ALBERT

Description: The carbon content, pH and 14C concentration of humic acids were determined for three soil series of Arctic and Subarctic ecosystems. The measured 14C ages were interpreted in the light of an equilibrium model of humus formation and of mineralization processes in recent soils, and the coefficient of renovation, Kr, was calculated for humic acids. The comparison of Kr for series formed under different climatic conditions suggested that global warming could accelerate decomposition of soil organic matter and possibly increase productivity of ecosystems of the Arctic region.

Description: Δ14C records are reported for post-bomb corals from three sites in the tropical Atlantic Ocean. In corals from 18°S in the Brazil Current, Δ14C values increased from ca. −58% in the early 1950s to +138% by 1974, then decreased to 110‰ by 1982. Shorter records from 8ºS off Brazil and from the Cape Verde Islands (17°N) showed initially higher Δ14C values before 1965 than those at 18ºS, but showed lower rates of increase of Δ14C during the early 1960s. There is general agreement between the coral results and Δ14C of dissolved inorganic carbon (DIC) measured in seawater previously for locations in the tropical Atlantic Ocean. Δ14C values at our tropical ocean sites increased at a slower rate than those observed previously in the temperate North Atlantic (Florida and Bermuda), owing to the latter's proximity to the bomb 14C input source in the northern, hemisphere. Model results show that from 1960–1980 the Cape Verde coral and selected DIG Δ14C values from the North Equatorial Current agree with that calculated for the North Atlantic based on an isopycnal mixing model with a constant water mass renewal rate between surface and subsurface waters. This is in contrast to Δ14C values in Bermuda corals that showed higher post-bomb values than those predicted using a constant water mass renewal rate, hence indicating that ventilation in the western north Atlantic Ocean had decreased by a factor of 3 during the 1960s and 1970s (Druffel 1989).

Description: Although soil contains about three times the amount of carbon present in the preindustrial atmosphere, determining how perturbations (e.g., changing land use, CO2 fertilization, changing climate and anthropogenic nitrogen deposition) alter soil carbon storage and influence atmospheric CO2 levels has proved elusive. Not knowing the soil carbon turnover times causes part of this uncertainty. I outline a strategy for using radiocarbon measurements to estimate soil organic matter turnover times and inventories in native soil. The resulting estimates of carbon exchange produce reasonable agreement with measurements of CO2 fluxes from soil. Furthermore, derivatives of the model are used to explore soil carbon dynamics of cultivated and recovering soil. Because the models can reproduce observed soil 14C measurements in native, cultivated, and recovering ecosystems (i.e., the underlying assumptions appear reasonable), the native model was modified to estimate the potential rate of additional carbon storage because of CO2 fertilization. This process may account for 45–65% of the “missing CO2 sink.”

Description: Recently, in an intercomparison of the Hohenheim German oak chronology (Becker 1993) and the Göttingen chronology (Leuschner and Delorme 1988), an error was discovered in the former (Leuschner, in preparation). Due to an error in adding sections at 5241 bc, 41 yr are missing in the published Hohenheim chronology. After correction of the error, the two chronologies synchronize over their entire common length, back to 7200 bc.

Description: Extensive measurements of radiocarbon have been used in New Zealand since the mid-1960s to follow carbon (C) movement and turnover in soils. We present here unpublished radiocarbon (14C) measurements on a range of eight New Zealand soils with details of the sites, ecosystems, climates, soil descriptions and associated analytical data. An overview is also given of published 14C measurements on soils, and the use of these measurements to model soil C turnover.

Description: We report here the remainder of the Hamburg University dates on thin soil layers (HAM 1652–3129).

Description: In order to verify sediment trap samples as indicators of upper ocean 14C concentrations, particulate inorganic radiocarbon (PICΔ14C) collected by time-series sediment traps in the Sea of Okhotsk and the Bering Sea was measured by accelerator mass spectrometry (AMS). All of the PICΔ14C measurements were 〈 0‰, in contrast to GEOSECS 14C data in the upper ocean from the northwestern North Pacific. This difference is attributed to the upwelling of deepwater that contains low Δ14C of dissolved inorganic carbon (DICΔ14C) and to the decrease over time of surface DICΔ14C owing to the decrease of atmospheric Δ14C values. In addition, PICΔ14C values showed significant seasonal variability: PICΔ14C collected in the fall was the greatest (-22‰ on average), whereas PICΔ14C collected in winter showed an average minimum of −48‰. It is likely that this difference was caused by changes in mixed layer thickness. Although some uncertainties remain, further study on PICΔ14C will enable us to estimate seasonal variability in DICΔ14C and air-sea CO2 exchange rate.

Description: For several undisturbed sites in Germany, 14C data are reported for soil organic matter (SOM) (4 sites), soil CO2 (10 sites) and dissolved organic carbon (DOC) (1 site). With the assumption of a fast degradable component (lifetime ca. 1 yr) and a slow degradable component (lifetime ca. 100 yr), a range between 0.6 and 1.6 mm yr-1 has been determined for the downward migration rates of soil organic carbon at the sampling sites from the soil 14C data. The soil CO2 measurements show that in deciduous forests the fast degradable component is ca. 60% and the slow degradable component is ca. 40% of the SOM. In coniferous forests this ratio is reversed. The 14C results for DOC could not be explained with the assumption of a first-order decay process. The removal of soil organic carbon by DOC is of minor importance for the estimation of carbon budgets for the investigated site.

Description: Measurements of cosmogenic nuclides made in situ in the Earth's surface are being used to help resolve a wide range of geologic and chronologic questions. Cosmogenic nuclides (3He, 10Be, 14C, 21Ne, 26Al 36C1 are presently used) can reveal rock exposure history information leading to estimates of timing of surface forming events, rates and styles of erosion, and timing and durations of episodes of burial. Depending on the problems being tackled, a significant source of error (±10–25%) for any cosmogenic nuclide method is the present uncertainty in the spatial and temporal variability of the rates of production of these in-situ nuclides.

Description: We present a compilation of 14C soil dates measured at the University of Hamburg through 1984 (HAM-1597).

Description: At the University of Miami Tritium Laboratory and the University of Washington Quaternary Isotope Laboratory, more than 1000 large-volume Pacific Ocean radiocarbon samples were measured for the WOCE program. Here we present a comprehensive data set, and a brief discussion of our findings.

Description: Bomb 14C from nuclear tests in the atmosphere has proved to be a particularly useful tool in the study of the carbon cycle. We provide here a ca. 30-yr time series of 14C concentrations in the atmosphere between 28°N and 71°N and in the ocean surface between 45°S and 45°N. More recently (since 1990), a north-south profile also has been obtained for 14C in the surface waters of the Atlantic Ocean. The measurements were performed using the conventional technique of beta counting or large samples (4 to 5 liter CO2) in CO2 proportional counters. These data show that the 14C concentration in the atmosphere is leveling off with a time constant of 0.055 yr-1, and is now approaching that of the ocean surface at lower latitudes.Additional tracer studies have been concerned especially with the penetration of bomb 14C into the deep ocean. The Norwegian and Greenland seas are of interest as a sink for atmospheric CO2 and also a source of water for the deep Atlantic Ocean. During the last five years, several 14C depth profiles have been measured from the Fram Strait (79°N) to south of Iceland (62°N), using the AMS technique available at the University of Arizona AMS Facility. We considered it important to repeat and compare a few of the profiles with those produced by the GEOSECS expedition in 1972 and the TTO expedition in 1981. The profiles show that water descending to the deep Atlantic Ocean is originating mainly from intermediate and surface depths in the Nordic Seas. However, the ventilation rate of the Norwegian Sea deepwater is too slow to be an important component in the transfer of water over the Greenland-Scotland Ridge.

Description: The Geological Survey of Canada (GSC) has developed, over the past decade, a user-oriented database, Date Locator File, of Canadian samples dated by the 14C technique. This database presently contains 〉3500 soil and soil-related dates. The primary category in this suite of dates is peat, as a large portion of the Canadian landscape is covered with this type of organic soil. The data is available gratis to all researchers in a large variety of formats from simple lists to complex tables for inclusion in publications. The site localities can also be plotted on base maps suitable for publication. The database is actively augmented on an ongoing basis, but to continue to be relevant, it depends largely on the altruism of the scientific community.

