ALBERT

Description: We experimentally study the formation of cusps at the free surfaces of viscous fluids in three simple cases that portray possible natural or industrial procceses. Two cases concern levelling driven by gravity: the case (A) of a sinusoidal surface and the case (B) of a single groove. In the third case (C), an initially sinusoidal surface evolves under the action of a fast enough lateral compression that the effects of gravity are negligible. Case (A) shows a critical aspect ratio above which the cusps form. Case (B) allows a more detailed study of the evolution of the cuspidal structure, which does not change in shape but reduces its size according to a simple power law dependence in time. In case (C), cusps form even for small initial aspect ratios.

Description: The yield conditions for the displacement of three-dimensional fluid droplets from solid boundaries are studied through a series of numerical computations. The study considers low-Reynolds-number shear flows over plane boundaries and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, capillary number Ca and advancing and receding contact angles, θ A and θ R . This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(λ, θ A , Δθ) where Δθ = θ A - θ R is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on solid surfaces. Additional constraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While good agreement is found in the joint asymptotic limits Δθ ≪ θ A ≪ 1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets.

Description: The formation time scale of axisymmetric vortex rings is studied numerically for relatively long discharge times. Experimental findings on the existence and universality of a formation time scale, referred to as the 'formation number', are confirmed. The formation number is indicative of the time at which a vortex ring acquires its maximal circulation. For vortex rings generated by impulsive motion of a piston, the formation number was found to be approximately four, in very good agreement with experimental results. Numerical extensions of the experimental study to other cases, including cases with thick shear layers, show that the scaled circulation of the pinched-off vortex is relatively insensitive to the details of the formation process, such as the velocity programme, velocity profile, vortex generator geometry and the Reynolds number. This finding might also indicate that the properly scaled circulation of steady vortex rings varies very little. The formation number does depend on the velocity profile. Non-impulsive velocity programmes slightly increase the formation number, while non-uniform velocity profiles may decrease it significantly. In the case of a parabolic velocity profile of the discharged flow, for example, the formation number decreases by a factor as large as four. These findings indicate that a major source of the experimentally found small variations in the formation number is the different evolution of the velocity profile of the discharged flow.

Description: The characteristics of turbulence caused by three-dimensional breaking of internal gravity waves beneath a critical level are investigated by means of high-resolution numerical simulations. The flow evolves in three stages. In the first one the flow is two-dimensional: internal gravity waves propagate vertically upwards and create a convectively unstable region beneath the critical level. Convective instability leads to turbulent breakdown in the second stage. The developing three-dimensional mixed region is organized into shear-driven overturning rolls in the plane of wave propagation and into counter-rotating streamwise vortices in the spanwise plane. The production of turbulent kinetic energy by shear is maximum. In the last stage, shear production and mechanical dissipation of turbulent kinetic energy balance. The evolution of the flow depends on topographic parameters (wavelength and amplitude), on shear and stratification as well as on viscosity. Here, only the implications of the viscosity for the instability structure and evolution in terms of the Reynolds number are considered. Smaller viscosity leads to earlier onset of convective instability and overturning waves. However, viscosity retards the onset of smaller-scale three-dimensional instabilities and leads to a reduced momentum transfer to the mean flow below the critical level. Hence, the formation of secondary overturning rolls is sustained by lower viscosity. The budgets of total kinetic and potential energies are calculated. Although the domain-averaged turbulent kinetic energy is less than 1% of the total kinetic energy, it is strong enough to form a patchy and intermittent turbulent mixed layer below the critical level.

Description: The reflection of a straight-crested gravity wave by a non-secular perturbation h1(x) in depth relative to an otherwise flat bottom of depth h0 is calculated through an expansion in ε ∝ h1/h0. Explicit results are developed up to second order for the sinusoidal patch h1 = -b sin(mπx/l), 0 〈 x 〈 l, and reduced for Bragg resonance. Trapped modes are absent at first order but appear at second order and contribute O(ε2)/O(ε3) to the maximum (Bragg-resonant) reflection coefficient for odd/even m. A third-order approximation that includes the dominant contributions of the third-order components of the resonant peak of the reflection coefficient for large m, but neglects the trapped modes, predicts resonant peaks that agree with the values measured by Davies & Heathershaw (1984).

Description: The modulation of isotropic turbulence by particles has been investigated using direct numerical simulation (DNS). The particular focus of the present work is on the class of dilute flows in which particle volume fractions and inter-particle collisions are negligible. Gravitational settling is also neglected and particle motion is assumed to be governed by drag with particle relaxation times ranging from the Kolmogorov scale to the Eulerian time scale of the turbulence and particle mass loadings up to 1. The velocity field was made statistically stationary by forcing the low wavenumbers of the flow. The calculations were performed using 963 collocation points and the Taylor-scale Reynolds number for the stationary flow was 62. The effect of particles on the turbulence was included in the Navier-Stokes equations using the point-force approximation in which 963 particles were used in the calculations. DNS results show that particles increasingly dissipate fluid kinetic energy with increased loading, with the reduction in kinetic energy being relatively independent of the particle relaxation time. Viscous dissipation in the fluid decreases with increased loading and is larger for particles with smaller relaxation times. Fluid energy spectra show that there is a non-uniform distortion of the turbulence with a relative increase in small-scale energy. The non-uniform distortion significantly affects the transport of the dissipation rate, with the production and destruction of dissipation exhibiting completely different behaviours. The spectrum of the fluid-particle energy exchange rate shows that the fluid drags particles at low wavenumbers while the converse is true at high wavenumbers for small particles. A spectral analysis shows that the increase of the high-wavenumber portion of the fluid energy spectrum can be attributed to transfer of the fluid particle covariance by the fluid turbulence. This in turn explains the relative increase of small-scale energy caused by small particles observed in the present simulations as well as those of Squires & Eaton (1990) and Elghobashi & Truesdell (1993).

Description: Unsteady boundary-layer development over moving walls in the limit of infinite Reynolds number is investigated using both the Eulerian and Lagrangian formulations. To illustrate general trends, two model problems are considered, namely the translating and rotating circular cylinder and a vortex convected in a uniform flow above an infinite flat plate. To enhance computational speed and accuracy for the Lagrangian formulation, a remeshing algorithm is developed. The calculated results show that unsteady separation is delayed with increasing wall speed and is eventually suppressed when the speed of the separation singularity approaches that of the local mainstream velocity. This suppression is also described analytically. Only 'upstream-slipping' separation is found to occur in the model problems. The changes in the topological features of the flow just prior to the separation that occur with increasing wall speed are discussed.

Description: The linear stability of doubly diffusive convection is considered for a two-dimensional, Boussinesq fluid in a tall thin slot. For a variety of boundary conditions on the slot walls, instability sets in through zero wavenumber over a wide range of physical conditions. Long-wave equations governing the nonlinear development of the instability are derived. The form of the long-wave equations sensitively depends on the thermal and salt boundary conditions; the possible long-wave theories are catalogued. Finiteamplitude solutions and their stability are studied. In some cases the finite-amplitude solutions are not the only possible attractors and numerical solutions presenting the alternatives are given. These reveal temporally complicated dynamics.

Description: Marangoni convection in a differentially heated binary mixture is studied numerically by continuation. The fluid is subject to the Soret effect and is contained in a two-dimensional small-aspect-ratio rectangular cavity with one undeformable free surface. Either or both of the temperature and concentration gradients may be destabilizing; all three possibilities are considered. A spectral-element time-stepping code is adapted to calculate bifurcation points and solution branches via Newton's method. Linear thresholds are compared to those obtained for a pure fluid. It is found that for large enough Soret coefficient, convection is initiated predominantly by solutal effects and leads to a single large roll. Computed bifurcation diagrams show a marked transition from a weakly convective Soret regime to a strongly convective Marangoni regime when the threshold for pure fluid thermal convection is passed. The presence of many secondary bifurcations means that the mode of convection at the onset of instability is often observed only over a small range of Marangoni number. In particular, two-roll states with up-flow at the centre succeed one-roll states via a well-defined sequence of bifurcations. When convection is oscillatory at onset, the limit cycle is quickly destroyed by a global (infinite-period) bifurcation leading to subcritical steady convection.

Description: X-ray lithography and micro-machining have been used to study gas-assisted liquid atomization in which a liquid film was impinged by a large number of sonic micro-gas jets. Three distinct breakup regimes were demonstrated. Two of these regimes share characteristics with previously observed atomization processes: a bubble bursting at a free surface (Newitt et al. 1954; Boulton-Stone & Blake 1993) and liquid sheet disintegration in a high gas/liquid relative velocity environment (Dombrowski & Johns 1963). The present work shows that suitable control of the gas/liquid interface creates a third regime, a new primary atomization mechanism, in which single liquid droplets are ejected directly from the liquid film without experiencing an intermediate ligament formation stage. The interaction produces a stretched liquid sheet directly above each gas orifice. This effectively pre-films the liquid prior to its breakup. Following this, surface tension contracts the stretched film of liquid into a sphere which subsequently detaches from the liquid sheet and is entrained by the gas jet that momentarily pierces the film. After droplet ejection, the stretched liquid film collapses, covering the gas orifice, and the process repeats. This new mechanism is capable of the efficient creation of finely atomized sprays at low droplet ejection velocities (e.g. 20 μm Sauter mean diameter methanol sprays using air at 239 kPa, with air-to-liquid mass ratios below 1.0, and droplet velocities lower than 2.0 m s-1). Independent control of the gas and the liquid flows allows the droplet creation process to be effectively de-coupled from the initial droplet momentum, a characteristic not observed with standard gas-assisted atomization mechanisms.

Description: The unsteady flow of a viscous incompressible fluid in a circular tube with an elastic insertion is studied numerically. The deformation of the elastic membrane is obtained by the theory of finite elasticity whose equations are solved simultaneously with the fluid equations in the axisymmetric approximation. The elastic wall expands outwards due to the positive transmural pressure and represents an idealized model for the response of pathologies in large arteries. It is found that if either the fluid discharge or the reference pressure are imposed downstream of the insertion, the fluid-wall interaction develops travelling waves along the membrane whose period depends on membrane elasticity; these are unstable in a perfectly elastic membrane and are stabilized by viscoelasticity. In the reversed system, when the fluid discharge is imposed on the opposite side, the stable propagation phenomenon remains the same because of symmetry arguments. Such arguments do not apply to the originally unstable behaviour. In this case, even when the membrane is perfectly elastic, propagation is damped and two natural fluctuations appear in the form of stationary waves. In all cases the resonance of the fluid-wall interaction has been analysed. Comparisons with previously observed phenomena and with results of analogous studies are discussed.

Description: A generalized similarity formulation extending the work of Terrill (1967) for Couette-Poiseuille flow in the annulus between concentric cylinders of infinite extent is given. Boundary conditions compatible with the formulation allow a study of the effects of inner and outer cylinder transpiration, rotation, translation, stretching and twisting, in addition to that of an externally imposed constant axial pressure gradient. The problem is governed by η, the ratio of inner to outer radii, a Poiseuille number, and nine Reynolds numbers. Single-cylinder and planar problems can be recovered in the limits η → 0 and η → 1, respectively. Two coupled primary nonlinear equations govern the meridional motion generated by uniform mass flux through the porous walls and the azimuthal motion generated by torsional movement of the cylinders; subsidiary equations linearly slaved to the primary flow govern the effects of cylinder translation, cylinder rotation, and an external pressure gradient. Steady solutions of the primary equations for uniform source/sink flow of strength F through the inner cylinder are reported for 0 ≤ η ≤ 1. Asymptotic results corroborating the numerical solutions are found in different limiting cases. For F 〈 0 fluid emitted through the inner cylinder fills the gap and flows uniaxially down the annulus; an asymptotic analysis leads to a scaling that removes the effect of η in the pressure parameter β, namely β = π2R*2, where R* = F(1 - η)/(1 + η). The case of sink flow for F 〉 0 is more complex in that unique solutions are found at low Reynolds numbers, a region of triple solutions exists at moderate Reynolds numbers, and a two-cell solution prevails at large Reynolds numbers. The subsidiary linear equations are solved at η = 0.5 to exhibit the effects of cylinder translation, rotation, and an axial pressure gradient on the source/sink flows.

