ALBERT

Description: The radiocarbon activity of carbon collected by vacuum distillation from a single partially saturated tuff began to decline after approximately 60% of the water and carbon had been extracted. Disproportionate changes in 14C activity and δ13C during distillation rule out simple isotopic fractionation as a causative explanation. Additional phenomena such as matrix diffusion and ion exclusion in micropores may play a role in altering the isotopic value of extracted carbon, but neither can fully account for the observed changes. The most plausible explanation is that distillation recovers carbon from an adsorbed phase that is depleted in 14C relative to DIC in the bulk pore water.

Description: Since 1996, the Mongolian-American Expedition to Northern Mongolia has been excavating in the Egiin Gol Valley. The goal of this research has been to examine the competing hypotheses explaining the emergence of pastoral nomadism and the evolution of nomadic complexity. The chronological placement of burials and sites in the survey area has been a key facet of this research. At present, these investigations have generated 10 radiocarbon dates from archaeological contexts. Presented here are the previoulsy unpublished 14C dates and some comments on their significance.

Description: A first-order ΔR correction value for marine samples is presented based on 3 radiocarbon determinations of known-age marine shells from Samoa.

Description: A 23-yr record of the measuring accuracy of the Copenhagen radiocarbon dating laboratory has retrospectively been provided through a true blind test. A total of 92 samples of oak from old tree trunks were dated in the period 1971 to 1993 and their dendrochronological age determined independently. The 14C activity of the dendrochronological samples measured in the Copenhagen radiocarbon laboratory was compared to the activity of the tree rings of the same age measured by Stuiver and Pearson (1993) for calibration purposes. The average difference was found to be 54 ± 72 14C yr. The results further indicate that the actual standard deviation is only 7% higher than that quoted by the laboratory. The investigation has shown a long-term stability of laboratory accuracy with no systematic laboratory variations either with respect to sample age or to the time of measurement from 1971 to 1993.

Description: We have studied 6 reference sections of bog and lake sediments in the Leningrad and Novgorod provinces to develop a geochronological scale for vegetational and paleoclimatic changes in northwestern Russia during the Late Glacial and Holocene. Every 10-cm layer along the peat and gyttja sections (4–8.5 m thick) was investigated palynologically and the great majority of them were radiocarbon dated. Using the data obtained, standard palynological diagrams were plotted and vegetation history reconstructed. The palynozones indicated on the diagrams were related to the climatic periods and subperiods (phases) of the Blytt-Sernander scheme. On the basis of 230 14C dates obtained, we derived the geochronology of climatic periods and phases, as well as the chronology for the appearance and areal distribution of forest-forming tree species. The uppermost peat layers were dated by using the “bomb effect”. We compared the stages of Holocene vegetation and paleoclimatic changes discovered for the Leningrad and Novgorod provinces with the those obtained for Karelia, which we had studied earlier using the same methodology.

Description: Since 1994, the Institute of Geological Sciences has undertaken an environmental monitoring program to measure radiocarbon levels in territory adjacent to active nuclear power plants (NPP). We determined 14C concentrations in natural objects from areas contiguous to Ignalina NPP as well as 14C background concentration in areas remote from the NPP. In the environs of the Ignalina station comparatively elevated levels of 14C were observed in vegetation and waters of Lake Drisvyaty. This appears to be a consequence of release of carbon radioisotope into the atmosphere and probably into waters of the lake during operation of the nuclear reactor.

Description: We studied a 12.6-m-long sequence from Lake Abiyata (Central Ethiopia) to establish a reliable and accurate chronology for use in global paleoclimatic reconstructions. The 26 accelerator mass spectrometry radiocarbon (AMS 14C) ages, performed on carbonates and organic matter, define 2 parallel chronologies, representing the complete Holocene period. However, these chronologies show a significant discrepancy from 500 to 900 BP in depth; ages obtained on carbonates were always older than those on organic matter. The hydrogeological and geochemical behavior of the Lake Abiyata basin has shed light on this discrepancy. We found that the carbonate crystallization is due mainly to the mixing of lake waters with groundwaters from the multi-layered aquifer contained in the 600-m-thick basement of the lake. The 14C activity of total dissolved inorganic carbon (TDIC) measured by AMS from bottom and surface lake waters (111.4 and 111.8 pMC, respectively) confirms that the mixing occurs at the water-sediment interface. This evidence of groundwater participation in the carbonate crystallization calls into question the current paleoclimatic reconstructions based on inorganic carbonates in lakes. Specific attention should thus be given to the respective proportions of each end-member in the mixing for the quantitative estimation of the groundwater input. This will help to validate the paleoenvironmental reconstructions and to highlight an eventual diagenetical evolution of inorganic carbonates during burial, via the study of pore waters.

Description: A method is presented for calibrating radiocarbon ages based on statistical analysis of a large number of randomly distributed dates. One interesting feature of this method is that it is internal; that is, it allows one to extend a known calibration curve further back in time by using only 14C dates, with no reference to any other dating technique. A serious difficulty in implementing this method lies in assembling a sample of dates with the correct statistical properties.

Description: Measurements on same-age tree-ring samples from proximal Ural Mountain trees by the Ioffe Institute research group and at the University of Arizona demonstrate a variance corresponding to a standard deviation of ±5.1% for Ioffe compared to ±2.1% for Tucson. There is also a calibration difference of 4.3 ±1.2 ‰. Comparison of the same years measured in Seattle on wood from the Pacific Northwest shows an offset of 2.2 ± 0.5 ‰. This is not a calibration error, but rather is expected from the well-documented evidence for divergence and upwelling of 14C-depleted CO2 along the west coast of North America.

Description: Accelerator mass spectrometry (AMS) radiocarbon measurements were made on 120 samples collected between Antarctica and South Africa along 30°E during the WOCE-France CIVA1 campaign in February 1993. Our principal objective was to complement the Southern Ocean's sparse existing data set in order to improve the 14C benchmark used for validating ocean carbon-cycle models, which disagree considerably in this region. Measured 14C is consistent with the θ-S characteristics of CIVA1. Antarctic Intermediate Water (AAIW) forming north of the Polar Front (PF) is rich in 14C, whereas surface waters south of the PF are depleted in 14C. A distinct old 14C signal was found for the contribution of the Pacific Deep Water (PDW) to the return flow of Circumpolar Deep Waters (CDW). Comparison to previous measurements shows a 14C decrease in surface waters, consistent with northward displacement of surface waters, replacement by old deep waters upwelled at the Antarctic Divergence, and atmospheric decline in 14C. Conversely, an increase was found in deeper layers, in the AAIW. Large uncertainties, associated with previous methods for separating natural and bomb 14C when in the Southern Ocean south of 45°S, motivated us to develop a new approach that relies on a simple mixing model and on chlorofluorocarbon (CFC) measurements also taken during CIVA1. This approach leads to inventories for CIVA1 that are equal to or higher than those calculated with previous methods. Differences between old and new methods are especially high south of approximately 55°S, where bomb 14C inventories are relatively modest.

Description: We describe a new methodology for separating organic temper from archaeological ceramics from Brazilian Amazonia. These experimental procedures were designed to directly date ceramic samples by accelerator mass spectrometry (AMS). An evaluation of the total carbon indicates the samples’ potential for dating.

