ALBERT

Description: The dislodging by dynamic pressure forces of a drop adhered by surface tension to a plane is analysed. An integro-differential equation describing the drop shape is solved numerically and the critical Weber number as a function of contact angle hysteresis is found. © 1988, Cambridge University Press. All rights reserved.

Description: A corrugated plate is translating in a rotating fluid. Assuming low Reynolds number and small amplitudes compared to the Ekman thickness, a perturbation solution is found to second order. The resistance and power due to drag depend on the relative orientation of the corrugations with the motion. In certain instances, it is easier to move a corrugated plate than a flat plate in a rotating fluid. © 1988, Cambridge University Press. All rights reserved.

Description: A theoretical and experimental investigation is reported dealing with the onset of buckling in a horizontal layer of highly viscous liquid. The layer floats on a heavier liquid with negligible viscosity, and at rest is stabilized by gravity and surface tension. When sheared at a sufficient rate, the flat configuration of the layer becomes unstable; and the aim of the investigation is to establish the relation between critical values of the shearing rate and values of the layer’s thickness and other physical parameters. A primitive theory based on membrane approximations is first reviewed and its deficiencies are appreciated. Then a more reliable theory is developed, providing estimates of values taken by a dimensionless shear stress f at the threshold of instability. The values fcare found to depend primarily on a dimensionless number H proportional to the thickness of the layer. Experiments on sheared layers of silicone oil with various high viscosities are then described. Measured values offc plotted against H over a wide range are shown to be in satisfactory agreement with the theory. Finally, discrepancies between previous experimental results and ours are discussed. © 1988, Cambridge University Press. All rights reserved.

Description: This paper deals with the Soret separation of a binary mixture in a cylinder subjected to an axial temperature gradient. The study is connected to an experiment designed to measure the Soret coefficient of an Agl-KI mixture corresponding to a moderate Prandtl number (Pr = 0.6) and a high Schmidt number (Sc = 60). In such an experiment the species separation is often hidden by a mixing effect due to the buoyancy-driven convection generated by a horizontal temperature gradient induced by some defect of the heating system. Here, such a defect is simulated by a slight misorientation of the cell with respect to the vertical; a small inclination (y = 1°) of the cell has been considered, but the results can be generalized for any other small y. For situations corresponding to a top heating and a positive Soret parameter, S, two quite different regimes have been exhibited depending on the value of S, For moderate S, the induced solutal buoyancy balances the imposed thermal buoyancy, slowing down the flow and giving a good separation rate. For small S this balance does not exist (except in the centre), leading to a remixing of the species and thus to poor separation (the separation would be still worse for negative S). The smaller the (positive) Soret parameter is, the smaller the cell misorientation y has to be to allow a good separation rate. © 1988, Cambridge University Press. All rights reserved.

Description: The consolidation or concentration of suspended particulate solids under the influence of gravitational forces is a problem of widespread practical and theoretical interest. The literature, which is scattered over several fields, contains most of the elements necessary for a complete understanding of gravity settling, but considerable controversy and confusion persists about their synthesis. Here we propose to construct a quantitative theory covering the full range of processes from transient settling of large, stable particles to the slow consolidation of flocculated suspensions of submicron particles. Conditions for the existence of shocks are identified and the basic equations describing the phenomena are solved numerically for several Peclet numbers. © 1988, Cambridge University Press. All rights reserved.

Description: We have investigated the modulated and unmodulated travelling azimuthal waves appearing on the toroidal Taylor—Görtler (TG) vortices in a fluid contained between two concentric spheres with the inner sphere rotating. For smaller-clearance cases, toroidal TG vortices appear at the equator, just as in the flow between two concentric cylinders with the inner cylinder rotating. When the Reynolds number of the flow increases quasi-statically, spiral TG vortices appear in addition to toroidal TG vortices, and no modulation occurs, even if the Reynolds number further increases quasi-statically. However, when the Reynolds number is increased from zero to a particular value with a specific acceleration of the inner sphere, modulated wavy toroidal TG vortices appear. We found that the necessary condition for occurrence of modulation is the prevention of spiral TG vortices. Using simultaneous flow-visualization and spectral techniques, and measuring the fluctuation of sinks and sources of vortex boundaries, we obtained the frequency f1 of travelling azimuthal waves passing a fixed point in the laboratory and the modulation frequencies f2and f2of these waves, as determined by an observer in the laboratory and an observer fixed in a reference frame that rotates in phase with the travelling azimuthal waves, respectively. The relationship among the characteristic frequencies, f1, f2 and f2, obtained by modal analysis and the experimental results, is (f2+ kfjm)/f2 = — 1, where k and m are a modulation parameter and the wavenumber of travelling azimuthal waves, respectively. © 1988, Cambridge University Press. All rights reserved.

Description: The stability of eddies with three-layer stratification is examined experimentally. When the difference in density between the upper two layers is much greater (or less) than that between the lower two layers baroclinic instability on two different lengthscales (the Rossby radii associated with the upper and the lower interfaces) is possible. The vortices are created using modifications of two techniques described by Griffiths & Linden (1981) in their study of two-layer eddies. ‘Constant-flux’ eddies are generated by the release of a constant flux of buoyant fluid from a small source positioned a t the surface of a two-layer fluid. In a second variation of this experiment, the source is positioned at the interface between two layers and fluid of intermediate density is injected. As the horizontal lengthscale increases, the vortices evolve from a stable to an unstable state. It is showns that the size a t which the vortices become unstable may be significantly altered by the presence of a second interface. The results agree qualitatively with the conclusions of a linear stability analysis of quasi-geostrophic three-layer flow in a channel (Smeed 1988), but it is necessary to examine the effects of horizontal shear and Ekman dissipation to explain the experimental results. ‘Constant-volume’ eddies are produced by the release of a volume of buoyant fluid, initially contained within a cylindrical barrier, a t the surface of a two-layer fluid. After the barrier is removed, the buoyant fluid spreads a distance of the order of the Rossby radius. Similarly, vortices are created by releasing a volume of fluid of density intermediate between the initial two layers. Within a few rotation periods the vortices become unstable to disturbances similar to those observed in two-layer experiments. Qualitative agreement is found between the observed wavelength and the fastest growing mode predicted by the linear stability theory (Smeed 1988). When the disturbances reach large amplitude a change in lengthscale is often observed. © 1988, Cambridge University Press. All rights reserved.

Description: An experimental investigation of double-diffusive convection in a two-layer, saltstratified solution destabilized by lateral heating and cooling has been performed. Initially, diffusive regime phenomena are observed as the two uniform salinity layers are thermally driven and behave somewhat independently. As salt is transferred across the interface separating the layers, salinity stabilization decreases and complicated flow structure is observed a t the interface. In the final stages before mixing, the stabilizing salinity gradient becomes small, the thermal/hydrodynamic boundary layers on the heated and cooled sidewalls penetrate the salinity interface and mixing, in the finger regime, occurs. The dimensionless mixing time is described with parameters associated with thermal and salinity buoyancy forces and the enclosure aspect ratio. Careful selection of the experimental conditions allows dimensionless interfacial salinity transport rates to be correlated with appropriatedimensionless parameters. © 1988, Cambridge University Press. All rights reserved.

Description: A theoretical and numerical investigation of strearnwise-oriented Dean vortices in curved channel flow is presented. The principal results are obtained from three-dimensional pseudospectral simulations of the incompressible time-dependent Navier-Stokes equations. With increasing Reynolds number, a sequence of transitions similar to that observed in non-turbulent Taylor-Couette flow is found. The transition from laminar curved channel Poiseuille flow to axisymmetric Dean vortex flow is studied using linear and weakly nonlinear analyses; these results are compared to the full simulations. Using the code, two transitions that cause the axisymmetric vortices to develop waves travelling in the streamwise direction at higher Reynolds numbers are discovered. The linear stability of axisymmetric Dean vortex flow to non-axisymmetric perturbations is examined. Associated with the two transitions are two different non-axisymmetric flows: undulating and twisting Dean vortex flow. Undulating vortices are similar to wavy Taylor vortices. Twisting vortices, with a much shorter streamwise wavelength, are dissimilar; to our knowledge, they have no counterpart in the Taylor-Couette problem. At sufficiently high Reynolds numbers, linear growth rates associated with twisting vortices far exceed those associated with undulating vortices. For the channel curvatures studied, angular speeds of both kinds of travelling waves are only weakly dependent on Reynolds number and wavenumber. A bifurcation limits the vortex spacings that can be examined and suggests an Eckhaus stability boundary. The development of wavy vortex flows from small-amplitude disturbances shows that full development of undulating vortices may require a streamwise distance greater than one circumference, whereas for sufficiently large Reynolds numbers, twisting vortices reach equilibrium amplitude within half this distance and are therefore more likely to be observed experimentally. We suggest twisting vortices are due to a shear instability. © 1988, Cambridge University Press. All rights reserved.

Description: Saffman argues that in decaying two-dimensional turbulence approximate discontinuities of vorticity will form, and the energy spectrum will fall off as fc-4. Saffman assumes that these discontinuities are well separated in this paper, we examine how accumulation points of such discontinuities may give an energy spectrum of between k4 and k~3. In particular we examine the energy spectra of spiral structures which form round the coherent vortices that are observed in numerical simulations of decaying two-dimensional turbulence. If the filaments of the spiral are assumed to be passively advected, the instantaneous energy spectrum has a k~11/z range. Thus we come some way to reconciling the argument of Saffman and the k~z energy spectrum predicted by models of quasi-equilibrium two-dimensional turbulence based on a cascade of enstrophy in Fourier space. © 1988, Cambridge University Press. All rights reserved.

