Numerical simulations of the collapse and spreading of granular columns onto a horizontal plane using the Contact Dynamics method are presented. The results are in agreement with previous experimental work. The final shape of the deposit appears to depend only on the initial aspect ratio a of the column. The normalized runout distance has a power-law dependence on the aspect ratio a, a dependence incompatible with a simple friction model. The dynamics of the collapse is shown to be mostly controlled by a free fall of the column. Energy dissipation at the base of the column can be described simply by a coefficient of restitution. Hence the energy available for the sideways flow is proportional to the initial potential energy $E_0$. The dissipation process within the sideways flow is approximated well by basal friction, unlike the behaviour of the runout distance. The proportion of mass ejected sideways is shown to play a determining role in the spreading process: as a increases, the same fraction of initial potential energy $E_0$ drives an increasing proportion of the initial mass against friction. This gives a possible explanation for the power-law dependence of the runout distance on a. We propose a new scaling for the runout distance that matches the data well, is compatible with a friction model, and provides a qualitative explanation of the column collapse.

A derivation of the mean second-order short-crested wave pattern and associated wave kinematics, conditional on a given magnitude of the wave crest, is presented. The analysis is based on the second-order Sharma and Dean finite-water wave theory. A comparison with a measured extreme wave profile, the Draupner New Year Wave, shows a good agreement in the mean, indicating that this second-order wave can be a good identifier of the shape and occurrence of extreme wave events. A discussion on its use as an initial condition for a fully nonlinear three-dimensional surface wave analysis is given.

The displacement of a three-dimensional immiscible droplet subject to gravitational forces in a duct is studied with the lattice Boltzmann method. The effects of the contact angle and capillary number (the ratio of viscous to surface forces) on droplet dynamics are investigated. It is found that there exists a critical capillary number for a droplet with a given contact angle. When the actual capillary number is smaller than the critical value, the droplet moves along the wall and reaches a steady state. When the capillary number is greater than the critical value, one or more small droplets pinch off from the wall or from the rest of the droplet, depending on the contact angle and the specific value of the capillary number. As the downstream part of the droplet is pinching off, a bottleneck forms and its area continues decreasing until reaching zero. The general trend found in a previous two-dimensional study that the critical capillary number decreases as the contact angle increases is confirmed. It is shown that at a fixed capillary number above the critical value, increasing the contact angle results in a larger first-detached portion. At a fixed contact angle, increasing the capillary number results in an increase of the size of the first detached droplet for $\theta\,{=}\,78^\circ$ and $\theta\,{=}\,90^\circ$, but a decrease for $\theta\,{=}\,118^\circ$. It is also found that the droplet is stretched longer as the capillary number becomes larger. For a detaching droplet, the maximal velocity value occurs near the bottleneck between the up-and downstream parts of the droplet and the shear stress there reaches a local maximum. The three-dimensional effects are most clearly seen for $\theta\,{=}\, 90^\circ$, where the wetted length and wetted area vary in the opposite direction and the shape of the interface between the wall and the droplet is distorted severely from the original round shape.

The interaction between a fixed particle and decaying homogeneous isotropic turbulence is studied numerically using an overset grid that provides resolution of all scales of fluid motion. A description of the numerical technique and validation of the solution procedure are presented. An ensemble of 64 simulations with the particle in different regions of the flow is computed. The particle diameter in the simulations is approximately twice the size of the unladen Kolmogorov length scale, and the maximum value of the particle Reynolds numbers due to the turbulent fluctuations is close to 20. Ensemble averages of quantities from the numerical solutions are used to investigate the turbulence modification and the fluid forces on the particle. Volume-averaged profiles of the turbulent kinetic energy and dissipation rate from the overset grid simulations reveal that the displacement of fluid by the particle and the formation of the boundary layer at the particle surface lead to turbulence modification in a local region. Time histories of the force applied to the particle from each overset grid simulation are compared to those predicted by a particle equation of motion. The particle equation of motion is shown to underpredict the root mean square (RMS) force applied to the particle by the turbulence. RMS errors between the forces from the overset grid simulation and those predicted by the particle equation of motion are shown to be between 15% and 30% of the RMS force on the particle. The steady viscous drag force is shown to be the dominant term in the particle equation of motion while the history integral term is negligible.

