A flowing granular material can behave like a collection of individual interacting grains or like a continuum fluid, depending in large part on the energy imparted to the grains. As yet, however, we have no general understanding of how or under what conditions the fluid limit is reached. Marston, Li & Thoroddsen (J. Fluid Mech., this issue, vol. 704, 2012, pp. 5–36) use high-speed imaging to investigate the ejection of grains from a granular bed due to the impact of a spherical projectile. Their high temporal resolution allows them to study the very fast processes that take place immediately following the impact. They demonstrate that for very fine grains and high impact energies, the dynamics of the ejecta is both qualitatively and quantitatively similar to what is seen in analogous experiments with fluid targets.

We present results from an experimental study of the speed and shape of the ejecta formed when a solid sphere impacts onto a granular bed. We use high-speed imaging at frame rates up to 100 000 f.p.s. to provide direct measurement of individual grain velocities and ejecta angles as well as the overall evolution of the granular ejecta. For larger grain sizes, the emergence velocities of the grains during the ‘early stage flow’, i.e. before the main ejecta ‘curtain’ forms, increase with the kinetic energy of the impacting sphere but are inversely proportional to the time from impact. We also observe that the fastest grains, which can obtain velocities up to five times that of the impacting sphere (), generally emerge at the earliest times and with the lowest ejection angles. As the grain size is decreased, a more ‘fluid-like’ behaviour is observed whereby the ejected material first emerges as a thin sheet of grains between the sphere and the bed surface, which is also seen when a sphere impacts a liquid pool. In this case, the sheet velocity is approximately double that of the impacting sphere () and independent of the bulk packing fraction. For the finest grains we provide evidence of the existence of a vortex ring inside the ejecta curtain where grains following the air flow are entrained through the curtain. In contrast to predictions from previous studies, we find that the temporal evolution of the ejecta neck radius is not initially quadratic but rather approaches a square-root dependence on time, for the finest grains with the highest impact kinetic energy. The evolution therefore approaches that seen for the crown evolution in liquid drop impacts. By using both spherical glass beads and coarse sands, we show that the size and shape distribution are critical in determining the post-impact dynamics whereby the sands exhibit a qualitatively different response to impact, with grains ejected at lower speeds and at later times than for the glass beads.

In this paper, we present the first laboratory experiments that show the generation of internal solitary waves by the impingement of a quasi-two-dimensional internal wave beam on a pycnocline. These experiments were inspired by observations of internal solitary waves in the deep ocean from synthetic aperture radar (SAR) imagery, where this so-called mechanism of ‘local generation’ was argued to be at work, here in the form of internal tidal beams hitting the thermocline. Nonlinear processes involved here are found to be of two kinds. First, we observe the generation of a mean flow and higher harmonics at the location where the principal beam reflects from the surface and pycnocline; their characteristics are examined using particle image velocimetry (PIV) measurements. Second, we observe internal solitary waves that appear in the pycnocline, detected with ultrasonic probes; they are further characterized by a bulge in the frequency spectrum, distinct from the higher harmonics. Finally, the relevance of our results for understanding ocean observations is discussed.

Propeller crashback is an off-design operating condition where a propeller rotates in the reverse direction. Experiments (Bridges 2004, Tech Rep. MSSU-ASE-04-1, Department of Aerospace Engineering, Mississippi State University) have shown that the presence of an upstream hull significantly increases the side force on a propeller in crashback below an advance ratio of . Large-eddy simulation (LES) is performed for a propeller with and without a hull at two advance ratios, and . LES reproduces the experimentally observed behaviour and shows good quantitative agreement. Time-averaged flow fields are investigated for a qualitative understanding of the complex flow resulting from the interaction of the upstream hull with the propeller blades. At , two noticeable flow features are found for the case with the hull – a recirculation zone upstream in the vicinity of the propeller and a vortex ring much closer to the propeller. In contrast, at , there is a much smaller recirculation zone which is further upstream due to the increased reverse flow. As a result, the hull does not make much difference in the immediate vicinity of the propeller at . For both advance ratios, side force is mainly generated from the leading-edge separation on the suction side. However, high levels of side force are also generated from trailing-edge separation on the suction side at .

