ALBERT

Description: Fully three dimensional surface gravity waves in deep water are investigated in the limit in which the length of the wave crests become long. We describe an analytic solution to fourth order in wave steepness, which matches onto known short crested wave solutions on the one hand and onto the well known two dimensional progressive wave solution on the other. In the progressive wave limit a particular solution in which the wave crests are semi finite is given to sixth order accuracy. These solutions are part of a more general set of solutions which are found from a nonlinear Schrodinger equation. © 1983, Cambridge University Press. All rights reserved.

Description: Using a ‘ clean ’ jet facility the relationship between the jet flow and its radiation field was studied experimentally in the Mach-number range 0.05 〈 Mj〈 0.20 and a Reynolds-number range 6 × 104〈 ReD〈 2.3 × 105. The various acoustic source parameters such as strength, frequency and Mach number were varied systematically, and the far-field pressure measured simultaneously. On the basis of these measurements the nature of the sources in the initial shear layer could be characterized. The principal results, equally valid for unexcited and excited jets, are as follows: the acoustic sources are not convected but are located within a confined volume fixed with respect to the nozzle even though they are being generated by moving disturbances in the jet; they are associated with the nonlinear saturation of the unstable wave amplitudes of the shear layer occurring at the vortex-pairing locations; the radiation intensity varies nonlinearly with the source strength and is highly directional, exponential in character. © 1983, Cambridge University Press. All rights reserved.

Description: Density currents in a rotating fluid are produced by releasing a volume of buoyant fluid from a lock at one end of a long rotating channel. Coriolis forces hold the current against one wall. It is observed that the velocity and depth of the nose decrease exponentially in time, implying that the nose can effectively come to a halt at a finite distance from the lock. In reality though, the flow regime eventually changes and a viscous wedge-shaped intrusion continues. The high-Reynolds-number currents contain three-dimensional turbulence a short distance behind the nose, but the influence of rotation causes this to become quasi-two-dimensional further upstream. The intrusion and turbulent motions represent a forcing to the lower layer that produces vortex and wave-like motions which penetrate deep into the lower-layer fluid. It is shown that the exponential decay can be attributed to radiation of momentum by these inertial waves. The width l of the turbulent current varies with distance behind the nose, from 0.6 times the local time-dependent deformation radius at the ‘head’ to l ≈ R0far upstream, where R0is the initial deformation radius in the lock. The nose of the boundary current is unstable, with billows appearing near the tip of the intruding nose and leading to an intermittent breakup of the ‘head ’ structure and oscillations of the nose velocity. These oscillations are rapid, often having frequencies much greater than f (where f = 2Ω is the Coriolis parameter), and, along with the production of the turbulence that is so characteristic of the currents, are attributed to a Kelvin-Helmholtz instability. Rotationally dominated baroclinic waves appear only a very large distance behind the nose. © 1983, Cambridge University Press. All rights reserved.

Description: Experiments were conducted with three delta wings and two rectangular wings to investigate the evolution of trailing vortices in stratified and unstratified water. The vortex trajectories were determined as a function of the normalized time V0t/b0stratification parameter Nb0/V0and an effective vortex core size re/b0. The results have shown that the vortices rise only to a finite height as they decay gradually at first and rapidly thereafter under the influence of turbulence, sinusoidal instability, and core bursting. The effect of stratification is to reduce the lifespan of vortices and the maximum height attained by them. © 1983, Cambridge University Press. All rights reserved.

Description: Variations in wave amplitude and steepness across current jets have been described for broad smooth-sided jets. Here, narrow jets are modelled by a top-hat pattern; an approximate solution is found for linear waves, based on conservation of wave action and vertical averaging. The results join remarkably well to those for a broad cosine-shaped jet (cf. McKee 1975). For jet widths less than about a third of a wavelength, there is little change in amplitude; the enhancement predicted by a WKB analysis is suppressed owing to interference with reflections from the far side of the jet. For directional spectra not too different from cos2θ, some suppression occurs near the middle of jets of all scales, owing to exclusion of the glancing wave components by reflection from the near side; this suppression can be significant for jets more than a wavelength wide. For monochromatic waves, maximum amplitudes occur some distance outside the jet owing to interference; the net reflection appears to have a positive phase shift along the near caustic of about ¼cycle. © 1983, Cambridge University Press. All rights reserved.

Description: In a wind tunnel designed for flow-acoustic measurements, the wall-pressure fluctuations beneath a turbulent boundary layer have been investigated. The measurements were carried out with variously sized pressure transducers (19 ≤ d+≤ 333) and with an array of four small transducers (separation distance Δx+= 75). It is shown that the dimensionless diameter d+= 19 of the transducers is sufficient to resolve the essential structures of the turbulent pressure fluctuations. The power spectrum Φ(ω+) measured with the smallest transducer d+= 19 partly exhibits power-law decay F ~ ω⅞, which has been theoretically predicted for locally isotropic turbulence. By visual analysis and signal averaging in the time domain, pressure structures with high amplitudes could be detected which have the shape of short wavetrains or pulses. Their characteristic frequency and longitudinal wavelength have the mean values ω+= 0.52 and A+= 145 respectively, and their mean convection velocity amounts to uc/u∞= 0.53. It was calculated from the measured probability density that these characteristic structures play an important role, although the probability of their occurrence is low. The sources of these wall-pressure structures can be located in the buffer layer of the boundary layer. © 1983, Cambridge University Press. All rights reserved.

Description: When a tsunami is incident on a plane shelf, the waveform and amplitude of the reflected wave differ significantly from those of the incident wave, and, when the shelf has the extremely irregular character of real continental shelves, the contrast between the two amplitudes is even greater. Here we quantify these statements and we also give an account of the manner in which the run-up depends on the obliquity and on other parameters characterizing the incident wave. © 1983, Cambridge University Press. All rights reserved.

Description: In this paper we investigate nonlinear interactions of narrowband, Gaussian-random, inhomogeneous wavetrains. Alber studied the stability of a homogeneous wave spectrum as a function of the width a of the spectrum. For vanishing bandwidth the deterministic results of Benjamin & Feir on the instability of a uniform wavetrain were rediscovered whereas a homogeneous wave spectrum was found to be stable if the bandwidth is sufficiently large. Clearly, a threshold for instability is present, and in this paper we intend to study the long-time behaviour of a slightly unstable modulation by means of a multiple-timescale technique. Two interesting cases are found. For small but finite bandwidth a the amplitude of the unstable modulation shows initially an overshoot, followed by an oscillation around the time-asymptotic value of the amplitude. This oscillation damps owing to phase mixing except for vanishing bandwidth because then the well-known Fermi-Pasta-Ulam recurrence is found. For large bandwidth, however, no overshoot is found since the damping is overwhelming. In both cases the instability is quenched because of a broadening of the spectrum. © 1983, Cambridge University Press. All rights reserved.

Description: Laser-Doppler velocity measurements were obtained in water between finite rotating disks, with and without throughflow, in four cases: ω1= ω2= 0; ω2/ω1=- 1; ω2/ω1=0; ω2/ω1= 1. The equilibrium flows are unique, and at mid-radius they show a high degree of independence from boundary conditions in r. With one disk rotating and the other stationary, this mid-radius ‘limiting flow ’ is recognized as the Batchelor profile of infinite-disk theory. Other profiles, predicted by this theory to coexist with the Batchelor profile, were neither observed experimentally nor were they calculated numerically by the finite-disk solutions, obtained here via a Galerkin, B-spline formulation. Agreement on velocity between numerical results and experimental data is good at large values of the ratio RQ/Re, where Rq= Q/2πVs Is the throughflow Reynolds number and Re= R22ω/v is the rotational Reynolds number. © 1983, Cambridge University Press. All rights reserved.

Description: Force measurements were conducted in a pressurized wind tunnel from subcritical up to transcritical Reynolds numbers 2.3 x 104≤ Re ≤7.1 x 106without changing the experimental arrangement. The steady and unsteady forces were measured by means of a piezobalance, which features a high natural frequency, low interferences and a large dynamic range. In the critical Reynolds-number range, two discontinuous transitions were observed, which can be interpreted as bifurcations at two critical Reynolds numbers. In both cases, these transitions are accompanied by critical fluctuations, symmetry breaking (the occurrence of a steady lift) and hysteresis. In addition, both transitions were coupled with a drop of the CDvalue and a jump of the Strouhal number. Similar phenomena were observed in the upper transitional region between the super- and the transcritical Reynolds-number ranges. The transcritical range begins at about Re ≈ 5 x 106, where a narrow-band spectrum is formed with Sr(Re = 7.1 X 106) = 0.29. © 1983, Cambridge University Press. All rights reserved.

Description: An analysis is given of the initial development of the lift on an aerofoil in inviscid starting flow. It is shown that because of the spiral shape of the vortex sheet shed initially from the trailing edge the lift and drag are both singular at the start of impulsive motion. This result is in contrast with the prediction of finite forces by methods that assume the vortex sheet to be initially planar. The effect of a steady rate of change of incidence following the sudden onset of transverse (heaving) motion of an aerofoil in a steady stream is also discussed. © 1983, Cambridge University Press. All rights reserved.

