We present a mainly theoretical study of high-Reynolds-number planar gravity currents in a uniformly flowing deep ambient. The gravity currents are generated by a constant line source of fluid, and may also be supplied with a source of horizontal momentum and a source of particles. We model the motion using a shallow-water approximation and represent the effects of the ambient flow by imposing a Froude-number condition in a moving frame. We present analytic and numerical expressions for the threshold ambient flow speed above which no upstream propagation can occur at long times. For homogeneous gravity currents in an ambient flow below threshold, we find similarity solutions in which the up- and downstream fronts spread at a constant rate and the current propagates indefinitely in both directions. For gravity currents consisting of both interstitial fluid of a different density to the ambient and a sedimenting particle load, we find long-time asymptotic solutions for ambient flow strengths below threshold. These consist of a steady particle-rich near-source region, in which settling and advection of particles balance, and an effectively particle-free frontal region. The homogeneous behaviour of the fronts ensures that they also spread at a constant rate and therefore can propagate upstream indefinitely. For gravity currents driven solely by a sedimenting particle load, we find numerically that a single regime exists for ambient flow strengths below threshold. In these solutions, settling balances advection near the source leading to a steady region, which joins on to a complex frontal boundary layer. The upstream front progressively decelerates. Our solutions for homogeneous and particle-driven gravity currents compare well with published experimental results.

Experimental results are presented of the mean flow and turbulence characteristics in the near field of a plane wall jet issuing from a nozzle onto flat and concave walls consisting of fixed sand beds. This is a flow configuration of interest for sediment erosion, also referred to as scouring. The measurements were made with an acoustic profiler that gives access to the three components of the instantaneous velocities. For the flat-wall flow, it is shown that the outer-layer spatial growth rate and the maxima of the Reynolds stresses approach the values accepted for the far field of a wall jet at a downstream distance x/b0 ≈ 8. These maxima are only about half the values of a plane free jet. This reduction in Reynolds stresses is also observed in the shear-layer region, x/b0 < 6, where the Reynolds shear stress is about half the value of a free shear layer. At distances x/b0 > 11, the maximum Reynolds shear stress approaches the value of a plane free jet. This change in Reynolds stresses is related to the mean vertical velocity that is negative for x/b0 < 8 and positive further downstream. The evolution of the inner region of the wall jet is found to be in good agreement with a previous model that explicitly includes the roughness length.

On the concave wall, the mean flow and the Reynolds stresses are drastically changed by the adverse pressure gradient and especially by the development of Görtler vortices. On the downslope side of the scour hole, the flow is nearly separating with the wall shear stress tending to zero, whereas on the upslope side, the wall-friction coefficient is increased by a factor of about two by Görtler vortices. These vortices extend well into the outer layer and, just above the wall, cause a substantial increase in Reynolds shear stress.

A series of Southern Hemisphere experiments have been performed to study turbulent convection on a continental shelf–slope placed in a large rotating tank filled with fresh water. Dense salt water was uniformly released at the fluid surface above the shelf. The resulting negatively buoyant bottom flow travelled over the shelf and then downslope. The non-dimensional reduced gravity of the dense downslope flow was found to scale as for a fixed slope angle, where B0 the buoyancy flux at the surface, g′ the reduced gravity of the bottom flow, h the water depth above the shelf and W the width of the dense water source. Under rotation, a bottom Ekman layer, with superimposed roll waves, propagated down the slope and towards the left sidewall when looking down the slope. As the modified natural Rossby number P decreased, where P = (B0/f3h2)1/2W/h and f is the Coriolis parameter, the appearance of the bottom layer had four different forms: laminar flow, the continuous formation of waves, the periodic release of wave groups, and the periodic generation of eddies. Vortices generated on the surface were cyclonic, suggesting, but not proving, that eddies in the dense bottom layer as originally formed were anticyclonic.

With a canyon cut from the middle of the shelf to the bottom of the slope, G′ values measured in dense flows to the left of the canyon, were significantly reduced. The canyon channelled a large amount of dense fluid with a buoyancy considerably larger than that of dense flows on the slope. However, the flow regime criteria remained basically unchanged with eddies and downslope Ekman layer being able to partially cross the canyon.