Description: Alongshore propagating low-frequency 0(0.01 Hz) waves related to the direction and intensity of the alongshore current were first observed in the surf zone by OltmanShay, Howd & Birkemeier (1989). Based on a linear stability analysis, Bowen & Holman (1989) demonstrated that a shear instability of the alongshore current gives rise to alongshore propagating shear (vorticity) waves. The fully nonlinear dynamics of finite-amplitude shear waves, investigated numerically by Allen, Newberger & Holman (1996). depend on α, the non-dimensional ratio of frictional to nonlinear terms, essentially an inverse Reynolds number. A wide range of shear wave environments are reported as a function of α, from equilibrated waves at larger α to fully turbulent flow at smaller α. When α is above the critical level αc, the system is stable. In this paper, a weakly nonlinear theory, applicable to α just below αc, is developed. The amplitude of the instability is governed by a complex Ginzburg-Landau equation. For the same beach slope and base-state alongshore current used in Allen et al. (1996), an equilibrated shear wave is found analytically. The finite-amplitude behaviour of the analytic shear wave, including a forced second-harmonic correction to the mean alongshore current, and amplitude dispersion, agree well with the numerical results of Allen et al. (1996). Limitations in their numerical model prevent the development of a side-band instability. The stability of the equilibrated shear wave is demonstrated analytically. The analytical results confirm that the Allen et al. (1996) model correctly reproduces many important features of weakly nonlinear shear waves.

Description: This paper studies the impinging of two ideal fluid jets. The usual two-dimensional model of jet flow uses an ideal, incompressible, weightless fluid to describe these impinging jets, so that the problem becomes one of complex analysis which seems to have an infinite number of analytical solutions, except for direct jet impacts. The new approach presented here is based on the construction of a dividing line between the two jets. It gives an efficient procedure for solving this problem numerically when the jets flow in arbitrary directions and the solution obtained seems to be unique.

Description: We consider a sheet flow in which heavy grains near a packed bed interact with a unidirectional turbulent shear flow of a fluid. We focus on sheet flows in which the particles are supported by their collisional interactions rather than by the velocity fluctuations of the turbulent fluid and introduce what we believe to be the simplest theory for the collisional regime that captures its essential features. We employ a relatively simple model of the turbulent shearing of the fluid and use kinetic theory for the collisional grain flow to predict profiles of the mean fluid velocity, the mean particle velocity, the particle concentration, and the strength of the particle velocity fluctuations within the sheet. These profiles are obtained as solutions to the equations of balance of fluid and particle momentum and particle fluctuation energy over a range of Shields parameters between 0.5 and 2.5. We compare the predicted thickness of the concentrated region and the predicted features of the profile of the mean fluid velocity with those measured by Sumer et al. (1996). In addition, we calculate the volume flux of particles in the sheet as a function of Shields parameter. Finally, we apply the theory to sand grains in air for the conditions of a sandstorm and calculate profiles of particle concentration, velocity, and local volume flux.

Description: Previous research has suggested that isolated, initially non-axisymmetric vortices in two-dimensional flows tend to become axisymmetric, in a coarse-grained sense, by purely inviscid mechanisms. That research, however, considered only vortices with broadly distributed vorticity. In this paper, it is shown that vortices with sufficiently steep edge gradients behave in a radically different way; in particular they can remain non-axisymmetric, apparently indefinitely. Such vortices, it is argued, are more typical in inviscid two-dimensional flows than the broadly distributed vortices previously considered, and hence the tendency for vortices to become axisymmetric is not generic to these flows.

Description: A previously unreported shock feature associated with the scouring of a horizontal granular bed by a dam-break wave is discussed. Near the wave centre, the present study shows, the free surface breaks backward and a hydraulic jump forms. This behaviour is described from the standpoint of shallow-water theory, suitably extended to deal with non-equilibrium sediment transport. The shock formation involves a particularly strong coupling between flow free-surface evolution and bed morphodynamics. Support for our conclusions is sought through experimental and numerical approaches. In order to magnify the observed phenomena, measurements were performed for the case of light bed particles moving in sheet and debris flow modes. A detailed picture of the transient two-phase flow is presented, based on whole field acquisition of the grain motions by particle tracking techniques. Corresponding shallow-water solutions are constructed numerically using a shock capturing scheme. Finally, an interpretation of the jump formation is proposed based on the theory of characteristics.

Description: Two free waves propagating in a parallel shear flow generate a critical layer when their nonlinear interaction induces a perturbation whose phase velocity matches the basic-state velocity somewhere in the flow domain. The condition necessary for this to occur may be interpreted as a resonance condition for a triad formed by the two waves and a (singular) mode of the continuous spectrum associated with the shear. The formation of the critical layer is investigated in the case of freely propagating Rossby waves in a two-dimensional inviscid flow in a β-channel. A weakly nonlinear analysis based on a normal-mode expansion in terms of Rossby waves and modes of the continuous spectrum is developed; it leads to a system of amplitude equations describing the evolution of the two Rossby waves and of the modes of the continuous spectrum excited during the interaction. The assumption of weak nonlinearity is not however self-consistent: it breaks down because nonlinearity always becomes strong within the critical layer, however small the initial amplitudes of the Rossby waves. This demonstrates the relevance of nonlinear critical layers to monotonic, stable, unforced shear flows which sustain wave propagation. A nonlinear critical-layer theory is developed that is analogous to the well-known theory for forced critical layers. Differences arise because of the presence of the Rossby waves: the vorticity in the critical layer is advected in the cross-stream direction by the oscillatory velocity field due to the Rossby waves. An equation is derived which governs the modification of the Rossby waves that results from their interaction; it indicates that the two Rossby waves are undisturbed at leading order. An analogue of the Stewartson-Warn-Warn analytical solution is also considered.

Description: Turbulent-shear-induced coagulation of monodisperse particles was examined experimentally in the nearly isotropic, spatially decaying turbulence generated by an oscillating grid. The 3.9 μm polystyrene microspheres used in the experiments were made neutrally buoyant and unstable by suspending them in a density-matched saline solution. In this way, particle settling, double-layer repulsion and particle inertia were negligible and the effect of turbulent shear was isolated. The coagulation rate was measured by monitoring the loss of singlet particles as a function of time and reactor turbulence intensity. By restricting consideration to experimental conditions where the singlet concentration was in excess, the effect of higher-order aggregate (i.e. triplet) formation was negligible and nonlinear regression using an integral rate expression that included terms for doublet formation and breakup was used to obtain the turbulent coagulation rate constant. The strength of the van der Waals attractions was characterized with the Hamaker constant obtained from Brownian coagulation experiments. Since particle bulk mixing was fast compared to the coagulation rate, the observed coagulation rate constants were averages over the local coagulation rates within the grid-stirred reactor. Knowledge of the spatial variation of turbulence within the reactor was necessary for quantitative prediction of the experiments because model predictions for the coagulation rate are nonlinear functions of shear rate. The investigation was conducted with particles smaller than the length scales of turbulence and since the smallest turbulent length scales, the Kolmogorov scales, have the highest shear rate they controlled the rate of particle aggregation. The distribution of the Kolmogorov shear rate at various grid oscillation frequencies was obtained by measuring the turbulent kinetic energy (E) using acoustic Doppler velocimetry and relating E to the Kolmogorov shear rate using scaling arguments. The experimentally measured turbulent coagulation rate constants were significantly lower than theoretical predictions that neglect interparticle interactions; however, simulations that included particle interactions showed excellent agreement with the experimental results. The favourable comparison provides evidence that the computer simulations capture the important physics of turbulent coagulation. That is, particle transport on length scales comparable to the particle radius controls the rate of turbulent shear coagulation and particle interactions are significant.

Description: The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as 'axisymmetrization', is studied asymptotically and numerically. The vortex is perturbed at t = 0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t-λ with λ = 1 + (n2 + 8)1/2, where n is the azimuthal wavenumber of the perturbation and n ≥ 1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).

Description: A detailed two-part computational investigation is conducted into the dynamical evolution of two-dimensional miscible porous media flows in the quarter five-spot arrangement of injection and production wells. High-accuary direct numerical simulations are performed that reproduce all dynamically relevant length scales in solving the vorticity-streamfunction formulation of Darcy's law. The accuracy of the method is demonstrated by a comparison of simulation data with linear stability results for radial source flow. Within this part, Part 1 of the present investigation, a series of simulations is discussed that demonstrate how the mobility ratio and the dimensionless flow rate denoted by the Péclet number Pe affect both local and integral features of homogeneous displacement processes. Mobility ratios up to 150 and Pe-values up to 2000 are investigated. For sufficiently large Pe-values, the flow near the injection well gives rise to a vigorous viscous fingering instability. As the unstable concentration front approaches the central region of the domain, nonlinear interactions between the fingers similar to those known from unidirectional flows are observed, such as merging, partial merging, and shielding, along with secondary tip-splitting and side-branching instabilities. At large Pe-values, several of these fingers compete for long times, before one of them accelerates ahead of the others and leads to the breakthrough of the front. In contrast to unidirectional flows, the quarter five-spot geometry imposes both an external length scale and a time scale on the flow. The resulting spatial non-uniformity of the potential base flow is observed to lead to a clear separation in space and time of large and small scales in the flow. Small scales occur predominantly during the early stages near the injection well, and at late times near the production well. The central domain is dominated by larger scales. Taken together, the results of the simulations demonstrate that both the mobility ratio and Pe strongly affect the dynamics of the flow. While some integral measures, such as the recovery at breakthrough, may show only a weak dependence on Pe for large Pe-values, the local fingering dynamics continue to change with Pe. The increased susceptibility of the flow to perturbations during the early stages provides the motivation to formulate an optimization problem that attempts to maximize recovery, for a constant overall process time, by employing a time-dependent flow rate. Within the present framework, which accounts for molecular diffusion but not for velocity-dependent dispersion, simulation results indeed indicate the potential to increase recovery by reducing the flow rate at early times, and then increasing it during the later stages.

