The free-surface flow of very steep forced and unforced solitary waves is considered. The forcing is due to a distribution of pressure on the free surface. Four types of forced solution are identified which all approach the Stokes-limiting configuration of an included angle of $12{0}^{\circ } $ and a stagnation point at the wave crests. For each type of forced solution the almost-highest wave does not contain the most energy, nor is it the fastest, similar to what has been observed previously in the unforced case. Nonlinear solutions are obtained by deriving and solving numerically a boundary integral equation. A weakly nonlinear approximation to the flow problem helps with the identification and classification of the forced types of solution, and their stability.

We study the effect of turbulence on marine life by performing numerical simulations of motile micro-organisms, modelled as prolate spheroids, in isotropic homogeneous turbulence. We show that the clustering and patchiness observed in laminar flows, linear shear and vortex flows, are significantly reduced in a three-dimensional turbulent flow mainly because of the complex topology; elongated micro-organisms show some level of clustering in the case of swimmers without any preferential alignment whereas spherical swimmers remain uniformly distributed. Micro-organisms with one preferential swimming direction (e.g. gyrotaxis) still show significant clustering if spherical in shape, whereas prolate swimmers remain more uniformly distributed. Due to their large sensitivity to the local shear, these elongated swimmers react more slowly to the action of vorticity and gravity and therefore do not have time to accumulate in a turbulent flow. These results show how purely hydrodynamic effects can alter the ecology of micro-organisms that can vary their shape and their preferential orientation.

We show that optimal perturbations artificially forced in parallel wakes can be used to completely suppress the absolute instability and to reduce the maximum temporal growth rate of the inflectional instability. To this end we compute optimal transient energy growths of stable streamwise uniform perturbations supported by a parallel wake for a set of Reynolds numbers and spanwise wavenumbers. The maximum growth rates are shown to be proportional to the square of the Reynolds number and to increase with spanwise wavelengths with sinuous perturbations slightly more amplified than varicose ones. Optimal initial conditions consist of streamwise vortices and the optimally amplified perturbations are streamwise streaks. Families of nonlinear streaky wakes are then computed by direct numerical simulation using optimal initial vortices of increasing amplitude as initial conditions. The stabilizing effect of nonlinear streaks on temporal and spatiotemporal growth rates is then determined by analysing the linear impulse response supported by the maximum amplitude streaky wakes profiles. This analysis reveals that at $\mathit{Re}= 50$, streaks of spanwise amplitude ${A}_{s} \approx 8\hspace{0.167em} \% {U}_{\infty } $ can completely suppress the absolute instability, converting it into a convective instability. The sensitivity of the absolute and maximum temporal growth rates to streak amplitudes is found to be quadratic, as has been recently predicted. As the sensitivity to two-dimensional (2D, spanwise uniform) perturbations is linear, three-dimensional (3D) perturbations become more effective than the 2D ones only at finite amplitudes. Concerning the investigated cases, 3D perturbations become more effective than the 2D ones for streak amplitudes ${A}_{s} \gtrsim 3\hspace{0.167em} \% {U}_{\infty } $ in reducing the maximum temporal amplification and ${A}_{s} \gtrsim 12\hspace{0.167em} \% {U}_{\infty } $ in reducing the absolute growth rate. However, due to the large optimal energy growths they experience, 3D optimal perturbations are found to be much more efficient than 2D perturbations in terms of initial perturbation amplitudes. Despite their lower maximum transient amplification, varicose streaks are found to be always more effective than sinuous ones in stabilizing the wakes, in accordance with previous findings.

Steady streaming flow from oscillating sessile bubbles at walls is the centrepiece of many microstreaming experiments. A complete asymptotic theory of the flow is developed, requiring only the oscillatory driving frequency and material parameters as input, and properly accounting for bubble and wall boundary conditions. It is shown that mixed-mode streaming of neighbouring bubble oscillation modes is responsible for the robustness of the generic ‘fountain’ vortex pair flow pattern, and that the pattern reverses for high frequencies when wall-induced streaming becomes dominant. The far-field flow and its dependence on control parameters are in agreement with experimental data and can be understood considering just a few asymptotic coefficients.