We report theoretical and experimental investigations of the flow instability responsible for mushy-layer convection with chimneys, drainage channels devoid of solid, during steady-state solidification of aqueous ammonium chloride. Under certain growth conditions a state of steady mushy-layer growth with no flow is unstable to the onset of convection, resulting in the formation of chimneys. We present regime diagrams to quantify the state of the flow as a function of the initial liquid concentration, the porous-medium Rayleigh number, and the sample width. For a given liquid concentration, increasing both the porous-medium Rayleigh number and the sample width drove a transition from a weakly convecting chimney free state to a state of mushy-layer convection with fully developed chimneys. Increasing the concentration ratio stabilized the system and suppressed the formation of chimneys. As the initial liquid concentration increased, onset of convection and formation of chimneys occurred at larger values of the porous-medium Rayleigh number, but the critical cell widths for chimney formation are far less sensitive to the liquid concentration. At the highest liquid concentration, the mushy-layer mode of convection did not occur in the experiment. The formation of multiple chimneys and the morphological transitions between these states are discussed. The experimental results are interpreted in terms of a previous theoretical analysis of finite amplitude convection with chimneys, with a single value of the mushy-layer permeability consistent with the liquid concentrations considered in this study.

Macroscale description of streaming-potential phenomena in the thin-double-layer limit, and in particular the associated electro-viscous forces, has been a matter of long-standing controversy. In part 1 of this work (Yariv, Schnitzer & Frankel, J. Fluid Mech., vol. 685, 2011, pp. 306–334) we identified that the product of the Hartmann () and Péclet () numbers is , being the dimensionless Debye thickness. This scaling relationship defines a one-family class of limit processes appropriate to the consistent analysis of this singular problem. In that earlier contribution we focused on the generic problems associated with moderate and large , where the streaming-potential magnitude is comparable to the thermal voltage. Here we consider the companion generic limit of moderate Péclet numbers and large Hartmann numbers, deriving the appropriate macroscale model wherein the Debye-layer physics is represented by effective boundary conditions. Since the induced electric field is asymptotically smaller, calculation of these conditions requires higher asymptotic orders in analysing the Debye-scale transport. Nonetheless, the leading-order electro-viscous forces are of the same relative magnitude as those previously obtained in the large- limit. The structure of these forces is different, however, first because the small Maxwell stresses do not contribute at leading order, and second because salt polarization results in a dominant diffuso-osmotic slip. Since the salt distribution is governed by an advection–diffusion equation, this slip gives rise to electro-viscous forces which are nonlinear in the driving flow. The resulting scheme is illustrated by the calculation of the electro-viscous excess drag in the prototype problem of a translating sphere.

Direct numerical simulations of subcritical rotating, stratified and magneto-hydrodynamic wall-bounded flows are performed in large computational domains, focusing on parameters where laminar and turbulent flow can stably coexist. In most cases, a regime of large-scale oblique laminar-turbulent patterns is identified at the onset of transition, as in the case of pure shear flows. The current study indicates that this oblique regime can be shifted up to large values of the Reynolds number by increasing the damping by the Coriolis, buoyancy or Lorentz force. We show evidence for this phenomenon in three distinct flow cases: plane Couette flow with spanwise cyclonic rotation, plane magnetohydrodynamic channel flow with a spanwise or wall-normal magnetic field, and open channel flow under stable stratification. Near-wall turbulence structures inside the turbulent patterns are invariably found to scale in terms of viscous wall units as in the fully turbulent case, while the patterns themselves remain large-scale with a trend towards shorter wavelength for increasing . Two distinct regimes are identified: at low Reynolds numbers the patterns extend from one wall to the other, while at large Reynolds number they are confined to the near-wall regions and the patterns on both channel sides are uncorrelated, the core of the flow being highly turbulent without any dominant large-scale structure.

