Many insects and spiders can travel on the water surface by rapid synchronized movements of the legs. It has been found that frictional forces, capillary waves and the creation of water jets and eddies beneath the fluid surface are all involved in the mechanisms of propulsion used by these creatures. Elaborate adaptations of the body structure mediate the wetting and de-wetting of the body surface to provide support and differential adhesion. Flow visualization as well as theoretical modelling have led to an understanding of the mechanisms invoked by water striders and spiders for water walking with a variety of gaits and speeds.

We present the results of a combined experimental and theoretical investigation of the dynamics of water-walking insects and spiders. Using high-speed videography, we describe their numerous gaits, some analogous to those of their terrestrial counterparts, others specialized for life at the interface. The critical role of the rough surface of these water walkers in both floatation and propulsion is demonstrated. Their waxy, hairy surface ensures that their legs remain in a water-repellent state, that the bulk of their leg is not wetted, but rather contact with the water arises exclusively through individual hairs. Maintaining this water-repellent state requires that the speed of their driving legs does not exceed a critical wetting speed. Flow visualization reveals that the wakes of most water walkers are characterized by a series of coherent subsurface vortices shed by the driving stroke. A theoretical framework is developed in order to describe the propulsion in terms of the transfer of forces and momentum between the creature and its environment. The application of the conservation of momentum to biolocomotion at the interface confirms that the propulsion of water walkers may be rationalized in terms of the subsurface flows generated by their driving stroke. The two principal modes of propulsion available to small water walkers are elucidated. At driving leg speeds in excess of the capillary wave speed, macroscopic curvature forces are generated by deforming the meniscus, and the surface behaves effectively as a trampoline. For slower speeds, the driving legs need not substantially deform the surface but may instead simply brush it: the resulting contact or viscous forces acting on the leg hairs crossing the interface serve to propel the creature forward.

Tomographic particle image velocimetry was used to quantitatively visualize the three-dimensional coherent structures in a supersonic (Mach 2) turbulent boundary layer in the region between y/δ = 0.15 and 0.89. The Reynolds number based on momentum thickness Reθ = 34000. The instantaneous velocity fields give evidence of hairpin vortices aligned in the streamwise direction forming very long zones of low-speed fluid, consistent with Tomkins & Adrian (J. Fluid Mech., vol. 490, 2003, p. 37). The observed hairpin structure is also a statistically relevant structure as is shown by the conditional average flow field associated to spanwise swirling motion. Spatial low-pass filtering of the velocity field reveals streamwise vortices and signatures of large-scale hairpins (height > 0.5δ), which are weaker than the smaller scale hairpins in the unfiltered velocity field. The large-scale hairpin structures in the instantaneous velocity fields are observed to be aligned in the streamwise direction and spanwise organized along diagonal lines. Additionally the autocorrelation function of the wall-normal swirling motion representing the large-scale hairpin structure returns positive correlation peaks in the streamwise direction (at 1.5δ distance from the DC peak) and along the 45° diagonals, which also suggest a periodic arrangement in those directions. This is evidence for the existence of a spanwise–streamwise organization of the coherent structures in a fully turbulent boundary layer.

An adaptive numerical method is employed to simulate shear instabilities in open-ocean internal solitary-like gravity waves. The method is second-order accurate, employs block-structured adaptive mesh refinement, solves the incompressible Navier–Stokes equations and allows for the simulation of all of the length scales of interest by dynamically tracking important regions with recursively-nested finer grids. Two-dimensional simulations are performed over a range of parameters, which allows us to assess the conditions under which the shear instabilities in the waves occur, including a method to evaluate the critical Richardson number for instability based on the bulk wave parameters. The results show that although the minimum Richardson number is well below the canonical value of 1/4 in all simulations, this value is not a sufficient condition for instability, but instead a much lower Richardson number of 0.1 is required. When the Richardson number falls below 0.1, shear instabilities develop and grow into two-dimensional billows of the Kelvin–Helmholtz type. A linear stability analysis with the Taylor–Goldstein equation indicates that an alternate criterion for instability is given by σiTw > 5, where σi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to a Richardson number of less than 1/4. A third criterion for instability requires that Lw/L > 0.86, where Lw is half the length of the region in which the Richardson number falls below 1/4 and L is the solitary wave half-width. An eigendecomposition of the rate-of-strain tensor is performed to show that the pycnocline thickness increases within the wave because of pycnocline-normal strain and not because of diffusion, which has important ramifications for stability. A three-dimensional simulation indicates that the primary instability is two-dimensional and that secondary, three-dimensional instabilities occur thereafter and lead to strong dissipation and mixing.

