ALBERT

Description: The linear stability of the spiral motion induced between concentric cylinders by an axial pressure gradient and independent cylinder rotation is studied numerically and experimentally for a wide-gap geometry. A three-dimensional disturbance is considered. Linear stability limits in the form of Taylor numbers TaLare computed for the rotation ratios µ = 0, 0.2, and — 0.5 and for values of the axial Reynolds number Re up to 100. Depending on the values of µand Re, the disturbance which corresponds to TaLcan have a toroidal vortex structure or a spiral form. Aluminium-flake flow visualization is used to determine conditions for the onset of a secondary motion and its structure at finite amplitude. The experimental results agree with the predicted values of TaLfor µ ≥ 0, and low Reynolds number. For other cases in which agreement is only fair, apparatus length is shown to be a contributing influence. The comparison between experimental and predicted wave forms shows good agreement in overall trends. © 1981, Cambridge University Press. All rights reserved.

Description: Convection flows have been systematically observed in a layer of fluid between two isothermal horizontal boundaries. The working fluid was a nematic liquid crystal, which exhibits a liquid-liquid phase change at which latent heat is released and the density changed. In addition to ordinary Rayleigh–Béxnard convection when either phase is present alone, there exist two distinct types of convective motions initiated by the unstable density difference. When a thin layer of heavy fluid is present near the top boundary, hexagons with downgoing centres exist with no imposed thermal gradient. When a thin layer of light fluid is brought on near the lower boundary, the hexagons have upshooting centres. In both cases, the motions are kept going once they are initiated by the instability due to release of latent heat. Relation of the results to applicable theories is discussed. © 1981, Cambridge University Press. All rights reserved.

Description: In this paper we present a rather personal view of the important developments in double-diffusive convection, a subject whose evolution has been the result of a close interaction between theoreticians, laboratory experimenters and sea-going oceanographers. More recently, applications in astrophysics, engineering and geology have become apparent. In the final section we attempt to draw some general conclusions and suggest that further progress will again depend on a close collaboration between fluid dynamicists and other scientists. © 1981, Cambridge University Press. All rights reserved.

Description: This article gives a review of six areas of current activity and importance in aero-acoustics, including (i) the generation of sound and vorticity by vorticity and sound, respectively, (ii) the basis for, and consequences of, the application of a Kutta condition in unsteady leading- and trailing-edge flows, and (iii)the suppression or amplification of broadband hydrodynamic and acoustic fields in a jet under the influence of weak discrete tone forcing. The intention is also to promote acceptance once again of acoustics as a serious branch of fluid mechanics. © 1981, Cambridge University Press. All rights reserved.

Description: Two topics are discussed in order to illustrate the author's own enjoyment of fluid mechanics. The first and longer discourse is about splashes. It makes no attempt at completeness but includes a little new research. The second part deals briefly with many variations on the theme of flow in pipes. © 1981, Cambridge University Press. All rights reserved.

Description: This is in no way intended as a review of turbulence - the subject is far too big for adequate treatment within a reasonably finite number of pages; the monumental treatise of Monin & Yaglom (1971, 1975) bears witness to this statement. It is rather a discourse on those aspects of the problem of turbulence with which I have myself had contact over the last twenty years or so. My choice of topics therefore has a very personal bias - but that is perhaps consistent with the style and objectives of this rather unusual issue of JFM. I have approached the dynamical problem of turbulence via two simpler (but nevertheless far from trivial) problems - viz the convection and diffusion of a passive scalar field and of a passive vector field by turbulence of known statistical properties. Particular emphasis is given to the method of successive averaging (a simplified version of the renormalization-group technique) which seems to me to have considerable potential. The difficulty of extending this method to the dynamical problem is discussed. In a final section (§6) I have allowed myself the luxury of discussing a somewhat esoteric topic - the problem of inviscid invariants and their relationship with the topological structure of a complex vorticity field. The helicity invariant is the prototype; it is identifiable with the Hopf invariant (1931) and it may be obtained from appropriate manipulation of Noether's theorem (Moreau 1977). A suggestion is made concerning possible measurement of this fundamental measure of ‘lack of reflexional symmetry’ in a turbulent flow. © 1981, Cambridge University Press. All rights reserved.

