We discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, J. Appl. Mech. Tech. Phys., vol. 9, 1968, pp. 190–194). Using Krasitskii's canonical transformation we derive general expressions for the second-order wavenumber and frequency spectrum and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea-state bias as seen by a radar altimeter. According to the present approach but in contrast with results obtained by Barrick & Weber (J. Phys. Oceanogr., vol. 7, 1977, pp. 11–21), in deep water second-order effects on the wavenumber spectrum are relatively small. However, in shallow water in which waves are more nonlinear, the second-order effects are relatively large and help to explain the formation of the observed second harmonics and infra-gravity waves in the coastal zone. The second-order effects on the directional-frequency spectrum are as a rule more important; in particular it is shown how the Stokes-frequency correction affects the shape of the frequency spectrum, and it is also discussed why in the context of the second-order theory the mean-square slope cannot be estimated from time series. The kurtosis of the wave field is a relevant parameter in the detection of extreme sea states. Here, it is argued that in contrast perhaps to one's intuition, the kurtosis decreases while the waves approach the coast. This is related to the generation of the wave-induced current and the associated change in mean sea level.

The breaking of periodic progressive two-layer interfacial waves at a Gaussian ridge is investigated through laboratory experiments. Length scales of the incident wave and topography are used to parameterize when and how breaking occurs. Qualitative observations suggest both shear and convection play a role in the instability of waves breaking at the ridge. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields from which the local gradient Richardson number Rig is calculated. The transition to breaking occurred when 0.2 ≤ Rig ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Rig than for similar waves breaking through shear instability in deep water (Troy & Koseff, J. Fluid Mech., vol. 543, 2005b, p. 107). Due to the effects of unsteadiness, nonlinear shoaling and flow separation, the canonical Rig > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Rig in waves between finite depth layers using scales of the incident wave scale and topography. The observed breaking transition corresponds with a constant estimated value of Rig from the model, suggesting that interfacial shear plays an important role in initial wave instability. For wave amplitudes above the initial breaking transition, convective breaking events are also observed.

In a previous publication, Dietze, Leefken & Kneer (J. Fluid Mech., vol. 595, 2008, p. 435) showed that flow separation takes place in the capillary wave region of falling liquid films. That investigation focused on the mechanistic explanation of the phenomenon mainly on the basis of numerical data. The present publication for the first time provides clear experimental evidence of the phenomenon obtained by way of highly resolving velocity measurements in a specifically designed optical test set-up. Characteristically, the refractive index of the working fluid was matched to that of the glass test section to provide optimal access to the cross-section of the film for the employed optical velocimetry techniques, namely, laser doppler velocimetry (LDV) and particle image velocimetry (PIV). Using LDV, time traces of the streamwise velocity component were recorded in high spatial (0.025 mm) and temporal resolutions (0.4 ms) showing negative velocity values in the capillary wave region. In addition, simultaneous film thickness measurements were performed using a Confocal Chromatic Imaging (CCI) technique enabling the correlation of velocity data and wave dynamics. Further, using PIV the spatio-temporal evolution of the velocity field in the cross-section of the film was measured with high spatial (0.02 mm) and temporal (0.5 ms) resolutions yielding insight into the topology of the flow. Most importantly these results clearly show the existence of a separation eddy in the capillary wave region. Due to the high temporal resolution of the PIV measurements, enabled by the use of a high-speed camera with a repetition rate of up to 4500 Hz, the effect of wave dynamics on the velocity field in all regions of the wavy film was elucidated. All experiments were performed using a dimethylsulfoxide (DMSO)–water solution and focused on laminar vertically falling liquid films with externally excited monochromatic surface waves. Systematic variations of both the Reynolds number (Re = 8.6–15.0) and the excitation frequency (f = 16–24 Hz) were performed. Results show that an increase in the wavelength of large wave humps, produced either by an increase in the Reynolds number or a decrease in the excitation frequency, leads to an increase in the size of the capillary separation eddy (CSE). Thereby, the CSE is shown to grow larger than the local film thickness, assuming an open shape with streamlines ending at the free surface.