Description: We study the stability of the interface between (a) two adjacent viscous layers flowing due to gravity through an inclined or vertical channel that is confined between two parallel plane walls, and (b) two superimposed liquid films flowing down an inclined or vertical plane wall, in the limit of Stokes flow. In the case of channel flow, linear stability analysis predicts that, when the fluids are stably stratified, the flow is neutrally stable when the surface tension vanishes and the channel is vertical, and stable otherwise. This behaviour contrasts with that of the gravity-driven flow of two superimposed films flowing down an inclined plane, where an instability has been identified when the viscosity of the fluid next to the plane is less than that of the top fluid, even in the absence of fluid inertia. We investigate the nonlinear stages of the motion subject to finite-amplitude two-dimensional perturbations by numerical simulations based on boundary-integral methods. In both cases of channel and film flow, the mathematical formulation results in integral equations for the unknown interface and free-surface velocity. The properties of the integral equation for multi-film flow are investigated with reference to the feasibility of computing a solution by the method of successive substitutions, and a deflation strategy that allows an iterative procedure is developed. In the case of channel flow, the numerical simulations show that disturbances of sufficiently large amplitude may cause permanent deformation in which the interface folds or develops elongated fingers. The ratio of the viscosities and densities of the two fluids plays an important role in determining the morphology of the emerging interfacial patterns. Comparing the numerical results with the predictions of a model based on the lubrication approximation shows that the simplified approach can only describe a limited range of motions. In the case of film flow down an inclined plane, we develop a method for extracting the properties of the normal modes, including the ratio of the amplitudes of the free-surface and interfacial waves and their relative phase lag, from the results of a numerical simulation for small deformations. The numerical procedure employs an adaptation of Prony's method for fitting a signal described by a time series to a sum of complex exponentials; in the present case, the signal is identified with the cosine or sine Fourier coefficients of the interface and free-surface waves. Numerical simulations of the nonlinear motion confirm that the deformability of the free surface is necessary for the growth of small-amplitude perturbations, and show that the morphology of the interfacial patterns developing subject to finite-amplitude perturbations is qualitatively similar to that for channel flow.

Description: In this paper the two-layer flow past a cylinder in a rotating frame is modelled numerically. This study follows on from the many theoretical and experimental examinations of the one-layer flow past a cylinder. The main aim of this study is to investigate the geophysically relevant effect on the flow of varying the vertical shear, while keeping the other parameters the same in both layers. This will include an examination of the contour plots of vorticity and streamfunction and measuring the size of the separated region and the position of separation as a function of the vertical shear. The most interesting result is that the separated region in the slower layer grows much larger than that in the faster layer as the vertical shear is increased.

Description: The general solution of the particle momentum equation for unsteady Stokes flows is otained analytically. The method used to obtain the solution consists of applying a fractional-differential operator to the first-order integro-differential equation of motion in order to transform the original equation into a second-order non-homogeneous equation, and then solving this last equation by the method of variation of parameters. The fractional differential operator consists of a three-time-scale linear operator that stretches the order of the Riemann-Liouville fractional derivative associated with the history term in the equation of motion. In order to illustrate the application of the general solution to particular background flow fields, the particle velocity is calculated for three specific flow configurations. These flow configurations correspond to the gravitationally induced motion of a particle through an otherwise quiescent fluid, the motion of a particle caused by a background velocity field that accelerates linearly in time, and the motion of a particle in a fluid that undergoes an impulsive acceleration. The analytical solutions for these three specific cases are analysed and compared to other solutions found in the literature.

Description: Several important issues pertaining to dispersion and polydispersity of droplets in turbulent flows are investigated via direct numerical simulation (DNS). The carrier phase is considered in the Eulerian context, the dispersed phase is tracked in the Lagrangian frame and the interactions between the phases are taken into account in a realistic two-way (coupled) formulation. The resulting scheme is applied for extensive DNS of low-Mach-number, homogeneous shear turbulent flows laden with droplets. Several cases with one- and two-way couplings are considered for both non-evaporating and evaporating droplets. The effects of the mass loading ratio, the droplet time constant, and thermodynamic parameters, such as the droplet specific heat, the droplet latent heat of evaporation, and the boiling temperature, on the turbulence and the droplets are investigated. The effects of the initial droplet temperature and the initial vapour mass fraction in the carrier phase are also studied. The gravity effects are not considered as the numerical methodology is only applicable in the absence of gravity. The evolution of the turbulence kinetic energy and the mean internal energy of both phases is studied by analysing various terms in their transport equations. The results for the non-evaporating droplets show that the presence of the droplets decreases the turbulence kinetic energy of the carrier phase while increasing the level of anisotropy of the flow. The droplet streamwise velocity variance is larger than that of the fluid, and the ratio of the two increases with the increase of the droplet time constant. Evaporation increases both the turbulence kinetic energy and the mean internal energy of the carrier phase by mass transfer. In general, evaporation is controlled by the vapour mass fraction gradient around the droplet when the initial temperature difference between the phases is negligible. In cases with small initial droplet temperature, on the other hand, the convective heat transfer is more important in the evaporation process. At long times, the evaporation rate approaches asymptotic values depending on the values of various parameters. It is shown that the evaporation rate is larger for droplets residing in high-strain-rate regions of the flow, mainly due to larger droplet Reynolds numbers in these regions. For both the evaporating and the non-evaporating droplets, the root mean square (r.m.s.) of the temperature fluctuations of both phases becomes independent of the initial droplet temperature at long times. Some issues relevant to modelling of turbulent flows laden with droplets are also discussed.

Description: The existing model equations governing the accelerated motion of a spherical particle are examined and their predictions compared with the results of the numerical solution of the full Navier-Stokes equations for unsteady, axisymmetric flow around a freely moving sphere injected into an initially stationary or oscillating fluid. The comparison for the particle Reynolds number in the range of 2 to 150 and the particle to fluid density ratio in the range of 5 to 200 indicates that the existing equations deviate considerably from the Navier-Stokes equations. As a result, we propose a new equation for the particle motion and demonstrate its superiority to the existing equations over a range of Reynolds numbers (from 2 to 150) and particle to fluid density ratios (from 5 to 200). The history terms in the new equation account for the effects of large relative acceleration or deceleration of the particle and the initial relative velocity between the fluid and the particle. We also examine the temporal structure of the near wake of the unsteady, axisymmetric flow around a freely moving sphere injected into an initially stagnant fluid. As the sphere decelerates, the recirculation eddy size grows monotonically even though the instantaneous Reynolds number of the sphere decreases.

Description: The interaction of oblique monochromatic incident waves with a horizontal flexible membrane is investigated in the context of two-dimensional linear hydro-elastic theory. First, analytic diffraction and radiation solutions for a submerged impermeable horizontal membrane are obtained using an eigenfunction expansion method. Secondly, a multi-domain boundary element method (BEM) is developed to confirm the analytic solutions. The inner solution based on a discrete membrane dynamic model and simple-source distribution over the entire fluid boundaries is matched to the outer solution based on an eigenfunction expansion. The numerical solutions are in excellent agreement with the analytic solutions. The theoretical prediction was then compared to a series of experiments conducted in a two-dimensional wave tank at Texas A & M University. The measured reflection and transmission coefficients reasonably follow the trend of predicted values. Using the computer program developed, the performance of surface-mounted or submerged horizontal membrane wave barriers is tested with various system parameters and wave characteristics. It is found that the horizontal flexible membrane can be an effective wave barrier if properly designed.

Description: This paper presents a numerical study of the dynamics and stability of two-dimensional thermal plumes in a significantly stratified layer. Motivated by stellar envelope convection in which radiative cooling at the star's photosphere drives vigorous down flows, we examine cool plumes descending through an adiabatically stratified layer of increasing density with depth. Such flows are inaccessible by laboratory experiments, yet are important to the understanding of heat and momentum transport, magnetic field generation, and acoustic excitation in stars like the Sun. We find that the structure of thermal plumes in a stratified compressible medium is significantly different from that in an incompressible one, with pressure perturbations playing an important dynamical role. Additionally, we find that the plumes are subject to vigorous secondary instabilities even in a quiescent background medium. While the flows studied are not fully turbulent but transitional, the nature of the compressive instabilities and their influence on subsequent flow evolution suggests that advective detrainment of fluid from the plume region results. Simplified plume models assuming a hydrostatic pressure distribution and velocity-proportional entrainment may thus be inappropriate in this context.

Description: The motion of membrane-bound objects is important in many aspects of biology and physical chemistry. A hydrodynamic model for this Fconfiguration was proposed by Saffman & Delbrück (1975) and here it is extended to study the translation of a disk-shaped object in a viscous surface film overlying a fluid of finite depth H. A solution to the flow problem is obtained in the form of a system of dual integral equations that are solved numerically. Results for the friction coefficient of the object are given for a complete range of the two dimensionless parameters that describe the system: the ratio of the sublayer (η) to membrane (ηm) viscosities, Λ=ηR/ηmh (where R and h are the object radius and thickness of the surface film, respectively), and the sublayer thickness ratio, H/R. Scaling arguments are presented that predict the variation of the friction coefficient based upon a comparison of the different length scales that appear in the problem: the geometric length scales H and R, the naturally occurring length scale [lscr ]m=ηmh/η, and an intermediate length scale [lscr ]H= (ηmhH/η)1/2. Eight distinct asymptotic regimes are identified based upon the different possible orderings of these length scales for each of the two limits Λ[Lt ]1 and Λ[Gt ]1. Moreover, the domains of validity of available approximations are established. Finally, some representative surface velocity fields are given and the implication of these results for the characterization of hydrodynamic interactions among membrane-bound proteins adjacent to a finite-depth sublayer is discussed briefly.

Description: The resonant scattering of topographically trapped, low-mode progressive edge waves by longshore periodic topography is investigated using a multiple-scale expansion of the linear shallow water equations. Coupled evolution equations for the slowly varying amplitudes of incident and scattered edge waves are derived for small-amplitude, periodic depth perturbations superposed on a plane beach. In ‘single-wave scattering’, an incident edge wave is resonantly scattered into a single additional progressive edge wave having the same or different mode number (i.e. longshore wavenumber), and propagating in the same or opposite direction (forward and backward scattering, respectively), as the incident edge wave. Backscattering into the same mode number as the incident edge wave, the analogue of Bragg scattering of surface waves, is a special case. In ‘multi-wave scattering’, simultaneous forward and backward resonant scattering results in several (rather than only one) new progressive edge waves. Analytic solutions are obtained for single-wave scattering and for a special case of multi-wave scattering involving mode-0 and mode-1 edge waves, over perturbed depth regions of both finite and semi-infinite longshore extent. In single-wave backscattering with small (subcritical) detuning (i.e. departure from exact resonance), the incident and backscattered wave amplitudes both decay exponentially with propagation distance over the periodic bathymetry, whereas with large (supercritical) detuning the amplitudes oscillate with distance. In single-wave forward scattering, the wave amplitudes are oscillatory regardless of the magnitude of the detuning. Multi-wave solutions combine aspects of single-wave backward and forward scattering. In both single- and multi-wave scattering, the exponential decay rates and oscillatory wavenumbers of the edge wave amplitudes depend on the detuning. The results suggest that naturally occurring rhythmic features such as beach cusps and crescentic bars are sometimes of large enough amplitude to scatter a significant amount of incident low-mode edge wave energy in a relatively short distance (O(10) topographic wavelengths).

Description: The initial value problem for the motion of an intense, quasi-geostrophic, equivalent-barotropic, singular vortex near an infinitely long escarpment is studied in three parts. First, for times small compared to the topographic wave timescale the motion of the vortex is analysed by deriving an expression for the secondary circulation caused by the advection of fluid columns across the escarpment. The secondary circulation, in turn, advects the primary vortex and integral expressions are found for its velocity components. Analytical expressions in terms of integrals are found for the vortex drift velocity components. It is found that, initially, cyclones propagate away from the deep water region and anticyclones propagate away from the shallow water region. Asymptotic evaluation of the integrals shows that both cyclones and anticyclones eventually propagate parallel to the escarpment with shallow water on their right at a steady speed which decays exponentially with distance from the escarpment. Secondly, it is shown that for times comparable to, and larger than, the wave timescale, the vortex always resonates with the topographic wave field. The flux of energy in the topographic waves leads to a loss of energy in the vortex and global energy and momentum arguments are used to derive an equation for the distance (or, equivalently, the vortex velocity) of the vortex from the escarpment. It is shown that cyclones, provided they are initially within an 0(1) distance (here a unit of distance is dimensionally equivalent to one Rossby radius of deformation) from the escarpment, drift further away from the deep water (i.e. toward higher ambient potential vorticity), possibly crossing the escarpment and accumulate at a distance of ≈ 1.2 on the shallow side of the escarpment. For distances larger than 1.2 there is essentially no drift of the vortex perpendicular to the escarpment. Anticyclones display similar behaviour except they drift in the opposite direction, i.e. away from the shallow water or toward lower ambient potential vorticity. Third, the method of contour dynamics is used to describe the evolution of the vortex and the interface representing the initial potential vorticity jump between the shallow and deep water regions. The contour dynamic results are in good quantitative agreement with the analytical results.