We present a study of the forces during free-surface water entry of spheres of varying masses, diameters, and surface treatments. Previous studies have shown that the formation of a subsurface air cavity by a falling sphere is conditional upon impact speed and surface treatment. This study focuses on the forces experienced by the sphere in both cavity-forming and non-cavity-forming cases. Unsteady force estimates require accurate determination of the deceleration for both high and low mass ratios, especially as inertial and hydrodynamic effects approach equality. Using high-speed imaging, high-speed particle image velocimetry, and numerical simulation, we examine the nature of the forces in each case. The effect of mass ratio is shown, where a lighter sphere undergoes larger decelerations and more dramatic trajectory changes. In the non-cavity-forming cases, the forces are modulated by the growth and shedding of a strong, ring-like vortex structure. In the cavity-forming cases, little vorticity is shed by the sphere, and the forces are modulated by the unsteady pressure required for the opening and closing of the air cavity. A data-driven boundary-element-type method is developed to accurately describe the unsteady forces using cavity shape data from experiments.

The spatial distribution, the velocity statistics and the dispersion of the gas phase have been investigated experimentally in a homogeneous swarm of bubbles confined within a thin gap. In the considered flow regime, the bubbles rise on oscillatory paths while keeping a constant shape. They are followed by unstable wakes which are strongly attenuated due to wall friction. According to the direction that is considered, the physical mechanisms are totally different. In the vertical direction, the entrainment by the wakes controls the bubble agitation, causing the velocity variance and the dispersion coefficient to increase almost linearly with the gas volume fraction. In the horizontal direction, path oscillations are the major cause of bubble agitation, leading to a constant velocity variance. The horizontal dispersion, which is lower than that in the vertical direction, is again observed to increase almost linearly with the gas volume fraction. It is however not directly due to regular path oscillations, which are unable to generate a net deviation over a whole period, but results from bubble interactions which cause a loss of the bubble velocity time correlation.

We report an experimental study on the dynamics of a thin film of polymer solution coating a vertical fibre. The liquid film has first a constant thickness and then undergoes the Plateau–Rayleigh instability, which leads to the formation of sequences of drops, separated by a thin film, moving down at a constant velocity. Different polymer solutions are used, i.e. xanthan solutions and polyacrylamide (PAAm) solutions. These solutions both exhibit shear-rate dependence of the viscosity, but for PAAm solutions, there are strong normal stresses in addition to the shear thinning effect. We characterize experimentally and separately the effects of these two non-Newtonian properties on the flow on the fibre. Thus, in the flat film observed before the emergence of the drops, only the shear-thinning effect plays a role, and tends to thin the film compared to the Newtonian case. The effect of the non-Newtonian rheology on the Plateau–Rayleigh instability is then investigated through the measurements of the growth rate and the wavelength of the instability. Results are in good agreement with linear stability analysis for a shear-thinning fluid. The effect of normal stress can be taken into account by considering an effective surface tension, which tends to decrease the growth rate of the instability. Finally, the dependence of the morphology of the drops on normal stress is investigated, and a simplified model including the normal stress within the lubrication approximation provides good quantitative results on the shape of the drops.

This paper represents a continuation of the work by Tempelmann et al. (J. Fluid Mech., vol. 646, 2010b, pp. 5–37) on spatial optimal growth in incompressible boundary layers over swept flat plates. We present an extension of the methodology to compressible flow. Also, we account for curvature effects. Spatial optimal growth is studied for boundary layers over both flat and curved swept plates with adiabatic and cooled walls. We find that optimal growth increases for higher Mach numbers. In general, extensive non-modal growth is observed for all boundary layer cases even in subcritical regions, i.e. where the flow is stable with respect to modal crossflow disturbances. Wall cooling, despite stabilizing crossflow modes, destabilizes disturbances of non-modal nature. Curvature acts similarly on modal as well as non-modal disturbances. Convex walls have a stabilizing effect on the boundary layer whereas concave walls have a destabilizing effect. The physical mechanisms of optimal growth in all studied boundary layers are found to be similar to those identified for incompressible flat-plate boundary layers.