Description: Acid hydrolysis is used to fractionate the soil organic carbon pool into relatively slow- and fast-cycling compartments on soils from Arizona, the Great Plains states and Michigan collected for carbon isotope tracer studies related to soil carbon sequestration, for studies of shifts in C3/C4 vegetation, and for “pre-bomb” soil-carbon inventories. Prior to hydrolysis, soil samples are first treated with cold 0.5–1N HCl to remove soil carbonates if necessary. Samples are then dispersed in a concentrated NaCl solution (ρ≍1.2 g cm-3) and floated plant fragments are skimmed off the surface. After rinsing and drying, all remaining recognizable plant fragments are picked from the soil under 20x magnification. Plant-free soils, and hot, 6N HCl acid-hydrolysis residue and hydrolyzate fractions are analyzed for carbon content, δ13C and 14C age, and the carbon distribution is verified within 1–2% by stable-carbon isotope mass balance. On average, the recalcitrant residue fraction is 1800 yr older and 2.6% more 13C-depleted than total soil organic carbon. A test of hydrolysis with fresh plant fragments produced as much as 71–76% in the acid-hydrolysis residue pool. Thus, if plant fragments are not largely removed prior to hydrolysis, the residue fraction may date much younger than it actually is.

Description: In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.

Description: The influence of different boundary conditions applied in the contact line region on the outer meniscus shape is analysed by means of a finite-element numerical simulation of the steady movement of a liquid-gas meniscus in a capillary tube. The free-surface steady shape is obtained by solving the unsteady creeping-flow approximation of the Navier-Stokes equations starting from some initial shape. Comparisons of the outer solutions obtained using two different inner models, together with that published by Lowndes (1980), indicate the relative insensitivity of the outer solution to the type of model utilized in the contact line region.

Description: The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios Γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on Γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1,2,3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with Γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Description: Since Bödewadt's (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bödewadt's self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savaş. (1983, 1987) has studied the end wall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bödewadt layers. For certain conditions, Savaş has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bödewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further. Here, the impulsive spin-down problem is examined numerically for Savaş' (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.

Description: It is well known that subgrid models such as Smagorinsky's cannot be used for the spatially growing simulation of the transition to turbulence of flat-plate boundary layers, unless large-amplitude perturbations are introduced at the upstream boundary: they are over-dissipative, and the flow simulated remains laminar. This is also the case for the structure-function model (SF) of Métais & Lesieur (1992). In the present paper we present a sequel to this model, the filtered-structure-function (FSF) model. It consists of removing the large-scale fluctuations of the field before computing its second-order structure function. Analytical arguments confirm the superiority of the FSF model over the SF model for large-eddy simulations of weakly unstable transitional flows. The FSF model is therefore used for the simulation of a quasi-incompressible (M∞ = 0.5) boundary layer developing spatially over an adiabatic flat plate, with a low level of upstream forcing. With the minimal resolution 650 × 32 × 20 grid points covering a range of streamwise Reynolds numbers Rex1 ε [3.4 × 105, 1.1 × 106], transition is obtained for 80 hours of time-processing on a CRAY 2 (whereas DNS of the whole transition takes about ten times longer). Statistics of the LES are found to be in acceptable agreement with experiments and empirical laws, in the laminar, transitional and turbulent parts of the domain. The dynamics of low-pressure and high-vorticity distributions is examined during transition, with particular emphasis on the neighbourhood of the critical layer (defined here as the height of the fluid travelling at a speed equal to the phase speed of the incoming Tollmien–Schlichting waves). Evidence is given that a subharmonic-type secondary instability grows, followed by a purely spanwise (i.e. time-independent) mode which yields peak-and-valley splitting and transition to turbulence. In the turbulent region, flow visualizations and local instantaneous profiles are provided. They confirm the presence of low- and high-speed streaks at the wall, weak hairpins stretched by the flow and bursting events. It is found that most of the vorticity is produced in the spanwise direction, at the wall, below the high-speed streaks. Isosurfaces of eddy viscosity confirm that the FSF model does not perturb transition much, and acts mostly in the vicinity of the hairpins.

Description: The interaction between a zero-pressure-gradient turbulent boundary layer and a pair of strong, common-flow-down, streamwise vortices with a sizeable velocity deficit is studied by large-eddy simulation. The subgrid-scale stresses are modelled by a localized dynamic eddy-viscosity model. The results agree well with experimental data. The vortices drastically distort the boundary layer, and produce large spanwise variations of the skin friction. The Reynolds stresses are highly three-dimensional. High levels of kinetic energy are found both in the upwash region and in the vortex core. The two secondary shear stresses are significant in the vortex region, with magnitudes comparable to the primary one. Turbulent transport from the immediate upwash region is partly responsible for the high levels of turbulent kinetic energy in the vortex core; its effect on the primary stress (úú) is less significant. The mean velocity gradients play an important role in the generation of (úú) in all regions, while they are negligible in the generation of turbulent kinetic energy in the vortex core. The pressure - strain correlations are generally of opposite sign to the production terms except in the vortex core, where they have the same sign as the production term in the budget of (úú). The results highlight the limitations of the eddy-viscosity assumption (in a Reynolds-averaged context) for flows of this type, as well as the excessive diffusion predicted by typical turbulence models.

Description: The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

Description: The effects of molecular diffusivities of heat and mass on the counter-gradient scalar and momentum transfer in strongly stable stratification are experimentally investigated in unsheared and sheared stratified water mixing-layer flows downstream of turbulence-generating grids. Experiments are carried out in two kinds of stably stratified water flows. In the case of thermal stratification, the difference between the turbulent fluxes of an active scalar (heat with the Prandtl number of Pr ≈ 6) and a passive scalar (mass with the Schmidt number of Sc ≈ 600) is investigated. In the case of salt stratification, the effects of the molecular diffusion of the active scalar (salt) with a very high Schmidt number of Sc ≈ 600 on the counter-gradient scalar transfer is studied. Comparisons of the effects of molecular diffusivities are also made between thermally stratified water and air (Pr ≈ 0.7) flows. Further, the effects of mean shear on the counter-gradient scalar and momentum transfer are investigated for both stratified cases. Instantaneous temperature, concentration and streamwise and vertical velocities are simultaneously measured using a combined technique with a resistance thermometer, a laser-induced fluorescence method, and a laser-Doppler velocimeter with high spatial resolution. Turbulent scalar fluxes, joint probability density functions, and cospectra are estimated.The results of the first case show that both active heat and passive mass develop counter-gradient fluxes but that the counter-gradient flux of passive mass is about 10% larger than that of active heat, mostly due to molecular diffusion effects at small scales. The counter-gradient scalar transfer mechanism in stable stratification can be explained by considering the relative balance between the available potential energy and the turbulent kinetic energy as in Schumann (1987). In thermally and salt-stratified water mixing-layer flows with the active scalars of high Prandtl and Schmidt numbers, the buoyancy-induced motions with finger-like structures first contribute to the counter-gradient scalar fluxes at small scales, and then the large-scale motions, which bring fluid back to its original levels, generate the counter-gradient fluxes at large scales. The contribution of the small-scale motions to the counter-gradient fluxes in stratified water flows is quite different from that in stratified air flows. The higher Prandtl or Schmidt number of the active scalar generates both the stronger buoyancy effects and the longer time-oscillation period of the counter-gradient scalar fluxes. The time-oscillation occurs at large scales but the counter-gradient fluxes at small scales persist without oscillating. The mean shear acts to reduce the counter-gradient scalar and momentum transfer at large scales, and therefore the counter-gradient fluxes in sheared stratified flows can be seen only in very strong stratification. The behaviour of the counter-gradient momentum flux in strong stratification is quite similar to that of the counter-gradient scalar flux.

Description: The improvement of heat transfer conditions in liquid-metal magnetohydrodynamic (MHD) flows is of prime importance for self-cooled fusion blanket design concepts. For many years the research was based on stationary inertialess assumptions since it was expected that time-dependent inertial flows would be suppressed by strong electromagnetic damping, especially in the extreme range of fusion relevant parameters. In the present analysis the stationary inertialess assumptions are abandoned. Nevertheless, the classical ideas usually used to obtain inertialess asymptotic solutions are drawn on. The basic inertial equations are reduced to a coupled two-dimensional problem by analytical integration along magnetic field lines. The magnetic field is responsible for a quasi-two-dimensional flow; the non-uniform distribution of the wall conductivity creates a wake-type profile, the MHD effect reducing to a particular forcing and friction. The solution for the two-dimensional variables, the field aligned component of vorticity, the stream function, and the electric potential are obtained by numerical methods. In a flat channel with non-uniform electrical wall conductivity, time-dependent solutions similar to the Kármán vortex street behind bluff bodies are possible. The onset of the vortex motion, i.e. the critical Reynolds number depends strongly on the strength of the magnetic field expressed by the Hartmann number. Stability analyses in viscous hydrodynamic wakes often use the approximation of a unidirectional flow which does not take into account the spatial evolution of the wake. The present problem exhibits a wake-type basic flow, which does not change along the flow path. It represents, therefore, an excellent example to which the simple linear analysis on the basis of Orr-Sommerfeld-type equations applies exactly. Once unstable, the flow first exhibits a regular time periodic vortex pattern which is rearranged further downstream. One can observe an elongation, pairing, or sometimes more complex merging of vortices. All these effects lead to larger flow structures with lower frequencies. The possibility for a creation and maintenance of time-dependent vortex-type flow pattern in MHD flows is demonstrated.