The flow of a viscous compressible fluid in a circular tube generated by a sudden impulse at a point on the axis is studied on the basis of the linearized Navier–Stokes equations. A no-slip boundary condition is assumed to hold on the wall of the tube. Owing to the finite velocity of sound the flow behaviour differs qualitatively from that of an incompressible fluid. The flow velocity and the pressure disturbance at any fixed point different from the source point vanish at short time and decay at long times with a t−3/2 power law.

We have performed direct numerical simulations of turbulent flows in a square duct considering a range of Reynolds numbers spanning from a marginal state up to fully developed turbulent states at low Reynolds numbers. The main motivation stems from the relatively poor knowledge about the basic physical mechanisms that are responsible for one of the most outstanding features of this class of turbulent flows: Prandtl's secondary motion of the second kind. In particular, the focus is upon the role of flow structures in its generation and characterization when increasing the Reynolds number. We present a two-fold scenario. On the one hand, buffer layer structures determine the distribution of mean streamwise vorticity. On the other hand, the shape and the quantitative character of the mean secondary flow, defined through the mean cross-stream function, are influenced by motions taking place at larger scales. It is shown that high velocity streaks are preferentially located in the corner region (e.g. less than 50 wall units apart from a sidewall), flanked by low velocity ones. These locations are determined by the positioning of quasi-streamwise vortices with a preferential sign of rotation in agreement with the above described velocity streaks' positions. This preferential arrangement of the classical buffer layer structures determines the pattern of the mean streamwise vorticity that approaches the corners with increasing Reynolds number. On the other hand, the centre of the mean secondary flow, defined as the position of the extrema of the mean cross-stream function (computed using the mean streamwise vorticity), remains at a constant location departing from the mean streamwise vorticity field for larger Reynolds numbers, i.e. it scales in outer units. This paper also presents a detailed validation of the numerical technique including a comparison of the numerical results with data obtained from a companion experiment.

We consider a periodic array of relatively small roughness elements whose spanwise separation is of the order of the local boundary-layer thickness and construct a local asymptotic high-Reynolds-number solution that is valid in the vicinity of the roughness. The resulting flow decays on the very short streamwise length scale of the roughness, but the solution eventually becomes invalid at large downstream distances and a new solution has to be constructed in the downstream region. This latter result shows that the roughness-generated wakes can persist over very long streamwise distances, which are much longer than the distance between the roughness elements and the leading edge. Detailed numerical results are given for the far wake structure.

Closed-form solutions describing the behaviour of buoyant axisymmetric turbulent rising plumes and fountains, emitted vertically from area sources in unconfined quiescent environments of uniform density, are proposed in a form that is universally applicable to Boussinesq and non-Boussinesq plumes. This paper, thereby, generalizes the results obtained separately for steady Boussinesq and non-Boussinesq plumes, including asymptotic virtual source corrections. The flux balance parameter Γ = Γ(z), a local Richardson number, is instrumental in describing the behaviour of steady plumes and the initial rise behaviour of fountains with height z. Non-dimensional graphs (cf. the ‘scale diagrams’ of Morton & Middleton, J. Fluid Mech., vol. 58, 1973, pp. 165–176) are plotted, showing certain characteristic heights for different source conditions, characterized by one single source flux balance parameter, giving a unique representation of the behaviour of Boussinesq fountains and both Boussinesq and non-Boussinesq plumes. Finally, a length scale has been identified that characterizes the height over which non-Boussinesq effects are important for lazy plumes rising from area sources.

