We consider the evolution of an isolated elliptical vortex in a weakly dissipative fluid. It is shown computationally that a spatially smooth vortex relaxes inviscidly towards axisymmetry on a circulation timescale as the result of filament generation. Heuristically, we derive a simple geometrical formula relating the rate of change of the aspect ratio of a particular vorticity contour to its orientation relative to the streamlines (where the orientation is defined through second-order moments). Computational evidence obtained with diagnostic algorithms validates the formula. By considering streamlines in a corotating frame and applying the new formula, we obtain a detailed kinematic understanding of the vortex's decay to its final state through a primary and a secondary breaking. The circulation transported into the filaments although a small fraction of the total, breaks the symmetry and is the chief cause of axisymmetrization.

A large-scale circulation velocity, often called the ‘wind’, has been observed in turbulent convection in the Rayleigh–Bénard apparatus, which is a closed box with a heated bottom wall. The wind survives even when the dynamical parameter, namely the Rayleigh number, is very large. Over a wide range of time scales greater than its characteristic turnover time, the wind velocity exhibits occasional and irregular reversals without a change in magnitude. We study this feature experimentally in an apparatus of aspect ratio unity, in which the highest attainable Rayleigh number is about 1016. A possible physical explanation is attempted.

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.

Single drop impact onto liquid films is simulated numerically. Surface tension and gravity are taken into account, whereas viscosity and compressibility are neglected. This permits recourse to a boundary-integral method, based on an integral equation for a scalar velocity potential. Calculations are performed for normal impacts resulting in axisymmetric flows.

For times that are small compared to the characteristic time of impact 2R/w0 (R being the drop radius, w0 its initial velocity towards the liquid film), it is found that a disk-like jet forms at the neck between the drop and the pre-existing liquid film, if the impact Weber number is high enough. This jet can pinch off a torus-shaped liquid volume at its tip or reconnect with the pre-existing liquid film, thus entraining a torus- shaped bubble. In reality, both the torus-shaped bubble and liquid torus will decay according to Rayleigh's capillary instability, thus breaking the cylindrical symmetry. This mechanism of bubble entrainment differs from those described in literature.

For times that are comparable to or larger than the characteristic time of impact, capillary waves on the film, or the well-known crowns, are obtained again according to whether the impact Weber number is low or high enough.

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.

We apply a new analytic technique, namely the homotopy analysis method, to give an analytic approximation of temperature distributions for a laminar viscous flow over a semi-infinite plate. An explicit analytic solution of the temperature distributions is obtained in general cases and recurrence formulae of the corresponding constant coefficients are given. In the cases of constant plate temperature distribution and constant plate heat flux, the first-order derivative of the temperature on the plate at the 30th order of approximation is given. The convergence regions of these two formulae are greatly enlarged by the Padé technique. They agree well with numerical results in a very large region of Prandtl number 1[les ]Pr[les ]50 and therefore can be applied without interpolations.

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 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.

Conditionally sampled measurements with particle image velocimetry (PIV) of a turbulent round submerged liquid jet in a laboratory have been taken at Re = 2 × 103 between 60 and 100 nozzle diameters from the nozzle in order to investigate the dynamics and transport processes at the continuous and well-defined bounding interface between the turbulent and non-turbulent regions of flow. The jet carries a fluorescent dye measured with planar laser-induced fluorescence (LIF), and the surface discontinuity in the scalar concentration is identified as the fluctuating turbulent jet interface. Thence the mean outward ‘boundary entrainment’ velocity is derived and shown to be a constant fraction (about 0.07) of the the mean jet velocity on the centreline. Profiles of the conditional mean velocity, mean scalar and momentum flux show that at the interface there are clear discontinuities in the mean axial velocity and mean scalar and a tendency towards a singularity in mean vorticity. These actual or asymptotic discontinuities are consistent with the conditional mean momentum and scalar transport equations integrated across the interface. Measurements of the fluxes of turbulent kinetic energy and enstrophy are consistent with computations by Mathew & Basu (Phys. Fluids, vol. 14, 2002, pp. 2065–2072) in showing that for a jet flow (without forcing) the entrainment process is dominated by small-scale eddying at the highly sheared interface (‘nibbling’), with large-scale engulfing making a small (less than 10%) contribution consistent with concentration measurements showing that the interior of the jet is well mixed. (Turbulent jets differ greatly from the free shear layer in this respect.) To explain the difference between velocity and scalar profiles, their conditional mean gradients are defined in terms of a local eddy viscosity and eddy diffusivity and the momentum and scalar fluxes inside the interface. Since the eddy diffusivity is larger than the eddy viscosity, the scalar profile is flatter inside the interface so that the scalar discontinuity is relatively greater than the mean velocity discontinuity. Theoretical arguments, following Hunt, Eames & Westerweel (in Proc. of the IUTAM Symp. on Computational Physics and New Perspectives in Turbulence, ed. Y. Kaneda, vol. 4, 2008, pp. 331–338, Springer), are proposed for how the vortex sheet develops, how the internal structure of the interface layer relates to the inhomogeneous rotational and irrotational motions on each side and why the dominant entrainment process of jets and wakes differs from that of free shear layers.