In this paper we extend linear momentum actuator disc theory to consider two rows of tidal turbines placed in a centred or staggered arrangement. The extensions assume a streamwise spacing between rows that is sufficient for pressure equalization, but is not too large for significant mixing of the upstream turbine wake before the second row. We first consider a given number of turbines in a tidal channel; in this case the average power for a staggered arrangement over two rows is found to be higher than that for a centred arrangement, but lower than can be obtained by placing all turbines side-by-side in one row (if all turbines have the same local resistance). Furthermore, staggered arrangements extract power more efficiently than centred arrangements, but less efficiently than a single row with the same number of turbines, and this has implications for ranking different arrangements of tidal turbines. We also use the extended actuator disc models (together with an argument of scale separation) to consider some example arrangements of tidal turbines in laterally unconfined flow. Specifically, it is shown that locally staggering a fixed number of turbines in an array to form a tidal farm generates less power than placing the same number of turbines side-by-side. However, if more than one row of turbines is adopted (perhaps to keep the farm spatially compact) then the optimum turbine spacing within a row increases significantly with addition of a second row. This trend suggests that multi-row tidal turbine farms would require wide turbine spacing within each row to maximize the power per turbine, similarly to existing offshore wind farms.

Large eddy simulations at $Re= 23\hspace{0.167em} 000$ are used to investigate the drag on a two-dimensional elongated cylinder caused by rear-edge periodic actuation, with particular focus on an optimum open-loop configuration. The 3.64 (length/thickness) aspect-ratio cylinder has a rectangular cross-section with rounded leading corners, representing the two-dimensional cross-section of the now generic Ahmed-body geometry. The simulations show that the optimum drag reduction occurs in the forcing Strouhal number range of $0. 09\leq S{t}_{act} \leq 0. 135$, which is approximately half of the Strouhal number corresponding to shedding of von Kármán vortices into the wake for the natural case. This result agrees well with recent experiments of Henning et al. (Active Flow Control, vol. 95, 2007, pp. 369–390). A thorough transient wake analysis employing dynamic mode decomposition is conducted for all cases, with special attention paid to the Koopman modes of the wake flow and vortex progression downstream. Two modes are found to coexist in all cases, the superimposition of which recovers the majority of features observed in the flow. Symmetric vortex shedding in the near wake, which effectively extends the mean recirculation bubble, is shown to be the major mechanism in lowering the drag. This is associated with opposite-signed vortices reducing the influence of natural vortex shedding, resulting in an increase in the pressure in the near wake, while the characteristic wake antisymmetry returns further downstream. Lower-frequency actuation is shown to create larger near-wake symmetric vortices, which improves the effectiveness of this process.

In this work, we study theoretically the thickness of a liquid film (typically made of a surfactant solution) pulled out of a bath at constant speed in the absence of gravity, when it features a viscous or an elastic interfacial rheology. We show that a purely viscous rheology does not lead to the extraction of a steady state film of constant thickness. In contrast, the thickness of the film is well defined in the elastic case, which allows us to compute it. This thickness depends on the capillary number of the experiment, and on the elasticity of the interface. It is always lower than or equal to that obtained for an incompressible interface predicted by Frankel (Mysels, Shinoda and Frankel, Soap Films: Studies of their Thinning and a Bibliography, 1959), which is recovered in the limit of an arbitrary large elasticity.

