A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity.

The work presented here focuses on the analysis of a turbulent boundary layer saturated with saltating particles. Experiments were carried out in a wind tunnel 15m long and 0.6m wide at the University of Aarhus in Denmark with sand grains 242 μm in size for wind speeds ranging from the threshold speed to twice its value. The saltating particles were analysed using particle image velocimetry (PIV) and particle-tracking velocimetry (PTV), and vertical profiles of particle concentration and velocity were extracted. The particle concentration was found to decrease exponentially with the height above the bed, and the characteristic decay height was independent of the wind speed. In contrast with the logarithmic profile of the wind speed, the grain velocity was found to vary linearly with the height. In addition, the measurements indicated that the grain velocity profile depended only slightly on the wind speed. These results are shown to be closely related to the features of the splash function that characterizes the impact of the saltating particles on a sandbed. A numerical simulation is developed that explicitly incorporates low-velocity moments of the splash function in a calculation of the boundary conditions that apply at the bed. The overall features of the experimental measurements are reproduced by simulation.

Thermal convection in a rotating cylinder near onset is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime dominated by the Coriolis force. For thermal driving too small to support convection throughout the entire cell, convection sets in as alternating pairs of hot and cold plumes in the sidewall boundary layer, the so-called wall modes of rotating convection. We subject the wall modes to small amplitude harmonic modulations of the rotation rate over a wide range of frequencies. The modulations produce harmonic Ekman boundary layers at the top and bottom lids as well as a Stokes boundary layer at the sidewall. These boundary layers drive a time-periodic large-scale circulation that interacts with the wall-localized thermal plumes in a non-trivial manner. The resultant phenomena include a substantial shift in the onset of wall-mode convection to higher temperature differences for a broad band of frequencies, as well as a significant alteration of the precession rate of the wall mode at very high modulation frequencies due to the mean azimuthal streaming flow resulting from the modulations.

We have succeeded in experimentally investigating the effects of a moderate Damköhler number, Da (defined as the ratio between a characteristic time of fluid motion and that of a chemical reaction), for various Péclet numbers, Pe, on miscible viscous fingering involving a decrease in the viscosity of the displaced liquid due to a chemical reaction in Hele-Shaw cells. We achieved this by using a chemical reaction between a polymer solution and metal ions. Main analysis has been done for the radial fingering. In the range of Pe employed here, the fingering patterns without the reaction (Da = 0) were independent of Pe. The fingering patterns with the reaction depended on the single parameter, Da, and the area occupied by the fingering pattern near the injection hole increased with Da in the range of Da employed here. The ratio of the area occupied by the fingering pattern within the circle radius of which is the length of longest finger to the area of the circle increased with Da in the range of Da employed here. This result is opposite to that of Nagatsu et al. (J. Fluid Mech., vol. 571, 2007, p. 475), in which the area was decreased by the reaction decreasing the viscosity involving significantly high Da. Experiments in the linear geometry show that the shape of a single finger also depended on the single parameter, Da, and the finger width increased near the base with Da. This result is also opposite to that in the previous case in which the width of a single finger was considered to be decreased by the reaction. These results, interestingly, show that the effects of the decrease in the displaced liquid's viscosity due to chemical reaction on the fingering pattern for moderate Da are opposite to those for significantly high Da. A mechanism for the opposite effects on the fingering pattern depending on Da is discussed.

The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time ± where − is comprised of a union of disjoint space–time ‘anomalies’. If − is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom ± needed to resolve the regions in ± satisfies , where u = uL/ν is a Reynolds number dependent on the local velocity field u(x, t).

The complex hydrodynamics of water entry by a spinning sphere are investigated experimentally for low Froude numbers. Standard billiard balls are shot down at the free surface with controlled spin around one horizontal axis. High-speed digital video sequences reveal unique hydrodynamic phenomena which vary with spin rate and impact velocity. As anticipated, the spinning motion induces a lift force on the sphere and thus causes significant curvature in the trajectory of the object along its descent, similar to a curveball pitch in baseball. However, the splash and cavity dynamics are highly altered for the spinning case compared to impact of a sphere without spin. As spin rate increases, the splash curtain and cavity form and collapse asymmetrically with a persistent wedge of fluid emerging across the centre of the cavity. The wedge is formed as the sphere drags fluid along the surface, due to the no-slip condition; the wedge crosses the cavity in the same time it takes the sphere to rotate one-half a revolution. The spin rate relaxation time plateaus to a constant for tangential velocities above half the translational velocity of the sphere. Non-dimensional time to pinch off scales with Froude number as does the depth of pinch-off; however, a clear mass ratio dependence is noted in the depth to pinch off data. A force model is used to evaluate the lift and drag forces on the sphere after impact; resulting forces follow similar trends to those found for spinning spheres in oncoming flow, but are altered as a result of the subsurface air cavity. Images of the cavity and splash evolution, as well as force data, are presented for a range of spin rates and impact speeds; the influence of sphere density and diameter are also considered.