Description: An experimental study of the influence of liquid viscosity and viscoelasticity on flow-generated waves on a compliant surface has been conducted in a rotating-disk geometry. Over the entire range of liquid properties studied, each test gave a well-defined critical onset flow velocity above which waves were present and below which no waves were observed. This onset velocity increased with increasing fluid viscosity, and for sufficiently high viscosities the onset occurred when the flow on the disk was laminar rather than turbulent. The effects of liquid viscoelasticity were examined in the turbulent flow using dilute solutions of high-molecular-weight polymers. This type of viscoelasticity had little influence on the onset flow velocity in these circumstances, but did make the wave structure on the surface more regular in appearance than when the liquid was Newtonian. In all cases the wave structure produced a dramatic increase in drag similar to that expected for a rough surface. For the viscoelastic fluid, however, the increase in drag was much less than for a viscous fluid of the same viscosity. © 1983, Cambridge University Press. All rights reserved.

Description: The output statistics of a laser anemometer operating in a low particle density are discussed. A rigorous derivation is given for the influence of two popular data-handling algorithms on these statistics. In particular it is shown that the measured statistics can differ from those of the flow statistics and from the particle-arrival statistics. The variables that control the statistical regime are derived and quantitative estimates are given for their ranges of influence. The first system discussed is a sample-and-hold system where the output is a piecewise-continuous signal obtained by holding the last processor measurement until a new one is obtained. The second system is one where an attempt is made to store all the measurements for processing, but which contains a rate-limiting device. Because of this device, some measurements may be lost when the particle rate is high. This system is referred to as a saturable system. In both cases it is found that the statistics of the output depend on the product of the mean particle rate and the flow correlation time as well as the flow statistics. The statistics of the saturable system also depend on the ratio of the mean particle rate to the maximum rate at which measurements can be accepted by the system. Because of this, the statistics of both systems depend on the particle density. Attainable conditions are demonstrated, where the output velocity measurement statistics are essentially identical with the flow statistics. © 1983, Cambridge University Press. All rights reserved.

Description: A two-dimensional Stokes flow close to the line of contact of two touching cylinders or three-dimensional axisymmetric Stokes flow close to the point of contact of two touching bodies is shown in general to separate into infinite sets of eddies with angles of separation from the bodies which tend to 58.61° as the line or point of contact is approached. The flow near the vertex of a conical cusp is shown to be a system of nested toroidal vortices and the separation angles tend to 45.25° as the vertex is approached. Stokes flow between parallel planes or within a circular cylinder is shown in general to separate far from the generating disturbances with cellular eddy structure and separation angles which tend to 58.61° and 45.25° respectively. The mathematical equivalence of the various problems is established. © 1983, Cambridge University Press. All rights reserved.

Description: The evolution of a weakly nonlinear, weakly dispersive gravity wave in water of depthd over a bottom of gradual slope δ and Chezy friction coefficient Cfis studied. It is found that an initially sinusoidal wave evolves into a periodic sequence of solitary waves with relative amplitude a/d = α1—15δ/4Cfif α1〈 αb, where αbis the relative amplitude above which breaking occurs. This prediction is supported by observations (Wells 1978) of the evolution of swell over mudflats. © 1983, Cambridge University Press. All rights reserved.

Description: A porous-wavemaker theory is developed to analyse small-amplitude surface waves on water of finite depth, produced by horizontal oscillations of a porous vertical plate. Analytical solutions in closed forms are obtained for the surface-wave profile, the hydrodynamic-pressure distribution and the total force on the wavemaker. The influence of the wave-effect parameter C and the porous-effect parameter G, both being dimensionless, on the surface waves and on the hydrodynamic pressures is discussed in detail. © 1983, Cambridge University Press. All rights reserved.

Description: A planar liquid layer is bounded below by a rigid plate and above by an interface with a passive gas. A steady shear flow is set up by imposing a temperature gradient along the layer and driving the motion by thermocapillarity. This dynamic state is susceptible to two types of thermal-convective instabilities: (i) stationary longitudinal rolls, which involve the classical Marangoni instability studied by Pearson; and (ii) unsteady hydrothermal waves, which involve a new mechanism of instability deriving its energy from the horizontal temperature gradients. Thermal stability characteristics for liquid layers with and without return-flow profiles are presented as functions of the Prandtl number of the liquid and the Biot number of the interface. Comparisons are made with available experimental observations. © 1983, Cambridge University Press. All rights reserved.

Description: Corrections to Stokes' law are determined to first order in a/b and a/h for a sphere of radius a in a one-dimensional array of identical spheres having centre-to-centre-spacing b and translating a distance h from a no-slip wall. When h/b is small the drag is greater than that given by Stokes' law; as h/b increases, the drag generally decreases and becomes less than that given by Stokes' law. Stability of the array is examined. Motion along the line of centres is found to be stable, but the other two motions are unstable. The wall is a stabilizing influence when motion is toward the wall and a destabilizing influence when motion is away from the wall. For motion parallel to the wall, the presence of the wall shifts the region of maximum instability to smaller wavelengths. Crowley’s results, which neglect any influence of the wall, are approached for h/b greater than about 5. © 1983, Cambridge University Press. All rights reserved.

Description: The 2.5-dimensional model of the turbulent field near a wall, proposed by Hatzia-vramidis & Hanratty (1979) and modified by Chapman & Kuhn (1981), has been used to test the subgrid models of Schumann (1973, 1975) and Moin & Kim (1982). The results are disquieting, both trends and orders of magnitude sometimes being seriously in error. It also appears that the contribution of the subgrid energy to the pseudopressure calculated in large-eddy simulations can be large, although this contribution is usually neglected. On the positive side, Leonard's model for the Leonard stress is extremely good, and Schumann’s synthetic boundary condition is also found to be reliable. These results must be taken with a grain of salt, since the tests reported in §5 show that the 2.5-dimensional model cannot reproduce important characteristics of the turbulence in the neighbourhood of y+. = 40. © 1983, Cambridge University Press. All rights reserved.

Description: A study has been made of how initially planar shocks in air propagate around 90° bends in channels of nearly rectangular cross-section. In shallow bends for which the radius of curvature R is much greater than the radius r of the channel, the shock recovers from a highly curved profile at the start of the bend to regain planarity towards the end of the bend. This occurs on account of the acceleration of the triple point across the channel following its interaction with the expansion waves generated at the convex wall. In sharp bends the shock profiles retain their pronounced curvature for some distance downstream of the bend. At the start of a shallow bend (R/r ≈ 6) the shock at the concave wall, initial Mach number M0, accelerates to Mw. = 1.15M0. and remains at this value until towards the end of the bend it begins to attenuate. At the convex wall, shocks of M0〉1.7 attenuate to Mw= 0.7 M0and propagate at this value for some distance around the bend. In the early stages of a sharper bend (R/r ≈ 3) the shock at the concave wall strengthens to Mw= 1.3M0, remaining at this value for some distance downstream of the bend. At the convex wall the shock decelerates to 0.6M0. Whitham’s (1974) ray theory is shown to predict with reasonable accuracy the Mach numbers of the wall shocks at both surfaces for both bends tested and the range of incident shock velocities used, 1.2 〈 M0〈 3. The agreement between the theory and experimental results is particularly close for stronger shocks propagating along the inner bend. Predictions from 3-shock theory (Courant & Friedrichs 1948) of the Mach number at the outer wall are consistently higher than those from Whitham’s analysis. In turn, the latter tends to slightly overestimate the strength of the wall shock. A model is developed, based on an extension of Whitham’s analysis, and is shown to predict the length of the Mach stem produced by shocks of M0. 〉 2 over the initial stages of the bend. © 1983, Cambridge University Press. All rights reserved.

Description: This paper discusses the stability of the flow of a low-Prandtl-number liquid contained in a shallow slot with differentially heated vertical endwalls. The effect of thermally insulating boundaries at the top and bottom of the container on wavelength selection is emphasized. Stability calculations indicate that, for Prandtl number Pr in the range 0.015 〈 Pr 〈 0.27, the first perturbations to grow are overstable (oscillatory) longitudinal rolls with axes perpendicular to the endwalls, and with very large cross-stream wavelengths of about 9 to 15 layer depths. Previous studies using thermally conducting boundaries predict critical wavelengths of about three layer depths. The new results are in substantial agreement with an experiment using a differentially heated layer of mercury with aspect ratio (depth/length) 0.047 in both horizontal directions. The implications of the long-wavelength instability for the interpretation of thermal oscillations observed in other smaller-aspect-ratio configurations is discussed. © 1983, Cambridge University Press. All rights reserved.

Description: We study the primary electroviscous effect in a suspension of spheres when the double layer thickness k−1. is small compared with the particle radius a. The case of a 1–1 symmetric electrolyte is examined using the methods of Dukhin & coworkers (1974), whilst the asymmetric electrolyte is studied along lines similar to those of O'Brien (1983). Sherwood's (1980) asymptotic results for high surface potentials and high Hartmann numbers are extended and complemented. © 1983, Cambridge University Press. All rights reserved.

Description: The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission; however, reflection is not total owing to the finiteness of the obstacle. Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution. © 1983, Cambridge University Press. All rights reserved.