Description: Oscillatory translational and rotational motions of small particles in viscous fluids are studied for two cases: (i) circular disks and (ii) nearly spherical particles. For circular disks, four motions are treated: broadside and edgewise oscillatory translations and out-of-plane and in-plane oscillatory rotations. In each case the unsteady Stokes equations are reduced to dual integral equations and solved exactly for all frequencies. Streamline portraits of the flow fields are used to understand the evolution of the velocity and pressure fields. The motions of nearly spherical particles are then studied using the reciprocal theorem. Asymptotic formulae for the hydrodynamic resistance tensors are derived and discussed.

Description: An experimental study of a turbulent boundary layer at Rθ ≈ 1070 and Rτ ≈ 543 was conducted. Detailed measurements of the velocity vector and the velocity gradient tensor within the near-wall region were performed at various distances from the wall, ranging from approximately y+ = 14 to y+ = 89. The measured mean statistical properties of the fluctuating velocity and vorticity components agree well with previous experimental and numerically simulated data. These boundary layer measurements were used in a joint probability density analysis of the various component vorticity and vorticity-velocity gradient products that appear in the instantaneous vorticity and enstrophy transport equations. The vorticity filaments that contribute most to the vorticity covariance ΩxΩy in this region were found to be oriented downstream with angles of inclination to the wall, when projected on the streamwise (x, y)-plane, that decrease with distance moving from the buffer to the logarithmic layer. When projected on the planview (x, z)- and cross-stream (y, z)-planes, the vorticity filaments that most contribute to the vorticity covariances ΩxΩz and ΩyΩz have angles of inclination to the z-ordinate axis that increase with distance from it. All the elements of the ΩiΩj∂Ui/∂xj term in the enstrophy transport equation, i.e. the term that describes the rate of increase or decrease of the enstrophy by vorticity filament stretching or compression by the strain-rate field, have been examined. On balance, the average stretching of the vorticity filaments is greater than compression at all y+ locations examined here. However, some individual velocity gradient components compress the vorticity filaments, on average, more than they stretch them.

Description: We discuss the linear stability of a cross-doubly-diffusive fluid layer with surface tension variation along the free surface. Two limiting cases of the mass flux basic state are considered in the presence of non-zero Soret and Dufour diffusivities. The first case, which has remained largely unexplored, is one where a temperature difference, ΔT̄, and a concentration difference, ΔC̄, are both imposed across the layer. The second Case, which has greater significance to thermosolutal systems, is that where the imposed ΔT̄ across the layer induces a ΔC̄. We rescale the problem in the absence of buoyancy, which leads to a more concise representation of neutral stability results in and near the limit of zero gravity. We obtain exact solutions for stationary stability in both cases. One important consequence of the exact solutions is the validation of recently published numerical results in the limit of zero gravity. Moreover, the precise location of asymptotes in relevant parameter (Smc, Mac) space are computed from exact solutions. Both numerical and exact solutions are used to further examine stability behaviour. We also derive algebraic expressions for stationary stability, oscillatory stability, frequency, and codimension two point from a one-term Galerkin approximation. The one-term solutions qualitatively reflect the stability behaviour of the system over the parameter ranges in our investigation. A practical consequence is that the nature of the stability (oscillatory or stationary) for a given set of parameter values can be determined approximately, without solving the numerical eigenvalue problem.

Description: A linear stability analysis has been implemented for hydromagnetic dissipative Couette flow, a viscous electrically conducting fluid between rotating concentric cylinders in the presence of a uniform axial magnetic field. The small-gap equations with respect to non-axisymmetric disturbances are derived and solved by a direct numerical procedure. Both types of boundary conditions, conducting and non-conducting walls, are considered. A parametric study covering wide ranges of μ, the ratio of angular velocity of the outer cylinder to that of inner cylinder, and Q, the Hartmann number which represents the strength of axial magnetic field, is conducted. Results show that the stability characteristics depend on the conductivity of the cylinders. For the case of non-conducting walls, it is found that the critical disturbance is a non-axisymmetric mode as the value of μ is sufficiently negative and the domain of Q where non-axisymmetric instability modes prevail is limited. Similar results are obtained for conducting walls at low Hartmann number. In addition, the transition of the onset of instability from non-axisymmetric modes to axisymmetric modes for the case μ = - 1 with increasing strength of magnetic field are discussed in detail. For high values of the Hartmann number, the critical disturbance is always the axisymmetric stationary mode for non-conducting walls but not for conducting walls. For - 1 ≤ 〈 1, it is demonstrated that non-axisymmetric instability modes prevail in a wide range of Q for conducting walls and axisymmetric oscillatory modes may, in fact, become more critical than both of the non-axisymmetric and axisymmetric stationary modes at higher values of the Hartmann number.

Description: The differential equation describing the one-point joint probability density function for the wind velocity given by Lundgren (1967) in neutral turbulent flows is extended by a term which also takes into consideration the pressure-mean strain interaction. For the new equation a solution is given describing the one-point probability density function for the wind velocity fluctuations if the profile of the mean wind velocity is logarithmic. The properties of this solution are discussed to identify the differences to a Gaussian having the same first and second moments.

Description: We study the respective effects of shear rate and of external field intensity and direction on the contribution to the bulk stress of Brownian dipolar axisymmetric particles suspended in a steady macroscopically homogeneous shear flow of an incompressible Newtonian fluid. Towards this end we obtain the steady orientational distribution and make use of existing general dynamic theories of dilute suspensions. The calculation focuses on the limit of weak rotary diffusion. Thus, unlike previous analyses, the present contribution is not restricted to weak shear effects. Explicit results are presented for the bulk stress when the external field acts in the plane of the simple shear flow. In cases when the deterministic rotary motion possesses a single sufficiently stable node a simple unified description of the respective effects of both the intensity and azimuthal direction of the external field is provided by the boundary-layer approximation. This approximation enables a qualitative explanation of existing numerical results as well as furnishing quantitatively accurate analytical results at relatively moderate values of the rotary Peclét number and the Langevin parameter (∼ 10). Furthermore, at still larger values of these parameters use of the present asymptotic approximation is clearly preferable since the numerical schemes rapidly deteriorate when steep orientational gradients appear. Singularities of the bulk stress are rationalized in terms of the corresponding deterministic rotary motion. This is particularly interesting because some of these singular phenomena (e.g. those associated with an 'intermediate regime' of the field intensity and direction, for which more than one stable attractor exists in the deterministic problem) have no counterparts in suspensions of dipolar spheres or torque-free axisymmetric particles. Finally, the present results obtained for the orientational distribution are also applicable to the study of other aspects of the macroscale description of suspensions of dipolar axisymmetric particles. In this context we mention the extension of continuum modelling of suspensions of swimming micro-organisms so as to enable the analysis of fully developed bioconvection.

Description: The damping rates, natural frequencies and amplitudes of parametrically excited, standing, water waves in a partially filled, right circular cylinder are measured and compared to existing theoretical models that assume wave slopes are small. The water surfaces were covered by insoluble monomolecular (surfactant) films of oleyl alcohol, lecithin, diolein, cholesterol, and arachidyl alcohol whose concentrations were varied from zero (clean) to saturation; wave slopes were varied from about 0.1 to 1.2. Measured damping rates increased with increasing film concentration as predicted using films of oleyl alcohol, lecithin, and diolein, even when wave slopes were about one. Measured damping rates increased with increasing film concentration as predicted, using films of cholesterol and arachidyl alcohol when wave slopes were small, but not when wave slopes were large. In fact, the measured damping rates for large-slope waves on these films were equivalent to those of waves on a clean surface. Measured natural frequencies varied as predicted for all films, but were about 5% larger. Contact-line effects are incorporated, using an empirical value for contact-line speed, to account for discrepancies between measurements and predictions of damping rates and natural frequencies. Measured steady-state amplitudes agreed well with predictions that used measured damping rates and natural frequencies in the calculations for all films except lecithin and arachidyl alcohol for which there was significant disagreement.

Description: When droplets are expelled at a high velocity by a spray, a strong vertical air jet is induced throughout which the smallest droplets are disperse (their Reynolds numbers associated with their relative motion being small). In our analysis we focus on the interaction between an external cross-flow and this spray jet. This interaction and the distances by which the spray jet and, over a longer distance, the large droplets are deflected are found to depend largely on the ratio of the cross-wind speed to the induced air speed U0/Uj. Using a multi-zone analysis we show that with a weak cross-flow (U0/Uj ≲ 0.1), in the region immediately below the nozzle the spray entrains the external cross-flow and acts like a line sink; the streamlines close to the spray curve inwards to the centre, while further away the sink flow is weak and the streamlines follow the cross-wind. The external flow stagnates at a certain distance from the spray centreline which depends on U0/Uj. When U0/Uj ≳ 0.1 the cross-section of the spray jet and its velocity distribution change in the same way as a fluid jet in a cross-flow, whose inertia causes the deflection of the external flow around it and whose surface vorticity causes a pair of axial vortices on the downwind side of the spray. These vortices have a significant effect on the spray because they induce a back flow which reduces the tendency of the small droplets to leave the spray. When the cross-wind is strong (U0/Uj 〉 0.3; U0 ≳ 10 m s-1) the flow is too strong to be entrained; in this limit the main effect of the larger spray droplets is simply to resist the cross-flow which causes the cross-flow to slow down as it passes through the spray and to divert some of the cross-flow around the spray jet. Since the cross-flow now passes through the spray it carries the smallest droplets downwind. In this paper analytical models have been developed for all the practical ranges of the ratio of the jet speed to the cross-wind speed. This enables spray drift to be calculated. These models require very little computer time and can be run interactively. Spray droplet trajectories can be plotted straightforwardly for both axisymmetric and flat-fan sprays.

Description: Recent work on single-bubble sonoluminescence (SBSL) has shown that many features of this phenomenon, especially the dependence of SBSL intensity and stability on experimental parameters, can be explained within a hydrodynamic approach. More specifically, many important properties can be derived from an analysis of bubble wall dynamics. This dynamics is conveniently described by the Rayleigh-Plesset (RP) equation. Here we derive analytical approximations for RP dynamics and subsequent analytical laws for parameter dependences. These results include (i) an expression for the onset threshold of SL, (ii) an analytical explanation of the transition from diffusively unstable to stable equilibria for the bubble ambient radius (unstable and stable sonoluminescence), and (iii) a detailed understanding of the resonance structure of the RP equation. It is found that the threshold for SL emission is shifted to larger bubble radii and larger driving pressures if surface tension is increased, whereas even a considerable change in liquid viscosity leaves this threshold virtually unaltered. As an enhanced viscosity stabilizes the bubbles to surface oscillations, we conclude that the ideal liquid for violently collapsing, surface-stable SL bubbles should have small surface tension and large viscosity, although too large viscosity (ηl ≳ 40ηwater) will again preclude collapses.