This paper revisits the problem of forces on obstacle arrays in combined waves and an in-line steady current. The intended application is the design and reassessment of offshore platforms. A series of experiments are performed on planar grids moved in both steady and oscillatory motion through otherwise stationary water. Detailed comparisons are made to a wave-current–structure interaction model recently presented by Taylor, Santo & Choo (Ocean Engng, vol. 57, 2013, pp. 11–24). We present new features of the model and test these against the experimental data. For relatively small current speed (${u}_{c} $) compared with oscillatory velocity amplitude (${u}_{w} $) with phase angle ($\omega t$), the drag force time history on grids with solid area ($A$) and projected frontal area (${A}_{f} $) is well approximated by a summation of the wave drag and the current drag components independently, so there is no ${u}_{w} \times {u}_{c} $ cross-term. The wave drag component is proportional to $\cos \omega t\vert \cos \omega t\vert $, while the current drag component to $\vert \cos \omega t\vert $, i.e. it is phase-locked to the oscillatory wave crests. The form of the predicted time history is new, so much of this paper is occupied in testing the adequacy of this theoretical form both in terms of an improved Morison-type formulation and also in the precise variation of the experimental drag force in time. We show that the measured crest and trough peak values of the drag force are consistent with the force peaks and troughs of the model prediction. The odd frequency harmonics of the measured drag force scale as the square of the oscillatory velocity amplitude $({ u}_{w}^{2} )$ and on the total hydrodynamic area (${C}_{d} A$). The shape of the odd harmonics is very similar to that for a pure oscillatory motion without steady current, but there are also even frequency harmonics associated with the current component. The even harmonics of the force scale as the square of the current speed $({ u}_{c}^{2} )$ and on the ${A}_{f} $, not on the ${C}_{d} A$. All of the above features are identified within the experimental data, and provide considerable support for the new current blockage model.

The new model is also shown to fit the entire force time history well for a wide range of individual cases, with different blockage ratio ($A/ {A}_{f} $) and number of grids, requiring only calibration of the Morison-type drag and inertia coefficients. In contrast, the industry-standard form of the Morison equation can only be matched at a single instant of the oscillation cycle, so present practice should be regarded as seriously inadequate for combined steady current and oscillatory flow acting on obstacle arrays.

Steady, incompressible flow down a slowly curving circular pipe is considered. Both real and complex solutions of the Dean equations are found by analytic continuation of a series expansion in the Dean number, $K$. Higher-order Hermite–Padé approximants are used and the results compared with direct computations using a spectral method. The two techniques agree for large, real $K$, indicating that previously reported asymptotic behaviour of the series solution is incorrect, and thus resolving a long-standing paradox. It is further found that a second solution branch, known to exist at high Dean number, does not appear to merge with the main branch at any finite $K$, but appears rather to bifurcate from infinity. The convergence of the series is limited by a square-root singularity on the imaginary $K$-axis. Four complex solutions merge at this point. One corresponds to an extension of the real solution, while the other three are previously unreported. This bifurcation is found to coincide with the breaking of a symmetry property of the flow. On one of the new branches the velocity is unbounded as $K\rightarrow 0$. It follows that the zero-Dean-number flow is formally non-unique, in that there is a second complex solution as $K\rightarrow 0$ for any non-zero $\vert K\vert $. This behaviour is manifested in other flows at zero Reynolds number. The other two complex solutions bear some resemblance to the two solution branches for large real $K$.

This paper deals with nearly inviscid, capillary–gravity, modulated waves parametrically excited by monochromatic horizontal vibrations in liquid containers whose width and depth are both large compared with the wavelength of the excited waves. A general linear amplitude equation is derived with appropriate boundary conditions that provides the threshold acceleration and associated spatiotemporal patterns, which compare very well with experimental measurements and visualizations. The primary instability is associated with a pair of complex Floquet multipliers that are close to (but strictly different from) −1, meaning that the instability is not strictly (2:1) subharmonic. The resulting (quasi-periodic) waves are generally oblique, not perpendicular to the vibrating endwalls. The extension of the theory to other confined systems such as vibrating containers of arbitrary shape and vibrating drops is also considered.