Description: A mathematical representation has been developed for the electromagnetic force field and the fluid-flow field in a coreless induction furnace. The fluid flow field was represented by writing the axisymmetric turbulent Navier-Stokes equations, containing the electromagnetic body-force term. The electromagnetic body force field was calculated by using a technique of mutual inductances. The k-e model was employed for evaluating the turbulent viscosity, and the resultant differential equations were solved numerically. The theoretically predicted velocity fields were in reasonably good agreement with the experimental measurements reported by Hunt & Moore; furthermore, the agreement regarding the turbulence intensities was essentially quantitative. These results indicate the k-e model does provide a good engineering representation of the turbulent recirculating flows occurring in induction furnaces. At this stage it is not clear whether the discrepancies between measurements and the predictions, which were not very great in any case, are attributable either to the model or to the measurement techniques employed. © 1983, Cambridge University Press. All rights reserved.

Description: Uniform flow in curved, wide, erodible-bed channels is formulated on the basis of the conservation of flux of moment-of-momentum, to obtain relations for the vertical distributions of radial-plane velocity and radial shear stress. The expression for the radial stress exerted on the bed is utilized in a force-equilibrium analysis of the moving bed layer to obtain relations for the average transverse slope of the bed and for the radial bed profile. The reduction of primary bed shear stress due to the net radial transport of streamwise momentum toward the outer (concave) bank is then calculated, by introducing the derived expressions for the velocity components into the momentum equation for the primary-flow direction. It is found that the stress reductions in deep, narrow channels can exceed 50%. Bed profiles and velocity distributions measured in natural and laboratory streams are found to be in good conformity with those calculated from this moment-based theory.

Description: The effect of a small amount of Brownian diffusion on shear-induced coagulation of spherical particles has been calculated. This has been accomplished by considering the binary collision process between a test sphere and identical spheres interacting with the test sphere through induced-dipole attraction, electrostatic repulsion and hydrodynamically induced forces. The effect of diffusion is found by means of an expansion in inverse Péclet number. Specific calculations were performed for uniaxial extension and for laminar shear flow. It is found that Brownian diffusion, the effect of which is nonlinearly coupled with flow type and strength, can act to increase or decrease the coagulation rate. © 1983, Cambridge University Press. All rights reserved.

Description: The time-dependent evolution of sheared Rossby waves starting from an initial disturbance is studied for the simple case in which the shear is uniform. The uniform-shear assumption allows explicit solutions to be obtained which are useful in addressing the issue of the long-time asymptotic approach to normal modes and in assessing the relative importance of viscosity, nonlinearity and time-dependence in the evolution of Rossby waves in the presence of critical layers. © 1983, Cambridge University Press. All rights reserved.

Description: In this analysis, the translation of a liquid drop experiencing a strong non-uniform radial velocity has been investigated. The situation arises when a moving liquid drop experiences condensation, evaporation or material decomposition at the surface. By simultaneously treating the flow fields inside and outside the drop, we have obtained physical results relevant to the problem. The magnitude of the radial velocity is allowed to be very large, but the drop motion is restricted to slow translation. The solution to the problem has been developed by considering a uniform radial flow with the translatory motion introduced as a perturbation. The role played by the inertial terms due to the strong radial field has been clearly delineated. The study has revealed several interesting features. An inward normal velocity on a slowly moving drop increases the drag. An increasing outward normal velocity decreases the drag up to a minimum beyond which it increases. The total drag force not only consists of contributions from the viscous and the form drags but also from the momentum transport at the interface. Since the liquid drop admits a non-zero tangential velocity, the tangential momentum convected by the radial velocity forms a part of this drag force. The circulation inside the drop decreases (increases) with an outward (inward) normal velocity. A sufficiently large non-uniform outward velocity causes the circulation to reverse. In the limit of the internal viscosity becoming infinite, our analysis collapses to the simple case of a translating rigid sphere experiencing a large non-uniform radial velocity. By letting the radial velocity become vanishingly small the Stokes-flow solution is recovered. An important contribution of the present study is the identification of a new singularity in the flow description. It accounts for both the inertial and the viscous forces and displays Stokeslet-like characteristics at infinity. © 1983, Cambridge University Press. All rights reserved.

Description: This paper describes a laboratory experiment designed to compare measurements with published theoretical ideas of the mixed-layer growth of a two-layer system in which the turbulence is induced by an oscillating grid. Experimental results show excellent agreement with an earlier theory by one of us (Long), in which the mixed-layer depth D* measured from a virtual origin is given by $D_{*}sim V_0^{-frac{7}{11}}K^{frac{9}{11}}t^{frac{2}{11}}$, where K is action, t is time and V0 is a characteristic velocity of the problem. The experiments also verify Long's theoretical entrainment relation E = α2Ri−7/4, where E is the entrainment coefficient and $Ri = D^3_{*}Delta b/K^2$, and Δb is the buoyancy difference between the two layers. The interfacial-layer thickness was observed to be proportional to the depth of the mixed layer, as also predicted by Long. After a certain depth, the entrainment law tends to deviate from Long's theory. The deviation may be due to wall effects.

Description: Experiments were conducted to determine the effective diffusivity for axial transport through a tube of circular cross-section of a contaminant gas in oscillatory flow. Results were compared with the theoretical predictions of Watson (1983) and found to be in excellent agreement. The experiments differ from the theoretical situation in that the oscillations are superimposed upon a steady flow due to a constant infusion of tracer gas, and a buoyancy-induced flow associated with spatial variations in gas density. The influence of both artifacts is found to be negligible.

Description: A novel, digital, hot-wire anemometer technique for the simultaneous measurement of the instantaneous streamwise and lateral velocity fields in high-intensity turbulent flows is discussed. It involves the use of a three-wire probe comprising two 45° slanted hot wires and a normal hot wire. A comprehensive and systematic examination of several factors that can affect the fidelity of the streamwise and lateral velocity waveforms is developed to assess the performance of the new technique as well as hot-wire systems generally. These factors are: (i) rectification, which stems from the inherent insensitivity of hot wires to the direction of the instantaneous (total) velocity vector in a turbulent flow; (ii) spanwise velocity fluctuations; (iii) axial cooling of hot wires; (iv) unpredictable variations in one of four hot-wire calibration parameters; (v) random hot-wire calibration errors; (vi) spanwise separation of the hot wires. Relevant hot-wire anemometer-response equations relating instantaneous anemometer output voltages to instantaneous flow velocities were established on the basis of extensive voltage-velocity calibration data pertaining to hot wires orientated with respect to the calibration flow velocity at various yaw and pitch angles ranging from 0° to 90°. Simulated Gaussian (streamwise, lateral and spanwise) velocity fields appropriate to flows with turbulence intensity levels varying between 5 and 80% and Reynolds shear-stress coefficients varying between 0.1 and 0.5 were generated by means of a digital computer, and the associated anemometer-voltage signals computed in accordance with the response equations subject to different combinations of the first four of the aforementioned factors. In order to take into account the effects of the last two factors, viz calibration errors and spanwise wire separation, uncorrelated Gaussian ‘noise’ fluctuations were superimposed on the above voltage signals. Estimates of the known (simulated) streamwise and lateral velocity signals were then determined by simultaneous solution of (a) the actual instantaneous response equations, (b) approximate versions of them, and (c) linearized versions of them. The results indicate that reasonably accurate estimates of velocity signals from a turbulent flow can be obtained by means of conventional hot-wire anemometer techniques – which assume that anemometer voltage fluctuations are linear functions of corresponding velocity fluctuations – only if the turbulent intensity level of the flow does not exceed about 20%. In marked contrast, the 3-wire anemometer technique introduced here can be used to measure streamwise and lateral velocity signals simultaneously with a high degree of accuracy for turbulence-intensity levels of up to 40%. In addition, this technique is capable of yielding high-fidelity streamwise velocity waveforms for levels in excess of 70%.

Description: Two-dimensional flow past a cylindrical body of arbitrary profile at small Reynolds numbers is studied theoretically. The asymptotic flow field at large distances from an immersed body is shown to depend only upon the force acting on the cylinder. This universal field is determined by solving the Navier-Stokes equation numerically. The result enables us to evaluate the force acting on the body as a function of the flow Reynolds number. A detailed calculation is made of the drag coefficients of a circular cylinder and a flat plate. Results compare favourably with existing experimental and numerical data. © 1983, Cambridge University Press. All rights reserved.

Description: Previous experimental studies are reviewed and those whose data are deemed reliable are identified. New experiments at larger scale are described and the results are reported. These are combined with the reliable previous studies to form a data set covering heights from 3.66 to 50m and gasflow rates from 0.0002 to 0.59 normal m3/s. These wide-ranging data are combined with an integral theory for bubble plumes to determine functional relationships between local plume properties and the entrainment coefficient and the fraction of the momentum flux that is carried in the turbulent velocity fluctuations. These relationships together with the integral theory provide a set of equations that are suitable for numerical solution for the mean flow properties of any round bubble plume. Examples of the numerical solutions are presented and a comparison of one of these with existing experimental data is given. The relationships between the local plume properties and the entrainment coefficient and the momentum flux carried by the turbulence are interpreted to provide a qualitative understanding of the parameters involved and their influences on the plume. © 1983, Cambridge University Press. All rights reserved.

Description: Hamilton's principle of mechanics has special advantages as the beginning point for approximations. First, it is extremely succinct. Secondly, it easily accommodates moving disconnecting fluid boundaries. Thirdly, approximations-however strong-that maintain the symmetries of the Hamiltonian will automatically preserve the corresponding conservation laws. For example, Hamilton’s principle allows useful analytical and numerical approximations to the equations governing the motion of a homogeneous rotating fluid with free boundaries. © 1983, Cambridge University Press. All rights reserved.

