Showing papers in "Journal of Fluid Mechanics in 1989"

The motion of a finite mass of granular material down a rough incline

Abstract: Rock, snow and ice masses are often dislodged on steep slopes of mountainous regions. The masses, which typically are in the form of innumerable discrete blocks or granules, initially accelerate down the slope until the angle of inclination of the bed approaches the horizontal and bed friction eventually brings them to rest. The present paper describes an initial investigation which considers the idealized problem of a finite mass of material released from rest on a rough inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth-averaged equations of motion are derived; they bear a superficial resemblance to the nonlinear shallow-water wave equations. Two similarity solutions are found for the motion. They both are of surprisingly simple analytical form and show a rather unanticipated behaviour. One has the form of a pile of granular material in the shape of a parabolic cap and the other has the form of an M-wave with vertical faces at the leading and trailing edges. The linear stability of the similarity solutions is studied. A restricted stability analysis, in which the spread is left unperturbed shows them to be stable, suggesting that mathematically both are possible asymptotic wave forms. Two numerical finite-difference schemes, one of Lagrangian, the other of Eulerian type, are presented. While the Eulerian technique is able to reproduce the M-wave similarity solution, it appears to give spurious results for more general initial conditions and the Lagrangian technique is best suited for the present problem. The numerical predictions are compared with laboratory experiments of Huber (1980) involving the motion of gravel released from rest on a rough inclined plane. Although in these experiments the continuum approximation breaks down at large times when the gravel layer is only a few particle diameters thick, the general features of the development of the gravel mass are well predicted by the numerical solutions.

Oblique and Parallel Modes of Vortex Shedding in the Wake of a Circular Cylinder at Low Reynolds Numbers

Abstract: Two fundamental characteristics of the low-Reynolds-number cylinder wake, which have involved considerable debate, are first the existence of discontinuities in the Strouhal-Reynolds number relationship, and secondly the phenomenon of oblique vortex shedding. The present paper shows that both of these characteristics of the wake are directly related to each other, and that both are influenced by the boundary conditions at the ends of the cylinder, even for spans of hundreds of diameters in length. It is found that a Strouhal discontinuity exists, which is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode. This transition is explained by a change from one mode where the central flow over the span matches the end boundary conditions to one where the central flow is unable to match the end conditions. In the latter case, quasi-periodic spectra of the velocity fluctuations appear; these are due to the presence of spanwise cells of different frequency. During periods when vortices in neighbouring cells move out of phase with each other, ‘vortex dislocations’ are observed, and are associated with rather complex vortex linking between the cells. However, by manipulating the end boundary conditions, parallel shedding can be induced, which then results in a completely continuous Strouhal curve. It is also universal in the sense that the oblique-shedding Strouhal data (S_θ) can be collapsed onto the parallel-shedding Strouhal curve (S_0) by the transformation, S_0 = S_θ/cosθ, where θ is the angle of oblique shedding. Close agreement between measurements in two distinctly different facilities confirms the continuous and universal nature of this Strouhal curve. It is believed that the case of parallel shedding represents truly two-dimensional shedding, and a comparison of Strouhal frequency data is made with several two-dimensional numerical simulations, yielding a large disparity which is not clearly understood. The oblique and parallel modes of vortex shedding are both intrinsic to the flow over a cylinder, and are simply solutions to different problems, because the boundary conditions are different in each case.

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

Abstract: An experimental study of Rayleigh-Benard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 107 < Ra < 6 × 1012) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra. However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu, proportional to . A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

Coherent structure of the convective boundary layer derived from large-eddy simulations