Description: A promising species for tropical dendrochronology is Pinus lagunae, a pine tree found in Baja California Sur (Mexico) around lat 23.5°N. In 1995, we sampled a total of 27 wood cores from 13 Pinus lagunae trees in Sierra La Victoria (23°36'N, 109°56'W), just north of Sierra La Laguna, at an elevation of 1500–1600 m. Selected trees were locally dominant, but their ring-width patterns could not be crossdated. To test the hypothesis that visible growth layers in Pinus lagunae are formed annually, we measured radiocarbon amounts in individual rings by means of accelerator mass spectrometry (AMS). Twenty-three 14C measurements were used to trace the location of the 1963–64 “bomb spike” in 3 wood increment cores. By comparing the location of that δ14C extreme with the number of visible radial wood increments, it was possible to conclude that 2 cores had a number of locally absent rings, while the 3rd one included a few years with more than one growth layer. Therefore, ring-width patterns of sampled Pinus lagunae were not consistent from one tree to another, most likely because of climatic regime in combination with microsite features. While the possibility of generating Pinus lagunae tree-ring chronologies cannot entirely be ruled out, the development of dendrochronological proxy records of climate from coniferous species in tropical North America should focus on species and sites that experience a more pronounced seasonality.

Description: We have experimentally examined the effects of bubble size (0.4 ≤ λ ≤ 2.0), inclination angle (0° ≤ α ≤ 90°), and tube material on suspended gas bubbles in flows in tubes for a range of Weber (0 ≤ We ≤ 3.6), Reynolds (0 ≤ Re ≤ 1200), and Froude (0 ≤ Frα ≤ 1) numbers. Flow rates and associated pressure differences which allow the suspension of bubbles in glass and acrylic tubes are measured. Due to contact angle hysteresis, bubbles which dry the tube wall (i.e. form a gas-solid interface) may remain suspended over a range of flows while non-drying bubbles remain stationary for a single flow rate depending on experimental conditions. Stationary bubbles increase the axial pressure gradient with larger bubbles and steeper inclination angles leading to the greatest increase in the pressure gradient. Both the suspension flow range and pressure difference modifications are strongly dependent upon gas/liquid/solid material interactions. Stronger contact forces, i.e. smaller spreading coefficients, cause dried bubbles in acrylic tubes to remain stationary over a wider range of suspension flows than bubbles in glass tubes. Bubble deformation is governed by the interaction of interfacial, contact, and flow-derived forces. This investigation reveals the importance of bubble size, tube inclination, and tube material on gas bubble suspension.

Description: We have performed a numerical study on the transition of a cylindrical pipe flow under the influence of a localized disturbance in the form of periodic suction and blowing (PSB) applied at the pipe wall. We focus here on the so-called receptivity problem where the spatial evolution of this disturbance is studied as it travels downstream through the pipe. The study is carried out by means of two techniques: an eigenmode expansion solution (EES) and a full nonlinear direct numerical simulation (DNS). The EES is based on an analytical expansion in terms of the eigenfunctions of the linear operator which follows from the equations of motion expressed in a cylindrical coordinate system. The DNS is formulated in terms of a spectral element method. We restrict ourselves to a so-called subcritical disturbance, i.e. the flow does not undergo transition. For very small amplitudes of the PSB disturbance the results of the EES and DNS techniques agree excellently. For larger amplitudes nonlinear interactions come into play which are neglected in the EES method. Nevertheless, the results of both methods are consistent with the following transition scenario. The PSB excites a flow perturbation that has the same angular wavenumber and frequency as the imposed disturbance itself. This perturbation is called the fundamental mode. By nonlinear self-interaction of this fundamental mode higher-order harmonics, both in the angular wavenumber and frequency, are generated. It is found that the harmonic with angular wavenumber 2, i.e. twice the wavenumber of the fundamental mode, and with zero frequency grows strongly by a linear process known as transient growth. As a result the (perturbed) pipe flow downstream of the disturbance region develops extended regions of low velocity, known as low-speed streaks. At large disturbance amplitudes these low-speed streaks show the development of high wavenumber oscillations and it is expected that at even higher disturbance amplitudes these oscillations become unstable and turbulent flow will set in. Our result agrees (at least qualitatively) with the transition scenario in a plane Poiseuille flow as discussed by Reddy et al. (1998) and Elofson & Alfredson (1998).

Description: An experimental study was conducted in a turbulent spray flame in which droplets were produced ultrasonically at low velocity relative to the host gas. In this fashion, injector-specific effects on the two-phase flow were minimized and a scenario generally characteristic of the far field of practical spray systems could be simulated. Close to the burner exit, the spray flame appeared as a dense column of drops burning with an envelope flame. Further downstream, it opened up slowly in the radial direction and developed a turbulent 'brush' appearance. Measurements of the size, velocity and concentration of the droplets, and of gas-phase velocity and temperature were made by combining a Phase-Doppler interferometric technique with Stokes/anti-Stokes Raman thermometry. The experimental data were used to derive scaling and self-similarity for the Reynolds-averaged continuity and momentum equations using suitable transformations. Results showed three distinct regions, on the basis of the behaviour of the gas axial velocity in the spray flame. In the lower part of the flame, the gas momentum increased because of vaporization. In the intermediate region of the spray flame, the axial velocity decayed along the centreline as an inverse power of the distance from the virtual origin, with exponents smaller than unity. In the upper part of the spray flame, the flow field recovered the axial velocity decay that is typical of incompressible jets, namely as an inverse of the axial distance. Self-similar behaviour held for the axial velocity throughout the intermediate region. The vapour source term in the gas continuity equation scaled approximately as an inverse power of axial distance, and exhibited self-similarity throughout the spray flame. As a result, a simple model of the Reynolds stress term could be formulated, in which two competing contributions appear: one, that is due to turbulent transport, tends to increase the value of the velocity correlation; another, that is due to the vaporization term, tends to reduce the value of the velocity correlation and can be construed as a vaporization-induced tendency towards relaminarization. The first term is modelled by a classic gradient-transport approach yielding an empirical mixing length relating the velocity correlation to the average velocity gradient. Model and experiments are found to be in good agreement, especially sufficiently far from the injector, where one-way coupling between the two phases holds.

Description: The linear response of an inviscid two-dimensional Couette flow disturbed by a time-periodic forcing is studied under the assumption that the forcing is distributed along a straight line. When the forcing is tilted against the shear, the disturbance streamfunction and energy are shown to be locally amplified downstream of the source before decaying at large distance. This spatially localized amplification is interpreted as an analogue of the transient growth phenomenon studied in the context of unforced intial-value problems. The self-consistency of the linear approximation and the instability of the disturbance are also examined.

Description: The effect of entrained bubbles on the structure of a vortex ring is studied using particle image velocimetry. Quantitative information on the velocity and vorticity distribution within the vortex core is obtained from multiple images recorded with two 65 frames per second, 35 mm cameras. Bubble trajectories and velocities are also determined from these images. It is demonstrated that for certain combinations of vortex strengths and bubble diameters, a few microscopic bubbles, at very low overall void fraction, shift and macroscopically deform the structure of the vortex. For example, five 512 μm diameter bubbles, entrained by a vortex with core diameter of 2 cm and strength of 160 cm2 s−1, displace the core by 3.5 mm and fragment the core into two regions with peak vorticities that are 20% higher than the original maximum vorticity. The same phenomenon is observed with laminar, transitional and turbulent vortices. Dimensional analysis along with the experimental data show that the distortion is maximum when the bubbles settle, following entrainment by the vortex, in a region located between 20% and 40% of core radius. The governing dimensionless parameters and trends are identified and discussed. The vortex distortions are explained in terms of changes to the liquid momentum caused by the entrainment of the bubbles. It is argued and proven in detail in Appendix A that the change to the liquid momentum due to the presence of the bubble is equal to the bubble volume multiplied by the local stresses that exist in the absence of the bubble. These stresses include the gravity-induced (buoyancy) and hydrodynamic pressure gradients as well as viscous stresses. The buoyancy displaces the core of the vortex upward whereas the force due to hydrodynamic pressure gradients reduces the core size and as a result increases the vorticity. Estimated distortions agree with the experimental data.

