The lattice Boltzmann equation method in two dimensions was used to analyse natural convective flows. The method was validated with experiments in an open cavity with one of the vertical walls divided into two parts, the lower part conductive, the upper part and all the other walls adiabatic. An upward thermal boundary layer formed near the conductive wall. This layer gave way to a wall plume. The numerical results compared well with experiments in the laminar ($Ra\,{=}\,2.0\,{\times}\,10^9$) and transition ($Ra\,{=}\,4.9\,{\times}\,10^9$) regimes. The behaviour of the starting plume was numerically studied for Rayleigh numbers Ra from $10^6$ to $4.9\times 10^9$. The wall plume grows in three stages: in the first with constant acceleration, in the second with constant ascending velocity and in the third with negative acceleration due to the presence of the top boundary layer. The acceleration of the first stage and the velocity of the second both scale with the Rayleigh number.

The effect of Coriolis force on the onset of thermal convection in a shallow layer of viscous liquid is studied in the limit of rapid rotation of the layer. Existing results which apply when the fluid lies between free boundaries are extended to the case where the boundaries are rigid.

The structure of normal shock waves in a gas composed of rigid sphere molecules is investigated by numerical experiments with a simulated gas on a digital computer. The non-equilibrium between the temperatures based on the longitudinal and lateral velocity components is studied and the results compared with the theory of Yen (1966). Details of the velocity distribution function are presented for a shock of Mach number 10. The distribution functions for both the longitudinal and lateral velocity components are plotted for a number of locations in the shock profile and are compared with the equilibrium distribution.

The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A high-order analytic perturbation solution, through terms of order Re7, is obtained for low Reynolds numbers. For larger Reynolds numbers an approximate Legendre polynomial series representation is used to reduce the governing system of equations to a non-linear ordinary differential equation boundary-value problem which is solved numerically. The resulting flow pattern and the torque required to rotate the spheres are presented for various cases considered.

Fully three-dimensional computations of flow through a long pipe demand a huge number of degrees of freedom, making it very expensive to explore parameter space and difficult to isolate the structure of the underlying dynamics. We therefore introduce a ‘2+ε-dimensional’ model of pipe flow, which is a minimal three-dimensionalization of the axisymmetric case: only sinusoidal variation in azimuth plus azimuthal shifts are retained; yet the same dynamics familiar from experiments are found. In particular the model retains the subcritical dynamics of fully resolved pipe flow, capturing realistic localized ‘puff-like’ structures which can decay abruptly after long times, as well as global ‘slug’ turbulence. Relaminarization statistics of puffs reproduce the memoryless feature of pipe flow and indicate the existence of a Reynolds number about which lifetimes diverge rapidly, provided that the pipe is sufficiently long. Exponential divergence of the lifetime is prevalent in shorter periodic domains. In a short pipe, exact travelling-wave solutions are found near flow trajectories on the boundary between laminar and turbulent flow. In a long pipe, the attracting state on the laminar–turbulent boundary is a localized structure which resembles a smoothened puff. This ‘edge’ state remains localized even for Reynolds numbers at which the turbulent state is global.

This study continues the work of Forbes (1986) on periodic waves beneath an elastic sheet floating on the surface of an infinitely deep fluid. The solution is sought as a Fourier series with coefficients that are computed numerically. Waves of extremely large amplitude are found to exist, and results are presented for waves belonging to several different nonlinear solution branches, characterized by different numbers of inflexion points in the wave profiles. The existence of multiple solutions, conjectured in the previous paper (Forbes 1986), is confirmed here by direct numerical computation.

A method of time-dependent conformal mapping is introduced to simplify the power-series solution procedure for time- and space-periodic standing waves in deep water. A solution has been found to 25th order in the wave amplitude. The values of certain coefficients are determined by the requirement that secular terms must be suppressed. Because the series for the wave profile is not always uniformly convergent, Padé approximants are used for summation. For very high waves, the slope of the surface has at least two relative maxima. The singularity structure of the solution is also discussed.

Streamwise oscillations of a circular cylinder in a steady uniform flow are investigated experimentally using a technique of high-image-density particle image velocimetry, in conjunction with instantaneous force measurements. This approach allows insight into the relationship between the loading and the patterns of vorticity and streamline topology in the near wake.