Abstract: Turbulence in the convective boundary layer (CBL) uniformly heated from below and topped by a layer of uniformly stratified fluid is investigated for zero mean horizontal flow using large-eddy simulations (LES). The Rayleigh number is effectively infinite, the Froude number of the stable layer is 0.09 and the surface roughness height relative to the height of the convective layer is varied between 10−6 and 10−2. The LES uses a finite-difference method to integrate the three-dimensional grid-volume-averaged Navier–Stokes equations for a Boussinesq fluid. Subgrid-scale (SGS) fluxes are determined from algebraically approximated second-order closure (SOC) transport equations for which all essential coefficients are determined from the inertial-range theory. The surface boundary condition uses the Monin–Obukhov relationships. A radiation boundary condition at the top of the computational domain prevents spurious reflections of gravity waves. The simulation uses 160 × 160 × 48 grid cells. In the asymptotic state, the results in terms of vertical mean profiles of turbulence statistics generally agree very well with results available from laboratory and atmospheric field experiments. We found less agreement with respect to horizontal velocity fluctuations, pressure fluctuations and dissipation rates, which previous investigations tend to overestimate. Horizontal spectra exhibit an inertial subrange. The entrainment heat flux at the top of the CBL is carried by cold updraughts and warm downdraughts in the form of wisps at scales comparable with the height of the boundary layer. Plots of instantaneous flow fields show a spoke pattern in the lower quarter of the CBL which feeds large-scale updraughts penetrating into the stable layer aloft. The spoke pattern has also been found in a few previous investigations. Small-scale plumes near the surface and remote from strong updraughts do not merge together but decay while rising through large-scale downdraughts. The structure of updraughts and downdraughts is identified by three-dimensional correlation functions and conditionally averaged fields. The mean circulation extends vertically over the whole boundary layer. We find that updraughts are composed of quasi-steady large-scale plumes together with transient rising thermals which grow in size by lateral entrainment. The skewness of the vertical velocity fluctuations is generally positive but becomes negative in the lowest mesh cells when the dissipation rate exceeds the production rate due to buoyancy near the surface, as is the case for very rough surfaces. The LES results are used to determine the root-mean-square value of the surface friction velocity and the mean temperature difference between the surface and the mixed layer as a function of the roughness height. The results corroborate a simple model of the heat transfer in the surface layer.

Optical and acoustic investigations of the dynamics of laser-produced cavitation bubbles near a solid boundary

Abstract: The dynamics of laser-produced cavitation bubbles near a solid boundary and its dependence on the distance between bubble and wall are investigated experimentally. It is shown by means of high-speed photography with up to 1 million frames/s that jet and counterjet formation and the development of a ring vortex resulting from the jet flow are general features of the bubble dynamics near solid boundaries. The fluid velocity field in the vicinity of the cavitation bubble is determined with time-resolved particle image velocimetry. A comparison of path lines deduced from successive measurements shows good agreement with the results of numerical calculations by Kucera & Blake (1988). The pressure amplitude, the profile and the energy of the acoustic transients emitted during spherical bubble collapse and the collapse near a rigid boundary are measured with a hydrophone and an optical detection technique. Sound emission is the main damping mechanism in spherical bubble collapse, whereas it plays a minor part in the damping of aspherical collapse. The duration of the acoustic transients is 20-30 ns. The highest pressure amplitudes at the solid boundary have been found for bubbles attached to the boundary. The pressure inside the bubble and at the boundary reaches about 2.5 kbar when the maximum bubble radius is 3.5 mm. The results are discussed with respect to the mechanism of cavitation erosion.

Turbulent oscillatory boundary layers at high Reynolds numbers

Abstract: This study deals with turbulent oscillatory boundary-layer flows over both smooth and rough beds. The free-stream flow is a purely oscillating flow with sinusoidal velocity variation. Mean and turbulence properties were measured mainly in two directions, namely in the streamwise direction and in the direction perpendicular to the bed. Some measurements were made also in the transverse direction. The measurements were carried out up to Re = 6 × 106 over a mirror-shine smooth bed and over rough beds with various values of the parameter a/ks covering the range from approximately 400 to 3700, a being the amplitude of the oscillatory free-stream flow and ks the Nikuradse's equivalent sand roughness. For smooth-bed boundary-layer flows, the effect of Re is discussed in greater detail. It is demonstrated that the boundary-layer properties change markedly with Re. For rough-bed boundary-layer flows, the effect of the parameter a/ks is examined, at large values (O(103)) in combination with large Re.

