ALBERT

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Description: The Darcy model with the Boussinesq approximations is used to study double-diffusive instability in a horizontal rectangular porous enclosure subject to two sources of buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while Dirichlet or Neumann boundary conditions on temperature and solute are imposed on the horizontal walls. The onset and development of convection are first investigated using the linear and nonlinear perturbation theories. Depending on the governing parameters of the problem, four different regimes are found to exist, namely the stable diffusive, the subcritical convective, the oscillatory and the augmenting direct regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy ratio, N, Lewis number, Le, normalized porosity of the porous medium, ε, aspect ratio of the enclosure, A, and the thermal and solutal boundary condition type, κ, applied on the horizontal walls. On the basis of the nonlinear perturbation theory and the parallel flow approximation (for slender or shallow enclosures), analytical solutions are derived to predict the flow behaviour. A finite element numerical method is introduced to solve the full governing equations. The results indicate that steady convection can arise at Rayleigh numbers below the supercritical value, indicating the development of subcritical flows. At the vicinity of the threshold of supercritical convection the nonlinear perturbation theory and the parallel flow approximation results are found to agree well with the numerical solution. In the overstable regime, the existence of multiple solutions, for a given set of the governing parameters, is demonstrated. Also, numerical results indicate the possible occurrence of travelling waves in an infinite horizontal enclosure.

Description: Flows with velocity profiles very different from the parabolic velocity profile can occur in the entrance region of a tube as well as in tubes with converging/diverging cross-sections. In this paper, asymptotic and numerical studies are undertaken to analyse the temporal stability of such 'non-parabolic' flows in a flexible tube in the limit of high Reynolds numbers. Two specific cases are considered : (i) developing flow in a flexible tube; (ii) flow in a slightly converging flexible tube. Though the mean velocity profile contains both axial and radial components, the flow is assumed to be locally parallel in the stability analysis. The fluid is Newtonian and incompressible, while the flexible wall is modelled as a viscoelastic solid. A high Reynolds number asymptotic analysis shows that the non-parabolic velocity profiles can become unstable in the inviscid limit. This inviscid instability is qualitatively different from that observed in previous studies on the stability of parabolic flow in a flexible tube, and from the instability of developing flow in a rigid tube. The results of the asymptotic analysis are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the developing flow could be unstable at much lower Reynolds numbers than the parabolic flow, and hence this instability can be important in destabilizing the fluid flow through flexible tubes at moderate and high Reynolds number. For flow in a slightly converging tube, even small deviations from the parabolic profile are found to be sufficient for the present instability mechanism to be operative. The dominant non-parallel effects are incorporated using an asymptotic analysis, and this indicates that non-parallel effects do not significantly affect the neutral stability curves. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

Description: An experimental configuration has been found which allows detailed observation of three-dimensional vortex-vortex and associated vortex-surface interactions which appear similar to those observed in fully turbulent flow. Hydrogen bubble visualization illustrates a complicated intertwining, or braiding, of two initially co-rotating vortices. It is observed that the three-dimensional interactions of the braided vortices induces a pair of local surface-fluid eruptions reminiscent of the 'bursting' behaviour characteristic of the near-wall regeneration process of fully turbulent boundary layers.

Description: Linear mode conversion is the partial transfer of wave energy from one wave type (a) to another (b) in a weakly non-uniform background state. For propagation in one dimension (x), the local wavenumber kj x of each wave (j = a,b) varies with x; if these are equal at some xR, the waves are locally in phase, and resonant energy transfer can occur. We model wave propagation in the Gulf of Guinea, where wave a is an equatorially trapped Rossby-gravity (Yanai) wave, and wave b is a coastal Kelvin wave along the (zonal) north coast of the Gulf, both propagating in zonal coordinate x. The coupling of the waves is due to the overlap of their eigenfunctions (normal modes in y, the meridional coordinate). We derive coupled mode equations from a variational principle, and obtain an analytic expression for the wave-energy conversion coefficient, in terms of the wave frequency and the scale length of the thermocline depth.

