Diffraction of regular waves by arrays of vertical bottom-mounted circular cylinders is investigated using theoretical, computational, and experimental methods. Experiments in an offshore wave basin are designed to measure free surface elevation η at multiple locations in the vicinity of a multi-column structure subjected to regular waves of frequency 0.449 < ka < 0.524 and steepness 0.122 < kA < 0.261, where k is the wavenumber, a the cylinder radius and A the wave amplitude. Results from regular wave data analysis for first-order amplitudes are compared with those from analytical linear diffraction theory, which is shown to be accurate for predicting incident waves of low steepness. Second- and third-order terms are also estimated from the measured time series, and the effects near a second-order near-trapping frequency are compared to semi-analytical second-order diffraction theory. Linear diffraction theory is shown to be very accurate at predicting the global surface elevation features, even for waves of high steepness. However, violent events and significant nonlinear interactions, including breaking induced by wave scattering, have been observed. Furthermore, second-order near-trapping was observed to affect the magnitude of local free surface oscillations as well as scattered far-field radiation.

Diffraction of irregular waves, focused wave groups, and random seas by an array of vertical bottom-mounted circular cylinders is investigated using theoretical, computational and experimental methods. This is an extension of our study of such an array in regular waves, reported in Part 1. Linear focused wave group theory is reviewed as a method for predicting the probable shape of extreme events from random wave spectra. Measurements are presented of the free surface elevation distribution in the vicinity of a multi-column structure in an offshore basin when subjected to irregular waves having peak frequencies and significant wave heights in the range 0.449 < kpa < 0.555 and 0.114 < Hs < 0.124 respectively, where a is the cylinder radius. Analytical linear diffraction theory is extended for application to focused wave groups and random seas. Experimental irregular wave data are analysed for comparison with this theory. Linear diffraction theory for random seas is shown to give an excellent prediction of incident wave spectral diffraction, while linear diffraction theory for focused wave groups works well for linearized extreme events. Due to the phase shifting of incident wave spectral components, diffraction is shown to generate focused wave groups in the vicinity of the cylinder array.

A systematic numerical analysis is performed for superharmonic excitations in a wake where a circular cylinder is rotationally oscillated in time. Emphasis is placed on identifying the secondary and tertiary lock-on in the forced wakes. The frequency responses are scrutinized by measuring the lift coefficient (CL). A direct numerical simulation has been conducted to portray the unsteady dynamics of wake flows behind a circular cylinder. The Reynolds number based on the diameter is Re = 106, and the forcing magnitude is 0.10 [les ] Ωmax [les ] 0.40. The tertiary lock-on is observed, where the shedding frequency (St0) is one third of the forcing frequency (Sf), i.e. the 1/3 subharmonic lock-on. The phase shift of CL with respect to the forcing frequency is observed. It is similar to that of the primary lock-on. However, in the secondary superharmonic excitation, modulated oscillations are observed, i.e. the lock-on does not exist. As Ωmax increases, St0 is gradually shifted from the natural shedding frequency (St*0) to lower values. The magnitudes and phases of Sf and St0 are analysed by the phase diagram. The vorticity contours are employed to examine the vortex formation mode against the forcing conditions.

The behaviour of turbulent shear flow over a mass-neutral permeable wall is studied numerically. The transpiration is assumed to be proportional to the local pressure fluctuations. It is first shown that the friction coefficient increases by up to 40% over passively porous walls, even for relatively small porosities. This is associated with the presence of large spanwise rollers, originating from a linear instability which is related both to the Kelvin–Helmholtz instability of shear layers, and to the neutral inviscid shear waves of the mean turbulent profile. It is shown that the rollers can be forced by patterned active transpiration through the wall, also leading to a large increase in friction when the phase velocity of the forcing resonates with the linear eigenfunctions mentioned above. Phase-lock averaging of the forced solutions is used to further clarify the flow mechanism. This study is motivated by the control of separation in boundary layers.

An analysis of the hydrodynamic stability of a fluid near its near critical point – initially at rest and in thermodynamic equilibrium – is considered in the Rayleigh–Bénard configuration, i.e. heated from below. The geometry is a two-dimensional square cavity and the top and bottom walls are maintained at constant temperatures while the sidewalls are insulated. Owing to the homogeneous thermo-acoustic heating (piston effect), the thermal field exhibits a very specific structure in the vertical direction. A very thin hot thermal boundary layer is formed at the bottom, then a homogeneously heated bulk settles in the core at a lower temperature; at the top, a cooler boundary layer forms in order to continuously match the bulk temperature with the colder temperature of the upper wall. We analyse the stability of the two boundary layers by numerically solving the Navier–Stokes equations appropriate for a van der Waals' gas slightly above its critical point. A finite-volume method is used together with an acoustic filtering procedure. The onset of the instabilities in the two different layers is discussed with respect to the results of the theoretical stability analyses available in the literature and stability diagrams are derived. By accounting for the piston effect the present results can be put within the framework of the stability analysis of Gitterman and Steinberg for a single layer subjected to a uniform, steady temperature gradient.

