Using the theory of homogenization we examine the correction to Darcy's law due to weak convective inertia of the pore fluid. General formulae are derived for all constitutive coefficients that can be calculated by numerical solution of certain canonical cell problems. For isotropic and homogeneous media the correction term is found to be cubic in the seepage velocity, hence remains small even for Reynolds numbers which are not very small. This implies that inertia, if it is weak, is of greater importance locally than globally. Existing empirical knowledge is qualitatively consistent with our conclusion since the linear law of Darcy is often accurate for moderate flow rates.

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

This paper examines the deformation of a solitary wave due to a slow variation of the bottom topography. Differential equations which determine the slow variation of the parameters of a solitary wave are derived by a certain averaging process applied to the exact in viscid equations. The equations for the parameters are solved when the bottom topography varies only in one direction, and when the wave evolves from a region of uniform depth. The variation of amplitude with depth is determined and compared with some recent experimental results.

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.

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.

The partial differential equations which describe steady flow of fluid in saturated homogeneous permeable solid material under non-isothermal conditions are stated. From these are derived the equations for flow of liquid (in particular, water) using suitable approximations and making use of empirical laws when necessary.

It is then postulated that the only ‘ponderomotive’ (i.e. massmoving) forces present are those due to thermal expansion effects. Free convection results. An approximate solution of the equations is attempted for plane flow by means of classical perturbation methods, the temperature and stream-function variables being represented by power series in a convection parameter proportional to the Rayleigh number.

A numberical example of the method, with boundary conditions based on a geothermal area at Wairakei, New Zealand, is given. The results show features which are in fair agreement with temperature measurements made in the area, and it appears that the convection parameter η is of the order of 10.

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.

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.

Computations of channel flow with rough walls comprising staggered arrays of cubes having various plan area densities are presented and discussed. The cube height h is 12.5% of the channel half-depth and Reynolds numbers (uτh/ν) are typically around 700 – well into the fully rough regime. A direct numerical simulation technique, using an immersed boundary method for the obstacles, was employed with typically 35 million cells. It is shown that the surface drag is predominantly form drag, which is greatest at an area coverage around 15%. The height variation of the axial pressure force across the obstacles weakens significantly as the area coverage decreases, but is always largest near the top of the obstacles. Mean flow velocity and pressure data allow precise determination of the zero-plane displacement (defined as the height at which the axial surface drag force acts) and this leads to noticeably better fits to the log-law region than can be obtained by using the zero-plane displacement merely as a fitting parameter. There are consequent implications for the value of von Kármán's constant. As the effective roughness of the surface increases, it is also shown that there are significant changes to the structure of the turbulence field around the bottom boundary of the inertial sublayer. In distinct contrast to two-dimensional roughness (longitudinal or transverse bars), increasing the area density of this three-dimensional roughness leads to a monotonic decrease in normalized vertical stress around the top of the roughness elements. Normalized turbulence stresses in the outer part of the flows are nonetheless very similar to those in smooth-wall flows.

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.

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.

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.

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.

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.

Using both the hydrogen-bubble flow-visualization technique and hot-wire-anemometry measurements, secondary vortices have been detected in the near-wake of circular cylinders and their characteristics studied over a Reynolds-number range of 1200–11000. The vortex-shedding characteristics of these secondary vortices clearly indicate that ‘transition waves’, detected in the cylinder near wake by Bloor (1964), and secondary vortices are identical phenomena. It is established that the non-dimensional shedding frequency of the secondary vortices demonstrates a 0.87 power-law relationship relative to Reynolds number, contrary to the 0.5 power-law reported by Bloor. The results suggest that the vortices result from a near-wake, free-shear instability which causes the separated cylinder boundary layer to roll up into the secondary vortices. Visual observations indicate that immediately following their formation the vortices undergo a strong three-dimensional distortion, which may provide the mechanism for the transition from laminar to turbulent Strouhal vortices.

Intense fluid motions can be generated by the solidification of a binary liquid. This review paper describes systematically some of the concepts involved in the fluid mechanics of solidification. It also presents quantitative calculations for the fluid motion, the rate of growth of solid and the evolution of both the thermal and the compositional fields in various geometries. The results of many of the calculations are favourably compared with data from laboratory experiments using aqueous solutions.

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).

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.