Description: This paper discusses numerical experiments in which an initially uniform columnar vortex is subject to several types of axisymmetric forcing that mimic the strain field of a turbulent flow. The mean value of the strain along the vortex axis is in all cases zero, and the vortex is alternately stretched and compressed. The emphasis is on identifying the parameter range in which the vortex survives indefinitely. This extends previous work in which the effect of steady single-scale non-uniform strains was studied. In a first series of experiments the effect of the unsteadiness of the forcing is analysed, and it is found that the vortex survives as a compact object if the ratio between the oscillation frequency and the strain itself is low enough. A theoretical explanation is given which agrees with the numerical results. The strain is then generalized to include several spatial scales and oscillation frequencies, with characteristics similar to those in turbulent flows. The largest velocities are carried by the large scales, while the highest gradients and faster time scales are associated with the shorter wavelengths. Also in these cases 'infinitely long' vortices are obtained which are more or less uniform and compact. Vorticity profiles averaged along their axes are approximately Gaussian. The radii obtained from these profiles are proportional to the Burgers' radius of the r.m.s. (small-scale) axial strain, while the azimuthal velocities are proportional to the maximum (large-scale) axial velocity differences. The study is motivated by previous observations of intense vortex filaments in turbulent flows, and the scalings found in the present experiments are consistent with those found in the turbulent simulations.

Description: Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.

Description: The dynamics of a passive tracer in the velocity field of four identical point vortices, moving under the influence of their self-induced advection, is investigated. Of interest is the change in mixing and transport properties of the tracer for the three different classes of vortex motion : periodic, quasi-periodic and chaotic. As a consequence of conservation laws, the vortex motion is confined to a finite region of phase space; therefore, the tracer phase space can be partitioned into an inner and an outer region. We find that in the case of quasi-periodic vortex motion the tracer phase space exhibits a well-defined barrier to transport between the central chaotic region and the outer region, where the trajectories are regular. In the case of chaotic vortex motion the barrier becomes permeable. The particle dynamics goes through an intermittent behaviour, where forays into the central region alternate with trapping in outer annular orbits. In the far field, an estimate of diffusion rates is made through a multipole expansion of the tracer velocity field. We make use of a specific stochastic model for the tracer velocity field which predicts no diffusion for the case of quasiperiodic vortex motion and, for the case of chaotic vortex motion, a diffusion rate that goes to zero at large distances from the vortex cluster.

Description: A premixed ducted flame, burning in the wake of a bluff-body flame-holder, is considered. For such a flame, interaction between acoustic waves and unsteady combustion can lead to self-excited oscillations. The concept of a time-invariant turbulent flame speed is used to develop a kinematic model of the response of the flame to flow disturbances. Variations in the oncoming flow velocity at the flame-holder drive perturbations in the flame initiation surface and hence in the instantaneous rate of heat release. For linear fluctuations, the transfer function between heat release and velocity can be determined analytically from the model and is in good agreement with experiment across a wide frequency range. For nonlinear fluctuations, the model reproduces the flame surface distortions seen in schlieren films. Coupling this kinematic flame model with an analysis of the acoustic waves generated in the duct by the unsteady combustion enables the time evolution of disturbances to be calculated. Self-excited oscillations occur above a critical fuel-air ratio. The frequency and amplitude of the resulting limit cycles are in satisfactory agreement with experiment. Flow reversal is predicted to occur during part of the limit-cycle oscillation and the flame then moves upstream of the flame-holder, just as in experimental visualizations. The main nonlinearity is identified in the rate of heat release, which essentially 'saturates' once the amplitude of the velocity fluctuation exceeds its mean. We show that, for this type of nonlinearity, describing function analysis can be used to give a good estimate of the limit-cycle frequency and amplitude from a quasi-nonlinear theory.

Description: Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

Description: A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg-de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of the Korteweg-de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Kortewegde Vries equation. The two types of solution are then used to quantitatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.

