Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a ${k}^{- 3} \log k$ spectrum in the transient stage, before the Kolmogorov ${k}^{- 5/ 3} $ asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of transient decaying turbulence.

The Lagrangian drift in an O(ϵ) monochromatic wave field on a shear flow, whose characteristic velocity is O(ϵ) smaller than the phase velocity of the waves, is considered. It is found that although shear has only a minor influence on drift in deep-water waves, its influence becomes increasingly important as the depth decreases, to the point that it plays a significant role in shallow-water waves. Details of the shear flow likewise affect the drift. Because of this, two temporal cases common in coastal waters are studied, viz. stress-induced shear, as would arise were the boundary layer wind-driven, and a current-driven shear, as would arise from coastal currents. In the former, the magnitude of the drift (maximum minus minimum) in shallow-water waves is increased significantly above its counterpart, viz. the Stokes drift, in like waves in otherwise quiescent surroundings. In the latter, on the other hand, the magnitude decreases. However, while the drift at the free surface is always oriented in the direction of wave propagation in stress-driven shear, this is not always the case in current-driven shear, especially in long waves as the boundary layer grows to fill the layer. This latter finding is of particular interest vis-à-vis Langmuir circulations, which arise through an instability that requires differential drift and shear of the same sign. This means that while Langmuir circulations form near the surface and grow downwards (top down), perhaps to fill the layer, in stress-driven shear, their counterparts in current-driven flows grow from the sea floor upwards (bottom up) but can never fill the layer.

We derive boundary conditions at a rigid wall for a granular material comprising rough, inelastic particles. Our analysis is confined to the rapid flow, or granular gas, regime in which grains interact by impulsive collisions. We use the Chapman–Enskog expansion in the kinetic theory of dense gases, extended for inelastic and rough particles, to determine the relevant fluxes to the wall. As in previous studies, we assume that the particles are spheres, and that the wall is corrugated by hemispheres rigidly attached to it. Collisions between the particles and the wall hemispheres are characterized by coefficients of restitution and roughness. We derive boundary conditions for the two limiting cases of nearly smooth and nearly perfectly rough spheres, as a hydrodynamic description of granular gases comprising rough spheres is appropriate only in these limits. The results are illustrated by applying the equations of motion and boundary conditions to the problem of plane Couette flow.

Small-amplitude wave motion in an inert gas confined between a moving piston and a fixed cylinder endwall is studied using the unsteady Euler equations. The waves, generated by either initial disturbances or piston motion, reflect back and forth in the cylinder on the acoustic timescale. The accumulated effect of these waves controls the bulk variations of velocity and thermodynamic variables on the longer piston timescale. Perturbation methods, based on the small ratio of acoustic to piston time, are employed to formulate the gasdynamic problem. The application of multiple timescaling allows the gasdynamic wave field to be separated from the bulk response of the gas. The evolution of the wave phenomena, including nonlinear wave deformation and weak shock formation during the piston passage time, is described in terms of time-dependent Fourier series solutions, whose coefficients are computed from a truncated system of coupled nonlinear ordinary differential equations. The long-time asymptotic flow field after shock formation is sawtooth-like, in which case the Fourier modes become decoupled. A remarkably simple relation between the shock amplitude and piston velocity is discovered. It is demonstrated that (i) the wave amplitude and frequency strongly depend on the piston motion; (ii) shock waves can be damped in a significant way by internal dissipation; and (iii) the mathematical approach developed in this study possesses certain advantages over the more traditional method of characteristics.