Non-Oberbeck–Boussinesq (NOB) effects on the flow organization in two-dimensional Rayleigh–Bénard turbulence are numerically analysed. The working fluid is water. We focus on the temperature profiles, the centre temperature, the Nusselt number and on the analysis of the velocity field. Several velocity amplitudes (or Reynolds numbers) and several kinetic profiles are introduced and studied; these together describe the various features of the rather complex flow organization. The results are presented both as functions of the Rayleigh number Ra (with Ra up to 108) for fixed temperature difference Δ between top and bottom plates and as functions of Δ (‘non-Oberbeck–Boussinesqness’) for fixed Ra with Δ up to 60K. All results are consistent with the available experimental NOB data for the centre temperature Tc and the Nusselt number ratio NuNOB/NuOB (the label OB meaning that the Oberbeck–Boussinesq conditions are valid). For the temperature profiles we find – due to plume emission from the boundary layers – increasing deviations from the extended Prandtl–Blasius boundary layer theory presented in Ahlers et al. (J. Fluid Mech., vol. 569, 2006, p. 409) with increasing Ra, while the centre temperature itself is surprisingly well predicted by that theory. For given non-Oberbeck–Boussinesqness Δ, both the centre temperature Tc and the Nusselt number ratio NuNOB/NuOB only weakly depend on Ra in the Ra range considered here.

Beyond Ra ≈ 106 the flow consists of a large diagonal centre convection roll and two smaller rolls in the upper and lower corners, respectively (‘corner flows’). Also in the NOB case the centre convection roll is still characterized by only one velocity scale. In contrast, the top and bottom corner flows are then of different strengths, the bottom one being a factor 1.3 faster (for Δ = 40K) than the top one, due to the lower viscosity in the hotter bottom boundary layer. Under NOB conditions the enhanced lower corner flow as well as the enhanced centre roll lead to an enhancement of the volume averaged energy based Reynolds number of about 4% to 5% for Δ = 60K. Moreover, we find , with β the thermal expansion coefficient and Tm the arithmetic mean temperature between top and bottom plate temperatures. This corresponds to the ratio of the free fall velocities at the respective temperatures. By artificially switching off the temperature dependence of β in the numerics, the NOB modifications of ReE is less than 1% even at Δ = 60K, revealing the temperature dependence of the thermal expansion coefficient as the main origin of the NOB effects on the global Reynolds number in water.

Damping mechanisms, damping rates and the dissipative modal structure of internal waves in stratified rotating circular basins are investigated analytically. The damping is shown to be due to a combination of the internal-wave cancelling, where waves emitted by the oscillatory boundary layers destructively interact with the parent wave and drain energy from it, and spin-down modified by the periodicity, where the energy is drained by the sinks and sources at the bottom corner caused by a discontinuity in the Ekman transport. It is shown that super-inertial Poincaré waves and sub-inertial Kelvin waves are damped predominantly by the internal-wave cancelling and modified spin-down, respectively. These processes also modify the internal-wave structure; for super-inertial waves, the boundary-layer-generated waves intensify the interior flow in the lower part of the water column and delay the phase relative to the isopycnal displacements, but for sub-inertial waves, the Ekman pumping and the corner sinks and sources add a horizontal circular flow that slants the crest and trough backwards near the wall.