We present the results of a combined experimental and theoretical investigation of droplets falling onto a horizontal soap film. Both static and vertically vibrated soap films are considered. In the static case, a variety of behaviours were observed, including bouncing, crossing and partial coalescence. A quasi-static description of the soap film shape yields a force–displacement relation that provides excellent agreement with experiment, and allows us to model the film as a nonlinear spring. This approach yields an accurate criterion for the transition between droplet bouncing and crossing. Moreover, it allows us to rationalize the observed constancy of the contact time and scaling for the coefficient of restitution in the bouncing states. On the vibrating film, a variety of bouncing behaviours were observed, including simple and complex periodic states, multi-periodicity and chaos. A simple theoretical model is developed that captures the essential physics of the bouncing process, reproducing all observed bouncing states. The model enables us to rationalize the observed coexistence of multiple periodic bouncing states by considering the dependence of the energy transferred to the droplet on the phase of impact. Quantitative agreement between model and experiment is deduced for simple periodic modes, and qualitative agreement for more complex periodic and chaotic bouncing states. Analytical solutions are deduced in the limit of weak forcing and dissipation, yielding insight into the contact time and periodicity of the bouncing states.

The thermal effect on axisymmetry breaking and transition to turbulence in the wake of a fixed heated sphere is investigated in the mixed convection configurations commonly known as ‘assisting’ and ‘opposing’ flows in which the buoyancy tends, respectively, to accelerate and decelerate the flow. The study is carried out in the Ri–Re parameter plane (Ri being the mixed convection parameter – the Richardson number) for two values of Prandtl number – 0.72 (≈air and many gases) and 7 (≈water). We show that convection affects considerably the transition (as compared to that observed in the wake of an unheated sphere) even at moderate Richardson numbers. The latter are taken to be positive in assisting flow and negative in opposing one. In this notation, it can be said that convection shifts the primary-instability threshold to higher Reynolds numbers with increasing Richardson number. In assisting flow, the primary bifurcation is always regular, but at Ri ≥ 0.6 it appears in azimuthal subspaces associated with higher azimuthal wavenumbers m > 1. The transition scenario is characterized by a large variety of regimes explainable by nonlinear interactions between different azimuthal subspaces. On the side of higher (positive) Richardson numbers the axisymmetric flow is found stable up to Re = 1400 at Pr = 0.72 and Ri = 0.7. In opposing flow, the m = 1 subspace is always the most unstable, but the regular bifurcation gives way to a Hopf one at Ri < −0.1. Close to the junction of both bifurcations a similar variety of regimes precedes the transition to chaos as in assisting flow. On the side of negative Richardson numbers the primary (Hopf) bifurcation threshold is found as low as Re = 100 at Ri = −0.25 and at both investigated Prandtl numbers. After a primary periodic regime characterized by vortex shedding with a symmetry plane, the transition proceeds via a series of increasingly irregular helical regimes.

We study the dynamics of a single two-dimensional elliptical particle sedimenting in a Newtonian fluid using numerical simulations. The main emphasis in this work is to study the effect of boundaries on the flow patterns observed during sedimentation. The simulations were performed using a multi-block lattice Boltzmann method as well as a finite-element technique and the results are shown to be consistent. We have conducted a detailed study on the effects of density ratio, aspect ratio and the channel blockage ratio on the flow patterns during sedimentation. As the channel blockage ratio is varied, our results show that there are five distinct modes of sedimentation: oscillating, tumbling along the wall, vertical sedimentation, horizontal sedimentation and an inclined mode where the particle sediments with a non-trivial orientation to the vertical. The inclined mode is shown to form a smooth bridge between the vertical and horizontal modes of sedimentation. For narrow channels, the mode of sedimentation is found to be sensitively dependent on the initial orientation of the particle. We present a phase diagram showing the transitions between the various modes of sedimentation with changing blockage ratio of the channel.