Supercritical ship internal wave wakes with $Fr=U/c_{0}\sim 4{-}12$ (where $U$ is the ship speed and $c_{0}$ is the linear internal long-wave speed) are calculated by a strongly nonlinear two-layer model in three dimensions, accounting for the complex ship geometry, and compared with field measurements. The degree of nonlinearity, defined by the ship draught ($d_{0}$) to average depth of pycnocline ($h_{0}$) ratio, is explored in the range $d_{0}/h_{0}\sim 0.1{-}1.2$, comparing nonlinear and linear calculations. For $d_{0}/h_{0}\sim 1$, the wave amplitude far downstream is overpredicted by up to 50 % by the linear method. The nonlinear trough amplitudes decay algebraically in the lateral coordinate with decay exponents in the range 0.16–0.61. The nonlinear leading trough is systematically somewhat ahead of the classical asymptotic pattern, and its amplitude and forerunning slope are appreciable, while the linear counterparts are very small. The calculated and asymptotic patterns are close for large $Fr$. Field measurements in a Canadian fjord of the internal wave wake of a 221 m long cargo ship of dead weight 43 000 tonnes moving at $Fr=6.6$ document a series of waves of 1–2 m isopycnal displacements at an off-track distance of 700 m. The nonlinear computations of a similar ship predict a wave train of height 1.5 m at a similar position, matching the field observation, whereas linear calculations give a wave train of height 2.3 m. Two- and three-layer theoretical models predict speeds that match the observed speeds of the second and third wave troughs. The observed leading signature of the diverging wave wake is moving at supercritical speed. This may be explained by the position and speed of the nonlinear wave slope moving ahead of the leading trough. Nonlinear computations of the surface velocity and strain rate compare well with measurements in the Loch Linnhe experiment of Watson et al. (J. Geophys. Res., vol. 97 (C6), 1992, pp. 9689–9703). The calculated nonlinear wave resistance of the observed cargo ship is comparable to the frictional force for $Fr\sim 4{-}6$, exceeds the surface wave resistance and increases the total drag by 40 %. A linear force prediction is useless when $d_{0}/h_{0}\sim 1$. The results show that nonlinearity of the dead water wake depends on the relative ship volume as well as the relative ship draught, with the pycnocline depth as the relevant length scale.

Naturally developing and acoustically excited shear layers at the Reynolds numbers $Re_{\unicode[STIX]{x1D703}_{0}}=U\unicode[STIX]{x1D703}_{0}/\unicode[STIX]{x1D708}=85{-}945$ are studied using the hot-wire (HW) anemometry and particle image velocimetry (PIV), with a focus on the shear/rotation competition during the initial Kelvin–Helmholtz (KH) roll-up. Velocity spectra and the spatial linear stability (LST) analysis characterize the fundamental ($f_{n}$) and its subharmonic ($f_{n}/2$) mode interacting due to the vortex pairing. For $276\leqslant Re_{\unicode[STIX]{x1D703}_{0}}\leqslant 780$, the root-mean-square (r.m.s.) of the streamwise turbulence intensity shows a double-peaking phenomenon, i.e. major and minor peaks of the $u_{rms}$ coexist towards the high-speed (HS) and the low-speed (LS) sides, respectively. The single/double-peaked $u_{rms}$ profiles are found to be correlated with the scattered/organized distribution of the shear/rotation, demonstrating a transitioning character with the downstream distance, $Re_{\unicode[STIX]{x1D703}_{0}}$ and the upstream turbulence levels. The rotating vortex cores and the corresponding peripheral shear regions, demonstrate the phase reversal of the velocity fluctuations with respect to the HS and the LS sides. Excitation at $f_{n}$ increases the vortex count by 21 %, advances the location of the first KH roll-up and hence also the minor peak formation location. Due to the enhanced pairing at the $f_{n}/2$ forcing, the vortex count reduces by 23 %. Before merging into the downstream rotation core, the upstream vortex is shifted towards the HS side and the major peak is accordingly augmented. Actuation advances the transition to the nonlinear state, as well as the saturation of the amplification factor. The volumetric topologies of the shear/rotation loops tracked in consecutive phases during the period of the acoustic excitation, separate from the edge and grow in time–space due to the viscous diffusion. The shearing and rotating loops are found to be associated with the thinning (elongation) and expansion (accumulation) of the vorticity, respectively.

The model proposed by Smith & Stewartson (1973), to describe the separated boundary layer induced by strong injection over a finite length of a flat plate in a supersonic mainstream, is shown to provide the basis for a fully consistent solution of the Navier–Stokes equations for this problem, valid in the limit of infinite Reynolds number. The solution takes the form of asymptotic expansions in each of a large number of overlapping regions of the flow field, which are consistently matched across areas of common validity.