Description: The first volume of the Journal of Fluid Mechanics contained nine articles (of 39) on shock waves. Some of these pioneered new branches of fluid mechanics. Others dealt with older problem areas. Surprising is one’s realization that important elements of all topics are still of current interest. The subjects treated were shock structure, diffraction, refraction, waves in supersonic and hypersonic flows, large-amplitude acoustic and blast waves, and astrophysical processes. The subsequent addition of work on chemically reactive flows, radiating and laser-induced shocks, the effects of electric and magnetic fields on shock propagation in ionized media and the development of computer-based methods of analysis have greatly broadened the scope of shock wave investigations during the ensuing twenty-five years. The paper traces some of the principal lines of investigation from early motivations to the present state of understanding and application. Motivation is not often consciously expressed in the scientific literature. Usually an external motivation in terms of identifiable needs for better understanding for the solution of practical problems can be identified; though much excellent work must be ascribed to that ubiquitous trait curiosity. The topics covered in this article were chosen as representative of the basic elements of shock wave interactions and effects. They are: shock structure, refraction, diffraction, shocks in liquid helium, and condensation and liquefaction shocks. The paper closes with an assessment of how approximate and computational methods developed for handling complex flow problems fare when applied to some of the basic shock interactions considered here. Most of the emphasis will be on shock waves in gases, for which knowledge of an equation of state has been key to the significant advances made during the last twenty-five years. For liquids and solids, shock waves have been used the other way around; to study state properties. © 1981, Cambridge University Press. All rights reserved.

Description: The present study is to investigate the structure of space-time correlations of bursting motions, such as ejections and sweeps in an open-channel flow, by a new conditional sampling analysis of the instantaneous velocity and Reynolds-stress signals measured simultaneously by two dual-sensor hot-film probes. One probe was fixed near the edge of the buffer layer, while the other probe was moved in the streamwise, vertical and spanwise directions. The sorted instantaneous Reynolds-stress signals obtained from the fixed probe were used as a detecting information of the occurrences of ejection or sweep events. The streamwise and vertical spatial characteristics of the ejection-sweep motions, and their convection process are investigated in detail. Also, the spanwise spatial properties of the high- and low-speed streaks in the bursting motions are examined experimentally by the present conditional sampling method. Next, a qualitative model is proposed which attempts to explain the space-time structures of the bursting phenomenon, on the basis of the above anemometry information and other visual observations. © 1981, Cambridge University Press. All rights reserved.

Description: In a high-speed subsonic jet impinging on a flat plate, the surface pressure fluctuations have a broad spectrum due to the turbulent nature of the high-Reynolds-number jet. However, these pressure fluctuations dramatically change their pattern into almost periodic waves, if the plate is placed close to the nozzle (x0/d 〈 7.5). In the present study extensive measurements of the near-field pressure provide solid support for the hypothesis that a feedback mechanism is responsible for the sudden change observed in the pressure fluctuations at the onset of resonance. The feedback loop consists of two elements: the downstream-con vected coherent structures and upstream-propagating pressure waves generated by the impingement of the coherent structures on the plate. The upstream-propagating waves and the coherent structures are phase-locked at the nozzle exit. The upstream-propagating waves excite the thin shear layer near the nozzle lip and produce periodic coherent structures. The period is determined by the convection speed of the coherent structures, the speed of the upstream-propagating waves as well as the distance between the nozzle and the plate. An instability process, herein referred to as the collective interaction’, was found to be critical in closing the feedback loop near the nozzle lip. © 1981, Cambridge University Press. All rights reserved.

Description: We examine the Galerkin (including single-mode and Lorenz-type) equations for convection in a sphere to determine which physical processes are neglected when the equations of motion are truncated too severely. We test our conclusions by calculating solutions to the equations of motion for different values of the Rayleigh number and for different values of the limit of the horizontal spatial resolution. We show how the gross features of the flow such as the mean temperature gradient, central temperature, boundary-layer thickness, kinetic energy and temperature variance spectra, and energy production rates are affected by truncation in the horizontal direction. We find that the transitions from steady-state to periodic, and then to aperiodic convection depend not only on Rayleigh number but also very strongly on the horizontal resolution of the calculation. All of our models are well resolved in the vertical direction, so the transitions do not appear to be due to poorly resolved boundary layers. One of the effects of truncation is to enhance the high-wavenumber end of the kinetic energy and thermal variance spectra. Our numerical examples indicate that, as long as the kinetic energy spectrum decreases with wavenumber, a truncation gives a qualitatively correct solution. © 1981, Cambridge University Press. All rights reserved.

Description: Experiments are described which show that large-scale coherent structures exist in the wakes behind three-dimensional blunt bodies. Using a ‘flying hot-wire’ apparatus, the vortex-shedding cycle has been described by phase-averaged vector fields allowing an animation of large-scale motions to be produced. It is found that the large-scale structures retain their identity for long streamwise distances and contribute significantly to the Reynolds stress. Using critical-point theory as described recently by Perry, Lim & Chong (1980), the effect of phase ‘jitter’ on ‘washout’ has been analysed. Furthermore, it is found from critical-point theory that the large-scale motions possess the same geometrical features as the low-Reynolds number (unsteady laminar flow) wake results of Perry & Lim (1978) and Perry et al. (1980). © 1981, Cambridge University Press. All rights reserved.