The stability of a two-layer return thermocapillary flow in the presence of an inclined temperature gradient is investigated. Both a linear stability analysis and nonlinear simulations have been performed for an air–water system. It is found that a rather weak deviation of the mean temperature gradient from the vertical direction suppresses Pearson's instability mechanism and leads to the appearance of oblique hydrothermal waves. In a certain region of parameters, transverse convective rolls drifting with the mean flow appear.

When deep water surface waves cross an area with variable current, refraction takes place. If the group velocity of the waves is much larger than the current velocity we show that the curvature of a ray, χ, is given by the simple formula χ = ζ/vg. Here ζ is the vertical component of the current vorticity and vg is the group velocity.

The effects of passive scalar anisotropy on subgrid-scale (SGS) physics and modelling for large-eddy simulations are studied experimentally. Measurements are performed across a moderate Reynolds number wake flow generated by a heated cylinder, using an array of four X-wire and four cold-wire probes. By varying the separation distance among probes in the array, we obtain filtered and subgrid quantities at three different filter sizes. We compute several terms that comprise the subgrid dissipation tensor of kinetic energy and scalar variance and test for isotropic behaviour, as a function of filter scale. We find that whereas the kinetic energy dissipation tensor tends towards isotropy at small scales, the SGS scalar-variance dissipation remains anisotropic independent of filter scale. The eddy-diffusion model predicts isotropic behaviour, whereas the nonlinear (or tensor eddy diffusivity) model reproduces the correct trends, but overestimates the level of scalar dissipation anisotropy. These results provide some support for so-called mixed models but raise new questions about the causes of the observed anisotropy.

The flow-induced deformation of an inviscid bubble occupied by a compressible gas and suspended in an ambient viscous liquid is considered at low Reynolds numbers with particular reference to the pressure developing inside the bubble. Ambient fluid motion alters the bubble pressure with respect to that established in the quiescent state, and requires the bubble to expand or contract according to an assumed equation of state. When changes in the bubble volume are prohibited by a global constraint on the total volume of the flow, the ambient pressure is modified while the bubble pressure remains constant during the deformation. A numerical method is developed for evaluating the pressure inside a two-dimensional bubble in an ambient Stokes flow on the basis of the normal component of the interfacial force balance involving the capillary pressure, the normal viscous stress, and the pressure at the free surface on the side of the liquid; the last is computed by evaluating a strongly singular integral. Dynamical simulations of bubble deformation are performed using the boundary integral method properly implemented to remove the multiplicity of solutions due to the a priori unknown rate of expansion, and three particular problems are discussed in detail: the shrinkage of a bubble at a specified rate, the deformation of a bubble subject to simple shear flow, and the deformation of a bubble subject to a purely elongational flow. In the case of shrinkage, it is found that the surface tension plays a critical role in determining the behaviour of the bubble pressure near the critical time when the bubble disappears. In the case of shear or elongational flow, it is found that the bubble contracts during an initial period of deformation from the circular shape, and then it expands to obtain a stationary shape whose area is higher than that assumed in the quiescent state. Expansion may destabilize the bubble by raising the capillary number above the critical threshold under which stationary shapes can be found.

The effect of high-amplitude forcing on a laminar wall jet over a heated flat plate is analysed. Highly accurate direct numerical simulations (DNS) are used to investigate the dominant transport mechanisms. When forcing is applied, the skin friction is reduced markedly and the wall heat transfer is increased. Detailed examination of the unsteady flow field showed that the concepts of eddy viscosity and eddy thermal diffusivity, usually applied to turbulent flows, can be applied to the analysis of unsteady laminar flows to explain the effect of highly unsteady phenomena.

We consider flow in a thin film generated by partially submerging an inclined rigid plate in a reservoir of ethanol– or methanol–water solution and wetting its surface. Evaporation leads to concentration and surface tension gradients that drive flow up the plate. An experimental study indicates that the climbing film is subject to two distinct instabilities. The first is a convective instability characterized by flattened convection rolls aligned in the direction of flow and accompanied by free-surface deformations; in the meniscus region, this instability gives rise to pronounced ridge structures aligned with the mean flow. The second instability, evident when the plate is nearly vertical, takes the form of transverse surface waves propagating up the plate.