In the quest to understand the dynamics of distributed systems on time-dependent spatial domains, we study experimentally the response to domain deformations by Faraday wave patterns – standing waves formed on the free surface of a liquid layer due to its vertical vibration – chosen as a paradigm owing to their historical use in testing new theories and ideas. In our experimental set-up of a vibrating water container with controlled positions of lateral walls and liquid layer depth, the characteristics of the patterns are measured using the Fourier transform profilometry technique, which allows us to reconstruct an accurate time history of the pattern three-dimensional landscape and reveal how it reacts to the domain dynamics on various length and time scales. Analysis of Faraday waves on growing, shrinking and oscillating domains leads to a number of intriguing results. First, the observation of a transverse instability – namely, when a two-dimensional pattern experiences an instability in the direction orthogonal to the direction of the domain deformation – provides a new facet to the stability picture compared to the one-dimensional systems in which the longitudinal (Eckhaus) instability accounts for pattern transformation on time-varying domains. Second, the domain evolution rate is found to be a key factor dictating the patterns observed on the path between the initial and final domain aspect ratios. Its effects range from allowing the formation of complex sequences of patterns to impeding the appearance of any new pattern on the path. Third, the shrinkage–growth process turns out to be generally irreversible on a horizontally evolving domain, but becomes reversible in the case of a time-dependent liquid layer depth, i.e. when the dilution and convective effects of the domain flow are absent. These experimentally observed enigmatic effects of the domain size variations in time are complemented here with appropriate theoretical insights elucidating the dynamics of two-dimensional pattern evolution, which proves to be more intricate compared to one-dimensional systems.

We present results of numerical simulations of leading-edge acoustic receptivity for acoustic waves impinging on the leading edge of a finite-thickness flat plate. We use both compressible and incompressible flow solvers fitted with high-order high-accuracy numerical methods and independent methods of estimating the receptivity coefficient. The results show that the level of acoustic receptivity in the existing literature appears to be one order of magnitude too high. Our review of previous numerical simulations and experiments clearly identifies some contradictory trends. In the limit of an infinitely thin flat plate, our results are consistent with asymptotic theory and numerical simulations.

A nonlinear theory is developed for a recent wave energy converter design inspired by the mechanics of animal arteries. The device is a long and hollow rubber tube immersed beneath the sea surface. Excited by passing water waves outside, pressure waves are resonated inside the tube and advance toward the stern to spin a turbine for power production. To account for significant magnification, the classical linear theory of blood vessels is modified. Diffraction is neglected but nonlinearity and wall friction are included. The spatial evolution of harmonic amplitudes is shown to be governed by a dynamical system similar to that in nonlinear optics. The maximum available power flux is predicted as a function of the tube length and other structural parameters. It is hoped that the theory may assist further development of the novel device.

A theoretical study of the interaction term in the two-fluid model equations is presented. The relevant Navier–Stokes equation is volume-integrated in a control volume fixed in a field of dispersed two-phase flow; then it is time-integrated. An expression for the interaction term is obtained in the limit of infinitesimal control volume, which rigorously fits to the two-fluid model equation based on time averaging. The interaction term is then analysed for dispersed two-phase flow with homogeneous particle size. The mathematical expression of the resulting interaction term clearly shows its property, which has been overlooked for more than 40 years. It can be decomposed into the conventional interaction term and an additional ‘virtual force’ term. The virtual force term is evaluated approximately for two types of dispersed multiphase flow in order to demonstrate its effectiveness. The first is solid–liquid dispersed two-phase flow with spherical solid particles undergoing creeping flow, and the second is gas–liquid dispersed two-phase flow with large bubbles in a highly turbulent flow field. For solid spherical particles in a homogeneous creeping flow, the term vanishes, as was found numerically by Ten Cate and Sundaresan (Intl J. Multiphase Flow, vol. 32, 2006, pp. 106–131), although the term could be significant for a flow field with velocity gradient. For large bubbles in bubble columns in a recirculating turbulent flow regime, the term is significant. The two-fluid model equations are corrected by the introduction of these virtual force terms, which in some circumstances are important in simulating the macroscopic properties of dispersed two-phase flow with spatial variations.

