In connection with the recent investigations of the instability of unbounded elliptical flows, some methods are discussed for the study of the instability of bounded flows. The stability of a ‘basic flow’ which is two-dimensional and rotating, with elliptical streamlines similar to the elliptical section of an experimentally studied cavity, is investigated in the framework of linear theory (for circular rotation, the flow discussed is stable). The regions of instability for three-dimensional disturbances are found in the plane of the parameters defining the geometry of the system (the height of the ellipsoidal cavity and the degree of ellipticity). It is shown that two types of instability exist, characterized by either monotone or oscillatory growth of the amplitudes of small disturbances.

The influence of the Coriolis force field on this instability mechanism is also studied. Rotation of the system as a whole changes the regions of instability in parameter space characterizing the geometry of the cavity and the wavenumbers of unstable disturbances. As a result, the Coriolis force may stabilize or destabilize the basic flow for a given geometry.

The instability of rotating density-stratified flow with elliptical streamlines is also considered.

We present observations of travelling-wave convection in ethanol/water mixtures in a rectangular container using shadowgraphic flow visualization along the axes of the convective rolls. The entrainment of solute into the centre of the fluid layer from the boundary layers at the top and bottom of the cell is clearly observed in the shadowgraph images as sharp features at the roll boundaries. The shadowgraph images are integrated to yield a map of the two-dimensional refractive-index field. The contributions to this field from the convective temperature and the concentration fields have sufficiently different spatial character to be reliably separated using image-processing techniques. This has allowed us to produce a map of these two flow fields that is in good quantitative agreement with numerical integrations of the full Navier–Stokes equations in the Boussinesq approximation. We also present qualitative visualizations of transients caused by abruptly triggering and quenching travelling-wave convection. These experiments illustrate the formation of concentration boundary layers and reveal the existence of large-scale currents which are an inherent part of travelling-wave convection.

A new cavity model that can explain the interaction between viscous effects including vortices and cavitation bubbles is presented in this study. This model, which is named a bubble two-phase flow (BTF) model, treats the inside and outside of a cavity as one continuum by regarding the cavity as a compressible viscous fluid whose density changes greatly. Navier–Stokes equations including cavitation bubble clusters are solved in finite-difference form by a time-marching scheme, where the growth and collapse of a bubble cluster is given by a modified Rayleigh's equation. Computation was made on a two-dimensional flow field around a hydrofoil NACA0015 at angles of attack of 8° and 20°. The Reynolds number was 3 × 105. The experiments were also performed at the same Reynolds number for comparison. The computed results by the BTF cavity model can express the feature of cloud-type cavitation shed from the trailing edge of the attached cavities when the angle of attack is 8°. It shows the mechanism of cavitation cloud generation and large-scale vortices. The boundary layer separates at the cavity leading edge. Then it rolls up and produces the cavitation cloud. In other words, the instability of the shear layer may produce the cavitation cloud. When the angle of attack is 20°, the flow was fully separated from the leading edge of the hydrofoil and vortex cavitation occurs in the separated region. The BTF cavity model can also express the generation of such vortex cavitation and the effect of cavitation nuclei in the uniform flow.

Results are given for experiments on water-lubricated pipelining of 6.01 P cylinder oil in a vertical apparatus in up- and downflow in regimes of modest flow rates, less than 3 ft/s. Measured values of the flow rates, holdup ratios, pressure gradients and flow types are presented and compared with theoretical predictions based on ideal laminar flow and on the predictions of the linear theory of stability. New flow types, not achieved in horizontal flows, are observed: bamboo waves in upflow and corkscrew waves in downflow. Nearly perfect core–annular flows are observed in downflows and these are nearly optimally efficient with values close to the ideal. The holdup ratio in upflow and fast downflow is a constant independent of the value and the ratio of values of the flow rates of oil and water. A vanishing holdup ratio can be achieved by fluidizing a long lubricated column of oil in the downflow of water. The results of experiments are compared with computations from ideal theory for perfect core–annular flow and from the linear theory of stability. Satisfactory agreements are achieved for the celerity and diagnosis of flow type. The wave is shown to be nearly stationary, convected with the oil core in this oil and all oils of relatively high viscosity. These results are robust with respect to moderate changes in the values of viscosity and surface tension. The computed wavelengths are somewhat smaller than the average length of the bamboo waves which are observed. This is explained by stretching effects of buoyancy and lubrication forces induced by the wave. Other points of agreement and disagreement are reviewed.