Description: In this paper we consider the causal response of the inviscid shear-layer flow over an elastic surface to excitation by a time-harmonic line force. In the case of uniform flow, Brazier-Smith & Scott (1984) and Crighton & Oswell (1991) have analysed the long-time limit of the response. They find that the system is absolutely unstable for sufficiently high flow speeds, and that at lower speeds there exist certain anomalous neutral modes with group velocity directed towards the driver (in contradiction of the usual radiation condition of out-going disturbances). Our aim in this paper is to repeat their analysis for more realistic shear profiles, and in particular to determine whether or not the uniform-flow results can be regained in the limit in which the shear-layer thickness on a length scale based on the fluid loading, denoted ∈, becomes small. For a simple broken-line linear shear profile we find that the results are qualitatively similar to those for uniform flow. However, for the more realistic Blasius profile very significant differences arise, essentially due to the presence of the critical layer. In particular, we find that as ∈ → 0 the minimum flow speed required for absolute instability is pushed to considerably higher values than was found for uniform flow, leading us to conclude that the uniform-flow problem is an unattainable singular limit of our more general problem. In contrast, we find that the uniform-flow anomalous modes (written as exp(ikx - iωt), say) do persist for non-zero shear over a wide range of ∈, although now becoming non-neutral. Unlike the case of uniform flow, however, the k-loci of these modes can now change direction more than once as the imaginary part of ω is increased, and we describe the connection between this behaviour and local properties of the dispersion function. Finally, in order to investigate whether or not these anomalous modes might be realizable at a finite time after the driver is switched on, we evaluate the double Fourier inversion integrals for the unsteady flow numerically. We find that the anomalous mode is indeed present at finite time, once initial transients have propagated away, not only for impulsive start-up but also when the forcing amplitude is allowed to grow slowly from a small value at some initial instant. This behaviour has significant implications for the application of standard radiation conditions in wave problems with mean flow.

Description: A new mechanism is proposed for the generation of Tollmien-Schlichting (T-S) waves by free-stream turbulence. For definiteness and self-consistency, the mechanism is described mathematically by using a triple-deck formalism. The free-stream turbulence is represented by convecting gusts consisting of the so-called vortical and entropy waves of small amplitude. We show that suitable convecting gusts can interact with sound waves in the free stream to produce a forcing that has the same time and length scales as those of the T-S waves, thereby exciting such waves in the vicinity of the lower branch of the neutral stability curve. The T-S waves so produced have the order of magnitude of ε2R5/16, where ε is the amplitude of the free-stream disturbance and R the global Reynolds number. The scale conversion is achieved without resorting to any non-homogeneity on the wall, and hence the mechanism operates in a flat boundary layer. Furthermore, the T-S waves so generated do not undergo any immediate decay, as they may do in some other receptivity processes. For homogeneous isotropic free-stream turbulence, the spectrum of the T-S waves is obtained. The efficiency of the receptivity mechanism is assessed by parametric studies.

Description: A complete analytical study is presented of the reflection and transmission of surface gravity waves incident on ice-covered ocean. The ice cover is idealized as a plate of elastic material for which flexural motions are described by the Timoshenko-Mindlin equation. A suitable non-dimensionalization extracts parameters useful for the characterization of ocean-wave and ice-sheet interactions, and for scaled laboratory studies. The scattering problem is simplified using Fourier transforms and the Wiener-Hopf technique; the solution is eventually written down in terms of some easily evaluated quadratures. An important feature of this solution is that the physical conditions at the edge of the ice sheet are explicitly built into the analysis, and power-flow theorems provide verification of the results. Asymptotic results for large and small values of the non-dimensional parameters are extracted and approximations are given for general parameter values.

Description: The tri-diagonal determinant for Faraday resonance of a viscous liquid subject to an externally imposed vertical oscillation is expressed in terms of the surface-wave impedance of the liquid and developed as a continued fraction to obtain a systematic sequence of analytical approximations for the threshold acceleration. The impedance is calculated for either a clean or a fully contaminated surface on the assumption that the capillary, gravitational and viscous length scales are small compared with the breadth and depth of the liquid. Limiting approximations for weak and strong viscosity are constructed.

Description: Laser-induced fluorescence (LIF) and laser Doppler velocimetry (LDV) are used to explore the structure of a turbulent boundary layer over a wall made up of two-dimensional square cavities placed transversely to the flow direction. There is strong evidence of occurrence of outflows of fluid from the cavities as well as inflows into the cavities. These events occur in a pseudo-random manner and are closely associated with the passage of near-wall quasi-streamwise vortices. These vortices and the associated low-speed streaks are similar to those found in a turbulent boundary layer over a smooth wall. It is conjectured that outflows play an important role in maintaining the level of turbulent energy in the layer and enhancing the approach towards self-preservation. Relative to a smooth wall layer, there is a discernible increase in the magnitudes of all the Reynolds stresses and a smaller streamwise variation of the local skin friction coefficient. A local maximum in the Reynolds shear stress is observed in the shear layers over the cavities.

Description: The fully developed flow in a vertical cavity or duct subject to horizontal heating is considered. Solutions of the Boussinesq equations are obtained for rectangular and elliptic sections, in terms of Fourier series and polynomials, respectively. Both generalize the familiar odd-symmetric cubic profile of the plane cavity. Uniqueness is demonstrated under the restriction that the flow is independent of height. For cavities with rectangular sections, it is predicted and verified that the flow in the plane of spanwise symmetry is practically independent of the span if this exceeds 1.7 times the breadth.

Description: An experimental investigation on the initial instability of nonlinear deep-water wave trains including wind effects is reported. The experiment was conducted at the Ocean Engineering Laboratory wind-wave facility (50m long, 4.2m wide, 2.1m deep), with a fully computer-controlled mechanical wave generator to explore the parameter space: steepness; sideband frequency; wind speed. The estimated growth rates of the Benjamin-Feir instability from seeded wind-free experiments agreed well with the theoretical prediction derived from Krasitskii's four-wave reduced equation as computed here. Wind was added to the same wave system ; the growth rates of the sidebands were reduced for weak, and enhanced for strong wind forcing. Experiments with naturally selected sidebands, i.e. unseeded, were conducted as well; measurements showed that wind did not inhibit the growth of sidebands in the case of either two-dimensional or three-dimensional instabilities. A comparison of the results with earlier work suggests that there are two independent effects of wind: first, the alteration of the inviscid growth for a given modulational frequency as shown by comparison with the seeded experiments without wind; second, a change in the natural modulational frequency appearing in the presence of wind which is a function of the wave age, as observed in unseeded experiments. Both effects combined will determine whether the modulational instability is enhanced or suppressed; comparison of experimental results with theoretical predictions suggests that the effect of wind on the natural selection of the modulational frequency is the dominant effect. It was shown that for moderate to old waves, the net effect of wind on the modulational instability is small. For all the experiments except a few unseeded cases with weak breakers, the modulation was small and no breaking was observed within the tank.

Description: Using the continuum model of Pedley, Hill & Kessler (1988) for bioconvection in a suspension of swimming, gyrotactic micro-organisms, we investigate the existence and stability of a two-dimensional plume in tall, narrow chambers with stress-free sidewalls. The system is governed by the Navier-Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved numerically using a conservative finite-difference scheme. In sufficiently deep chambers, the plume is always unstable to both varicose and meandering modes. A linear stability analysis for an infinitely long plume predicts the growth rates of these instabilities, explains the mechanisms, and is in good agreement with the numerical results.