We demonstrate that the observed longitudinal rolls are driven by the combined influence of surface deformations and alcohol concentration gradients. Guided by the observation that the rolls are flattened, we develop a quasi-two-dimensional theoretical model for the instability of the film, based on lubrication theory, which includes the effects of gravity, capillarity and Marangoni stresses at the surface. We develop stability criteria for the film which are in qualitative agreement with our experimental observations. Our analysis yields an equation for the shape of the interface which is solved numerically and reproduces the salient features of the observed flows, including the slow lateral drift and merging of the ridges.

The effects of fluid inertia, geometry and flow confinement upon the dynamics of neutrally buoyant elliptical and non-elliptical cylinders over a wide range of aspect ratios in simple shear are studied experimentally for moderate shear-based Reynolds numbers Re. Unlike circular cylinders, elliptical cylinders of moderate aspect ratio cease to rotate, coming to rest at a nearly horizontal equilibrium orientation above a critical Reynolds number Recr (‘stationary behaviour’). Simple dynamics arguments are proposed to explain the effects of aspect ratio and flow confinement upon critical Reynolds number and particle dynamics. Experiments confirm results from previous numerical simulations that the normalized rotation period for Re < Recr (‘periodic behaviour’) is proportional to (Recr − Re)−0.5 for small Recr − Re. For periodic behaviour, maximum and minimum angular cylinder speeds both decrease, and period increases, as Recr − Re decreases. For stationary behaviour, the cylinder rotates until it achieves a nearly horizontal equilibrium orientation, which increases as the Reynolds number approaches the critical value. The experimental results are in good agreement with previous lattice-Boltzmann simulations for a 0.5 aspect ratio cylinder.

Variation in angular speed over a rotation period decreases as aspect ratio increases, while Recr increases as flow confinement and aspect ratio increase. A non-elliptical cylinder of 0.33 aspect ratio also ceases to rotate above a certain Reynolds number. Although Recr is different from the corresponding elliptical case, the scaling of the normalized rotation period for this body as Recr → Re is identical to that for the elliptical cylinder, suggesting that this scaling is independent of particle shape (i.e. ‘universal’, as conjectured in previous numerical studies). The results also demonstrate that a variety of centrosymmetric bodies with aspect ratios below unity transition from periodic to stationary behaviour.

Linearly stratified salt solutions of different Prandtl number were subjected to turbulent stirring by a horizontally oscillating vertical grid in a closed laboratory system. The experimental set-up allowed the independent direct measurement of a root mean square turbulent lengthscale Lt, turbulent diffusivity for mass Kρ, rate of dissipation of turbulent kinetic energy ε, buoyancy frequency N and viscosity v, as time and volume averaged quantities. The behaviour of both Lt and Kρ was characterized over a wide range of the turbulence intensity measure, ε/vN2, and two regimes were identified.

In the more energetic of these regimes (Regime E, where 300 < ε/vN2 < 105), Lt was found to be a function of v, κ and N, whilst Kρ was a function of v, κ and (ε/vN2)1/3. From these expressions for Lt and Kρ, a scaling relation for the root mean square turbulent velocity scale Ut was derived, and this relationship showed good agreement with direct measurements from other data sets.

In the weaker turbulence regime (Regime W, where 10 < ε/vN2 < 300) Kρ was a function of v, κ and ε/vN2.

For 10 < ε/vN2 < 1000, our directly measured diffusivities, Kρ, are approximately a factor of 2 different to the diffusivity predicted by the model of Osborn (1980). For ε/vN2 > 1000, our measured diffusivities diverge from the model prediction. For example, at ε/vN2 ≈ 104 there is at least an order of magnitude difference between the measured and predicted diffusivities.

The in uence of the thermal conductivity of rigid boundaries is analysed for the case when the onset of convection occurs in the form of a time-dependent mode. A new form of long-wavelength convection is found which is disjunct from the high-wavenumber thermal Rossby wave in the presence of infinitely conducting boundaries usually considered.

