Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the forced oscillations and the initial impulsive motion of a two-dimensional stationary disturbance. The measurements of the wave configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth. The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects. A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions. A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution.

In this paper, we present a detailed investigation of the size, scale and dynamics of macro-turbulent flow structures in gravel-bed rivers. We used an array of seven electromagnetic current meters with high resolution in both space and time to measure the streamwise velocity fluctuations in a gravel-bed river. The array was deployed successively in various configurations in order to quantify the vertical, lateral and longitudinal extent of the flow structures and to estimate their advecting velocities. To depict the spatial and temporal properties of the flow structures, we used space–time velocity matrices, space–time correlation analysis and coherent-structure detection schemes. The results show that the large-scale turbulent flow structures in a gravel-bed river occupy the entire depth of the flow and that they are elongated and narrow. The length of the structures is 3 to 5 times the flow depth while the width is between 0.5 and 1 times flow depth. In spite of the high roughness of the bed, these values are similar to those reported in the literature for laboratory experiments on large-scale turbulent flow structures. The dynamics of the large-scale turbulent flow structures investigated using flow visualization highlight the interactions between the outer flow region and the near-bed region. Our evidence suggests that large-scale flow incursions trigger ejections in the near-bed region that can develop into megabursts that can reach the water surface.

Three types of homogeneous anisotropic turbulence were produced by the plane distortion, axisymmetric expansion and axisymmetric contraction of grid-generated turbulence, and their behaviour in returning to isotropy was experimentally studied using hot-wire anemometry. It was found that the turbulence trajectory after the plane distortion was highly nonlinear, and did not follow Rotta's linear model in returning to isotropy. The turbulence wanted to become axisymmetric even more than it wanted to return to isotropy. In order to show the rate of return to isotropy of homogeneous turbulence, a map of the ratio of the characteristic time scale for the decay of turbulent kinetic energy to that of the return to isotropy was constructed. This demonstrated that the rate of return to isotropy was much lower for turbulence with a greater third invariant of the anisotropy tensor. The invariant technique was then applied to the experimental results to develop a new turbulence model for the return-to-isotropy term in the Reynolds stress equation which satisfied the realizability conditions. The effect of the Reynolds number on the rate of return to isotropy was also investigated and the results incorporated in the proposed model.

The structure of turbulence and its role in the breaking wave dynamics within the surf zone have been investigated through laboratory experiments using several flow visualization techniques and a fibre-optic LDV system. The results indicate that there exists a characteristic structure of large-scale eddies referred to here as ‘horizontal eddies’ and ‘obliquely descending eddies’, which has a significant role in the generation of Reynolds stress and thus affects the deformation of the mean flow field. The experiments also reveal that these eddies caused by the wave breaking bring a large amount of vorticity (with non-zero average) into otherwise almost irrotational velocity fields, resulting in the generation of vorticity-related mean flow fields as well as turbulence (vorticity-containing velocity fluctuation). This means that the breaking waves in the surf zone can be regarded as pseudowaves which consist of irrotational velocity components as ‘wave motion’ and appreciable amounts of rotational mean velocity components as ‘eddying motion’ (with non-zero mean vorticity) together with turbulence. It is found that the generation of the mean rotational velocity component due to wave breaking causes considerable increase in mass and momentum transport, as compared with ordinary non-breaking waves, and thus a decrease in wave height.

The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.

The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.

In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

A general theory is presented to account for the small, free, lateral motions of a flexible, slender, cylindrical body immersed in fluid flowing parallel to the position of rest of its axis. The cylinder is either clamped or pinned at both ends, or clamped at the upstream end and free at the other; it lies in a horizontal plane wherein all motion is considered to be confined. It is shown that for sufficiently large flow velocities the cylinder may be subject to buckling and oscillatory instabilities in its first and higher flexural modes, respectively. It is shown that for cylinders with both ends supported the oscillatory instabilities are specifically caused by lateral frictional forces, and that in the absence of hydrodynamic-drag effects only buckling is possible. The same applies for cylinders supported at the upstream end and with a very long, gradually tapering free end. The critical conditions of stability, expressed in dimensionless form, are evaluated extensively for clamped-free and pinned-pinned cylinders, illustrating the effect of the various system parameters on stability.