The motion of a spherical inclusion released in a vertical solid-body rotation flow is investigated theoretically and experimentally. Solid spheres and bubbles are considered. The particle Reynolds number, the Taylor number, the Weber number and the capillary number are smaller than unity. The motion equations of the inclusion are obtained by revisiting the hydrodynamic equations. The axial (vertical) motion and the horizontal motion are uncoupled, even though they are sensitive to the rotation rate of the flow. Analytical solutions of the particle motion equation are compared to experimental results obtained by releasing a particle in a rotating tank filled with silicone oil. For solid spheres and bubbles, both the terminal velocity and the particle ejection rate (or trapping rate) predicted by the theory agree with experiments, without any empirical adjustment. In particular, the experimental device enables us to check the validity of various theories involving solid or gaseous inclusions with or without inertia or history effects. It is observed that the mobility tensor obtained by writing the fluid motion equations in the rotating frame accurately predicts the horizontal particle trajectory, like the Boussinesq-Basset equation obtained by writing the fluid motion equations in the non-rotating frame and neglecting the horizontal contribution of inertia effects.

The influence of large-scale outer-region motions on the properties and processes of the inner layer remains an open issue in wall turbulence research. In the present work, two-dimensional power spectra of the streamwise component are measured in streamwise-spanwise planes throughout the logarithmic region of smooth flat-plate turbulent boundary layers at $Re_\theta\,{=}\,1015$ and 7705. The spectra are based on PIV measurements with a wide spanwise view ($z/\delta\,{>}\, 2.5$), and the spanwise energy distribution is emphasized. The spectra reveal that the mode associated with the spacing of the low-speed streaks near the wall, $\lambda_z^+ \,{\approx}\, 100$, contains surprisingly little energy relative to modes in the range $\lambda_z^+ \,{\approx}\,$200–400 at $y^+ \,{=}\, 21$. This result is consistent with measurements in a channel flow (Liuet al. 1996) at a similar height. Further from the wall, large-scale structures that scale with outer variables organize with spacing $\lambda_z/\delta\,{=}\,$0.75–0.9, and these motions dominate the spanwise distribution of streamwise energy throughout the logarithmic region. The large spanwise modes are associated with the large streamwise modes on average, as the median energetic spanwise mode increases roughly linearly with increasing streamwise mode up to approximately $\lambda_{z,\mathrm{med}}/\delta \,{\approx}\, 0.8$, and then remains roughly constant for larger streamwise modes. The aspect ratio $\lambda_x / \lambda_{z,\mathrm{med}}$ decreases with increasing distance from the wall, suggesting that the most streaky structures remain buried near the wall.

Turbulent shear flow beneath a flat free surface driven by a surface stress is simulated numerically to gain a better understanding of the hydrodynamic processes governing the scalar transfer across the air-water interface. The simulation is posed to mimic the subsequent development of a wind-driven shear layer as in a previous experiment except that the initiation of the surface waves is inhibited, thus focusing on the boundary effect of the stress-imposed surface on the underlying turbulent flow and vice versa. Despite the idealizations inherent in conducting the simulation, the computed flow exhibits the major surface features, qualitatively similar to those that appear in the laboratory and field experiments. Two distinct surface signatures, namely elongated high-speed streaks and localized low-speed spots, are observed in the simulated flow. Including temperature as a passive tracer and describing an upward heat flux at the surface, we obtain high-speed streaks that are colder and low-speed spots that are warmer than the surrounding regions. The high-speed streaks, arranged with somewhat equal cross-spacing of centimetres scale, are formed by an array of streamwise jets within the viscous sublayer immediately next to the surface. Beneath the streaks, counter-rotating streamwise vortex pairs are observed among other prevailing elongated vortices. However, they are significantly shorter in length and more irregular than their corresponding high-speed streaks at the surface. Accompanying the more organized high-speed streaks, localized regions of low streamwise velocity emerge randomly on the surface. These low-speed spots are attributed to strong upwelling flows which disrupt the viscous sublayer and also bring up the submerged fluids of low streamwise velocity. The occasional interruptions of the streamwise high-speed jets by the upwelling flows account for bifurcation or dislocation of the surface streaks. Statistics of the turbulence are presented and their implications for the formation of the flow structures are discussed.

We present the results of an experimental investigation of the effect of a magnetic field on the stability of convection in a liquid metal. A rectangular container of gallium is subjected to a horizontal temperature gradient and a uniform magnetic field is applied separately in three directions. The magnetic field suppresses the oscillation most effectively when it is applied in the vertical direction and is least efficient when applied in the direction of the temperature gradient. The critical temperature difference required for the onset of oscillations is found to scale exponentially with the magnitude of the magnetic field for all three orientations. Comparisons are made with available theory and qualitative differences are discussed.