Description: Experiments have been performed, in capillary tubes, on the displacement of a viscous fluid (glycerine) by a less viscous one (a glycerine-water mixture) with which it is miscible in all proportions. A diagnostic measure of the amount of viscous fluid left behind on the tube wall has been found, for both vertical and horizontal tubes, as a function of the Péclet (Pe) and Atwood (At) numbers, as well as a parameter that is a measure of the relative importance of viscous and gravitational effects. The asymptotic value of this diagnostic quantity, for large Pe and an At of unity, has been found to agree with that found in immiscible displacements, while the agreement with the numerical results of Part 2 (Chen & Meiburg 1966), over the whole range of At, is very good. At values of the average Pe greater than 1000 a sharp interface existed so that it was possible to make direct comparisons between the present results and a prior experiment with immiscible fluids, in particular an effective surface tension at the diffusing interface could be evaluated. The effect of gravity on the amount of viscous fluid left on the tube wall has been investigated also, and compared with the results of Part 2. A subsidiary experiment has been performed to measure both the average value of the diffusion coefficient between pure glycerine and several glycerine-water mixtures, in order to be able to calculate a representative Péclet number for each experiment, and the local value as a function of the local concentration of glycerine, in the dilute limit.

Description: Thermoacoustic refrigeration occurs in periodic flow in a duct with heat transfer within the fluid and to the tube. This study considers the periodic limit cycle with large pressure oscillations that is obtained in a tube when prescribed, phase-shifted, periodic velocities at the tube ends, at frequencies lower than acoustic eigenmodes, sweep a length comparable to the tube length. The temperature differences between the two ends are of arbitrary magnitude, heat transfer in the transverse direction within the fluid is assumed to be very effective and the thermal mass of the wall is large. The geometry is two-dimensional, axisymmetric, and conduction is accounted for, not only in the fluid, but also with and within the tube wall. A perturbation solution valid in a local near-isothermal limit determines the equilibrium longitudinal temperature profile that is reached at the periodic regime, the pressure field including longitudinal gradients, and the longitudinal enthalpy flux. Results are presented for tubes open at both ends and also with one end closed. In the latter case, a singularity occurs in the temperature at the closed end, with behaviour identical to Rott's result for acoustic flow with small pressure amplitude. Other new results obtained for tubes open at both ends show that when velocities at both ends are in opposite phase, internal singularities in the temperature profiles may occur.

Description: Following the study in Part 1 of cross-flow and other non-symmetric effects on vortex/wave interactions in boundary layers, the present Part 2 applies the ideas of Part 1 and related works to an incident axisymmetric flow supplemented by a small swirl or azimuthal velocity. This is with a view to possibly increasing understanding of vortex breakdown. The wave components involved are predominantly inviscid Rayleigh-like ones. The presence of the swirl leads to extra features and complications associated mainly with extra logarithmic contributions but for the dominant interactions essentially the same equations as in Part 1 are found. These dominant nonlinear interactions must be based on azimuthal wavenumbers of ± 1 in the case of the Squire jet with swirl. In contrast to Part 1, which consisted mainly of an analysis of the quasi-bounded solutions, a representative set of numerical solutions of the full integro-differential amplitude equations is presented, for realistic axial and swirl velocity profiles. The work points also to the influence of further increases in the incident swirl.

Description: An asymptotic analysis for the long-time unsteady laminar far wake of a bluff body due to a step change in its travelling velocity from U1 to U2 is presented. For U1 ≥ 0 and U2 〉 0, the laminar wake consists of a new wake of volume flux Q2 corresponding to the current velocity U2, an old wake of volume flux Q1 corresponding to the original velocity U1, and a transition zone that connects these two wakes. The transition zone acts as a sink (or a source) of volume flux (Q2 - Q1) and is moving away from the body at speed U2. Streamwise diffusion is negligible in the new and old wakes but a matched asymptotic expansion that retains the streamwise diffusion is required to determine the vorticity transport in the transition zone. A source of volume flux Q2 located near the body needs to be superposed on the unsteady wake to form the global flow field around the body. The asymptotic predictions for the unsteady wake velocity, unsteady wake vorticity, and the global flow field around the body agree well with finite difference solutions for flow over a sphere at finite Reynolds numbers. The long-time unsteady flow structures due to a sudden stop (U2 = 0) and an impulsive reverse (U1 U2 〈 0) of the body are analysed in detail based on the asymptotic solutions for the unsteady wakes and the finite difference solutions. The elucidation of the long-time behaviour of such unsteady flows provides a framework for understanding the long-time particle dynamics at finite Reynolds number.

Description: A theory and simulation code are developed to study non-steady sources as means to control sonic booms of supersonic aircraft. A key result is that the source of sonic boom pressure is not confined to the length of the aircraft but occupies an extensive segment of the flight path. An aircraft in non-steady flight functions as a synthetic aperture antenna, generating complex acoustic waves with no simple relation to instantaneous volume or lift distributions. The theory applies linear acoustics to slender non-steady sources but requires no farfield approximation. The solution for pressure contains a term not seen in Whitham's theory for sonic booms of distant supersonic aircraft. The term describes a pressure field that decays algebraically behind the Mach cone and, in the case of steady flight, integrates to a ground load equal to the weight of the aircraft. The algebraic term is separate from those that describe the sonic boom. Two non-steady source phenomena are evaluated: periodic velocity changes (surge), and periodic longitudinal lift redistribution (slosh). Surge can attenuate a sonic boom and covert it into prolonged weak reverberation, but accelerations needed to produce the phenomenon seem too large for practical use. Slosh may be practical and can alter sonic booms but does not, on average, result in boom attenuation. The conclusion is that active sonic boom abatement is possible in theory but maybe not practical.

Description: The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics. The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg-de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg-de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg-de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant. Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.

Description: Experiments and simulations of a travelling wave state of incompressible Newtonian flow in a curved duct of square cross-section are presented. The travelling wave mode develops from the well-documented steady four-cell flow state and is characterized by oscillations of the two Dean vortices near the centre of the outer wall. The oscillations were induced by a carefully positioned pin at 5° from the inlet of the curved section along the symmetry line of the cross-section. It was shown that the travelling wave state is characteristic for curved duct flow and that the pin made it possible to observe the oscillations within the 270° long curved duct. Travelling waves were observed at flow rates above Dn = 170 (Dn = Re/(R/a)l/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension. The curvature ratio, R/a, is 15.1). If no other disturbances are imposed, the oscillations are the result of the selective amplification of random disturbances in the flow, leading to a broad frequency spectrum. The travelling wave was found to lock in to an imposed periodic disturbance at a selected frequency. The flow structure of the locked state was investigated in detail, using flow visualization and a one-component laser Doppler anemometer to measure streamwise or spanwise velocities. Direct numerical simulations using the package CFDS-FLOW3D are in very good agreement with the experiments and confirm the existence of a fully developed, streamwise-periodic travelling wave state. The inflow region between the two Dean vortices, which transports low-speed fluid away from the outer wall, creates strongly inflectional spanwise profiles of the streamwise velocity. Similarities with twisting vortices in a curved channel and sinuous oscillations of Görtler vortices show that the travelling waves observed here result from a secondary shear instability of these spanwise inflectional profiles.

Description: The anisotropic behaviour of density-gradient fluctuations in stably stratified grid turbulence and the consequences for simplified (isotropic) estimates of scalar dissipation rates χ were experimentally studied in a thermally stratified wind tunnel at moderate Reynolds numbers (Reλ ≃ 20). Strong stable stratifications were attained, with Brunt-Väisälä frequency N as high as 4 rad s-1. The correlation method was used to estimate the mean-square cross-stream and streamwise density gradients. Cross-stream gradients were measured using two cold wires. The mean-square vertical gradients were found to become larger than the streamwise gradients by as much as a factor of 2.2 for the largest dimensionless buoyancy times (Nt = 7). This corresponds to a 40% error in the scalar dissipation estimates based on ∂θ/∂x alone, and assuming the validity of the isotropic relations. Gradient spectral relations show that this buoyancy-induced anisotropy persists at all length scales. Better closure of the scalar variance balance was attained than in previously reported measurements by other researchers. This is attributed to our use of cold-wire temperature sensors having larger length-to-diameter ratio than used in the previous measurements.

Description: A two-dimensional Boussinesq equation, utt-uxx+ 3(u2)xx - uxxxx - uyy = 0, is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation. A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

Description: Individual travelling cavitation bubbles were examined as they interacted with the flow over a two-dimensional hydrofoil. Each bubble was produced from a single nucleus created upstream of the hydrofoil, and the flow near the hydrofoil was visualized using particle imaging velocimetry (PIV). Travelling bubbles were observed to generate a local region of turbulence as they passed close to an unstable laminar boundary layer. By producing a locally turbulent region, the bubbles could temporarily sweep away a portion of attached cavitation at the foil midchord. Also, the bubbles were observed to strongly interact with a turbulent boundary layer, producing local regions of patch cavitation.