Description: A three-dimensional mathematical model based on the Brinkman extended Darcy equation has been used to study double-diffusive natural convection in a fluidsaturated porous cubic enclosure subject to opposing and horizontal gradients of temperature and concentration. The flow is driven by conditions of constant temperature and concentration imposed along the two vertical sidewalls of the cubic enclosure, while the remaining walls are impermeable and adiabatic. The numerical simulations presented here span a wide range of porous thermal Rayleigh number, buoyancy ratio and Lewis number to identify the different steady-state flow patterns and bifurcations. The effect of the governing parameters on the domain of existence of the three-dimensional flow patterns is studied for opposing flows (N 〈 0). Comprehensive Nusselt and Sherwood number data are presented as functions of the governing parameters. The present results indicate that the double-diffusive flow in enclosures with opposing buoyancy forces is strictly three-dimensional for a certain range of parameters. At high Lewis numbers multiple dipole vortices form in the transverse planes near the horizontal top and bottom surfaces, which the two-dimensional models fail to detect. The dipolar vortex structures obtained are similar to those created in laboratory experiments by the injection of fluid into a stratified medium.

Description: A coherent-vortex analysis is made of a computational solution for the free decay of homogeneous, Charney-isotropic geostrophic turbulence at large Reynolds number. The method of analysis is a vortex detection and measurement algorithm that we call a vortex census. The census demonstrates how, through non-conservative interactions among closely approaching vortices, the vortex population evolves towards fewer, larger, sparser, and more weakly deformed vortices. After emergence from random initial conditions and a further period of population adjustment, there is a period of approximately self-similar temporal evolution in the vortex statistics. This behaviour is consistent with a mean-vortex scaling theory based on the conservation of energy, vortex extremum, and vortex aspect ratio. This period terminates as the population approaches a late-time non-turbulent end-state vortex configuration. The end state develops out of merger and alignment interactions among like-sign vortices, and even during the scaling regime, local clusters of nearly aligned vortices are common.

Description: When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2πrm and width Δ ≪ rm around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that Δ ∝ rm3 and rm ∼ (tγ/πη) ln [tγ/(ηR)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A significantly different structure is found in which the outer-fluid forms a toroidal bubble of radius Δ ∝ rm3/2 at the meniscus and rm ∼ (tγ/4πη) ln [tγ/(ηR)]. This basic difference is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for rm(t) ≪ 1 involves matching of lengthscales of order rm2 rm3/2, rm, 1 and probably rm7/4.

Description: The influence of inertia and elasticity on the onset and stability of Taylor-vortex flow (TVF) is examined for an Oldroyd-B fluid. The Galerkin projection method is used to obtain the departure from Couette flow (CF). Only axisymmetric flow is examined. The solution is capable of capturing the dynamical behaviour observed experimentally for viscoelastic fluids in the inertio-elastic and purely elastic ranges. For flow with dominant inertia, the bifurcation picture is similar to that for a Newtonian fluid. However, transition from CF to TVF is oscillatory because of fluid elasticity. Steady TVF sets in, via supercritical bifurcation, as Re reaches a critical value, Rec. The critical Reynolds number decreases with fluid elasticity, and is strongly influenced by fluid retardation. As elasticity exceeds a critical level, a subcritical bifurcation emerges at Rec, similar to that predicted by the Landau-Ginzburg equation. It is found that slip along the axial direction tends to be generally destabilizing. The coherence of the formulation is established under steady and transient conditions through comparison with exact linear stability analysis, experimental measurements, and flow visualization. Good agreement is obtained between theory and the measurements of Muller et al. (1993) in the limit of purely elastic overstable TVF.

Description: In this paper we investigate the linear stability of detonations in which the underlying steady one-dimensional solutions are of the pathological type. Such detonations travel at a minimum speed, which is greater than the Chapman-Jouguet (CJ) speed, have an internal frozen sonic point at which the thermicity vanishes, and the unsupported wave is supersonic (i.e. weak) after the sonic point. Pathological detonations are possible when there are endothermic or dissipative effects present in the system. We consider a system with two consecutive irreversible reactions A→B→C, with an Arrhenius form of the reaction rates and the second reaction endothermic. We determine analytical asymptotic solutions valid near the sonic pathological point for both the one-dimensional steady equations and the equations for linearized perturbations. These are used as initial conditions for integrating the equations. We show that, apart from the existence of stable modes, the linear stability of the pathological detonation is qualitatively the same as for CJ detonations for both one- and two-dimensional disturbances. We also consider the stability of overdriven detonations for the system. We show that the frequency of oscillation for one-dimensional disturbances, and the cell size based on the wavenumber with the highest group velocity for two-dimensional disturbances, are both very sensitive to the detonation speed for overdriven detonations near the pathological speed. This dependence on the degree of overdrive is quite different from that obtained when the unsupported detonation is of the CJ type.

Description: This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson & Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these 'supercritical leaps' can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.

Description: The transient evolution of the bubble-size probability density functions resulting from the breakup of an air bubble injected into a fully developed turbulent water flow has been measured experimentally using phase Doppler particle sizing (PDPA) and image processing techniques. These measurements were used to determine the breakup frequency of the bubbles as a function of their size and of the critical diameter Dc defined as Dc = 1.26 (σ/ρ)3/5∈-2/5, where ∈ is the rate of dissipation per unit mass and per unit time of the underlying turbulence. A phenomenological model is proposed showing the existence of two distinct bubble size regimes. For bubbles of sizes comparable to Dc, the breakup frequency is shown to increase as (σ/ρ)-2/5 ∈3/5 √D/Dc-1, while for large bubbles whose sizes are greater than 1.63Dc, it decreases with the bubble size as ∈1/3 D-2/3. The model is shown to be in good agreement with measurements performed over a wide range of bubble sizes and turbulence intensities.

Description: Direct numerical simulations of turbulence resulting from Kelvin-Helmholtz instability in stratified shear flow are used to examine the geometry of the dissipation range in a variety of flow regimes. As the buoyancy and shear Reynolds numbers that quantify the degree of isotropy in the dissipation range increase, alignment statistics evolve from those characteristic of parallel shear flow to those found previously in studies of stationary, isotropic, homogeneous turbulence (e.g. Ashurst et al. 1987; She et al. 1991; Tsinober et al. 1992). The analysis yields a limiting value for the mean compression rate of scalar gradients that is expected to be characteristic of all turbulent flows at sufficiently high Reynolds number. My main focus is the value of the constant q that appears in both the Batchelor (1959) and Kraichnan (1968) theoretical forms for the passive scalar spectrum. Taking account of the effects of time-dependent strain, I propose a revised estimate of q, denoted qe, which appears to agree with spectral shapes derived from simulations and observations better than do previous theoretical estimates. The revised estimate is qe = 7.3±0.4, and is expected to be valid whenever the buoyancy Reynolds number exceeds O(102). The Kraichnan (1968) spectral form, in which effects of intermittency are accounted for, provides a better fit to the DNS results than does the Batchelor (1959) form.

Description: The weakly nonlinear theory for modelling flows away from the bifurcation point developed by the authors in their previous work (Suslov & Paolucci 1997) is generalized for flows of variable-density fluids in open systems. It is shown that special treatment of the continuity equation is necessary to perform the analysis of such flows and to account for the potential total fluid mass variation in the domain. The stability analysis of non-Boussinesq mixed convection flow of air in a vertical channel is then performed for a wide range of temperature differences between the walls, and Grashof and Reynolds numbers. A cubic Landau equation, which governs the evolution of a disturbance amplitude, is derived and used to identify regions of subcritical and supercritical bifurcations to periodic flows. Equilibrium disturbance amplitudes are computed for regions of supercritical bifurcations.