In analogy with the classical locked-on state arising from transverse oscillations of a cylinder in uniform flow, it is possible to attain either locked-on or quasi-locked-on states due to streamwise oscillations. In these cases, however, the repetitive signature of the transverse force is not sinusoidal; rather, it is strongly modulated and the corresponding spectra can exhibit several sharply defined peaks. The predominant peak can vary over a remarkably wide range, extending from the subharmonic to the third harmonic of the cylinder oscillation frequency; for certain locked-on states of the transverse force signature, the spectral peak at the cylinder oscillation frequency is actually suppressed. Corresponding instantaneous traces and spectra of the in-line force simply show dominance of the spectral peak at the cylinder oscillation frequency. Further interpretation of the loading is provided in terms of Lissajous patterns of the transverse and in-line force coefficients.

All of these features are related to the instantaneous patterns of vortex formation in the near wake. During a typical cycle of the cylinder oscillation, these patterns can be divided into two broad categories: Kármán-like shedding; and a nearly ‘frozen’ array of shed vortices. The order of occurrence of these basic patterns during an oscillation cycle dictates the instantaneous signatures and time-averaged spectra of the transverse force.

We analyse the wind and boundary layer properties of turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio one for Prandtl number and Rayleigh numbers () up to by means of highly resolved direct numerical simulations. We identify time periods in which the orientation of the large-scale circulation (LSC) is nearly constant in order to perform a statistical analysis of the LSC. The analysis is then reduced to two dimensions by considering only the plane of the LSC. Within this plane the LSC is treated as a wind with thermal and viscous boundary layers developing close to the horizontal plates. Special focus is on the spatial development of the wind magnitude and the boundary layer thicknesses along the bottom plate. A method for the local analysis of the instantaneous boundary layer thicknesses is introduced which shows a dramatically changing wind magnitude along the wind path. Furthermore a linear increase of the viscous and thermal boundary layer thickness along the wind direction is observed for all considered while their ratio is spatially constant but depends weakly on . A possible explanation is a strong spatial variation of the wind magnitude and fluctuations in the boundary layer region.

Wave-power absorption by an oscillating water column in a channel is studied within linear theory, and for different boundary conditions regarding the regions between the absorber and the channel walls. Particular attention is given to the effect of placing projecting sidewalls in front of the column.

Several interesting features of the flow field in free-molecule flow through an orifice are discussed. An estimate is then made of the deviation of the mass flow $\dot{m}$ through the orifice from its limiting free-molecule value $\dot{m}$ for small departures from the limit. Using an iteration method proposed by Willis, it is shown that this deviation is of the first order in ε, the inverse Knudsen number, defined as the ratio of the radius of the hole to the mean free path in the gas at upstream infinity. An estimate of the coefficient is obtained making some reasonable assumptions about the three-dimensional nature of the flow, and the value so derived, giving $\dot{m}=\dot{m}(1+0.25\epsi)$, shows fair agreement with the measurements of Liepmann. It appears that ‘nearly’ free-molecular conditions prevail up to ε ∼ 1.0.

We study the dynamics of neutrally buoyant particles with diameters varying in the range [1, 45] in Kolmogorov scale units (η) and Reynolds numbers based on Taylor scale (Reλ) between 590 and 1050. One component of the particle velocity is measured using an extended laser Doppler velocimetry at the centre of a von Kármán flow, and acceleration is derived by differentiation. We find that the particle acceleration variance decreases with increasing diameter with scaling close to (D/η)−2/3, in agreement with previous observations, and with a hint for an intermittent correction as suggested by arguments based on scaling of pressure spatial increments. The characteristic time of acceleration autocorrelation increases more strongly than previously reported in other experiments, and possibly varying linearly with D/η. Further analysis shows that the probability density functions of the acceleration have smaller wings for larger particles; their flatness decreases as well, as expected from the behaviour of pressure increments in turbulence when intermittency corrections are taken into account. We contrast our measurements with previous observations in wind-tunnel turbulent flows and numerical simulations.