Lagrangian statistics from direct numerical simulations of isotropic turbulence

Abstract: A comprehensive study is reported of the Lagrangian statistics of velocity, acceleration, dissipation and related quantities, in isotropic turbulence. High-resolution direct numerical simulations are performed on 643 and 1283 grids, resulting in Taylor-scale Reynolds numbers Rλ in the range 38-93. The low-wavenumber modes of the velocity field are forced so that the turbulence is statistically stationary. Using an accurate numerical scheme, of order 4000 fluid particles are tracked through the computed flow field, and hence time series of Lagrangian velocity and velocity gradients are obtained.The results reported include: velocity and acceleration autocorrelations and spectra; probability density functions (p.d.f.'s) and moments of Lagrangian velocity increments; and p.d.f.'s, correlation functions and spectra of dissipation and other velocity-gradient invariants. It is found that the acceleration variance (normalized by the Kolmogorov scales) increases as R½λ - a much stronger dependence than predicted by the refined Kolmogorov hypotheses. At small time lags, the Lagrangian velocity increments are distinctly non-Gaussian with, for example, flatness factors in excess of 10. The enstrophy (vorticity squared) is found to be more intermittent than dissipation, having a standard-deviation-to-mean ratio of about 1.5 (compared to 1.0 for dissipation). The acceleration vector rotates on a timescale about twice the Kolmogorov scale, while the timescales of acceleration magnitude, dissipation and enstrophy appear to scale with the Lagrangian velocity timescale.

Elliptic jets. I - Characteristics of unexcited and excited jets

Abstract: Experimental studies of incompressible elliptic jets of different aspect ratios and initial conditions are summarized along with the effects of excitations at selected frequencies and amplitudes. The experimental facilities and procedures are described and jet spread and decay are discussed. The instability of elliptic shear layers, the behavior of the jet column under controlled excitation, and the time-average measures of unexcited jets are addressed.

Reynolds-number effects on the structure of a turbulent channel flow

Abstract: A high resolution, two component laser-Doppler anemometer has been used for turbulence measurements at a high data rate in a channel flow of water. Measurements of the velocity components in the stream direction and in a direction normal to the wall are reported over the Reynolds number range of 3000–40000. The combination of high spatial resolution and high data rates enabled accurate reconstruction of time dependent velocity traces. Long-time statistical averages of these signals clearly show that profiles of the dimensionless turbulence quantities such as turbulence intensities and Reynolds stress are strongly Reynolds-number dependent over a large part of the channel flow. For instance, in the Reynolds-number range of this investigation, it is shown that the fluctuating turbulence quantities do not scale with wall variables even as close as 15 viscous lengths from the wall. The velocity traces and associated power spectra exposed two phenomena which may explain the Reynolds number dependencies.

Characteristic-eddy decomposition of turbulence in a channel

Abstract: The proper orthogonal decomposition technique (Lumley's decomposition) is applied to the turbulent flow in a channel to extract coherent structures by decomposing the velocity field into characteristic eddies with random coefficients. In the homogeneous spatial directions, a generaliztion of the shot-noise expansion is used to determine the characteristic eddies. In this expansion, the Fourier coefficients of the characteristic eddy cannot be obtained from the second-order statistics. Three different techniques are used to determine the phases of these coefficients. They are based on: (1) the bispectrum, (2) a spatial compactness requirement, and (3) a functional continuity argument. Results from these three techniques are found to be similar in most respects. The implications of these techniques and the shot-noise expansion are discussed. The dominant eddy is found to contribute as much as 76 percent to the turbulent kinetic energy. In both 2D and 3D, the characteristic eddies consist of an ejection region straddled by streamwise vortices that leave the wall in the very short streamwise distance of about 100 wall units.

Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid

Abstract: In this paper we examine some general features of the time-dependent dynamics of drop deformation and breakup at low Reynolds numbers. The first aspect of our study is a detailed numerical investigation of the ‘end-pinching’ behaviour reported in a previous experimental study. The numerics illustrate the effects of viscosity ratio and initial drop shape on the relaxation and/or breakup of highly elongated droplets in an otherwise quiescent fluid. In addition, the numerical procedure is used to study the simultaneous development of capillary-wave instabilities at the fluid-fluid interface of a very long, cylindrically shaped droplet with bulbous ends. Initially small disturbances evolve to finite amplitude and produce very regular drop breakup. The formation of satellite droplets, a nonlinear phenomenon, is also observed.

Geometry of self-propulsion at low Reynolds number

Abstract: The problem of swimming at low Reynolds number is formulated in terms of a gauge field on the space of shapes. Effective methods for computing this field, by solving a linear boundary-value problem, are described. We employ conformal-mapping techniques to calculate swimming motions for cylinders with a variety of crosssections. We also determine the net translationl motion due to arbitrary infinitesimal deformations of a sphere.

The viscous flow on surfaces with longitudinal ribs

Abstract: The viscous sublayer of a turbulent boundary layer on a surface with fine longitudinal ribs (riblets) is investigated theoretically. The mean flow constituent of this viscous flow is considered. Using conformal mapping, the velocity distributions on various surface configurations are calculated. The geometries that were investigated include sawtooth profiles with triangular and trapezoidal grooves as well as profiles with thin blade-shaped ribs, ribs with rounded edges and ribs having sharp ridges and U-shaped grooves. (This latter riblet configuration is also found on the tiny scales of fast sharks.) Our calculations enable us to determine the location of the origin of the velocity profile that lies somewhat below the tips of the ridges. The distance between this origin and the tip of the ridge we call ‘protrusion height’. The upper limit for the protrusion height is found to be 22% of the lateral rib spacing; the coefficient 0.22 being the value of the expression π−1 In 2. This limit is valid for two-dimensional riblet geometries. Analogous experiments with an electrolytic tank are carried out as an additional check on the theoretical calculations. This is also an easy way to determine experimentally the location of the origin of the velocity profile for arbitrary new riblet geometries. A possible connection between protrusion height and drag reduction in a turbulent boundary layer flow is discussed. Finally, the present theory also produces an orthogonal grid pattern above riblet surfaces which may be utilized in future numerical calculations of the whole turbulent boundary layer.

Inertial migration of a sphere in Poiseuille flow

Abstract: The inertial migration of a small sphere in a Poiseuille flow is calculated for the case when the channel Reynolds number is of order unity. The equilibrium position is found to move towards the wall as the Reynolds number increases. The migration velocity is found to increase more slowly than quadratically. These results are compared with the experiments of Segre & Silberberg (1962 a, b).

Spectral approach to non-isotropic turbulence subjected to rotation

Abstract: The non-isotropic effects of solid-body rotation on homogeneous turbulence are investigated in this paper. A spectral formalism using eigenmodes introduces the spectral Coriolis effects more easily and leads to simpler expressions for the integral quadratic terms which come mostly from classical two-point closures. The analysis is then applied to a specific eddy damped quasi-normal Markovian model, which includes the inertial waves regime in the evaluation of triple correlations. This procedure allows for a departure from isotropy by external rotation effects. When started with rigorously isotropic initial data, the various trends observed on the Reynolds stresses and the integral lengthscales remain in accordance with the results from direct simulations; moreover they reflect a very specific spectral angular distribution. Such an angular dependence allows a drain of spectral energy from the parallel to the normal wave vectors (with respect to the rotation axis), and thus appears consistent with a trend toward two-dimensionality.

Near-wall structure of a turbulent boundary layer with riblets

Abstract: A detailed wind tunnel study has been carried out on the near-wall turbulence structure over smooth and riblet wall surfaces under zero pressure gradient. Time-average quantities as ‘well as conditionally sampled profiles were obtained using hotwire/film anemometry, along with a simultaneous flow visualization using the smoke-wire technique and a sheet of laser light. The experimental results indicated a significant change of the structure in the turbulent boundary layer near the riblet surface. The change was confined within a small volume of the flow close to the wall surface. A conceptual model for the sequence of the bursts was then proposed based on an extensive study of the flow visualization, and was supported by the results of conditionally sampled velocity fields. A possible mechanism of turbulent drag reduction by riblets is discussed.