A time-dependent extension of the reduced wave equation of Berkhoff is developed for the case of waves propagating over a bed consisting of ripples superimposed on an otherwise slowly varying mean depth which satisfies the mild-slope assumption. The ripples are assumed to have wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter along with the bottom slope. The theory is verified by showing that it reduces to the case of plane waves propagating over a patch of sinusoidal ripples, which vary in one direction and extend to ± ∞ in the transverse direction, studied recently by Davies & Heathershaw and Mei. We then formulate and use coupled parabolic equations to study propagation over patches of arbitrary form in order to study wave reflection.

The situation of a small-diameter liquid jet exposed to a large-diameter high-speed gas jet (gas-to-liquid nozzle area ratio of order 100 to 1000) is investigated experimentally. Flow visualization and particle-sizing techniques are employed to examine the initial jet breakup process and primary liquid atomization. Observations of the initial breakup of the liquid jet in the near-nozzle region, combined with droplet-size mea-surements, are used in an effort to elucidate the dominant mechanism of primary breakup of the liquid. It is shown that for large aerodynamic Weber numbers, the bulk of the liquid atomization is completed within a few gas-jet diameters of the nozzle exit, inside of the potential cone of the gas flow. Breakup is therefore completed within the zone of constant ambient gas velocity. It is argued that the mechanism of initial jet breakup is similar to that of a liquid drop suddenly exposed to a high-speed gas stream. A phenomenological breakup model is proposed for the initial droplet size, based upon the accelerative, secondary destabilization (via Rayleigh–Taylor instability) of the liquid wave crests resulting from the primary Kelvin–Helmholtz instability of the liquid jet surface. Primary mean droplet sizes are shown to scale well on the most unstable Rayleigh–Taylor wavelength, and the dependence of the droplet diameter on both the atomizing gas velocity and the liquid surface tension are successfully captured by the proposed breakup model.

A reciprocating oscillatory turbulent flow in a rectangular duct is investigated experimentally by making use of a laser-Doppler velocimeter, hot-wire anemometers as well as electronic digital sampling and processing equipments.

The profiles of the mean velocity, the turbulence intensities, the Reynolds stress and the turbulent-energy production rate are compared for the accelerating and decelerating phases.

The characteristics of such a flow are quite different from wall turbulence which is steady in the mean. In the accelerating phase, turbulence is triggered by the shear instability at a slight distance from the wall but is suppressed and cannot develop. However, with the beginning of flow deceleration, turbulence grows explosively and violently and is maintained by the bursting type of motion.

The turbulent-energy production becomes exceedingly high in the decelerating phase, but the turbulence is reduced to a very low level at the end of the decelerating phase and in the accelerating stage of reversal flow. Spectra and spatial correlations for the various phases are compared. The spectral decay in the high-frequency range for the decelerating phase with high turbulence is far steeper than that of Kolmogorov's −5/3 power law, indicating remarkably high energy dissipation by high-frequency turbulence.

Notwithstanding the great difference between the ensemble-averaged characteristics of the oscillatory flow and those of steady wall turbulence, its basic processes such as ejection, sweep and interactions directed towards and away from the wall are the same as those of ‘steady’ wall turbulence.

This paper reports on an extensive experimental study of the flows due to under-expanded axisymmetric jets impinging on flat plates. The range of plate locations extends to a point where the jet is just subsonic but the main emphasis is on the behaviour in the first shock cell. Plate inclinations from 90° to 30° were investigated by means of comprehensive surface pressure measurements and shadowgraph pictures. Wherever possible, the main features of the results have been reconstructed using inviscid analyses of the wave interactions.

The flows are shown to be extremely complex due to the local structure of the free jet and, particularly, due to interactions between shock waves in the free jet and those created by the plate. In the near field, these interactions tend to be the controlling factors but at larger distances from the nozzle, mixing effects become increasingly important.

The maximum pressure on the plate when it is inclined can be very much larger than when the plate is perpendicular, owing to the possibility of high pressure recoveries through multiple shock systems. Correlations are presented for some of the main features on perpendicular plates and it is shown that the integrated pressure loads for both normal and inclined plates can be predicted well by a simple momentum balance.