Description: The problem of gas motion in a tube closed at one end and driven at the other by an oscillating piston is studied theoretically. When the piston vibrates with a finite amplitude at the first acoustic resonance frequency, periodic shock waves are generated, travelling back and forth in the tube. A perturbation method, based on a small Mach number, M and a global mass conservation condition, is employed to formulate a solution of the problem in the form of two standing waves separated by a jump (shock front). By expanding the equations of motion in a series of a small parameter ε = M1/2, all hydrodynamic properties are predicted with an accuracy to second-order terms, i.e. to ε2. It is found that the first-order solution coincides with the previous theories of Betchov (1958) and Chester (1964), while additional terms predict a non-homogeneous time-averaged pressure along the tube. This prediction compares favourably with experimental results from the literature. The importance of the phenomenon is discussed in relation to different transport processes in resonance tubes.

Description: Two-dimensional mixing driven by an instability mechanism which is concentrated near one of the boundaries is considered, with particular application to Langmuir circulations driven by a wave spectrum. The question of how to define the equivalent of the Rayleigh number is attacked using the energy balance equations and simple truncated models of the instability. Given a particular horizontal wavelength for the disturbance, the strength of the forcing on the cells, and thus the growth rate, is determined by a tradeoff between maximizing the depth-averaged forcing and maximizing the depth of penetration. As a result of this tradeoff, long-wavelength cells grow more slowly, but penetrate more deeply and have a larger equivalent Rayleigh number. At finite amplitude, these long-wavelength cells come to dominate the flow field. The depth of penetration of, and density transport accomplished by, Langmuir cells is considered as a function of the mean stratification and diffusion. An application to oceanic mixed layers is considered assuming the Mellor-Yamada 2 1/2-level turbulence closure model to define the background level of turbulent mixing. For many realistic cases, Langmuir cells are predicted to dominate the vertical transport of momentum and density.

Description: A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.

Description: A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant - unlike the deep-water case - in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities 'contaminate' most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves 'close' to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d = 1; the study is performed down to depth d = 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of shortcrested wave fields will not be observable, although most of the short-crested wave fields will be.

Description: In a series of laboratory experiments the growth of double-diffusive salt fingers from an initial configuration of two homogeneous reservoirs with salt in the lower and sugar in the upper layer was investigated. For most of the experiments the stability ratio was between 2.5 and 3, where the latter value is at the upper limit (the ratio of salt to sugar diffusivities) for which fingers can exist. In these experiments long slender fingers are generated at the interface. Essentially all theories or physical bases for models of salt fingers presuppose such a configuration of long fingers. Our measurements show that the length of fingers at high stability ratio increases with time like i'/2, with a coefficient that is consistent with the diffusive spread of the faster diffusing component (salt). When the initial stability ratio is closer to unity, fingers penetrate into the reservoirs very rapidly carrying with them large anomalies of salt and sugar which give rise to convective overturning of the reservoirs. The convection sweeps away the ends of the fingers, and when it is intense enough (as it is when the sugar anomaly is large) it can reduce the finger height to a value less than the width. After this initial phase the finger length grows linearly with time as has been found in previous studies. These results show that salt fingers can evolve in quite different ways depending on the initial stability ratio and must cast doubt on the use of simple similarity arguments to parameterize the heat and salt fluxes produced by fingers.

Description: We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure-velocity correlation to the single-point pressure-strain tensor, is also derived. This law shows that the two-point pressure-velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity-enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with εω 2/3 in the enstrophy inertial range, εω being the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to the k-1 law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).

Description: Simulations of decaying two-dimensional turbulence suggest that the one-point vorticity density has the self-similar form ρω ∼ t f(ωt) implied by Batchelor's (1969) similarity hypothesis, except in the tails. Specifically, similarity holds for |ω| 〈 ωm, while ρω falls off rapidly above. The upper bound of the similarity range, ωm, is also nearly conserved in high-Reynolds-number hyperviscosity simulations and appears to be related to the average amplitude of the most intense vortices (McWilliams 1990), which was an important ingredient in the vortex scaling theory of Carnevale et al. (1991). The universal function f also appears to be hyperbolic, i.e. f(x) ∼ c/2|x|1+qc, for |x| 〉 x*, where qc = 0.4 and x* = 70, which along with the truncated similarity form implies a phase transition in the vorticity moments matrix presented between the self-similar 'background sea' and the coherent vortices. Here cq and c are universal. Low-order moments are therefore consistent with Batchelor's similarity hypothesis whereas high-order moments are similar to those predicted by Carnevale et al. (1991). A self-similar but less well-founded expression for the energy spectrum is also proposed. It is also argued that ωs = x*/t represents 'mean sea-level', i.e. the (average) threshold separating the vortices and the sea, and that there is a spectrum of vortices with amplitudes in the range (ωs, ωm). The total area occupied by vortices is also found to remain constant in time, with losses due to mergers of large-amplitude vortices being balanced by gains due to production of weak vortices. By contrast, the area occupied by vortices above a fixed threshold decays in time as observed by McWilliams (1990).

Description: Motivated by the study of blood flow in the major coronary arteries, which are situated on the outer surface of the pumping heart, we analyse flow of an incompressible Newtonian fluid in a tube whose curvature varies both along the tube and with time. Attention is restricted to the case in which the tube radius is fixed and its centreline moves in a plane. Nevertheless, the governing equations are very complicated, because the natural coordinate system involves acceleration, rotation and deformation of the frame of reference, and their derivation forms a major part of the paper. Then they are applied to two particular, relatively simple examples: a tube of uniform but time-dependent curvature; and a sinuous tube, representing a small-amplitude oscillation about a straight pipe. In the former case the curvature is taken to be small and to vary by a small amount, and the solution is developed as a triple power series in mean curvature ratio δ0, curvature variation ε and Dean number D. In the latter case the Reynolds number is taken to be large and a linearized solution for the perturbation to the flow in the boundary layer at the tube wall is obtained, following Smith (1976a). In each case the solution is taken far enough that the first non-trivial effects of the variable curvature can be determined. Results are presented in terms of the oscillatory wall shear stress distribution and, in the uniform curvature case, the contribution of steady streaming to the mean wall shear stress is calculated. Estimation of the parameters for the human heart indicates that the present results are not directly applicable, but point the way for future work.

Description: The aim of the paper is to present a new transport process which is likely to have great importance for understanding the internal constitution of the stars.In order to set the problem in context, we first give a short presentation of the physical properties of the Sun and stars, described usually under the names Standard Solar Model or Standard Stellar Models (SSM). Next we show that an important shortcoming of SSM is that they do not explain the age dependence of the lithium deficiency of stars of known age: stars of galactic clusters and the Sun. It was suggested a long time ago that the presence of a macroscopic diffusion process in the radiative zone should be assumed, below the surface convective zone of solar-like stars. It is then possible for the lithium present in the convective zone to be carried to the thermonuclear burning level below the convective zone. The first assumption was that differential rotation generates turbulence and therefore that a turbulent diffusion process takes place. However, this model predicts a lithium abundance which is strongly rotation dependent, contrary to the observations. Furthermore, as the diffusion coefficient is large all over the radiative zone, it prevents the possibility of gravitational separation by diffusion and consequently leads to the impossibility of explaining the difference in helium abundance between the surface and the centre of the Sun. The consequence is obviously that we need to take into account another physical process.Stars having a mass M 〈 1.3M[odot ] have a convective zone which begins close to the stellar surface and extends down to a depth which is an appreciable fraction of the stellar radius. In the convective zone, strong stochastic motions carry, at least partially, heat transfer. These motions do not vanish at the lower boundary and generate internal waves into the radiative zone. These random internal waves are at the origin of a diffusion process which can be considered as responsible for the diffusive transport of lithium down to the lithium burning level. This is certainly not the only physical process responsible for lithium deficiency in main sequence stars, but its properties open the way to a completely consistent analysis of lithium deficiency.The model of generation of gravity waves is based on a model of heat transport in the convective zone by diving plumes. The horizontal component of the turbulent motion at the boundary of the convective zone is assumed to generate the horizontal motion of internal waves. The result is a large horizontal component of the diffusion coefficient, which produces in a short time an horizontally uniform chemical composition. It is known that gravity waves, in the absence of any dissipative process, cannot generate vertical mixing. Therefore, the vertical component of the diffusion coefficient is entirely dependent on radiative damping. It decreases quickly in the radiative zone, but is large enough to be responsible for lithium burning.Owing to the radial dependence of velocity amplitude, the diffusion coefficient increases when approaching the stellar centre. However, very close to the centre, nonlinear dissipative and radiative damping of internal waves become large and the diffusion coefficient vanishes at the very centre.