Description: We investigate the temporal evolution of the geometrical distribution of a passive scalar injected continuously into the far field of a turbulent water jet at a scale d smaller than the local integral scale of the turbulence. The concentration field is studied quantitatively by a laser-induced-fluorescence technique on a plane cut containing the jet axis. Global features such as the scalar dispersion from the source, as well as the fine structure of the scalar field, are analysed. In particular, we define the volume occupied by the regions whose concentration is larger than a given concentration threshold (support of the scalar field) and the surface in which this volume is enclosed (boundary of the support). The volume and surface extents, and their respective fractal dimensions are measured as a function of time t, and the concentration threshold is normalized by the initial concentration Cs/C0 for different injection sizes d. All of these quantities display a clear dependence on t, d and Cs, and their evolutions rescale with the variable ξ = (ut/d)(Cs/C0), the fractal dimension being, in addition, scale dependent. The surface-to-volume ratio and the fractal dimension of both the volume and the surface tend towards unity at large ξ, reflecting the sheet-like structure of the scalar at small scales. These findings suggest an original picture of the kinetics of turbulent mixing.

Description: History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier-Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

Description: We consider the reflection of oblique compression waves from a two-dimensional, steady, laminar boundary layer on a flat, adiabatic plate at free-stream pressures such that dense-gas effects are non-negligible. The full Navier-Stokes equations are solved through use of a dense-gas version of the Beam-Warming implicit scheme. The main fluids studied are Bethe-Zel'dovich-Thompson (BZT) fluids. These are ordinary gases which have specific heats large enough to cause the fundamental derivative of gasdynamics to be negative for a range of pressures and temperatures in the single-phase vapour regime. It is demonstrated that the unique dynamics of BZT fluids can result in a suppression of shock-induced separation. Numerical tests performed reveal that the physical mechanism leading to this suppression is directly related to the disintegration of any compression discontinuities originating in the flow. We also demonstrate numerically that the interaction of expansion shocks with the boundary layer produces no adverse effects.

Description: This paper describes the evolution of an incompressible turbulent boundary layer on the flat wall of an 'S'-shaped wind tunnel test section under the influence of changing streamwise and spanwise pressure gradients. The unit Reynolds number based on the mean velocity at the entrance of the test section was fixed to 106 m-1 resulting in Reynolds numbers Reδ2, based on the streamwise momentum thickness and the local freestream velocity, between 3.9 and 11 x 103. The particular feature of the experiment is the succession of two opposite changes of core flow direction which causes a sign change of the spanwise pressure gradient accompanied by a reversal of the spanwise velocity component near the wall, i.e. by the formation of so-called cross-over velocity profiles. The aim of the study is to provide new insight into the development of the mean and fluctuating flow field in three-dimensional pressure-driven boundary layers, in particular of the turbulence structure of the near-wall and the cross-over region. Mean velocities, Reynolds stresses and all triple correlations were measured with a newly developed miniature triple-hot-wire probe and a near-wall hot-wire probe which could be rotated and traversed through the test plate. Skin friction measurements were mostly performed with a wall hot-wire probe. The data from single normal wires extend over wall distances of y+ ≳ 3 (in wall units), while the triple-wire probe covers the range y+ ≳ 30. The data show the behaviour of the mean flow angle near the wall to vary all the way to the wall. Then, to interpret the response of the turbulence to the pressure field, the relevant terms in the Reynolds stress transport equations are evaluated. Finally, an attempt is made to assess the departure of the Reynolds stress profiles from local equilibrium near the wall.

Description: Three-dimensional large-amplitude oscillations of a mercury drop were obtained by electrical excitation in low gravity using a drop tower. Multi-lobed (from three to six lobes) and polyhedral (including tetrahedral, hexahedral, octahedral and dodecahedral) oscillations were obtained as well as axisymmetric oscillation patterns. The relationship between the oscillation patterns and their frequencies was obtained, and it was found that polyhedral oscillations are due to the nonlinear interaction of waves. A mathematical model of three-dimensional forced oscillations of a liquid drop is proposed and compared with experimental results. The equations of drop motion are derived by applying the variation principle to the Lagrangian of the drop motion, assuming moderate deformation. The model takes the form of a nonlinear Mathieu equation, which expresses the relationships between deformation amplitude and the driving force's magnitude and frequency.