Description: The nonlinear evolution of wavepackets in a laminar boundary layer has been studied experimentally. The packets were generated by acoustic excitations injected into the boundary layer through a small hole in the plate. Various packets with different phases relative to the envelope were studied. It was found that for all the packets the nonlinearity involved the appearance of oblique modes of frequency close to the subharmonic of the dominant two-dimensional wave. Moreover, the results confirmed that the phase had a strong influence on the strength of the nonlinear interaction. The experimental observations also indicated that although a subharmonic resonance appeared to be present in the process, it alone could not explain the nonlinear behaviour. The experiment demonstrated that the process must also involve a mechanism that generates oblique waves of frequency lower than the Tollmien - Schlichting band.

Description: Cavitation experiments performed in the near field of a 50mm diameter (D) jet at ReD = 5 × 105, showed inception in the form of inclined 'cylindrical' bubbles at axial distances (x/D) less than 0.55, with indices of 2.5. On tripping the boundary layer, cavitation inception occurred at x/D ≈2, as distorted 'spherical' bubbles with inception indices of 1.7. To investigate these substantial differences, the near field of the jet was measured using PIV. Data on the primary flow, the strength distribution of the 'streamwise' vortices and the velocity profiles within the initial boundary layers were obtained. The untripped case showed a direct transition to three-dimensional flow in the near field (x/D 〈 0.7) even before rolling up to distinct vortex rings. Strong 'streamwise' vortices with strengths up to 25% of the jet velocity times the characteristic wavelength were seen. Cavitation inception occurred in the core of these vortices. In contrast, in the tripped jet the vortex sheet rolled up to the familiar Kelvin-Helmholtz vortex rings with weak secondary vortices. Using the measured nuclei distribution, strengths and straining of the 'streamwise' structures, the rates of cavitation events were estimated. The estimated results match very well the measured cavitation rates. Also, the Reynolds stresses in the near field of the jet show similar trends and magnitudes to those of Browand & Latigo (1979) and Bell & Mehta (1990) for a plane shear layer.

Description: In this paper, we present a new two-dimensional viscometer, and the hydrodynamic calculations used to obtain the surface viscosities from the measurements. In order to interpret the experiments, performed with solutions of sodium dodecyl sulfate (SDS) and also with monolayers of insoluble surfactants, we develop various hydrodynamic models of soluble Gibbs monolayers and of incompressible Langmuir monolayers, that describe well the experimental results. In the case of SDS solutions, the calculations allow the determination of the surface shear viscosity, and its value is in good agreement with previous studies.

Description: This paper concerns a relaxation of the assumption of uniform mixture composition in the interior of sonoluminescence bubbles. Intense temperature and pressure gradients within the bubble drive relative mass diffusion which overwhelms diffusion driven by concentration gradients. This thermal and pressure diffusion results in a robust compositional inhomogeneity in the bubble which lasts several orders of magnitude longer than the temperature peak or light pulse at the main collapse of the bubble. This effect has important consequences for control of sonoluminescence, gas dynamics, sonochemistry, and the physics of light production.

Description: Elliptic jets have decided advantages for technological applications over circular jets; this paper explores further advantages achieved by jet forcing due to self-excitation. Using hot-wire measurements and flow visualization, we have studied an elliptic whistler (i.e. self-excited) air jet of 2 :1 aspect ratio which, in contrast to an elliptic jet issuing from a contoured nozzle, displays no axis switching, but significantly increased spread in the major-axis plane. Its near-field mass entrainment is considerably higher (by as much as 70%) than that of a non-whistling jet. Flow visualization reveals unexpected dynamics of the elliptic vortical structures in the whistler jet compared to that in the non-whistling jet. Vortices rolled up from the lip of the elliptic pipe impinge onto the collar, producing secondary vortices; interaction of these two opposite-signed vortices is shown to cause the different behaviour of the whistler jet.

Description: The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, ω) = 0; where k is a wavenumber, ω is a frequency and D(k, ω) is the dispersion relation function.The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability – as predicted by some model equations – is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem.The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.

Description: A theory is developed for the hydrodynamic interactions of surfactant-covered spherical drops in creeping flows. The surfactant is insoluble, and flow-induced changes of surfactant concentration are small, i.e. the film of adsorbed surfactant is incompressible. For a single surfactant-covered drop in an arbitrary incident flow, the Stokes equations are solved using a decomposition of the flow into surface-solenoidal and surface-irrotational components on concentric spherical surfaces. The surface-solenoidal component is unaffected by surfactant; the surface-irrotational component satisfies a slip-stick boundary condition with slip proportional to the surfactant diffusivity. Pair hydrodynamic interactions of surfactant-covered bubbles are computed from the one-particle solution using a multiple-scattering expansion. Two terms in a lubrication expansion are derived for axisymmetric near-contact motion. The pair mobility functions are used to compute collision efficiencies for equal-size surfactant-covered bubbles in linear flows and in Brownian motion. An asymptotic analysis is presented for weak surfactant diffusion and weak van der Waals attraction. In the absence of surfactant diffusion, collision efficiencies for surfactant-covered bubbles are higher than for rigid spheres in straining flow and lower in shear flow. In shear flow, the collision efficiency vanishes for surfactant diffusivities below a critical value if van der Waals attraction is absent.

Description: For the first time since Lord Kelvin's original conjectures of 1875 we address and study the time evolution of vortex knots in the context of the Euler equations. The vortex knot is given by a thin vortex filament in the shape of a torus knot Jp,q (p 〉 1, q 〉 1; p, q co-prime integers). The time evolution is studied numerically by using the Biot-Savart (BS) induction law and the localized induction approximation (LIA) equation. Results obtained using the two methods are compared to each other and to the analytic stability analysis of Ricca (1993, 1995). The most interesting finding is that thin vortex knots which are unstable under the LIA have a greatly extended lifetime when the BS law is used. These results provide useful information for modelling complex structures by using elementary vortex knots.

Description: The coupled Kuramoto-Sivashinsky (CKS) equations for multilayer downflowing films are derived and explored. The CKS equations exhibit a wealth of dynamical behaviour, displaying travelling periodic waves, regular and chaotic-like patterns, coexistence of different attractors, and perfect and imperfect synchronization of the interfaces. New physical effects are found, such as suppression of the Rayleigh-Taylor instability for heavy-top stratified films, and new surface-tension-driven instability.

Description: Surface-tension-driven Bénard convection in low-Prandtl-number fluids is studied by means of direct numerical simulation. The flow is computed in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal directions and either a free-slip or no-slip bottom wall using a pseudospectral Fourier - Chebyshev discretization. Deformations of the free surface are neglected. The smallest possible domain compatible with the hexagonal flow structure at the linear stability threshold is selected. As the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary hexagonal convection pattern becomes quickly time-dependent and eventually reaches a state of satio-temporal chaos. No qualitative difference is observed between the zero-Prandtlnumber limit and a finite Prandtl number corresponding to liquid sodium. This indicates that the zero-Prandtl-number limit provides a reasonable approximation for the prediction of low-Prandtl-number convection. For a free-slip bottom wall, the flow always remains three-dimensional. For the no-slip wall, two-dimensional solutions are observed in some interval of Marangoni numbers. Beyond the Marangoni number for onset of inertial convection in two-dimensional simulations, the convective flow becomes strongly intermittent because of the interplay of the flywheel effect and three-dimensional instabilities of the two-dimensional rolls. The velocity field in this intermittent regime is characterized by the occurrence of very small vortices at the free surface which form as a result of vortex stretching processes. Similar structures were found with the free-slip bottom at slightly smaller Marangoni number. These observations demonstrate that a high numerical resolution is necessary even at moderate Marangoni numbers in order to properly capture the small-scale dynamics of Marangoni convection at low Prandtl numbers.

