The computations predict particle paths representing passive tracers from experiments and their motion illustrates overall flow patterns. Figures are presented comparing recorded experimental data with numerically generated results for a number of particle paths. The principal results show good qualitative agreement with the main characteristics of flows observed in the experimental study of Pettitt (2001).

We report experimental results on highly supercritical thermal convection in a rapidly rotating hemispherical shell with parabolic gravity. Using silicone oil as the working fluid and an Ekman number $Ek \,{=}\, 4.7 \,{\times}\, 10^{-6}$ we reach Rayleigh numbers up to $1.2 \,{\times}\, 10^{10}$, over 600 times critical. In-situ temperature measurements show that, at these highly supercritical states where convective heat transfer becomes dominant, the time-averaged temperature in the fluid becomes nearly uniform except in a thin thermal boundary layer near the inner spherical boundary. Heat transfer measurements show that Nusselt number $\hbox{\it Nu}$ increases with Rayleigh number $\hbox{\it Ra}$ as $\hbox{\it Nu} \,{\propto}\, Ra^{0.4}$. The measured amplitudes of temperature fluctuations scale well with a model of geostrophic convective turbulence. We also examine convection in a two-layer fluid in the same geometry, using layers of water and silicone oil to produce a stable density stratification. We determine the dependence of heat transfer on the thickness ratio of the layers.

Results from experiments with four different ducts are reported when magnitudes of the field strength and mean velocity are such that the Hartmann number and interaction parameter are large. The first is a straight, circular, highly conducting wall duct situated in a non-uniform transverse magnetic field. Results suggest that as a first approximation the flow may be regarded as being fully developed throughout. In fact there is a slight distortion of the flow in the non-uniform field region revealed by hot-film probe measurements of the streamwise velocity which varies in a novel but readily explicable manner. The second duct is similar except that its wall is weakly conducting. A pressure drop across the non-uniform field region suggests that the behaviour of the flow is weakly reminiscent of that in a non-conducting duct. The two other ducts also have weakly conducting walls but contain either one or two 90° bends and are situated in a uniform field. Symmetry of each duct about its mid-point leads to symmetric potential distributions which indicate the existence of two symmetrically arranged recirculating current flows and these lead to pressure drops across the bends. In the duct with two bends, part of it, the offset, lies parallel to the field lines and a surprising prediction relating the pressure drop across the offset to N finds some support.

The unsteady two-dimensional heat-transfer problem of melting around a horizontal heated cylinder is studied numerically. The cylinder is heated isothermally. A physical model is introduced which accounts for the effects of density change upon melting and subcooling effect, as well as natural convection. Most previous work has assumed that the density number (ratio of solid density to liquid density) is unity. In practice, all solid materials exhibit some density change upon melting. If the density number is greater than one, this induces a blowing effect at the phase-change boundary. If the density number is less than one, a suction effect is produced. This study indicates that the density-change effect on heat transfer during melting is minor. Subcooling results when the solid is at a temperature below the melting temperature. When the melting process begins, some of the available thermal energy must be used as sensible heat, to raise the solid's temperature to the melting point. As a result, less thermal energy is available for melting. Subcooling effects are found to have a substantial effect on the heat-transfer process. The effects of natural convection have been clearly documented, and indicate that natural convection must be included in any realistic model of the melting process. Detailed predictions of the effects of density change and subcooling on the melting process are given. Information on the temperature and flow fields for representative values of Stefan, Rayleigh, Prandtl, subcooling and density number is given. Further results from the numerical solutions include information on local and average heat-transfer rates and sensibleheat gain as well as melt volume as a function of time. Comparisons are made with earlier numerical and analytical results.

This study investigates the propagation of a semi-infinite buoyancy-driven hydraulic fracture in situations when the fluid is able to solidify along the crack walls. Such problems occur when hot magma ascends from a chamber due to buoyancy forces and solidifies by interacting with colder rock. In the model, the solidification rate is calculated assuming a one-dimensional heat transfer problem, in which case it becomes mathematically equivalent to Carter’s leak-off model, which is commonly used to describe the fluid leak-off from a hydraulic fracture into a porous rock formation. In order to construct a mathematical model for a buoyancy-driven hydraulic fracture with solidification, the aforementioned thermal problem is combined with (i) linear plane-strain elasticity to ensure equilibrium of the rock surrounding the fracture, (ii) linear elastic fracture mechanics to determine the fracture propagation, (iii) lubrication theory to capture the viscous fluid flow inside the crack and to account for the effect of buoyancy, and (iv) volume balance of the magma. To address the problem, the governing equations are first rewritten in terms of one integral equation with a non-singular kernel, which significantly simplifies the analysis and the procedure for obtaining a numerical solution. The latter solution is shown to obey a multiscale behaviour near the fracture tip that is fully resolved by the numerical scheme. In order to understand the structure of the solution and to quantify the regimes of propagation (and the associated transitions), a thorough analysis of the problem has been performed. Finally, the developments are applied to investigate the non-steady propagation of a buoyancy-driven fracture that is fed by a constant flux.