Description: Strongly nonlinear vortex-Tollmien-Schlichting-wave interaction equations are derived for the case where the undisturbed motion represents the developing flow in a circular pipe. The effect upon the equations of moving the wave input position further downstream is investigated and the development of the flow is found to be accelerated by increasing the size of the wave disturbance. Numerical solutions of the three-dimensional interaction equations are presented and indicate that the form of interaction considered here appears to promote the three-dimensionality as the flow develops downstream. It is shown that one of the interactions considered here can develop within an initially two-dimensional Blasius boundary layer.

Description: It is well known that in any conservative system that admits resonant triad interactions, a uniform (test) wavetrain that participates in a single triad is unstable if it has the highest frequency in the triad, and neutrally stable otherwise. We show that this result changes significantly in the presence of coupled triads : with coupling, the test wave can be unstable to a high-frequency perturbation. The coupling sends energy from the (weak) high-frequency source into particular low-frequency waves that grow even though they had zero amplitudes initially. This mechanism thereby selects these low-frequency waves from the spectrum of low-frequency waves available for triad interactions. Moreover, the instability persists in the presence of weak damping, provided the wave amplitudes exceed two thresholds. First, the initial amplitude of the test wavetrain must be large enough for the instability to dominate the damping. Secondly, the (small) initial amplitudes of the high-frequency perturbations must exceed a threshold in order for the low-frequency waves to grow to a prescribed amplitude.

Description: The spectral mechanisms of the differential diffusion of pairs of passive scalars with different molecular diffusivities are studied in stationary isotropic turbulence, using direct numerical simulation data at Taylor-scale Reynolds number up to 160 on 1283 and 2563 grids. Of greatest interest are the roles of nonlinear triadic interactions between different scale ranges of the velocity and scalar fields in the evolution of spectral coherency between the scalars, and the effects of mean scalar gradients. Analysis of single-scalar spectral transfer (extending the results of a previous study) indicates a robust local forward cascade behaviour at high wavenumbers, which is strengthened by both high diffusivity and mean gradients. This cascade is driven primarily by moderately non-local interactions in which two small-scale scalar modes are coupled via a lower-wavenumber velocity mode near the peak of the energy dissipation spectrum. This forward cascade is coherent, tending to increase the coherency between different scalars at high wavenumbers but to decrease it at lower wavenumbers. However, at early times coherency evolution at high wavenumbers is dominated by de-correlating effects due to a different type of non-local triad consisting of two scalar modes with a moderate scale separation and a relatively high-wavenumber velocity mode. Consequently, although the small-scale motions play little role in spectral transfer, they are responsible for the rapid de-correlation observed at early times. At later times both types of competing triadic interactions become important over a wider wavenumber range, with increased relative strength of the coherent cascade, so that the coherency becomes slow-changing. When uniform mean scalar gradients are present, a stationary state develops in the coherency spectrum as a result of a balance between a coherent mean gradient contribution (felt within about 1 eddy-turnover time) and the net contribution from scale interactions. The latter is made less de-correlating because of a strengthened coherent forward cascade, which is in turn caused by uniform mean gradients acting as a primarily low-wavenumber source of scalar fluctuations with the same spectral content as the velocity field.

Description: The effect of wall cooling on hypersonic boundary-layer separation near a compression ramp is considered. Two cases are identified corresponding to the value of the average Mach number M̄ across the upstream boundary layer approaching the compression ramp. The flow is referred to as supercritical for M̄ 〉 1 and subcritical for M̄ 〈 1. The interaction is described by triple-deck theory, and numerical results are given for both cases for various ramp angles and levels of wall cooling. The effect of wall cooling on the absolute instability described recently by Cassel, Ruban & Walker (1995) for an uncooled wall is of particular interest; a stabilizing effect is observed for supercritical boundary layers, but a strong destabilizing influence occurs in the subcritical case. Wall cooling also influences the location and size of the separated region. For supercritical flow, progressive wall cooling reduces the size of the recirculating-flow region, the separation point moves downstream, and upstream influence is diminished. In contrast for the subcritical case downstream influence is reduced with increased cooling. In either situation, a sufficient level of wall cooling eliminates separation altogether for the ramp angles considered. The present numerical results closely confirm the strong wall cooling theory of Kerimbekov, Ruban & Walker (1994).

Description: A number of new experiments have been performed on the rise of air bubbles in clean mixtures of distilled water and pure, reagent grade, glycerine covering a range of the relevant parameter, the Morton number, Mo = gv4ρ3/σ3, of 1013. Here g is the acceleration due to gravity, v the kinematic viscosity, ρ the density and σ the surface tension of the mixture. In these careful measurements several scaling regimes have been found that have not been discussed before in the extensive literature on the subject. The transitions between these regimes have been delineated and attempts made to discuss the dynamical processes that might be important in each of them.

Description: A potential flow model has been formulated for scallop swimming. Under the small-disturbance approximation, the problem of the unsteady flow past the wing-like configuration of a scallop is separated into two linear sub-problems: the steady lifting problem and the unsteady symmetric thickness problem. The latter is associated with the expansion and contraction of the boundary surface of the scallop due to the shell opening and closing. A quasi-two-dimensional analytical solution of the thickness problem was obtained to give the time-dependent fluid forces acting on the outer surfaces of the shells. In addition to the added-mass effect, which has been widely accepted in the hydrodynamics of aquatic locomotion, there are two other mechanisms in the fluid reaction: flow-induced pseudo-elasticity and pseudo-viscosity. The pseudoelasticity provides a force proportional to the gape angle displacement, and will assist shell opening but resist shell closing. The pseudo-viscosity force is proportional to the angular velocity of the gape, and benefits both shell opening and closing. Their roles are discussed through comparison with those of shell inertia, hinge ligament elasticity and hinge damping. At 10°C the hinge damping in the scallop was found to be almost compensated by the flow pseudo-viscosity. The unsteady fluid reaction may have a significant effect on the operation of the dynamic swimming system of scallops.

Description: The linear stability of elliptic pipe flow is considered for finite aspect ratios thereby bridging the gap between the small-aspect-ratio analysis of Davey & Salwen (1994) and the large-aspect-ratio asymptotics of Hocking (1977). The flow is found to become linearly unstable above an aspect ratio of about 10.4 to the spanwise-modulated analogue of the Orr-Sommerfeld mode to which plane Poiseuille flow first loses stability. This disturbance is found to possess a series of intense vortices along its critical layer at lateral stations far removed from the central minor axis. The critical Reynolds number appears to fall from infinity as the aspect ratio increases above 10.4, ultimately approaching Hocking's (1977) asymptotic result at much larger aspect ratios.

Description: In this paper, we have identified a new mechanism which can promote rapid growth of three-dimensional disturbances. The mechanism involves the interaction between a planar mode and an oblique mode, or a pair of oblique modes, which are phase-locked in the sense that they have the same phase speed. This allows a powerful nonlinear interaction to take place within the common critical layer(s). The disturbance is not required to form a subharmonic resonant triad, and hence the mechanism operates under much less restrictive conditions than does subharmonic resonance (although it is somewhat less powerful). We show that the quadratic interaction between the planar mode and the oblique modes drives an exceptionally large forced mode with the difference frequency, which in turn interacts with the planar mode to contribute the dominant nonlinear effect. This interaction can cause the oblique modes to grow super-exponentially provided that their magnitude is sufficiently small. As a result of the super-exponential growth, the oblique mode may soon become strong enough to produce a feedback effect on the planar mode, so that the interactions eventually become fully coupled. This subsequent stage takes slightly different forms depending on whether a single or a pair of oblique modes is present. Both cases are investigated. Particular attention is paid to symmetric plane shear layers, e.g. a planar wake or jet, for which subharmonic resonance of sinuous modes is inactive.

Description: We have investigated the lift force on a small isolated particle which is attached to a flat smooth surface and embedded within the viscous sublayer of the turbulent boundary layer over this surface. We have developed a novel experimental technique with which it is possible to measure both the mean and fluctuating lift force by gluing the particle on top of a silicium cantilever. The deflection of this cantilever is measured with a focused laser beam. The sensitivity of the focus detection system allows us to measure a lift force with an average value around 10-8N and with a standard deviation of approximately 5% of the mean. This means that our device is at least a factor of 100 more sensitive than previous devices and at the same time able to measure the lift forces on smaller particles. Data for the mean lift force (FL+) as a function of the particle radius (a+), where both parameters have been non-dimensionalized with the kinematic viscosity v and the friction velocity u*, are obtained in the range 0.3 〈 a+ 〈 2. The data support the relationship: FL+ = (56.9 + 1.1) (a+)1.87±0.04. Also results on the fluctuating lift force have been obtained. We find that the ratio of the r.m.s. to the mean lift force is approximately 2.8.