Description: Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described by (SV) streamwise vortices ⇒ streamwise streaks ⇒ streak breakdown ⇒ transition. The oblique wave scenario is described by (OW) oblique waves ⇒ streamwise vortices ⇒ streamwise streaks ⇒ streak breakdown ⇒ transition. The purpose of this paper is to investigate the streak breakdown phase of the above scenarios by a linear stability analysis and compare threshold energies for transition for the above scenarios with those for transition initiated by Tollmien-Schlichting waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical Reynolds numbers. We find that if instability occurs, it is confined to disturbances with streamwise wavenumbers α0 satisfying 0 〈 αmin 〈 |α0| 〈 αmax. In these unstable cases, the least stable mode is located near the centre of the channel with a phase velocity approximately equal to the centreline velocity of the mean flow. Growth rates for instability increase with streak amplitude. For Couette flow streak breakdown is inhibited by mean shear. Using the linear stability analysis, we determine lower bounds on threshold amplitude for transition for scenario (SV) that are consistent with thresholds determined by direct numerical simulations. In the final part of the paper we show that the threshold energies for transition in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and (OW) are two orders of magnitude lower than the threshold for transition initiated by Tollmien-Schlichting waves (TS) and an order of magnitude lower than that for (2DOPT). Scenarios (SV) and (OW) occur on a viscous time scale. However, even when transition times are taken into account, the threshold energy required for transition at a given time for (SV) and (OW) is lower than that for the (TS) and (2DOPT) scenarios at Reynolds number 1500.

Description: Statistical properties of the subgrid-scale stress tensor, the local energy flux and filtered velocity gradients are analysed in numerical simulations of forced three-dimensional homogeneous turbulence. High Reynolds numbers are achieved by using hyperviscous dissipation. It is found that in the inertial range the subgrid-scale stress tensor and the local energy flux allow simple parametrization based on a tensor eddy viscosity. This parametrization underlines the role that negative skewness of filtered velocity gradients plays in the local energy transfer. It is found that the local energy flux only weakly correlates with the locally averaged energy dissipation rate. This fact reflects basic difficulties of large-eddy simulations of turbulence, namely the possibility of predicting the locally averaged energy dissipation rate through inertial-range quantities such as the local energy flux is limited. Statistical properties of subgrid-scale velocity gradients are systematically studied in an attempt to reveal the mechanism of local energy transfer.

Description: The paper is devoted to an investigation of convective turbulence. A simplified approach is used for this purpose. It considers an isolated turbulent pulsation as the eigensolution to the corresponding equations of thermohydrodynamics. Turbulence is generated by nonlinear interaction of pulsations: not all interactions, but only the most probable of them are investigated. It is assumed that during convection these are interactions of cells located along the gravity vector, i.e. lying in a vertical line, and lateral interaction of the cells is ignored. This assumption allows one to consider the process of the evolution and interaction of cells as axially symmetric. It is also assumed that the vertical scales of convective cells are larger than their horizontal scales. Therefore, the Boussinesq equations simplified in accordance with the theory of vertical boundary layers can be used. The fact that buoyancy forces, in addition to diffusion, influence the increase of the vertical scales, serves as a basis for this assumption. These assumptions make it possible to obtain the analytical and numerical-analytical solutions, which qualitatively describe the evolution and interaction of convective cells of two essentially different scales: (i) centimetre-scale convective pulsations and (ii) thermals and convective clouds, and to reduce the problem to the solution of nonlinear equations (equations of the Burgers type). Two opposite tendencies are revealed, manifested in the interaction of convective cells. First, there is coagulation of cells and fine nonlinear effects associated with it, which are known from observations and supported by the theory. Secondly, there is destruction of a strong rising cell through its collision with a weak descending 'cold' cell. It is assumed that the destruction of cells corresponds to the absence of solutions, when some parameters reach their critical values. A numerical solution to a more accurate problem without simplifications of the vertical boundary layer serves as a basis for this hypothesis. It shows that at critical values of the parameters the process of 'wave turnover' begins. It is accompanied by entrainment of the motions of the cold surrounding air into a system of convection and fast dissipation of a cell. In the simplified model, this dissipation is considered to be instantaneous and is called destruction. When the cells are sufficiently strong vertically, weak random fluctuations in the fields of meteorological elements cause their destruction. These results make it possible to propose a hypothesis which relates the degree of instability of cells with the probability of their existence, and to construct functions of cell distributions.

Description: Martel et al. (1998) have shown that interior damping may be comparable with boundary-layer damping for surface waves in small cylinders and that its incorporation yields predictions in agreement with the experimental results of Henderson & Miles (1994) for non-axisymmetric waves on a clean surface with a fixed contact line. In the present note, Henderson & Miles's boundary-layer calculation is supplemented by a calculation of interior damping based on Lamb's dissipation integral for an irrotational flow. The analysis, which omits second-order boundary-layer effects, is simpler than that of Martel et al. (which includes these effects and is based on an expansion in an inverse Reynolds number), but yields results of comparable accuracy within the parametric domain of the experiments. The corresponding calculations for a fully contaminated (inextensible) surface reduce the discrepancy between calculation and experiment but, in contrast to the results for a clean surface, leave a significant residual discrepancy. An unexplained discrepancy also remains for axisymmetric waves on either a clean or a contaminated surface.

Description: Two identical circular cylinders are submerged to the same depth in tandem in a stream. There are separation distances between the cylinder centres such that the combination makes no downstream waves, and hence is subject to zero net wave drag. In general there is then a non-zero equal and opposite horizontal force on each cylinder. However, there are special depths of submergence such that this interaction force between the cylinders also vanishes, and hence each cylinder is separately free of horizontal force. The parameter range for this phenomenon is explored both by linear theory for cylinders of small radius, and by a fully nonlinear computer program. For example, a configuration with a separation distance of approximately one half-wavelength gives zero force on each cylinder when the depth of submergence is approximately three-quarters of the separation distance.

Description: Results are presented from an experimental study into the fine-scale structure of generic, Sc ≈ 1, dynamically passive, conserved scalar fields in turbulent shear flows. The investigation was based on highly resolved, two-dimensional imaging of laser Rayleigh scattering, with measurements obtained in the self-similar far field of an axisymmetric coflowing turbulent jet of propane issuing into air at local outer-scale Reynolds numbers Reδ ≡ uδ/v of 11000 and 14000. The resolution and signal quality of these measurements allowed direct differentiation of the scalar field data ζ(x, t) to determine the instantaneous scalar energy dissipation rate field (ReSc)-1▽ζ·▽ζ(X, t). Results show that, as for large-Sc scalars (Buch & Dahm 1996), the scalar dissipation rate field consists entirely of strained, laminar, sheet-like diffusion layers, despite the fact that at Sc ≈ 1 the scale on which these layers are folded by vorticity gradients is comparable to the layer thickness. Good agreement is found between the measured internal structure of these layers and the self-similar local solution of the scalar transport equation for a spatially uniform but time-varying strain field. The self-similar distribution of dissipation layer thicknesses shows that the ratio of maximum to minimum thicknesses is only 3 at these conditions. The local dissipation layer thickness is related to the local outer scale as λD/δ ≡ ΛReδ -3/4 Sc-1/2, with the average thickness found to be 〈Λ〉 = 11.2, with both the largest and smallest layer thicknesses following Kolmogorov (Reδ -3/4) scaling.

Description: The detailed structure of the aqueous surface sublayer flow immediately adjacent to the wind-driven air-water interface is investigated in a laboratory wind-wave flume using particle image velocimetry (PIV) techniques. The goal is to investigate quantitatively the character of the flow in this crucial, very thin region which is often disrupted by microscale breaking events. In this study, we also examine critically the conclusions of Okuda, Kawai & Toba (1977), who argued that for very short, strongly forced wind-wave conditions, shear stress is the dominant mechanism for transmitting the atmospheric wind stress into the water motion - waves and surface drift currents. In strong contrast, other authors have more recently observed very substantial normal stress contributions on the air side. The availability of PIV and associated image technology now permits a timely re-examination of the results of Okuda et al., which have been influential in shaping present perceptions of the physics of this dynamically important region. The PIV technique used in the present study overcomes many of the inherent shortcomings of the hydrogen bubble measurements, and allows reliable determination of the fluid velocity and shear within 200 μm of the instantaneous wind-driven air-water interface. The results obtained in this study are not in accord with the conclusions of Okuda et al. that the tangential stress component dominates the wind stress. It is found that prior to the formation of wind waves, the tangential stress contributes the entire wind stress, as expected. With increasing distance downwind, the mean tangential stress level decreases marginally, but as the wave field develops, the total wind stress increases significantly. Thus, the wave form drag, represented by the difference between the total wind stress and the mean tangential stress, also increases systematically with wave development and provides the major proportion of the wind stress once the waves have developed beyond their early growth stage. This scenario reconciles the question of relative importance of normal and tangential stresses at an air-water interface. Finally, consideration is given to the extrapolation of these detailed laboratory results to the field, where the present findings suggest that the sea surface is unlikely to become fully aerodynamically rough, at least for moderate to strong winds.

Description: Axisymmetric gravity currents in a system rotating around a vertical axis, that result when 3 dense fluid intrudes horizontally under a less dense ambient fluid, are studied. Situations for which the density difference between the fluid is due either to compositional differences or to suspended particulate matter are considered. The fluid motion is described theoretically by the inviscid shallow-water equations. A 'diffusion' equation for the volume fraction in the suspension is derived for the particle-driven case, and two different models for this purpose are presented. We focus attention on situations in which the apparent importance of the Coriolis terms relative to the inertial terms, represented by the parameter script c sign (the inverse of a Rossby number), is not large. Numerical and asymptotic solutions of the governing equations clarify the essential features of the flow field and particle distribution, and point out the striking differences from the non-rotating case (Bonnecaze, Huppert & Lister 1995). It is shown that the Coriolis effects eventually become dominant; even for small script c sign, Coriolis effects are negligible only during an initial period of about one tenth of a revolution. Thereafter the interface of the current acquires a shape which has a downward decreasing profile at the nose and its velocity of propagation begins to decrease to zero more rapidly than in the non-rotating situation. This relates the currents investigated here to the previously studied quasi-steady oceanographic structures called rings, eddies, vortices or lenses, and may throw additional light on the dynamics of their formation. The theoretical results were tested by some preliminary experiments performed in a rotating cylinder of diameter 90 cm filled with a layer of water of depth 10 cm in which a cylinder of heavier saline fluid of diameter 9.4 cm was released.

Description: The unsteady fully nonlinear free-surface flow due to an impulsively started submerged point sink is studied in the context of incompressible potential flow. For a fixed (initial) submergence h of the point sink in otherwise unbounded fluid, the problem is governed by a single non-dimensional physical parameter, the Froude number, ℱ = Q/4π(gh5)1/2, where Q is the (constant) volume flux rate and g the gravitational acceleration. We assume axisymmetry and perform a numerical study using a mixed-Eulerian-Lagrangian boundary-integral-equation scheme. We conduct systematic simulations varying the parameter ℱ to obtain a complete quantification of the solution of the problem. Depending on ℱ, there are three distinct flow regimes: (i) ℱ 〈 ℱ1 ≈ 0.1924 - a 'sub-critical' regime marked by a damped wave-like behaviour of the free surface which reaches an asymptotic steady state; (ii) ℱ1 〈 ℱ 〈 ℱ2 ≈ 0.1930 - the 'trans-criticar regime characterized by a reversal of the downward motion of the free surface above the sink, eventually developing into a sharp upward jet; (iii) ℱ 〉 ℱ2 - a 'super-critical' regime marked by the cusp-like collapse of the free surface towards the sink. Mechanisms behind such flow behaviour are discussed and hydrodynamic quantities such as pressure, power and force are obtained in each case. This investigation resolves the question of validity of a steady-state assumption for this problem and also shows that a small-time expansion may be inadequate for predicting the eventual behaviour of the flow.