We present experimental observations of the disk of air caught under a drop impacting onto a solid surface. By imaging the impact through an acrylic plate with an ultra-high-speed video camera, we can follow the evolution of the air disk as it contracts into a bubble under the centre of the drop. The initial size and contraction speed of the disk were measured for a range of impact Weber and Reynolds numbers. The size of the initial disk is related to the bottom curvature of the drop at the initial contact, as measured in free-fall. The initial contact often leaves behind a ring of micro-bubbles, marking its location. The air disk contracts at a speed comparable to the corresponding air disks caught under a drop impacting onto a liquid surface. This speed also seems independent of the wettability of the liquid, which only affects the azimuthal shape of the contact line. For some impact conditions, the dynamics of the contraction leaves a small droplet at the centre of the bubble. This arises from a capillary wave propagating from the edges of the contracting disk towards the centre. As the wave converges its amplitude grows until it touches the solid substrate, thereby pinching off the micro-droplet at the plate, in the centre of the bubble. The effect of increasing liquid viscosity is to slow down the contraction speed and to produce a more irregular contact line leaving more micro-bubbles along the initial ring.

In this paper we consider buoyancy-driven flow and directional solidification of a ternary alloy in two dimensions. A steady flow can be established by forcing liquid downward at an average rate V through a temperature gradient that is fixed in the laboratory frame of reference and spans both the eutectic and liquidus temperature of the material being solidified. Our results include both a linear stability analysis and numerical solution of the governing equations for finite-amplitude steady states. The ternary system is characterized by two distinct mushy zones – a primary layer with dendrites composed of a single species and, beneath the primary layer, a secondary layer with a dendritic region composed of two species. The two layers have independent effective Rayleigh numbers, which allows for a variety of convection scenarios.

The influence of turbulence on the orientation state of a dilute density matched suspension of stiff fibres at high Reynolds number in a planar contraction is investigated. High-speed imaging and laser-Doppler velocimetry techniques are used to quantify fibre orientation distribution and turbulent characteristics. A nearly homogeneous isotropic grid-generated turbulent flow is introduced at the contraction inlet. Flow Reynolds number and inlet turbulent characteristics are varied in order to determine their effects on orientation distribution. The orientation anisotropy is shown to be accurately modelled by a Fokker-Planck type equation. Results show that rotational diffusion is highly influenced by inlet turbulent characteristics and decays exponentially with convergence ratio. Furthermore, the effect of turbulent energy production in the contraction is shown to be negligible. Also, the results show that the flow Reynolds number has negligible effect on the development of orientation anisotropy, and the influence of turbulence on fibre rotation is negligible for rotational Péclet number $\gt$10.

The spreading of suspension drops on a flat surface is studied using mixtures of liquid and density-matched particles. As expected from an energy balance model, the spreading factor is reduced with increasing particle volume fraction. This decrease is quantitatively understood through an effective viscosity coefficient. However, for large drop Reynolds number, the particles are not uniformly distributed into a spread drop but form an annulus. For higher impact velocities, a particle-induced break-up of the drop is observed.

From visualizations and velocity measurements, we study the formation of new spiral patterns at the periphery of a rotor-stator cavity when an axial inward throughflow is superimposed. We determine the transition diagram for a given intermediate aspect ratio in the plane (${\hbox {\it Re}},V_z$), where ${\hbox {\it Re}}$ is the rotational Reynolds number and $V_z$ the axial velocity in the radial gap between the rotating disk and the shroud. Both techniques are used to characterize the frequency, the azimuthal wavenumber, the inclination angle of the spirals and to localize them axially and radially. We also show that this new instability is a crossflow instability due to the strong competition between rotation and throughflow.

The large-scale vortex structures under spilling and plunging breakers are investigated, using a fully three-dimensional large-eddy simulation (LES). When an overturning jet projecting from the crest in a breaking wave rebounds from the water surface ahead, the vorticity becomes unstable in a saddle region of strain between the rebounding jet and a primary spanwise vortex, resulting in spanwise undulations of the vorticity. The undulations are amplified on a braid in this saddle region, leading to a vortex loop with counter-rotating vorticity. This vortex loop consequently envelops adjacent primary vortices, to form a typical rib structure. This rib component (the stretched vortex loop) in the large-scale vortex structure, which intensifies in the strains associated with the multiple primary vortices generated throughout the splash-up cycle, appears to be the previously found obliquely descending eddy.

We study the impact and subsequent retraction of liquid droplets upon high-speed impact on hydrophobic surfaces. Extensive experiments show that the drop retraction rate is a material constant and does not depend on the impact velocity. We show that on increasing the Ohnesorge number, $\Oh\,{=}\,\eta/\sqrt{\rho R_{\rm I} \gamma}$, the retraction, i.e. dewetting, dynamics crosses from a capillary-inertial regime to a capillary-viscous regime. We rationalize the experimental observations by a simple but robust semi-quantitative model for the solid-liquid contact line dynamics inspired by the standard theories for thin-film dewetting.

