The evolution of drift with time at a plane normal to the passage of a body from far right to far left is investigated. It is shown that the measurable part of the drift volume as a function of time is uniquely determined, and that the indeterminacy of Darwin's drift integral simply does not matter to the physical world, in which there is a Darwin theorem stating that the drift mass is equal to the added mass. The method for determining the shape of the drift surface is also given.

In a previous study of the effects of cooling and solidification on flows issuing onto a horizontal plane and spreading under gravity we considered the case of a viscous fluid that solidifies to form a thin surface crust with a finite yield strength. In that case, the coupling of solidification and viscous stresses in the flow led to a sequence of flow regimes or styles of flow and crustal deformation. Here, we study the spreading, from a small source, of a plastic material having a yield strength before cooling. In this case the fluid again begins to freeze as it spreads radially under gravity, and forms a dome having a surface crust which is stronger than the extruded fluid. If cooling is sufficiently rapid compared to gravity-driven spreading, the flow is found to be controlled by solidification. The flow again takes on one of a number of flow regimes depending on the pace of solidification relative to the rate of lateral flow, or extrusion rate. However, these flow regimes are quite different from those for the viscous extrusions, implying that the internal yield stress has a strong influence on the behaviour. Styles of flow ranged from inflation of an axisymmetric dome to irregular extrusion of lateral lobes and vertical spines. These qualitatively different regimes have much in common with the eruption styles of volcanic lava domes produced by effusion of extremely viscous silicic magmas which may possess a yield strength, and the model provides information about the factors influencing the morphology and hazards of such volcanic flows.

The mathematical consequences of a few simple scaling assumptions regarding the effects of compressibility are explored using a singular perturbation idea and the methods of statistical fluid mechanics. Representations for the pressure–dilatation and dilatational dissipation appearing in single-point moment closures for compressible turbulence are obtained. The results obtained, in as much as they come from the same underlying procedure, represent a unified development for both dilatational covariances. While the results are expressed in the context of a statistical turbulence closure they provide, with very few phenomenological assumptions, an interesting and clear mathematical model for the ‘scalar’ effects of compressibility. For homogeneous turbulence with quasi-normal large scales the expressions derived are – in the small turbulent Mach number squared isotropic limit – exact. The expressions obtained contain constants that have a precise physical significance and are defined in terms of integrals of the longitudinal velocity correlation. The pressure–dilatation covariance is found to be a non-equilibrium phenomenon related to the time rate of change of the kinetic energy and internal energy of the turbulence; it is seen to scale with α2M2t εs [Pk/ε−1] (Sk/εs)2. Implicit in the scaling is a dependence on the square of a gradient Mach number, S[lscr ]/c. A new feature indicated by the analysis is the appearance of the Kolmogorov scaling coefficient, α, suggesting that large-scale quantities embodied in the well-established ε∼u˜3/[lscr ] relationship provide a link to the structural dependence of the effects of compressibility. The expressions for the dilatational dissipation are found to depend on the turbulent Reynolds number and scale as M4t (Sk/εs)4R−1t. The scalings for the pressure–dilatation are found to produce an excellent collapse of the pressure–dilatation data from direct numerical simulation.

The critical-layer analysis of the nonlinear resonant-triad interaction by Goldstein & Lee (1992) is extended to include viscous effects. A generalized scaling which is valid both for the quasi-equilibrium and non-equilibrium critical-layer analyses in zero- or non-zero-pressure-gradient boundary layers is obtained. A system of partial differential equations which governs the fully coupled non-equilibrium critical-layer dynamics is obtained and it is solved by using a numerical method. Amplitude equations and their viscous limits are also presented. The parametric-resonance growth rate of the non-equilibrium critical-layer solution with finite viscosity is larger than that of the viscous-limit quasi-equilibrium solution. The viscosity delays both the onset of the fully coupled interaction and the ultimate downstream location of the singularity. The difference between the non-equilibrium critical-layer solution and the corresponding quasi-equilibrium critical-layer solution becomes smaller, at least in the parametric resonance region, as the viscosity parameter becomes large. However, the non-equilibrium solution with finite viscosity always ends in a singularity at a finite downstream position unlike the viscous-limit solution.