A possible fundamental physical mechanism by which instability modes generate sound waves in subsonic jets is presented in the present paper. It involves a wavepacket of a pair of helical instability modes with nearly the same frequencies but opposite azimuthal wavenumbers. As the wavepacket undergoes simultaneous spatial–temporal development in a circular jet, the mutual interaction between the helical modes generates a strong three-dimensional, slowly modulating ‘mean-flow distortion’. It is demonstrated that this ‘mean field’ radiates sound waves to the far field. The emitted sound is of very low frequency, with characteristic time and length scales being comparable with those of the envelope of the wavepacket, which acts as a non-compact source. A matched-asymptotic-expansion approach is used to determine, in a self-consistent manner, the acoustic field in terms of the envelope of the wavepacket and a transfer factor characterizing the refraction effect of the background base flow. For realistic jet spreading rates, the nonlinear development of the wavepacket is found to be influenced simultaneously by non-parallelism and non-equilibrium effects, and so a composite modulation equation including both effects is constructed in order to describe the entire growth–attenuation–decay cycle. Parametric studies pertaining to relevant experimental conditions indicate that the acoustic field is characterized by a single-lobed directivity pattern beamed at an angle about 45°–60° to the jet axis and a broadband spectrum centred at a Strouhal number St ≈ 0.07–0.2. As the nonlinear effect increases, the radiation becomes more efficient and the noise spectrum broadens, but the gross features of the acoustic field remain robust, and are broadly in agreement with experimental observations.

The asymptotic behaviour of large-scale velocity statistics in an homogeneous turbulent shear flow is investigated using direct numerical simulations (DNS) of the incompressible Navier–Stokes equations on a 5123 grid, and with viscous rapid distortion theory (RDT). We use a novel pseudo-spectral algorithm that allows us to set the initial value of the shear parameter in the range 3–30 without the shortcomings of previous numerical approaches. We find there is an explicit dependence of the early-time behaviour on the initial value of the shear parameter. Moreover, the long-time asymptotes of large-scale quantities such as the ratio of the turbulent kinetic energy production rate over dissipation rate, the Reynolds stress anisotropic tensor and the shear parameter itself depend sensitively on the initial value of the shear parameter over the range of Reynolds number we could achieve (26 ≤ Rλ ≤ 63) with the stringent resolution requirements that were satisfied. To gain further insight into the matter, we analyse the full viscous RDT. While inviscid RDT has received a great deal of attention, viscous RDT has not been fully analysed. Our motivation for considering viscous RDT is so that the energy dissipation rate enters the problem, enabling the shear parameter to be defined. We show asymptotic expansions for the short-time behaviour and numerically evaluate the integrals to determine the long-time prediction of viscous RDT. The results are in quantitative agreement with DNS for short times; however, at long times viscous RDT predicts the turbulent energy decays to zero. Through an analysis of the pressure–strain terms, we show that the nonlinear ‘slow’ terms are essential for rearranging turbulent energy from the streamwise direction to the mean shear direction, and this sustains the indefinite growth of the kinetic energy at long times. In effect, the nonlinear pressure–strain correlation maintains the three-dimensionality of the turbulence, countering the tendency of the mean shear to project the turbulence onto the two-dimensional plane of the mean-flow streamlines. We postulate that the predictions of viscous RDT at long times could be improved by introducing a model for the ‘slow’ pressure–strain term, along the lines of the Rotta model.

The behaviour of spherical Brownian particles in a near-wall shear flow is explored using Langevin simulations and experimental measurements, focusing on the effects of anisotropic hindered particle mobility and the formation of a particle depletion layer due to repulsive forces. The results are discussed in the context of particle velocity distributions obtained by near-wall image-based velocimetry. It is observed that the shear force and dispersion dominate at high Péclet number (Pe > 3), and the asymmetric shapes of particle velocity distributions are attributed to broken symmetry due to the presence of the wall. Furthermore, the excursions outside the observation depth between image acquisitions and the shear-induced slowdowns of tracer particles cause significant measurement bias for long and short inter-frame time intervals, respectively. Also impeding the measurement accuracy is the existence of a near-wall particle depletion layer that leads to an overestimation of the fluid velocity. An analytical protocol to infer the correct fluid velocity from biased measurements is presented.

Linear refraction of waves on inhomogeneous current is known to provoke extreme waves. We investigate the effect of nonlinearity on this phenomenon, with respect to the variation of significant wave height, kurtosis and occurrence of freak waves. Monte Carlo simulations are performed employing a modified nonlinear Schrödinger equation that includes the effects of a prescribed non-potential current. We recommend that freak waves should be defined by a local criterion according to the wave distribution at each location of constant current, not by a global criterion that is either averaged over, or insensitive to, inhomogeneities of the current. Nonlinearity can reduce the modulation of significant wave height. Depending on the configuration of current and waves, the kurtosis and probability of freak waves can either grow or decrease when the wave height increases due to linear refraction. At the centre of an opposing current jet where waves are known to become large, we find that freak waves should be more rare than in the open ocean away from currents. The largest amount of freak waves on an opposing current jet is found at the jet sides where the significant wave height is small.