Results are presented from a high-resolution computational study of particle-driven gravity currents in a plane channel. The investigation was conducted in order to obtain better insight into the energy budget and the mixing behaviour of such flows. Two- and three-dimensional simulations are discussed, and the effects of different factors influencing the flow are examined in detail. Among these are the aspect ratio of the initial suspension reservoir, the settling speed of the particles, and the initial level of turbulence in the suspension. While most of the study is concerned with the lock-exchange configuration, where the initial height of the suspension layer is equal to the height of the channel, part of the analysis is also done for a deeply submerged case. Here, the suspension layer is only one-tenth of the full channel height. Concerning the energy budget, a careful analysis is undertaken of dissipative losses in the flow. Dissipative losses arising from the macroscopic fluid motion are distinguished from those due to the microscopic flow around each sedimenting particle. It is found that over a large range of settling velocities and suspension reservoir aspect ratios, sedimentation accounts for roughly half of all dissipative losses. The analysis of the mixing behaviour of the flow concentrates on the mixing between interstitial and ambient fluid, which are taken to be of identical density. The former is assumed to carry a passive contaminant, whose dispersion with time is analysed qualitatively and quantitatively by means of Lagrangian markers. The simulations show the mixing between interstitial and ambient fluid to be more intense for larger values of the particle settling velocity. Finally, the question is addressed of whether or not initial turbulence in the suspension may exert a significant effect on the flow evolution. To this end, results from three simulations with widely different levels of initial kinetic energy are compared. While the initial turbulence level strongly affects the mixing within the current, it has only a small influence on the front velocity and the overall sedimentation rate.

Two-layer rotating exchange flows through channels of rectangular cross-section are modelled using semi-geostrophic, zero-potential-vorticity theory. For a given channel cross-section the full range of possible flow states is considered. The interface always has a uniform slope across the channel, but may separate from one or both of the sidewalls to attach to the upper or lower boundary. The flow may be subcritical, critical or supercritical. These different states are identified in a pseudo-Froude-number plane analogous to that developed by Armi (1986) for non-rotating flows. If the ratio of the channel width to the Rossby radius is constant along the length of the channel, then the solution may be traced along the entire channel using a single diagram. Several examples of maximal and submaximal exchanges are considered. This graphical method of solution is contrasted with the functional approach of Dalziel (1988, 1990).

The exchange flux is determined as a function of the channel geometry, the strength of rotation and the difference in Bernoulli potential between the two layers.

Motivated by devices such as the atomic force microscope, we compute the drag experienced by a cylindrical body of circular or rectangular cross-section oscillating at small amplitude near a plane wall. The body lies parallel to the wall and oscillates normally to it; the body is assumed to be long enough for the dominant flow to be two-dimensional. The flow is parameterized by a frequency parameter $\gamma^2$ (a Strouhal number) and the wall–body separation $\Delta$ (scaled on body radius). Numerical solutions of the unsteady Stokes equations obtained using finite-difference computations in bipolar coordinates (for circular cross-sections) and boundary-element computations (for rectangular cross-sections) are used to determine the drag on the body. Numerical results are validated and extended using asymptotic predictions (for circular cylinders) obtained at all extremes of $(\gamma,\Delta)$-parameter space. Regions in parameter space for which the wall has a significant effect on drag are identified.

The slug model provides a formula for the circulation of vortex rings formed by starting jets ejected from a nozzle or orifice in terms of the time history of the jet velocity. For rapidly initiated jets, however, the model under-predicts circulation by an amount that remains nearly constant as circulation increases. To remedy this shortcoming, the present study derives an equation for circulation directly from the governing equations for the flow, illustrating that the circulation can be expressed as the sum of a flux term, $\Gamma_U$, determined by the jet centreline velocity and a ‘source’ term, $\Gamma_p$, determined by over-pressure at the jet centreline during vortex ring formation. For impulsively or nearly impulsively initiated jets, $\Gamma_p$ is modelled using potential flow analysis since over-pressure is significant primarily during the rapid jet initiation in these cases. The effects of the jet initiation and boundary-layer growth on the jet centreline velocity are also considered to provide a realistic model for $\Gamma_U$. Models for nozzle and orifice configurations are developed. In the case of a nozzle-type generator, the need to model the jet centreline velocity leads to an approximate solution for starting flow near the exit of an infinitely long pipe. For the nozzle case, the model results agree with available circulation measurements to within ${\pm}$6.2% for a wide range of conditions, except for jet velocity programs that are nearly triangular. In the orifice case, the model agrees with available results to within ${-}$13%. Both cases show substantial improvement over the slug model, which is in error by ${-}$21% to ${-}$67% for the data considered.