Description: This paper reports a fundamental study of the fluid dynamics inside a triangular (attic-shaped) enclosure with cold upper wall and warm horizontal bottom wall. The study was undertaken in three distinct parts. In the first part, the flow and temperature fields in the cavity are determined theoretically on the basis of an asymptotic analysis valid for shallow spaces (H/L→0, where H and L are the attic height and length). It is shown that in the H/L→0 limit the circulation consists of a single elongated cell driven by the cold upper wall. The net heat transfer in this limit is dominated by pure conduction. In the second part of the study, the transient behaviour of the attic fluid is examined, based on a scaling analysis. The transient phenomenon begins with the sudden cooling of the upper sloped wall. It is shown that both walls develop thermal and viscous layers whose thicknesses increase towards steady-state values. The criterion for the existence of distinct thermal layers in the steady state is (H/L)1/2. RaH−1/4. 〉 1, where RaH. is the Rayleigh number based on attic height. The corresponding criterion for distinct viscous wall jets is (H/L)1/2. RaH1/4. Pr−1/2. 〉 1, where Pr is the Prandtl number. The third phase of this study focused on a complete sequence of transient numerical simulations covering the ranges H/L = 0.2, 0.4, 1; RaH/Pr = 10, 103, 105; Pr = 0.72, 6. The numerical experiments verify the flow features described theoretically in the first two parts of the study. The effect of thermal convection on the net heat transfer between the bottom and top walls is illustrated numerically. Finally, the transient numerical experiments show that in the present parametric domain the single-cell circulation pattern is stable with respect to the Benard instability expected in fluid layers heated from below. © 1983, Cambridge University Press. All rights reserved.

Description: Measurements of a steady free-stream, nominally two-dimensional, separating turbulent boundary layer have been reported in earlier parts of this work. Here measurements are reported that show the effects of frequency on sinusoidal unsteadiness of the free-stream velocity on this separating turbulent boundary layer at reduced frequencies of 0.61 and 0.90. The ratio of oscillation amplitude to mean velocity is about 1/3 for each flow. Upstream of flow detachment, hot-wire anemometer measurements were obtained. A surface hot-wire anemometer was used to measure the phase-averaged skin friction. Measurements in the detached-flow zone of phase-averaged velocities and turbulence quantities were obtained with a directionally sensitive laser anemometer. The fraction of time that the flow moves downstream was measured by the LDV and by a thermal flow-direction probe. Upstream of any flow reversal or backflow, each flow behaves in a quasisteady manner, i.e. the phase-averaged flow is described by the steady free-stream flow structure. The semilogarithmic law-of-the-wall velocity profiles applies at each phase of the cycle. The Perry & Schofield (1973) velocity-profile correlations fit the mean and ensemble-averaged velocity profiles near detachment. After the beginning of detachment, large amplitude and phase variations develop through each flow. Unsteady effects produce hysteresis in relationships between flow parameters. As the free-stream velocity during a cycle begins to increase, the detached shear layer decreases in thickness, and the fraction of time [formula omitted] that the flow moves downstream increases as backflow fluid is washed downstream. As the free-stream velocity nears the maximum value in a cycle, the increasingly adverse pressure gradient causes progressively greater near-wall backflow at downstream locations while [formula omitted] remains high at the upstream part of the detached flow. After the free-stream velocity begins to decelerate, the detached shear layer grows in thickness, and the location where flow reversal begins moves upstream. This cycle is repeated as the free-stream velocity again increases. In both unsteady flows, the ensemble-averaged detached-flow velocity profiles agree with steady free-stream profiles for the same [formula omitted] value near the wall when [formula omitted]. However, the reduced-frequency k = 0.90 flow has much larger hysteresis in ensemble-averaged velocity profile shapes when [formula omitted]. Larger and negative values of the profile shape factor Ĥ occur for this flow during phases when the non-dimensional backflow is greater and [formula omitted]. © 1983, Cambridge University Press. All rights reserved.

Description: The nonlinear time-dependent adjustment of a homogeneous rotating-channel flow to the sudden obtrusion of an obstacle is studied. Solutions are obtained using a Lax-Wendroff numerical scheme which allows rotating breaking bores and jumps to form and be maintained. The flow upstream of the obstacle is found to be completely blocked, partially blocked (and hydraulically controlled), or unobstructed depending upon the height of the obstacle. Partial blockage is accomplished through the excitation of a combination of nonlinear Kelvin waves, some of which steepen into interfacial shocks. Riemann invariants for the Kelvin waves are found, and jump conditions on mass, momentum and potential vorticity for the shocks are discussed. The shocks are surrounded by dispersive regions of Rossby deformation scale, and the potential vorticity of passing fluid is altered at a rate proportional to the differential rate of energy dissipation along the line of breakage. For the special case of initially uniform potential vorticity the asymptotic state is found as a function of the initial conditions. © 1983, Cambridge University Press. All rights reserved.

Description: Experiments are performed on steady and impulsively started flow in an approximately two-dimensional closed channel, with one wall locally indented. In plan the indentation is a long trapezium which halves the channel width; the inclination of the sloping walls is approximately 5.7°, and these tapered sections merge smoothly into the narrowest section via rounded corners. The Reynolds number Re = a0u0/v (ao= unindented channel width, u0= steady mean velocity in the unindented channel) lies in the range 300 ^ Re ^ 1800. In steady flow, flow visualization reveals that separation occurs on the lee slope of the indentation, at a distance downstream of the convex corner which decreases (tending to a non-zero value) as Re increases. There is no upstream separation, and there is some evidence of three-dimensionality of the flow in the downstream separated eddy. Pressure measurements agree qualitatively but not quantitatively with theoretical predictions. Unsteady flow visualization reveals that, as in external flow, wall-shear reversal occurs over much of the lee slope (at dimensionless time τ = ū0t/a0≈ 4) before there is any evidence of severe boundary-layer thickening and breakaway. Then, at τ ≈ 5.5, a separated eddy develops, and its nose moves gradually upstream from the downstream end of the indentation to its eventual (τ ≈ 75) steady-state position on the lee slope. At about the same time as the wall-shear reversal, wavy vortices appear at the edge of the boundary layer on both walls of the channel, and (for Re 〈 750) subsequently disappear again; these are interpreted as manifestations of inflection-point instability and not as intrinsic aspects of boundary-layer separation. Pressure measurements are made to investigate the discrepancy between the actual pressure drop across the lee slope and that predicted on the assumption that energy dissipation is quasi-steady. This discrepancy has a maximum value of approximately 1.5ρū20 (ρ = fluid density), and decays to zero by the time τ ≈ 7. © 1983, Cambridge University Press. All rights reserved.

Description: An analytical solution of a system of linearized equations for a gas-particle mixture is obtained for steady periodic motions. A finite volume fraction of particles and a continuous distribution of particle sizes are taken into consideration. It is shown that the effect of a continuous distribution of particle radii on the acoustic motions of a dusty gas is incorporated through only four integral quantities containing the relaxation times of the particle velocity and temperature, integrated over all particle sizes. This solution is applied to the problem of acoustic reflection, absorption, and transmission by a screen of dusty gas. © 1983, Cambridge University Press. All rights reserved.

Description: A general expression is obtained for the longitudinal dispersion coefficient for a passive contaminant in a varying channel. This expression reveals the upstream memory character of the dispersion coefficient. Simple examples are used to illustrate the effects of: A sudden change in breadth, centrifugally driven secondary flows, and changes in the depth profile. © 1983, Cambridge University Press. All rights reserved.

Description: A function ϕ is derived which is constant along the orthogonal trajectories of streamlines in two-dimensional flow. In irrotational flows, ϕ reduces to the velocity potential. The pair of functions ϕ and Ψ, where Ψ is the stream function, are used to define a coordinate system in rotational fluid flows. Tensor methods are used to transform the equations of motion of a turbulent fluid and the equations for second moments of turbulent fluctuations to this coordinate system. Explicit extra terms appear in the transformed equations embodying the effects of streamline curvature and mean flow acceleration. These extra terms are characterized by two lengthscales which arise naturally from the transformation: the local radius of curvature of the streamline and the ‘e-folding’ distance of the mean streamwise velocity. © 1983, Cambridge University Press. All rights reserved.

Description: The temporal and spatial evolution of large-scale modulations of weakly nonlinear edge waves on a uniformly sloping beach is studied using the full water-wave formulation for beach angles α=π/2N. Equations governing the evolution of envelopes of edge waves, excited by resonant interactions with incident wavetrains, are derived. It is deduced that a uniform train of free periodic edge waves is always unstable to large-scale variations, so that envelope solitons will develop; the resulting three-dimensional solitons are described in detail. In addition, it is shown that steady-state standing subharmonic edge waves, excited by incident wavetrains on a long, mildly sloping beach, can be unstable to large-scale modulations. The possible physical significance of these findings is discussed. © 1983, Cambridge University Press. All rights reserved.