On the structure of pressure fluctuations in simulated turbulent channel flow

Abstract: Pressure fluctuations in a turbulent channel flow are investigated by analyzing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared to the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure-strain correlations are also examined by evaluating the integral associated with Green's function representations of them. In the wall region where the pressure-strain terms are large, most contributions to the pressure-strain terms are from the wall region (i.e., local), whereas away from the wall where the pressure-strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.

Structure of the turbulent flow field under breaking waves in the surf zone

Abstract: The structure of turbulence and its role in the breaking wave dynamics within the surf zone have been investigated through laboratory experiments using several flow visualization techniques and a fibre-optic LDV system. The results indicate that there exists a characteristic structure of large-scale eddies referred to here as ‘horizontal eddies’ and ‘obliquely descending eddies’, which has a significant role in the generation of Reynolds stress and thus affects the deformation of the mean flow field. The experiments also reveal that these eddies caused by the wave breaking bring a large amount of vorticity (with non-zero average) into otherwise almost irrotational velocity fields, resulting in the generation of vorticity-related mean flow fields as well as turbulence (vorticity-containing velocity fluctuation). This means that the breaking waves in the surf zone can be regarded as pseudowaves which consist of irrotational velocity components as ‘wave motion’ and appreciable amounts of rotational mean velocity components as ‘eddying motion’ (with non-zero mean vorticity) together with turbulence. It is found that the generation of the mean rotational velocity component due to wave breaking causes considerable increase in mass and momentum transport, as compared with ordinary non-breaking waves, and thus a decrease in wave height.

On the three families of instability waves of high-speed jets

Abstract: An analytical and computational study of the normal-mode small-amplitude waves of high-speed jets is presented. Three families of instability waves have been identified: (1) the familiar Kelvin-Helmholtz instability waves; (2) supersonic instability waves; and (3) subsonic waves. It is demonstrated that the computed wave patterns and propagation characteristics of these three wave types are consistent with the findings of Oetel (1979, 1980, 1982). The subsonic waves are shown to be unstable only for jets with mixing layers of finite thickness.

The generation and collapse of a foam layer at the roof of a basaltic magma chamber

Abstract: Basaltic volcanoes erupt in several different regimes which have not been explained. At Kilauea (Hawaii), eruption can take the form of either fire fountaining, where gas-rich jets propel lava clots to great heights in the atmosphere, or quiet effusive outflow of vesicular lava. Another regime is commonly observed at Stromboli, where large gas slugs burst intermittently at the vent. In an attempt to provide a unifying framework for these regimes, we investigate phenomena induced by degassing in a reservoir which empties into a small conduit. Laboratory experiments are done in a cylindrical tank topped by a thin vertical tube. Working liquids are silicone oils and glycerol solutions to investigate a range of viscosity and surface tension. Gas bubbles are generated at the tank bottom with known bubble diameter and total gas flux. The bubbles rise through the tank and accumulate in a foam layer at the roof. Depending on the behaviour of this foam layer, three different regimes can be distinguished: (i) steady horizontal flow of the foam leading to bubbly flow in the conduit; (ii) alternating regimes of foam build-up and collapse leading to the eruption of a single, large gas pocket; (iii) flow of the foam partially coalesced into larger gas pockets leading to intermittent slug flow in the conduit. These regimes have natural counterparts in basaltic volcanoes.A simple theory is proposed to explain regimes (i) and (ii). The bubbles in contact with the roof deform under the action of buoyancy forces, developing flat contact areas whose size increases as a function of foam thickness. Maximum deformation corresponds to a critical thickness hc = 2σ/eρlgR, where σ is the coefficient of surface tension, ρl the liquid density, g the acceleration due to gravity, R the bubble radius and e the gas volume fraction in the foam. The foam thickness is determined by a balance between the input of bubbles from below and the output into the conduit, and is proportional to (μl Q/e2 ρlg)¼, where μl is the liquid viscosity and Q the gas flux. A necessary and sufficient condition for collapse is that it exceeds the critical value hc. In a liquid of given physical properties, this occurs when the gas flux exceeds a critical value which depends on viscosity, surface tension and bubble size. Experimental determinations of the critical gas flux and of the time between two events of foam collapse are in agreement with this simple theory.