Description: Reheat buzz is a low-frequency instability of afterburners. It is caused by the interaction of longitudinal acoustic waves and unsteady combustion. Similar combustion instabilities occur in laboratory rigs. A theory is developed to determine the frequency and mode shape of the instability and is tested by comparison with the experimental results described in Part 1. The predicted and measured frequencies are found tobe within 6 Hz (7 %) of each other. The theory is able to predict the observed variation of frequency with equivalence ratio, inlet Mach number and geometry. © 1988, Cambridge University Press. All rights reserved.

Description: Detailed measurements throughout the separated region behind a flat plate placed normal to a turbulent stream are reported. A long, central, downstream splitter plate prevented vortex shedding and led to a relatively extensive reversed flow region. Mean flow and turbulence data are compared with results obtained in the (nominal) absence of free-stream turbulence, and attention is concentrated on the changes in the shear-layer structure resulting from the different nature of the upstream flow. Many aspects of the results confirm those obtained recently by other workers. Free-stream turbulence enhances shear-layer entrainment rates, reduces the distance to reattachment and modifies the relatively low-frequency ‘flapping’ motion of the shear layer. In addition, however, extensive use of pulsed wire anemometry has allowed detailed measurements of the turbulence structure throughout the flow and it is shown that this is also modified significantly by the stream turbulence. © 1988, Cambridge University Press. All rights reserved.

Description: A two-dimensional numerical large-eddy simulation of a temporal mixing layer submitted to a white-noise perturbation is performed. It is shown that the first pairingof vortices having the same sign is responsible for the formation of a continuous spatial longitudinal energy spectrum of slope between After two successive pairings this spectral range extends to more than 1 decade. The vorticity thickness, averaged over several calculations differing by the initial white-noise realization, is shown to grow linearly, and eventually saturates. This saturation is associated with the finite size of the computational domain. We then examine the predictability of the mixing layer, considering the growth of decorrelation between pairs of flows differing slightly at the first roll-up. The inverse cascade of error through the kinetic energy spectrum is displayed. The error rate is shown to grow exponentially, and saturates together with the levelling-off of the vorticity thickness growth. Extrapolation of these results leads to theconclusion that, in an infinite domain, the two fields would become completely decorrelated. It turns out that the two-dimensional mixing layer is an example of flow that is unpredictable and possesses a broadband kinetic energy spectrum, though composed mainlyof spatially coherent structures. It is finally shown how this two-dimensional predictability analysis can be associated with the growth of a particular spanwise perturbationdeveloping on a Kelvin-Helmholtz billow this is done in the framework of a one-mode spectral truncation in the spanwise direction. Within this analogy, the loss of two-dimensional predictability would correspond to a return to three-dimensionality and a loss of coherence. We indicate also how a new coherent structure could then be recreated, usingan eddy-viscosity assumption and the linear instability of the mean inflexional shear. © 1988, Cambridge University Press. All rights reserved.

Description: A brief review of the relevant papers and an examination of the current statusof research in the field of the acoustics of gas-particle suspensions are given. Nonstationary momentum, mass and energy transfer processes between a gas (vapour) and dispersed particles (droplets) under high-frequency acoustic perturbations are considered. A comparative evaluation of characteristic times and temperature differences for gas-particle and vapour-droplet mixtures subjected to acoustic perturbation is given. General dispersion equations to describe the propagation of weak monochromatic waves for a wide range of frequencies complying with the requirements of the acoustic homogeneity of the medium are derived. Frequency dependences of propagation velocity and attenuation coefficient of weak waves in water vapour-droplet mixtures are investigated. Frequency ranges are indicated over which different types of approximate theories are valid. © 1988, Cambridge University Press. All rights reserved.

Description: The interaction between shear and buoyancy effects for Bénard convection in plane Couette flow is studied by performing direct numerical simulations. At moderate Rayleigh numbers (≈ 10000–50000), shear tends to organize the flow into quasi-two-dimensional rolls parallel to the mean flow and can enhance heat transfer, while at higher Rayleigh numbers (〉 150000), shear tends to disrupt the formation of convective plumes and can reduce heat transfer. A significant temporal oscillation in the local Nusselt number was consistently observed at high Rayleigh numbers, a factor that may contribute to the scatter seen in experimental data. This effect, plus the time-varying reversal of the mean temperature gradient in the middle of the channel, is consistent with a flow model in which the dynamics of large-scale, quasi-two-dimensional, counter-rotating vortical cells are alternately driven by buoyancy and inertial effects. An analysis of the energy balance in the flow shows that the conservative pressure diffusion term, which has been frequently neglected in turbulence models, plays a very important dynamical role in the flow evolution and should be more carefully modelled. Most of the turbulent energy production due to mean shear is generated in the boundary layers, while the buoyant production occurs mainly in the relatively uniform convective core. The simulations and the laboratory experiments of Deardorff & Willis (1967) are in very reasonable qualitative agreement, suggesting that the basic dynamics of the flow are being accurately simulated. © 1988, Cambridge University Press. All rights reserved.

Description: The development of Gortler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel-flow analysis impossible except in the high-wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using a Fourier series expansion whilst the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact the equilibrium state reached is qualitatively similar to the large-amplitude high-wavenumber solution described asymptotically by Hall (1982b). In general it is found that the nonlinear interactions are dominated by a ‘mean field’ type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of our calculations is the finding that the downstream velocity field develops a strongly inflexional character as the flow moves downstream; the latter result suggests that the major effect of Görtler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability. © 1988, Cambridge University Press. All rights reserved.

Description: This study is devoted to the onset of convection in differentially heated cylinders under gravity modulation. It specifically concerns the case of a vertical cylinder of infinite length, when a negative temperature gradient is maintained in the upward direction. The effect of modulation on the stability limits given by linear theory in the standard steady case is analysed. A method based on Floquet theory is proposed in the case of small values of the modulation amplitude ε, for a fixed value of the frequency ω. A general technique, called matrix method, which can easily be adapted to various kinds of geometries and boundary conditions, has been developed. Analytical approaches have been derived in some cases. Finally, an asymptotic analysis is presented for large ω, under very general boundary conditions and periodic constraints, for finite ε. An asymptotic relation is established for the onset of convection under periodic gravity modulation for large ω values, when ε ≪ ω the mathematical and physical foundations of this inequality are discussed. © 1988, Cambridge University Press. All rights reserved.

Description: We study the evolution of unsteady two-dimensional vorticity structures surrounded by fluid at rest. The flow is initiated by a short fluid impulse in a horizontal layer of mercury and is constrained to be two-dimensional by a vertical uniform magnetic field. The impulse is generated by an electric pulse between two electrodes, and a flow circulation can be produced by diverting part of the current through the external frame. The velocity field is measured from the streaks of small particles floating on the free upper surface, and the vorticity is then obtained by means of an analytical interpolation and differentiation. The flow always evolves toward a set of independent steady structures with symmetry which are either circular vortices (monopoles) or couples (dipoles). The latter have a linear or circular steady motion depending on the flow circulation around them. The region of non-zero vorticity is always close to a circle. The steadiness is confirmed by plotting the vorticity versus the stream function in the frame of reference moving with the couple. We obtain a curve, as appropriate for a steady solution of the Euler equation. The slope of this curve is either constant or has no maximum. We suggest that this result could correspond to a general stability condition. The interaction between two symmetric couples at various angles of incidence yields two new couples by exchange of their vortices. Oscillations of the resulting couples are often damped by releasing a circular vortex. © 1988, Cambridge University Press. All rights reserved.

Description: We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier-Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system. © 1988, Cambridge University Press. All rights reserved.

Description: We extend the recent work on Bragg scattering of water waves by one-dimensional parallel bars of sinusoidal profile to two-dimensional, doubly sinusoidal bed waves. The resonance condition governing the phase matching between the incident, scattered and bed waves is now more complicated and a much richer variety of resonant reflection can occur. In particular, for one normally incident wave there can be two reflected waves forming a standing wave in the transverse direction. Solutions for a wide strip of bed waves are discussed for incident water waves satisfying approximately the Bragg resonance condition. Modifications for a two-dimensional array of hemispheroids are also given. Possible application to the design of submerged breakwaters is suggested. © 1988, Cambridge University Press. All rights reserved.

Description: Similarity solutions have been found for steady two-dimensional laminar flow in which dense fluid is emitted upwards from a horizontal plane into a laminar shear flow or into a uniform flow. The solutions also apply to a light fluid released at an upper horizontal surface. The Navier-Stokes equations and the diffusion-advection equation are simplified by making the Boussinesq approximation and the boundary-layer approximation, which here also implies that pressure is hydrostatic.For an oncoming linear shear flow representing flow near a solid surface, a similarity solution is obtained with depth proportional to $x^{frac{1}{3}}$ where x is the horizontal coordinate. Horizontal velocity and concentration of dense fluid both increase as $x^{frac{1}{3}}$, so that the solution represents fluid propagating upstream along the surface, and diffusing vertically to be swept downstream again. Numerical solutions for vertical profiles of velocity and concentration are presented for a Schmidt or Prandtl number σ between 0.71 and infinity. Two alternative sets of boundary conditions are possible. In one set, the pressure above the boundary layer is unchanged but the velocity profile is displaced upwards. In the second, this displacement is forced to be zero with the result that a pressure gradient is generated in the outer flow. These two boundary conditions are known to apply to disturbances in a laminar boundary-layer on horizontal lengthscales respectively greater or smaller than the triple-deck scale.With a uniform velocity upstream and a stress-free boundary, representing flow at a free surface, similarity solutions exist only for a plume growing downstream from the source of a buoyancy flux B, with depth increasing as x½ and concentration decreasing as x−½. When gravity has negligible effects, so that B = 0, the solution is a Gaussian plume. With finite B, there is an adverse gradient of hydrostatic pressure and the plume is decelerated so that it is deeper than in neutral flow. Numerical solutions for σ = 0.71 reveal that there is a maximum buoyancy flux Bcrit above which no similarity solution exists. This occurs with a non-zero value of the surface velocity. For B 〈 Bcrit it is found that there are in fact two possible solutions. One has surface velocity greater than at the critical flux and tends to the passive Gaussian plume as B → 0. In the other, surface velocity decreases from Bcrit, reaching zero at a non-zero value of B. Similar behaviour is found in an asymptotic solution for very large σ.

