This paper is concerned with calculations regarding a collection of small gas bubbles rising under buoyancy in a clear liquid. For dilute mixtures interactions can be restricted to those between two bubbles. In the analysis of binary interactions it is assumed that the Reynolds number for relative motion between bubbles and liquid is large, that surfactants, if present, do not modify the condition of zero tangential stress at the bubble-liquid interface, and that bubbles bounce at an encounter.

A two-bubble probability density is derived from the analysis, valid on a short timescale associated with the interaction. It is shown that on a long timescale, based on viscous dissipation, clustering together of pairs takes place, most likely even when triple encounters are allowed for. An analysis is given of the vertical motion of pairs, followed by a calculation of the mean vertical bubble velocity with help of the (short timescale) probability density function. The result is compared with experimental data.

This investigation addresses the unsteady wake from a blunt trailing edge subjected to controlled perturbations. The relationship between the structure of the near wake, the surface loading on the edge, and the motion of the edge is characterized by flow visualization in conjunction with velocity and pressure measurements. The response of the near wake can be classified into two general categories: a modulated wake, characterized by ordered variations in the near-wake flow structure over a number of cycles of oscillation of the trailing edge; and a phase-locked wake, whereby the near-wake structure does not change from cycle to cycle of the edge oscillation. For the modulated wake, there are large, repetitive excursions of the near-wake vortex pattern in the stream wise direction due to coexistence of the self-excited global instability of the wake and the applied excitation. These excursions can have an amplitude two orders of magnitude larger than the amplitude of the edge motion. The duration of these excursions, in relation to the cyclic motion of the trailing edge, is deterministic. For the phase-locked wake, small changes of the edge oscillation frequency produce large changes in the phase shift of the initially formed vortex from the edge. These phase shifts are due to changes in the times required for vortex formation and departure from the near wake. The corresponding mechanisms are interpreted in terms of the crucial topological features of the near wake and a phase clock concept.

The steady hydraulics of a continuously stratified fluid flowing from a stagnant reservoir through a horizontal contraction was studied experimentally and theoretically. As the channel narrows, the flow accelerates through a succession of virtual controls, at each of which the flow passes from sub-critical to supercritical with respect to a particular wave mode. When the narrowest section acts as a control, the flow is asymmetric about the narrowest section, supercritical in the divergent section and self- similar throughout the channel. With increased flow rate a new enclosed self-similar solution was found with level isopycnals and velocity uniform with depth. This flow is only symmetric in the immediate neighbourhood of the narrowest section, and in the divergent section remains supercritical with respect to higher internal modes, has separation isopycnals and splits into one or more jets separated by regions of stagnant, constant-density fluid. Flows which are subcritical with respect to lowest modes can also be asymmetric about the narrowest section for higher internal modes. The experiments are interpreted using steady, inviscid hydraulic theory. Solutions require separation isopycnals and regions of stationary, constant-density fluid in the divergent section.

The propagation of guided internal waves on non-uniform large-scale flows of arbitrary geometry is studied within the framework of linear inviscid theory in the WKB-approximation. Our study is based on a set of Hamiltonian ray equations, with the Hamiltonian being determined from the Taylor-Goldstein boundary-value problem for a stratified shear flow. Attention is focused on the fundamental fact that the generic smooth non-uniformities of the large-scale flow result in specific singularities of the Hamiltonian. Interpreting wave packets as particles with momenta equal to their wave vectors moving in a certain force field, one can consider these singularities as infinitely deep potential holes acting quite similarly to the ‘black holes’ of astrophysics. It is shown that the particles fall for infinitely long time, each into its own ‘black hole‘. In terms of a particular wave packet this falling implies infinite growth with time of the wavenumber and the amplitude, as well as wave motion focusing at a certain depth. For internal-wave-field dynamics this provides a robust mechanism of a very specific conservative and moreover Hamiltonian irreversibility.