Description: The critical Taylor number, phenomena accompanying the transition to turbulence, and the cellular structure of Taylor-Görtler vortex in the flow between two concentric spheres, of which the inner one is rotating and the outer is stationary, are investigated using three kinds of flow-visualization technique. The critical Taylor number generally increases with the ratio β of clearance to inner-sphere radius. For β ≤ 0.08, the critical Taylor number in spherical Couette flow is smaller than in circular Couette flow, but vice versa for β 〉 0.08. A pair of toroidal Taylor-Görtler vortices occurs first around the equator at the critical Reynolds number Rec. (or critical Taylor number Tc). More Taylor-Görtler vortices are added with increasing Reynolds number Re. After reaching the maximum number of vortex cells, as Reis increased, the number of vortex cells decreases along with the various transition phenomena of Taylor-Görtler vortex flow, and the vortex finally disappears for very large Re, where the turbulent basic flow is developed. The instability mode of Taylor-Görtler vortex flow depends on both ft and Re. The vortex flows encountered as Reis increased are toroidal, spiral, wavy, oscillating (quasiperiodic), chaotic and turbulent Taylor-Görtler vortex flows. Fourteen different flow regimes can be observed through the transition from the laminar basic flow to the turbulent basic flow. The number of toroidal and/or spiral cells and the location of toroidal and spiral cells are discussed as a means to clarify the spatial organization of the vortex. Toroidal cells are stationary. However, spiral cells move in relation to the rotating inner sphere, but in the reverse direction of its rotation and at about half its speed. The spiral vortices number about six, and the spiral angle is 2-10°. © 1983, Cambridge University Press. All rights reserved.

Description: Three modes of instabilities in the slit-jet flow field are recognized. Additional evidence for the universality of the Strouhal number for the second mode ST. =fw/U0. = 0.43 and additional information on the wavelength (A) between, and convection speed uc. of the symmetrically placed, large-scale motions that result from this instability are presented. Specifically λ/w≈1.2 and uc/U0≈ 0.51. The third instability mode is initiated at a Reynolds number U0w/v of approximately 1600; this instability results in a loss of the regular pattern associated with the large-scale motions. © 1983, Cambridge University Press. All rights reserved.

Description: The linear stability of the stationary Ekman-layer flow near a plane boundary is considered. Analytical formulas for the eigenfunctions are derived by a spectral analysis. Standard optimization algorithms are used to calculate critical points, maximum growth rates and neutral-stability curves. The new approach provides a better basis for both a linear and a nonlinear stability analysis than the well-known methods have done. The method may also be applied to other boundary-layer problems. © 1983, Cambridge University Press. All rights reserved.

Description: It is shown that the advection—diffusion equation for contaminant dispersion in water of variable depth admits of an inverse approach in which the stochastic mean concentration at a fixed sensitive location is calculated for arbitrary discharge sites. A ray method is used to obtain simple approximate solutions for the inverse problem when the sensitive location is situated at the shoreline. This makes explicit the considerable improvements in shoreline pollution levels which can be gained by siting effluent outfalls further away from the shore. © 1983, Cambridge University Press. All rights reserved.

Description: The growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Görtler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a). © 1983, Cambridge University Press. All rights reserved.

Description: When the flow over a submerged, round, upright cylinder, situated in a large ocean, is forced by a train of plane waves, linear theory (Yamamuro 1981) shows that the response can be abnormally large for certain forcing frequencies. The aim of this paper is to present a weakly nonlinear theory, where wave interactions, arising from the quadratic terms in the free-surfaee boundary conditions, can yield abnormally large responses. A specific interaction will be considered between a flow at a subharmonic frequency and a flow at the driving frequency. The reason for considering such an interaction derived from a consideration of some experimental results of Barnard, Pritchard & Provis (1981). © 1983, Cambridge University Press. All rights reserved.

Description: The flow in a partially filled, strongly rotating cylinder with a differentially rotating endcap was studied both experimentally and numerically. The cylindrical container was mounted with a vertical axis of rotation, partially filled with an incompressible fluid, and rotated at a sufficiently high angular velocity that the fluid formed a film of essentially uniform thickness on the sidewall of the container. An axial circulation in this fluid film was induced by the differential rotation of one of the container endcaps. A laser-Doppler velocimeter was used to measure the axial and azimuthal velocity components. The experimental results were compared with a finite-difference model of the flow, and the agreement between the two was good. Boundary layers of thickness proportional to E⅓, where E — v/ΩL2 is the Ekman number, are found both at the lateral wall and at the vertical free surface. The existence of an E⅓ boundary layer along the free surface is due to the invariant structure of the E1/2 Ekman layers on the horizontal surfaces with respect to a free surface. The radial transport in the Ekman layers of a partially filled rotating cylinder is essentially the same as that in a completely filled container. The axial transport, which in a completely filled container would have occurred in the volume now occupied by an empty core, is instead confined to a thin boundary layer along the free surface. © 1983, Cambridge University Press. All rights reserved.

Description: Free-surface and internal stationary waves in a meandering stream are treated, and analytical solutions given. It is shown that for each category there is an infinite number of Froude numbers, depending on the wavenumber of the meander, at which resonance occurs, and the amplitude of one of the wave components becomes infinite, according to the linear theory. These critical Froude numbers are interpreted physically. Furthermore, variable depth is treated for the case of free-surface waves, and in this treatment it is shown, incidentally, how the eigenvalues of a singular differential equation can be found under the requirement that the eigenfunction be non-singular. Finally, an attempt is made to explain the self-induced, non-stationary waves in water flowing between corrugated vertical walls, found by Binnie (1960), by an instability mechanism proposed by Yih (1976). There is strong evidence that this mechanism is at work, at least when a sloshing mode is involved in the wave-triad interaction. © 1983, Cambridge University Press. All rights reserved.

Description: Liquids held by surface tension forces can bridge the gap between two solid bodies placed not too far apart from each other. The equilibrium conditions and stability criteria for static, cylindrical liquid bridges are well known. However, the behaviour of an unstable liquid bridge, regarding both its transition toward breaking and the resulting configuration, is a matter for discussion. The dynamical problem of axisymmetric rupture of a long liquid bridge anchored at two equal coaxial disks is treated in this paper through the adoption of one-dimensional theories which are widely used in capillary jet problems. © 1983, Cambridge University Press. All rights reserved.

Description: A numerical study is conducted on the pressure-driven laminar flow of an incompressible viscous fluid through a rectangular channel subjected to a spanwise rotation. The full nonlinear time-dependent Navier-Stokes equations are solved by a finite-difference technique for various rotation rates and Reynolds numbers in the laminar regime. At weak rotation rates, a double-vortex secondary flow appears in the transverse planes of the channel. For more rapid rotation rates, an instability occurs in the form of longitudinal roll cells in the interior of the channel. Further increases in the rotation rate leads to a restabilization of the flow to a Taylor—Proudman regime. It is found that the roll-cell and Taylor—Proudman regimes lead to a substantial distortion of the axial-velocity profiles. The specific numerical results obtained are shown to be in excellent agreement with previously obtained experimental measurements and theoretical predictions. © 1983, Cambridge University Press. All rights reserved.

Description: The equilibrium configurations of a liquid spreading on a rough solid surface are derived by making expansions in terms of the characteristic slope e of the surface roughness, which is assumed to be very small. It is also assumed that the microscopic contact angle is a constant and that the liquid-air interface is planar at large distances from the contact line. Expressions for the value of the macroscopic contact angle and a discussion of the existence of contact-angle hysteresis and of stick-jump behaviour of the contact line are given for (i) surfaces with parallel grooves, (ii) surfaces with periodicity in two perpendicular directions and (iii) general non-period surfaces. © 1983, Cambridge University Press. All rights reserved.

Description: The occurrence of new unstable modes of quasigeostrophic baroclinic oscillation of rotating stratified shear flow over a wavy bottom is examined. To obtain a tractable mathematical problem, the bottom topography is considered as a perturbation modifying the oscillations. It is found that combinations of a top-intensified and a bottom-intensified Eady mode, each stable without topography, can be destabilized by topography if certain resonant conditions are met. These are that (i) the two modes possess the same wavefrequency, and (ii) topography possesses a wavenumber c bridging the gap between the wavenumbers of the modes a, b, i.e. c = a — b. Growth rate of this instability (called type A) is proportional to the amplitude of the topographic component. There are two special cases: (i) when one of the basic modes is a marginally neutral mode - according to the classical analysis without topography - the instability is stronger (type M) with growth rate proportional to the ⅔-power of topographic amplitude; (ii) when both modes are marginally neutral the instability is even stronger (type M2) with growth rate proportional to the square root of topographic amplitude. These topographic instabilities, like classical baroclinic instability, draw their energy from the available potential by transporting buoyancy down the mean gradient associated with the geostrophic shear flow. © 1983, Cambridge University Press. All rights reserved.

Description: The transition process of mechanically generated regular waves into wind waves is investigated in a wind-wave tunnel. Modulations of characteristic quantities of the waves are examined in both space and time using wave-gauge arrays. The transition begins with the occurrence of wave breaking, and it is associated with the following two processes: (i) the irregularization, i.e. the generation and amplification of a random modulation whose wave height and period are in phase; and (ii) the frequency shift to the lower side by the mutual coalescence of waves through the amplification of the modulation, the coalescence occurring at the troughs of the period modulation. The modulational properties under the effect of wind with wave breaking are thus in part different from the theoretical prediction of modulational instability of a freely travelling wavetrain. It is concluded that the mechanism of the transition is the modulational instability coupled with the wind effects. © 1983, Cambridge University Press. All rights reserved.