Description: We consider a plane, steady, homogeneous flow of circular disks. The disks are identical, smooth, and inelastic. We adopt the assumption of molecular chaos and introduce an anisotropic Maxwellian velocity distribution function based on the full second moment of the velocity fluctuations. In the limits of dilute and dense flows, we determine approximate analytic solutions of the balance law for the second moment that result in stresses whose qualitative behaviour and magnitudes are in good agreement with numerical simulations.

Description: We present calculations of the change in phase speed of one train of water waves in the presence of another. We use a general method, based on Zakharov's (1968) integral equation. It is shown that the change in phase speed of each wavetrain is directly proportional to the square of the amplitude of the other. This generalizes the work of Longuet-Higgins & Phillips (1962) who considered gravity waves only.In the important case of gravity-capillary waves, we present the correct form of the Zakharov kernel. This is used to find the expressions for the changes in phase speed. These results are then checked using a perturbation method based on that of Longuet-Higgins & Phillips (1962). Agreement to 6 significant digits has been obtained between the calculations based on these two distinct methods. Full numerical results in the form of polar diagrams over a wide range of wavelengths, away from conditions of triad resonance, are provided.

Description: We solve the problem of magnetic field generation by a laminar flow of conducting fluid with helical (screw-like) streamlines for large magnetic Reynolds numbers, Rm. Asymptotic solutions are obtained with help of the singular perturbation theory. The generated field concentrates within cylindrical layers whose position, the magnetic field configuration and the growth rate are determined by the distribution of the angular, Ω and longitudinal, Vz, velocities along the radius. The growth rate is proportional to Rm2. When Ω and Vz are identically distributed along the radius, the asymptotic forms are of the WKB type for different distributions, singular—layer asymptotics of the Prandtl type arise. The solutions are qualitatively different from those obtained for solid—body screw motion. The generation threshold strongly depends on the velocity profiles. © 1988, Cambridge University Press. All rights reserved.

Description: The stability of Couette flow in Hell is considered by an analysis of the HVBK equations. These equations are based on the Landau two-fluid model of Hell and include mutual friction between the normal and superfluid components, and the vortex tension due to the presence of superfluid vortices. We find that the vortex tension strongly affects the nature of the Taylor instability at temperatures below ≈ 2.05 K. The effect of the vortex tension is to make non-axisymmetric modes the most unstable, and to make the critical axial wavelength very long. We compare our results with experiments. © 1988, Cambridge University Press. All rights reserved.

Description: A numerical study has been conducted to determine the various modes of Taylor—Couette flow that exist between concentric vertical cylinders, as the aspect ratio F (height to gap width, H/d) and the Reynolds number Re (based on the inner cylinder speed) are varied. Furthermore, the effects of the introduction of buoyancy on the development of the flow are examined. This is accomplished by considering both cylinders to be isothermal, with the rotating inner cylinder at a higher temperature than the stationary outer cylinder. Results are presented for a wide range of the Grashof number Gr (based on the temperature difference ΔT across the annular gap). The structure of the Taylor vortices is observed to be distorted considerably with the buoyant flows, and the nature of the onset and subsequent development of the vortices is altered. The hysteresis between the different modes of cellular flow, characteristic of the bifurcation phenomena, is also substantially modified. © 1988, Cambridge University Press. All rights reserved.

Description: Substructures within a turbulent spot which develops in a slightly heated laminar boundary layer have been identified using arrays of cold wires aligned in either a streamwise direction or in a direction normal to the wall. At any given streamwise distance from the spot origin, histograms of the number of detected substructures exhibit a peak, defining the most probable spot or the spot with the most likely number of substructures. The number of substructures in the most probable spot increases with streamwise distance but all substructures are convected at approximately the same velocity for any given distance from the wall. This velocity is approximately equal to that of the leading edge of the spot and increases slightly with distance from the wall. The increase in the number of substructures accounts for the streamwise growth of the spot. A simple relation is derived for determining the number of substructures at a particular streamwise station and a geometrical construction is proposed for identifying the origin of a new substructure. There is sufficient evidence for suggesting that the new substructures are formed near the trailing edge of the spot. The convection velocity, inclination and lengthscales of the substructures compare favourably with the corresponding characteristics of hairpin vortices. © 1988, Cambridge University Press. All rights reserved.

Description: The instabilities of barotropic and baroclinic, quasi-geostrophic, f-plane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depth-varying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that large-scale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases. the other two kinds of instability are nonlinearly destabilized. © 1988, Cambridge University Press. All rights reserved.

Description: The finite-amplitude evolution of circular two-layer quasi-geostrophic vortices with piecewise uniform potential vorticity in each layer (also termed ‘heton’ clouds by Hogg & Stommel 1985a and Pedlosky 1985) is studied using the contour dynamics method. The numerical investigations are preceded by a linear stability analysis which shows the stabilizing influence of deepening the lower layer. Net barotropic flow may be either stabilizing or destabilizing. The contour dynamics calculations for baroclinic vortices show that supercritical (i.e. linearly unstable) conditions may lead to explosive break up of the vortex via the generation of continuous hetons at the cloud boundary. The number of vortex pairs is equal to the azimuthal mode number of the initial disturbance. An additional weakly supercritical regime in which amplitude vacillation occurs, but not explosive growth, is identified. Vortices with net barotropic circulation behave similarly except that the layer with vorticity opposite to the barotropic circulation will break up first. Strong barotropic circulation can inhibit the development of hetons. The stronger layer may eject thin filaments, but remain mostly intact. Calculations for initial conditions composed of several unstable modes show that the linearly most unstable mode dominates at finite amplitude. © 1988, Cambridge University Press. All rights reserved.

Description: The laminar pulsatile flow over a semi-infinite flat plate, on which is located a small (steady) surface distortion is investigated; triple-deck theory provides the basis for the study. The problem is of direct relevance to the externally imposed acoustic excitation of boundary layers. The investigation is primarily numerical and involves the solution of the nonlinear, unsteady boundary-layer equations which arise from the lower deck. The numerical method involves the use of finite differencing in the transverse direction, Crank-Nicolson marching in time, and Fourier transforms in the streamwise direction, and as such is an extension of the spectral method of Burggraf & Duck (1982). Supersonic and incompressible flows are studied. A number of the computations presented suggest that the small surface distortion can excite a large-wavenumber, rapidly growing instability, leading to a breakdown of the solution, with the wall shear at a point seeming to increase without bound as a finite time is approached. Rayleigh modes for the basic (undisturbed) velocity profile are computed and there is some correlation between the existence and magnitude of the growth rate of these unstable modes, and the occurrence of the apparent singularity. Streamline plots indicate that this phenomenon is linked to the formation of closed (or ‘cats-eye’) eddies in the main body of the boundary layer, away from the wall. Tollmien-Schlichting instabilities are clearly seen in the case of incompressible flows. © 1988, Cambridge University Press. All rights reserved.

Description: A general expression is derived for the fluid force on a body of simple shape moving with a velocity v through inviscid fluid in which there is an unsteady non-uniform rotational velocity field u0(x, t) in two or three dimensions. It is assumed that the radius is small compared with the scale over which the strain rate changes, though for the sphere it is also assumed that the changes in the ambient velocity field over the scale of the sphere are small compared with the velocity of the body relative to the flow. Given these approximations it is shown that the effects of the rate of change of the vorticity of the ambient flow is of second order and can be neglected. However the rate of change of the irrotational straining motion is included in the analysis. It is shown that the inertial forces derived by many authors for irrotational flow can be simply added to a generalization of the lift force derived by Auton (1987) in a companion paper. It is shown how this lift force is made up of a rotational and an inertial or added-mass component. For three-dimensional bluff bodies the latter is generally larger (by a factor of three for a sphere), and can be simply calculated from the added-mass coefficient. For illustration, the general expression is used to derive formulae for (i) the motion of a spherical bubble in a steady non-uniform flow to contrast with the motion in an unsteady flow, and (ii) the motion of rigid volumes of neutral density across an inviscid shear flow. These results show how added-mass (and lift) forces lead to different motions for a sphere and a cylinder. The general expression is useful in two-phase flow calculations, and for indicating the forces and motions of ‘lumps of fluid’ in turbulent flows. © 1988, Cambridge University Press. All rights reserved.

Description: The problem has been examined using a kinematic model for wall pliability, wherein a kinematic postulation of the wall boundary conditions is made. A form of the normalized wall-displacement and its phase are used as additional parameters in an extended eigenvalue problem. Using this technique the entire gamut of possibilities regarding stability of flow past (normally) pliable walls can be examined, yet without recourse to any specific material properties for the wall. Rather, the results based on the kinematic model can be used to back-calculate the material properties corresponding to any chosen model for the dynamics of the wall. A sample back calculation is discussed herein for the Benjamin—Landahl wall model, and based on this some predictions are made regarding both stabilization of the flow and physical realizability of modes. It is believed that the kinematic model will prove useful in further understanding of the problem, and in the design of stabilizing coatings. The results show that there are three important mode classes’ (distinct from ‘modes’), namely the Tollmien—Schlichting (TS), resonant (R) and Kelvin-Helmholtz (KH). Whereas the TS and R mode classes broadly agree with modes bearing similar names as found by earlier workers, the present KH mode class is difficult to classify based on earlier work. Moreover, there are also important transitional mode classes in the regions of bifurcations of the regular mode classes. Two important concepts evolve in connection with the TS and R mode classes, namely the existence of ‘stable pockets for the former and ‘ unstable pockets’ for the latter. It is also confirmed herein that there are conflicting requirements on the damping d to stabilize TS and R modes. Considering these points it has been suggested that TS and R modes be avoided by keeping soft surfaces as compliant coatings. However, this in turn leads to instabilities from one of the transitional mode classes. It is also seen that a soft surface that is also marginally active (i.e. having a small negative value of d) could render even better stabilization. © 1988, Cambridge University Press. All rights reserved.