This phenomenon was previously studied for the simplest model of the flow non-uniformity, parallel shear flow (Badulin, Shrira & Tsimring 1985), where the term ‘trapping’ for it was introduced and the basic features were established. In the present paper we study the case of arbitrary flow geometry. Our main conclusion is that although the wave dynamics in the general case is incomparably more complicated, the phenomenon persists and retains its most fundamental features. Qualitatively new features appear as well, namely, the possibility of three-dimensional wave focusing and of ‘non-dispersive’ focusing. In terms of the particle analogy, the latter means that a certain group of particles fall into the same hole.

These results indicate a robust tendency of the wave field towards an irreversible transformation into small spatial scales, due to the presence of large-scale flows and towards considerable wave energy concentration in narrow spatial zones.

Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number Mt of the upstream turbulence and the shock strength M21 — 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 ≤ Mt ≤ 0.110 and 1.05 ≤ M1 ≤ 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations.

Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M2t < a(M21 — 1) with a ≈ 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term.

Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for M2t < a(M21— 1) with a ≈ 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with M2t > a(M21— 1 with a ≈ 0.1, shock waves no longer had well-defined fronts: shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.

We develop an analytical model to describe the generation of vapour as water moves through a hot porous rock, as occurs in hot, geothermal reservoirs. Typically the isotherms in the liquid lag behind the water-vapour interface and so water is supplied to the interface at the interface temperature. This temperature is lower than that in the rock far ahead of the interface. Therefore, as the hot porous rock is invaded with water, it cools and the heat released is used to vaporize some of the water. At low injection rates, vapour formed from the injected liquid may readily move ahead of the advancing liquid-vapour interface and so the interfacial pressure remains close to that in the far field ahead of the interface. The mass fraction that vaporizes is then limited by the superheat of the rock. For larger injection rates, the interfacial vapour pressure becomes considerably greater than that in the far field in order to drive the vapour ahead of the moving interface. As a result, the interfacial temperature increases. The associated reduction in the thermal energy available for vaporization results in a decrease in the mass fraction of vapour produced.

Since the vapour is compressible, the motion of the vapour ahead of the interface is governed by a nonlinear diffusion equation. Therefore, the geometry of injection has an important effect upon the mass fraction of water that vaporizes. We show that with a constant supply of water from (i) a point source, the mass fraction of water which vaporizes increases towards the maximum permitted by the superheat of the rock; (ii) a line source, a similarity solution exists in which the mass fraction vaporizing is constant; and (iii) a planar source, the liquid-vapour interface steadily translates through the rock with a very small fraction of the injected water vaporizing.

The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning point Xt of the WKBJ approximation, while its quantization is provided by the connection of solutions across Xt. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.

An incompressible inviscid flow theory for single and two-element airfoils experiencing trailing-edge stall is presented. For the single airfoil the model requires a simple sequence of conformal transformations to map a Joukowsky airfoil, partially truncated on the upper surface, onto a circle over which the flow problem is solved. Source and doublet singularities are used to create free streamlines simulating shear layers bounding the near wake. The model's simplicity permits extension of the method to airfoil-flap configurations in which trailing-edge stall is assumed on the flap. Williams’ analytical method to calculate the potential flow about two lifting bodies is incorporated in the Joukowsky-arc wake-singularity model to allow for flow separation. The theoretical pressure distributions from these models show good agreement with wind-tunnel measurements.

A linear stability analysis of laterally confined swirling flow is given, of the type described by Long's equation in the inviscid limit or by the von Kármán similarity equations in the absence of lateral confinement. The flow of interest involves identical counterflowing fluid streams injected with equal velocity W0 through opposing porous disks, rotating with angular velocities Ω and ±Ω, respectively, about a common normal axis. By means of mass transfer experiments on an aqueous system of this type we have detected an apparent hydrodynamic instability having the appearance of an inviscid supercritical bifurcation at a certain |Ω| > 0. As an attempt to elucidate this phenomenon, linear stability analyses are performed on several idealized flows, by means of a numerical Galerkin technique. An analysis of high-Reynolds-number similarity flow predicts oscillatory instability for all non-zero Ω. The spatial structure of the most unstable modes suggests that finite container geometry, as represented by the confining cylindrical sidewalls, may have a strong influence on flow stability. This is borne out by an inviscid stability analysis of a confined flow described by Long's equation. This analysis suggests a novel bifurcation of the inviscid variety, which serves qualitatively to explain the results of our mass transfer experiments.