Description: Gravity-driven granular flow of slightly frictional particles down an inclined, bumpy chute is studied. A modified kinetic model which includes the frictional energy loss effects is used, and the boundary conditions for a bumpy wall with small friction are derived by ensuring the balance of momentum and energy. At the free surface, the condition of vanishing of the solid volume fraction is used. The mean velocity, the fluctuation kinetic energy and the solid volume fraction profiles are evaluated. It is shown that steady granular gravity flow down a bumpy frictional chute could be achieved at arbitrary inclination angles. The computational results also show that the slip velocity may vary considerably depending on the granular layer height, the surface boundary roughness, the friction coefficient and the inclination angles. The model predictions are compared with the existing experimental and simulation data, and good agreement is observed. In particular, the model can well predicate the features of the variation of solid volume fraction and fluctuation energy profiles for different particle-wall friction coefficients and wall roughnesses.

Description: Experimental investigations in the three-dimensional boundary layer of a swept flat plate with the pressure gradient induced from outside are aimed at enhancing knowledge of the transition process in the presence of pure crossflow instability. The development of disturbances is characterized by the occurrence of both stationary and travelling instability modes, by early nonlinear development and by complex dependence upon the environmental conditions. Experiments under natural conditions of transition showed a good correspondence of the identified modes with those predicted by local linear stability theory. The disturbance growth, however, is generally overpredicted. Controlled excitation of crossflow vortices allowing measurements closer to the linear range of amplification confirmed this result. Nonlinear effects such as interaction between stationary disturbances and base flow and between travelling and stationary modes have already been observed when the naturally excited instabilities become of measurable size. The most striking feature of the disturbance development is the complex dependence on initial conditions. Experiments under systematically varied environments showed that surface roughness represents the key parameter responsible for the initiation of stationary crossflow vortices. In contrast to two-dimensional boundary layers, free-stream turbulence influences the transition process indirectly. Only for turbulence levels Tu 〉 0.2% and smooth surfaces do the travelling instability waves dominate. The location of the final breakdown of laminar flow is clearly determined by the saturation amplitude of crossflow vortices. The receptivity to sound, two-dimensional surface roughness and non-uniformities of the test-section mean flow was found to be very weak.

Description: Large-eddy simulation (LES) was used to study mixing of turbulent, coannular jets discharging into a sudden expansion. This geometry resembles that of a coaxial jet-combustor, and the goal of the calculation was to gain some insight into the phenomena leading to lean blow-out (LBO) in such combustion devices. This is a first step in a series of calculations, where the focus is on the fluid dynamical aspects of the mixing process in the combustion chamber. The effects of swirl, chemical reactions and heat release were not taken into account. Mixing of fuel and oxidizer was studied by tracking a passive scalar introduced in the central jet. The dynamic subgrid-scale (DM) model was used to model both the subgrid-scale stresses and the subgrid-scale scalar flux. The Reynolds number was 38000, based on the bulk velocity and diameter of the combustion chamber. Mean velocities and Reynolds stresses are in good agreement with experimental data. Animated results clearly show that intermittent pockets of fuel-rich fluid (from the central jet) are able to cross the annular jet, virtually undiluted, into the recirculation zone. Most of the fuel-rich fluid is, however, entrained into the recirculation zone near the instantaneous reattachment point. Fuel trapped in the recirculation zone is, for the most part, entrained back into the step shear layer close to the base of the burner.

Description: It is shown that a train of long waves can suppress a short-wave field due to four-wave resonance interactions. These interactions lead to the diffusion (in Fourier space) of the wave action of the short-wave field, so that the wave action is transported to the regions of higher wavenumbers, where it dissipates more effectively. The diffusion equation is derived.

Description: A new exact solution of the nonlinear shallow-water equations is presented. The solution corresponds to divergent and non-divergent free oscillations in an infinite straight channel of parabolic cross-section on the rotating Earth. It provides a description of the one-dimensional subclass of shallow-water flows in paraboloidal basins considered by Ball (1964), Thacker (1981), Cushman-Roisin (1987) and others in which the velocity field varies linearly and the free-surface displacement varies quadratically with the spatial coordinates. In contrast to the previous exact solutions describing divergent oscillations in circular and elliptic paraboloidal basins, the oscillation frequency of the divergent oscillation in the parabolic channel is found to depend, in part, on the amplitudes of the relative vorticity and free-surface curvature. This result is consistent with Thacker's (1981) numerical finding that when the free surface in parabolic channel flow is curved, the oscillation frequency depends on the amplitude of the motion. Solutions for parcel trajectories are also presented. The exact solution provides a rare description of a class of nonlinear flows and is potentially valuable as a validation test for numerical shallow-water models in Eulerian and Lagrangian frameworks.

Description: A series of experiments has been carried out on low-viscosity fluid in a right-circular cylinder that rotates rapidly at a constant speed about its axis of symmetry. This axis in turn is made to undergo less rapid precession about a second axis passing through the centroid of the cylinder. The linear inviscid response of the fluid to such forcing can be expressed as a spectrum of inertial wave modes. However, there are several interesting features of the problem that are associated with nonlinear and viscous effects. One such phenomenon is the appearance of an azimuthal flow under conditions that are related to the underlying linear inertial wave behaviour. Results are presented concerning the manner in which this flow depends on the various experimental parameters. Dynamical properties of the circulation following the onset of forcing have also been investigated. The flow at forcing frequencies close to the fundamental inertial wave resonance was found to have a vortex-like structure, and this led to data that suggest that hydrodynamic instabilities may play a part in the observed breakdown to turbulent motion in regimes of strong forcing.

Description: The dynamic model for large-eddy simulation of turbulence samples information from the resolved velocity field in order to optimize subgrid-scale model coefficients. When the method is used in conjunction with the Smagorinsky eddy-viscosity model, and the sampling process is formulated in a spatially local fashion, the resulting coefficient field is highly variable and contains a significant fraction of negative values. Negative eddy viscosity leads to computational instability and as a result the model is always augmented with a stabilization mechanism. In most applications the model is stabilized by averaging the relevant equations over directions of statistical homogeneity. While this approach is effective, and is consistent with the statistical basis underlying the eddy-viscosity model, it is not applicable to complex-geometry inhomogeneous flows. Existing local formulations, intended for inhomogeneous flows, are most commonly stabilized by artificially constraining the coefficient to be positive. In this paper we introduce a new dynamic model formulation, that combines advantages of the statistical and local approaches. We propose to accumulate the required averages over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to inhomogeneous flows in complex geometries. We analyse direct numerical simulation data to document the effects of such averaging on the Smagorinsky coefficient. The characteristic Lagrangian time scale over which the averaging is performed is chosen based on measurements of the relevant Lagrangian autocorrelation functions, and on the requirement that the model be purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are slightly superior to those of the spatially (planar) averaged dynamic model. The relationship between the model and vortical structures in isotropic turbulence, as well as ejection events in channel flow, is investigated. Computational overhead is kept small (about 10% above the CPU requirements of the spatially averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

Description: The entrainment of fluids from two streams into the shear region of an incompressible mixing layer is dominated by the evolution of large coherent structures. However, fine-scale mixing of the entrained fluids mainly occurs at the interfaces of the small-scale turbulence. In this investigation, experiments were conducted to understand the properties of the small scales and to explore a method for controlling the population of the fine-scale turbulence. Furthermore, a dissipation scale, ζ, is found from the zero-crossing of the time derivative of the velocity fluctuations. This scale characterizes the most probable size of fine-scale turbulence, which produces most of the viscous dissipation.

Description: The equivalents of the classical theorems of hydrodynamic stability are derived for inviscid flow through a flexible tube. An important difference between flows in plane and cylindrical geometries is that the Squire transformation, which states that two-dimensional perturbations in plane parallel flows are always more unstable than three-dimensional perturbations, is not valid for tube flows. Therefore, it is necessary to analyse both axisymmetric and non-axisymmetric perturbations in flows in cylindrical geometries. Perturbations of the form υi = υ̃i exp [ik(x-ct)+inΦ] are imposed on a steady axisymmetric mean flow V(r), and the stability of the mean velocity profiles and bounds for the phase velocity of the unstable modes are determined. Here r, Φ and x are the radial, polar and axial directions, and k and c are the wavenumber and phase velocity. The flexible wall is represented by a standard constitutive equation which contains inertial, elastic and dissipative terms. Results for general velocity profiles are derived in two limiting cases : axisymmetric flows (n = 0) and highly non-axisymmetric flows (n ≫ k). The results indicate that axisymmetric perturbations are always stable for (V″-r-1V′) V ≥ 0 and could be unstable for (V″-r-1V′) V 〈 0, while highly non-axisymmetric perturbations are always stable for (V″+r-1V′) V ≥ 0 and could be unstable for (V″+r-1V′) V 〈 0. In addition, bounds on the real part (cr) and imaginary part (ci) of the phase velocity are also derived. For the practically important case of Hagen-Poiseuille flow, the present analysis indicates that axisymmetric perturbations are always stable, while highly non-axisymmetric perturbations could be unstable. This is in contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones.