An analysis is presented of the three-dimensional creeping flow in and around a porous sphere, modelled as a generalized Brinkman medium, near a smooth plane where the sphere (i) translates uniformly without rotating in an otherwise quiescent Newtonian fluid, (ii) rotates uniformly without translating in an otherwise quiescent Newtonian fluid, and (iii) is fixed in a shear field, which is uniform in the far field and has a linearly increasing velocity profile with increasing distance from the plane. The linear superposition of these three flow regimes is also considered for the special case of the free translational and rotational motion of a neutrally buoyant porous sphere in a shear field that is uniform in the far field. Exact series solutions to the momentum equations are derived for the velocity and pressure fields in the Brinkman and Stokes-flow regions. Coefficients in the series solutions for each flow regime are determined using recursion relations derived from the continuity equations in the Brinkman and Stokes-flow regions, from the interfacial boundary conditions on the porous spherical surface, and from the no-slip condition on the plane. Results are presented in terms of the drag force on the porous sphere and torque about the sphere centre as a function of the dimensionless clearance distance between the sphere and the rigid plane for several values of the dimensionless hydraulic permeability of the Brinkman medium. The free motion of the neutrally buoyant sphere is calculated by requiring that the net hydrodynamic drag force and torque acting on the sphere vanish. Results for this case are presented in terms of the dimensionless translational and rotational speeds of the porous sphere relative to the far-field shear rate as a function of the dimensionless clearance distance for several values of the dimensionless hydraulic permeability. The work is motivated by insights it offers into the behaviour of porous agglomerates, and by its potential utility in industrial, biological, biophysical, medicinal and environmental applications wherever gas or liquid suspensions of porous agglomerates might arise.

A particle with a density slightly larger than that of the fluid in which it is immersed will sediment. However, if the particle's temperature is higher than that of the fluid, the terminal velocity of sedimentation will be smaller and can even change sign. When the terminal velocity is zero we say there is thermal levitation. Thermal levitation can also occur when the density and temperature of the particle are smaller than those of the fluid. Using a two-component thermal lattice Boltzmann equation method, we study this phenomenon and show it can be stable or unstable.

The breakdown of the no-slip condition at fluid–solid interfaces generates a host of interesting fluid-dynamical phenomena. In this paper, we consider such a scenario by investigating the low-Reynolds-number hydrodynamics of a novel ‘slip–stick’ spherical particle whose surface is partitioned into slip and no-slip regions. In the limit where the slip length is small compared to the size of the particle, we first compute the translational velocity of such a particle due to the force density on its surface. Subsequently, we compute the rotational velocity and the response to an ambient straining field of a slip–stick particle. These three Faxén-type formulae are rich in detail about the dynamics of the particles: chiefly, we find that the translational velocity of a slip–stick sphere is coupled to all of the moments of the force density on its surface; furthermore, such a particle can migrate parallel to the velocity gradient in a shear flow. Perhaps most important is the coupling we predict between torque and translation (and force and rotation), which is uncharacteristic of spherical particles in unbounded Stokes flow and originates purely from the slip–stick asymmetry.

We present an experimental study of turbulent Rayleigh–Bénard convection (RBC) in which the input energy that drives the turbulent flow is in the form of periodical pulses. A surprising finding of the study is that in this ‘kicked’ thermal turbulence the heat transfer efficiency is enhanced compared to both constant and sinusoidally modulated energy inputs. For the apparatus used in the present study, an enhancement of 7% of the dimensionless Nusselt number Nu has been achieved. The enhancement is found to depend on two factors. One is the synchronization of the kicking period of energy input with the intrinsic time scale of the turbulent flow. When the repetition period of the input energy pulse equals half of the large-scale flow turnover time, a resonance or optimization of the enhancement is achieved. The other factor is the pulse shape (the inverse square of the energy input duty cycle). We find that a spiky pulse is more efficient for heat transfer than a flatter one of the same energy. It is found that in this kicked thermal turbulence there exist appropriate ranges of the kicking strength A and the kicking frequency f in which the Rayleigh number Ra grows to a saturation level and that the saturated Ra fluctuates between a lower saturation level and an upper saturation level . For large enough saturated Ra, power-law dependences on f and A are found: and . The scaling law for is found to agree quantitatively with the prediction of a mean-field theory of kicked turbulence (Lohse, Phys. Rev. E vol. 62, 2000, p. 4946) when the latter is appropriately extended to the case of kicked thermal turbulence. It is further found that a large-scale circulatory flow (LSC) still exists in the kicked RBC, and that its Reynolds number has the same scaling with Ra as in the steadily driven case, i.e. Ref ∝ Ra0.46±0.01. The present study provides an example of achieving enhanced heat transfer in a convective system by first triggering the emission of clustered thermal plumes via an active control and then synchronizing the transport of the plume clusters with an internal time scale.