Description: Natural convection in a rectangular cavity is examined, utilizing the second law of thermodynamics. Through an application of the second law the rate of entropy generation associated with the convective pattern changes is evaluated for the onset of natural convection in a cavity with free boundaries, for which an exact solution is sought, as well as with rigid boundaries which is studied numerically. Entropy to be generated from the perturbed temperature and velocity fields is shown to depend on AR (aspect ratio of the cavity), Rac (the critical Rayleigh number) and a non-dimensional parameter, Ω, which is related to the ratio of entropy generation by viscous friction to that by thermal transport. The convective pattern change is related to a change in the spatial distributions of the rate of entropy generation due to heat transfer and due to dissipation, demonstrating that an application of the second law helps examine convective pattern changes quantitatively by dealing with temperature and velocity fields in a unified manner.

Description: Large-scale structures in a plane turbulent mixing layer are studied through the use of the proper orthogonal decomposition (POD). Extensive experimental measurements are obtained in a turbulent plane mixing layer by means of two cross-wire rakes aligned normal to the direction of the mean shear and perpendicular to the mean flow direction. The measurements are acquired well into the asymptotic region. From the measured velocities the two-point spectral tensor is calculated as a function of separation in the cross-stream direction and spanwise and streamwise wavenumbers. The continuity equation is then used for the calculation of the non-measured components of the tensor. The POD is applied using the cross-spectral tensor as its kernel. This decomposition yields an optimal basis set in the mean square sense. The energy contained in the POD modes converges rapidly with the first mode being dominant (49% of the turbulent kinetic energy). Examination of these modes shows that the first mode contains evidence of both known flow organizations in the mixing layer, i.e. quasi-two-dimensional spanwise structures and streamwise aligned vortices. Using the shot-noise theory the dominant mode of the POD is transformed back into physical space. This structure is also indicative of the known flow organizations.

Description: A theoretical framework is developed to predict the rate of geometric collision and the collision velocity of small size inertialess particles in general turbulent flows. The present approach evaluates the collision rate for small size, inertialess particles in a given instantaneous flow field based on the local eigenvalues of the rate-of-strain tensor. An ensemble average is then applied to the instantaneous collision rate to obtain the average collision rate. The collision rates predicted by Smoluchowski (1917) for laminar shear flow and by Saffman & Turner (1956) for isotropic turbulence are recovered. The collision velocities presently predicted in both laminar shear flow and isotropic turbulence agree well with the results from numerical simulations for particle collision in both flows. The present theory for evaluating the collision rate and the collision velocity is also applied to a rapidly sheared homogeneous turbulence to assess the effect of strong anisotropy on the collision rate. Using (ε/ν)1/2, in which ε is the average turbulence energy dissipation rate and ν is the fluid kinematic viscosity, as the characteristic turbulence shear rate to normalize the collision rate, the effect of the turbulence structure on the collision rate and collision velocity can be reliably described. The combined effects of the mean flow shear and the turbulence shear on the collision rate and collision velocity are elucidated.

Description: We show that the class of constitutive relations for turbulence models put forward by Wang (1997) in this journal conflicts with dimensional analysis, unless the turbulent Reynolds stresses were to be tied to the molecular viscous stresses everywhere in the flow. We then reiterate, using counter-examples, that the controversial postulate of material frame-indifference is unfounded for turbulence, and is counter-productive in the quest for accuracy. We add a comment on the role of acceleration.

