A direct numerical simulation at resolution 2403 is used to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150). The energy spectrum displays an inertial subrange. The velocity derivative distribution, known to be strongly non-Gaussian, is found to be close to, but not, exponential. The nth-order moments of this distribution, as well as the velocity structure functions, do not scale with n as predicted by intermittency models. Visualization of the flow confirms the previous finding that the strongest vorticity is organized in very elongated thin tubes. The width of these tubes is of the order of a few dissipation scales, while their length can reach the integral scale of the flow.

A continuum model based on the critical-state theory of soil mechanics is used to generate stress, density, and velocity profiles, and to compute discharge rates for the flow of granular material in a mass flow bunker. The bin–hopper transition region is idealized as a shock across which all the variables change discontinuously. Comparison with the work of Michalowski (1987) shows that his experimentally determined rupture layer lies between his prediction and that of the present theory. However, it resembles the former more closely. The conventional condition involving a traction-free surface at the hopper exit is abandoned in favour of an exit shock below which the material falls vertically with zero frictional stress. The basic equations, which are not classifiable under any of the standard types, require excessive computational time. This problem is alleviated by the introduction of the Mohr–Coulomb approximation (MCA). The stress, density, and velocity profiles obtained by integration of the MCA converge to asymptotic fields on moving down the hopper. Expressions for these fields are derived by a perturbation method. Computational difficulties are encountered for bunkers with wall angles θw [ges ] 15° these are overcome by altering the initial conditions. Predicted discharge rates lie significantly below the measured values of Nguyen et al. (1980), ranging from 38% at θw = 15° to 59% at θw = 32°. The poor prediction appears to be largely due to the exit condition used here. Paradoxically, incompressible discharge rates lie closer to the measured values. An approximate semi-analytical expression for the discharge rate is obtained, which predicts values within 9% of the exact (numerical) ones in the compressible case, and 11% in the incompressible case. The approximate analysis also suggests that inclusion of density variation decreases the discharge rate. This is borne out by the exact (numerical) results – for the parameter values investigated, the compressible discharge rate is about 10% lower than the incompressible value. A preliminary comparison of the predicted density profiles with the measurements of Fickie et al. (1989) shows that the material within the hopper dilates more strongly than predicted. Surprisingly, just below the exit slot, there is good agreement between theory and experiment.

Amplitude equations (including nonlinear damping terms) are derived which describe the evolution of patterns in large-aspect-ratio driven capillary wave experiments. For drive strength just above threshold, a reduction of the number of marginal modes (from travelling capillary waves to standing waves) leads to simpler amplitude equations, which have a Lyapunov functional. This functional determines the wavenumber and symmetry (square) of the most stable uniform state. The original amplitude equations, however, have a secondary instability to transverse amplitude modulation (TAM), which is not present in the standing-wave equations. The TAM instability announces the restoration of the full set of marginal modes.

We have performed high-resolution numerical simulations of supersonic slip surfaces to confirm and illuminate earlier analytic nonlinear stability calculations of such structures. This analytic work was in turn inspired by earlier computer simulations reported in Woodward (1985) and Woodward et al. (1987). In particular Artola & Majda (1987) examined the response of a supersonic slip surface to an incident train of small-amplitude nonlinear sound waves. They found analytic solutions which indicate that nonlinear resonance occurs at three angles of incidence which depend upon the Mach number of the relative motion. The two-dimensional simulations described here numerically solve this problem for a Mach-4 flow using the piecewise-parabolic method (Colella & Woodward 1984; Woodward & Colella 1984). The simulations show that sound waves incident at a predicted resonance angle excite nonlinear behaviour in the slip surface. At these angles the amplitude of the reflected waves is much greater than the incident wave amplitude (i.e. a shock forms). The observed resonance is fairly broad, but the resonance narrows as the strength of the incident waves is reduced.

The nature of the nonlinear kink modes observed in the simulations is similar to that discussed by Artola & Majda. Most of the modes move in either direction with speeds near the predicted value. Speeds of other than this value are observed, but the disagreement is not serious in view of the strongly nonlinear behaviour seen in the simulations but not treated in the analytic work. The stationary modes seen in the analytic results are perhaps observed as transient structures. They may eventually dominate the flow at late times (Woodward et al. 1987).