Description: Exact solutions of the equations of motion for an inviscid fluid are rare. Using the formalism of John (1953), this paper presents a class of exact zero-gravity flows in which the free surface assumes the form of an ellipse having arbitrary but time-constant aspect ratio. The dynamically important region beneath the overturning crest of a breaking gravity wave is examined and the profile is found to be remarkably well approximated by a A √3 aspect-ratio ellipse. The range of examples presented includes high-resolution computations in both deep and shallow water, and also the plunger-generated laboratory waves of Miller (1976). The ellipse solution is shown to model qualitatively certain essential features of the numerical waves. A recent self-similar solution due to Longuet-Higgins (1981, 1982), in which the free surface is a parametric cubic curve, is also discussed. © 1983, Cambridge University Press. All rights reserved.

Description: We focus attention on an idealized granular material comprised of identical, smooth, imperfectly elastic, spherical particles which is flowing at such a density and is being deformed at such a rate that particles interact only through binary collisions with their neighbours. Using general forms of the probability distribution functions for the velocity of a single particle and for the likelihood of binary collisions, we derive local expressions for the balance of mass, linear momentum and fluctuation kinetic energy, and integral expressions for the stress, energy flux and energy dissipation that appear in them. We next introduce simple, physically plausible, forms for the probability densities which contain as parameters the mean density, the mean velocity and the mean specific kinetic energy of the velocity fluctuations. This allows us to carry out the integrations for the stress, energy flux and energy dissipation and to express these in terms of the mean fields. Finally, we determine the behaviour of these fields as solutions to the balance laws. As an illustration of this We consider the shear flow maintained between two parallel horizontal plates in relative motion. © 1983, Cambridge University Press. All rights reserved.

Description: If different contaminant species are subject to different transverse drift rates (e.g. gravitational settling), then there is a tendency for the species to separate out. The efficiency of this separation depends upon the relative shapes of the longitudinal concentration distributions. Jayaraj & Subramanian (1978) have drawn attention to the disparity between their computed skew concentration distributions and the symmetric Gaussian distributions predicted by one-dimensional diffusion models. Here it is shown that a one-dimensional delay-diffusion model yields suitably skew predictions. The model equation is used to investigate the extent to which the separation of different contaminant species can be improved by pre-treating the sample (i.e. allowing differential drift) in a stationary fluid before being eluted into the shear flow. Pretreatment is found to be very effective for plane Poiseuille flow but not for the thermogravitational columns. © 1983, Cambridge University Press. All rights reserved.

Description: Nonlinear electrohydrodynamic Rayleigh-Taylor instability is investigated. A charge-free surface separating two semi-infinite dielectric fluids influenced by a normal electric field is subjected to nonlinear deformations. We use the method of multiple-scale perturbations in order to obtain uniformly valid expansions near the cutoff wavenumber separating stable from unstable flows. We obtain two nonlinear Schrodinger equations by means of which we can deduce the cutoff wavenumber and analyse the stability of the system. It is found that if a finite-amplitude wave exists then its small modulation is stable. We also obtain the surface elevation for such waves. The electric field plays a dual role in the stability criterion and the dielectric constant plays a distinctive role in this analysis. If the dielectric constant of the upper fluid is smaller than that of the lower fluid the field has a destabilizing effect for large wavenumbers. For relatively smaller wavenumbers the electric field stabilizes considerable parts of the first and second subharmonic regions in the stability diagrams; a result which is in contrast with the linear theory. If the dielectric constant of the upper fluid is larger than that of the lower fluid, then the field is stabilizing for larger values of the wavenumber K' when p is small (p is the density ratio) and destabilizing for smaller values of K'. © 1983, Cambridge University Press. All rights reserved.

Description: This paper describes the flow of a homogeneous fluid contained in a rapidly rotating cylinder. The upper part of the cylinder rotates slightly faster, giving rise to a discontinuity in the sidewall velocity. The Stewartson-layer structure arising at the sidewall is essentially affected by this discontinuity. In contrast with previously studied problems, the E¼ layer (E is the Ekman number) is unable to perform the matching of the interior flow to the sidewall. It is shown that this matching is carried out partially by the E¼ layer and partially by the E⅓ layer, the latter accounting for the jump discontinuity. This paper also presents an analytical description of the flow in the singularity region near the sidewall discontinuity. © 1983, Cambridge University Press. All rights reserved.

Description: The pattern of dispersion and uptake of an inhaled slug of tissue-soluble gas is examined within a branching model of the bronchial airways of the human lung, considered as an assembly of segments from infinitely long, straight rigid tubes with absorbing walls of finite thickness. The model is based on the first three (time-dependent) spatial moments of the solute distribution in such tubes, determined by the Aris method of moments. Poiseuille flow in each airway is assumed, and the solute distribution is taken to be initially zero in the tissue and radially uniform in the gas. First, the time dependence of axial velocity and mixing coefficient of the advancing solute in infinitely long tubes is shown and the mechanisms responsible are discussed. Transit times, uptake, uptake efficiency and mixing coefficient predicted from the model are then shown for different flow rates and solubilities, as functions of the generation of branching. As is expected, greater penetration is found for lower-solubility gases. However, of greater interest is the model prediction that uptake decreases with increasing flow rate whereas uptake efficiency increases, a result consistent with experimental indications. Finally, the mixing coefficient is shown to fall, with distance into the lung, to a value which may be much smaller than the molecular diffusivity, depending on the solubility. © 1983, Cambridge University Press. All rights reserved.

Description: Elementary calculations indicate that the effect of the Earth’s rotation is likely to be important in the dynamics of most internal waves in oceans, lakes and the atmosphere. Here we present measurements of the structure and properties of one class of such waves, namely solitary internal Kelvin waves, in which the Coriolis force generated by wave motion in a stratified fluid is opposed by a pressure gradient and hence change in wave amplitude along its crest. We confirm that the wave speed is independent of the rate at which the system rotates and depends only on the stratification and maximum wave amplitude. However, rotation is shown to have a large effect on both the rate at which the amplitude varies with time and the ‘cross-stream’ structure of the wave. In accordance with well-established theory, the amplitude transverse to the direction of propagation varies exponentially. This results in a decreasing wave speed with increasing distance from the wall, which in turn requires the wave front be curved backwards in order for the wave as a whole to propagate at a speed given by its maximum amplitude. Such a front curvature is not contained within the available theories. The rapid decay of wave amplitude is found to be due to the generation of inertial waves in the homogeneous fluid above and below the internal wave, and a reasonably successful scaling of this effect has been found. We also discuss the adjustment of the waves to geostropic balance and comment on applications of our results to natural systems. © 1983, Cambridge University Press. All rights reserved.

Description: In this paper we concern ourselves with the theoretical description of curved converging shock waves, where nonlinear interaction effects, between the shock fronts and the flow behind them, and refraction effects are equally important. In a non-viscous, isoenergetic and isentropic flow the problem can be described by a nonlinear wave equation for the pressure field. This equation then admits an analytical solution with the help of the method of strained coordinates provided that the nonlinear terms contain only derivatives with respect to two independent variables. This restrictive condition is approximately fulfilled if the incoming wave is only slightly curved. Replacing in the solution the strained coordinates - which themselves depend on the solution - by physical coordinates, we get an accurate description of the transition from the shock pattern obtained by the geometric-acoustics approach (very weak shocks) to the pattern determined by Whitham’s shock dynamics (strong shocks). Furthermore, the solution describes the complete flow field and agrees very favourably with experimental data by Sturtevant & Kulkarny. © 1983, Cambridge University Press. All rights reserved.

Description: We present an asymptotic study of steady two-dimensional radial flow between converging plane walls (Jeffery-Hamel flow) when the viscosity ρ and density ρ vary with the angular coordinate θ. Two representative situations are considered, the first being a two-layer system (in which ρ and p are uniform except for discontinuities at an interface θ = θI), and the other involving a fluid for which μ and ρ vary continuously with θ. The flow is analysed in the asymptotic limit when a parameter c related to the wall pressure gradient is large; this corresponds to converging flow at large Reynolds number. Solutions are derived for the boundary layers at the walls and for the shear layer at the interface; the results are shown to agree well with some exact (numerical) profiles. The solutions obtained are not unique, though for given c they represent the ‘simplest’ type of profile, and the one that seems most likely to be stable. We demonstrate the non-uniqueness by deriving in §3 an alternative solution for the interfacial shear layer. This solution, however, can exist for only restricted ranges of values of the density and viscosity ratios, and involves an outgoing jet, suggesting that it is likely to be unstable. © 1983, Cambridge University Press. All rights reserved.

Description: The steady motion of an eccentrically positioned sphere in a circular cylindrical tube filled with viscous fluid is considered as a regular perturbation of the axisymmetric problem. A sequence of boundary-value problems is formulated involving Stokes equations and some linear boundary conditions. Solutions of the first- and second-order problems yield the leading terms in the perturbation series of the additional drag and the torque on the spheres. The results are found to be in good agreement with the previous off-axis solutions. © 1983, Cambridge University Press. All rights reserved.