The flow of a viscous incompressible fluid in a circular tube generated by a sudden impulse 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. At short time the flow is irrotational and may be described by a potential which varies with the square root of time. At later times there is a sequence of moving and decaying vortex rings. At long times the flow velocity decays with an algebraic long-time tail. The impulse generates a time-dependent pressure difference between the ends of the tube.

The problem of unsteady behaviour of a floating thin plate is solved. The simultaneous motion of the plate and the fluid is considered within the framework of linear shallow-water theory. It is assumed that the bottom is not uniform in depth under the heterogeneous plate represented by an infinitely extended strip of finite width. The elastic deflection of the plate is expressed by a superposition of modal functions of a homogeneous beam with free edge conditions. The time-dependent unknown amplitudes are determined from the solution of a linear set of ordinary differential equations with constant coefficients. The eigenvalues of this set are determined numerically. Proposed method is used for the solution of three unsteady problems: the scattering of localized surface wave by an elastic plate, decay of the initial deformation of the plate in the fluid at rest and the action of a periodic load on a plate. Numerical calculations are performed for the ice sheet with the variable thickness and various bottom topographies.

A uniformly valid balanced model that represents the quasi-geostrophic model's counterpart in the equatorial region is derived. The quasi-geostrophic model itself fails in the equatorial region because it is only valid where the dominant balance is geostrophic, i.e. where the Rossby number is small. The smallness of the Rossby number is assumed in the quasi-geostrophic model's standard derivation and therefore this derivation cannot be repeated for the equatorial region. An alternative derivation of the quasi-geostrophic model that is independent of the Rossby number was presented by Leith in 1980, using the geometric framework of nonlinear normal mode initialization. Its independence of the Rossby number allows it to be repeated for the equatorial region, leading to an equatorial balanced model that thus represents the equatorial counterpart of the quasi-geostrophic model. As such it also coincides with the quasi-geostrophic model sufficiently far away from the equator. Its dispersion relation can be expressed in an explicit analytic form and, compared to that of other balanced models of similar simplicity, approximates that of the shallow water equations strikingly well.

Turbulent combustion processes are inherently unsteady and, thus, a source of acoustic radiation. While prior studies have extensively characterized their total sound power, their spectral characteristics are not well understood. This work investigates these acoustic spectral features, including the flame's low- and high-frequency characteristics and the scaling of the frequency of peak acoustic emissions. The spatiotemporal characteristics of the flame's chemiluminescence emissions, used as a marker of heat release fluctuations, were measured and used to determine the heat release spectrum, spatial distribution and spatial coherence characteristics. These heat release characteristics were then used as inputs to an integral solution of the wave equation and compared to measured acoustic spectra obtained over a range of conditions and burners and at several spatial locations. The spectral characteristics of the flame's acoustic emissions are controlled by two processes: the underlying spectrum of heat release fluctuations that are ultimately the combustion noise source, and the transfer function relating these heat release and acoustic fluctuations. An important result from this work is the clarification of the relative roles of these two processes in controlling the shape of the acoustic spectrum. This transfer function is primarily controlled by the spatiotemporal coherence characteristics of the heat release fluctuations which are, in turn, strongly influenced by burner configuration/geometry and operating conditions. Low-frequency acoustic emissions are controlled by the heat release spectrum essentially independent of flame geometry. Both the heat release spectrum and heat release-acoustic transfer function are important at intermediate and high frequencies. An important feature of the investigated geometry that controls the heat release phase dynamics is the high-velocity flow relative to the flame speed and anchored character of the flame. This leads to convection of flame sheet disturbances (i.e. heat release fluctuations) along the front that dominates the high frequency and peak frequency scaling of the flame's acoustic emissions.