This work brings together experimental and theoretical studies of nonlinear stages aimed at the K-regime in boundary-layer transition, and some combined theoretical and experimental results are discussed. It is shown that the initial stages in the formation of so-called spikes, observed in many experiments, may be described very well by the asymptotic theory. These flashes-spikes are shown to be (in certain regimes) possible solitons of the boundary layer and governed by the integral-differential Benjamin-Ono equation. Properties of the spike-solitons, obtained both theoretically and experimentally in the quasi-planar stages of their development, are presented. Features of the disturbance behaviour connected with the subsequent development of three-dimensionality are also discussed, as are the effects of viscosity and shorter lengthscales. The main conclusion of the work concerns the hypothesis of the possible soliton nature of the flashes-spikes (within limits), which seems reliably corroborated by the good agreement found between the theory and the experimental data.

Steady axially invariant (fully developed) incompressible laminar flow of a Newtonian fluid in helical pipes of constant circular cross-section with arbitrary pitch and arbitrary radius of coil is studied. A loose-coiling analysis leads to two dominant parameters, namely Dean number, Dn = Reλ½, and Germano number, Gn = Reη, where Re is the Reynolds number, λ is the normalized curvature ratio and η is the normalized torsion. The Germano number is embedded in the body-centred azimuthal velocity which appears as a group in the governing equations. When studying Gn effects on the helical flow in terms of the secondary flow pattern or the secondary flow structure viewed in the generic (non-orthogonal) coordinate system of large Dn, a third dimensionless group emerges, γ = η/(λDn)½. For Dn < 20, the group γ* = Gn Dn-2 = η/(λRe) takes the place of γ.

Numerical simulations with the full Navier-Stokes equations confirmed the theoretical findings. It is revealed that the effect of torsion on the helical flow can be neglected when γ ≤ 0.01 for moderate Dn. The critical value for which the secondary flow pattern changes from two vortices to one vortex is γ* > 0.039 for Dn < 20 and γ > 0.2 for Dn ≥ 20. For flows with fixed high Dean number and A, increasing the torsion has the effect of changing the relative position of the secondary flow vortices and the eventual formation of a flow having a Poiseuille-type axial velocity with a superimposed swirling flow. In the orthogonal coordinate system, however, the secondary flow generally has two vortices with sources and sinks. In the small-γ limit or when Dn is very small, the secondary flow is of the usual two-vortex type when viewed in the orthogonal coordinate system. In the large-γ limit, the appearance of the secondary flow in the orthogonal coordinate system is also two-vortex like but its orientation is inclined towards the upper wall. The flow friction factor is correlated to account for Dn, A and γ effects for Dn ≤ 5000 and γ < 0.1.

This paper examines the large-scale dynamics of a layer of stratified fluid on the β-plane. A three-dimensional asymptotic system is derived which governs geostrophic flows with large displacement of isopycnal surfaces. This is then reduced to a two-dimensional set of equations which describe the interaction of a baroclinic ‘quasi-mode’ with arbitrary vertical profile and barotrophic motion. The baroclinic instability of large-amplitude zonal flows with vertical shear is studied within the framework of these equations. In the case where the displacement of isopycnal surfaces is small, the results obtained should overlap with the ‘traditional’ baroclinic instability of quasi-geostrophic (small-amplitude) flows. In order to compare the two types of instability, the quasi-geostrophic boundary-value problem is solved asymptotically for the case of long-wave disturbances and weak β-effect (the latter limit of quasi-geostrophic theory has not been considered previously). The instability that is found is linked to the Hamiltonian structure of the governing equations. The equations derived are generalized for the case of more than one baroclinic quasi-mode.

When a train of gravity waves encounters an opposing current, the wavelength is shortened and the waves may be reflected. If capillarity is included, the shortened waves may be reflected for a second time and experience further shortening. By this process the initially long gravity waves can be damped by viscosity quickly without breaking. In this paper a boundary-layer approximation is obtained close to the reflection points, and is matched to the ray approximations outside. This is then applied to the propagation of a wavepacket. Damping is accounted for in the ray solution and the result is compared to the undamped solution. The case where the two reflection points coalesce is also considered. It is found that as the separation between the reflection points decreases, the wavepacket appears to remain longer in the region of reflections relative to the width of this region.