Direct numerical simulation of stratified homogeneous turbulent shear flows

Abstract: The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances. The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri [approximate] 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy. Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.

Chaotic advection by laminar flow in a twisted pipe

Abstract: The appearance of chaotic particle trajectories in steady, laminar, incompressible flow through a twisted pipe of circular cross-section is demonstrated using standard dynamical systems diagnostics and a model flow based on Dean's perturbation solutions. A study is performed to determine the parameters that control fluid stirring in this mixing device that has no moving parts. Insight into the chaotic dynamics are provided by a simple one-dimensional map of the pipe boundary onto itself. The results of numerical experiments illustrating the stretching of material lines, stirring of blobs of material, and the three-dimensional trajectories of fluid particles are presented. Finally, enhanced longitudinal particle dispersal due to the coupling between chaos in the transverse direction and the non-uniform longitudinal transport of particles is shown.

Frequency selection and asymptotic states in laminar wakes

Abstract: A better understanding of the transition process in open flows can be obtained through identification of the possible asymptotic response states in the flow. In the present work, the asymptotic states in laminar wakes behind circular cylinders at low supercritical Reynolds numbers are investigated. Direct numerical simulation of the flow is performed, using spectral-element techniques. Naturally produced wakes, and periodically forced wakes are considered separately.It is shown that, in the absence of external forcing, a periodic state is obtained, the frequency of which is selected by the absolute instability of the time-average flow. The non-dimensional frequency of the vortex street (Strouhal number) is a continuous function of the Reynolds number. In periodically forced wakes, however, non-periodic states are also possible, resulting from the bifurcation of the natural periodic state. The response of forced wakes can be characterized as: (i) lock-in, if the dominant frequency in the wake equals the excitation frequency, or (ii) non-lock-in, when the dominant frequency in the wake equals the Strouhal frequency. Both types of response can be periodic or quasi-periodic, depending on the combination of the amplitude and frequency of the forcing. At the boundary separating the two types of response transitional states develop, which are found to exhibit a low-order chaotic behaviour. Finally, all states resulting from the bifurcation of the natural state can be represented in a two-parameter space inside ‘resonant horn’ type of regions.

Transition to chaos in a differentially heated vertical cavity

Abstract: We investigate numerically the transition from laminar to chaotic flow of a Boussinesq fluid with Pr = 0.71 in two-dimensional closed, differentially heated, vertical cavities having aspect ratios near unity. The cavities have rigid conducting sidewalls, and rigid insulating top and bottom walls. The physical nature of the resulting flow is a function of the aspect ratio and Rayleigh number.It is shown that an oscillatory approach to steady-state, oscillatory instabilities, quasi-periodic flow, and chaotic flow exist for the flow regimes investigated. We find that for aspect ratios of approximately three or larger the the first transition from steady-state is due to instability of the sidewall boundary layers, while for small aspect ratios, but larger than ½, it is due to internal waves near the departing corners. For both instabilities we obtain the critical Rayleigh number as a function of aspect ratio and write expressions relating the fundamental frequencies of the oscillatory flow to the Rayleigh number and aspect ratio. When Ra is increased significantly above the first critical value, the flow becomes complex since both types of instabilities can be present. With a further increase in Rayleigh number the flow becomes chaotic and eventually turbulent. The above results are illustrated for different Rayleigh numbers and aspect ratios using time histories, spectral analysis, and streamlines at different values of time.