Gas flow through wire–plate electrostatic precipitators is influenced by a secondary flow of electrical origin known as electric wind. This phenomenon arises when significant momentum is transferred from corona-generated ions to the gas. Electric wind can produce turbulence and recirculation. This complex flow field was characterized in a simple, three-wire precipitator by flow visualization, electrostatic and fluid dynamic numerical modelling, and laser-Doppler anemometry (LDA).

Velocity of seeded smoke was measured by two-component LDA. Coulomb effects were regionally eliminated by performing measurements along symmetry axes where the electric field and streamwise velocity component were mutually perpendicular. Coulomb drift velocities were also estimated from field charging theory to allow interpretation of measured transverse velocities. For a low inlet velocity (0.5 m/s), mean flow recirculation was evident and turbulence intensities as high as 50 % were measured. Higher inlet velocities (1.0, 2.0 m/s) yielded no flow recirculation and lower turbulence levels that were polarity-dependent. Measured profiles of streamwise velocity showed that flow acceleration zones occurred upstream of each wire and also between wires near the collecting plate. The induced turbulence displayed significant inhomogeneity and anisotropy.

A combined finite-element, finite-difference electrostatic model was developed to yield ion density and electric field distributions within the precipitator. These predictions were used to incorporate an electric body force into a two-dimensional, turbulent fluid dynamic model based upon the k–ε formulation. The model predicted recirculating mean flow and turbulent diffusivities that were consistent with the smoke flow visualizations and LDA measurements.

The purpose of the present study is to characterize the process of laminar–turbulent transition at Reynolds numbers which are subcritical from the two-dimensional linear point of view. The development of a point-like disturbance was studied in an air flow channel with hot-wire anemometry at a Reynolds number of 1600. Localized disturbances were triggered at one of the walls and their development followed downstream by traversing the hot-wire probe in the streamwise direction over a distance of 90 half channel heights, as well as in the normal and spanwise directions. The disturbance evolved into elongated streaky structures with strong spanwise shear (i.e. normal vorticity) which grew in amplitude and streamwise extension and thereafter either decayed or gave rise to a turbulent spot. The results indicate that the mechanism underlying the initial growth is a linear one resulting from the coupling between the normal velocity and the normal vorticity, as described by the three-dimensional linear equations. The nonlinear development of the structure leads to the formation of intense normal shear layers and the appearance of oscillations and ‘spikes’, which multiply and form the rear or a turbulent spot.

Bethe–Zel'dovich–Thompson fluids are ordinary, single-phase fluids in which the fundamental derivative of gasdynamics is negative over a finite range of temperatures and pressures. We examine the steady transonic flow of these fluids over two-dimensional thin wings and turbine blades. The free-stream state is taken to be in the neighbourhood of one of the zeros of the fundamental derivative. It is shown that a modified form of the classical transonic small-disturbance equation governs such flows. Critical Mach number estimates are provided which take into account the non-monotone variation of the Mach number with pressure predicted in previous investigations. Critical Mach numbers well over 0.95 are predicted for conventional airfoil sections. Numerical solutions reveal substantial reductions in the strength of the compression shocks occurring in supercritical flows. Further new results include the prediction of expansion and compression shocks in the same flow and compression bow shocks in flows with subsonic free streams.