Measurements of turbulent flow through a constricted tube with a contoured 75% constriction have been performed with a laser-Doppler anemometer. One motivation for these studies was the problem of flow through arterial stenoses and consequently a range of relatively low Reynolds numbers (5000–15000) was employed. Velocity profiles, r.m.s. turbulence velocities, and energy spectra were recorded along with determinations of the wall pressure variation and length of recirculation region. Results show extremely high levels of turbulence and considerable turmoil within the so-called recirculation region. A method for determining the reattachment point in an intensely turbulent flow is given which should offer improvement over flow visualization techniques. Similarity in turbulence energy spectra measured at various radial positions is found at several axial locations, but does not occur in the region immediately downstream of separation. The downstream recovery of the turbulence velocity to upstream values takes place slowly. A secondary motivation of providing experimental data useful in evaluating turbulence models for computational fluid dynamics led to extensive measurements of radial and axial r.m.s. turbulence velocity components at a Reynolds number of 15000.

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

In this paper, we report experimental results concerning a three-dimensional short-wave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le Dizés & Laporte (2002) concerning elliptic instability of co-rotating vortices show very good agreement.

At later stages of the flow, when the perturbation amplitude becomes sufficiently large, the two vortices merge into a single structure. This happens for smaller cores sizes than in the case of two-dimensional merging. The three-dimensional merging leads to a final vortex characterized by turbulent small-scale motion, whose size appears to be larger than it would have been without instability. The vorticity profile of the final vortex is non-Gaussian after both two-dimensional and three-dimensional merging. The profile contains more vorticity outside the inner core than a Gaussian vortex, resulting from the ejection of vorticity filaments during the merging stage.

The stability of a periodic internal wave has been investigated experimentally and theoretically. From the analysis it is found that if a primary wave, with wave-number k0 and frequency ω0, is perturbed by two infinitesimal wave-like disturbances with wave-numbers k1 and k1 + k0 and frequencies ω1 and ω1 + ω0, exponential growth of these disturbances will take place under certain conditions. The analysis also indicates which resonantly interacting disturbances can induce an instability and, when viscous dissipation is accounted for, predicts the minimum amplitude for which a wave is unstable. Experimental results demonstrate that this type of instability can cause the breakdown of a first mode internal wave propagating in a fluid composed of two layers of uniform density separated by a thin region in which the density varies continuously.

The history force is one of the hydrodynamic forces that act on a particle moving through a fluid. It is an integral over the full time history of the particle’s motion and significantly complicates the equations of motion (accordingly it is often neglected). We present here a study of the influence of this force on particles moving in a turbulent flow, for a wide range of particle parameters. It is shown that the magnitude of the history force can be significant and that it can have a considerable effect on the particles’ slip velocity, acceleration, preferential concentration and collision rate. We also investigate the parameter dependence of the strength of these effects.

Taylor's hypothesis, relating temporal to spatial fluctuations in turbulent flows is investigated using powerful numerical computations by del Álamo & Jiménez (J. Fluid Mech., 2009, this issue, vol. 640, pp. 5–26). Their results cast doubt on recent interpretations of bimodal spectra in relation to very large-scale turbulent structures in experimental measurements in turbulent shear flows.

By focusing a laser pulse inside a capillary partially filled with liquid, a vapour bubble is created that emits a pressure wave. This pressure wave travels through the liquid and creates a fast, focused axisymmetric microjet when it is reflected at the meniscus. We numerically investigate the formation of this microjet using axisymmetric boundary integral simulations, where we model the pressure wave as a pressure pulse applied on the bubble. We find a good agreement between the simulations and experimental results in terms of the time evolution of the jet and on all parameters that can be compared directly. We present a simple analytical model that accurately predicts the velocity of the jet after the pressure pulse and its maximum velocity.

The problem of waves propagating on the surface of a two-dimensional ideal fluid of infinite depth bounded above by an elastic sheet is studied with asymptotic and numerical methods. We use a nonlinear elastic model that has been used to describe the dynamics of ice sheets. Particular attention is paid to forced and unforced dynamics of waves having near-minimum phase speed. For the unforced problem, we find that wavepacket solitary waves bifurcate from nonlinear periodic waves of minimum speed. When the problem is forced by a moving load, we find that, for small-amplitude forcing, steady responses are possible at all subcritical speeds, but for larger loads there is a transcritical range of forcing speeds for which there are no steady solutions. In unsteady computations, we find that if the problem is forced at a speed in this range, very large unsteady responses are obtained, and that when the forcing is released, a solitary wave is generated. These solitary waves appear stable, and can coexist within a sea of small-amplitude waves.