Description: The spatial stability of the plane, two-dimensional jet flow to infinitesimal disturbances is investigated by taking into account the effects of transverse velocity component and the streamwise variations of the basic flow and of the disturbance amplitude, wavenumber and spatial growth rate. This renders the growth rate dependent on the flow variable as well as on the transverse and streamwise co-ordinates. Growth rates for the energy density of the disturbance and the associated neutral curves are provided as a function of the streamwise co-ordinate. Variation of growth rate of the disturbance stream function and streamwise component of velocity with the transverse co-ordinate is also given for different disturbance frequencies and streamwise locations. Results are compared with those for the parallel-flow stability analysis, and also with those for an analysis that accounts for only some of the non-parallel effects. It is found that the critical Reynolds number based on the growth of energy density of the disturbance depends on the streamwise co-ordinate and lies within the range (around 20) found experimentally, while the parallel-flow theory yields a rather low value of 4.0. © 1981, Cambridge University Press. All rights reserved.

Description: An approximate solution of two-dimensional convection in the limit of low Prandtl number is presented in which the buoyancy force is balanced by the inertial terms. The results indicate that inertial convection becomes possible when the Rayleigh number exceeds a critical value of about 7 × 103. Beyond this value the velocity and temperature fields become independent of the Prandtl number except in thin boundary layers. The convective heat transport approaches the law Nμ = 0.175 R1/4for the Nusselt number Nμ. These results are in reasonably close agreement with the numerical results described in the preceding paper by Clever & Busse (1980). © 1981, Cambridge University Press. All rights reserved.

Description: Steady solutions in the form of two-dimensional rolls are obtained numerically for convection in a horizontal layer of a low-Prandtl-number fluid heated from below. Prandtl numbers in the range 0.001 ≤ P ≤ 0.71 are investigated for Rayleigh numbers between the critical value, R = 1708, and R = 20,000 in the case of rigid boundaries. The calculations reveal that the convective heat transport is relatively independent of the Prandtl number at Rayleigh numbers greater than a finite critical value R2 of the order of 5 × 1O3. At R = 10,000 the convective heat transport varies by only about 30% for Prandtl numbers in the range investigated. As the Rayleigh number is increased above the critical value R2, the streamlines of the convection flow become circular, independent of the horizontal wavelength as long as the latter is larger than or about equal to twice the height of the layer. © 1981, Cambridge University Press. All rights reserved.

Description: The effect of suspended sediment on the modification of individual dunes in unsteady channel flow is analyzed. The model is an extension of that developed by Fredsoe (1979), in which the change of the dune dimensions was analyzed assuming that the transport of sediment mainly occurs as bed load. By including suspended sediment, it is possible to describe dune behaviour at high sediment transport rates, for which the dunes are washed out, so the bed at sufficiently high bed shear stress becomes plane. The model is compared with laboratory experiments and the agreement is reasonable. The effect of temperature variations on dune dimensions is incorporated in the analysis. © 1981, Cambridge University Press. All rights reserved.

Description: Perfect-fluid theory is applied to the description of steady motions that can be generated as the outflow into a horizontal channel from a large reservoir of incompressible heavy fluid whose density is an arbitrary decreasing function of height. A particular aim is to pinpoint the significance of an already known class of flows, called self similar, which satisfy the approximate (shallow-water) equations applicable when the horizontal scale of the motion greatly exceeds its vertical scale, but which have not until now been shown to match the downstream conditions that primarily determine the motion in practice. New variational principles are introduced characterizing the class of self-similar flows: in §2 there is a characterization in terms of flow force among parallel flows realized asymptotically in a uniform channel, in §3 among a wider range of possibilities including periodic flows, and in §6 among supercritical flows realized in a convergent-divergent channel. Aspects of general flows in channels of gradually varying breadth are treated in §§4 and 5, including the remarkable fact, proven in §5, that every steady flow outside but close to the self-similar class must somewhere undergo a local crisis unaccountable by the shallow-water approximation. Practical interpretations afforded by the theoretical results are noted in §7. © 1981, Cambridge University Press. All rights reserved.