The paper considers the flow field during spin-up from rest of liquid metal in a cylindrical stationary cavity due to a rotating transverse uniform low-frequency magnetic field. It is assumed that the Ekman and the magnetic Reynolds numbers are small. An approximate model, based on matching of Bödewadt-type layers with an inviscid core, with possible influence from the sidewall, for laminar flow is developed. It is shown that the angular velocity in the core is a function of time only. Analytical solutions for the angular velocity and the meridional flow in the core are presented, and supplemented by finite-difference results to show the sidewall effects. The spin-down following the switch-off of the magnetic forcing, the influence of the axial variations of the magnetic field, and the relevance to turbulent flows are discussed.

A fully nonlinear, diffusive, and weakly dispersive wave equation is derived for describing gravity surface wave propagation in a shallow porous medium. Darcy's flow is assumed in a homogeneous and isotropic porous medium. In deriving the general equation, the depth of the porous medium is assumed to be small in comparison with the horizontal length scale, i.e. O(μ2) =O(h0/L)2[Lt ]1. The order of magnitude of accuracy of the general equation is O(μ4). Simplified governing equations are also obtained for the situation where the magnitude of the free-surface fluctuations is also small, O(ε)=O(a/h0)[Lt ]1, and is of the same order of magnitude as O(μ2). The resulting equation is of O(μ4, ε2) and is equivalent to the Boussinesq equations for water waves. Because of the dissipative nature of the porous medium flow, the damping rate of the surface wave is of the same order magnitude as the wavenumber. The tide-induced groundwater fluctuations are investigated by using the newly derived equation. Perturbation solutions as well as numerical solutions are obtained. These solutions compare very well with experimental data. The interactions between a solitary wave and a rectangular porous breakwater are then examined by solving the Boussinesq equations and the newly derived equations together. Numerical solutions for transmitted waves for different porous breakwaters are obtained and compared with experimental data. Excellent agreement is observed.

We study the linear stability of surface-tension-driven unidirectional shear flow in an unbounded electrically conducting liquid layer heated from the side and subjected to a uniform magnetic field in the plane of the layer. The threshold of convective instability with respect to oblique travelling waves is calculated depending on the strength and orientation of the magnetic field. For longitudinal waves the critical Marangoni number and the corresponding wavelength are found to increase directly with the induction of a sufficiently strong magnetic field. In general, a coplanar magnetic field causes stabilization of all disturbances except those aligned with the field, which are not influenced at all. With increase of the magnetic field this effect results in the alignment of the most unstable disturbance along the magnetic flux lines. The maximal stabilization is ensured by the magnetic field being imposed spanwise to the basic flow. The corresponding critical Marangoni number is found to be almost insensitive to the thermal properties of the bottom. The strength of the magnetic field necessary to attain the maximal stabilization for a thermally well-conducting bottom is considerably lower than that for an insulating bottom. The basic return flow is found to be linearly stable with respect to purely hydrodynamic disturbances. This effect determines the stability of the basic state with respect to transverse hydrothermal waves at Prandtl number Pr<Prc=0.018. For such a small Pr no alignment of the critical perturbation with a spanwise magnetic field is possible, and the critical Marangoni number can be increased almost directly with the strength of the magnetic field without limit.