Description: The simple shear-flow model of Stern & Adam (1973), in which a layer of uniform vorticity and depth overlies an infinitely deep fluid, is here extended by the addition of an upper fluid layer of uniform thickness and constant velocity. In this way many experimentally observed velocity profiles can be approximated. The normal mode instabilities of such a model can be found analytically, and their properties calculated through the solution of a quartic polynomial equation. The dispersion relation is here determined and illustrated in its dependence on the Froude number and on the ratio H1/H2, where H1, and H2 denote the mean depths of the surface layer and the base of the shear layer, respectively. It is found that two branches of instability which are distinct when H1/H2 is moderate or small can become merged when H1/H2 ≥ 0.4924. Also calculated are the fastest-growing modes, and their wavelengths. The results are applied to some examples of surface flows generated by towed bodies, and to steady spilling breakers.

Description: The stability of uniform straining flow in a semi-infinite body of viscous fluid subjected to surface cooling is examined. The viscosity of the fluid is assumed to be a prescribed function of temperature. If the viscosity variations caused by the cooling are sufficiently large the straining flow is linearly unstable to a mode in which the rate of extension of the viscous thermal boundary layer becomes localized. The parameters of the problem are the viscosity contrast in the fluid and a dimensionless measure of the rate of strain relative to the rate of cooling. The conditions under which instability occurs are determined and the physical mechanisms responsible are examined. The results are applied to discuss the formation of some surface features in lava flows.

Description: Properties of the flow generated by a continuous source of dense fluid on a slope in a rotating system are investigated with a variety of laboratory experiments. The dense fluid may initially flow down the slope but it turns (under the influence of rotation) to flow along the slope, and initial geostrophic adjustment gives it an anticyclonic velocity profile. Some of the dense fluid drains downslope in a viscous Ekman layer, which may become unstable to growing waves. Provided that the viscous draining is not too strong, cyclonic vortices form periodically in the upper layer and the dense flow breaks up into a series of domes. Three processes may contribute to the formation of these eddies. First, initial downslope flow of the dense current may stretch columns of ambient fluid by the 'Taylor column' process (which we term 'capture'). Secondly, the initial geostrophic adjustment implies lower-layer collapse which may stretch the fluid column, and thirdly, viscous drainage will progressively stretch and spin up a captured water column. Overall this last process may be the most significant, but viscous drainage has contradictory effects, in that it progressively removes dense lower-layer fluid which terminates the process when the layer thickness approaches that of the Ekman layer. The eddies produced propagate along the slope owing to the combined effects of buoyancy - Coriolis balance and 'beta-gyres'. This removes fluid from the vicinity of the source and causes the cycle to repeat. The vorticity of the upper-layer cyclones increases linearly with Γ = Lα/D (where L is the Rossby deformation radius, α the bottom slope and D the total depth), reaching approximately 2f in the experiments presented here. The frequency at which the eddy/dome structures are produced also increases with Γ, while the speed at which the structures propagate along the slope is reduced by viscous effects. The flow of dense fluid on slopes is a very important part of the global ocean circulation system and the implications of the laboratory experiments for oceanographic flows are discussed.

Description: The effect of wall inertia on the self-excited oscillations in a collapsible channel flow is investigated by solving the full coupled two-dimensional membrane-flow equations. This is the continuation of a previous study in which self-excited oscillations were predicted in an asymmetric channel with a tensioned massless elastic membrane (Luo & Pedley 1996). It is found that a different type of self-excited oscillation, a form of flutter, is superposed on the original large-amplitude, low-frequency oscillations. Unlike the tension-induced oscillations, the flutter has high frequency, and grows with time from a small amplitude until it dominates the original slower mode. The critical value of tension below which oscillations arise (at fixed Reynolds number) is found to increase as the wall inertia is increased. The rate at which energy is (a) dissipated in the flow field and (b) transferred to the wall during the flutter is discussed, and results at different parameter values are compared with those of a massless membrane. There is also a discussion of whether the onset of flutter, or that of the slower oscillations, is correlated with the appearance of flow limitation, as is thought to be the case in the context of wheezing during forced expiration of air from the lungs.

Description: The effects of riblets on one wall of a channel bounding fully developed turbulent flow are investigated. Various perturbation elements including wires, fins and slots are modelled in order to understand the effects of riblets. It is found that widely spaced riblets, fins and wires create a substantial increase in turbulent activity just above the element. These elements are also found to produce a remarkable pattern of secondary mean flows consisting of matched pairs of streamwise vortices. The secondary flows occur only if the bulk flow is turbulent and their characteristics depend on element geometry. It is suggested that these secondary flows are strongly linked with the increase in drag experienced by widely spaced riblets in experimental studies. The secondary flows are probably caused by two-dimensional spanwise sloshing of the flow, inherent in a turbulent boundary layer, interacting with the stream-aligned element. This two-dimensional mechanism is investigated with a series of two-dimensional simulations of sloshing flow over isolated elements. Grid resolution and domain size checks are made throughout the investigation.

Description: The stability of supercritical Couette flow has been studied extensively, but few measurements of the velocity field of flow have been made. Particle image velocimetry (PIV) was used to measure the axial and radial velocities in a meridional plane for non-wavy and wavy Taylor-Couette flow in the annulus between a rotating inner cylinder and a fixed outer cylinder with fixed end conditions. The experimental results for the Taylor vortex flow indicate that as the inner cylinder Reynolds number increases, the vortices become stronger and the outflow between pairs of vortices becomes increasingly jet-like. Wavy vortex flow is characterized by azimuthally wavy deformation of the vortices both axially and radially. The axial motion of the vortex centres decreases monotonically with increasing Reynolds number, but the radial motion of the vortex centres has a maximum at a moderate Reynolds number above that required for transition. Significant transfer of fluid between neighbouring vortices occurs in a cyclic fashion at certain points along an azimuthal wave, so that while one vortex grows in size, the two adjacent vortices become smaller, and vice versa. At other points in the azimuthal wave, there is an azimuthally local net axial flow in which fluid winds around the vortices with a sense corresponding to the axial deformation of the wavy vortex tube. These measurements also confirm that the shift-and-reflect symmetry used in computational studies of wavy vortex flow is a valid approach.

Description: It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.

Description: The buoyancy-induced laminar flow and temperature fields associated with a line source of heat in an unbounded environment are described by numerically solving the non-dimensional Boussinesq equations with the appropriate boundary conditions. The solution is given for values of the Prandtl number, the single parameter, ranging from zero to infinity. The far-field form of the solution is well known, including a self-similar thermal plume above the source. The analytical description close to the source involves constants that must be evaluated with the numerical solution. These constants are used when calculating the free convection heat transfer from wires (or cylinders of non-circular shape) at small Grashof numbers. We find two regions in the flow field: an inner region, scaled with the radius of the wire, where the effects of convection can be neglected in first approximation, and an outer region where, also in first approximation, the flow and temperature fields are those due to a line source of heat. The cases of large and small Prandtl numbers are considered separately. There is good agreement between the Nusselt numbers given by the asymptotic analysis and by the numerical analysis, which we carry out for a wide range of Grashof numbers, extending to very small values the range of existing numerical results; there is also agreement with the existing correlations of the experimental results. A correlation expression is proposed for the relation between the Nusselt and Grashof numbers, based on the asymptotic forms of the relation for small and large Grashof numbers.

Description: Cross-stream migration and stable orientations of elliptic particles falling in an Oldroyd-B fluid in a channel are studied. We show that the normal component of the extra stress on a rigid body vanishes; lateral forces and torques are determined by the pressure. Inertia turns the long side of the ellipse across the stream and elasticity turns it along the stream; tilted off-centre falling is unstable. There are two critical numbers: the elasticity and Mach numbers. When the elasticity number is smaller than critical the fluid is essentially Newtonian with broadside-on falling at the centreline of the channel. For larger elasticity numbers the settling turns the long side of the particle along the stream in the channel centre for all velocities below a critical one, identified with a critical Mach number of order one. For larger Mach numbers the ellipse flips into broadside-on falling again. The critical numbers are functions of the channel blockage ratio, the particle aspect ratio and the retardation/relaxation time ratio of the fluid. Two ellipses falling near to each other, attract, line-up vertically and straighten-out with long sides vertical. Stable, off-centre tilting is found for ellipses falling in shear-thinning fluids and for cylinders with flat ends in which particles tend to align their longest diameter with gravity.

Description: The Chapman - Enskog expansion is generalized in order to derive constitutive relations for flows of inelastically colliding spheres in three dimensions - to Burnett order. To this end, the pertinent (nonlinear) Boltzmann equation is perturbatively solved by performing a (double) expansion in the Knudsen number and the degree of inelasticity. One of the results is that the normal stress differences and the 'temperature anisotropy', characterizing granular fluids, are Burnett effects. The constitutive relations derived in this work differ, both qualitatively and quantitatively, from those obtained in previous studies. In particular, the Navier-Stokes (order) terms have a different dependence on the degree of inelasticity and the number density than in previously derived constitutive relations; for instance, the expression for the heat flux contains a term which is proportional to ∈∇ log n, where ∈ is a measure of the degree of inelasticity and n denotes the number density. This contribution to the heat flux is of zeroth order in the density; a similar term, i.e. one that is proportional to ∈∇n, has been previously obtained by using the Enskog correction but this term is O(n) and it vanishes in the Boltzmann limit. These discrepancies are resolved by an analysis of the Chapman Enskog and Grad expansions, pertaining to granular flows, which reveals that the quasi-microscopic rate of decay of the temperature, which has not been taken into account heretofore, provides an important scale that affects the constitutive relations. Some (minor) quantitative differences between our results and previous ones exist as well. These are due to the fact that we take into account an isotropic correction to the leading Maxwellian distribution, which has not been considered before, and also because we consider the full dependence of the corrections to the Maxwellian distribution on the (fluctuating) speed.

Description: The proper orthogonal decomposition (POD) is applied to the minimal flow unit (MFU) of a turbulent channel flow. Our purpose is to establish a numerical validation of low-dimensional models based on the POD. The simplest (two-mode) model possible is built for the simplified flow in the minimal unit. The dynamical behaviour predicted by the model is compared with that actually occurring in the direct numerical simulation of the flow. The various modelling assumptions which underlie the construction of low-dimensional models are examined and confronted with numerical evidence. The relationship between intermittency in the MFU and intermittent low-dimensional parameters is investigated closely. The agreement observed is quite satisfactory, especially given the crudeness of the truncation considered. To further demonstrate the adequacy of the model, we develop a dynamical filtering procedure to recover information from realistic (partial) measurements. The success obtained illustrates the versatility of the low-dimensional paradigm.