Phase-averaged and directionally triggered digital particle image velocimetry measurements were taken in longitudinal and transverse planes in the near field of the flow emerging from a fluidic precessing jet nozzle. Measurements were performed at nozzle inlet Reynolds and Strouhal numbers of 59000 and 0.0017, respectively. Results indicate that the jet emerging from the nozzle departs with an azimuthal component in a direction opposite to that of the jet precession. In addition, the structure of the ‘flow convergence’ region, reported in an earlier study, is better resolved here. At least three unique vortex-pair regions containing smaller vortical ‘blobs’ are identified for the first time. These include a vortex-pair region originating from the foci on the downstream face of the nozzle centrebody, a vortex-pair region shed from the edge of the centrebody and a vortex-pair region originating from the downstream surface of the nozzle exit lip.

A freely floating motion-trapping structure can be defined as one or more rigid bodies floating on the surface of a fluid which extends to infinity in at least one direction, whose free motion under its natural hydrostatic restoring force is coupled to that of the surrounding fluid in such a way that no waves are radiated to infinity. The resulting local time-harmonic oscillation of the structure and the surrounding fluid is called a motion-trapped mode. Such a structure would, if displaced slightly from its equilibrium position and released, ultimately oscillate indefinitely at the trapped-mode frequency. Previous examples of motion-trapping structures have been devised using an inverse approach in which the shape of pairs of such structures is determined implicitly by sketching certain streamlines. In this paper an alternative direct approach to the construction of motion-trapping structures in the form of a pair of identical floating cylinders of rectangular cross-section in two dimensions is presented. It is also shown that a thick-walled axisymmetric heaving circular cylinder can act as a motion-trapping structure.

The forces and torques governing the planar zigzag motion of thick, slightly buoyant disks rising freely in a liquid at rest are determined by applying the generalized Kirchhoff equations to experimental measurements of the body motion performed for a single body-to-fluid density ratio ρs/ρf ≈ 1. The evolution of the amplitude and phase of the various contributions is discussed as a function of the two control parameters, i.e. the body aspect ratio (the diameter-to-thickness ratio χ = d/h ranges from 2 to 10) and the Reynolds number (100 < Re < 330), Re being based on the rise velocity and diameter of the body. The body oscillatory behaviour is found to be governed by the force balance along the transverse direction and the torque balance. In the transverse direction, the wake-induced force is mainly balanced by two forces that depend on the body inclination, i.e. the inertia force generated by the body rotation and the transverse component of the buoyancy force. The torque balance is dominated by the wake-induced torque and the restoring added-mass torque generated by the transverse velocity component. The results show a major influence of the aspect ratio on the relative magnitude and phase of the various contributions to the hydrodynamic loads. The vortical transverse force scales as fo = (ρf − ρs)ghπd2 whereas the vortical torque involves two contributions, one scaling as fod and the other as f1d with f1 = χfo. Using this normalization, the amplitudes and phases of the vortical loads are made independent of the aspect ratio, the amplitudes evolving as (Re/Rec1 − 1)1/2, where Rec1 is the threshold of the first instability of the wake behind the corresponding body held fixed in a uniform stream.