Localized regions of turbulence, or turbulent clouds, in a stratified fluid are the subject of this study, which focuses on the edge dynamics occurring between the turbulence and the surrounding quiescent region. Through laboratory experiments and numerical simulations of stratified turbulent clouds, we confirm that the edge dynamics can be subdivided into materially driven intrusions and horizontally travelling internal wave-packets. Three-dimensional visualizations show that the internal gravity wave-packets are in fact large-scale pancake structures that grow out of the turbulent cloud into the adjacent quiescent region. The wave-packets were tracked in time, and it is found that their speed obeys the group speed relation for linear internal gravity waves. The energetics of the propagating waves, which include waveforms that are inclined with respect to the horizontal, are also considered and it is found that, after a period of two eddy turnover times, the internal gravity waves carry up to 16 % of the cloud kinetic energy into the initially quiescent region. Turbulent events in nature are often in the form of decaying turbulent clouds, and it is therefore suggested that internal gravity waves radiated from an initial cloud could play a significant role in the reorganization of energy and momentum in the atmosphere and oceans.

We examine the structure of the turbulence boundary of a temporal plane jet at $\mathit{Re}= 5000$ using statistics conditioned on the enstrophy. The data is obtained by direct numerical simulation and threshold values span 24 orders of magnitude, ranging from essentially irrotational fluid outside the jet to fully turbulent fluid in the jet core. We use two independent estimators for the local entrainment velocity ${v}_{n} $ based on the enstrophy budget. The data show clear evidence for the existence of a viscous superlayer (VSL) that envelopes the turbulence. The VSL is a nearly one-dimensional layer with low surface curvature. We find that both its area and viscous transport velocity adjust to the imposed rate of entrainment so that the integral entrainment flux is independent of threshold, although low-Reynolds-number effects play a role for the case under consideration. This threshold independence is consistent with the inviscid nature of the integral rate of entrainment. A theoretical model of the VSL is developed that is in reasonably good agreement with the data and predicts that the contribution of viscous transport and dissipation to interface propagation have magnitude $2{v}_{n} $ and $- {v}_{n} $, respectively. We further identify a turbulent core region (TC) and a buffer region (BR) connecting the VSL and the TC. The BR grows in time and inviscid enstrophy production is important in this region. The BR shows many similarities with the turbulent–non-turbulent interface (TNTI), although the TNTI seems to extend into the TC. The average distance between the TC and the VSL, i.e. the BR thickness is about 10 Kolmogorov length scales or half a Taylor length scale, indicating that intense turbulent flow regions and viscosity-dominated regions are in close proximity.

We consider the linear stability of an inviscid parallel shear flow of air over water with gravity and capillarity. The velocity profile in the air is monotonically increasing upwards from the sea surface and is convex, while the velocity in the water is monotonically decreasing from the surface and is concave. An archetypical example, the ‘double-exponential’ profile, is solved analytically and studied in detail. We show that there are two types of unstable mode which can, in some cases, co-exist. The first type is the ‘Miles mode’ resulting from a resonant interaction between a surface gravity wave and a critical level in the air. The second unstable mode is an interaction between surface gravity waves and a critical level in the water, resulting in the growth of ripples. The gravity–capillary waves participating in this second resonance have negative intrinsic phase speed, but are Doppler shifted so that their actual phase speed is positive, and matches the speed of the base-state current at the critical level. In both cases, the Reynolds stresses of an exponentially growing wave transfer momentum from the vicinity of the critical level to the zone between the crests and troughs of a surface wave.