Description: A photographic study of the wakes of slugs rising in tubes of 19 mm and 52 mm internal diameter is presented. The dependence of the flow pattern in the wake upon the Reynolds number of the rising slug, R is established for different slug lengths. Values of R covered in this study are in the range 25 to 1.3 x 104. For low values of R the flow pattern in the wake is laminar and axisymmetric and values of wake length and wake volume could be determined from the photographs these values were correlated with the other variables in the system by means of dimensional analysis. © 1988, Cambridge University Press. All rights reserved.

Description: This paper describes how the flows around three equal circular cylinders arranged in an equilateral—triangular manner interact at different angles of incidence a and spacing ratios l/d, Some vortex-shedding-frequency data evaluated from flow visualization experiments conducted at Reynolds numbers of 2.1 x 10 3 and 3.5 x 10 3 , based on the diameter of a single cylinder, using a dye-injection technique, are presented. In order to provide additional insight to the understanding of the flow structure around this particular cylinder array, some photographs indicating the typical flow patterns for various arrangements are also presented. The investigation indicates that the flows interact in a complex fashion for spacing ratios smaller than 2.29 and it also reveals that, at this range of spacing ratios and at a = 0°, bistable flow characteristic exists. Moreover, for l/d approximately smaller than 4.65 there always exists an angle at which the vortex shedding behind an upstream cylinder is suppressed by a nearest downstream cylinder. This angle is found not to remain constant but increases as the spacing ratio increases. For illustration and comparisons, some numerical results obtained from the application of the surface-vorticity method have also been presented. © 1988, Cambridge University Press. All rights reserved.

Description: In this paper, we are concerned with the linear stability of zero pressure-gradient laminar boundary-layer flow over compliant walls which are composed of one or more layers of isotropic viscoelastic materials and backed by a rigid base. Wall compliance supports a whole host of new instabilities in addition to the Tollmien-Schlichting mode of instability, which originally exists even when the wall is rigid. The perturbations in the flow and the compliant wall are coupled at their common interface through the kinematic condition of velocity continuity and the dynamical condition of stress continuity. The disturbance modes in the flow are governed by the Orr–Sommerfeld equation using the locally-parallel flow assumption, and the response of the compliant layers is described using a displacement-stress formalism. The theoretical treatment provides a unified formulation of the stability eigenvalue problem that is applicable to compliant walls having any finite number of uniform layers; inclusive of viscous sublayer. The formulation is well suited to systematic numerical implementation. Results for single-and multi-layer walls are presented. Analyses of the eigenfunctions give an insight into some of the physics involved. Multi-layering gives a measure of control over the stability characteristics of compliant walls not available to single-layer walls. The present study provides evidence which suggests that substantial suppression of disturbance growth may be possible for suitably tailored compliant walls. © 1988, Cambridge University Press. All rights reserved.

Description: Disturbances in the form of pressure fields, source distributions and time-dependent bottom topographies are discussed and found to produce similar wave patterns. Results obtained for wide channels are discussed in the light of the features of soliton reflection at a wall. Comparison with experiments shows excellent agreement. The introduction of radiation conditions enables long-time simulation of the development of wave patterns in infinite and semi-infinite fluids. A stationary wave pattern is also found to emerge for slightly supercritical Froude numbers, but contrary to linear results the leading divergent waves may originate ahead of the disturbance. This behaviour is due to nonlinear interactions similar to those governing collisions between solitons. This study on wave generation by a moving disturbance is based on numerical solutions of Boussinesq-type equations. The equations in their most general form are integrated by an implicit difference method. Strongly supercritical cases are described by a simplified set of equations which is solved by a semi-implicit difference scheme. © 1988, Cambridge University Press. All rights reserved.

Description: A vortex pair, impulsively generated from a planar nozzle, is shown to have a degree of vorticity concentration in good agreement with inviscid theory, providing wellposed initial conditions for interaction with basic types of bodies (cylinders and plates). The scale of these bodies ranges from the same order as, to over an order of magnitude smaller than, the scale (distance between centres) of the incident vortex pair.The fundamental case of a (primary) vortex pair symmetrically incident upon a very small cylinder shows rapid growth of a secondary vortex pair. These secondary vortices quickly attain a circulation of the same order as that of the corresponding primary vortices within a distance smaller than the lengthscale of the primary vortex pair. At this location, the temporal variation of integrated vorticity of primary and secondary vortices attains a maximum simultaneously. This zero phase shift between arrival of vorticity maxima provides the basis for formation of counter-rotating, primary-secondary vortex pairs, where both the primary and secondary vortices move at the same phase speed. Visualization shows that the mode of secondary vortex formation is highly sensitive to the degree of symmetry of the initial encounter of the incident vortex pair with the body. The symmetrical mode of (in-phase) secondary vortex formation shows very rapid growth of large-scale secondary vortices ; their development is relatively independent of the particulars of body shape and scale. On the other hand, the antisymmetrical mode takes two basic forms : large-scale secondary vortex formation, with the phase shift between their formation determined by the lengthscale of the body ; and small-scale, antisymmetrical shedding of secondary vortices from the body occurring for a body lengthscale an order of magnitude smaller than that of the incident vortex pair. Correspondingly, there are several types of distortion of the cores and trajectories of the primary (incident) vortices. © 1988, Cambridge University Press. All rights reserved.

Description: Herein we show that the modal description of deep-water waves on the sea surface (Watson & West 1975) is independent of any reference surface around which expansions of the velocity potential and the surface velocity are done. We demonstrate by direct construction that the interaction between long and short waves does not lead to divergent terms in the equations of motion when this formalism is used. © 1988, Cambridge University Press. All rights reserved.

Description: The propagation of weak shock waves (Ms= 1.007, 1.03 and 1.1) through a statistically uniform random medium has been investigated experimentally in a shock tube. The wave-front geometry, rise time and amplitude of initially plane shocks which have propagated through a random mixture of helium and refrigerant 12 are measured. The effect of shock propagation on the properties of the random medium is visualized with schlieren and shadow photography. The pressure histories of the distorted shock waves reflecting from a normal end wall are observed to be both peaked and rounded. In the rounded case the perturbed shock is found to be made up of a succession of weak, slightly curved fronts with a total effective rise time orders of magnitude greater than the classical Taylor thickness. The radius of curvature of the weakest shocks after propagating through the random medium is inferred from observations at two downstream stations to be about 7 times the integral scale of the gas inhomogeneities. It is concluded that the observed distortions of the wave fronts can best be explained in terms of random focusing and defocusing of the front by the inhomogeneities in the medium. A ray-tracing calculation has been used to interpret the experimental observations. It is found that geometrical considerations are sufficient to account for many of the effects observed on the shocks. © 1988, Cambridge University Press. All rights reserved.

Description: Viscoelastic theory is used to describe the response of a floating ice sheet to a moving vehicle. We adopt a two-parameter memory function to describe the behaviour of the ice, subjected to a steadily moving line or point load. The viscoelastic dissipation produces an asymmetric quasi-static response at subcritical speed, renders a finite response at the critical speed, and damps the shorter leading waves rather more severely than the longer trailing waves at supercritical speed. We extend earlier asymptotic theory to consider the anisotropic damping of the flexural waves. There is enhanced agreement between theory and experiment. © 1988, Cambridge University Press. All rights reserved.

Description: The wakes of a pair of circular cylinders are grossly unsteady when the cylinders are separated in a direction normal to the approaching flow by less than one cylinder diameter. The wakes flop randomly between two asymmetric states. The time—scale for the flopping is several orders of magnitude longer than the timescale of vortex shedding, and also several orders of magnitude longer than the timescale for instability of the separating shear layers. When a splitter plate is positioned suitably on the centreline of the cylinders, the flopping can be stopped and the flow made to assume either of the asymmetric states, or a symmetric steady state. For a range of plate positions a new, periodic oscillation occurs. Acoustic excitation can also destroy the flopping mean flow, replacing it by a symmetric flow. © 1988, Cambridge University Press. All rights reserved.

Description: A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and FjØrtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions. © 1988, Cambridge University Press. All rights reserved.

Description: Various types of moments of velocity and scalar fluctuations of the first to the fourth order have been measured and analysed. First, an orthogonal series expansion for the three-dimensional joint probability density function (p.d.f.) is developed using the cumulants and Hermite polynomials. This p.d.f. is found to provide satisfactory predictions for the statistical characteristics, including triple products, of turbulent momentum and scalar transfer. Next, the conditional sampling and averaging technique is employed to investigate the statistical characteristics of coherent turbulent transfer processes of momentum and scalar. Conditional p.d.f.s are developed for various moments of velocity and scalar up to the third order. It is shown that the present p.d.f.s can predict the detailed role of coherent motions in the dynamics of wall turbulent shear flows and in the relevant process of scalar transport by turbulence. In particular, the importance of coherent motions in the turbulent diffusion process of Reynolds-stresscomponents and scalar fluxes is demonstrated for the first time by the present theory. © 1988, Cambridge University Press. All rights reserved.