The velocity field and turbulence structure within the surf and swash zones forced by a laboratory-generated plunging breaking wave were investigated using a particle image velocimetry measurement technique. Two-dimensional velocity fields in the vertical plane from 200 consecutive monochromatic waves were measured at four cross-shore locations, shoreward of the breaker line. The phase-averaged mean flow fields indicate that a shear layer occurs when the uprush of the bore front interacts with the downwash flow. The turbulence characteristics were examined via spectral analysis. The larger-scale turbulence structure is closely associated with the breaking-wave- and the bore-generated turbulence in the surf zone; then, the large-scale turbulence energy cascades to smaller scales, as the turbulent kinetic energy (TKE) evolves from the outer surf zone to the swash zone. Smaller-scale energy injection during the latter stage of the downwash phase is associated with the bed-generated turbulence, yielding a −1 slope in the upper inertial range in the spatial spectra. Depth-integrated TKE budget components indicate that a local TKE equilibrium exists during the bore-front phases and the latter stage of the downwash phases in the outer surf zone. The TKE decay resembles the decay of grid turbulence during the latter stage of the uprush and the early stage of the downwash, as the production rate is small because of the absence of strong mean shear during this stage of the wave cycle as well as the relatively short time available for the growth of the bed boundary layer.

We investigate here what happens beyond the onset of motion of a droplet on a wall by the action of an imposed shear flow, accounting for inertial effects and contact-angle hysteresis. A diffuse-interface method is used for this purpose, which alleviates the shear stress singularity at a moving contact line, resulting in an effective slip length. Various flow regimes are investigated, including steadily moving drops, and partial or entire droplet entrainment. In the regime of quasi-steadily moving drops, the drop speed is found to be linear in the imposed shear rate, but to exhibit an apparent discontinuity at the onset of motion. The results also include the relation between a local maximum angle between the interface and the wall and the instantaneous value of the contact-line speed. The critical conditions for the onset of entrainment are determined for pinned as well as for moving drops. The corresponding critical capillary numbers are found to be in a rather narrow range, even for quite substantial values of a Reynolds number. The approach to breakup is then investigated in detail, including the growth of a ligament on a drop, and the reduction of the radius of a pinching neck. A model based on an energy argument is proposed to explain the results for the rate of elongation of ligaments. The paper concludes with an investigation of detachment of a hydrophobic droplet from the solid wall.

Patterned surfaces with large effective slip lengths, such as super-hydrophobic surfaces containing trapped gas bubbles, have the potential to greatly enhance electrokinetic phenomena. Existing theories assume either homogeneous flat surfaces or patterned surfaces with thin double layers (compared with the texture correlation length) and thus predict simple surface-averaged, isotropic flows (independent of orientation). By analysing electro-osmotic flows over striped slip-stick surfaces with arbitrary double-layer thickness, we show that surface anisotropy generally leads to a tensorial electro-osmotic mobility and subtle, nonlinear averaging of surface properties. Interestingly, the electro-osmotic mobility tensor is not simply related to the hydrodynamic slip tensor, except in special cases. Our results imply that significantly enhanced electro-osmotic flows over super-hydrophobic surfaces are possible, but only with charged liquid–gas interfaces.

A theoretical analysis of equilibrium magnetohydrodynamic flows in annular channels is performed from the perspective of establishing required conditions for liquid metal magnetorotational instability (MRI) experiments. Two different types of fluid rotation are considered: electrically driven flow in an annular channel and Taylor–Couette flow between rotating cylinders. The structure of these flows is studied within a unified approach as a function of the Hartmann and Reynolds numbers. The parameters appropriate for realization of MRI experiments are determined.