The terminal velocities, drag coefficients, and orientations of single cylinders falling in a large tank of viscous liquid have been determined for Reynolds numbers Re ranging from > 0·01 to 1000 to 1000. Stationary eddies appear in the wakes of cylinders at 26 > Re > 50, and the shedding of a Kármán vortex street at Re > 50 causes them to oscillate as they fall.

The interaction of two long, thin cylinders, which is largely determined by their initial relative positions, has been studied in some detail. One interesting result is that two equal cylinders, initially non-parallel and vertically separated by about 50 diameters, catch up and slide along each other until they bisect at right angles. When several cylinders are released in random orientations, they tend to cluster and then separate into pairs crossed at right angles and into trip-lets in the form of a symmetrical ++.

Cones with a flat base fall with apex upwards if the vertex angle θ < ¼π and with apex downwards if θ > ¼π. Double cones, cemented base to base, and with apex angles θ1 and θ2(θ2 > θ1), fall with acute apex upwards if $2\theta _1 + \theta_2 \textless \frac {3}{2} \pi$, the stable orientation being apex downwards if this condition is not satisfied. At Re ≃ 100, the shedding of a Kármán vortex street causes the cones to oscillate, while at Re > 800, the flow becomes highly turbulent and they tumble as they fall.

Stereoscopic particle image velocimetry (PIV) measurements are made in streamwise–spanwise and inclined cross-stream planes (inclined at $45^\circ$ and $135^\circ$ to the principal flow direction) of a turbulent boundary layer at moderate Reynolds number ($\hbox{\it Re}_\tau\,{\sim} 1100$). Two-point spatial velocity correlations computed using the PIV data reveal results that are consistent with an earlier study in which packets of hairpin vortices were identified by a feature-detection algorithm in the log region, but not in the outer wake region. Both streamwise–streamwise ($R_{\hbox{\scriptsize\it uu}}$) and streamwise–wall-normal ($R_{uw}$) correlations are significant for streamwise displacements of more than 1500 wall units. Zero crossing data for the streamwise fluctuating component $u$ reveal that streamwise strips between zero crossings of 1500 wall units or longer occur more frequently for negative $u$ than positive $u$, suggesting that long streamwise correlations in $R_{\hbox{\scriptsize\it uu}}$ are dominated by slower streamwise structures. Additional analysis of $R_{ww}$ correlations suggests that the long streamwise slow-moving regions contain discrete zones of strong upwash over extended streamwise distances, as might occur within packets of angled hairpin vortices. At a wall-normal location outside of the log region ($z/\delta \,{=}\, 0.5$), the correlations are shorter in the streamwise direction and broader in the spanwise direction. Correlations in the inclined cross-stream plane data reveal good agreement with the streamwise–spanwise plane. $R_{\hbox{\scriptsize\it uu}}$ in the $45^\circ$ plane is more elongated along the in-plane wall-normal direction than in the $135^\circ$ plane, which is consistent with the presence of hairpin packets with a low-speed region lifting away from the wall.

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

The detonation stability problem is studied by a normal mode approach which greatly simplifies the calculation of linear instability of detonation in contrast to the Laplace transform procedure used by Erpenbeck. The method of solution, for an arbitrary parameter set, is a shooting method which can be automated to generate easily the required information about instability. The condition on the perturbations applied at the end of the reaction zone is shown to be interpreted as either a boundedness condition or an acoustic radiation condition. Continuous and numerically exact neutral stability curves and boundaries are given as well as growth rates and eigenfunctions which are calculated for the first time. Our calculations include the Chapman–Jouguet (CJ) case which presents no special difficulty. We give representative results for our detonation model and summarize the one-dimensional stability behaviour in parameter space. Comparison with previous results for the neutral stability boundaries and approximations to the unstable discrete spectrum are given. Parametric studies of the unstable, discrete spectrum's dependence on the activation energy and the overdrive factor are given with the implications for interpreting the physical mechanism of instability observed in experiments. This first paper is restricted to the case of one-dimensional linear instability. Extensions to transverse disturbances will be treated in a sequel.