Description: The budgets for the Reynolds stresses and for the dissipation rate of the turbulence kinetic energy are computed using direct simulation data of a turbulent channel flow. The budget data reveal that all the terms in the budget become important close to the wall. For inhomogeneous pressure boundary conditions, the pressure-strain term is split into a return term, a rapid term and a Stokes term. The Stokes term is important close to the wall. The rapid and return terms play different roles depending on the component of the term. A split of the velocity pressure-gradient term into a redistributive term and a diffusion term is proposed, which should be simpler to model. The budget data are used to test existing closure models for the pressure-strain term, the dissipation rate, and the transport rate. In general, further work is needed to improve the models. © 1988, Cambridge University Press. All rights reserved.

Description: When coils carrying high-frequency currents are placed in the neighbourhood of a stream of liquid metal (or other electrically conducting fluid), the magnetic pressure on the liquid surface causes a deflection of the stream. This effect is studied for a two-dimensional stream on the assumptions that the width of the stream is small compared with the scale characterizing the applied magnetic pressure distribution, and that the effect of gravity may be neglected over this scale. The relationship between the angle of deflection of the stream and the power supplied to the perturbing currents is determined. More complex deformations associated with distributed current sources are considered. Experiments are performed in which a thin sheet of mercury is deflected by two antiparallel line currents. The agreement between theory and experiment is reasonable, despite a tendency towards three-dimensionality in the latter. A second configuration is considered in which a thin current-carrying circular jet is deflected by a vertical line current. The path of the deflected jet is calculated. The limitations of the analysis are briefly discussed. © 1988, Cambridge University Press. All rights reserved.

Description: A simple phenomenological drag relation is used to characterize weak dynamic coupling of a liquid membrane to an adjacent solid substrate. With this linear velocity-dependent drag relation, the inertialess equations of motion for membrane flow are easily solved for steady translation and rotation of a disk-like particle. The resulting drag coefficients exhibit functional dependencies on the dimensionless particle size very similar to the relations obtained for particle motion in a membrane bounded by semi-infinite liquid domains (Hughes et al. 1981), although the scaling of particle size is different. Within this phenomenological approach, diffusivities of molecular probes in membranes can be used to investigate the intrinsic molecular drag at a solid--liquid membrane interface and to estimate properties of thin lubricating liquid layers between membrane and substrate. © 1988, Cambridge University Press. All rights reserved.

Description: A numerical method employing an upwind finite-difference technique is adopted for an investigation of peristaltic pumping in circular cylindrical tubes, such as some organs in the living body. Various peristaltic flows are calculated under conditions of finite wave amplitudes, finite wavelengths and finite Reynolds numbers, and the influence of the magnitude of these quantities on the flow is investigated. The fluid mechanics of peristaltic mixing and transport are studied in detail by analysing the reflux and the trapping phenomena. The mechanical efficiency of peristaltic pumping is also discussed, with reference to engineering and physiological applications. It is shown that quantitative differences are observed between the results obtained for flows in a circular cylindrical tube and a two-dimensional plane channel. However, for both cases the appearance of peristaltic reflux depends upon the Reynolds number and the wavenumber (mean tube radius/wavelength). Much greater peristaltic mixing and transport are realized in a circular tube than in a plane channel. © 1988, Cambridge University Press. All rights reserved.

Description: A series of visualizations of non-cavitating and cavitating unsteady flows around an oscillating hydrofoil has been carried out in order to investigate the effect of unsteadiness on attached cavitation. The major conclusion of the present experimental analysis is that the strong interaction that was previously pointed out in the case of steady cavitation between an attached cavity and the boundary layer which develops upstream cavity detachment, still plays a prominent role in unsteady cavitation. We propose to generalize for the case of unsteady attached cavitation the two following points which were initially established under steady conditions and which constitute a cavitation detachment criterion: (i) a cavity detaches behind laminar separation of the boundary layer; (ii) transition to turbulence sweeps away an attached cavity. © 1988, Cambridge University Press. All rights reserved.

Description: A basis is developed for a similarity treatment of the coupled mean fields in turbulent open-channel flows over a flat sand bed in equilibrium with a suspension of sand. Hypotheses involving multiple scales, asymptotic similarity, and matching over an overlapping region are made instead of the traditional dependence on eddy diffusion and Reynolds analogies. Velocity measurements using the laser-Doppler technique in experiments with well-sorted natural sands showed that, contrary to the implications of previous models, the effect of suspended sediment on the mean velocity profile may, over a variety of laboratory conditions, be confined to a layer adjacent to the bed. This is interpreted within the similarity framework in terms of the existence of an inner lengthscale, differing from either the viscous or the grain scale, associated with the presence of sediment. The suggested lengthscales and concentration scales are examined in light of the experimental results. © 1988, Cambridge University Press. All rights reserved.

Description: A method for calculating the effect of the curvature of a solid wall bounding a viscous fluid upon the quasi-static Stokes force F and torque T experienced by a spherical particle performing arbitrarily directed translational and rotational motions in close proximity to the wall is given. The results presented herein are valid for values of the ratio = a/d (a — sphere radius, d = shortest perpendicular distance from the sphere centre to the wall) over the entire range 0 provided that ß= «1 and, simultaneously, = ß/k« 1 = characteristic radius of curvature of the wall). Unlike existing wall-effect theories, our results are valid for =0(1). It is shown that to the first-order in ß (and, concomitantly, in, wall curvature effects upon F and T depend linearly upon two scalar principal curvature coefficients of the wall at the foot of the shortest normal to the wall from the sphere centre. This single-particle analysis is used to resolve a ‘paradox’ relating to macroscopic slip boundary conditions prevailing at a wall bounding a dilute ferrofluid suspension undergoing rotation relative to a magnetic field. © 1988, Cambridge University Press. All rights reserved.

Description: Motivated by considerations of the solar toroidal magnetic field we have studied the behaviour of a layer of uniform magnetic field embedded in a convectively stable atmosphere. Since the field can support extra mass, such a configuration is top-heavy and thus instabilities of the Rayleigh-Taylor type can occur. For both static and rotating basic states we have followed the evolution of the interchange modes (no bending of the field lines) by integrating numerically the nonlinear compressible MHD equations. The initial Rayleigh-Taylor instability of the magnetic field gives rise to strong shearing motions, thereby exciting secondary Kelvin-Helmholtz instabilities which wrap the gas into regions of intense vorticity. The subsequent motions are determined primarily by the strong interactions between vortices which are responsible for the rapid disruption of the magnetic layer. © 1988, Cambridge University Press. All rights reserved.

Description: ‘Bioconvection’ is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. ‘Gyrotaxis describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are nonspherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamy-domonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5—6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth. © 1988, Cambridge University Press. All rights reserved.

Description: The flow field in a plane, turbulent mixing layer disturbed by a small oscillating flap was investigated. Three experiments were carried out: one in which the flap oscillated sinusoidally at a single frequency: a second in which the flap oscillated at two frequencies, a fundamental and a subharmonic, but the ensuing motion was dominated by the fundamental perturbation; and a third in which the amplitude of the subharmonic perturbation was increased until a distortion in the mean flow was noticeable. Two velocity components were measured at all phase angles relative to the subharmonic oscillation of the flap at densely spaced intervals. The data were used to map the phase-locked spanwise vorticity component and the phase-locked streakline patterns for the purpose of assessing the relevance of the latter to the understanding of the dynamical process involved. © 1988, Cambridge University Press. All rights reserved.

Description: When a temperature gradient is imposed on the free surface of a thin liquid layer, fluid motion can develop due to thermocapillarity. Previous work using linear theory has shown that the layer can become unstable to a pair of obliquely travelling hydrothermal waves. Here, we shall study the nonlinear behaviour of this system to determine possible equilibrium waveforms for the instability when the critical point from the linear theory is slightly exceeded. We find that for all Prandtl numbers and small Biot numbers, possible waveforms are composed of only one of the unstable linear waves. For small Prandtl number and larger Biot numbers, a combination of the two linear waves is a possible waveform. Further analysis of these equilibrium states shows that both exhibit the Eckhaus and Benjamin—Feir sideband instability and a corresponding phase instability. Thus, they become modulated on long length-and timescales as the system develops. © 1988, Cambridge University Press. All rights reserved.

Description: Experimental and theoretical studies of vibration-induced flow and mixing of dry granular materials are described. Tests were performed on rounded polystyrene beads contained in a rectangular box having transparent front and back walls and a flexible, nominally horizontal bottom which could be driven at various frequencies and amplitudes. The amplitude of the bottom vibrations was a maximum at the centre and decreased towards the vertical sidewalls. Slow recirculating flows were observed; they had the form of two vortices in which the velocity was upwards at the vertical centreline and downwards along the vertical sidewalls. The streaming velocities were measured as a function of bed vibration frequency and displacement amplitude. An explanation proposed for the recirculating flows is that the vibrating base sends ‘acoustic’ waves upwards through the bed. These waves ‘fluidize’ the granular material but are in turn attenuated because of the dissipative nature of the collisions between the ‘fluidized’ particles. Thus the slow recirculating flows in the granular material are analogous to the more familiar ‘acoustic streaming’ in air. An approximate analysis of these streaming motions is developed by making use of a modification of the constitutive theory of Jenkins & SAVAGE (1983). A number of simplifying assumptions are introduced to make the analysis tractable. The general flow patterns of the streaming motions are predicted, but the velocities are overestimated as a result of the simplifying assumptions. The analysis is restricted to a rather narrow range of conditions. © 1988, Cambridge University Press. All rights reserved.