Description: The flickering candle: transition to a global oscillation in a thermal plumeJournal of Fluid Mechanics, vol. 390 (1999), pp. 297–323On page 302, the fourth and third lines from the bottom of the page, for the word ‘important’ read ‘unimportant’.

Description: We present simulation results of vortex-induced vibrations of an infinitely long flexible cylinder at Reynolds number Re = 1000, corresponding to a 'young' turbulent wake (i.e. exhibiting a small inertial subrange). The simulations are based on a new class of spectral methods suitable for unstructured and hybrid grids. To obtain different responses of the coupled flow-structure system we vary the structure's bending stiffness to model the behaviour of a vibrating inflexible (rigid) cylinder, a cable, and a beam. We have found that unlike the laminar flow previously studied, the amplitude of the cross-flow oscillation is about one diameter for the cable and the beam, close to experimental measurements, but is lower for the rigid cylinder. We have also found that for the latter case the flow response corresponds to parallel shedding, but for the beam and cable with free endpoints a mixed response consisting of oblique and parallel shedding is obtained, caused by the modulated travelling wave motion of the structure. This mixed shedding pattern which alternates periodically along the span can be directly related to periodic spatial variation of the lift force. In the case of structures with pinned endpoints a standing wave response is obtained for the cylinder; lace-like flow structures are observed similar to the ones seen in the laminar regime. Examination of the frequency spectra in the near wake shows that at Re = 1000 all cases follow a -5/3 law in the inertial range, which extends about half a decade in wavenumber. However, these spectra are different in all three cases both in low and high frequencies, with the exception of the beam and cable, for which the high-frequency portion is identical despite the differences in the displacement time history and the large-scale features of the corresponding flow.

Description: It is shown that there is an abstract subgrid model that is in all senses ideal. An LES using the ideal subgrid model will exactly reproduce all single-time, multi-point statistics, and at the same time will have minimum possible error in instantaneous dynamics. The ideal model is written as an average over the real turbulent fields whose large scales match the current LES field. But this conditional average cannot be computed directly. Rather, the ideal model is the target for approximation when developing practical models, though no new practical models are presented here. To construct such models, the conditional average can be formally approximated using stochastic estimation. These optimal formulations are presented, and it is shown that a relatively simple but general class of one-point estimates can be computed from twopoint correlation data, and that the estimates retain some of the statistical properties of the ideal model. To investigate the nature of these models, optimal formulations were applied to forced isotropic turbulence. A variety of optimal models of increasing complexity were computed. In all cases, it was found that the errors between the real and estimated subgrid force were nearly as large as the subgrid force itself. It is suggested that this may also be characteristic of the ideal model in isotropic turbulence. If this is the case, then it explains why subgrid models produce reasonable results in actual LES while performing poorly in a priori tests. Despite the large errors in the optimal models, one feature of the subgrid interaction that is exactly represented is the energy transfer to the subgrid scales by each wavenumber.

Description: We have investigated the role of elasticity in the stability of air-fluid interfaces during fluid displacement flows. Our investigations of the stability of coating flows with an eccentric cylinder geometry for both a viscous Newtonian fluid and ideal elastic Boger fluids are discussed in terms of three classes of phenomena. To begin, we have documented several new features in traditional fingering instabilities in elastic displacement flows. These include a very strong elastic destabilization of forward roll coating: a destabilization which can be correlated directly with the elasticity of the coating fluid and which appears to be present even in the absence of diverging channel walls. Moreover, elastic effects are shown to create a novel saw-toothed cusped pattern in the eccentric cylinder roll-and-plate geometry. Secondly, we have found that purely elastic bulk flow instabilities in the neighbourhood of air-fluid interfaces can cause surface deformations if the secondary flow is of sufficient strength. Finally, flows created by the displacement of less viscous air by a more viscous elastic fluid are found to display a new class of purely elastic instabilities which appear to be independent of traditional viscous fingering instabilities and elastic bulk flow instabilities. Thus interfaces which are stable for Newtonian fluids are unstable via purely elastic mechanisms. We have found that indeed elasticity has a dramatic effect on the stability of interfaces, not only changing the critical conditions, but also changing the manifestation of traditional fingering instabilities, and causing new purely elastic interfacial instabilities.

Description: Based on amplitude expansions developed in Part 1 (Suslov & Paolucci 1999), we examine the mean flow characteristics of non-Boussinesq mixed convection flow of air in a vertical channel in the vicinity of bifurcation points for a wide range of temperature differences between the walls, and Grashof and Reynolds numbers. The constant mass flux and constant pressure gradient formulations are shown to lead to qualitatively similar, but quantitatively different, results. The physical nature of the distinct shear and buoyancy disturbances is investigated, and detailed mean flow and energy analyses are presented. The variation of the total mass of fluid in a flow domain as disturbances develop is discussed. The average Nusselt number and mass flux are estimated for supercritical regimes for a wide range of governing parameters.

Description: This paper describes an experimental investigation of mixing due to Rayleigh-Taylor instability between two miscible fluids. Attention is focused on the gravitationally driven instability between a layer of salt water and a layer of fresh water with particular emphasis on the internal structure within the mixing zone. Three-dimensional numerical simulations of the same flow are used to give extra insight into the behaviour found in the experiments. The two layers are initially separated by a rigid barrier which is removed at the start of the experiment. The removal process injects vorticity into the flow and creates a small but significant initial disturbance. A novel aspect of the numerical investigation is that the measured velocity field for the start of the experiments has been used to initialize the simulations, achieving substantially improved agreement with experiment when compared with simulations using idealized initial conditions. It is shown that the spatial structure of these initial conditions is more important than their amplitude for the subsequent growth of the mixing region between the two layers. Simple measures of the growth of the instability are shown to be inappropriate due to the spatial structure of the initial conditions which continues to influence the flow throughout its evolution. As a result the mixing zone does not follow the classical quadratic time dependence predicted from similarity considerations. Direct comparison of external measures of the growth show the necessity to capture the gross features of the initial conditions while detailed measures of the internal structure show a rapid loss of memory of the finer details of the initial conditions. Image processing techniques are employed to provide a detailed study of the internal structure and statistics of the concentration field. These measurements demonstrate that, at scales small compared with the confining geometry, the flow rapidly adopts self-similar turbulent behaviour with the influence of the barrier-induced perturbation confined to the larger length scales. Concentration power spectra and the fractal dimension of iso-concentration contours are found to be representative of fully developed turbulence and there is close agreement between the experiments and simulations. Other statistics of the mixing zone show a reasonable level of agreement, the discrepancies mainly being due to experimental noise and the finite resolution of the simulations.