The flow over a circular cylinder at ReD = 3900 and 10000 and M = 0.4 is considered a platform to study the aero-optical distortions by separated shear layers and turbulent wakes. The flow solution is obtained by large eddy simulation (LES) and validated against previous experimental and numerical results. The fluctuating refractive index obtained from LES is used in a ray-tracing calculation to determine wavefront distortions after the beam passes through the turbulent region. Free-space propagation to the far field is computed using Fourier optics. The optical statistics are analysed for different conditions in terms of optical wavelength, aperture size and the beam position. It is found that there exists an optimal wavelength which maximizes the far-field peak intensity. Optical results at both Reynolds numbers are compared. The optical distortion by the downstream turbulent wake is found to be Reynolds number insensitive. However, due to their different transition mechanisms, distortions by the near wake regions are different in the two flows. The aero-optical effects of different flow scales are examined using filtering and grid refinement. Through a grid convergence study it is confirmed that an adequately resolved LES can capture the aero-optics of highly aberrating flows without requiring additional subgrid scale model for the optics.

We examine the interaction of a spherical foreign particle with a propagating solidification front in a binary alloy. Depending on the material properties and the speed of the front, the particle may be pushed ahead of the front, or engulfed and incorporated into the solid phase. We apply numerical boundary integral and continuation methods to determine the critical speed for particle capture, as a function of the system parameters. We reconcile the differing predictions of previous theoretical works, and show that many typical systems may obey a new scaling of the critical speed, as obtained here. We show that due to constitutional undercooling, the presence of solute decreases particle speeds by an order of magnitude below those for a single-component system. We briefly consider the case of spherical bubbles, where thermocapillary and solutocapillary effects play a large role.

In this study, we consider the effects of linear and nonlinear instability waves on the near-field dynamics and aeroacoustics of two-dimensional laminar compressible mixing layers. Through a combination of direct computations, linear and nonlinear stability calculations, we demonstrate the significant role of nonlinear mechanisms in accurately describing the behaviour of instability waves. In turn, these processes have a major impact on sound generation mechanisms such as Mach wave radiation and vortex pairing sound. Our simulations show that the mean flow correction, which is required in order to accurately describe the dynamics of large-scale vortical structures, is intrinsically tied to the nonlinear modal interactions and accurate prediction of saturation amplitudes of instability waves. In addition, nonlinear interactions are largely responsible for the excitation and development of higher harmonics in the flow which contribute to the acoustic radiation. Two flow regimes are considered: In supersonic shear layers, where the far-field sound is determined by the instability wave solution at sufficiently high Mach numbers, it is shown that these nonlinear effects directly impact the Mach wave radiation. In subsonic shear layers, correctly capturing the near-field vortical structures and the interactions of the subharmonic and fundamental modes become critical due to the vortex pairing sound generation process. In this regime, a method is proposed to combine the instability wave solution with the Lilley–Goldstein acoustic analogy in order to predict far-field sound.

Motivated by the geological sequestration of carbon dioxide (CO2), we study the propagation of gravity currents in a porous medium bounded by a thin layer of much lower permeability. We formulate a model for drainage assuming that the fluid remains simply connected throughout. Using this model we examine the propagation of both two-dimensional and axisymmetric currents numerically. We find that for the fixed-flux situation solutions approach a steady state which is described analytically. The approach to this final solution depends on both the permeability contrast and thickness of the thin layer, and in many cases the current first overshoots before relaxing back to its ultimate steady state. Finally, we examine propagation along multiple thin, lower permeability layers as a reduced-order model of the plume of CO2 currently being injected at Sleipner in the North Sea.

We report experiments wherein groups of particles were allowed to sediment in an otherwise quiescent fluid contained in a large tank. The Reynolds number of the particles, defined as Re = aU/ν, ranged from 93 to 425; here, a is the radius of the spherical particle, U its settling velocity and ν the kinematic viscosity of the fluid. The characteristic size of a cluster, in a plane transverse to gravity, was measured by a ‘cluster variance’(〈r2t〉); the latter is defined as the mean square of the transverse coordinates of all constituent particles, averaged over a series of runs. The cluster variance, when plotted as a function of time, exhibited two regimes. There was a quadratic growth in the variance at short times(〈r2t〉 ∝ t2), while for long times, the cluster variance exhibited a slower sublinear growth with 〈r2t〉 ∝ t0.67. A theory, based on isotropic repulsive hydrodynamic interactions between particles, predicts the cluster variance to grow as t2/3 in the limit of long times. The theoretical framework was originally proposed to describe the long-time self-similar evolution of dilute clusters in the limit Re ≪ 1 Subramanian & Koch (J. Fluid Mech., vol. 603, 2008, p. 63), when the probability of wake-mediated interactions between particles remains asymptotically small; the latter requirement is satisfied for homogeneous spherical clusters larger than a critical radius, and is evidently satisfied for planar clusters oriented transversely to gravity. The isotropy of the interactions therefore stems from the isotropy, at large distances, of the disturbance velocity field produced by a single sedimenting particle outside its wake(which contains the compensating inflow to satisfy mass conservation). Herein, the theory is extended to large Re using an empirical correlation for the drag on a sedimenting particle. This allows one to predict, as a function of Re, the numerical prefactors in the expressions for the cluster variance of both spherical and planar clusters; the predictions for the growth exponent remain unchanged. The agreement between the theoretical and experimental growth exponents supports the hypothesis of a self-similar expansion at long times. The prefactor determined from the experimental observations is found to lie between the theoretical predictions for planar and spherical clusters.