Description: The form of steep capillary waves is of interest as a possible initial condition for the formation of air bubbles at a free surface. In this paper the limiting forms of pure capillary waves and of quasi-capillary waves are studied analytically. Crapper's finite-amplitude solution is expressed in a simple form, and is shown to be one of several exact elementary solutions to the pure-capillary free-surface condition. Among others are the solution z = w + sinh w, where w is the velocity potential, and also z= ω3. The latter solution, though it represents a self-intersecting flow, can be used as the first in a sequence of approximations to the form of the steepest wave. Hence it is shown that the influence of gravity on the shape of the limiting ‘bubble’ is very small. The result is confirmed by an examination of Hogan's numerical calculations of limiting capillary—gravity waves. In the crest of a limiting wave the particle velocity is almost constant and equal to the phase speed. This property makes it possible to apply a quasi-static approximation so as to determine the form of the crest, and hence to find an expression for the complete profile of a capillary—gravity wave of limiting steepness. It appears that there exists a solitary wave of capillary—gravity type on deep water. © 1988, Cambridge University Press. All rights reserved.

Description: The macroscopic dispersion of tracer in microscopically disordered fluid flow can ultimately, at large times, be described by an advection-diffusion equation. But before this asymptotic regime is reached there is an intermediate regime in which first and second spatial moments of the distribution are proportional to tv. Conventional advection-diffusion (which applies at large times) has v = 1 but in the intermediate regime v 〈 1. This phenomenon is referred to as ‘anomalous diffusion’ and this article discusses the special case v = 1/2 in detail. This particular value of v results from tracer dispersion in a central pipe with many stagnant side branches leading away from it. The tracer is “held up’ or ‘arrested’ when it wanders into the side branches and so the dispersion in the central duct is more gradual than in conventional advection-diffusion (i.e. v = 1/2 〈 1). This particular example serves as an entry point into a more general class of models which describe tracer arrest in closed pockets of recirculation, permeable particles, etc. with an integro-differential equation. In this view tracer is arrested and detained at a particular site for a random period. A quantity of fundamental importance in formulating a continuum model of this interrupted random walk is the distribution of stopping times at a site. Distributions with slowly decaying tails (long sojourns) produce anomalous diffusion while the conventional model results from distributions with short tails. © 1988, Cambridge University Press. All rights reserved.

Description: Mechanisms of turbulent mixing are explored by numerical simulations of onedimensional and two-dimensional mixing with Pr 〈 1. The simulations suggest that the local rate of strain γ mixes the scalar field by a t least two interacting mechanisms: the mechanism of generation, pinching and splitting of extrema proposed by Gibson (1968a)w hich acts along lines where the scalar-gradient magnitude is small; and a new mechanism of alignment, pinching and amplification of the gradients which acts along lines where the scalar-gradient magnitude is large. After extrema are generated, they split to form new extrema of the same sign, and saddle points. These zero-gradient points are connected by minimal-scalar-gradient lines which continuously stretch at rates of order γ, becoming longer than the viscous scale LK. For Pr 〈 1, this extends the influence of the local rate of strain to lengths of a t least the order of the inertial-diffusive scale LC 〉 LK; that is, larger than the maximum assumed possible by Batchelor, Howells & Townsend (1959). Roughly orthogonal maximal-scalar-gradient lines are also embedded in the fluid, and compressive mixing along these lines also reflects the magnitude and direction of the local rate of strain over distances larger than LK. Because the two rate-of-strain mixing mechanisms act along lines, they can be modelled by one-dimensional numerical simulation. Both are Prandtl-number independent and together they provide a plausible physical basis for the universal scalar similarity hypothesis of Gibson (1968b) that turbulent mixing depends on γ for all Pr. © 1988, Cambridge University Press. All rights reserved.

Description: The velocity fields beneath an air-water interface have been determined in a laboratory facility for the cases of wind-generated waves, with wind speeds ranging from 1.5 to 13.1 m/s, and of wind-ruffled mechanically generated waves of about 22 mm amplitude and 1 Hz frequency, with wind speeds ranging from 1.7 to 6.2 m/s. The velocity was measured in a fixed frame of reference with a two-component, laser-Doppler anemometer. It was possible to determine the lengthscales and evaluate the behaviour of the mean, wave-related and turbulent components of the flows. The waves affect the mean flows, even though the profiles remain essentially logarithmic and the wave field conforms generally with the results of linear theory. In the wind-wave cases the turbulent quantities behave similarly to those in flows over flat plates. They have different trends in the mechanical-wave cases, suggesting a coupling between waves and turbulence. Finally, measured values of the mean wave-induced shear stress were negative, leading to an energy transfer from the waves to the mean flow. © 1988, Cambridge University Press. All rights reserved.

Description: We consider the situation where a deep-water wavetrain approaching from infinity forms a circular caustic, is glancingly reflected at the caustic, and then propagates on out to infinity. At every point in the wavefield there are two wavetrains, the incident and reflected waves. Thus the wavefield can be treated as a slowly varying field of short-crested waves. This work generalizes that of Peregrine (1981) who considered a wavefield of incident waves only. The problem is formulated using the averaged-Lagrangian variational approach of Whitham (1974). Owing to the circular symmetry of the problem, the governing differential equations can be reduced to a set of algebraic equations at each radius. Results for the wave steepness and wavenumber are presented. These indicate that the nonlinear caustic occurs at a larger radius than does the linear caustic, and that the ray paths are no longer straight but curve away from the caustic. It is found that the slowly varying assumption is invalid at the caustic radius. To overcome this we derive, by the method of multiple scales, a modified nonlinear Schrodinger equation which is valid in this region. The solution of this equation, involving the second Painleve transcendent, is then asymptotically matched to the slowly varying solution to provide a complete description of the wavefield. © 1988, Cambridge University Press. All rights reserved.

Description: The laminar—turbulent transition of the Taylor-Görtler (TG) vortex flow in the clearance between two concentric spheres with only the inner sphere rotating (spherical Couette flow) is investigated by velocity measurement and simultaneous spectral and flow-visualization measurements by measuring the intensity of light scattered by the aluminium flakes used in flow visualization in the case of a relatively small ratio of the clearance to inner-sphere radius (clearance ratio β = 0.14). An azimuthal velocity component has been measured by a constant-temperature hot-wire anemometer at two different colatitudes (meridian angles) θ θ = 80° and 90° (the equator). A critical Reynolds number, some transition Reynolds numbers, flow regimes and flow states are obtained by the simultaneous spectral and flow-visualization measurements. The flow state is expressed by the number of toroidal TG vortex cells N, that of spiral TG vortex pairs SP, the wavenumber of the travelling azimuthal waves on the toroidal TG vortices m and the wavenumber of shear waves SH. The mean velocity distribution and the characteristic values of the fluctuating velocity, such as autocorrelation coefficient, power spectrum and turbulence intensity (r.m.s. value), are considered over a great range of Reynolds number Re. Three kinds of fundamental frequencies of the velocity fluctuation are discovered and their characteristics are clarified by means of the velocity measurement and the simultaneous spectral and flow-visualization measurements. The three kinds of fundamental frequencies expressed by fS, fW and fH correspond to the spiral TG vortices, the travelling azimuthal waves and the shear waves, respectively. These fundamental frequencies are independent of both θ and wall distances from the inner sphere, but depend strongly on Re. Although the rotation frequency of the travelling azimuthal waves (or wave speed) in the circular Couette flow decreases monotonically with increasing Reynolds number until it reaches a plateau, the values of the rotation frequencies of the spiral TG vortices, the travelling azimuthal waves and the shear waves in the spherical Couette flow, FS/Sp, FW/m and FH/SH, are nearly constant as the Reynolds number is increased, and differ slightly from one another. © 1988, Cambridge University Press. All rights reserved.

Description: The present work investigates the dynamics of two-dimensional, steady bubbly flows over a surface and inside a symmetric channel with sinusoidal profiles. Bubble dynamics effects are included. The equations of motion for the average flow and the bubbleradius are linearized and a closed-form solution is obtained. Energy dissipation due toviscous, thermal and liquid compressibility effects in the dynamics of the bubbles is included, while the relative motion of the two phases and viscous effects at the flow boundaries are neglected. The results are then generalized by means of Fourier synthesis to the case of surfaces with slender profiles of arbitrary shape. The flows display various flow regimes (subsonic, supersonic and superresonant) with different properties according to the value of the relevant flow parameters. Examples are discussed in order to show the effects of the inclusion of the various energy dissipation mechanisms on theflows subject to harmonic excitation. Finally the results for a flow over a surface with a Gaussian-shaped bump are presented and the most important limitations of the theoryare briefly discussed. © 1988, Cambridge University Press. All rights reserved.

Description: The rotating flow of a separating mixture within an axisymmetric container is considered with emphasis on the pure fluid layer adjacent to the inclined ‘bottom’ boundary from which particles are removed by centrifugal buoyancy. Within the framework of ‘mixture’ (‘diffusion’) model and when the relative density difference, the Ekman number E and particle Taylor number β are small, it is shown that the behaviour of that layer is governed by ≡ = E12α1,|cotγB|/β (where α1 is the volume fraction of the dispersed particles and yB is the elevation angle of the bottom wall with respect to the centrifugal force). If ≡ is small the layer thickens quickly into an inviscid core, in accordance with previous studies. However, novel features show up for ≡ large or O(1), w hen the viscosity-induced Ekman suction is able to counteract the separation velocity. In the former case the pure fluid layer is thin and quasisteady, and the remaining part of the interface is essentially perpendicular to the force field, in close apparent resemblance to the analog gravitational process. In the latter case, a thin quasi-steady layer and a continuously thickening core of pure fluid coexist in the same vessel. © 1988, Cambridge University Press. All rights reserved.