The role of the kink modes in the stability of slab jets is discussed, and it is argued that the stationary modes are more disruptive than the propagating modes.

Positions, velocities and rotations of individual particles obtained from high-speed motion pictures of essentially two-dimensional flows of plastic spheres in an inclined glass-walled chute were used to test critical assumptions of microstructural theories for the flow of granular materials. The measurements provide a well-defined set of observations for refining and validating computer simulations of granular flows, and point out some important limitations of physical experiments. Two nearly steady, uniform, collisional flows of 6-mm-diameter plastic spheres over a fixed bed of similar spheres inclined at 42.75° were analysed in detail. Particle fluxes were about 2230 particles s−1 and 1280 particles s−1. The nominal depth in both flows was about 18 particle diameters. Profiles of mean downstream velocity and mean rotations, translational temperature and rotational temperature, and bulk density in the flows show slip at the bed of 17 and 26% of the mean flow velocity for the high- and low-flux flows, respectively; mean rotation rates $\overline{\omega}_x$ and $\overline{\omega}_y$ less than 9% of $\overline{\omega}_z$ ($\hat{e}_x$ parallel to the bed, $\hat{e}_x$ normal to the sidewall); translational temperature nearly independent of distance from the bed; rotational temperature decreasing with distance from the bed; and density decreasing almost linearly with distance from the bed. The continuum hypothesis (i.e. small gradients in mean-flow properties) is satisfied throughout the flow except near the fixed bed, where large gradients in the mean rotation $\overline{\omega}_z$ and downstream velocity occur over a few particle diameters. The distributions of velocities and rotations are approximately Maxwellian, except near the fixed bed. Testing microstructural theories with physical experiments is severely hampered by limitations on material properties of particles, flow lengthscale and the spatial and temporal resolution of observations. Only a small volume of the parameter space for collision-dominated flows can reasonably be explored by physical experiment. Extraneous forces due to air drag, sidewall friction and electrical effects are not included in theories but must be addressed in physical experiments. Properly designed experiments are the essential link between computer simulations and theory, because they focus attention on particular features critical to testing the simulations, which in turn provide detailed particle-scale information needed to test theories.

Trapped or edge-wave modes are well-known in linear water-wave theory. They occur at discrete frequencies below a certain cutoff frequency and consist of local oscillations trapped near a long horizontal submerged body in finite or infinite depth or over a sloping beach. Less well known is the existence of trapped modes in certain problems in acoustics where the governing equation is the Helmholtz equation. Jones (1953) has proved the existence of such modes which correspond to point-eigenvalues of the spectrum of the differential operator satisfying certain boundary conditions in a semi-infinite region. In this paper we describe a constructive method for determining point-eigenvalues or trapped-mode frequencies in two specific problems in which the two-imensional Helmholtz equation is satisfied.

The problems arise from a consideration of the fluid motion in a long narrow wave tank with a free water surface which contains a vertical cylinder of uniform horizontal cross-section extending throughout the water depth. Separation of the depth dependence results in Helmholtz's equation with Neumann boundary conditions. By seeking solutions which are antisymmetric with respect to the centreline of the channel, trapped modes are constructed for the case of a cylinder of rectangular cross-section placed symmetrically in the centre of the channel and also for the case of a symmetric rectangular indentation in the tank walls. These problems do not appear to be covered directly by Jones’ theory and whilst the method described provides convincing numerical evidence, it falls short of a rigorous existence proof. Extensions to other purely acoustic problems having no water-wave interpretation, including problems which are covered by the general theory of Jones, are also discussed.

We study virtual mass and drag effects in a fluid suspension consisting of spherical particles immersed in an incompressible, nearly inviscid fluid. We derive average equations of motion for the fluid phase and the particle phase by the method of ensemble averaging. We show that the virtual mass and drag coefficients may be expressed exactly in terms of the dielectric constant of a corresponding dielectric suspension with the same distribution of particles. We make numerical predictions for the case of an equilibrium distribution of hard spheres.

A viscous linear stability analysis of a trailing line (Batchelor) vortex is presented. Employing a staggered Chebyshev spectral collocation technique, very accurate results were obtained. The destabilising role of viscous forces has been shown to produce two types of viscous instability modes. These viscous disturbances consist of an axisymmetric mode and an asymmetric mode. Both disturbances are long-wave instabilities with maximum growth rates which are orders of magnitude smaller than the inviscid modes which have been found by others. Comparison with experimental results and condensation trail observations are found to be in good qualitative agreement with the present study.