The zero-Prandtl-number limit of the Oberbeck—Boussinesq equations is compared to small-Prandtl-number Rayleigh—Bénard convection through numerical simulations. Both no-slip and free-slip boundary conditions, imposed at the top and bottom of a small-aspect-ratio, horizontally periodic box are considered. A rich variety of regimes is observed as the Rayleigh number is increased: supercritical oscillatory instabilities for various values of the aspect ratios, competition between two-dimensional rolls, squares and hexagonal patterns, competition between travelling and standing waves, transition to chaos, and scalings laws for the first Rayleigh-number decade. This multiplicity of regimes can be attributed to the close interaction between the stationary and oscillatory instabilities at zero Prandtl number.

Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increases as tα, then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t(1 + α)/3 (t(1 + α)/4). A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exists ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For α < ½ (α < 1), the diffusive lengthscale at the shock grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds as t½. The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. Additional forces are also considered: weak vertical gravity can also destroy the shock in finite time, while effects usually neglected from lubrication theory are important only early in spreading. Experiments have shown that the severe thinning of the film behind the shock can cause it to rupture: the dryout process is modelled by introducing van der Waals forces.

The linear-eddy turbulent mixing model, formulated to capture the distinct influences of turbulent convection and molecular transport on turbulent mixing of diffusive scalars, is applied to two mixing configurations in homogeneous flow: a scalar mixing layer and a two-line-source configuration. Finite-rate second-order chemical reactions are considered, as well as the limits of fast reaction and frozen flow. Computed results are compared to measurements in a reacting-scalar mixing layer and in a two-line-source configuration involving passive-scalar mixing. For each configuration, the adjustment of a single model parameter related to the turbulence integral scale yields computed results in good agreement with a variety of measured quantities. The results are interpreted with reference to a simpler model, based solely on large-scale flapping effects, that reproduces many qualitative trends. For three-stream mixing with finite-rate chemistry, a novel dependence of the reactant correlation coefficient on the chemical reaction rate is predicted.

The flow of a linearly stratified fluid past a sphere is considered experimentally in the Froude number Fi, Reynolds number Re, ranges 0.005 ≤ Fi ≤ 20 and 5 ≤ Re ≤ 10000. Flow visualization techniques and density measurements are used to describe the rich range of characteristic flow phenomena observed. These different flow patterns are mapped on a detailed Fi against Re flow regime diagram. In most instances the flow patterns were found to be very different from those observed in homogeneous fluids. Vortex shedding characteristics, for example, were found to be dramatically affected by the presence of stratification. Where possible, the results are compared with available analytical and numerical models.

Experiments of the type described by Phillips et al. (1986) were performed using different slopes with the aim of examining the slope dependence of the buoyancy and volume transports, particularly at small slopes. The new observations confirmed the general flow patterns described for experiments conducted at a fixed slope, but a reconsideration of the local balances suggest that the buoyancy flux at small slopes θ is proportional to $(\kappa N)^{\frac{3}{2}}\sin\theta $, where κ is the turbulent diffusivity. and the volume flux associated with the overall convergence flow is linear with depth and proportional to (κ3/2/N½h) sin θ, where h is the thickness of the pycnocline. These differ in their dependence on the slope suggested (but not tested) in Phillips et al., and are generally consistent with measurements over a range of slopes from 7.1° to 23.5°.

In this report, a model designed for the description of the flow of two miscible phases in a fluidized bed is discussed. Apart from basic problems of modelling accurately such multi-phase flows, little analytical progress had been achieved in the investigation of a certain standard model based on the theory of interacting continua. It turns out, however, that the model under consideration can be investigated with the help of bifurcation theory. In particular, the methods of the theory of bifurcation with symmetry can be applied owing to the symmetries of the system.

In general, a stationary homogeneous state exists in fluidized beds which can become unstable when the physical parameters of the system are varied. Then pattern formation takes place, e.g. in the form of one- and/or two-dimensional waves, bubbles, or convection patterns; also turbulent behaviour has been observed.