Motivated by experiments showing that a sessile drop of volatile perfectly wetting liquid initially advances over the substrate, but then reverses, we formulate the problem describing the contact region at reversal. Assuming a separation of scales, so that the radial extent of this region is small compared with the instantaneous radius $a$ of the apparent contact line, we show that the time scale characterizing the contact region is small compared with that on which the bulk drop is evolving. As a result, the contact region is governed by a boundary-value problem, rather than an initial-value problem: the contact region has no memory, and all its properties are determined by conditions at the instant of reversal. We conclude that the apparent contact angle $\theta $ is a function of the instantaneous drop radius $a$, as found in the experiments. We then non-dimensionalize the boundary-value problem, and find that its solution depends on one parameter $\mathscr{L}$, a dimensionless surface tension. According to this formulation, the apparent contact angle is well-defined: at the outer edge of the contact region, the film slope approaches a limit that is independent of the curvature of bulk drop. In this, it differs from the dynamic contact angle observed during spreading of non-volatile drops. Next, we analyse the boundary-value problem assuming $\mathscr{L}$ to be small. Though, for arbitrary $\mathscr{L}$, determining $\theta $ requires solving the steady diffusion equation for the vapour, there is, for small $\mathscr{L}$, a further separation of scales within the contact region. As a result, $\theta $ is now determined by solving an ordinary differential equation. We predict that $\theta $ varies as ${a}^{- 1/ 6} $, as found experimentally for small drops ($a\lt 1~\mathrm{mm} $). For these drops, predicted and measured angles agree to within 10–30 %. Because the discrepancy increases with $a$, but $\mathscr{L}$ is a decreasing function of $a$, we infer that some process occurring outside the contact region is required to explain the observed behaviour of larger drops having $a\gt 1~\mathrm{mm} $.

An experimental investigation was conducted to study the dynamical behaviour of a model valve in a pulsatile flow. The valve is modelled as a pair of curved, rectangular, flexible leaflets that open and close under a time-periodic flow. Using image analysis, the range of flow parameters for which a valve (of a particular geometry and material properties of the leaflets) works correctly were identified. A correct performance was considered to be when the valve opened in one direction but blocked the flow in the reversed direction. A model is proposed to predict the performance of the valves. Furthermore, an analysis of fluid strains is conducted for valves that operate correctly to identify the influence of the valve’s design on fluid stresses. The main purpose of this investigation is to gain insight for the design of future prosthetic heart valves.

This paper presents an experimental study of isothermal rarefied gas flow through a tube with sudden expansion in the slip flow regime. The measurements reported here are for nitrogen flowing at low pressures in conventional tubes with sudden expansion area ratios of 1.48, 3.74, 12.43 and 64. The flow is dynamically similar to gas flow in a microchannel as the Knudsen number $(0. 0001\lt \mathit{Kn}\lt 0. 075)$ falls in the slip flow regime; the Reynolds number in the smaller section (${\mathit{Re}}_{s} $) ranges between 0.2 and 837. The static pressure along the wall is measured for different mass flow rates controlled by a mass flow controller and analysed to understand the flow behaviour. The velocity profiles are obtained through a momentum balance and using the pressure measurements. A discontinuity in the slope of pressure at the sudden expansion junction is noted and given special attention. The absence of flow separation is another key feature observed from the measurements. The streamlines are found to be concave near the junction. It is demonstrated that the flow ‘senses’ the oncoming sudden expansion junction and starts adjusting itself much before reaching the junction; this interesting behaviour is attributed to an increased axial momentum diffusion and wall slip. The additional acceleration of the central core of the gas flow causes an increase in the wall shear stress and a larger pressure drop as compared with a straight tube. These results are not previously available and should help in improving understanding of gaseous slip flows.

A generalized Reynolds analogy (GRA) is proposed for compressible wall-bounded turbulent flows (CWTFs) and validated by direct numerical simulations. By introducing a general recovery factor, a similarity between the Reynolds-averaged momentum and energy equations is established for the canonical CWTFs (i.e. pipes, channels, and flat-plate boundary layers that meet the quasi-one-dimensional flow approximation), independent of Prandtl number, wall temperature, Mach number, Reynolds number, and pressure gradient. This similarity and the relationships between temperature and velocity fields constitute the GRA. The GRA relationship between the mean temperature and the mean velocity takes the same quadratic form as Walz’s equation, with the adiabatic recovery factor replaced by the general recovery factor, and extends the validity of the latter to diabatic compressible turbulent boundary layers and channel/pipe flows. It also derives Duan & Martín’s (J. Fluid Mech., vol. 684, 2011, pp. 25–59) empirical relation for flows at different physical conditions (wall temperature, Mach number, enthalpy condition, surface catalysis, etc.). Several key parameters besides the general recovery factor emerge in the GRA. An effective turbulent Prandtl number is shown to be the reason for the parabolic profile of mean temperature versus mean velocity, and it approximates unity in the fully turbulent region. A dimensionless wall temperature, that we call the diabatic parameter, characterizes the wall-temperature effects in diabatic flows. The GRA also extends the analysis to the fluctuation fields. It recovers the modified strong Reynolds analogy proposed by Huang, Coleman & Bradshaw (J. Fluid Mech., vol. 305, 1995, pp. 185–218) and explains the variation of the temperature–velocity correlation coefficient with wall temperature. Thus, the GRA unveils a generalized similarity principle behind the complex nonlinear coupling between the thermal and velocity fields of CWTFs.