Description: Hamiltonian approximation methods yields approximate dynamical equations that apply to nearly geostrophic flow at scales larger than the internal Rossby deformation radius. These equations incorporate fluid inertia with the same order of accuracy as the semi-geostrophic equations, but are nearly as simple (in appropriate coordinates) as the equations obtained by completely omitting the inertia.

Description: The plane unsteady problem of the deflection of a solid, slightly curved plate in collision with an ideal weakly compressible liquid is considered. In order to describe the impact process, the acoustic approximation and the method of normal modes are used. The analysis is focused on the supersonic stage of the impact when the liquid surface remains undisturbed outside the contact spot between the solid plate and the liquid. However, the positions of the contact points are unknown in advance, in contrast to the case of undeformable body impact, and have to be found together with the liquid flow, the pressure distribution, and the bottom deformations. It was shown that the duration of the supersonic stage depends on the entering body elasticity. A spray jet is formed earlier and the stage at which the liquid compressibility is a governing factor is shorter than under rigid-body impact. It is revealed that the elastic plate deflection is quite small and can be satisfactorily approximated by a few modes. On the other hand, the calculation of the bending stress distribution needs a much greater number of normal modes. The pressure distribution over the contact region is quite difficult to find by the mode method; an alternative approach is suggested.

Description: Steady, inviscid, incompressible two-dimensional flow in a quarter-circular cavity containing two vortex patches is investigated. A two-parameter family of solutions, characterized by any two out of the positions of the separation and reattachment points of the main eddy, the tangential velocity at separation and the ratio of the core vorticities, is identified and computed numerically. It is found that solutions can only be obtained for a rather narrow band of combinations of these parameters; the reasons for this constraint are discussed. Finally, we consider whether any of the coupled Batchelor flow solutions actually does represent the limit of high Reynolds number flow by comparing the inviscid results with those of earlier Navier-Stokes computations (Vynnycky & Kimura 1994). Agreement for the position of the dividing streamline and the location of the centre of the main core proves to be very encouraging, and suggestions are made as to the possible future development of such a two-eddy model.

Description: Thermal convection with a continuous finite bandwidth of modes in a porous layer with horizontal walls at different mean temperatures is considered when a spatially non-uniform temperature is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using multiple scales and perturbation techniques. The preferred flow solutions are determined by a stability analysis. It is found that for the case of near-resonant wavelength excitation regular or non-regular solutions in the form of superposition of small-scale multi-modal solutions with large-scale multi-modal (or non-modal) amplitude can become preferred, provided the wave vectors of the solutions are contained in the set of wave vectors due to the modal form of the boundary imperfections and the form of the large-scale part is the same as that due to the boundary imperfections. For the case of non-resonant wavelength excitation some three-dimensional solutions in the form of superposition of small-scale multi-modal solutions with large-scale multi-modal (or non-modal) amplitudes can be preferred, provided that the wavelength of the small-scale modulation is not too small. Largescale flow structures are quite different from the small-scale flow structures in a number of cases and, in particular, they can exhibit kinks and can be non-modal in nature. The resulting flow patterns are affected accordingly, and they can provide quite unusual and non-regular three-dimensional preferred patterns. In particular, they are multiples of irregular rectangular patterns, and they can be non-periodic.

Description: Wave diffraction, wave forces and wave drift damping due to a floating body performing a slow rotation about the vertical axis (yaw) is considered. The rotation angle of the body may be arbitrary. The angular velocity is assumed small compared to the wave frequency, however. The problem is formulated in the frame of reference following the slow rotation of the body, accounting for non-Newtonian forces. By applying the method of multiple timescales, the fluid flow is determined consistently to leading order in the slow angular velocity and to second order in the wave amplitude. Mathematical solution of the problem is obtained by means of integral equations that are applicable to geometries of arbitrary shape. The wave loads are found by applying conservation of linear and angular momentum. The wave drift damping is expressed by the far-field amplitudes of the wave field and the dipole moments of the time-averaged second-order potential. Numerical results are presented for a ship and a vertical cylinder describing a circular path in the horizontal plane. The results show that the wave drift damping due to a slow yaw motion of a floating body is one order of magnitude larger than the time-averaged forces and moment when there is no rotation. Wave drift damping due to slow rotation and slow translation are found to be of equal importance.

Description: We consider the slow axial motion of a symmetric particle or drop in a bounded rotating fluid for small Rossby and Ekman numbers, Ro and E. Previous investigations pointed out that the available linear-theory results, based on the assumption of a dominant geostrophic core and infinitesimally thin viscous layers, yield a drag force larger than the available relevant experimental results, and are unable to explain some of the observed flow-field properties, for both solid and deformable particles. Here we attempt to improve the drag calculation model and the interpretation of the flow field by incorporating shear effects in the core (outside the E1/2 Ekman and E1/3 Stewartson layers), first in the linear (Ro = 0) formulation, then with keeping some influential nonlinear inertial terms for small but finite Ro. The major equation for the angular velocity in the core, ω(r), was usually solved by a finite-differences method, because in the practical parameter range the available analytical results are sufficiently accurate only for a disk particle or for a bubble. Results for various ε = (1/2H)1/2E1/4, no-slip parameter of particle surface K, and half container height H are presented for both spherical and disk particles. The drag is below the geostrophic value, typically, by 25% for ε ≈ 0.1 and by 50% for ε ≈ 0.5. The inclusion of the inertial terms causes the lateral ('vertical') shear regions to contract and expand on the upstream and downstream sides, respectively, and an inertial sublayer appears in the latter when Ro ≈ O(E3/4), but the net contribution to the drag is smaller than expected. Compared with more accurate solutions and experiments the present results underestimate the drag (the reasons are discussed) but are qualitatively consistent in many respects, which indicates that many of the observed flow-field features that have been traditionally attributed to inertial effects (not sufficiently small Ro) are, rather, by-products of the lateral shear (finite value of ε).

Description: The far-field sound of an unstable wave packet undergoing transition in a low-Mach-number, flat-plate boundary layer is investigated in the framework of Lighthill's acoustic analogy. Detailed accounts of the wave packet evolution are obtained by solving the full incompressible Navier-Stokes equations at δ = 1000. The numerically simulated flow structures show qualitative agreement with experimental observations of the fundamental breakdown type. The acoustic calculations are focused on the quadrupole source functions arising from Reynolds stress fluctuations. The wave packet is shown to produce negligible sound throughout the primary and secondary instability stages. Dramatic amplification of the Reynolds stress quadrupoles occurs as a result of the disintegration of the detached high-shear layer and the associated vortex shedding near the boundary layer edge. The dominant frequency of source oscillations coincides with that of vortex shedding.

Description: A class of unsteady boundary layers that form on flat extensible surfaces of finite but increasing length in otherwise stagnant surroundings is considered. The surface length R̄ is assumed to grow with time as tp where p 〉 0 and the velocity at any location 0 ≤ r̄ ≤ R̄ on the surface as tp-np-1r̄n, where n ≥ 0. The problem is cast into similarity variables and the governing parabolic differential equation shown to exhibit, for various combinations of n and p, regions of mixed mathematical diffusivity and reversals in the direction of convection of vorticity. Equations depicting such behaviour are usually termed singular parabolic and are here classified as follows: type-0, in which the mathematical diffusivity may be either positive or mixed but in which there are no reversals in the direction of convection of vorticity; type-1, in which the mathematical diffusivity may be either positive or mixed but in which there are reversals in the direction of convection of vorticity. Both types are shown to occur. Moreover while type-0 flows occur only when n = 1 and form with an unsteady separated stagnation point at the origin, type-1 flows occur only for 0 ≤ n 〈 1 and form with a steady stagnation point at the origin. Type-1 flows are further characterized by boundary layers with zero displacement thickness both at the origin and leading edge. Because singular parabolic equations require two initial conditions plus boundary conditions to ensure uniqueness, they are here treated numerically in a manner akin to elliptic boundary value problems. A successive-approximation implicit scheme was thus used and a wide range of cases solved in the parameter range n ∈ [0,1], p ∈ (0,2]. Amongst other things, it is shown that type-0 flows have lower drag than their type-1 counterparts. It is further shown that the drag on a flat rigid surface of finite length moving in its own plane at constant velocity and being continuously produced at the origin is higher than on a corresponding length of either a semi-infinite surface likewise produced or a semi-infinite plate in an aligned uniform stream; however if the surface is extensible and n 〉 1/2 the converse is true.