Description: The dynamic effects of an alternating magnetic field on containers of conducting fluid are investigated in two special cases: (i) an infinitely long circular cylinder in a uniform magnetic field normal to the generators; (ii) a truncated circular cylinder in a uniform magnetic field parallel to the axis. Neglecting the motion effects in Maxwell’s equations, the problem is conveniently decoupled into electromagnetic and dynamic parts. Using either analytical or numerical solutions of the electromagnetic equations, the electromagnetic forces are calculated and introduced in the motion equations. In the first case, asymptotic solutions of the Navier-Stokes equations valid for high frequencies are calculated and compared with numerical solutions obtained for the same geometry. The second case has been studied numerically, and the solutions are presented and interpreted. © 1981, Cambridge University Press. All rights reserved.

Description: An investigation of the turbulent flow structure over a progressive water wave, as well as the structure of the wave-induced flow field in a transformed wave-following frame, is reported. Experimental results are given for a free-stream velocity of 2.4 ms-1over a 1 Hz mechanically generated deep-water wave. The velocity components were measured with a cross hot-film probe oscillating in a transformed wave-following frame. The amplitude and phase of the wave-induced velocity components are deduced by correlation to the generated water wave. The mean flow tends to follow the wave form so that the water wave should not be regarded as surface roughness. The mean velocity profile is basically log-linear and is similar to that over a smooth plate, because ripples riding on the waves do not produce sufficient roughness to interfere with the wind field. The wave-induced motion in the free stream is irrotational; but, in the boundary layer, it has strong shear behaviour related to the wave-associated Reynolds stress. The shear stress production as well as the energy production from the mean flow is concentrated near the interface. A phase jump of 180° in the wave-induced turbulent Reynolds stresses in the middle of the boundary layer was observed. The relationships between the induced turbulent Reynolds stresses and the induced velocities are of an eddy-viscosity type. © 1981, Cambridge University Press. All rights reserved.

Description: The purpose of the study was to examine the influence of a nearby wall on the lift experienced by spheres fixed in a fluid stream, and the decay of this influence with increasing separation from the wall. For gaps less than one quarter of the sphere diameter, wall effects were found to dominate all dependencies other than that on Reynolds number. At larger gap ratios the lift vector rotates around the flow axis, sometimes this way and sometimes that but only occasionally changes direction abruptly. The rotation seems to be associated with free-stream turbulence and is inhibited by shear or by uneven surface roughness. These effects are not influenced by sphere mounting, and sphere rotation does not seem important. Three series of experiments are described, each using different apparatus. In a Reynolds number range 3000–50000, calculated from sphere diameter and relative local velocity, repeatable lift values were observed in zones close to a flow boundary (the wind-tunnel wall), and remote from a boundary for smooth spheres. For rough spheres, results were erratic remote from the boundary. © 1981, Cambridge University Press. All rights reserved.

Description: A theoretical calculation is made of (an off-diagonal element of) the pressure-strain-rate term Po-1⟨p[∇u + (∇u)T]) for a simple turbulent shear flow at high Reynolds number. This calculation is described as follows. (1) An expression for the pressure-strain-rate term is analytically derived in terms of measurable quantities (velocity spectra) - this derivation makes use of a cumulant discard. (2) It is proved that, to the lowest order in the spectral anisotropy, the (nonlinear part of) the pressure-strain-rate term is linearly proportional to the Reynolds stress. (3) A formula is derived for the constant of this proportionality (the Rotta constant) in terms of arbitrary velocity spectra. (4) This formula is used to analytically calculate Rotta’s constant, Cxz, for a class of models of velocity spectra (the variation of Rotta’s constant caused by variations in the spectral shapes is examined). (5) It is found that Cxz is surprisingly insensitive to the large-wavelength part of the spectrum. This insensitivity suggests that Cxz should not vary much from one turbulence application to another provided that the Reynolds number is very large. However, it is also shown that Cxz is unexpectedly sensitive to the short-wavelength part of the spectrum, and varies with Reynolds number when the latter is less than about 30. The calculation is based on a straightforward solution of the Navier-Stokes equation to obtain formal expressions for u and p These expressions are then used to write the pressure-strain-rate in terms of a two-time fourth-order velocity correlation. The latter correlation is evaluated by a standard cumulant discard. Simplifying assumptions of the calculation are that average quantities vary little in space and time, and that the mean flow are unidirectional. These simplifications are made in order to emphasize the method of calculation and its details. © 1981, Cambridge University Press. All rights reserved.