The simulation method for prolate spheroids in Stokes flow introduced in a companion paper (Claeys & Brady 1993a) is extended to handle statistically homogeneous unbounded dispersions. The convergence difficulties associated with the slow decay of velocity disturbances at zero Reynolds number are overcome by applying O';Brien's renormalization procedure. The Ewald summation technique is employed to accelerate the evaluation of all mobility interactions. As a first application of this new method, the hydrodynamic transport properties of equilibrium hard-ellipsoid structures are calculated for aspect ratios ranging from 3 to 50. Calculated viscosities in the isotropic phase agree reasonably well with published experimental measurements.

The nonlinear interaction between an incident and a reflected Rossby wave produces a steady flow parallel to the (non-zonal) reflecting wall and a transient flow oscillating at twice the frequency of the incident-reflected pair. If the transient forcing is resonant, i.e. a free Rossby wave, the resonant response must have zero amplitude at the wall in order to fulfil the boundary condition there; a straightforward expansion predicts a linear growth of its amplitude in the offshore direction y. Resonance is possible only if 0 < |sin α| ≤ $\frac13$, where α is the angle between the wall and the easterly direction. This requirement is met by several boundaries in the ocean. A simple graphical method to find a resonant triad is described.

Using the method of multiple scales, it is shown that the wave amplitudes of the triad are slowly varying periodic functions of y, such that the energy flux of the triad through any plane parallel to the wall vanishes, as required by energy conservation. The waves participating in the resonant triad become wave packets. The three waves do not exchange energy in time due to the additional constraint on the motion imposed by the boundary condition at the wall. It is shown that the wave amplitudes cannot be slowly varying functions of v and time.

As a possible oceanic application of the theoretical findings, the distance from the wall where one would expect to find large semi-annual amplitudes if annual Rossby waves are impinging on the boundary is of the order of 100 km. Motivated by similar studies (Plumb 1977; Mysak 1978), there are speculations on what would happen if three incident-reflected Rossby wave pairs (or modes) are taken, allowing each mode amplitude to be slowly varying in time.

Spectral local isotropy tests are applied to direct numerical simulation data, mainly at the centreline of a fully developed turbulent channel flow. Despite the small Reynolds number of the simulation, the high-wavenumber behaviour of velocity and vorticity spectra is consistent with local isotropy. This consistency is verified by the relationship between streamwise wavenumber spectra and spanwise wavenumber spectra. The high-wavenumber behaviour of the pressure spectrum is also consistent with local isotropy and compares favourably with the calculation of Batchelor (1951), which assumes isotropy and joint normality of the velocity field at two points in space. The latter assumption is validated by the shape but not the magnitude of the quadruple correlation of the streamwise velocity fluctuation at small separations. There is only partial support for local spectral isotropy away from the centreline as the magnitude of the mean strain rate increases.

The laminar flow in a curved channel is studied numerically to analyse the initial formation, development and interaction phenomena of an array of centrifugally induced longitudinal vortices arranged across the span of the channel. Simulations employing streamwise periodic boundary conditions (temporal model) as well as inlet-outlet conditions (spatial model) are carried out. In the temporal approach the interactions (pairing of vortices and growth of new vortex pairs) of fully developed vortex pairs are time-dependent, whereas in the spatial approach these events are inherently steady and concern vortices not in their fully developed state. The initial spatial development of the vortices is in excellent agreement with results of a linear stability analysis up to fairly large disturbance amplitudes. In the nonlinear regime a good agreement with experimental results has also been found. The receptivity of the flow is very important in a convectively unstable situation such as the present one and different behaviour is found at fixed Reynolds number (equal to 2.43 times the critical value for the onset of Dean vortices): the flow can be either steady or undergo a continuous sequence of merging and splitting events, depending on the inlet conditions. In the latter situation decorrelated patterns of low- and high-speed streaks are produced in streamwise-spanwise planes and they bear several similarities to near-wall coherent structures of turbulent boundary layers.