The drag-reducing action of dilute solutions of long-chain polymers in a flat-plate turbulent boundary layer is studied using particle imaging velocimetry (PIV) and planar laser induced fluorescence (PLIF). The results are used to characterize and quantify the spatial distribution of the injected polymer solution and the downstream development of the DR along the flat plate. The two techniques were used simultaneously to document and study the spread of the injected polymer solution and the resulting changes in the structure and statistics of the turbulence in the boundary layer. The PLIF images provide a qualitative and quantitative measure of the dispersion of the injected polymer solution. The mean and root mean square (r.m.s.) concentration profiles obtained using PLIF showed that the polymer greatly suppressed the turbulent dispersion in the near-wall region. The quantitative concentration measurements across the boundary layer, combined with simultaneous velocity measurements, are used to obtain concentration flux measurements in the boundary layer and are used to study the effect of the turbulence on the dispersion of the injected polymer. The variation of the fluxes with concentration of the injected polymer solutions and with increasing downstream distance is also studied and documented. The action of the polymer is to reduce the streamwise fluxes in the boundary layer, the suppression increasing with concentration. Further, the fluxes are also used to estimate the turbulent Schmidt number (ScT) for the drag-reduced flow. For the polymer injection experiments, the ScT are all greater than unity with the highest magnitude measured to be around 6, with the magnitude increasing with increasing concentration of the injected solutions. However, for each experiment, the estimated ScT decreases along the length of the flat plate reflecting the loss of polymer effectiveness.

Simulations of steady two-dimensional stratified flow over an isolated obstacle are presented where the obstacle is tall enough so that the topographic Froude number, Nhm/Uo ≫ 1. N is the buoyancy frequency, hm the height of the topography from the channel floor and Uo the flow speed infinitely far from the obstacle. As for moderate Nhm/Uo (~1), a columnar response propagates far up- and downstream, and an arrested lee wave forms at the topography. Upstream, most of the water beneath the crest is blocked, while the moving layer above the crest has a mean velocity Um = UoH/(H−hm). The vertical wavelength implied by this velocity scale, λo = 2πUm/N, predicts dominant vertical scales in the flow. Upstream of the crest there is an accelerated region of fluid approximately λo thick, above which there is a weakly oscillatory flow. Downstream the accelerated region is thicker and has less intense velocities. Similarly, the upstream lift of isopycnals is greatest in the first wavelength near the crest, and weaker above and below. Form drag on the obstacle is dominated by the blocked response, and not on the details of the lee wave, unlike flows with moderate Nhm/Uo.

Directly downstream, the lee wave that forms has a vertical wavelength given by λo, except for the deepest lobe which tends to be thicker. This wavelength is small relative to the fluid depth and topographic height, and has a horizontal phase speed cpx = −Um, corresponding to an arrested lee wave. When considering the spin-up to steady state, the speed of vertical propagation scales with the vertical component of group velocity cgz = αUm, where α is the aspect ratio of the topography. This implies a time scale = tNα/2π for the growth of the lee waves, and that steady state is attained more rapidly with steep topography than shallow, in contrast with linear theory, which does not depend on the aspect ratio.

The onset of thermal convection in a finite rotating cylinder is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime in which spatio-temporal complexity is observed directly after onset. The system is examined in the non-physical limit of zero centrifugal force as well as with an experimentally realizable centrifugal force, leading to two different paths to Küppers–Lortz-like spatio-temporal chaos. In the idealized case, neglecting centrifugal force, the onset of convection occurs directly from a conduction state, resulting in square patterns with slow roll switching, followed at higher thermal driving by straight roll patterns with faster roll switching. The case with a centrifugal force typical of laboratory experiments exhibits target patterns near the theoretically predicted onset of convection, followed by a rotating wave that emerges via a Hopf bifurcation. A subsequent Hopf bifurcation leads to ratcheting states with sixfold symmetry near the axis. With increasing thermal driving, roll switching is observed within the ratcheting lattice before Küppers–Lortz-like spatio-temporal chaos is observed with the dissolution of the lattice at a slightly stronger thermal driving. For both cases, all of these states are observed within a 2% variation in the thermal driving.