Description: An experimental study of entrainment of sand-sized particles in turbulent boundary layers has been performed in a high-speed wind tunnel at square-pulse flow speeds of 27 to 101ms-1 and for soil bed lengths varying from 2.1 to 5.8m. Because of high particle drag-to-weight ratios (D/W = 100-1000) and friction velocities (uf) well above soil threshold friction velocities (uft;10 ≤ uf/uft ≤ 40), the present results correspond to the suspension regime of dust lofting, in contrast to low-speed saltation flows (1 ≤ uf/uft ≤ 5; D/W ≤ 15). Results are obtained characterizing particle entrainment for both a natural soil (White Sands Missile Range (WSMR) sand; 50% finer-by-weight diameter, D̄50 = 180μm) and a monosized sand sample (Ottawa sand, D̄50 = 250 μm). Measurements of local boundary layer velocities and dust densities were performed with traversing state-of-the-art diagnostics. Scouring rate data (0.015 ≤ m̄s ≤ 0.30gcm-2 s-1) and streamwise soil flux (10 ≤ Q ≤ 150gcm-1 s-1) as a function of bed length and velocity were determined. Scouring rates were found to increase as the 3/2-power of velocity, but decay as the inverse square root of dust bed length. Corresponding streamwise soil fluxes (also known as soil loss rates) increased to the 3/2-power of velocity in contrast to the cube power dependence for low-speed results (ufree-stream ≤ 15m s-1; Q ≤ 1.5gcm-1 s-1). Comparison of scouring rate data (from pre/post-test soil loss measurements) with derived data based on the rate of change of streamwise flux with distance was favourable. WSMR rates were always lower than Ottawa sand rates, a result consistent with the lower repose angle for the Ottawa sand sample. Both sets of soil data demonstrate that dust edges extend vertically to higher elevations than corresponding velocity edges. This result implies that the turbulent Schmidt number for the present flows is less than unity and of the order of 0.7. Favourable collapsing of the scouring rate data base was achieved when measured rates were normalized by the friction velocity mass flux, square root of edge Mach number and sand repose angle ratio. A universal rate of 0.3±0.1 correlated well with the bulk of the data.

Description: Stably stratified flows past three-dimensional orography have been investigated using a stratified towing tank. Flows past idealized axisymmetric orography in which the Froude number, Fh = U/Nh (where U is the towing speed, N is the buoyancy frequency and h is the height of the obstacle) is less than unity have been studied. The orography considered consists of two sizes of hemisphere and two cones of different slope. For all the obstacles measurements show that as Fh decreases, the drag coefficient increases, reaching between 2.8 and 5.4 times the value in neutral flow (depending on obstacle shape) for Fh ≲ 0.25. Local maxima and minima in the drag also occur. These are due to the finite depth of the tank and can be explained by linear gravity-wave theory. Flow visualization reveals a lee wave train downstream in which the wave amplitude is O(Fhh), the smallest wave amplitude occurring for the steepest cone. Measurements show that for all the obstacles, the dividing-streamline height, zs, is described reasonably well by the formula zs/h = 1 - Fh. Flow visualization and acoustic Doppler velocimeter measurements in the wake of the obstacles show that vortex shedding occurs when Fh ≲ 0.4 and that the period of the vortex shedding is independent of height. Based on velocity measurements in the wake of both sizes of hemisphere (plus two additional smaller hemispheres), it is shown that a blockage-corrected Strouhal number, S2c. = fL2/Uc, collapses onto a single curve when plotted against the effective Froude number, Fhc = Uc/Nh. Here, Uc is the blockage-corrected free-stream speed based on mass-flux considerations, f is the vortex shedding frequency and L2 is the obstacle width at a height zs/2. Collapse of the data is also obtained for the two different shapes of cone and for additional measurements made in the wake of triangular and rectangular flat plates. Indeed, the values of S2c for all these obstacles are similar and this suggests that despite the fact that the obstacle widths vary with height, a single length scale determines the vortex-street dynamics. Experiments conducted using a splitter plate indicate that the shedding mechanism provides a major contribution to the total drag (∼ 25%). The addition of an upstream pointing 'verge region' to a hemisphere is also shown to increase the drag significantly in strongly stratified flow. Possible mechanisms for this are discussed.

Description: The reflection of asymmetric shock waves in steady flows is studied both theoretically and experimentally. While the analytical model was two-dimensional, three-dimensional edge effects influenced the experiments. In addition to regular and Mach reflection wave configurations, an inverse-Mach reflection wave configuration, which has been observed so far only in unsteady flows (e.g. shock wave reflection over concave surfaces or over double wedges) has been recorded. A hysteresis phenomenon similar to the one that exists in the reflection of symmetric shock waves has been found to also exist in the reflection of asymmetric shock waves. The domains and transition boundaries of the various types of overall reflection wave configurations are analytically predicted.