Description: In the idealized problem of homogeneous isotropic stationary inertial-range turbulence the rate of relative dispersion of an ensemble of tracer pairs can be characterized by a constant C0. In order to compute this constant with random-flight equations, however, it is necessary first to know the values of two other constants, C1 and C2, that occur in the two-particle velocity-component relations of Lagrangian tracers (Faller 1992). C1 and C2 are found by an elaborate trial and error procedure in a new two-tracer random-flight model of dispersion that matches input and output values of these two variates. The constant C0 is then computed using the Lagrangian relations and is found to be significantly smaller than when the Eulerian Kármán/Howarth correlations are used. The probability density distribution of tracer separations has a kurtosis slightly larger than that of a comparable Gaussian distribution. At small spacings the frequency of tracer spacings is six to ten times larger than would be expected from a Gaussian distribution. The distribution function for the speed of separation of the Lagrangian tracers has a negative skewness similar to that found for two-point Eulerian velocities.

Description: Shock wave propagation arising from steady one-dimensional motion of a piston in a granular gas composed of inelastically colliding particles is treated theoretically. A self-similar long-time solution is obtained in the strong shock wave approximation for all values of the upstream gas volumetric concentration v0. Closed form expressions for the long-time shock wave speed and the granular pressure on the piston are obtained. These quantities are shown to be independent of the particle collisional properties, provided their impacts are accompanied by kinetic energy losses. The shock wave speed of such non-conservative gases is shown to be less than that for molecular gases by a factor of about 2. The effect of particle kinetic energy dissipation is to form a stagnant layer (solid block), on the surface of the moving piston, with density equal to the maximal packing density, vM. The thickness of this densely packed layer increases indefinitely with time. The layer is separated from the shock front by a fluidized region of agitated (chaotically moving) particles. The (long-time, constant) thickness of this layer, as well as the kinetic energy (granular temperature) distribution within it are calculated for various values of particle restitution and surface roughness coefficients. The asymptotic cases of dilute (v0 ≪ 1) and dense (v0 ∼ vM) granular gases are treated analytically, using the corresponding expressions for the equilibrium radial distribution functions and the pertinent equations of state. The thickness of the fluidized region is shown to be independent of the piston velocity. The calculated results are discussed in relation to the problem of vibrofluidized granular layers, wherein shock and expansion waves were registered. The average granular kinetic energy in the fluidized region behind the shock front calculated here compared favourably with that measured and calculated (Goldshtein et al. 1995) for vibrofluidized layers of spherical granules.

Description: An experimental study of the effect of riblets on three-dimensional nonlinear structures, the so-called A-vortices on laminar-turbulent transition showed that riblets delay the transformation of the A-vortices into turbulent spots and shift the point of transition downstream. This result is opposite to the negative influence of such ribbed surfaces on two-dimensional linear Tollmien-Schlichting waves (the linear stage of transition). Thus, the ribbed surface influences laminar-turbulent transition structures differently: a negative influence on the linear-stage transition structures and a positive influence on the nonlinear-stage transition structures. It is demonstrated that transition control by means of riblets requires special attention to be paid to the choice of their location, taking into account the stage of transition.

Description: We have studied numerically the stability of a two-dimensional Couette flow in a polytropic fluid subjected to a localized shear, using a pseudo-spectral method (Fourier-Chebyshev). The polytropic index has been chosen equal to 2 and a radial force (pseudo-gravity) is introduced in order to perform comparisons with the shallow water experimental results. When the Reynolds number is not too low, the initial flow which is purely azimuthal becomes unstable and a stable rotating pattern is formed, with a number of azimuthal modes which decreases when the Mach number increases. A qualitative agreement is found with the experimental results, although the spatial resolution constraint strongly limits the numerical Reynolds and Mach numbers. From the variation of the linear growth rate of the unstable modes with the Mach number, we are able to show the transition between a flow subjected to Kelvin-Helmholtz instability towards one essentially driven by a centrifugal instability, which is efficient for rotating supersonic flows if the angular momentum decreases outwards. The latter situation may occur for some flows in astrophysical disks.

Description: The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as 'pseudogravitational'. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field, and the analogous potential of the 'modified vorticity field', the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In §4, the (Arnold) techniques of nonlinear stability are developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean-square vector potential of the magnetic field.

Description: A weakly nonlinear analysis is presented of the small oscillations of nearly inviscid liquid bridges subjected to almost resonant axial vibrations of the disks. An amplitude equation is derived for the evolution of the complex amplitude of the oscillations that exhibits hysteresis and period doublings. We also analyse the steady streaming in the bulk forced by the oscillatory boundary layers near the disks; the boundary layer near the free surface produces no forcing terms. In particular some experimentally observed patterns are explained, and some new, non-observed ones are predicted. We conclude that the structure of this steady flow is not the appropriate one to counterbalance steady thermocapillary convection, but our results indicate how to get the desired counterbalancing effect.

Description: Numerical simulations are presented which, in conjunction with the accompanying experimental investigation by Petitjeans & Maxworthy (1996), are intended to elucidate the miscible flow that is generated if a fluid of given viscosity and density displaces a second fluid of different such properties in a capillary tube or plane channel. The global features of the flow, such as the fraction of the displaced fluid left behind on the tube walls, are largely controlled by dimensionless quantities in the form of a Péclet number Pe, an Atwood number At, and a gravity parameter. However, further dimensionless parameters that arise from the dependence on the concentration of various physical properties, such as viscosity and the diffusion coefficient, result in significant effects as well. The simulations identify two distinct Pe regimes, separated by a transitional region. For large values of Pe, typically above O(103), a quasi-steady finger forms, which persists for a time of O(Pe) before it starts to decay, and Poiseuille flow and Taylor dispersion are approached asymptotically. Depending on the strength of the gravitational forces, we observe a variety of topologically different streamline patterns, among them some that leak fluid from the finger tip and others with toroidal recirculation regions inside the finger. Simulations that account for the experimentally observed dependence of the diffusion coefficient on the concentration show the evolution of fingers that combine steep external concentration layers with smooth concentration fields on the inside. In the small-Pe regime, the flow decays from the start and asymptotically reaches Taylor dispersion after a time of O(Pe). An attempt was made to evaluate the importance of the Korteweg stresses and the consequences of assuming a divergence-free velocity field. Scaling arguments indicate that these effects should be strongest when steep concentration fronts exist, i.e. at large values of Pe and At. However, when compared to the viscous stresses, Korteweg stresses may be relatively more important at lower values of these parameters, and we cannot exclude the possibility that minor discrepancies observed between simulations and experiments in these parameter regimes are partially due to these extra stresses.

Description: The magnetohydrodynamic (MHD) flow through sharp 90° bends of rectangular cross-section, in which the flow turns from a direction almost perpendicular to the magnetic field to a direction almost aligned with the magnetic field, is investigated experimentally for high values of the Hartmann number M and of the interaction parameter N. The bend flow is characterized by strong three-dimensional effects causing a large pressure drop and large deformations in the velocity profile. Since such bends are basic elements of fusion reactors, the scaling laws of magnetohydrodynamic bends flows with the main flow parameters such as M and N as well as the sensitivity to small magnetic field inclinations are of major importance. The obtained experimental results are compared to those of an asymptotic theory. In the case where one branch of the bend is perfectly aligned with the magnetic field good agreement between the results obtained by the asymptotic model and by the experiments was found at high M ≈ 8 × 103 and N ≈ 105 for pressure as well as for electric potentials on the duct surface. At lower values of N a significant influence of inertia has been detected. The pressure drop due to inertial effects was found to scale with N-1/3. The same - 1/3-power dependency on N has been found in the vicinity of the bend for the electric potentials at walls aligned with the magnetic field. At walls with a significant normal component of the field an influence neither of the Hartmann number nor of the interaction parameter has been found. This suggests that the inertial part of the pressure drop arises from inertial side layers, whereas the core flow remains inertialess and inviscid. A variation of the Hartmann number is of negligible influence compared to inertia effects with respect to pressure drop and surface potential distribution. The viscous part of the pressure drop scales with M-1/2. Changes of the magnetic field orientation with respect to the bend lead in general to different flow patterns in the duct, because the electric current paths are changed. The inertia-electromagnetic interaction determines the magnitude of the inertial part of the pressure drop, which scales with N-1/3 for any magnetic field orientation. The dependence of the pressure drop on M remains proportional to M-1/2. With increasing M and N the measured data tend to those predicted by the asymptotic model. Local measurements within the liquid metal exhibit discrepancies with the model predictions for which no adequate explanation has been found. But they show that below a critical interaction parameter flow regions exist in which the flow is time dependent. These regions are highly localized, whereas the flow in the rest of the bend remains steady.