Description: Many algorithms have been proposed and evaluated for dealing with laser-anemo-meter measurements in low particle densities since the original paper of McLaughlin & Tiederman (1973). All of them describe the effect of making a measurement whenever a particle is visible to the measurement instrument. However, in many circumstances, one wishes to sample the velocity at a constant rate-for instance, in measurement of velocity fluctuation spectra. It is shown here that the measurement statistics for this case are different from those previously discussed in the literature. The product of the particle density and the sample interval is the controlling parameter for the statistical description of the measurements. The asymptotic forms for low and high particle density-sample time products are derived. © 1981, Cambridge University Press. All rights reserved.

Description: Flow patterns near conical stagnation points in supersonic flow have been investigated on the basis of potential flow. Near the conical stagnation point the nonlinear equation for the conical velocity potential reduces to the equation of Laplace. Solutions of the equation of Laplace for incompressible plane flow are then used as a guide to generate conical stagnation-point solutions. Apart from known types of streamline patterns, such as nodes and saddle points, new types are found. Among them are oblique saddle points, saddle-nodes, topological nodes and topological saddle points. They may be used to clarify certain questions in a number of practical conical-flow problems. The oblique saddle point may be used to describe the inviscid flow associated with flow separation and also certain features of the flow over an external corner. The saddle-node, being structurally unstable, may fall apart into a saddle and a node. It may then be used to interpret the lift-off phenomenon of the singularity in the flow around a circular cone at incidence as a bifurcation. Similarly, this may be done for the appearance of a dividing streamline in the same flow at still higher angles of incidence, where a vortex system is formed at the leeward side of the cone. © 1981, Cambridge University Press. All rights reserved.

Description: The mixed convection in a two-dimensional line heat source is studied for the situations where buoyancy effects are favourable or adverse with respect to the oncoming vertical stream. The problem is analysed in terms of two co-ordinate expansions, direct and inverse, valid for small and large values of streamwise distance from the heat source. The solution for the first eleven and seven terms in direct and inverse co-ordinate expansions, respectively, are obtained. The direct expansion, when suitably transformed by Euler transformation and other techniques, predicts the velocity and temperature to two-digit accuracy for all values of streamwise coordinates, with a maximum error of 0.1 % for velocity, 0.8% for temperature and 2.2% for displacement thickness far downstream from the source. © 1981, Cambridge University Press. All rights reserved.

Description: A floating horizontal cylinder of infinite length is performing simple harmonic oscillations of small amplitude in the free surface of a uniform inviscid fluid under gravity. The cylinder intersects the mean free surface at right angles, and the fluid is bounded below by a fixed horizontal plane. Let the corresponding two-dimensional velocity potential be expressed as a distribution of simple wave sources over the boundary of the body. Then it is known that the source density satisfies a Fredholm integral equation of the second kind which has a unique solution except at those frequencies (the irregular frequencies) at which the Fredholm determinant vanishes. The present work is concerned with the irregular frequencies. Let the simple wave sources be replaced by a fundamental solution which consists of a simple wave source together with additional wave singularities inside the cylinder. It is shown how irregular frequencies can be eliminated by an appropriate choice of these singularities. © 1981, Cambridge University Press. All rights reserved.

Description: The Rayleigh theory of oscillation of liquid drops is extended to include the effects of viscosity and a uniform external electric field. The resonant frequencies of the modes of the drop are shown to be shifted by the electric field. The magnitude and sign of the frequency shift depends on the dielectric constant of the drop. The condition for instability of drops in large electric fields is given and found to differ from that given by previous workers. This difference is attributed to the assumption by previous workers that the drops, under the influence of an electric field, distort into ellipsoids of revolution about the field direction. The dynamical equations are derived and the solution for small oscillations is given in an oscillating field and in an amplitude-modulated optical field. © 1981, Cambridge University Press. All rights reserved.