The interaction between a turbulent gas flow and particle motion was investigated by numerical simulations of gas–particle turbulent downward flow in a vertical channel. In particular the effect of inter-particle collision on the two-phase flow field was investigated. The gas flow field was obtained by large-eddy simulation (LES). Particles were treated by a Lagrangian method, with inter-particle collisions calculated by a deterministic method. The spatial resolution for LES of gas–solid two-phase turbulent flow was examined and relations between grid resolution and Stokes number are presented. Profiles of particle mean velocity, particle wall-normal fluctuation velocity and number density are flattened as a result of inter-particle collisions and these results are in good agreement with experimental measurements. Calculated turbulence attenuation by particles agrees well with experimental measurements for small Stokes numbers, but not for large Stokes number particle. The shape and scale of particle concentrations calculated considering inter-particle collision are in good agreement with experimental observations.

The instability of a weakly sheared density-stratified two-dimensional wavy flow to longitudinal vortices is considered. The instability mechanism is Craik–Leibovich type 2, or CL2, and the problem is posited in the context of Langmuir circulations beneath irrotational wind-driven surface waves. Of interest is the influence to the instability of Prandtl Pr and Richardson Ri numbers according to linear theory. The basis for the study is an initial value problem posed by Leibovich & Paolucci (1981) in which the liquid substrate is of semi-infinite extent and the wind-driven current is permitted to grow in the presence of neutral waves. In the present work Pr is varied from zero to infinity, and both stabilizing and destabilizing Ri are considered; so too are monochromatic and measured wave fields, and laminar and turbulent velocity profiles. Only the Ri = 0 results recover those of Leibovich & Paolucci. For stabilizing Ri, it is found in general that diminishing Pr are destabilizing to Langmuir circulations (LCs), and thus that LCs can be present or absent at the same Langmuir number La provided Ri ≠ 0. It is further found that two branches of neutral curves occur for some combinations of Pr and Ri, and that minor changes in either parameter permit the preferred spacing to switch from one branch to the other. In consequence the preferred spanwise spacing may change from smaller than the wavelength of the dominant waves to larger than it. Furthermore, although LCs will not form at inverse La below a global lower bound given by an energy stability analysis, the actual value of La at onset is found to depend greatly upon local details of the wave and shear fields. Interestingly although this global lower bound is independent of Ri and Pr for Ri [ges ] 0, that is not the case for Ri < 0, where it approaches zero as Ri → −∞, indicating that the CL2 instability is viable even at low Reynolds numbers.

In the first part of this paper, we investigate passive scalar or tracer advection–diffusion in frozen, two-dimensional, non-circular symmetric vortices. We develop an asymptotic description of the scalar field in a time range 1 [Lt ] t/T [Lt ] Pe1/3, where T is the formation time of the spiral in the vortex and Pe is a Péclet number, assumed much larger than 1. We derive the leading-order decay of the scalar variance E(t) for a singular non-circular streamline geometry,

The variance decay is solely determined by a geometrical parameter μ and the exponent β describing the behaviour of the closed streamline periods. We develop a method to predict, in principle, the variance decay from snapshots of the advected scalar field by reconstructing the streamlines and their period from just two snapshots of the advected scalar field.

In the second part of the paper, we investigate variance decay in a periodically moving singular vortex. We identify three different regions (core, chaotic and KAM-tori). We find fast mixing in the chaotic region and investigate a conjecture about mixing in the KAM-tori region. The conjecture enables us to use the results from the first section and relates the Kolmogorov capacity, or box-counting dimension, of the advected scalar to the decay of the scalar variance. We check our theoretical predictions against a numerical simulation of advection–diffusion of scalar in such a flow.

Nearly two decades ago, Couder (1981) and Gharib & Derango (1989) used soap films to perform classical hydrodynamics experiments on two-dimensional flows. Recently soap films have received renewed interest and experimental investigations published in the past few years call for a proper analysis of soap film dynamics. In the present paper, we derive the leading-order approximation for the dynamics of a flat soap film under the sole assumption that the typical length scale of the flow parallel to the film surface is large compared to the film thickness. The evolution equations governing the leading-order film thickness, two-dimensional velocities (locally averaged across the film thickness), average surfactant concentration in the interstitial liquid, and surface surfactant concentration are given and compared to similar results from the literature. Then we show that a sufficient condition for the film velocity distribution to comply with the Navier–Stokes equations is that the typical flow velocity be small compared to the Marangoni elastic wave velocity. In that case the thickness variations are slaved to the velocity field in a very specific way that seems consistent with recent experimental observations. When fluid velocities are of the order of the elastic wave speed, we show that the dynamics are generally very specific to a soap film except if the fluid viscosity and the surfactant solubility are neglected. In that case, the compressible Euler equations are recovered and the soap film behaves like a two-dimensional gas with an unusual ratio of specific heat capacities equal to unity.