We compare the turbulence statistics of the canopy/roughness sublayer (RSL) and the inertial sublayer (ISL) above. In the RSL the turbulence is more coherent and more efficient at transporting momentum and scalars and in most ways resembles a turbulent mixing layer rather than a boundary layer. To understand these differences we analyse a large-eddy simulation of the flow above and within a vegetation canopy. The three-dimensional velocity and scalar structure of a characteristic eddy is educed by compositing, using local maxima of static pressure at the canopy top as a trigger. The characteristic eddy consists of an upstream head-down sweep-generating hairpin vortex superimposed on a downstream head-up ejection-generating hairpin. The conjunction of the sweep and ejection produces the pressure maximum between the hairpins, and this is also the location of a coherent scalar microfront. This eddy structure matches that observed in simulations of homogeneous-shear flows and channel flows by several workers and also fits with earlier field and wind-tunnel measurements in canopy flows. It is significantly different from the eddy structure educed over smooth walls by conditional sampling based only on ejections as a trigger. The characteristic eddy was also reconstructed by empirical orthogonal function (EOF) analysis, when only the dominant, sweep-generating head-down hairpin was recovered, prompting a re-evaluation of earlier results based on EOF analysis of wind-tunnel data. A phenomenological model is proposed to explain both the structure of the characteristic eddy and the key differences between turbulence in the canopy/RSL and the ISL above. This model suggests a new scaling length that can be used to collapse turbulence moments over vegetation canopies.

Navier–Stokes turbulence subject to solid-body rotation is studied by high-resolution direct numerical simulations (DNS) of freely decaying and stationary flows. Setups characterized by different Rossby numbers are considered. In agreement with experimental results strong rotation is found to lead to anisotropy of the direct nonlinear energy flux, which is attenuated primarily in the direction of the rotation axis. In decaying turbulence the evolution of kinetic energy follows an approximate power law with a distinct dependence of the decay exponent on the rotation frequency. A simple phenomenological relation between exponent and rotation rate reproduces this observation. Stationary turbulence driven by large-scale forcing exhibits k⊥−2-scaling in the rotation-dominated inertial range of the one-dimensional energy spectrum taken perpendicularly to the rotation axis. The self-similar scaling is shown to be the cumulative result of individual spectral contributions which, for low rotation rate, display k⊥−3-scaling near the k∥ = 0 plane. A phenomenology which incorporates the modification of the energy cascade by rotation is proposed. In the observed regime the nonlinear turbulent interactions are strongly influenced by rotation but not suppressed. Longitudinal two-point velocity structure functions taken perpendicularly to the axis of rotation indicate weak intermittency of the k∥ = 0 (2D) component of the flow while the intermittent scaling of k∥ ≠ 0 (3D) fluctuations is well captured by a modified She–Lévêque intermittency model which yields the expression ζp = p/6 + 2(1 − (2/3)p/2) for the structure function scaling exponents.

A numerical simulation is performed to study the velocity, streamlines, vorticity and shear stress distributions in viscous water waves with different wave steepness in intermediate and deep water depth when the average wind velocity is zero. The numerical results present evidence of ‘clockwise’ and ‘anticlockwise’ rotation of the fluid at the trough and crest of the water waves. These results show thicker vorticity layers near the surface of water wave than that predicted by the theories of inviscid rotational flow and the low Reynolds number viscous flow. Moreover, the magnitude of vorticity near the free surface is much larger than that predicted by these theories. The analysis of the shear stress under water waves show a thick shear layer near the water surface where large shear stress exists. Negative and positive shear stresses are observed near the surface below the crest and trough of the waves, while the maximum positive shear stress is inside the water and below the crest of the water wave. Comparison of wave energy decay rate in intermediate depth and deep water waves with laboratory and theoretical results are also presented.

In the published article Monaghan et al. (2009) one of the authors' names was misspelled and should have read Catherine Mériaux.

The Press apologises to Dr Mériaux, her coauthors and the readers for this error.