Description: The effect of non-uniformity on the development of a modulated, weakly nonlinear wavepacket is studied. The non-uniformity, characterized by slowly varying wavenumber and frequency of the primary wave, may lead to significant modification of the stability properties compared with the uniform case. As a specific example we consider a modulated Stokes wave on deep water. In the uniform case such a wave proves to be definitely unstable (Benjamin & Feir 1967). In the non-uniform case, on the other hand, the wave may become stable under certain conditions. One of these is an increase of the local group velocity in the direction of wave propagation. Then the Benjamin - Feir instability mechanism is quenched on a time scale determined by the degree of non-uniformity. In addition, a sufficient degree of non-uniformity leads to stability of the wave to linear perturbations. However, when the local group velocity decreases in the direction of wave propagation, non-uniformity has a destabilizing effect. A comparison is made with experiments. It is also shown that the analysis, based on this specific example, is readily applied to a greater variety of non-uniform, dispersive waves.

Description: An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters-nonlinearity, the dispersion parameter, and the bottom slope-the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.

Description: From shallow-water gravity wave theories it is shown that the velocity field in the whole fluid domain can be reconstructed using an analytic transformation (a renormalization). The resulting velocity field satisfies the Laplace equation exactly, which is not the case for shallow-water approximations. Applying the renormalization to the first-order shallow-water solution of limited accuracy, gives accurate simple solutions for both long and short waves, even for large amplitudes. The KdV and Airy solutions are special limiting cases.

Description: A steady, two-dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial-temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov-Schmidt method.

Description: Experimental results from the Interface Configuration Experiment (ICE) performed aboard the Space Shuttle and the Mir Space Station are reported. The experiment concerns fluid interfaces in certain 'exotic' containers in a low-gravity environment. These containers are rotationally symmetric and have the property that for given contact angle and liquid volume, a continuum of distinct rotationally symmetric equilibrium configurations can appear, all of which have the same mechanical energy. These symmetric equilibrium configurations are unstable, in that deformations that are not rotationally symmetric can be shown mathematically to yield configurations with lower energy. It is found experimentally, in confirmation of mathematical results and of numerical computations, that distinct locally stable configurations can form that are not rotationally symmetric and have differing dynamic characteristics. It is found that this intriguing phenomenon of asymmetric local energy minimizers can occur even if conditions for an exotic container are not entirely met.

Description: In Part 1 (Woodley & Peake 1999) we described a method for predicting the occurrence of resonant states in a system comprising twin cascades in zero relative motion. We now demonstrate how that work can be extended to account for the case of more practical interest, in which the upstream cascade (rotor) is rotating in the transverse direction relative to the downstream cascade (stator). Time periodicity now forces the temporal frequency of any disturbance to be an integer multiple of the rotor passing frequency in the stator frame, and vice versa, and this leads to the requirement to sum over a discrete set of temporal modes, as well as over the spatial modes already described in Part 1. The mechanisms by which temporal and spatial modes are scattered by the blade rows is made clear by the analytical approach adopted here; the scattering of the incident pressure (and, for the stator, vorticity) fields by each row in its own frame is completed using results similar to those presented in Part 1, and the fields in the two frames then matched across the inter-row gap to provide a single matrix equation. Specimen results for the conditioning of this equation are given, and although it seems more difficult to obtain very strong excitation than it was for zero rotation, the significance of Parker resonance of the stator is again apparent.

Description: A two-dimensional model of flow and bed topography is proposed to investigate the effect of sediment heterogeneity on the development of alternate bars. Within the context of a linear stability theory the flow field, the bed topography and the grain size distribution function are perturbed leading to an integro-differential linear eigenvalue problem. It is shown that the selective transport of different grain size fractions and the resulting spatial pattern of sorting may appreciably affect the balance between stabilizing and destabilizing actions which govern bar instability. Theoretical results suggest that sediment heterogeneity leads to a damping of both growth rate and migration speed of bars, while bar wavelength is shortened with respect to the case of uniform sediment. The above findings conform, at least qualitatively, to the experimentally detected reduction of bar height, length and celerity. The observed tendency of coarser particles to pile up towards bar crests is also reproduced by theoretical results.

Description: We present experimental force and power measurements demonstrating that the power required to propel an actively swimming, streamlined, fish-like body is significantly smaller than the power needed to tow the body straight and rigid at the same speed U. The data have been obtained through accurate force and motion measurements on a laboratory fish-like robotic mechanism, 1.2 m long, covered with a flexible skin and equipped with a tail fin, at Reynolds numbers up to 106, with turbulence stimulation. The lateral motion of the body is in the form of a travelling wave with wavelength λ and varying amplitude along the length, smoothly increasing from the front to the tail end. A parametric investigation shows sensitivity of drag reduction to the non-dimensional frequency (Strouhal number), amplitude of body oscillation and wavelength λ, and angle of attack and phase angle of the tail fin. A necessary condition for drag reduction is that the phase speed of the body wave be greater than the forward speed U. Power estimates using an inviscid numerical scheme compare favourably with the experimental data. The method employs a boundary-integral method for arbitrary flexible body geometry and motions, while the wake shed from the fish-like form is modelled by an evolving desingularized dipole sheet.

Description: The flow between parallel walls driven by the time-periodic oscillation of one of the walls is investigated. The flow is characterized by a non-dimensional amplitude Δ and a Reynolds number R. At small values of the Reynolds number the flow is synchronous with the wall motion and is stable. If the amplitude of oscillation is held fixed and the Reynolds number is increased there is a symmetry-breaking bifurcation at a finite value of R. When R is further increased, additional bifurcations take place, but the structure which develops, essentially chaotic flow resulting from a Feigenbaum cascade or a quasi-periodic flow, depends on the amplitude of oscillation. The flow in the different regimes is investigated by a combination of asymptotic and numerical methods. In the small-amplitude high-Reynolds-number limit we show that the flow structure develops on two time scales with chaos occurring on the longer time scale. The chaos in that case is shown to be associated with the unsteady breakdown of a steady streaming flow. The chaotic flows which we describe are of particular interest because they correspond to Navier-Stokes solutions of stagnation-point form. These flows are relevant to a wide variety of flows of practical importance.

Description: The response of a Gaussian vortex to a weak time-dependent external strain field is studied numerically. The cases of an impulsive strain, an on-off step function, and a continuous random strain are considered. Transfers of enstrophy between mean and azimuthal components are observed, and the results are compared with an analogous passive scalar model and with Kida's elliptical vortex model. A 'rebound' phenomenon is seen: after enstrophy is transferred from mean to azimuthal component by the external straining field, there is a subsequent transfer of enstrophy back from the azimuthal component to the mean. Analytical support is given for this phenomenon using Lundgren's asymptotic formulation of the spiral wind-up of vorticity. Finally the decay of the vortex under a continuous random external strain is studied numerically and compared with the passive scalar model. The vorticity distribution decays more slowly than the scalar because of the rebound phenomenon.

Description: The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [O, N], N being the Brunt-Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0,N], It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not squareintegrable and the corresponding point spectrum is also empty.

Description: We consider the evolution under the action of surface tension of wedges and cones of viscous fluid whose initial semi-angles are close to π/2. A short time after the fluid is released from rest, there is an inner region, where surface tension and viscosity dominate, and an outer region, where inertia and viscosity dominate. We also find that the velocity of the tip of the wedge or cone is singular, of O(log(1/t)), as time, f, tends to zero. After a long time, the free surface asymptotes to a similarity form where deformations are of O(t2/3), and capillary waves propagate away from the tip. However, a distance of 0(t3/4) away from the tip, viscosity acts to damp out the capillary waves. We solve the linearized governing equations using double integral transforms, which we calculate numerically, and use asymptotic techniques to approximate the solutions for small and large times. We also compare the asymptotic solution for the inviscid fat wedge with a numerical solution of the nonlinear inviscid problem for wedges of arbitrary semi-angle.