In order to understand the occurrence of wave patterns and other phenomena as an inherent feature of the system, a finite, but periodically continued two-dimensional bed is investigated. While this suppresses certain boundary effects, it gives us thorough insight into the principal behaviour of this complicated system.

In particular, it allows us not only to perform easily a linear stability analysis of the basic state of uniform fluidization, but also to conclude that bifurcation of travelling waves occurs when this state becomes unstable. Well-known patterns like vertical and oblique travelling waves (OTW) of the form $u(x,y,t) = \tilde{u}(x-\omega t\pm ky)$, k [ges ] 0, are discovered. Owing to symmetry, the existence of standing travelling waves (STW) of the form $u(x,y,t) = \tilde{u}(x-\omega t, y)$ is also expected, but regrettably no mathematically rigorous proof of this last conjecture is presently available.

Bubble formation can also be approached via the instability of a vertical travelling plane wave train to transverse perturbations. Then a secondary stationary bifurcation to another kind of standing travelling waves takes place. This scenario is also in agreement with experimental observations. In addition, the occurrence of bifurcations of higher order, which lead to more and more complex wave patterns and are to be found on the route to turbulence, can be deduced.

A new singular-vortex theory is presented for geostrophic, beta-plane dynamics. The stream function of each vortex is proportional to the modified Bessel function Ko(pr), where p can be an arbitrary positive constant. If p−1 is equal to the Rossby deformation scale Rd, then the vortex is a point vortex; for p−1 ≠ Rd the relative vorticity of the vortex contains an additional logarithmic singularity. Owing to the β-effect, the redistribution of the background potential vorticity produced by the vortices generates a regular field in addition to the velocity field induced by the vortices themselves. Equations governing the joint evolution of singular vortices and the regular field are derived. A new invariant of the motion is found for this system. If the vortex amplitudes and coordinates are set in a particular way then the regular field is zero, and the vortices form a system moving along latitude circles at a constant speed lying outside the range of the phase velocity of linear Rossby waves. Each of the systems is a discrete two-dimensional Rossby soliton and, vice versa, any distributed Rossby soliton is a superposition of the singular vortices concentrated in the interior region of the soliton. An individual singular vortex is studied for times when Rossby wave radiation can be neglected. Such a vortex produces a complicated spiral-form regular flow which consists of two dipoles with mutually perpendicular axes. The dipoles push the vortex westward and along the meridian (cyclones move northward, and anticyclones move southward). The vortex velocity and trajectory are calculated and applications to oceanic and atmospheric eddies are given.

Laminar—turbulent transition in the decelerating boundary layer of Gad-el-Hak et al. (1984) is studied by solving the incompressible, time-dependent, three-dimensional Navier—Stokes equations numerically. The temporal and spatial evolution of the experimental flow structures is approximated with spatial periodicity and temporal evolution while maintaining mean-velocity profiles appropriate to the spatially-developing flow. Other than the mean flow, the initial flow fields include only small-amplitude white random noise. This and the large streamwise and spanwise dimensions of the numerical domain allow unstable waves to be selected by the dynamics, instead of being imposed arbitrarily as in previous numerical studies. In that sense, the transition is natural. In the early stages of transition, two-dimensional and slightly oblique waves grow rapidly owing to inflexional instability. Their subsequent nonlinear interactions trigger the breakdown and create a pattern of Λ vortices. The patterns of Λ vortices are more irregular than those found in ribbon-induced transition. The tips of the Λ vortices are rarely aligned with the flow direction, and they appear in local patches, consistent with the experimental visualizations of Gad-el-Hak et al. A simple model based on the interference of multiple modes of instability accounts for these features, but the specific pattern of Λ vortices depends on the random content of the initial flow field. A simulation of the later stages of transition is performed with higher numerical resolution, showing that the ‘naturally-born’ Λ vortices undergo breakdown processes similar to those of their ribbon-induced counterparts.