Vertical mixing due to the breaking of critical internal waves on sloping boundaries

Abstract: A laboratory experiment is used to examine the vertical mixing resulting from the breaking of internal waves on a sloping boundary in a continuously stratified fluid. Attention is confined to the case of critical waves when the slope of the group velocity vector of the incident waves is equal to the bottom slope. Along the sloping boundary a turbulent bottom boundary layer forms with a thickness dependent on the incident wave amplitude. The mixing efficiency, defined as the ratio of the increase in potential energy due to mixing to the loss of kinetic energy by the incident waves, is dependent upon the stability of the flow and has an upper bound of approximately 0.20.By examining the increase in potential energy of the fluid as a result of sustained mixing, we are able to compute the transition value of the dissipation etr below which no mixing occures. For mixing due to the breaking of critical internal waves on sloping boundaries we find that etr = (8±2)νN2. From comparisons with experiments with grid-generated turbulence, this suggests that while etr/νN2 = 0(10) in the available data sets, the specific value of etr may be mechanism dependent.

Slow spreading of a sheet of Bingham fluid on an inclined plane

Abstract: To study the dynamics of fluid mud with a high concentration of cohesive clay particles, we present a theory for a thin sheet of Bingham-plastic fluid flowing slowly on an inclined plane. The physics is discussed on the approximate basis of the lubrication theory. Because of the yield stress, the free surface need not be horizontal when the Bingham fluid is in static equilibrium, nor parallel to the plane bed when in steady flow. We then show that there is a variety of gravity currents that can advance at a constant speed and with the same profile. Experimental confirmation of one type is presented. By solving a nonlinear partial differential equation, transient flows due either to a steady upstream discharge or to the sudden release of a finite fluid mass on another fluid layer are studied. In the first case there is a mud front which ultimately propagates as a constant speed as a steady gravity current. In the second case, when the ambient layer is sufficiently shallow that there is no initial motion, the flow induced by the new fluid can terminate after the disturbance has travelled a finite distance. The extent of the final spread is examined. Disturbances due to an external pressure travelling parallel to the free surface are also examined. It is found in particular that a travelling localized pulse of pressure gradient not only generates a localized mud disturbance which travels along with the forcing pressure, but further leaves behind a permanent footprint.

Experiments on mixing due to chaotic advection in a cavity

Abstract: Chaotic mixing of fluids in slow flows is ubiquitous but incompletely understood. However, relatively simple experiments provide a wealth of information regarding mixing mechanisms and indicate the need for complementary theoretical developments in dynamical systems. In this work we presnt a versatile cavity flow apparatus, capable of producing a variety of two-dimensional velocity fields, and use it to conduct a detailed experimental study of mixing in low-Reynolds-number flows. Since the goal is detailed understanding, only two time-periodic co-rotating flows induced by wall motions are considered: one continuous and the other discontinuous. Both types of flows produce exponential growth of intermaterial area, as expected from chaotic flows, and a mixture of islands and chaotic regions. A procedure for identifying periodic points and determining their movements is presented as well as how to make meaningful comparisons between periodic flows. We observe that periodic points move very much as a planetary system; planets (hyperbolic points) have moons (elliptic points) with twice the period of the planets; furthermore the spatial arrangement of periodic points becomes symmetric at regular time intervals. Detailed analyses reveal complex behaviour: birth, bifurcation, and collapse of islands; formation and periodic motion of coherent structures, such as islands and large-scale folds. However, the richness and complexity of the results obtained indicate that these two-dimensional time-periodic systems are far from completely understood and that other wall motions might deserve a similar level of scrutiny.