Direct numerical simulations of unsteady channel flow were performed at low to moderate Reynolds numbers on computational boxes chosen small enough so that the flow consists of a doubly periodic (in x and z) array of identical structures. The goal is to isolate the basic flow unit, to study its morphology and dynamics, and to evaluate its contribution to turbulence in fully developed channels. For boxes wider than approximately 100 wall units in the spanwise direction, the flow is turbulent and the low-order turbulence statistics are in good agreement with experiments in the near-wall region. For a narrow range of widths below that threshold, the flow near only one wall remains turbulent, but its statistics are still in fairly good agreement with experimental data when scaled with the local wall stress. For narrower boxes only laminar solutions are found. In all cases, the elementary box contains a single low-velocity streak, consisting of a longitudinal strip on which a thin layer of spanwise vorticity is lifted away from the wall. A fundamental period of intermittency for the regeneration of turbulence is identified, and that process is observed to consist of the wrapping of the wall-layer vorticity around a single inclined longitudinal vortex.

The process of alignment, a new fundamental interaction between vortices in a stratified and rapidly rotating fluid, is defined and studied in detail in the context of the two-layer quasi-geostrophic model. Alignment occurs when two vortices in different density layers coalesce by reducing their horizontal separation. It is found that only vortices whose radii are comparable with or larger than the Rossby deformation radius can align. In the same way as the merger process (in a single two-dimensional layer) is related to the reverse energy cascade of two-dimensional turbulence, geostrophic potential vorticity alignment is related the barotropic-to-baroclinic energy cascade of geostrophic turbulence in two layers. It is also shown how alignment is intimately connected with the existence of two-layer doubly connected geostrophic potential vorticity equilibria (V-states), for which the analysis of the geometry of the stream function in the corotating frame is found to be a crucial diagnostic. The finite-area analogues of the hetons of Hogg & Stommel (1985) are also determined: they consist of a propagating pair of opposite-signed potential vorticity patches located in different layers.

This study deals with the flow around a circular cylinder placed near a plane wall and exposed to an oscillatory flow. The study comprises instantaneous pressure distribution measurements around the cylinder at high Reynolds numbers (mostly at Re ∼ 105) and a flow visualization study of vortex motions at relatively smaller Reynolds numbers (Re ∼ 103–104). The range of the gap-to-diameter ratio is from 0 to 2 for the pressure measurements and from 0 to 25 for the flow visualization experiments. The range of the Keulegan–Carpenter number KC is from 4 to 65 for the pressure measurements and from 0 to 60 for the flow visualization tests. The details of vortex motions around the cylinder are identified for specific values of the gap-to-diameter ratio and for the KC regimes known from research on wall-free cylinders. The findings of the flow visualization study are used to interpret the variations in pressure with time around the pipe. The results indicate that the flow pattern and the pressure distribution change significantly because of the close proximity of the boundary where the symmetry in the formation of vortices breaks down, and also the characteristic transverse vortex street observed for wall-free cylinders for 7 < KC < 13 disappears. The results further indicate that the vortex shedding persists for smaller and smaller values of the gap-to-diameter ratio, as KC is decreased. The Strouhal frequency increases with decreasing gap-to-diameter ratio. The increase in the Strouhal frequency with respect to its wall-free-cylinder value can be as much as 50% when the cylinder is placed very close to the wall with a gap-to-diameter ratio of O(0.1).

Within the framework of the study of coherent vortex structures as emerging in rotating, quasi-two-dimensional flows, the tripolar vortex is a relatively novel feature. It consists of a symmetric, linear arrangement of three patches of distributed vorticity of alternate signs, and the axis of this configuration rotates about the centre of the core vortex. This paper describes an experimental study of the formation of a tripole from an unstable axisymmetric vortex in a solidly rotating, homogeneous fluid. The flow is visualized by addition of dye, and is measured by streak photography of tracer particles. After digitization, the spatial distributions of the vorticity ω and the stream function ψ are calculated numerically, and 'scatter plots’ of ω versus ψ are presented for the various stages in the tripole formation process. Owing to viscous effects (spin-down by the bottom Ekman layer and lateral entrainment of ambient fluid) the tripole shows an exponential decay, both in its rotation speed and its internal, relative flow. The comparison of the observed flow characteristics with a simple point-vortex model shows reasonable quantitative agreement.