Description: In an effort to resolve some controversies regarding the turbulent mixing-layer structure, the near field of a large (18 cm diameter) air jet has been investigated for the jet exit speed of 30 m s−1. The smoke-laden axisymmetric mixing layer has been illuminated by a thin sheet of laser light in an azimuthal plane passing through the jet axis. High-speed visualization films of the mixing layer in the region of its self-preservation (of which a few picture sequences depicting space-time evolutions of the structure of the layer are presented) reveal that most of the time the mixing layer is in a state of disorganization, consisting of relatively smaller scale, random and diffuse turbulent motions; only occasionally are organized distinct large-scale coherent structures formed. The survival distances of the large-scale structures are found to be comparable to their average sizes. The survival time of these structures is about one ‘turnover’ time, each being roughly about five times the local characteristic time scale of the mixing layer. It is seen that tearing is as dominant a mode of large-scale interaction as pairing is; large-scale structures are continually sheared and typically fragmented due to a segment on the high-speed side being torn and swept away from the slower-moving outer portion. Evolution of the large structures occur not primarily through complete pairing as widely believed but quite frequently through ‘fractional pairing’ between segments which have been torn from different upstream large-scale coherent structures or through ‘partial pairing’ when one structure captures only a part of another. The movies show that along with entrainment of non-vortical ambient fluid, radially outward ejection of vortical fluid into the ambient is an important aspect of jet mixing. From aligned displays of ciné film frame sequences, space-time trajectories of identifiable vortical fluid elements have been traced. The convection velocity variation across the shear layer and even the overall structure convection velocity measured from these trajectories agree with those determined from the wave-number-celerity spectra, obtained from double-Fourier transformation of longitudinal velocity space-time correlation measurements with hot-wires.The visualization films do not bear out the two-street vortex ring model recently propounded by Lau. Based on our observations, we propose that tearing, ‘slippage’ and fractional and partial pairings are responsible for the observed radial variation of structure passage frequency, and the causes of the different coherent structures educed by Bruun on the high- and low-speed sides of the mixing layer and for Yule's failure in educing a coherent structure on the low-speed side of the layer.

Description: This paper describes mean-flow and turbulence measurements conducted in a round jet over a range of Mach numbers from 0.3 to 1.7 and jet-exit static temperatures from — 40 to over 400 °C. It is a continuation of an earlier work, reported by Lau, Morris & Fisher (1979), to try to map the distribution of the various flow characteristics in the jet flow field and to observe the effects of changing jet exit conditions. In the earlier study, the effort was confined to isothermal jets at a limited number of exit Mach numbers, and the laser velocimeter proved to be a particularly useful instrument, especially in situations where the more severe flow conditions made it impossible to extract fluctuating-velocity data by any other means. The present effort capitalizes on this aspect of the velocimeter and also its ability to measure mean velocities accurately; and the extended range and detail of jet conditions chosen for this study is intended to provide a clearer understanding of the effects of systematically changing the jet conditions. Corresponding Pitot and total temperature measurements are also carried out under a representative set of jet conditions specifically to try to shed light on the effect of jet heating. Based on the various axial and radial distributions which are obtained, a picture is constructed of the changing boundaries of the shear layer with changing jet conditions. © 1981, Cambridge University Press. All rights reserved.

Description: A method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed is presented. No analytical approximations are made. A finite Fourier series, similar, to Dean’s stream function series, is used to give a set of nonlinear equations which can be solved using Newton’s method. Application to laboratory and field situations is emphasized throughout. When compared with known results for wave speed, results from the method agree closely. Results for fluid velocities are compared with experiment and agreement found to be good, unlike results from analytical theories for high waves. The problem of shoaling waves can conveniently be studied using the present method because of its validity for all wavelengths except the solitary wave limit, using the conventional first-order approximation that on a sloping bottom the waves at any depth act as if the bed were horizontal. Wave period, energy flux and mass flux are conserved. Comparisons with experimental results show good agreement. © 1981, Cambridge University Press. All rights reserved.

Description: A systematic low-frequency theory is developed for the propagation of one-dimensional sound waves in a variable-area duct. The mean flow in the duct is assumed to be isentropic, compressible, and one-dimensional. Two applications are made of the theory. One concerns the reflexion coefficient from a pipe—nozzle combination, in which case comparisons are also made with some experimental data. In the second application, we consider the case of a sonic throat separating subsonic and supersonic flow. In this case, if the mean Mach number distribution in addition to being unity at the throat is also stationary at the throat, there is an axial‘boundary-layer’ region in which the impedance of the sound wave changes from a fundamentally unsteady (reflexion-free) value at the sonic throat to the quasi-steady value away from the throat. © 1981, Cambridge University Press. All rights reserved.

Description: Previous measurements in nearly homogeneous sheared turbulence with a uniform mean temperature gradient are here supplemented with data on the fine structure of the velocity and temperature fluctuation fields. The statistics of signal derivatives and of band-passed signals show that neither field is locally isotropic in the spectral range covered, possibly because of the insufficiently large turbulent Reynolds and Péclet numbers. Observed skewnesses of both velocity and temperature derivatives are explained qualitatively with the use of a kind of ‘mixing-length model. The flatness factors of the derivatives and of band-passed, high-frequency signals indicate appreciable departures from normality, consistent with the spatially ‘spotty’ fine structure. The temperature ilatnesses are a bit larger than those of the streamwise velocity. The homogeneous shear flow data are compatible with measurements in turbulent boundary layers at comparable Rλand Pλ θ. © 1981, Cambridge University Press. All rights reserved.