The nonlinear perturbation response of two families of vortices, the Norbury family of axisymmetric vortex rings and the Pierrehumbert family of two-dimensional vortex pairs, is considered. Members of both families are subjected to prolate shape perturbations similar to those previously introduced to Hill’s spherical vortex, and their response is computed using contour dynamics algorithms. The response of the entire Norbury family to this class of perturbations is considered, in order to bridge the gap between past observations of the behaviour of thin-cored members of the family and that of Hill’s spherical vortex. The behaviour of the Norbury family is contrasted with the response of the analogous two-dimensional family of Pierrehumbert vortex pairs. It is found that the Norbury family exhibits a change in perturbation response as members of the family with progressively thicker cores are considered. Thin-cored vortices are found to undergo quasi-periodic deformations of the core shape, but detrain no circulation into their wake. In contrast, thicker-cored Norbury vortices are found to detrain excess rotational fluid into a trailing vortex tail. This behaviour is found to be in agreement with previous results for Hill’s spherical vortex, as well as with observations of pinch-off of experimentally generated vortex rings at long formation times. In contrast, the detrainment of circulation that is characteristic of pinch-off is not observed for Pierrehumbert vortex pairs of any core size. These observations are in agreement with recent studies that contrast the formation of vortices in two and three dimensions. We hypothesize that transitions in vortex formation, such as those occurring between wake shedding modes and in vortex pinch-off more generally, might be understood and possibly predicted based on the observed perturbation responses of forming vortex rings or dipoles.

We present the first computational study of the dynamics of inertial particles in a shearless turbulence mixing layer. We parametrize our direct numerical simulations to isolate the effects of turbulence, Reynolds number, particle inertia, and gravity on the entrainment process. By analysing particle concentrations, particle and fluid velocities, particle size distributions, and higher-order velocity moments, we explore the impact of particle inertia and gravity on the mechanism of turbulent mixing. We neglect thermodynamic processes, including phase changes between the drops and surrounding air, which is equivalent to assuming the air is saturated (i.e. 100 % humidity). Entrainment is found to be governed by the large scales of the flow and is relatively insensitive to the Reynolds number over the range considered. Our results show that both fluid and particle velocities exhibit intermittency and that gravity and turbulent diffusion interact in unexpected ways to dictate particle dynamics. An analysis of the temporal evolution of fluid and particle statistics suggests that particle concentration profiles and velocities are self-similar under certain circumstances. We also observe large fluctuations in particle concentrations resulting from entrainment and introduce a model to estimate the impact these fluctuations have on the radial distribution function, a statistic that is often used to quantify inertial particle clustering. Our study is both a computational counterpart to and an extension of the wind tunnel experiments by Gerashchenko, Good & Warhaft (J. Fluid Mech., vol. 668, 2011, pp. 293–303) and Good, Gerashchenko, & Warhaft (J. Fluid Mech., vol. 694, 2012, pp. 371–398). We find good agreement between these experimental studies and our computational results. We anticipate that a better understanding of the role of gravity and turbulence in inertial particle entrainment will lead to improved cloud evolution predictions.

The stability of a core–annular fluid arrangement consisting of two concentric fluid layers surrounding a solid cylindrical rod on the axis of a circular pipe is examined when the interface between the two fluid layers is covered with an insoluble surfactant. The motion is driven either by an imposed axial pressure gradient or by the movement of the rod at a prescribed constant velocity. In the basic state the fluid motion is unidirectional and the interface between the two fluids is cylindrical. A linear stability analysis is performed for arbitrary layer thicknesses and arbitrary Reynolds number. The results show that the flow can be fully stabilized, even at zero Reynolds number, if the base flow shear rate at the interface is set appropriately. This result is confirmed by an asymptotic analysis valid when either of the two fluid layers is thin in comparison to the gap between the pipe wall and the rod. It is found that for a thin inner layer the flow can be stabilized if the inner fluid is more viscous than the outer fluid, and the opposite holds true for a thin outer layer. It is also demonstrated that traditional core–annular flow, for which the rod is absent, may be stabilized at zero Reynolds number if the annular layer is sufficiently thin. Finally, weakly nonlinear simulations of a coupled set of partial differential evolution equations for the interface position and surfactant concentration are conducted with the rod present in the limit of a thin inner layer or a thin outer layer. The ensuing dynamics are found to be sensitive to the size of the curvature of the undisturbed interface.