Description: Atmospheric and oceanic convection often occurs over areas occupied by many localized circulation elements known as plumes. The convective transports therefore may depend not only on the individual elements, but also on the interactions between plumes and the turbulent environment created by other plumes. However, many attempts to understand these plumes focus on individual isolated elements, and the behaviour of an ensemble is not understood. Geophysical convection may be influenced by rotation when the transit time of a convecting element is long compared to an inertial period (for example in deep oceanic convection). Much recent attention has been given to the effect of rotation on individual plumes, but the role of rotation in modifying the behaviour of an ensemble is not fully understood. Here we examine the behaviour of plumes within an ensemble, both with and without rotation, to identify the influence of rotation on ensemble plume dynamics. We identify the coherent structures (plumes) present in numerical solutions of turbulent Rayleigh-Bénard convection, a canonical example of a turbulent plume ensemble. We use a conditional sampling compositing technique to extract the typical structure in both non-rotating and rotating solutions. The dynamical balances of these composite plumes are evaluated and compared with entraining plume models. We find many differences between non-rotating and rotating plumes in their transports of mass, buoyancy and momentum. As shown in previous studies, the expansion of the turbulent plume by entrainment of exterior fluid is suppressed by strong rotation. Our most significant new result is quantification of the continuous mixing between the plume and ambient fluid which occurs at high rotation without any net changes in plume volume. This mixing is generated by the plume-plume interactions and acts to reduce the buoyancy anomaly of the plume. By contrast, in the non-rotating case, no such loss of buoyancy by mixing occurs. As a result, the total buoyancy transport by upwardly moving plumes diminishes across the layer in the rotating case, while remaining approximately constant in the non-rotating case. At high values of rotation, the net vertical acceleration is considerably reduced compared to the non-rotating case due to loss of momentum through entrainment and mixing and a decelerating pressure gradient which partially balances the buoyancy-driven acceleration of plumes. As a result of the dilution of buoyancy, the pressure-gradient deceleration and the loss of momentum due to mixina with the environment in the rotating solutions, the conversion of potential energy to kinetic energy is significantly less than that of non-rotating plumes. The combination of efficient lateral mixing and slow vertical movement by the plumes accounts for the unstable mean temperature gradient that occurs in rotating Rayleigh-Bénard convection, while the less penetrative convection found at low Rossby number is a consequence of the reduced kinetic energy transport. Within the ensemble of plumes identified by the conditional sampling algorithm, distributions of vertical velocity, buoyancy and vorticity mimic those of the volume as a whole. Plumes cover a small fraction of the total area, yet account for most of the vertical heat flux.

Description: The present paper is devoted to an analysis of tip vortex cavitation under confined situations. The tip vortex is generated by a three-dimensional foil of elliptical planform, and the confinement is achieved by flat plates set perpendicular to the span, at an adjustable distance from the tip. In the range of variation of the boundary-layer thickness investigated, no significant interaction was observed between the tip vortex and the boundary layer which develops on the confinement plate. In particular, the cavitation inception index for tip vortex cavitation does not depend significantly upon the length of the plate upstream of the foil. On the contrary, tip clearance has a strong influence on the non-cavitating structure of the tip vortex and consequently on the inception of cavitation in its core. The tangential velocity profiles measured by a laser-Doppler velocimetry (LDV) technique through the vortex, between the suction and the pressure sides of the foil, are strongly asymmetric near the tip. They become more and more symmetric downstream and the confinement speeds up the symmetrization process. When the tip clearance is reduced to a few millimetres, the two extrema of the velocity profiles increase. This increase results in a decrease of the minimum pressure in the vortex centre and accounts for the smaller resistance to cavitation observed when tip clearance is reduced. For smaller values of tip clearance, a reduction of tip clearance induces on the contrary a significant reduction in the maxima of the tangential velocity together with a significant increase in the size of the vortex core estimated along the confinement plate. Hence, the resistance to cavitation is much higher for such small values of tip clearance and in practice, no tip vortex cavitation is observed for tip clearances below 1.5mm. The cavitation number for the inception of tip vortex cavitation does not correlate satisfactorily with the lift coefficient, contrary to classical results obtained without any confinement. Owing to the specificity introduced by the confinement, the usual procedure developed in an infinite medium to estimate the vortex strength from LDV measurements is not applicable here. Hence, a new quantity homogeneous to a circulation had to be defined on the basis of the maximum tangential velocity and the core size, which proved to be better correlated to the cavitation inception data.