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.

A formal expansion method for analysis of the non-linear development of an oblique wave in a parallel flow is presented. The present approach constitutes an extension and modification of the method of Stuart and Watson. Results are obtained for plane Poiseuille flow, and for a combination of plane Poiseuille and plane Couette flow. The Poiseuille flow exhibits finite-amplitude subcritical instability, and relatively weak but finite disturbances markedly reduce the critical Reynolds number. The combined flow, which becomes stable to infinitesimal disturbances at all Reynolds numbers when the Couette component is sufficiently great, remains unstable to finite disturbances.

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.

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.

An experiment has been performed to determine the effect of yaw upon transition in the boundary layer formed on the windward face of a long cylinder. The china-clay-evaporation and surface-oil-flow techniques have been used to study the development of the fixed-wavelength stationary disturbances which are characteristic of cross-flow instability. It has been found that the boundary layer is also susceptible to time-dependent disturbances which grow to very large amplitudes prior to the onset of transition. These disturbances have been studied with a hot-wire anemometer. The conditions necessary for the onset and completion of transition have been determined by the use of surface Pitot tubes. Data from the experiment have been compared with the simple criteria for instability and transition which were proposed by Owen & Randall over thirty years ago. In general it has been found that these criteria are inadequate, and, where possible, improvements have been proposed. The raw data are presented in sufficient detail for them to be used to test, or calibrate, future theoretical models of the transition process in three-dimensional boundary-layer flows.

An orthogonal coordinate system along a generic spatial curve has been introduced, and the Navier-Stokes equations for a steady incompressible viscous flow have been explicitly written in this frame of reference. As an application the flow in a helical pipe has been studied, and, formdii 5f curvature and torsion small compared with the radius of the pipe, the flow has been considered as a perturbed Poiseuille flow. The result is that for curvatures and torsions of the same order and for low Reynolds number the curvature induces on the flow a first-order effect on the parameter ε =κa, where κ is the curvature and a the radius of the pipe, while the effect of the torsion on the flow is of the second order in E. This last result disagrees with those of Wang (1981), who, adopting a non-orthogonal coordinate system, found a first-order effect of torsion on the flow.

Turbulent flow over idealized water waves with varying wave slope ak and wave age c/u∗ is investigated using direct numerical simulations at a bulk Reynolds number Re = 8000. In the present idealization, the shape of the water wave and the associated orbital velocities are prescribed and do not evolve dynamically under the action of the wind. The results show that the imposed waves significantly influence the mean flow, vertical momentum fluxes, velocity variances, pressure, and form stress (drag). Compared to a stationary wave, slow (fast) moving waves increase (decrease) the form stress. At small c/u∗, waves act similarly to increasing surface roughness zo resulting in mean vertical velocity profiles with shorter buffer and longer logarithmic regions. With increasing wave age, zo decreases so that the wavy lower surface is nearly as smooth as a flat lower boundary. Vertical profiles of turbulence statistics show that the wave effects depend on wave age and wave slope but are confined to a region kz < 1 (where k is the wavenumber of the surface undulation and z is the vertical coordinate). The turbulent momentum flux can be altered by as much as 40% by the waves. A region of closed streamlines (or cat's-eye pattern) centred about the critical layer height was found to be dynamically important at low to moderate values of c/u∗. The wave-correlated velocity and flux fields are strongly dependent on the variation of the critical layer height and to a lesser extent the surface orbital velocities. Above the critical layer zcr the positions of the maximum and minimum wave-correlated vertical velocity ww occur upwind and downwind of the peak in zcr, like a stationary surface. The wave-correlated flux uwww is positive (negative) above (below) the critical layer height.

Simulations have been performed with a fairly narrow band numerical gravity wave model (higher-order nls type) and a computational domain of dimensions $128\times 128$ typical wavelengths. The simulations are initiated with $\thicksim6\times10^{4}$ fourier modes corresponding to truncated jonswap spectra and different angular distributions giving both short- and long-crested waves. A development of the spectra on the so-called benjamin–feir timescale is seen, similar to the one reported by dysthe et al. (J. Fluid Mech. vol. 478, 2003, p.1). The probability distributions of surface elevation and crest height are found to fit theoretical distributions found by tayfun (J. Geophys. Res. Vol. 85, 1980, p. 1548) very well for elevations up to four standard deviations (for realistic angular spectral distributions). moreover, in this range of the distributions, the influence of the spectral evolution seems insignificant. for the extreme parts of the distributions a significant correlation with the spectral change can be seen for very long-crested waves. For this case we find that the density of large waves increases during spectral change, in agreement with a recent experimental study by onorato et al. (J. Fluid Mech. 2004 submitted).

A modified version of the mild-slope equation is derived and its predictions of wave scattering by two-dimensional topography compared with those of other equations and with experimental data. In particular, the modified mild-slope equation is shown to be capable of describing known scattering properties of singly and doubly periodic ripple beds, for which the mild-slope equation fails. The new equation compares favourably with other models of scattering which improve on the mild-slope equation, in that it is widely applicable and computationally cheap.