Description: A numerical model based on the incompressible two-dimensional Navier-Stokes equations in the Boussinesq approximation is used to study mode-2 internal solitary waves propagating on a pycnocline between two deep layers of different densities. Numerical experiments on the collapse of an initially mixed region reveal a train of solitary waves with the largest leading wave enclosing an intrusional 'bulge'. The waves gradually decay as they propagate along the horizontal direction, with a corresponding reduction in the size of the bulge. When the normalized wave amplitude, a, falls below the critical value ac = 1.18, the wave is no longer able to transport mixed fluid as it propagates away from the mixed region, and a sharp-nosed intrusion is left behind. The wave structure is studied using a Lagrangian particle tracking scheme which shows that for small amplitudes the bulges have a well-defined elliptic shape. At larger amplitudes, the bulge entrains and mixes fluid from the outside while instabilities develop in the rear part of the bulge. Results are obtained for different wave amplitudes ranging from small-amplitude 'regular' waves with a = 0.7 to highly nonlinear unstable waves with a = 3.8. The dependence of the wave speed and wavelength on amplitude is measured and compared with available experimental data and theoretical predictions. Consistent with experiments, the wave speed increases almost linearly with amplitude at small values of a. As a becomes large, the wave speed increases with amplitude at a smaller rate, which gradually approaches the asymptotic limit for a two-fluid model. Results show that in the parameter range considered the wave amplitude decreases linearly with time at a rate inversely proportional to the Reynolds number. Numerical experiments are also conducted on the head-on collision of solitary waves. The simulations indicate that the waves experience a negative phase shift during the collision, in accordance with experimental observations. Computations are used to determine the dependence of the phase shift on the wave amplitude.

Description: In order to elucidate the mechanism of cavitation erosion, the dynamics of a single laser-generated cavitation bubble in water and the resulting surface damage on a flat metal specimen are investigated in detail. The characteristic effects of bubble dynamics, in particular the formation of a high-speed liquid jet and the emission of shock waves at the moment of collapse are recorded with high-speed photography with framing rates of up to one million frames/s. Damage is observed when the bubble is generated at a distance less than twice its maximum radius from a solid boundary (γ = 2, where γ = s/Rmax, s is the distance between the boundary and the bubble centre at the moment of formation and Rmax is the maximum bubble radius). The impact of the jet contributes to the damage only at small initial distances (γ ≤ 0.7). In this region, the impact velocity rises to 83 m s-1, corresponding to a water hammer pressure of about 0.1 GPa, whereas at γ 〉 1, the impact velocity is smaller than 25 m s-1. The largest erosive force is caused by the collapse of a bubble in direct contact with the boundary, where pressures of up to several GPa act on the material surface. Therefore, it is essential for the damaging effect that bubbles are accelerated towards the boundary during the collapse phases due to Bjerknes forces. The bubble touches the boundary at the moment of second collapse when γ 〈 2 and at the moment of first collapse when γ 〈 1. Indentations on an aluminium specimen are found at the contact locations of the collapsing bubble. In the range γ = 1.7 to 2, where the bubble collapses mainly down to a single point, one pit below the bubble centre is observed. At γ ≤ 1.7, the bubble shape has become toroidal, induced by the jet flow through the bubble centre. Corresponding to the decay of this bubble torus into multiple tiny bubbles each collapsing separately along the circumference of the torus, the observed damage is circular as well. Bubbles in the ranges γ ≤ 0.3 and γ = 1.2 to 1.4 caused the greatest damage. The overall diameter of the damaged area is found to scale with the maximum bubble radius. Owing to the possibility of generating thousands of nearly identical bubbles, the cavitation resistance of even hard steel specimens can be tested.

Description: An analysis is presented of some steady natural convection flows at large distances downstream of point heat sources on solid walls. These asymptotic self-similar flows depend only on the Prandtl number of the fluid. The flow induced by a localized source on an adiabatic wall that is vertical or facing downwards is described numerically, whereas the flow due to a localized source on a wall facing upwards separates and leads to a self-similar plume. When the wall is held at the same temperature as the ambient fluid far from the source, the flow is described by a self-similar solution of the second kind, with the algebraic decay of the temperature excess above the ambient temperature determined by a nonlinear eigenvalue problem. Numerical solutions of this problem are presented for two-dimensional and localized heat sources on a vertical wall, whereas the problem for a localized heat source under an inclined isothermal downwards-facing wall turns out to capture the Rayleigh - Taylor instability of the flow and could not be solved by the methods used in this paper. The analogous flows in fluid-saturated porous media are found to be the solutions of parameter-free problems. A conservation law similar to the one holding for a wall jet is found in the case of a two-dimensional source on an isothermal wall and numerical solutions are presented for the other cases.

Description: It is argued that because shallow water cyclones on a β-plane drift westward at a speed equal to an available Rossby wave phase speed, they must radiate energy and cannot, therefore, be steady. The form of the Rossby wave wake accompanying a quasi-steady cyclone is calculated and the energy flux in the radiated waves determined. Further, an explicit expression for the radiation-induced northward drift of the cyclone is obtained. A general method for determining the effects of the radiation on the radius and amplitude of the vortex based on conservation of energy and potential vorticity is given. An example calculation for a cyclone with a 'top-hat' profile is presented, demonstrating that the primary effect of the radiation is to decrease the radius of the vortex. The dimensional timescale associated with the decay of oceanic vortices is of the order of several months to a year.

Description: Cold-wire measurements of a scalar, temperature, its fluctuations and the axial and radial components of the scalar dissipation between two opposed turbulent jet flows, where one jet was slightly heated, show that the residence times of the scalar in the mixing layer were short, that the scalar fluctuations and their dissipation were strongly correlated and that the probability distributions of the conditional scalar dissipation components were log-normal at values of the dissipation larger than the mean. The first finding is consistent with the fact that the scalar turbulence was 'young', in the sense that residence times were shorter than the large-eddy turn-over time, so that the results are likely to be representative of scalar turbulence when scalar mixing first takes place between two streams, for example close to the stabilization region of turbulent diffusion flames. The second implies that the mean scalar dissipation, conditional on the stoichiometric mixture fraction, is larger than the unconditional mean by up to an order of magnitude. Dependence of the distributions of the mean and r.m.s. conditional scalar dissipation on the shape of the scalar p.d.f. was demonstrated by relating the largest conditional dissipation values to the rarest scalar fluctuations and it was found that this dependence was also valid in other flows where scalar dissipation has been measured. The third finding implies that the use of a log-normal distribution to describe the p.d.f. of the conditional scalar dissipation, in the context of flame extinction modelling, will be in error by only 15% provided that the mean and the r.m.s. conditional scalar dissipation are accurately known.

Description: The dynamics of a homogeneous turbulent flow subjected to a stable stratification are studied by means of direct numerical simulations (DNS) and by a two-point closure statistical EDQNM model, adapted for anisotropic flows by Cambon (1989). The purpose of this work is to investigate the validity of the anisotropic statistical model, which we refer to as the EDQNM2 model. The numerical simulations are of high resolution, 2563 which permits Reynolds numbers comparable to those of recent laboratory experiments. Thus, detailed comparisons with the wind-tunnel experiments of Lienhardt & Van Atta (1990) and Yoon & Warhaft (1990) are also presented. The initial condition is chosen so as to test the anisotropic closure assumption of the EDQNM2 model. This choice yields a ratio of kinetic to potential energy of 2 : 1. This important amount of initial potential energy drives the flow dynamics during the first Brunt Väisälä period. Because stronger transfer rates of potential energy than of kinetic energy occur toward small scales, the heat flux is (persistently) counter gradient at those small scales. The loss of potential energy at large scales is partly made up for by conversion of vertical kinetic energy, and this sets up a down-gradient heat flux at those scales, as if no or little potential energy were present at the initial time. Thus, common features with wind-tunnel experiments (in which there is relatively little potential energy just behind the grid) are found. Interestingly, only one quantity displays a similarity law in the DNS, in the EDQNM2 model and in the experiments of Lienhardt & Van Atta (1990) and Yoon & Warhaft (1990) as well: this is the ratio of the vertical heat flux to the dissipation rate of kinetic-energy, which can also be interpreted as an instantaneous mixing efficiency. Thus, this parameter seems to be independent of initial flow conditions. Our calculations simulate a longer evolution of the flow dynamics than laboratory experiments (in which the flow develops for at most one Brunt-Väisälä period). We find that the flow dynamics change from about 1.5 Brunt-Väisälä periods. At that time, the heat flux collapses while the dissipation rate of kinetic energy displays a self-similarity law attesting that this quantity becomes driven by buoyancy forces. This permits us to link the collapse of the largest scales of the flow with the smallest scales being influenced by the buoyancy force. We finally discuss the influence of a geometrical confinement effect upon the above results. The EDQNM2 model compares remarkably well with the DNS, with respect to previous statistical models of stably stratified turbulent flows. Insufficient decor-relation between the vertical velocity and the temperature fluctuations is however 3 observed, but with no dynamical significance. The vortex part of the flow is also overestimated by the EDQNM2 model, but the relative difference between the model prediction and the DNS does not exceed 15 % after 6 Brunt-Väisälä periods. The EDQNM2 model offers interesting perpectives because of its ability to predict the dynamics of stratified flows at high Reynolds numbers. Knowledge about small-scale behaviour will be especially useful, to build up parameterization of the subgrid scales for instance.

Description: The evolution characteristics of monopolar vortices in an irrotational annular shear flow were investigated both experimentally and theoretically. The background flow was generated in a rotating tank by an appropriate source-sink configuration, while the monopolar vortex was created by withdrawing fluid for a short time. Dyevisualization studies demonstrated the gradual destruction of the vortex through a process called 'vortex stripping', i.e. long filaments of passive tracers were being shed from the edge of the vortex. In contrast to uniform shear flows, these filaments were asymmetrically attached to the vortex core. Furthermore, the vortex was observed to evolve in a quasi-stationary manner until its final indefinite breaking. The asymmetric stripping process could be explained by modelling both the monopolar vortex and the ambient flow simply by point vortices, and by adopting the method of contour kinematics to trace material contours in the velocity field induced by the point vortices. Furthermore, the effect of a continuous spatial vorticity distribution was investigated by applying the contour dynamics technique, in which the vortex is represented by a stack of uniform vorticity patches. The observed vortex evolution could be well captured by this latter approach.

Description: The thermocapillary and shear-induced instabilities of a thin heated layer of liquid bounded from the top by a deformable free surface and at the bottom by a horizontally oscillating plate are studied for both Earth-bound and microgravity conditions. Finite-wavelength thermocapillary convection can be stabilized very significantly by the oscillatory shear, whereas shear-induced instabilities are greatly stabilized if the Marangoni number is negative. For long-wavelength thermocapillary convection, oscillatory shear can stabilize or destabilize the basic state, depending primarily on the imposed forcing frequency. With microgravity, significant stabilization of the dominant long-wavelength convection can be achieved by carefully selecting the imposed frequency.

Description: Nikolayeva & Tsimring's (1986) collisionless Boltzmann model for surface-wave generation by a slowly fluctuating wind U(z, t) is transformed to an equivalent steady flow in which the wind speed in the reference frame of the wave (of speed c) is given by V(z) = 〈(U - c)-2〉-1/2, where 〈 〉 signifies a Gaussian average. This leads to a Sturm-Liouville equation for the Gaussian-averaged, complex amplitude of the wave-induced pressure. The wind-to-wave energy transfer for a logarithmic wind profile with the mean friction velocity κŪ1 (κ = Kárman's constant), the standard deviation δŪ1, and the roughness length z0 = ΩŪ2 1/g is determined as a function of the parameters δ and Ω (Charnock's constant) through numerical integration of a Riccati equation (derived from the Sturm-Liouville equation). The energy transfer exceeds that predicted by the quasi-laminar model (Miles 1957; Conte & Miles 1959) by as much as 20-30% for δ ≈ 1 and c (wave speed) ≲ 6Ū1 but is decreased for c ≳ 8Ū1 and may be negative for sufficiently large c/Ū1. These predictions contrast with the order-of-magnitude increase predicted by Nikolayeva & Tsimring.