The development of turbulence is investigated in the presence of a mean plane shear flow (rate S) rotating with angular velocity vector (rate Ω) perpendicular to its plane. An important motivation was generalizing the work by Lee, Kim & Moin (1990) to rotating shear flow, in particular detailed comparisons of homogeneous rapid distortion theory (RDT) results and the databases of homogeneous and channel flow direct numerical simulations (DNS). Linear analysis and related RDT are used starting from the linearized equations governing the fluctuating velocity field. The parameterization based on the value of the Bradshaw–Richardson number B=R(1+R) (with R=−2Ω/S) is checked against complete linear solutions. Owing to the pressure fluctuation, the dynamics is not governed entirely by the parameter B, and the subsequent breaking of symmetry (between the R and −1 −R cases) is investigated. New analytical solutions for the ‘two-dimensional energy components’ [Escr ](l)ij =Eij(kl=0, t) (i.e. the limits at kl=0 of the one-dimensional energy spectra) are calculated by inviscid and viscous RDT, for various ratios Ω/S and both streamwise l=1 and spanwise l=3 directions. Structure effects (streak-like tendencies, dimensionality) in rotating shear flow are discussed through these quantities and more conventional second-order statistics. In order to compare in a quantitative way RDT solutions for single-point statistics with available large-eddy simulation (LES) data (Bardina, Ferziger & Reynolds 1983), an ‘effective viscosity’ model (following Townsend) is used, yielding an impressive agreement.

This paper presents a theoretical model of turbulence and mixing at a shear-free stable density interface. In one case (single-sided stirring) the interface separates a layer of fluid of density ρ in turbulent motion, with r.m.s. velocity uH and lengthscale LH, from a non-turbulent layer with density ρ+Δρ, while in the second case (double-sided stirring) the lower layer is also in turbulent motion. In both cases, the external Richardson number Ri=ΔbLH/ u2H (where Δb is the buoyancy jump across the interface) is assumed to be large. Based on the hypotheses that the effect of the interface on the turbulence is as if it were suddenly imposed (which is equivalent to generating irrotational motions) and that linear waves are generated in the interface, the techniques of rapid distortion theory are used to analyse the linear aspects of the distortion of turbulence and of the interfacial motions. New physical concepts are introduced to account for the nonlinear aspects.

To describe the spectra and variations of the r.m.s. fluctuations of velocity and displacements, a statistically steady linear model is used for frequencies above a critical frequency ωr/μc, where ωr(=Δb/2uH) is the maximum resonant frequency and μc<1. As in other nonlinear systems, observations below this critical frequency show the existence of long waves on the interface that can grow, break and cause mixing between the two fluid layers. A nonlinear model is constructed based on the fact that these breaking waves have steep slopes (which determines the form of the displacement spectrum) and on the physical argument that the energy of the vertical motions of these dissipative nonlinear waves should be comparable to that of the forced linear waves, which leads to an approximately constant value for the parameter μc. The model predictions of the vertical r.m.s. interfacial velocity, the interfacial wave amplitude and the velocity spectra agree closely with new and published experimental results.

An exact unsteady inviscid linear analysis is used to derive the growth rate of the full spectrum, which asymptotically leads to the growth of resonant waves and to the energy transfer from the turbulent region to the wave motion of the stratified layer. Mean energy flux into the stratified layer, averaged over a typical wave cycle, is used to estimate the boundary entrainment velocity for the single-sided stirring case and the flux entrainment velocity for the double-sided stirring case, by making the assumption that the ratio of buoyancy flux to dissipation rate in forced stratified layers is constant with Ri and has the same value as in other stratified turbulent flows. The calculations are in good agreement with laboratory measurements conducted in mixing boxes and in wind tunnels. The contribution of Kelvin–Helmholtz instabilities induced by the velocity of turbulent eddies parallel to the interface is estimated to be insignificant compared to that of internal waves excited by turbulence.