When a body of fluid with a vertical salinity and temperature gradient is bounded by a sloping boundary, convective instabilities are often observed. These can occur if the fluid is subjected to heating or the addition of solute at the boundary, or if the boundary is an insulator. These instabilities often take the form of long thin convection cells that are almost horizontal. We present a linear stability analysis of the background states associated with these different boundary conditions and derive criteria for their stability in terms of one non-dimensional parameter, Q. This parameter is related to the Rayleigh number and is a generalization of the similar parameter found by Kerr (1989) in his study of heating a salinity gradient from a vertical boundary. This analysis uses a quasi-static assumption that is valid when the vertical lengthscales of the instabilities are less than the horizontal lengthscales.

The problem of the interaction of large-scale vortices with small-scale homogeneous isotropic helical turbulence in a compressible medium is considered. Averaged equations are derived using a closure procedure which is based on the functional technique. It is shown that the averaged vorticity equation has solutions that grow exponentially in time and which describe the effect of amplification of large-scale helical vortices by turbulence (hydrodynamical α-effect). The dependence of the growth rate on the compressibility is analysed, the limiting cases of incompressible fluid and turbulence δ-correlated in time being considered. The applications of the hydrodynamical α-effect discussed include the Earth's atmosphere and interstellar gas of spiral galaxies.

Measurements were made of the wave height of periodic, quasi-periodic, and chaotic parametrically forced cross-waves in a long rectangular channel. In general, three frequencies (and their harmonics) may be observed: the subharmonic frequency and two slow temporal modulations — a one-mode instability associated with streamwise variation and a sloshing motion associated with spanwise variation. Their interaction, as forcing frequency, f, and forcing amplitude, a, were varied, produced a pattern of Arnold tongues in which two or three frequencies were locked. The overall picture of frequency-locked and -unlocked regions is explained in terms of the Arnold tongues predicted by the circle-map theory describing weakly coupled oscillators. Some of the observed tongues are apparently folded by a subcritical bifurcation, with the tips of the tongues lying on the unstable manifold folded under the observed stable manifold. Near the intersection of the neutral stability curves for two adjacent modes, a standing wave localized on one side of the tank was observed in agreement with the coupled-mode analysis of Ayanle, Bernoff & Lichter (1990). At large cross-wave amplitudes, the spanwise wave structure apparently breaks up, because of modulational instability, into coherent soliton-like structures that propagate in the spanwise direction and are reflected by the sidewalls.

Experimental results on flow-field statistics are presented for turbulent oscillatory flow in a circular pipe for the range of Reynolds numbers Reδ = U0δ/ν (U0 = amplitude of cross-sectional mean velocity, δ = (2ν/ω)½) = Stokes layer thickness) from 550 to 2000 and Stokes parameters Λ = R/δ (R = radius of the pipe) from 5 to 10. Axial and radial velocity components were measured simultaneously using a two-colour laser-Doppler anemometer, providing information on ensemble-averaged velocity profiles as well as various turbulence statistics for different phases during the cycle. In all flows studied, turbulence appeared explosively towards the end of the acceleration phase of the cycle and was sustained throughout the deceleration phase. During the turbulent portion of the cycle, production of turbulence was restricted to the wall region of the pipe and was the result of turbulent bursts. The statistics of the resulting turbulent flow showed a great deal of similarity to results for steady turbulent pipe flows; in particular the three-layer description of the flow consisting of a viscous sublayer, a logarithmic layer (with von Kármán constant = 0.4) and an outer wake could be identified at each phase if the corresponding ensemble-averaged wall-friction velocities were used for normalization. Consideration of similarity laws for these flows reveals that the existence of a logarithmic layer is a dimensional necessity whenever at least two of the scales R, u*/ω and ν/u* are widely separated; with the exact structure of the flow being dependent upon the parameters u*/Rω and u2*/ων. During the initial part of the acceleration phase, production of turbulence as well as turbulent Reynolds stresses were reduced to very low levels and the velocity profiles were in agreement with laminar theory. Nevertheless, the fluctuations retained a small but finite energy. In Part 2 of this paper, the major features observed in these experiments are used as a guideline, in conjunction with direct numerical simulations of the ‘perturbed’ Navier–Stokes equations for oscillatory flow in a channel, to identify the nature of the instability that is most likely to be responsible for transition in this class of flows.