When subject to sufficiently strong electric fields, particles and drops suspended in a weakly conducting liquid exhibit spontaneous rotary motion. This so-called Quincke rotation is a fascinating example of nonlinear symmetry-breaking phenomena. To illuminate the rotation of liquid drops we here analyse the asymptotic limit of large electric Reynolds numbers, $\mathit{Re}\gg 1$, within the framework of a two-dimensional Taylor–Melcher electrohydrodynamic model. A non-trivial dominant balance in this singular limit results in both the fluid velocity and surface-charge density scaling as $\mathit{Re}^{-1/2}$. The flow is governed by a self-contained nonlinear boundary-value problem that does not admit a continuous fore–aft symmetric solution, thus necessitating drop rotation. Furthermore, thermodynamic arguments reveal that a fore–aft asymmetric solution exists only when charge relaxation within the suspending liquid is faster than that in the drop. The flow problem possesses both mirror-image (with respect to the direction of the external field) and flow-reversal symmetries; it is transformed into a universal one, independent of the ratios of electric conductivities and dielectric permittivities in the respective drop phase and suspending liquid phase. The rescaled angular velocity is found to depend weakly upon the viscosity ratio. The corresponding numerical solutions of the exact equations indeed collapse at large $\mathit{Re}$ upon the asymptotically calculated universal solution.

We present a new approach – the entropy-viscosity method (EVM) – for numerical modelling of high Reynolds number flows and investigate its potential by simulating fully developed incompressible turbulent flow, first in a stationary pipe and subsequently in a flexible pipe. This method, which was first proposed by Guermond et al. (J. Comput. Phys., vol. 230 (11), 2011, pp. 4248–4267), introduces the concept of entropy viscosity, computed based on the nonlinear localized residual obtained from the energy equation. Specifically, this nonlinear viscosity based on the local size of entropy production is added to the spectral element discretization employed in our work for stabilization at insufficient resolution. Unlike its original formulation, which includes an ad hoc tuneable parameter $\unicode[STIX]{x1D6FC}$, here, we determine the value of $\unicode[STIX]{x1D6FC}$ by assuming that the entropy viscosity is analogous to the eddy viscosity of the Smagorinsky model. However, the overall approach has the flavour of the implicit large eddy simulation (ILES) instead of the standard large eddy simulation (LES). Given the empiricism of our approach, we have performed systematic studies of homogeneous isotropic turbulence for validation (see appendix A). We have also carried out a more complete numerical simulation study to investigate incompressible turbulent flow in a stationary pipe at $Re_{D}=5300$ and $Re_{D}=44\,000$, following the work of Wu & Moin (J. Fluid Mech., vol. 608, 2008, pp. 81–112) who performed very accurate direct numerical simulations (DNS) of these two cases. We found that the mean flow, turbulence fluctuations, and two-point correlations of the EVM-based LES are in good agreement with the DNS of Wu & Moin despite the fact that we employed grids with resolution two orders of magnitude smaller. If we instead use the standard Smagorinsky model in our simulations, the computations become unstable due to insufficient resolution of the smaller scales. Another important difference is that the entropy-viscosity model scales with the cube of the distance from the wall and approaches zero at the wall, which is theoretically correct, as shown by our a posteriori tests. Based on the validated EVM approach, we then simulated fully developed turbulent flow at $Re_{D}=5300$ in a flexible pipe subject to prescribed vibrations in the cross-flow plane corresponding to a standing wave of amplitude $A$ and wavelength $\unicode[STIX]{x1D706}=3D$, where $D=2R$ is the pipe diameter and $R$ is the radius. We have simulated 11 cases corresponding to increasing values of wave steepness $s_{o}=2A/\unicode[STIX]{x1D706}$, with $s_{o}\in [0,0.067]$. We found a quadratic dependence of the friction factor on $s_{o}$ with the minimum at approximately $s_{o}\approx 0.01$, so, surprisingly, we have a slight decrease in drag at first and then a substantial increase compared to the stationary pipe. To obtain the turbulence statistics, we averaged the simulated flow over twenty time periods at the nodes and anti-nodes separately. We found substantial changes in the mean velocity profile at distances $(1-r)^{+}>5$ while the peaks of turbulent intensities were amplified by 50 % in the axial direction and by 200 % in the normal and azimuthal directions at $s_{o}=0.067$. The peak shear stress at the node increased by more than 200 % whereas at the anti-node it attained negative values. Turbulent budgets revealed large changes close to the wall at $(1-r)^{+}<50$ while flow visualizations showed that many more strong worm-like vortices were generated in the near-wall regions compared to the stationary pipe. We have also computed various spatio-temporal correlations that show that the pressure fluctuations are very sensitive to the pipe vibration and scale quadratically with $s_{o}$. Both pressure and velocity correlations exhibit cellular patterns consistent with the standing-wave pipe motion.

The three-dimensional transition of the flow behind a circular cylinder with a near-wake wire disturbance has been investigated experimentally. The asymmetric placement of a wire in the near-wake region of the cylinder causes an unnatural mode of shedding to occur, namely mode C. We performed flow visualization and particle image velocimetry (PIV) experiments to investigate the influence of the wire on various properties of the flow, such as the dynamics of the streamwise secondary vortices. Experiments were performed at the Reynolds number range of Re = 165–300. From these experiments, it can be concluded that mode C structures are formed as secondary streamwise vortices around the primary von Kármán vortices. The spanwise wavelength of those mode C structures is determined to be approximately two cylinder diameters. The presence of the wire also triggered the occurrence of period doubling in the wake. Each new set of mode C structures is out of phase with the previous set, i.e. doubling the shedding period. This period-doubling phenomenon is due to a feedback mechanism between the consecutively shed upper vortices.