In this work we discuss a non-trivial effect of the interfacial curvature on the stability of uniformly and suddenly accelerated interfaces, such as liquid rims. The new stability analysis is based on operator and boundary perturbation theories and allows us to treat the Rayleigh–Taylor and Richtmyer–Meshkov instabilities as a single phenomenon and thus to understand the interrelation between these two fundamental instabilities. This leads, in particular, to clarification of the validity of the original Richtmyer growth rate equation and its crucial dependence on the frame of reference. The main finding of this study is the revealed and quantified influence of the interfacial curvature on the growth rates and the wavenumber selection of both types of instabilities. Finally, the systematic approach taken here also provides a generalization of the widely accepted ad hoc idea, due to Layzer (Astrophys. J., vol. 122, 1955, pp. 1–12), of approximating the potential velocity field near the interface.

Non-modal amplification of disturbances in streamwise-constant channel flows of Oldroyd-B fluids is studied from an input–output point of view by analysing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing equations are assumed to be harmonic in the spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number (Re) scaling of frequency responses from different forcing to different velocity components is developed, showing the same Re dependence as in Newtonian fluids. It is found that some of the frequency response components peak at non-zero temporal frequencies. This is in contrast to Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional time scales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The temporal frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio (ratio of solvent viscosity to total viscosity) and show maxima for non-zero elasticity number. Our analysis of the Reynolds–Orr equation demonstrates that the energy-exchange term involving the streamwise/wall-normal polymer stress component τxy and the wall-normal gradient of the streamwise velocity ∂yu becomes increasingly important relative to the Reynolds-stress term as the elasticity number increases and is thus the main driving force for amplification in flows with strong viscoelastic effects.

A novel pseudo-three-timescale asymptotic procedure is developed and implemented for obtaining accurate approximations to solutions of an evolution equation arising in thin-film free-surface viscous flow. The new procedure, which employs strained fast and slow timescales, requires considerably fewer calculations than its standard three-timescale counterpart employing fast, slow and slower timescales and may readily be applied to other evolution equations of fluid mechanics possessing wave-like solutions exhibiting exponential decay in amplitude and variations in phase over disparate timescales. The new method is validated on the evolution of free-surface waves on a thin, viscous film coating the exterior of a horizontal rotating cylinder and is shown to yield accurate solutions up to non-dimensional times exceeding by an order of magnitude those of previous related studies. Results of the new method applied to this test problem are demonstrated to be in excellent agreement, over large timescales, with those of corroborative spectrally accurate numerical integrations.

The detailed fluid mechanics of sperm propulsion are fundamental to our understanding of reproduction. In this paper, we aim to model a human sperm swimming in a microscope slide chamber. We model the sperm itself by a distribution of regularized stokeslets over an ellipsoidal sperm head and along an infinitesimally thin flagellum. The slide chamber walls are modelled as parallel plates, also discretized by a distribution of regularized stokeslets. The sperm flagellar motion, used in our model, is obtained by digital microscopy of human sperm swimming in slide chambers. We compare the results of our simulation with previous numerical studies of flagellar propulsion, and compare our computations of sperm kinematics with those of the actual sperm measured by digital microscopy. We find that there is an excellent quantitative match of transverse and angular velocities between our simulations and experimental measurements of sperm. We also find a good qualitative match of longitudinal velocities and computed tracks with those measured in our experiment. Our computations of average sperm power consumption fall within the range obtained by other authors. We use the hydrodynamic model, and a prototype flagellar motion derived from experiment, as a predictive tool, and investigate how sperm kinematics are affected by changes to head morphology, as human sperm have large variability in head size and shape. Results are shown which indicate the increase in predicted straight-line velocity of the sperm as the head width is reduced and the increase in lateral movement as the head length is reduced. Predicted power consumption, however, shows a minimum close to the normal head aspect ratio.