Description: The dynamics of the formation of the axisymmetric meniscus around a cone contacting a free liquid surface are discussed. An approximate phenomenological model is set up. In the case considered, where Re ≪ 1 and viscosity dominates the retarding forces, this leads to a differential equation relating the height of the circle of contact to time. Solutions are derived, involving one or more unknown parameters, which describe the time dependence of the height of the circle of contact. Experimental data, obtained from delayed flash photographs of the meniscus profiles of silicone fluid climbing over the surface of glass cones, provide general support for the model. The agreement between the predicted and observed height as a function of time is sufficiently close to justify the model as a useful description. © 1983, Cambridge University Press. All rights reserved.

Description: Laminar natural convection from a horizontal plate is studied by a finite-difference analysis and by experiments for Rayleigh numbers from 10 to 104. The plate with uniform surface temperature or concentration on one side and insulated on the other is situated in an ‘infinite’ fluid medium. The buoyancy near the surface is directed either outward or inward normal to the active surface - equivalent to a heated plate facing upward or downward. The effect of insulated vertical and horizontal extensions to the plate are also examined. Finite-difference solutions are obtained for a heated strip in a two-dimensional domain for a Prandtl number of 0–7. Mass-transfer experiments are performed with square naphthalene plates in air. Both numerical and experimental results justify a £-power law in the present range of Rayleigh number-i.e. Nusselt number or Sherwood number proportional to the Rayleigh number raised to the ⅕ power. The horizontal extensions cause a limited reduction in the transfer rate for the plate generating ‘ outward buoyancy ’, and a larger reduction with ‘ inward buoyancy ’. The vertical walls block the fluid flow directly, and thus greatly lower the transfer rate with either outward or inward buoyancy. © 1983, Cambridge University Press. All rights reserved.

Description: Exact expressions are derived for the centroid and variance as functions across the flow when there has been an initially uniform contaminant release in an oscillatory flow. Two examples are given to demonstrate that there can be a substantial region of the flow (where the velocity shear is relatively large) in which the contaminant distribution exhibits contraction after flow reversal. This effect, and the sensitivity of the variance to the precise time of discharge, is most marked when the flow oscillations are rapid relative to the timescale for cross-sectional mixing. © 1983, Cambridge University Press. All rights reserved.

Description: The evolution of temperature variance and heat flux in decaying grid turbulence with a linear cross-stream temperature gradient is studied by producing the temperature gradient by means of two different methods: (a) by placing a ‘mandoline’ (Warhaft & Lumley 1978) downstream from the grid but with its wires differentially heated for the present study, and (b) by differentially heating ribbons of nichrome (a ‘toaster’) placed in the plenum chamber of the wind tunnel. For the former method the initial thermal/mechanical lengthscale ratio LθL was varied by changing the mandoline configuration. For this method it is shown that the gradient causes Lθ/L to equilibrate to a value of about 0.9 regardless of its initial value, and that when this value is achieved the temperature variance increases approximately linearly with time. The toaster was used to produce a temperature gradient without the associated initial temperature variance (and initial thermal lengthscale) that is necessarily produced by the mandoline wires; for the toaster the temperature variance was produced solely by the action of turbulence against the temperature gradient. For this experiment too, the thermal variance grew linearly with time, and Lθ/L was approximately the same as the equilibrium value for the mandoline experiments. The equilibrium value of the ratio of temperature-variance production to temperature-variance dissipation was approximately 1.5 for all of the experiments. The ratio of the mechanical-dissipation/thermal-dissipation timescales was also found to equilibrate, but there was considerably more scatter in the data for this parameter. The values of the equilibrium length and timescale ratios were not affected by the magnitude of the temperature gradient, which was varied for both experiments. Good transverse homogeneity in the thermal field was achieved in all cases, in contrast with previous experiments (using heated grids). © 1983, Cambridge University Press. All rights reserved.

Description: Detailed measurements of the time-dependent velocities induced inside and outside the opening of an acoustically excited, two-dimensional Helmholtz resonator imbedded in a grazing flow are presented. The remarkably clear structure of the perturbation field which evokes a pulsating source and a coherently pulsating vortex-image pair is described. © 1983, Cambridge University Press. All rights reserved.

Description: Using experimental measurements, estimates are made of the efficiency of conversion of kinetic energy into potential energy through vertical mixing in a continuously stratified fluid. In the experiments kinetic energy was supplied continuously at a rate of e to a fundamental internal wave mode in a rectangularly bounded and initially linear stable stratification. Mixing resulted from the In stability of this wave and its consequent ‘breaking'. Potential energy was gained by the system at rate p through the gradual weakening of the stratification. The In stability is predictable using wave-interaction theory, affording a means of estimating the amount of kinetic energy lost (at rate ev) to laminar viscosity without first cascading to the fine scales characteristic of turbulent mixing. With account taken of this viscous loss the mixing efficiency p/εM, based upon the residual kinetic energy input εM(= e —ev) was found to be approximately constant, and not significantly correlated with the rate at which energy was supplied, nor with the estimated instantaneous minimum gradient Richardson number. The average value for eight separate experimental runs using two different experimental configurations was 0–26 with a sample standard deviation of 0–06. Measured density profiles also afforded an estimate of the effective vertical diffusivity Kdof density as a result of mixing. Vertically averaged values of the product of A:dand the squared local static buoyancy frequency N, KdV2, were found to have an average for seven runs of 0-24εM, with a standard deviation for the coefficient of 0.1, and no significant correlation with energy supply rate. These results, the first of their kind to correct for incidental losses, substantiate the values previously assumed in estimates of dissipation and vertical diffusion in the ocean and the atmosphere, and validate the assumption of similarity between buoyancy and mass transfer on which they are based. The efficiency value also agrees with the kinematic prediction for localized homogenization in small discrete volumes made in the companion paper (McEwan 1983). On the basis of that work it is inferred from the present results that the mixing efficiency is only weakly dependent upon Prandtl number provided that this is of order unity or greater. © 1983, Cambridge University Press. All rights reserved.

Description: The flow between two finite rotating disks enclosed by a cylinder is investigated both numerically and experimentally. For this finite geometry the full stationary Navier-Stokes equations are solved numerically without similarity assumptions. Experimental results are obtained by means of stereophotography of small tracer particles. The results are in good agreement with the numerical solution. Owing to the presence of the cylinder sidewall, the solution is found to be unique for all values of the parameters considered. When the disks rotate in opposite senses with counter-rotation above 15%, a stagnation point appears at the slower-rotating disk. This stagnation point is associated with a two-cell structure in the meridional plane and is experimentally observed as a ring of particles at the slower-rotating disk. Near the axis of rotation the solution is found to satisfy similarity demands; for weak counter-rotation the solution is of Batchelor type near the axis of rotation, but for strong counter-rotation a Stewartson profile is found to be more adequate for the description of the tangential velocity near the axis. © 1983, Cambridge University Press. All rights reserved.

Description: The present investigation considers the vortex wakes behind finite-length yawed cylinders that are stationary or vibrating transversly in a uniform flow. In the Reynolds-number range 160–103a number of three-dimensional and nominally two-dimensional wake flows are observed and attributed to the dominance of end conditions over rather large spans of the cylinder (aspect ratios up to 100). © 1983, Cambridge University Press. All rights reserved.

Description: Gravity currents or intrusions for which the fluid within the current contains a substance that gives it a diffusivity different from that of its surroundings are very common both in natural and technological applications. The interface between the two fluids can become the site of either a ‘fingering’ or a ‘diffusive’ type of In stability, and vigorous convection and material exchange occurs (Turner 1975). This transfer of material has several important effects upon the dynamics of the intrusion. Horizontal momentum can be transferred across the interface to create a stress which in many cases dominates the more conventional viscous stresses. Entrainment into the convective plume beneath the intrusion and, in a container of finite depth, the formation of a secondary, bottom-boundary current, creates a flow external to the main intrusion which modifies its behaviour even more. Two cases have been studied: the release of a fixed volume of fluid and the injection of fluid at a constant rate for both types of interface. Several experiments on the motion of an intrusion for which the interface is essentially non-diffusive are presented for comparison. © 1983, Cambridge University Press. All rights reserved.

Description: The Lighthill theory has been extended so that it may be used to determine the flow noise induced by a turbulent boundary layer over a plane homogeneous flexible surface. The influence of the surface properties and the mean flow on the sound generated is brought out explicitly through the use of a Green function. It is found that there is an analogy between the sound generated by turbulence and equivalent sources placed between a surface with the same compliance as the physical surface and a hypothetical vortex sheet positioned at the outer edge of the boundary layer. This analogy is used to determine the spectrum of the surface-pressure fluctuations under statistically stationary turbulence. The form of this wall-pressure spectrum is investigated in detail for three particular types of surface: rigid surfaces, bending plates and sound-absorbent liners. © 1983, Cambridge University Press. All rights reserved.

Description: The evolution of an internal wavebreaking event in a continuous stable stratification is examined using schlieren colour imagery combined with streak photography. For standing but strongly interacting wavemodes forced to breaking, the Richardson number defined by local gradients of density and velocity is critically low only in the immediate vicinity of the breaking region where convective overturning also occurs. Breaking and vertical mixing results from the rapid development of three-dimensional interleaving density microstructure within a confined volume which persists and gravitates to modify weakly the surrounding density distribution. Continual mixing through internal motion is seen as a widespread repetition of such events. Based on the observations, a general description of the process is proposed and applied in two types of simple kinematical model to provide an estimate of mixing efficiency, the ratio of potential energy gained through stratification weakening to kinetic energy expended in motions on the scales of the mixing event. These models do not rely on an assumed similarity between momentum and buoyancy transfer. They yield, for complete homogenization of small discrete volumes, an efficiency of ¼ in a linear stratification. Mixing across a density interface is predicted to have a lower efficiency, and lower efficiency is expected where homogenization is incomplete. © 1983, Cambridge University Press. All rights reserved.