A laboratory experimental study was performed to investigate turbulence, waves and mixing at a sharp density interface (with a jump in buoyancy Δb), subjected to shear-free turbulence induced by oscillating grids with typical velocity and length scales of uH and LH, respectively. The cases where turbulence is present on one side (single-sided stirring) or on both sides (double-sided stirring) of the interface were considered. Extensive flow visualization studies and quantitative measurements were performed on the motion field and mixing characteristics at the interface. It was found that, rather than any one mechanism controlling the mixing process, different mechanisms (namely engulfment, generation of waves and their breaking, eddy impingement and Kelvin–Helmholtz billows) play dominant roles over different ranges of the bulk Richardson number Ri(Ri =ΔbLH/u2H). For the Ri range where wave generation is significant, certain hypotheses and predictions of the companion paper by Fernando & Hunt (1997) were tested in detail, by flow visualization studies of the qualitative properties of interfacial motions and quantitative measurements of the r.m.s. fluctuations of interfacial velocity and displacement, the local gradient Richardson number within the stratified layer, the frequency spectra and the related fractal properties of the interface. The results are consistent with the hypothesis that, at high values of Ri(>35), the density interface consists of linear internal waves driven by turbulence at high frequencies and breaking waves with sharp horizontal gradients of density at low frequencies.

Fully developed intermittent flow in a strongly curved tube was numerically simulated using a numerical scheme based on the simpler method. Physiological pulsatile flow in the aorta was simulated as intermittent flow, with a waveform consisting of a pulse-like systolic flow period followed by a stationary diastolic period. Numerical simulations were carried out for the following conditions: Dean number κ=393, frequency parameter α=4–27, curvature ratio δ=1/2, 1/3 and 1/7, and intermittency parameter η=0–1/2, where η is the ratio of a systolic time to the cycle period. For α=18 and 27 the axial-flow profile in a systolic period becomes close to that of a sinusoidally oscillatory flow. At the end of the systole, a region of reversed axial velocity appears in the vicinity of the tube wall, which is caused by the blocking of the flow, similar to blocked flow in a straight tube. This area is enlarged near the inner wall of the bend by the curvature effect. Circumferential flow accelerated in a systole streams into the inner corner and collides at the symmetry line, which creates a jet-like secondary flow towards the outer wall. The region of reversed axial velocity is extended to the tube centre by the secondary flow. The development of the flow continues during the diastolic period for α higher than 8, and the flow does not completely dissipate, so that a residual secondary vortex persists until the next systole. Accordingly, the development of secondary flow in the following systolic phase is strongly affected by the residual vortex at the end of the previous diastolic phase, especially by stationary diastolic periods. Therefore, intermittent flow in a curved tube is strongly affected by the stationary diastolic period. For η=0 and 1/5, the induced secondary flow in a systole forms additional vortices near the inner wall, whereas for η=1/3 and 1/2 additional vortices do not appear. The characteristics of intermittent flow in a curved tube are also strongly affected by the length of the diastolic period, which represents a period of zero flow.

System rotation is known to substantially affect the mean flow pattern as well as the turbulence structure in rotating channel flows. In a numerical study of plane Couette flow rotating slowly about an axis aligned with the mean vorticity, Bech & Andersson (1996a) found that the turbulence level was damped in the presence of anticyclonic system rotation, in spite of the occurrence of longitudinal counter-rotating roll cells. Moreover, the turbulence anisotropy was practically unaffected by the weak rotation, for which the rotation number Ro, defined as the ratio of twice the imposed angular vorticity Ω to the shear rate of the corresponding laminar flow, was ±0.01. The aim of the present paper is to explore the effects of stronger anticyclonic system rotation on directly simulated turbulent plane Couette flow. Turbulence statistics like energy, enstrophy and Taylor lengthscales, both componental and directional, were computed from the statistically steady flow fields and supplemented by structural information obtained by conditional sampling.