The migration of a capsule enclosed by an elastic membrane in a wall-bounded linear shear is investigated using a front-tracking method. A detailed comparison with the migration of a viscous drop is presented varying the capillary number (in the case of a capsule, the elastic capillary number) and the viscosity ratio. In both cases, the deformation breaks the flow reversal symmetry and makes them migrate away from the wall. They quickly go through a transient evolution to eventually reach a quasi-steady state where the dynamics becomes independent of the initial position and only depends on the wall distance. Previous analytical theories predicted that for a viscous drop, in the quasi-steady state, the migration and slip velocities scale approximately with the square of the inverse of the drop–wall separation, whereas the drop deformation scales as the inverse cube of the separation. These power law relations are shown to hold for a capsule as well. The deformation and inclination angle of the capsule and the drop at the same wall separation show a crossover in their variation with the capillary number: the capsule shows a steeper variation than that of the drop for smaller capillary numbers and slower variation than the drop for larger capillary numbers. Using the Green’s function of Stokes flow, a semi-analytic theory is presented to show that the far-field stresslet that causes the migration has two distinct contributions from the interfacial stresses and the viscosity ratio, with competing effects between the two defining the dynamics. It predicts the scaling of the migration velocity with the capsule–wall separation, however, matching with the simulated result very well only away from the wall. A phenomenological correlation for the migration velocity as a function of elastic capillary number, wall distance and viscosity ratio is developed using the simulation results. The effects of different membrane hyperelastic constitutive equations – neo-Hookean, Evans–Skalak, and Skalak – are briefly investigated to show that the behaviour remains similar for different equations.

We present the results of an experimental investigation of a droplet walking on the surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating that the stable quantized orbits reported by Fort et al. (Proc. Natl Acad. Sci., vol. 107, 2010, pp. 17515–17520) arise only for a finite range of vibrational forcing, above which complex trajectories with multimodal statistics arise. We first present a detailed characterization of the emergence of orbital quantization, and then examine the system behaviour at higher driving amplitudes. As the vibrational forcing is increased progressively, stable circular orbits are succeeded by wobbling orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude about a stationary centre is punctuated by the orbital centre leaping approximately half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular trajectories emerge, characterized by a multimodal probability distribution that reflects the persistent dynamic influence of the unstable orbital states.

Deposition and resuspension mechanisms in particle-laden turbulent flows are dominated by the coherent structures arising in the wall region. These turbulent structures, which control the turbulent regeneration cycles, are affected by the roughness of the wall. The particle-laden turbulent flow in a channel bounded by irregular two-dimensional rough surfaces is analysed. The behaviour of dilute dispersions of heavy particles is analysed using direct numerical simulations (DNS) to calculate the three-dimensional turbulent flow and Lagrangian tracking to describe the turbophoretic effect associated with two-phase turbulent flows in a complex wall-bounded domain. Turbophoresis is investigated in a quantitative way as a function of the particle inertia. The analysis of the particle statistics, in term of mean particle concentration and probability density function (p.d.f.) of wall-normal particle velocity, shows that the wall roughness produces a completely different scenario compared to the classical smooth wall. The effect of the wall roughness on the particle mass flux is shown for six particle populations having different inertia.