In this paper, we derive a general relationship for the turbulent Prandtl number Prt for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Prt, is developed in terms of a mixing length scale LM and an overturning length scale LE, the ratio of the mechanical (turbulent kinetic energy) decay time scale TL to scalar decay time scale Tρ and the gradient Richardson number Ri. We show that our formulation for Prt is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio LM/LE ≈ 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, TL/Tρ. We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1–20; J. Fluid Mech., vol. 525, 2005, pp. 193–214) to propose a new parameterization for Prt in terms of the gradient Richardson number Ri. The formulation presented here provides a general framework for calculating Prt that will be useful for turbulence closure schemes in numerical models.

Hydraulic equations are derived for a stratified (two-layer) flow in which the horizontal velocity varies continuously in the vertical. Viscosity is included in the governing equations, and the effect of friction in hydraulically controlled flows is examined. The analysis yields Froude numbers which depend upon the integrated inverse square of velocity but reduce to the original layered Froude numbers when velocity is constant with depth. The Froude numbers reveal a critical condition for hydraulic control, which equates to the arrest of internal gravity waves.

Solutions are presented for the case of unidirectional flow through a lateral constriction, both with and without bottom drag. In the free-slip lower boundary case, viscosity transports momentum from the faster to the slower layer, thereby shifting the control point downstream and reducing the flux through the constriction. However, while the velocity shear at the interface between the two layers is reduced, the top-to-bottom velocity difference of the controlled solution is increased for larger values of viscosity. This counter-intuitive result is due to the restrictions placed on the flow at the hydraulic control point. When bottom drag is included in the model, the total flux may increase, in some cases exceeding that of the inviscid solution.

Direct stability analysis and numerical simulations have been employed to identify and characterize secondary instabilities in the wake of the flow around two identical circular cylinders in tandem arrangements. The centre-to-centre separation was varied from 1.2 to 10 cylinder diameters. Four distinct regimes were identified and salient cases chosen to represent the different scenarios observed, and for each configuration detailed results are presented and compared to those obtained for a flow around an isolated cylinder. It was observed that the early stages of the wake transition changes significantly if the separation is smaller than the drag inversion spacing. The onset of the three-dimensional instabilities were calculated and the unstable modes are fully described. In addition, we assessed the nonlinear character of the bifurcations and physical mechanisms are proposed to explain the instabilities. The dependence of the critical Reynolds number on the centre-to-centre separation is also discussed.

The paper extends a pilot study into a detailed investigation of properties of breaking waves and processes responsible for breaking. Simulations of evolution of steep to very steep waves to the point of breaking are undertaken by means of the fully nonlinear Chalikov–Sheinin model. Particular attention is paid to evolution of nonlinear wave properties, such as steepness, skewness and asymmetry, in the physical, rather than Fourier space, and to their interplay leading to the onset of breaking. The role of superimposed wind is also investigated. The capacity of the wind to affect the breaking onset is minimal unless the wind forcing is very strong. Wind is, however, important as a source of energy for amplification of the wave steepness and ultimately altering the breaking statistics. A detailed laboratory study is subsequently described. The theoretical predictions are verified and quantified. In addition, some features of the nonlinear development not revealed by the model (i.e. reduction of the wave period which further promotes an increase in steepness prior to breaking) are investigated. Physical properties of the incipient breaker are measured and examined, as well as characteristics of waves both preceding and following the breaker. The experiments were performed both with and without a superimposed wind, the role of which is also investigated. Since these idealized two-dimensional results are ultimately intended for field applications, tentative comparisons with known field data are considered. Limitations which the modulational instability mechanism can encounter in real broadband three-dimensional environments are highlighted. Also, substantial examination of existing methods of breaking onset detection are discussed and inconsistencies of existing measurements of breaking rates are pointed out.

We consider the steady-state statistics of turbulence in the inertial interval. The Kolmogorov flux relation (4/5-law) is shown to be a particular case of the general relation on the current–density correlation function. Using that, we derive an analogous flux relation for compressible turbulence and a new exact relation for incompressible turbulence.