Lubricated pipelining: Stability of core-annular flow

Abstract: The stability of core-annular flow (CAF) in pipes is analysed using the linear theory of stability. Attention is confined to the potentially stable case of lubricated pipelining with the less viscous liquid, say water, in the annulus. The effects of surface tension and density are included, but gravity is excluded. We find upper and lower branches of the neutral curve in a Reynolds number (ℝ) vs. wavenumber (α) plane. A window of parameters is identified in which CAF is stable to small disturbances. When ℝ is below the lower critical value, CAF is destabilized by surface tension and long waves break up into slugs and bubbles. The sizes of slugs and bubbles of oil in water observed by Charles, Govier & Hodgson (1961) are given by the wavelength of the fastest growing long wave. This long-wave instability is a capillary instability, modified by shear, which reduces to Rayleigh's instability in the appropriate limit. At higher ℝ, the capillary instability is stabilized by shear. At yet higher ℝ, above the upper critical value, the flow is unstable to generally shorter waves which leads to emulsification, water droplets in oil. The theory agrees with experiments. The analysis seems to be applicable to the design of lubricated pipelines; for example, there is an optimum viscosity ratio for stability, greater stability can be obtained by using heavy liquid as a lubricant when the flow is unstable to capillary modes on the lower branch and by using light liquids when the flow is unstable to emulsifying disturbances on the upper branch.

Further experiments on the evolution of turbulent stresses and scales in uniformly sheared turbulence

Abstract: Measurements of the Reynolds stresses, integral lengthscales and Taylor microscales are reported for several cases of uniformly sheared turbulent flows with shear values in a range substantially wider than those of previous measurements. It is shown that such flows demonstrate a self-preserving structure, in which the dimensionless Reynolds stress ratios and the dissipation over production ratio, e/ P , remain essentially constant. Flows with sufficiently large $k_{\rm s} = (1/\overline{U_{\rm c}}){\rm d}\overline{U_1}dx_2$ have exponentially growing stresses and e/ P ≈ 0.68; a linear relationship between the coefficient in the exponentiallaw and k s is shown to be compatible with measurements having k s > 3. The possibility of a self-preserving structure with asymptotically constant stresses and e/ P ≈ 1.0 is also compatible with measurements, corresponding to flows with small values of k s . The integral lengthscales appear to grow according to a power law with an exponent of about 0.8, independent of the mean shear, while the Taylor microscales, in general, approach constant values. Various attempts to scale the stresses and to predict their evolution are discussed and the applicability of Hasen's theory is scrutinized. Finally, an ‘exact’ expression for the pressure-strain rate covariance is derived and compared to some popular models.

Wavenumber spectra of short gravity waves

Abstract: The spectral balances involved in shaping the short gravity wave region of the ocean wave-height spectrum have been the subject of recent physical models. In terms of the wind friction velocity u*, gravitational acceleration g and local wavenumber k, these models predict a wavenumber dependence of , where k = |k|, and a linear dependence on u* for the equilibrium range of gravity waves above the spectral peak. In this paper we present the results of an experimental determination of the wavenumber spectrum for the wavelength range of 0.2−1.6 m, based on stereophotogrammetric determinations from an oil platform under open ocean conditions.From our observations, for this wavenumber range, the one-dimensional equilibrium wavenumber spectrum was determined as \[ \phi (k_i) \sim \left(\frac{u^2_*k}{g}\right)^{\gamma} k^{-3}_{i}\;\;\;\;\;\;\;(i=1,2 \;\;\; K = (k_1,k_2)) \] where γ = 0.09±0.09 at the 95% confidence level. These limits embrace wind-independent approximations to the observed one-dimensional and two-dimensional wavenumber spectra of the form \[ \phi (k_i) \sim B k^{-3}_i \;\;\; (i = 1,2), \] and \[ \psi(k_i) \sim A k^{-4}, \] respectively, with B ∼ 10−4 and A ∼ 0.3 × 10−4 for and k = |k| is expressed in cycles/metre.The present findings do not support the wavenumber dependence predicted by the recent models in this wavenumber range and are at variance with their predicted dependence on the friction velocity. However, our observations are generally consistent with the radar reflectivity dependence on wind direction and wind speed under Bragg scattering conditions within our wavenumber range. The experimental observations also point out the potentially important role of wave-breaking of longer wave components in influencing the spectral levels of short gravity wave components.