The objective of this numerical study is to increase the base pressure on a backward-facing step via linear feedback control, to be ultimately translated to a drag reduction on a blunt-based bluff body. Two backward-facing step cases are simulated: a laminar two-dimensional (2D) flow at a Reynolds number of , and a turbulent three-dimensional (3D) flow at using large-eddy simulation. The control is effected by a full-span slot jet with zero-net-mass-flux, and two jet locations are examined. Linear system identification is performed to characterize the flow response to actuation, used to synthesize a control law. The control strategy is based on the premise that an attenuation of the instantaneous pressure fluctuations on the base of the step should lead to an increase in the time-averaged base pressure. Open-loop harmonic forcing is examined within a broad frequency range for both the 2D and 3D flows, which are found to respond differently to actuation. The controllers based on disturbance attenuation lead to sensible increases in base pressure (up to 70 % in 2D and 20 % in 3D) with higher efficiency than the best results achieved in open-loop. The results support the conjecture about the link between the base pressure fluctuations and mean, although it is shown that such a black-box model approach is not suitable for optimization without further physical insight.

We present experimental results for the acoustic field of jets with Mach numbers between 0.35 and 0.6. An azimuthal ring array of six microphones, whose polar angle, , was progressively varied, allows the decomposition of the acoustic pressure into azimuthal Fourier modes. In agreement with past observations, the sound field for low polar angles (measured with respect to the jet axis) is found to be dominated by the axisymmetric mode, particularly at the peak Strouhal number. The axisymmetric mode of the acoustic field can be clearly associated with an axially non-compact source, in the form of a wavepacket: the sound pressure level for peak frequencies is found be superdirective for all Mach numbers considered, with exponential decay as a function of , where is the Mach number based on the phase velocity of the convected wave. While the mode spectrum scales with Strouhal number, suggesting that its energy content is associated with turbulence scales, the axisymmetric mode scales with Helmholtz number – the ratio between source length scale and acoustic wavelength. The axisymmetric radiation has a stronger velocity dependence than the higher-order azimuthal modes, again in agreement with predictions of wavepacket models. We estimate the axial extent of the source of the axisymmetric component of the sound field to be of the order of six to eight jet diameters. This estimate is obtained in two different ways, using, respectively, the directivity shape and the velocity exponent of the sound radiation. The analysis furthermore shows that compressibility plays a significant role in the wavepacket dynamics, even at this low Mach number. Velocity fluctuations on the jet centreline are reduced as the Mach number is increased, an effect that must be accounted for in order to obtain a correct estimation of the velocity dependence of sound radiation. Finally, the higher-order azimuthal modes of the sound field are considered, and a model for the low-angle sound radiation by helical wavepackets is developed. The measured sound for azimuthal modes 1 and 2 at low Strouhal numbers is seen to correspond closely to the predicted directivity shapes.

High-resolution direct numerical simulations are conducted to analyse turbulent states of the flow of an electrically conducting fluid in a duct of square cross-section with electrically insulating walls and imposed transverse magnetic field. The Reynolds number of the flow is and the Hartmann number varies from to . It is found that there is a broad range of Hartmann numbers in which the flow is neither laminar nor fully turbulent, but has laminar core, Hartmann boundary layers and turbulent zones near the walls parallel to the magnetic field. Analysis of turbulent fluctuations shows that each zone consists of two layers: the boundary layer near the wall characterized by small-scale turbulence and the outer layer dominated by large-scale vortical structures strongly elongated in the direction of the magnetic field. We also find a peculiar scaling of the mean velocity, according to which the reciprocal von Kármán coefficient grows nearly linearly with the distance to the wall.