Description: An experimental study of the three-dimensional turbulent boundary layer on a body of revolution is reported. The data correspond to axisymmetric flow as well as the flow at an angle of incidence of 15°, and include surface pressure distributions and the distribution of the magnitude and orientation of the velocity vector in the boundary layer. The results clearly exhibit most of the complexities encountered in practical three-dimensional boundary-layer flows, such as viscid-inviscid interaction, reversal of cross-flow, open separation and onset of longitudinal vortices. Major implications of these results on the development of computation procedures are discussed. © 1981, Cambridge University Press. All rights reserved.

Description: The paper presents a theoretical and experimental study of natural convection in a horizontal cavity which communicates laterally with a large reservoir. The cavity walls and the reservoir are at different temperatures. It is shown theoretically that the flow consists of a horizontal counterflow which penetrates the cavity over a distinct length. The penetration length is shown to be proportional to the cavity height and to the square root of the Rayleigh number based on cavity height and cavity-reservoir temperature difference. The validity of the theory is demonstrated on the basis of a flow visualization experiment. It is shown also that the Nusselt number for cavity-reservoir heat exchange is proportional to the square root of the Rayleigh number, and is relatively insensitive to the Prandtl number in the Pr range 0.7 to oo. The energy-engineering applications of the lateral penetration flow are discussed. © 1981, Cambridge University Press. All rights reserved.

Description: The structure of fully developed turbulence in a smooth circular tube has been studied for a Reynolds number of 69000 (based on centre-line velocity and radius) and at a distance from the wall of y+ — 70. The data were taken as correlations of the longitudinal component of turbulence in narrow frequency bands, the longitudinal and transverse separations being varied simultaneously. Fourier transformation of these correlations defines power spectral density functions with frequency ω and longit-dinal and transverse wavenumbers kx and kzas the independent variables. In this form the data show the distribution of convection velocity among waves of different size and inclination as well as defining the coherence lengths associated with such wave packets. Essential features of a geometrically similar wave description of the turbulence are discussed, such a model allowing considerable simplification in the description of the turbulence both for two-point and three-point space-time correlations of the velocity field. Morrison & Kronauer (1969) predicted that the wave convection velocity should depend only on total wavenumber k in a specific manner related to the mean velocity profile. The experimentally determined convection velocities contradict this prediction. An alternative formulation for convection velocity involving an additional empirical function of frequency S(ω), fits the data for the range of experimentation. Unfortunately the results provide no information on the functional dependence (if any) of convection velocity on distance from the wall. © 1981, Cambridge University Press. All rights reserved.

Description: A joint experiment to study microscale fluctuations of atmospheric pressure above surface gravity waves was conducted in the Bight of Abaco, Bahamas, during November and December 1974. Field hardware included a three-dimensional array of six wave sensors and seven air-pressure sensors, one of which was mounted on a wave follower. The primary objectives of the study were to resolve differences in previous field measurements by Dobson (1971), Elliott (1972b) and Snyder (1974), and to estimate the vertical profile of wave-induced pressure and the corresponding input of energy and momentum to the wave field. Analysis of a pre-experiment intercalibration of instruments and of 30 h of field data partially removes the discrepancy between the previous measurements of the wave-induced component of the pressure and gives a consistent picture of the profile of this pressure over a limited range of dimensionless height and wind speed. Over this range the pressure decays approximately exponentially without change of phase; the decay is slightly less steep than predicted by potential theory. The corresponding momentum transfer is positive for wind speeds exceeding the phase speed. Extrapolation of present results to higher frequencies suggests that the total transfer is a significant fraction of the wind stress (0.1 to 1.0, depending on dimensionless fetch). Analysis of the turbulent component of the atmospheric pressure shows that the ‘intrinsic’ downwind coherence scale is typically an order-of-magnitude greater than the crosswind scale, consistent with a ‘frozen’ turbulence hypothesis. These and earlier data of Priestley (1965) and Elliott (1972c) suggest a horizontally isotropic ‘intrinsic ’turbulent pressure spectrum which decays as k-v where k is the (horizontal) wave-number and v is typically —2 to — 3; estimates of this spectrum are computed for the present data. The implications of these findings for Phillips’ (1957) theory of wave growth are examined. © 1981, Cambridge University Press. All rights reserved.

Description: Steady, three-dimensional convection in rectangular boxes of fluid-saturated porous material with square horizontal cross-section heated from below is found to be non-unique. The properties of a special class of solutions exhibiting a high degree of symmetry are determined as a function of box size and Rayleigh number. The stability of these solutions to general three-dimensional perturbations is also determined. In some cases, when these solutions are found to be unstable, the alternative forms of three-dimensional convection are presented. Multiple three-dimensional steady states are given for a few particular values of box size and Rayleigh number. © 1981, Cambridge University Press. All rights reserved.