Description: The role that differential rotation plays in the hydromagnetic stability of rapidly rotating fluids has recently been investigated by Fearn & Proctor (1983) (hereinafter referred to as I) as part of a wider study related to the geodynamo problem. Starting with a uniformly rotating fluid sphere, the strength of the differential rotation was gradually increased from zero and several interesting features were observed. These included the development of a critical region whose size decreased as the strength of the shear increased. The resolution of the two-dimensional numerical scheme used in I is limited, and consequently it was only possible to consider small shear strengths. This is unfortunate because differential rotation is probably an important effect in the Earth’s core and a more detailed study at higher shear strengths is desirable. Here we are able to achieve this by studying a rapidly rotating Benard layer with imposed magnetic field B0= BMSs and shear U0= UMsΩ(z)§, where (S, Ø, z) are cylindrical polar coordinates. In the limit where the ratio q of the thermal to magnetic diffusivities vanishes (q —0), the governing equations are separable in two space dimensions and the problem reduces to a one-dimensional boundary-value problem. This can be solved numerically with greater accuracy than was possible in the spherical geometry of I. The strength of the shear is measured by a modified Reynolds number Rt= UMd/K, where the depth of the layer and K Is the thermal diffusivity, and the shear becomes important when Rt≥ 0(1). It is possible to compute solutions well into the asymptotic regime Rt≫ 1, and details of the behaviour observed are dependent on the nature of Ω(z). Specifically, two cases were considered: (a) Ω(z) has no turning point in 0 〈 z 〈 1, and (6) Ω(z) has a turning point at z = z?y0 〈 zT〈 1 (Ω‘(zT) = 0, Ω”(zT) 4‡ 0). In both cases, as Rt increases a critical layer centred at z —zLdevelops, with width proportional to (a) Rt-⅓, (b) Rt¼. In the case where Ω(z) has a turning point, the critical layer is located at the turning point (zL= zT). The critical Rayleigh number Rcincreases with (a) Beα Rt, (b) Rcα Rf, and the In stability is carried around with the fluid velocity at the critical layer. The relevance of these results to the geomagnetic secular variation is discussed. © 1983, Cambridge University Press. All rights reserved.

Description: The mechanism of peristaltic transport of an incompressible viscoelastic fluid by means of an infinite train of sinusoidal waves travelling along the wall of the duct is studied in the case of a plane flow. The main assumptions are that the relevant Reynolds number is small enough to neglect inertia forces, and that the ratio of the wavelength and the channel height is large, which implies that the pressure is constant over the cross-section. For sufficiently small values of the ratio of the wave amplitude and the mean height of the channel, details of the fluid motion are studied analytically within a second-order approximation with respect to the amplitude ratio. Under these conditions the integral constitutive equation of finite linear viscoelasticity is relevant. Particular attention is given to the pressure—discharge characteristics of the peristaltic pump and to the pumping efficiency. The results are influenced by specific values of the complex viscosity of the fluid, which can be determined using standard rheometers. In general, the rate of discharge turns out to be a non-monotonic function of the wave speed. This leads to an optimal wave speed, for which the memory of the fluid particles extends over several wave periods. From an energetic point of view, relatively small wave speeds are the best, where the fluid changes its state slowly such that the memory and with it the elasticity of the fluid do not influence the flow field at all. As the dimensionless memory parameter tends to zero, the analytical results reduce to the well-known case of a Newtonian fluid. © 1983, Cambridge University Press. All rights reserved.

Description: The no-slip condition on rigid boundaries necessitates a correction to the critical value of the Rayleigh-Darcy number for the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below. It is shown that the use of the Brinkman equation to obtain this correction is not justified, because of the limitations of that equation. These limitations are discussed in detail. An alternative procedure, based on a model in which the porous medium is sandwiched between two fluid layers, and the Beavers-Joseph boundary condition is applied at the interfaces, is described, and an expression for the correction is obtained. It is found that the correction can be of either sign, depending on the relative magnitudes of the parameters involved. © 1983, Cambridge University Press. All rights reserved.

Description: The validity of recent shallow-flow equations with bed curvature is examined. Subcritical, critical and supercritical steady-flow solutions are identified, and the point of critical flow on the bed located, in terms of a generalized Froude number. Experiments on steady flow over both a symmetric and unsymmetric bed profile show that the theory satisfactorily predicts the tree-surface and bed-pressure profiles over-2 ≲ Kh ≤ 0.54 (where K is the bed curvature and h is the flow depth normal to the bed). © 1983, Cambridge University Press. All rights reserved.

Description: Time series of amplitude, frequency, wavenumber and phase speed are measured in an unstable deep-water wavetrain using a Hilbert-transform technique. The modulation variables evolve from sinusoidal perturbations that are well described as slowly varying Stokes waves, through increasingly asymmetric modulations that finally result in very rapid jumps or ‘phase reversals ’. These anomalies appear to correspond to the ‘crest pairing’ described by Ramamonjiarisoa & Mollo-Christensen (1979). The measurements offer a novel local description of the In stability of deep-water waves which contrasts markedly with the description afforded by conventional Fourier decomposition. The measurements display very large local modulations in the phase speed, modulations that may not be anticipated from measurements of the phase speeds of individual Fourier components travelling (to leading order) at the linear phase speed (Lake & Yuen 1978). © 1983, Cambridge University Press. All rights reserved.

Description: Experiments were performed on concentrated suspensions of large (0•97-1•78 mm mean diameters) neutrally buoyant spherical particles sheared in a concentric-cylinder Couette-flow apparatus in which the inner cylinder rotated while the outer one was fixed. The variations of shear stress with apparent shear rate, concentration, particle diameter and wall roughness were studied, and the results are compared with related experiments of Bagnold. Generally the shear stresses measured in the present experiments were larger than those of Bagnold. The difference can be attributed to differences in the experimental arrangements; Bagnold’s flexible-walled inner cylinder was fixed while the outer cylinder rotated. A strong effect of wall roughness was observed. The higher stresses generated with rough walls imply that particle ‘slip’ may have occurred in the smooth wall tests. The larger stresses might also be due to an increase in strength of the interparticle collisions caused by the roughness. No dependence of stress upon particle diameter was observed for concentrations of about 0•3, but a strong dependence (〉 d2) was found at the highest concentrations with the rough walls. © 1983, Cambridge University Press. All rights reserved.

Description: An analysis is presented that describes a model of the flow field of a rotating compressible fluid in a cylinder with internal sources or sinks of mass, momentum or energy. A solution of the mathematical model is obtained using an expansion in eigenfunctions of the corresponding homogeneous equation. The internal sources or sinks produce countercurrent flows analogous to flows generated by boundary conditions in the classical analysis of the problem. The application of this model to the flow driven by a feed stream or a scoop is discussed. Some sample calculations are presented that illustrate the countercurrent flow produced by sources of mass, the three components of momentum, energy and a mass source/sink combination. Calculations simulating feed introduction and a tails-removal scoop have been performed and the fluid-dynamics solutions have been used to calculate the optimum separative performance of the example centrifuge. © 1983, Cambridge University Press. All rights reserved.

Description: Numerical solutions for the impulsively started spin-up from rest of a homogeneous fluid in a cylinder for small Ekman numbers are presented. The basic analytical theory for this spin-up flow is due to Wedemeyer (1964). Wedemeyer’s solution shows that the interior flow is divided into two regions by a moving front which propagates radially inward across the cylinder. The fluid ahead of the front remains non-rotating, while the fluid behind the front is being spun up. Experimental observations have shown that Wedemeyer’s model captures the essential dynamics of the azimuthal flow, but that it is not a quantitative model. Wedemeyer made several assumptions in formulating an Ekman compatibility condition, and inconsistencies exist between these assumptions and his solution. Later workers attempted to improve the analytical theory, but their work still included the same basic assumptions made by Wedemeyer. No previous work has provided a comprehensive and accurate set of three-dimensional flow-field data for this spin-up problem. We chose to acquire such data using a numerical model based on the Navier—Stokes equations. This model was first checked against accurate laser-Doppler measurements of the azimuthal flow for spin-up from rest. New flow-field data over a range of Ekman numbers 9.18 x 10 −6 ≤ E ≤ 9.18 x 10 –4 are presented. Diagnostic studies, which reveal the various contributions to spin-up of the separate inviscid and viscous terms as functions of radius and time, are also presented. The plots of the viscous-diffusion term reveal the moving front, which is identified as a layer of enhanced local viscous activity. Immediately after the impulsive start, viscous diffusion is seen to be the major contributor to spin-up, then the nonlinear radial advection term takes over, and, finally, when spin-up is well progressed, the linear Coriolis force dominates. In the vicinity of the front, the inward radial flow is a maximum, and the vertical velocity is very small. Strong radial gradients of the vertical velocity are observed across the front and behind the front at the edge of the Ekman layer, and the azimuthal flow behind the front shows strong departures from solid-body rotation. These results enable us to fill in details of the flow not accurately given by Wedemeyer’s model and its extensions. © 1983, Cambridge University Press. All rights reserved.