A localized, insoluble, surfactant monolayer, spreading under the action of surface-tension gradients over a thin liquid film, has at its leading edge an integrable stress singularity which renders conventional thin-film approximations locally non-uniform. Here high-Reynolds-number asymptotics are used to explore the quasi-steady two-dimensional developing flow near the monolayer tip, assuming that gravity keeps the free surface almost flat, that weak ‘contaminant’ surfactant regularizes the singularity and that the monolayer spreads fast enough for inertial effects to be important in a region which is long compared to the film depth but which is short compared to the length of the monolayer. It is shown how downward displacement of the inviscid core flow by the subsurface viscous boundary layer yields a non-uniform pressure distribution which, when the monolayer is spreading fast enough for cross-stream pressure gradients to be significant at its tip, creates a short free-surface hump which is the thin-film version of a Reynolds ridge. The ridge and other singular flow structures are smoothed as the monolayer slows and levels of contaminant are increased. The conditions under which lubrication theory provides a uniformly accurate approximation for this class of surfactant-spreading flows are established.

The multi-point Monin–Obukhov similarity (MMO) was recently proposed (Tong & Nguyen, J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) to address the issue of incomplete similarity in the framework of the original Monin–Obukhov similarity theory (MOST). MMO hypothesizes the following: (1) The surface-layer turbulence, defined to consist of eddies that are entirely inside the surface layer, has complete similarity, which however can only be represented by multi-point statistics, requiring a horizontal characteristic length scale (absent in MOST). (2) The Obukhov length $L$ is also the characteristic horizontal length scale; therefore, all surface-layer multi-point statistics, non-dimensionalized using the surface-layer parameters, depend only on the height and separations between the points, non-dimensionalized using $L$. However, similar to MOST, MMO was also proposed as a hypothesis based on phenomenology. In this work we derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations, which are equivalent to the two-point horizontal differences of these variables, using the spectral forms of the Navier–Stokes and the potential temperature equations. We show that, for the large-scale motions (wavenumber $k<1/z$) in a convective surface layer, the solution is uniformly valid with respect to $z$ (i.e. as $z$ decreases from $z>-L$ to $z<-L$), where $z$ is the height from the surface. However, for $z<-L$ the solution is not uniformly valid with respective to $k$ as it increases from $k<-1/L$ to $k>-1/L$, resulting in a singular perturbation problem, which we analyse using the method of matched asymptotic expansions. We show that (1) $-L$ is the characteristic horizontal length scale, and (2) the Fourier transforms satisfy MMO with the non-dimensional wavenumber $-kL$ as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for $z\ll -L$ in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for finite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale ($L$) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.

In the paper we consider harbour oscillations excited by wind-generated gravity waves. The analysis is based on the fact that waves propagate along rays (wave orthogonals). In this way the elliptic boundary-value problem is turned into an initial-value problem along each ray. When a ray strikes the boundary (the harbour walls), reflected rays are produced in accordance with the law of reflexion. When a ray strikes an edge point of the boundary (e.g. the tip of a breakwater) diffracted rays are produced and emitted in all directions into the harbour. Algorithms for the tracing of incident, multiply reflected and singly diffracted rays as well as the computation of the field on each ray are presented. Attenuation mechanisms (e.g. partial reflexion), which limit the number of rays needed to compute the field, are included. Numerical examples for a rectangular and an actual harbour are given. A comparison between the results obtained by ray methods and finite difference methods is included.

The fluid is incompressible, inviscid and non-diffusive. It has a uniform Brunt–Väisälä frequency, N, and is of constant depth, D. A body or wing moves horizontally through the fluid at velocity U in a straight line, exerting a vertical force during a given time interval. The force is constant, or oscillatory with frequency σ. The vertical average of the strain rate in a thin surface layer is calculated for a network of points behind the body.

The linearized analysis is first applied with tank walls, then modified for remote walls and a vertical force of long duration.

For moderately high velocity and forcing frequency (U/ND = 5, σ/N≅ 4−16) the recurring internal wave pattern just behind the body is well established in one cycle of the oscillatory force. A tank width one or two times the depth gives good agreement between tank and no-wall calculations for the chosen examples.

For a stationary wing (U/ND = 0) in a cubic tank with forcing frequency one-half the natural frequency (σ/N = ½) the strain rates after one cycle are 103 times greater than for the moving wing case. After five cycles the magnitudes are twenty times larger than after one cycle. Presumably these large increases are due to the continuous and efficient feeding of energy into a small fluid volume which occurs for the stationary wing. No-wall calculations for many cycles give amplitudes roughly one-half those for five cycles in the tank, showing the effect of escaping energy.

The relation of these developments to stationary phase analysis and preferred directions is discussed.