The stability of oscillatory channel flow to different classes of infinitesimal and finite-amplitude two- and three-dimensional disturbances has been investigated by direct numerical simulations of the Navier–Stokes equations using spectral techniques. All infinitesimal disturbances were found to decay monotonically to a periodic steady state, in agreement with earlier Floquet theory calculations. However, before reaching this periodic steady state an infinitesimal disturbance introduced in the boundary layer was seen to experience transient growth in accordance with the predictions of quasi-steady theories for the least stable eigenmodes of the Orr–Sommerfeld equation for instantaneous ‘frozen’ profiles. The reason why this growth is not sustained in the periodic steady state is explained. Two-dimensional infinitesimal disturbances reaching finite amplitudes were found to saturate in an ordered state of two-dimensional quasi-equilibrium waves that decayed on viscous timescales. No finite-amplitude equilibrium waves were found in our cursory study. The secondary instability of these two-dimensional finite-amplitude quasi-equilibrium states to infinitesimal three-dimensional perturbations predicts transitional Reynolds numbers and turbulent flow structures in agreement with experiments.

Swirling flow through a pipe is a highly complex turbulent flow and is still challenging to predict. An experimental investigation is performed to obtain systematic data about the flow and to understand its physics. A free-vortex-type swirling flow is introduced in a long straight circular pipe. The swirling component decays downstream as a result of wall friction. The velocity distributions are continuously changing as they approach fully developed parallel flow. The swirl intensity Ω, defined as a non-dimensional angular momentum flux, decays exponentially. The decay coefficients, however, are not constant as conventionally assumed, but depend on the swirl intensity. The wall shear stresses are measured by a direct method and, except in a short inlet region, are a function only of the swirl intensity and the Reynolds number. The velocity distributions and all Reynolds stress components are measured at various axial positions in the pipe. The structure of the tangential velocity profile is classified into three regions: core, annular and wall regions. The core region is characterized by a forced vortex motion and the flow is dependent upon the upstream conditions. In the annular region, the skewness of the velocity vector is noticeable and highly anisotropic so that the turbulent viscosity model does not work well here. The tangential velocity is expressed as a sum of free and forced vortex motion. In the wall region the skewness of the flow becomes weak, and the wall law modified by the Monin–Oboukhov formula is applicable. Data on the microscale and the spectrum are also presented and show quite different turbulence structures in the core and the outer regions.

An analysis that includes the effects of Basset and gravitational forces is presented for the dispersion of particles experiencing Stokes drag in isotropic turbulence. The fluid velocity correlation function evaluated on the particle trajectory is obtained by using the independence approximation and the assumption of Gaussian velocity distributions for both the fluid and the particle, formulated by Pismen & Nir (1978). The dynamic equation for particle motion with the Basset force is Fourier transformed to the frequency domain where it can be solved exactly. It is found that the Basset force has virtually no influence on the structure of the fluid velocity fluctuations seen by the particles or on particle diffusivities. It does, however, affect the motion of the particle by increasing (reducing) the intensities of particle turbulence for particles with larger (smaller) inertia. The crossing of trajectories associated with the gravitational force tends to enhance the effect of the Basset force on the particle turbulence. An ordering of the terms in the particle equation of motion shows that the solution is valid for high particle/fluid density ratios and to 0(1) in the Stokes number.

An experimental observation of the flow following the discharge of firearms has been carried out by means of ultra-high-speed visualization. The theory of similarity has been applied in order to define the rules governing the tests on models, chiefly for gun firing-air intake interference problems. When the blast effect predominates, no geometric similarity is required between the simulation gun and the simulated one, so the model and the simulation gun can have different scales. It is shown that the main parameter characterizing the blast effect is the energy rate at the muzzle which can be considered as a point source of energy caused by a very hot gas. So, the muzzle wave tends asymptotically toward the blast wave of a non-instantaneous intense point explosion. Specific experiments confirm this assertion. All previous results allow a theoretical modelling of gun-firing aerodynamic phenomena which will be presented in a separate paper.