The spatial structure of momentum sources and sinks (T > 0 and T < 0; where T is the streamwise component of the Lamb vector) is examined in a turbulent boundary layer by using dual-plane particle image velocimetry data obtained in streamwise–spanwise planes at two wall-normal locations (x2/δ = 0.1 and 0.5, where x2 is the wall-normal location and δ is the boundary layer thickness). Two-point correlations of T indicate that the size of source motions remains relatively constant while the size of sink motions increases with increasing wall-normal distance. The relative strength of sink motions also increases away from the wall. The velocity field in the vicinity of source/sink motions was explored by computing cross-correlations of T with the velocity components. Source-like motions are correlated with elongated low-momentum zones that possess regions of upwash embedded within them and appear to be the strongest in areas where these low-momentum zones meander in the spanwise direction. Momentum sinks appear to be located within low-speed regions that are within larger high-momentum zones. The velocity fluctuations undergo rapid transitions between quadrants in the vicinity of sinks (i.e. both streamwise and wall-normal velocity fluctuations change sign). The length scales, over which the fluctuations change sign, are much larger at x2/δ = 0.5 than at x2/δ = 0.1.

The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, with Frh < 0.2 and 0.1 < Frz < 10. Here Frh ≡ U/(NR) and Frz ≡ Ukz/N are the horizontal and vertical Froude numbers, U and R are the horizontal velocity and length scales of the vortices, N is the Brunt–Väisälä frequency, and 2π/kz is the vertical wavelength of a small initial perturbation. At early times with Frz < 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations with Frz > 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R ≈ 0.2/Frz for the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 < Frz < 0.8. Short perturbations with Frz > 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds number Re ≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.

We study the coalescence of a drop with its bulk phase in fluid–fluid demixing colloid–polymer mixtures. Such mixtures show behaviour analogous to molecular fluid–fluid systems, but the interfacial tension is between 105 to 107 times smaller than in the molecular case. Such an ultralow interfacial tension has several important consequences and offers significant advantages in the study of droplet coalescence. The coalescence process can be divided into three consecutive stages: (i) drainage of the continuous film between droplet and bulk phase, (ii) rupture of the film, and (iii) growth of the connection. These stages can be studied within a single experiment by optical microscopy thanks to the ultralow interfacial tension in colloid–polymer mixtures, which significantly changes the relevant characteristic length and time scales. The first stage is compared with existing theories on drainage, where we show several limiting theoretical cases. The experimental drainage curves of different colloid–polymer mixtures can be scaled and then show very similar behaviour. We observe that drainage becomes very slow and eventually the breakup of the film is induced by thermal capillary waves. The time it takes for a certain height fluctuation of the interface to occur, which turns out to be an important parameter for the kinetics of the process, can be directly obtained from experiment. During the third stage we observe that the radius of the connecting neck grows linearly with time both for gas bubbles and liquid droplets with an order of magnitude that is in good agreement with the capillary velocity. Finally, partially bleaching the fluorescent dye inside the liquid droplet reveals how the surface energy is transformed into kinetic energy upon coalescence. This opens the way for a more complete understanding of the hydrodynamics involved.

An exact similarity solution is obtained for the rise of a buoyant thermal in Stokes flow, in which both the rise height and the diffusive growth scale like t1/2 as time t increases. The dimensionless problem depends on a single parameter Ra = B/(νκ) – a Rayleigh number – based on the (conserved) total buoyancy B of the thermal, and the kinematic viscosity ν and thermal diffusivity κ of the fluid. Numerical solutions are found for a range of Ra. For small Ra there are only slight deformations to a spherically symmetric Gaussian temperature distribution. For large Ra, the temperature distribution becomes elongated vertically, with a long wake containing most of the buoyancy left behind the head. Passive tracers, however, are advected into a toroidal structure in the head. A simple asymptotic model for the large-Ra behaviour is obtained using slender-body theory. The width of the thermal is found to increase like (κt)1/2, while the wake length and rise height both increase like (RalnRa)1/2(κt)1/2, consistent with the numerical results. Previous experiments suggest that there is a significant transient regime.