The designation of the imposed system rotation as ‘high’ was associated with a reversal of the conventional Reynolds stress anisotropy so that the velocity fluctuations perpendicular to the wall exceeded those in the streamwise direction. It was observed that the anisotropy reversal was accompanied by an appreciable region of the mean velocity profile with slope ∼2Ω, i.e. the absolute mean vorticity tended to zero. It is particularly noteworthy that these characteristic features were shared by two fundamentally different flow regimes. First, the two-dimensional roll cell pattern already observed at Ro=0.01 became more regular and energetic at Ro=0.10 and 0.20, whereas the turbulence level was reduced by about 50%. Then, when Ro was further increased to 0.50, a disordering of the predominant roll cell pattern set in during a transient period until the flow field settled at a new statistically steady state substantially less affected by the roll cells. This was accompanied by a substantial amplification of the streamwise turbulent vorticity and an anomalous variation of the mean turbulent kinetic energy which peaked in the middle of the channel rather than near the walls. While the predominant flow structures of the non-rotating flow were longitudinal streaks, system rotation generated streamwise vortices, either ordered secondary flow or quasi-streamwise vortices. Eventually, at Ro=1.0, the turbulent fluctuations were completely suppressed and the flow field relaminarized.

The sound generated by the interaction between convected vortical and entropic disturbances and a blade row is a significant component of the total noise emitted by a modern aeroengine, and the blade geometry has an important effect on this process. As a first step in the development of a general prediction scheme, we model in this paper just the action of the blade mean loading by treating the blades as flat plates aligned at a non-zero incidence angle, δ, to the oncoming stream, and consider harmonic components of the incident field with reduced frequency k. We then use asymptotic analysis in the realistic limit k[Gt ]1, δ[Lt ]1 with kδ=O(1) to make a consistent asymptotic expansion of the compressible Euler equations. The flow is seen to consist of inner regions around each leading edge, in which sound is generated by the local gust–airfoil and gust–flow interactions, and an outer region in which both the incident gust is distorted according to rapid distortion theory and the out-going sound is refracted by the non-uniform mean flow. The complicated multiple interactions between the sound and the cascade are included to the appropriate asymptotic order, and analytical expressions for the forward radiation are derived. It is seen that even a relatively small value of δ can have a significant effect, thanks to both the O(δk1/2) change in the amplitudes and the O(kδ) change in the phases of the various radiation components, corresponding to the additional source mechanisms associated with the flow distortion around each leading edge and the effects of propagation through the non-uniform flow, respectively. Further work will extend this analysis to include the effects of camber and thickness.

An agglomerate made of solid particles held together by a viscous liquid phase when sheared in an otherwise dry granular material is observed to deform by stretching. This observation, based on experimental results, is confirmed in the present paper by means of a computer simulation model. Simulations as well as experimental results indicate that the degree of deformation by stretching, a critical factor influencing the stability of such agglomerates, is governed by a dimensionless parameter of the system, called the deformation Stokes number, Stdef. Two regimes, involving high and low characteristic degrees of deformation, can be identified based upon the value of this number. Simulation results indicate that for the range of conditions simulated, the value separating the two regimes, the critical deformation Stokes number, St*def, is relatively insensitive to the agglomerate size and other parameters of the system. This critical number defines the conditions below which forces inducing agglomerate breakage are low and above which they are high and result in agglomerate break-up. Calculation and/or measurement of this parameter is essential for prediction of equilibrium sizes of agglomerates in industrial granulation operations.

Euromech Colloquium 353, held at the University of Karlsruhe, Germany, 1–3 April 1996 brought together scientists working in the field of localized disturbances of flows in order to discuss new developments and the potential for application. The colloquium attracted a total of 56 participants from nine European countries, i.e. France, Germany, The Netherlands, Poland, Russia, Sweden, Switzerland, Ukraine and United Kingdom as well as from the US and Israel.

Lord Rayleigh argued that after a discontinuity develops in a one-dimensional compression wave in an ideal inviscid fluid some sort of motion must continue. Arguments are given in support of this view and a suggestion is made as to what that motion might be. The relationship of this motion to that proposed by Onsager for incompressible inviscid turbulent flows is discussed.