Description: Two plates placed in tandem in a duct flow shed vortices, which can excite and sustain an acoustic resonance associated with the duct. The sound can in turn ‘feed back and ‘lock’ the vortex shedding rate to the sound frequency. The experimental conditions under which loud resonant sound is sustained are described in this paper. The acoustic sources are predicted by combining a vortex model of the flow field with a finite-element solution of the sound field, and then using Howe's theory of aerodynamic sound to calculate the energy exchange between the flow and the sound field. Only in certain regions near the plates is substantial net energy exchange possible; the direction of energy transfer depends on the spacing of the plates. The region around the trailing edge of the upstream plate is found to be always a net acoustic source during resonance, while the region around the downstream plate is a net source or sink depending on the phase of the acoustic cycle at which vortices arrive there, which in turn depends on plate spacing and flow velocity. The net source region around the downstream plate is suppressed over a wide range of plate spacings by splitting this plate at midspan and rejoining it so that one half is offset in the flow direction by the distance a vortex travels in half asound cycle. © 1988, Cambridge University Press. All rights reserved.

Description: In a companion paper the existence of trapped waves over submerged cylinders has been analysed, and a necessary condition for their excitation was derived. In the present paper, this study is extended to obtain physically more important results. First we consider a more realistic geometry, namely a finite, although slender, cylinder. Second we derive the necessary and sufficient conditions for the excitation of trapped modes; and lastly, the induced resonant response is studied with the multiple-scales technique. It is shown then that the wave amplitude satisfies an equation similar to the resonant nonlinear oscillator. © 1988, Cambridge University Press. All rights reserved.

Description: In this paper the mathematical formulation associated with waves trapped by submerged cylinders is recast as a standard eigenvalue problem. In this way the existence is proven of trapped waves for every frequency Q and for arbitrary geometry of the submerged cylinder. At the same time a simple expression for the first eigenvalue and eigenmode, correct in the limits Ω→0 or Ω→ ∞, is derived. The expression can be a useful approximation for a structure relatively transparent to the wave action such as, for instance, a semisubmersible platform. © 1988, Cambridge University Press. All rights reserved.

Description: Two like-signed vorticity regions can pair or merge into one vortex. This phenomenon occurs if the original two vortices are sufficiently close together, that is, if the distance between the vorticity centroids is smaller than a certain critical merger distance, which depends on the initial shape of the vortex distributions. Our conclusions are based on an analytical/numerical study, which presents the first quantitative description of the cause and mechanism behind the restricted process of symmetric vortex merger. We use two complementary models to investigate the merger of identical vorticity regions. The first, based on a recently introduced low-order physical-space moment model of the two-dimensional Euler equations, is a Hamiltonian system of ordinary differential equations for the evolution of the centroid position, aspect ratio and orientation of each region. By imposing symmetry this system is made integrable and we obtain a necessary and sufficient condition for merger. This condition involves only the initial conditions and the conserved quantities. The second model is a high-resolution pseudospectral algorithm governing weakly dissipative flow in a box with periodic boundary conditions. When the results obtained by both methods are juxtaposed, we obtain a detailed kinematic insight into the merger process. When the moment model is generalized to include a weak Newtonian viscosity, we find a ‘metastable’ state with a lifetime depending on the dissipation timescale. This state attracts all initial configurations that do not merge on a convective timescale. Eventually, convective merger occurs and the state disappears. Furthermore, spectral simulations show that initial conditions with a centroid separation slightly larger than the critical merger distance initially cause a rapid approach towards this metastable state. © 1988, Cambridge University Press. All rights reserved.

Description: The symmetry-plane laminar boundary layer over an impulsively-started prolate spheroid of axes ratio 1/4 at various incidence is calculated in detail. Results agree with the steady solutions at large times. The most important one is concerned with the similarity between the distribution of the leeside skin friction at a fixed incidence, but varying in time, and that of the leeside skin friction for steady flows varying in incidence. The latter patterns led previously to the concept of an open and closed separation sequence for steady flows, likewise the newly found similarity suggests an unsteady open and closed sequence i.e. at low incidence, separation starts around the rear stagnation point and gradually expands upstream in time, but it is always of the closed type. At moderate to high incidence, closed separation prevails at small times, open separation develops at large times, but separation may either remain open at moderate incidence or return to closed at high incidence as the steady-state condition is approached. The rate of approach toward the steady-state condition increases with incidence. For a less slender spheroid there is no open separation involved; unsteady separation lines are all of the closed type. For bodies other than spheroids, similar ideas may be applied. © 1988, Cambridge University Press. All rights reserved.

Description: Groups of short waves within a narrow frequency band are known to be accompanied by second-order long waves travelling at the group velocity of the predominant short waves. When the short waves are refracted by bottom topography, new long waves can be further radiated and propagated away from the topography at the shallow-water speed. Since over a long submarine ridge there can be trapped modes of long-period waves, incident groups of short waves can excite the trapped waves through a second-order mechanism. In this paper we study such excitations over a rectangular shelf which scatters the first-order short waves. By employing asymptotic methods we examine the transient excitation of the trapped long wave by both sinusoidal wave groups and wave packets. The effects of a small angular spread of the incident waves are also included. © 1988, Cambridge University Press. All rights reserved.

Description: The flow field in a two-dimensional mixing layer, highly disturbed by a sinusoidally oscillating flap, was mapped in order to estimate the significance of the nonlinear processes associated with the large coherent structures existing in this flow. A mixing layer which does not diverge linearly in the direction of streaming is loosely defined as being highly disturbed. Two velocity components were measured throughout the flow field using a rake of X-wire probes. Streaklines were calculated from the phase-locked measured data and were compared to pictures of smoke injected into the flow, creating a link between flow visualization and quantitative experimental results. The phase-locked vorticity and the Reynolds stresses were calculated from these measurements. It was determined that fluctuations, locked in phase with the disturbance frequency, were not only responsible for the fast initial growth of the mixing layer but also for its contraction farther downstream (the occurrence of regions I and II in the parlance of Oster & WYGNANSKI 1982). The resumption of the growth of the mixing layer in region III is not controlled by the phase-locked oscillations. The first subharmonic of the imposed frequency was insignificant everywhere, and vortex amalgamation was not observed by visual means. Detailed comparisons between experimental results and theoretical calculations, based on a linear stability model, were carried out. The theory predicted very well the normalized, cross-flow distribution of any quantity that was measured, but failed to predict the amplification rates of these quantities in the direction of streaming. © 1988, Cambridge University Press. All rights reserved.

Description: We investigate numerically the time evolution of a two-dimensional flow submitted to a spatially periodic shear force. Initially, the flow is at equilibrium, the forcing balancing viscous stresses. At Reynolds numbers slightly above critical, a large-scale, linear instability drives the fluid towards a stable laminar state. At larger Reynolds number turbulence finally develops after several transient states. These transient states are described by measuring the divergence rate of linearized trajectories from the turbulent flow. This rate gives asymptotically a measure of the first Lyapunov exponent of the flow. We find that the first Lyapunov exponent scales as the characteristic frequency of the flow at large scale. We show here data on incompressible, isothermal and perfect gas (subsonic) two-dimensional flows with unit Prandtl number, and Reynolds number around 30. © 1988, Cambridge University Press. All rights reserved.

Description: Impulsively started, viscous, incompressible flows around a circular cylinder are simulated by a Lagrangian vortex solution of the vorticity equation using random walks for diffusion and the vortex-in-cell method for convection in a fractional-step scheme. Vortices are introduced around the surface at each timestep to satisfy the zero-slip condition. In the range of Reynolds numbers 2.5 x 102 to 105, comparisons with two analytical solutions, valid for small times (t 〈 1), show reasonable agreement. For somewhat longer times (t 〈 5), for a similar range of Reynolds numbers, comparisons are made with accurate Eulerian numerical solutions and with careful flow-visualization experiments. Agreement is good provided a sufficiently large number of vortices is introduced per timestep. The number required increases as Reynolds number increases. If too few are introduced, the vorticity in the wake tends to roll up too tightly. The vortex method remains stable, whereas Eulerian schemes have been reported to become eventually unstable unless upwind differencing is used, reducing accuracy. © 1988, Cambridge University Press. All rights reserved.

Description: Nonlinear oscillations and other motions of large axially symmetric liquid drops in zero gravity are studied numerically by a boundary-integral method. The effect of small viscosity is included in the computations by retaining first-order viscous terms in the normal stress boundary condition. This is accomplished by making use of a partial solution of the boundary-layer equations which describe the weak vortical surface layer. Small viscosity is found to have a relatively large effect on resonant-mode coupling phenomena. © 1988, Cambridge University Press. All rights reserved.

Description: A fluid-mechanical approach for the cleavage of biological cells is presented. The equations of motion were combined with concentration and orientation distribution balances, for active contractile filaments on the cell surface, to provide a dynamic evolution of interfacial forces and deformation. The resulting flow and the simultaneously developing surface-tension anisotropy provided a mechanism that facilitates the generation of a contractile ring at the cell equator: a major organelle in the establishment of cell furrow and the ultimate cleavage. The moving-boundary problem was solved numerically using boundary-integral representation for the Stokes equations which was modified to incorporate the anisotropic interfacial tensions. © 1988, Cambridge University Press. All rights reserved.

Description: The extension of the classic Rayleigh-Bénard problem of a horizontal layer heated from below to the three-dimensional convection in rectangular boxes is dealt with in this paper both numerically and experimentally. Also discussed is the influence of shear flows in tilted boxes and the transition to time-dependent oscillatory convection. Three-dimensional numerical simulations allow the calculation of stationary solutions and the direct simulation of oscillatory instabilities. We limited ourselves to laminar and transcritical flows. For studying the particular characteristics of three-dimensional convection in horizontal containers, we carefully selected two container geometries with aspect ratios of 10:4:1 and 4:2:1. The onset of steady cellular convection in tilted boxes is calculated by an iterative application of a combined finite-difference method and a Galerkin method. The appearance of longitudinal and transverse convection rolls is determined by means of interferometrical measuring techniques and is compared with the results of the linear stability theory. The spatial flow structure and the transition to oscillatory convection is calculated for selected examples in the range of supercritical Rayleigh numbers. Experimental investigations concerning the stability behaviour of the steady solutions with regard to time-dependent disturbances show a distinct influence of the Prandtl number and confirm the importance of nonlinear effects. © 1988, Cambridge University Press. All rights reserved.