Description: This paper deals with the theoretical and experimental investigation of acoustically levitated droplets. A method of calculation of the acoustic radiation pressure based on the boundary element method (BEM) is presented. It is applied to predict shapes of droplets levitated in an acoustic field (and as a result, deformed by it). The method was compared with several known exact and approximate analytical results for rigid spheres and shown to be accurate (and a widely used approximate formula for the acoustic levitation force acting on a rigid sphere was found to be inaccurate for sound wavelengths comparable with the sphere radius). The method was also compared with some other theoretical approaches known from the literature. Displacement of the droplet centre relative to the pressure node is accounted for and shown to be significant. The results for droplet shapes and displacements are compared with experimental data, and the agreement is found to be rather good. Furthermore, the experimental investigations reveal a unique relationship between the aspect ratio of an oblate droplet and the sound pressure level in the levitator. This relationship agrees well with the predicted shapes. A practical link between droplet shape or droplet displacement and sound pressure level in a levitator is therefore now available.

Description: We perform a detailed experimental study of large-scale vortices propagating in the rotating shallow-water layer in a paraboloidal vessel. A specific data acquisition technique is used in order to ensure precise measurements of the free-surface elevation. We find two qualitatively different types of vortex behaviour controlled by the relative elevation value. For small elevations we observe a standard quasi-geostrophic pattern with an asymmetric secondary circulation around an initially symmetric vortex which leads to a meridional drift and Rossby wave radiation. This type of behaviour is exhibited by both cyclonic and anticyclonic vortices. For relative elevations larger than 1 (nonlinear regime) the necessarily anticyclonic vortices are drifting strictly zonally maintaining their circular symmetry during the viscous decay. By varying the initial latitude of the vortex we were able to check that in the nonlinear regime the vortex lifetime is not sensitive to the beta-effect, while it is the case in the quasi-geostrophic regime. In the same way we show that the observed difference in cyclone-anticyclone lifetimes is not influenced by the beta-effect.

Description: We present a numerical study of steady convection in a two-dimensional mushy layer during the directional solidification of a binary mixture. The calculations reveal the internal structure of strongly nonlinear states featuring upflow which has been focused into solid-free regions known as chimneys. The mushy layer is modelled as a porous medium whose permeability is a function of the local solid fraction. The mushy layer is coupled to a chimney that is modelled as a narrow vertical channel where lubrication scalings are used to simplify the Navier-Stokes equations. We use these methods to exhibit solutions which give the detailed structure of the temperature, solute, flow and solid-fraction fields within the mushy layer. A key finding of the numerics is that there are two distinct chimney solutions at low Rayleigh numbers, presumably corresponding to stable and unstable portions of a subcritical solution branch. We also explore the relationship between convective solutions with and without chimneys.

Description: Hydrothermal-wave instabilities in thermocapillary convection are known to produce undesirable effects when they occur during the float-zone crystal-growth process, and perhaps in other situations. Suppression of the hydrothermal-wave instability produced in the model system of Part 1 (Riley & Neitzel 1998) is demonstrated through the sensing of free-surface temperature perturbations and the periodic addition of heat at the free surface along lines parallel to the crests of the hydrothermal waves.

Description: Solutions are determined for normally incident non-breaking linear gravity waves in a perfect fluid over a porous plane bed of arbitrary slope α both with and without bed friction. For simplicity, computations are restricted to α = π/2m,m ε N. Modifications to wave height transformations due to percolation and to friction are determined for a variety of slopes and coefficients. The effect on the reflection coefficient Rf is also studied and excellent qualitative agreement is found with recent work on damping and reflection by permeable structures. In particular, for a choice of parameters, the Rf response is determined in closed form.

Description: This is an experimental investigation of two turbulent boundary layers (cases 2 and 4) where the streamwise negative pressure gradient changes mean properties of the flow, e.g. mean velocity profiles and skin friction, so that they display laminar-like behaviour. The maximum acceleration parameter K ≤ 4 × 10-6 and the starting value of the Reynolds number is 862 or 2564. Relaminarization occurs in both boundary layers as a gradual change of the turbulence properties and is not catastrophic. Retransition, however, is a fast process due to the remaining turbulence structure and may be compared with bypass transition. Together with an extensive investigation of the turbulence structure as in the companion paper, Part 1, which describes two cases (1 and 3) of boundary layers which remain turbulent, spectra and integral length scales for all four boundary layers are discussed.

Description: We investigate the evolution of nearby like-sign vortices whose centres are at different vertical levels in a stably stratified rotating fluid. We employ two differently singularized representations of the potential vorticity distribution in the quasi-geostrophic equations (QG), in order to elucidate the pair-interaction behaviour previously seen in non-singular QG numerical solutions. The first is an analytically tractable conservative (Hamiltonian) elliptical-moment model (EM) for thin-core vortices, which exhibits a regime of very strong horizontal elongation of a vortex in response to the strain induced by its partner. We interpret this as an early evolutionary stage towards the irreversible dissipative merger and alignment interactions. This interpretation is strengthened by weakly dissipative numerical solutions of a thin-core contour-dynamics model (CD), which exhibit even further progress towards the completion of these vortex interactions in the same regime. In the EM model we classify the co-rotating stationary states which exist always for vertically offset thin-core vortices. However, the mutual strain field among the vortices cannot be balanced by co-rotation in a weakly elongated stationary state for a certain class of neighbouring, but substantially non-aligned, vortex configurations, and our interpretive assumption is that such configurations will rapidly evolve in non-singular QG solutions towards a more aligned configuration through significantly non-conservative reorganizations of the potential vorticity field. Both the EM and CD models show qualitatively similar regime boundaries between evolutions with weakly and strongly deformed vortices. In particular, there is a fairly close correspondence between the occurrence of strong vortex elongation in the EM solutions and significant filamentation and splitting in the CD solutions.

Description: The shape of liquid surfaces in regular N-pods in the absence of gravity is considered. A liquid volume in the vertex of a regular N-pod wets the adjacent faces if the sum of the liquid's contact angle γ with the faces and half the dihedral angle α between adjacent faces is smaller than π/2. A suggestion for why the surface shape in the wedge approaches its cylindrical shape at infinity exponentially is given. The range of this exponential decrease is related to the curvature of the meniscus and the angles α and γ: The decrement of the decrease generally shows a weak dependence on α+γ, predominantly depending on the liquid volume. Extremely close to the wetting limit, when α+γ approaches π/2, the decrement vanishes. The exponential meniscus shape leads to a similarity relation and allows small relative liquid volumes in polyhedrons to be split up into partial volumes ascribed to the corners and others ascribed to the wedges. The respective relations among volume, curvature, contact angle and corner geometry are obtained by numerical simulation and the limits of applicability are discussed. This greatly simplifies the calculation of liquid surfaces in the limit of small liquid volumes. The results obtained apply to liquid surfaces in a Space environment, e.g. to metallic melts in crucibles and to propellants and other technical fluids in tanks and reservoirs, as well as to liquid surfaces on Earth, e.g. to liquids trapped in polyhedral pores and to liquid foams, provided their characteristic length is sufficiently small compared to the capillary length.

Description: Steady incompressible flow past an aerofoil with a cavity in its upper surface is considered. The aerofoil and cavity shape is constructed to ensure that the high-Reynolds-number asymptotics of the flow is described by the Batchelor model.

Description: Using a barotropic model in spherical geometry, we construct new solutions for steadily travelling vortex pairs and study their stability properties. We consider pairs composed of both point and finite-area vortices, and we represent the rotating background with a set of zonal strips of uniform vorticity. After constructing the solution for a single point-vortex pair, we embed it in a rotating background, and determine the equilibrium configurations that travel at constant speed without changing shape. For equilibrium solutions, we find that the stability depends on the relative strength (which may be positive or negative) of the vortex pair to the rotating background: eastward-travelling pairs are always stable, while westward-travelling pairs are unstable when their speeds approach that of the linear Rossby-Haurwitz waves. This finding also applies (with minor differences) to the case when the vortices are of finite area; in that case we find that, in addition to the point-vortex-like instabilities, the rotating background excites some finite-area instabilities for vortex pairs that would otherwise be stable. As for practical applications to blocking events, for which the slow westward pairs are relevant, our results indicate that free barotropic solutions are highly unstable, and thus suggest that forcing mechanisms must play an important role in maintaining atmospheric blocking events.

Description: We study the pressure-driven transient motion of a periodic file of deformable liquid drops through a cylindrical tube with circular cross-section, at vanishing Reynolds number. The investigations are based on numerical solutions of the equations of Stokes flow obtained by the boundary-integral method. It is assumed that the viscosity and density of the drops are equal to those of the suspending fluid, and the interfaces have constant tension. The mathematical formulation uses the periodic Green's function of the equations of Stokes flow in a domain that is bounded externally by a cylindrical tube, which is computed by tabulation and interpolation. The surface of each drop is discretized into quadratic triangular elements that form an unstructured interfacial grid, and the tangential velocity of the grid-points is adjusted so that the mesh remains regular for an extended but limited period of time. The results illustrate the nature of drop motion and deformation, and thereby extend previous studies for axisymmetric flow and small-drop small-deformation theories. It is found that when the capillary number is sufficiently small, the drops start deforming from a spherical shape, and then reach slowly evolving quasi-steady shapes. In all cases, the drops migrate radially toward the centreline after an initial period of rapid deformation. The apparent viscosity of the periodic suspension is expressed in terms of the effective pressure gradient necessary to drive the flow at constant flow rate. For a fixed period of separation, the apparent viscosity of a non-axisymmetric file is found to be higher than that of an axisymmetric file. In the case of non-axisymmetric motion, the apparent viscosity reaches a minimum at a certain ratio of the drop separation to tube radius. Drops with large effective radii to tube radius ratios develop slipper shapes, similar to those assumed by red blood cells in flow through capillaries, but only for capillary numbers in excess of a critical value.

Description: The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

Description: Interactions of oblique waves have recently been investigated theoretically and numerically and found to give rise to rapid transition in flows subcritical to linear wave disturbances. The transition scenario consists of the formation and transient growth of streamwise streaks of high and low velocity and later a rapid growth of high-frequency disturbances leading to breakdown. The present study is the first extensive experimental investigation of oblique transition. The experiments were carried out in a plane Poiseuille flow air channel in which the oblique waves were generated, one at each wall, by vibrating ribbons and the development of the flow was mapped with hot-wire anemometry. The experiments consist both of low- and high-amplitude wave disturbances; in both cases streaky structures are created. For the low-amplitude case these structures decay, whereas for the high amplitude the flow goes towards breakdown. This study has confirmed and extended previous theoretical and numerical results showing that oblique transition may be an important transition scenario.

Description: An investigation is made into the dynamics involved in the movement of the contact line when a single liquid with an interface moves into a vacuum over a smooth solid surface. In order to remove the stress singularity at the contact line, it is postulated that slip between the liquid and the solid or some other mechanism occurs very close to the contact line. It is assumed that the flow produced is inertia dominated with the Reynolds number based on the slip length being very large. Following a procedure similar to that used by Cox (1986) for the viscous-dominated situation (in which the Reynolds number based on the macroscopic length scale was assumed very small) using matched asymptotic expansions, we obtain the dependence of the macroscopic dynamic contact angle on the contact line velocity over the solid surface for small capillary number and small slip length to macroscopic lengthscale ratio. These results for the inertia-dominated situation are then extended (at the lowest order in capillary number) to an intermediate Reynolds number situation with the Reynolds number based on the slip length being very small and that based on the macroscopic lengthscale being very large.