A small separation between reactants, not exceeding 10−8 − 10−7 cm, is the necessary condition for various chemical reactions. It is shown that random advection and stretching by turbulence leads to the formation of scalar-enriched sheets of strongly fluctuating thickness ηc. The molecular-level mixing is achieved by diffusion across these sheets (interfaces) separating the reactigants. Since the diffusion time scale is , knowledge of the probability density Q(ηc, Re) is crucial for evaluation of mixing times and chemical reaction rates. According to Kolmogorov–Batchelor phenomenology, the stretching time τeddy ≈ L/urms = O(1) is independent of large-scale Reynolds number Re = urmsL/ν and the diffusion time is very small. Therefore, in previous studies, molecular diffusion was frequently neglected as being too fast to contribute substantially to the reaction rates. In this paper, taking into account strong intermittent fluctuations of the scalar dissipation scales, this conclusion is re-examined. We derive the probability density Q(ηc, Re, Sc), calculate the mean scalar dissipation scale and predict transition in the reaction rate behaviour from to the high-Re asymptotics . These conclusions are compared with known experimental and numerical data.

We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.

In this paper, the low-Reynolds number (Re = 80) flow around a row of nine square cylinders placed normal to the oncoming flow is investigated using the lattice-Boltzmann method. The effects of the cylinder spacing on the flow are studied for spacing to diameter ratios of 0.3 to 12. No significant interaction between the wakes is observed with spacings greater than six times the diameter. At smaller spacings, the flow regimes as revealed by vorticity field and drag coefficient signal are: synchronized, quasi-periodic and chaotic. These regimes are shown to result from the interaction between primary (vortex shedding) and secondary (cylinder interaction) frequencies; the strength of the latter frequency in turn depends on the cylinder spacing. The secondary frequency is also related to transition between narrow and wide wakes behind a cylinder.

The mean drag coefficient and Strouhal number are found to increase rapidly with a decrease in spacing; correlations of these parameters with spacing are proposed. The Strouhal number based on gap velocity becomes approximately constant for a large range of spacings, highlighting the significance of gap velocity for this class of flows. It is also possible to analyse the vortex pattern in the synchronized and quasi-periodic regimes with the help of vorticity dynamics. These results, most of which have been obtained for the first time, are of fundamental significance.

Direct numerical simulations of the three-dimensional flow past rough surfaces with elements of different shapes are performed to create a database. Our main interest is in finding a new parameterization for turbulent rough flows, which, so far, has been based on the concept of equivalent sand grain height or on the net separation between k and d type roughnesses. The new parameterization permits us to find a simple expression for the roughness function and the root mean square of the normal velocity fluctuation at the plane of the crests. We also wish to find statistical quantities characterizing the effects of the different rough surfaces: one is the ratio between mean flow and turbulence time scales (Sq/ε), the other is the helicity density. Passive scalar visualizations evince a reduction of the wall streak coherence, and the absence of a signature of the rough surfaces on the passive scalar distribution. The tendency towards a flow isotropy near the roughness has been explained also through Sq/ε.

The local flow patterns and their bifurcations associated with non-simple degenerate critical points appearing away from boundaries are investigated under the symmetric condition about a straight line in two-dimensional incompressible flow. These flow patterns are determined via a bifurcation analysis of polynomial expansions of the streamfunction in the proximity of the degenerate critical points. The normal form transformation is used in order to construct a simple streamfunction family, which classifies all possible local streamline topologies for given order of degeneracy (degeneracies of order three and four are considered). The relation between local and global flow patterns is exemplified by a cavity flow.

Wunsch (1970) and Phillips (1970) (Deep-Sea Res. vol. 17, pp. 293, 435) showed that a temperature flux condition on a sloping non-slip surface in a stratified fluid can generate a slow steady upward flow along a thin ‘buoyancy layer’. Their analysis is extended here to the more-general case of steady flow in a contained fluid where buoyancy layers may expel or entrain fluid from their outer edge. A compatibility condition that relates the mass flux and temperature gradient along that edge is derived, and this allows the fluid recirculation and temperature perturbation to be determined in the broader-scale ‘outer flow’ region. The analysis applies when the Wunsch–Phillips parameter R is small, in the linear case for which the density variations are dominated by a constant vertical gradient.