Description: Motion pictures taken a t Duck Creek, a clear stream 6.5 m wide and 35 cm deep near Pinedale, Wyoming, provide detailed, quantitative information on both the modes of motion of individual bedload particles and the collective motions of large numbers of them. Bed shear stress was approximately 6 Pa (60 dynes cm-2)w, hich was about twice the threshold for movement of the 4mm median diameter fine gravel bed material ; and transport was almost entirely as bedload. The displacements of individual particles occurred mainly by rolling of the majority of the particles and saltation of the smallest ones, and rarely by brief sliding of large, angular ones. Entrainment was principally by rollover of the larger particles and liftoff of the smaller ones, and infrequently by ejection caused by impacts, whereas distrainment was primarily by diminution of fluid forces in the case of rolling particles and by collisions with larger bed particles in the case of saltating ones. The displacement times averaged about 0.2–0.4 s and generally were much shorter than the intervening repose times. The collective motions of the particles were characterized by frequent, brief, localized, random sweep- transport events of very high rates of entrainment and transport, which in the aggregate transported approximately 70% of the total load moved. These events occurred 9 % of the time a t any particular point of the bed, lasted 1-2 s, affected areas typically 20-50 cm long by 10-20 cm wide, and involved bedload concentrations approximately 10 times greater than background, The distances travelled during displacements averaged about 15 times the particle diameter. Despite the differences in their dominant modes of movement, the 8- 16 mm particles typically travelled only about 30 YO slower during displacement than the 2–4 mm ones, whose speeds averaged 21 cm2 s-l. Particles starting from the same point not only moved intermittently downstream but also dispersed both longitudinally and transversely, with diffusivities of 4.6 and 0.26 cm2 s-l, respectively. The bedload transport rates measured from the films were consistent with those determined conventionally with a bedload sampler. The 2–4 mm particles were entrained 6 times faster on finer areas of the bed, where 8-16 mm particles covered 6 YO of the surface area, than on coarser ones, where they covered 12 %, even though 2 4 and 4-8 mm particles covered practically the same percentage areas in both cases. The 4-8 and 8-16 mm particles, in contrast, were entrained a t the same rates in both cases. To within the statistical uncertainty, the rates of distrainment balanced the rates of entrainment for all three sizes, and were approximately proportional to the corresponding concentrations of bedload. © 1988, Cambridge University Press. All rights reserved.

Description: The interface at the top of a dilute sedimenting suspension of small particles which are not identical does not remain sharp but instead becomes increasingly diffuse as the sedimentation proceeds. For more concentrated suspensions, the self-sharpening effect of hindered settling leads to a considerable reduction in the observed spreading of the sedimenting interface. In order to quantify this spreading, a light extinction technique was used to measure the concentration profile in the interface of a suspension of particles with a small spread of sizes as it fell past a thin sheet of light. A particle volume-fraction range of 0.002 〈=ϕ0〈 =0.15 was examined, and each fluid-particle system had a particle Reynolds number less than 10—3 and a Péclet number greater than 107 so that inertia and colloidal effects were negligible. Calculations of the spreading arising from the small degree of polydispersity in particle sizes and the self-sharpening effect are presented. Surprisingly, the measured vertical thickness of the interface was found to be several times that predicted from this theory. It is proposed that the observed spreading may be attributed to hydrodynamic interactions between particles that lead to fluctuations in particle settling velocities about the mean. An analysis of the data shows that the measured interface thickness, after subtracting off that predicted from polydispersity and self-sharpening, increases approximately with the square root of the settling distance and may therefore be described as a diffusion process, termed‘self-induced hydrodynamic diffusion’. By scaling the hydrodynamic diffusivity as D = au1/2D(〈P0), where u1/2is the median hindered settling velocity, a is the median particle radius, and 〈ϕ0is the volume fraction of particles well below the interface, an approximate analysis of the data was used to infer that the dimensionless scaled diffusion coefficient, D, is between 1 and 2 for the smaller particle volume fractions examined, increases very rapidly with increasing concentration to a value between 10 and 15 for particle concentrations of a few percent by volume, and then levels off or declines slightly as the particle concentration is increased further. © 1988, Cambridge University Press. All rights reserved.

Description: Mass transfer between a flowing region and an adjacent stationary medium can greatly alter the overall contaminant dispersion. Here, an extension is given of Taylor’s (1953) method to encompass this class of complications. The only mathematical assumption made is that the mass flux transfer at the boundary depends linearly upon the concentration at earlier times. Expressions are derived for the longitudinal shear dispersion coefficient. Detailed results are presented for the effects both of reactions and of retention at the bed upon contaminant dispersion in turbulent open-channel flow. © 1988, Cambridge University Press. All rights reserved.

Description: The behaviour of an evolving, stably stratified turbulent shear flow was investigated in a ten-layer, closed-loop, salt-stratified water channel. Simultaneous single-point measurements of the mean and fluctuating density and longitudinal and vertical velocities were made over a wide range of downstream positions. For strong stability, i.e. a mean gradient Richardson number Rigreater than a critical value of Ricr0.25, there is no observed growth of turbulence and the buoyancy effects are similar to those in the unsheared experiments of Stillinger, Helland & VAN Atta (1983) and Itsweire, Helland & VAN Atta (1986). For values of Richardson number less than Ricrthe turbulence grows at a rate depending on Riand for large evolution times the ratio between the Ozmidov and turbulent lengthscale approaches a constant value which is also a function of Richardson number. Normalized velocity and density power spectra for the present experiments conform to normalized spectra from previous moderate to high-Reynolds-number studies. With increasing r = (x/Ū)(dŪ/dz) or decreasing stability, the stratified shear spectra exhibit greater portions of the universal non-stratified spectrum curve. The shapes of the shear-stress and buoyancy-flux cospectra confirm that they act as sources and sinks for the velocity and density fluctuations. © 1988, Cambridge University Press. All rights reserved.

Description: The problem of whether a stream of microscopic particles may be concentrated into a focal point by entrainment within a carrier gas is considered for dilute particles linearly coupled to the velocity field of an incompressible gas. Typically, the dynamical behaviour of the particles is governed by a so-called Stokes number S, the product of their relaxation time and a characteristic value of the velocity gradient in the suspending fluid. An inequality due to Robinson (1956) is used to illustrate the natural tendency of potential flows to concentrate the particles. For geometries with planar or axial symmetry, with errors cubic in their initial distance to the axis, the trajectories of identical particles originating near an axis of symmetry are shown to cross it at a common focal point provided they have some initial convergence and their Stokes number is larger than a critical value S*. The position of the focal point of supercritical particles depends on their Stokes number, tending to infinity as S approaches S*. Particle trajectories originating far from the axis of symmetry are seen to cross the centreline at defocused positions, in analogy with the optical geometric aberration effect. The focusing phenomenon is illustrated numerically for two-dimensional potential flows through nozzles of several geometries and also analysed in the proximity of the axis of symmetry. For these examples, the threshold value S* of the Stokes number for focusing is of order one, over an order of magnitude larger than typical values of the familiar critical Stokes number marking the onset of particle impaction on solid surfaces. The focal width may be made over two orders of magnitude smaller than the nozzle diameter by restricting the region where particles are seeded to a moderate angle away from the axis. This angle may be higher than 1/4π for the case of a jet exiting through a slit in an infinitely thin plate. There is also some discussion of the use of high-resolution focusing instruments. © 1988, Cambridge University Press. All rights reserved.

Description: Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied where N is equal to E in terms of the Ekman number E and Rossby number Ro. For this parameter range the boundary layer contains a singularity at the rear stagnation point. The asymptotic structure of this singularity is shown to consist of three distinct asymptotic regions, one of which is viscous while the others are inviscid. New accurate numerical solutions of the boundary-layer equation confirm this singularity structure. The use of Von Mises coordinates both simplifies the analysis, and enables numerical solutions to be found closer to the critical value N = 1, beneath which the flow separates upstream of the rear stagnation point. © 1988, Cambridge University Press. All rights reserved.

Description: A fully nonlinear model is developed for the unidirectional propagation of periodic gravity wave groups in deep water, in which the shape of the group envelopes changes cyclically. It is intended to describe the slow-time evolution of wave groups on the open ocean surface, and to generalize the cyclic recurrence that can occur during the sideband modulation of Stokes waves and Schrödinger wave groups. The weak nonlinear interactions are shown to concentrate the wave energy at the centre of each group at regular intervals, causing the waves there to be of greater height locally in space and time. This is suggested as one mechanism for the local wave breaking that is observed on the open ocean surface. The cyclically recurring wave groups may be interpreted as the limit-cycle stage in a progression from uniform wave groups to chaos on the forced, damped, ocean surface. © 1988, Cambridge University Press. All rights reserved.

Description: The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer • dissipation is compared with recent experimental results. © 1988, Cambridge University Press. All rights reserved.

Description: Unsteady effects in a separated recirculating eddy beneath a subsonic or supersonic mainstream are considered, with emphasis on nonlinear properties. The eddy is slender and predominantly inviscid, its length being comparable with, or less than, the chord of an airfoil, for instance. It is found, from the study of a family of integrodifferential equations, that the planar eddy can break up nonlinearly within a finite time, causing an eruption into the main stream and setting up a subsequent Euler stage in the unsteady motion. Comparisons with recent experiments and further applications of the theory are discussed. © 1988, Cambridge University Press. All rights reserved.