The turbulent stress tensor in large-eddy simulation is examined from a theoretical point of view. Realizability conditions for the components of this tensor are derived, which hold if and only if the filter function is positive. The spectral cut-off, one of the filters frequently used in large-eddy simulation, is not positive. Consequently, the turbulent stress tensor based on spectrally filtered fields does not satisfy the realizability conditions, which leads to negative values of the generalized turbulent kinetic energy k. Positive filters, e. g. Gaussian or top-hat, always give rise to a positive k. For this reason, subgrid models which require positive values for k should be used in conjunction with e. g. the Gaussian or top-hat filter rather than with the spectral cutoff filter. If the turbulent stress tensor satisfies the realizability conditions, it is natural to require that the subgrid model for this tensor also satisfies these conditions. With respect to this point of view several subgrid models are discussed. For eddy-viscosity models a lower bound for the generalized turbulent kinetic energy follows as a necessary condition. This result provides an inequality for the model constants appearing in a ‘Smagorinsky-type’ subgrid model for compressible flows.

It is shown that the surface wind drift in the ocean substantially reduces the maximum wave height ξx and wave orbital velocity that can be attained before breaking. If q is the magnitude of the surface drift at the point where the wave profile crosses the mean water level and c is the wave speed, then \[ \zeta_{\max} = \frac{c^2}{2g}\bigg(1-\frac{q}{c}\bigg)^2. \] Incipient breaking in a steady wave train is characterized by the occurrence of stagnation points at wave crests, but not necessarily by discontinuities in slope. After breaking, there is in the mean flow a stagnation point relative to the wave profile near the crest of the broken wave, on one side of which the water tumbles forward and behind which it recedes more smoothly to the rear. Some simple flow visualization studies indicate the general extent of the wake behind the breaking region.

The capillary instability of vertical liquid jets of different viscosities have been examined by imposing audio-frequency disturbances. Real time sequences of photographs allow a direct measurement of growth rates of disturbances of various wavelengths. Results show that in general non-linear effects dominate the growth processes. This is in agreement with Yuen's analysis. The growth rate of the difference between the neck and the swell, however, agrees well with the linearized analysis of Rayleigh and Chandrasekhar. The non-linear effect causes a liquid jet to disintegrate into drops with ligaments in between. The sizes of the ligaments decrease with increasing wave-number. The subsequent roll up of the ligament into droplet, the eventual coalescing of the droplet with the main drop and drop oscillation have also been studied.

In flow between concentric rotating circular cylinders, it was shown by Taylor (1923) that instability may occur in the form of toroidal vortices spaced regularly along the axis. When the vortex motion occurs additional torque is required to keep the cylinders in motion at given speeds. Stuart (1958) used an energy-balance method, in the case when the annular gap is small compared with the radius, to estimate the additional torque and the associated finite amplitude attained by the vortices. He included the effect of distortion of the mean motion, but ignored the generation of harmonics of the fundamental mode and the distortion of the velocity associated with the fundamental mode. It is now known that these are not valid mathematical approximations and a rigorous perturbation expansion is developed here to remedy the deficiency. The analysis is valid for any gap width and any angular speeds of the containing cylinders, but requires the amplification rate of the disturbance to be small.

Numerical results using a digital computer are obtained for the shape and amplitude of the vortices in three cases: (i) when the outer cylinder has twice the radius of the inner one and is kept at rest, (ii) when the gap is small and the cylinders rotate with nearly the same speeds, and (iii) when the gap is small and the outer cylinder is kept at rest. The equilibrium amplitude obtained in the last case is substantially the same as that found by Stuart.

The results for cases (i) and (iii) give close agreement with the experimental values obtained by Taylor (1936) and Donnelly (1958) for the torque required to keep the inner cylinder rotating with constant speed while the outer one is at rest, for a certain range of speeds. In the small-gap problem it is shown that the equilibrium amplitude is almost proportional to 1 − m, where m is the ratio of the angular speeds of the outer and inner cylinders.

The response of lean (ϕ ≤ 0.4) premixed hydrogen flames to maintained homogeneous isotropic turbulence is investigated using detailed numerical simulation in an idealised three-dimensional configuration over a range of Karlovitz numbers from 10 to 1562. In particular, a focus is placed on turbulence sufficiently intense that the flames can no longer be considered to be in the thin reaction burning regime. This transition to the so-called distributed burning regime is characterised through a number of diagnostics, and the relative roles of molecular and turbulent mixing processes are examined. The phenomenology and statistics of these flames are contrasted with a distributed thermonuclear flame from a related astrophysical study.