Description: In this paper we investigate the stability of a bilayer exposed to air flow. The bilayer consists of a viscoelastic solid layer (mucus), which rests on a viscous fluid film (serous fluid). The motivation behind this work is to examine the coupled, fluid/elastic instabilities related to mucus clearance in the lung where breathing and cough apply shear forces from the air flow onto the bilayer. Previous research on mucus transport due to air flow has not addressed the effects of the underlying serous layer nor those of surface tension at the mucus-air interface, two new features incorporated into the model. Surface tension effects are governed by the new parameter K′ = (σ/dG′) where σ is the air-mucus surface tension, G′ is the elastic shear modulus of the mucus, and d is a characteristic thickness of the bilayer. The model predictions for the onset of unstable waves as a function of the parameters are compared to previous theories and experiments to provide physical interpretations and to compare results. The comparison with experiments show good qualitative and quantitative agreement. The results are compared, also, to flow over a single, viscoelastic layer, with no viscous fluid underneath, to demonstrate the appearance of new wave behaviour when the viscous fluid is added.

Description: The tendency of granular materials in rapid shear flow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

Description: Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg-de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

Description: The intermittency of pressure and pressure gradient in stationary isotropic turbulence at low to moderate Reynolds numbers is studied by direct numerical simulation (DNS) and theoretically. The energy spectra scale in Kolmogorov units as required by the universal-equilibrium hypothesis, but the pressure spectra do not. It is found that the variances of the pressure and pressure gradient are larger than those computed using the Gaussian approximation for the fourth-order moments of velocity, and that the variance of the pressure gradient, normalized by Kolmogorov units, increases roughly as script R signλ1/2, where script R signλ is the Taylor microscale Reynolds number. A theoretical explanation of the Reynolds number dependence is presented which assumes that the small-scale pressure field is driven by coherent small-scale vorticity-strain domains. The variance of the pressure gradient given by the model is the product of the variance of Ui,jUj,i, the source term of the Poisson equation for pressure, and the square of an effective length of the small-scale coherent vorticity-strain structures. This length can be expressed in terms of the Taylor and Kolmogorov microscales, and the ratio between them gives the observed Reynolds number dependence. Formal asymptotic matching of the spectral scaling observed at small scales in the DNS with the classical scaling at large scales suggests that at high Reynolds numbers the pressure spectrum in these forced flows consists of three scaling ranges which are joined by two inertial ranges, the classical k-7/3 range and a k-5/3 range at smaller scale. It is not possible, within the classical Kolmogorov theory, to determine the length scale at which the inertial range transition occurs because information beyond the energy dissipation rate is required.

Description: Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

Description: In most real or numerically simulated turbulent flows, the energy dissipated at small scales is equal to that injected at very large scales, which are anisotropic. Despite this injection-scale anisotropy, one generally expects the inertial-range scales to be locally isotropic. For moderate Reynolds numbers, the isotropic relations between second-order and third-order moments for temperature (Yaglom's equation) or velocity increments (Kolmogorov's equation) are not respected, reflecting a non-negligible correlation between the scales responsible for the injection, the transfer and the dissipation of energy. In order to shed some light on the influence of the large scales on inertial-range properties, a generalization of Yaglom's equation is deduced and tested, in heated grid turbulence (Rλ = 66). In this case, the main phenomenon responsible for the non-universal inertial-range behaviour is the non-stationarity of the second-order moments, acting as a negative production term.

Description: A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

Description: Using a matched asymptotic expansion we analyse the two-dimensional, near-critical reflection of a weakly nonlinear internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the solution of Phillips (1966) in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/t. During the course of this scale reduction, the stratification is Overturned' and the Miles-Howard condition for stratified shear flow stability is violated. However, for all slope angles, the Overturning' occurs before the Miles-Howard stability condition is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process. In particular, the buoyancy field computed from the analytic solution is in good agreement with visualizations reported by Thorpe & Haines (1987). One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansion all of the apparently resonant nonlinear contributions cancel amongst themselves). However, the reconstructed fields do contain nonlinearly driven second harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.

Description: The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x,t), h(x,t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants. The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(X)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in 'modified' vorticity field introduced in Part 1 of this series. Finally, the general stability criterion is applied to a variety of classes of steady states {U(x),H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

Description: Axisymmetric pipeline transportation of oil and water is simulated numerically as an initial value problem. The simulations succeed in predicting the spatially periodic Stokes-like waves called bamboo waves, which have been documented in experiments of Bai, Chen & Joseph (1992) for up-flow. The numerical scheme is validated against linearized stability theory for perfect core-annular flow, and weakly nonlinear saturation to travelling waves. Far from onset conditions, the fully nonlinear saturation to steady bamboo waves is achieved. As the speed is increased, the bamboo waves shorten, and peaks become more pointed. A new time-dependent bamboo wave is discovered, in which the interfacial waveform is steady, but the accompanying velocity and pressure fields are time-dependent. The appearance of vortices and the locations of the extremal values of pressure are investigated for both up- and down-flows.

Description: Surface waves superimposed upon a larger-scale flow are blocked and reflected at the points where the group velocities balance the convection by the larger-scale flow. In this study, we first extended the theory of Shyu & Phillips (1990) to the situation when short deep-water gravity waves propagate obliquely upon a steady unidirectional irrotational current and are reflected by it. In this case, the uniformly valid solution and the WKBJ solution of the short waves were derived from the Laplace equation and the kinematical and dynamical boundary conditions. These solutions in terms of some parameters (the expressions for which have also been deduced in this case) take the same forms as those derived by Shyu & Phillips, which by referring to Smith's (1975) theory can even be proved to be valid for gravity waves in an intermediate-depth region and near a curved moving caustic induced by an unsteady multidirectional irrotational current. In this general case, the expressions for certain parameters in these solutions cannot be obtained so that their values must be estimated in a numerical calculation. The algorithm for estimates of some of these parameters that are responsible for the amplitude of the reflected wave not being equal to that of the incident wave in the vicinity of the caustic and therefore are crucial for the computer calculation of the ray solution to be continued after reflection, was illustrated through numerical tests. This algorithm can avoid the error magnification phenomenon that occurred in the previous estimates of the reflected wave in the vicinity of the caustic using the action conservation principle directly. The forms of the solutions have also been utilized to clarify the wave profiles near caustics in a general situation, which indicate that in storm conditions freak waves characterized by a steeper forward face preceded by a deep trough will probably occur in the caustic regions.

Description: The aerodynamic and acoustic properties of supersonic elliptic and circular jets are experimentally investigated. The jets are perfectly expanded with an exit Mach number of approximately 1.5 and are operated in the Reynolds number range of 25 000 to 50 000. The reduced Reynolds number facilitates the use of conventional hot-wire anemometry and a glow discharge excitation technique which preferentially excites the varicose or flapping modes in the jets. In order to simulate the high-velocity and low-density effects of heated jets, helium is mixed with the air jets. This allows the large-scale structures in the jet shear layer to achieve a high enough convective velocity to radiate noise through the Mach wave emission process. Experiments in the present work focus on comparisons between the cold and simulated heated jet conditions and on the beneficial aeroacoustic properties of the elliptic jet. When helium is added to the jet, the instability wave phase velocity is found to approach or exceed the ambient sound speed. The radiated noise is also louder and directed at a higher angle from the jet axis. In addition, near-field hotwire spectra are found to match the far-field acoustic spectra only for the helium/air mixture case. These results demonstrate that there are significant differences between unheated and heated asymmetric jets in the Mach 1.5 speed range, many of which have been found previously for circular jets. The elliptic jet was also found to radiate less noise than the round jet at comparable operating conditions.