This paper provides a unique, detailed evaluation of basic aeroacoustic theory applied to low-Mach-number (M = 0.08) cold jets. In contrast to most prior studies comparing theoretical predictions of jet noise with experimental results, our comparison uses a relatively complete knowledge of the flow field and employs vortex sound theory — an acoustic analogy which is shown to be insensitive to those aspects of the flow field about which our knowledge is incomplete. The primary result is that the measured sound field directivity of vortex ring pairing in circular jets is very similar to that predicted by theory: a stationary, axisymmetric, lateral quadrupole. This directivity is very unlike the monotonic polar dependence found in time-average measures of jet noise fields and unlike the directivity found in similar excited jet experiments. Although not perfect, the agreement between experiment and theory here is satisfyingly close in comparison to the discrepancies found by Huerre & Oighton (1983). Our result also proves that pairing of purely axisymmetric coherent structures is not the dominant sound source in low-Mach-number jets and that vortex asymmetry must be an essential aspect of the vortex motions which produce noise in such jets.

The aerodynamic and thermal aspects of the wall jet flow, formed after angled impingement of an axisymmetric jet, have been studied with emphasis on the large-scale transport of the passive scalar by the spatially coherent structures. Time-averaged and instantaneous structures of the turbulent flow were examined by visualization and local measurements of a jet arrangement with an impingement angle between the jet axis and the surface normal of 20°, a nozzle-to-plate distance to nozzle exit diameter ratio of 22, and a nozzle exit Reynolds number of 1.3 × 104.

The results show that the oblique impingement introduced vertical velocities so that boundary-layer approximations were inapplicable and led to the distribution of the time-averaged properties of the velocity and temperature field with strong azimuthal dependence, which increased gradually with angle of impingement to 40° where a sudden change of the orientation of the contours of surface pressure and temperature took place. It also led to instantaneous, spatially coherent structures which were most pronounced at an angle of 20°. These structures improved the large-scale transport of the passive scalar but, owing to the extreme regularity of their path, also led to an inactive zone near the vortex centre.

The inner region of the decelerating wall jet exhibited a momentum equilibrium layer extending to the point of radial velocity maximum and the intercept of the linear region of the semilogarithmic wall law varied in the local streamwise direction as for turbulent flows over rough walls with adverse pressure gradient. The thermal equilibrium layer had an invariant functional form but extended far beyond the point of maximum velocity.

The cell size or the wavenumbers of supercritical hexagonal convection cells in primarily surface-tension-driven convection on a uniformly heated plate has been studied experimentally in thermal equilibrium in thin layers of silicone oil of large aspect ratio. It has been found that the cell size decreases with increased temperature difference in the slightly supercritical range, and that the cell size is unique within the experimental error. It has also been observed that the cell size reaches a minimum and begins to increase at larger temperature differences. This reversal of the rate of change of the wavenumber with temperature difference is attributed to influences of buoyancy on the fluid motion. The consequences of buoyancy have been tested with three fluid layers of different depth.

Two-dimensional free-surface flows due to a pressure distribution moving at a constant velocity U at the surface of a fluid of infinite depth are considered. Both gravity g and surface tension T are included in the dynamic boundary condition. The velocity U is assumed to be smaller than (4gT/ρ)¼, so that there are no waves in the far field. Here ρ is the density of the fluid. The problem is solved numerically by a boundary integral equation technique. It is shown that for some values of U, four different flows are possible. Three of these flows are interpreted as perturbations of solitary waves in water of infinite depth. It is found that both elevation and depression solitary waves are possible in water of infinite depth. The numerical results for depression waves confirm and extend the solutions previously computed by Longuet-Higgins (1989).

The linear stability of a layer of compressible fluid, permeated by an oblique magnetic field, is discussed. It is shown that regardless of the system parameters, all bifurcations generically lead to travelling waves. Wave speeds and direction of the wave propagation are investigated. Symmetry arguments are used to show that when the field is almost vertical, waves with a wave vector aligned with the tilt are preferred over those with a wave vector perpendicular to the tilt. The nonlinear development of the travelling waves is explored using simple model equations.