Description: In recent comparative measurements using a burst-counter type laser velocimeter and a hot-wire anemometer to assess the capabilities of the velocimeter (e.g. Barnett & Giel 1976; Lau, Morris & Fisher 1979), it was found that the laser velocimeter held good promise as an instrument for turbulence research, especially in high speed, high temperature flows where a hot-wire cannot be used. The axial mean velocities obtained with the LV compared very well with hot-wire measurements. Similarly, the characteristic shapes of the spectra and probability density distributions of the velocity fluctuations were faithfully reproduced. The trends in the distributions of the various turbulence characteristics (e.g. turbulence intensity, velocity covariances, skewness and kurtosis) in a given flow field were identical to those obtained with hotwires. The one significant difference between LV and hot-wire results was the magnitudes of the turbulence level. Since the LV results were obtained with the help of the latest validation and discrimination techniques (Asher 1973), which have now become standard equipment (Durst, Melling & Whitelaw 1976), such a discrepancy was unexpected. The reason for the discrepancy is now fairly clear and a method has been suggested by Whiffen, Lau & Smith (1978) on how to eliminate the error. But the approach is lengthy and time-consuming. This paper describes a method which effectively accomplishes the same end with less effort. © 1981, Cambridge University Press. All rights reserved.

Description: A rough-wall turbulent boundary layer which grows into a temperature interface situated at the outer edge is investigated experimentally. Reynolds numbers based on boundary-layer thickness δ range from 4 x 103 to 104. Overall Richardson numbers Ri, defined in terms of the friction velocity and the boundary-layer thickness, are in the range 0 ≤ Ri 80. Measurements of the mean profiles and of the variances of the velocity fluctuations show that the interface acts in some respects like a moving wall: the velocity profile tends towards a turbulent Couette-type profile and the longitudinal r.m.s. turbulent velocity begins to be amplified at the base of the interfacial layer and reaches a maximum in about the centre. Time-lag correlations of fluctuating quantities taken just above the centre of the interfacial layer have a behaviour characteristic of internal waves, namely a 90° phase lag between vertical velocity and temperature fluctuations. These waves occur as short wave packets and propagate mainly horizontally. On the base of the interface the correlations exhibit the usual symmetric behaviour. © 1981, Cambridge University Press. All rights reserved.

Description: Three-dimensional finite-amplitude thermal convection in a fluid layer is considered in the case where the boundaries of the layer are much poorer conductors than the fluid. It can be shown that if the conductive heat flux through the layer is not too large, the horizontal scale of motion is much greater than the layer depth. Then a ‘shallow water theory’ approximation leads to a nonlinear evolution equation for the leading-order temperature perturbation, which can be analysed in terms of a variational principle. It is proved that the preferred planform of convection is a square cell tesselation, as found in a rather more restricted parameter range by Busse & Riahi (1980), in contrast to the roll solutions that obtain for perfectly conducting boundaries. It is also shown that the preferred wavelength of convection increases slowly with amplitude. © 1981, Cambridge University Press. All rights reserved.

Description: The thermal non-equilibrium boundary layer at the end wall of a shock tube in methane combustion initiated by a reflected shock, is investigated theoretically and experimentally. Time-dependent boundary conditions are caused by the shock-boundary-layer interaction and the combustion process outside the boundary layer, and are taken into account. Space- and time-resolved density measurements using a focused laser beam are in good agreement with results from numerical computation and show existence of similarity solutions for short and long times. Differences, so far unexplained, occur between measured and predicted heat fluxes to the wall. © 1981, Cambridge University Press. All rights reserved.

Description: Characteristics of the turbulent motion in a cylinder wake have been measured during passage through a distorting section of a wind tunnel, the overall effect of the distortion being considerable lateral extension and compression without a considerable change of flow velocity. However, the sectional area is not constant and rates of longitudinal extension are comparable with the rates of lateral straining. Hot-wire anemometers, mostly in X-configurations, are used to measure mean velocities, turbulent intensities, Reynolds stresses, intermittency factors and spectra, and velocity correlations have been calculated from digital recordings of the outputs from arrays of eight single-wire anemometers. In contrast to previous investigations, the direction of compression is parallel to the axis of the cylinder, and the original entrainment eddies of the plane wake are suppressed rather than amplified. © 1981, Cambridge University Press. All rights reserved.

Description: By using the technique of multiple scaling a theory of gas-bubble oscillations in a liquid is developed. Collections of bubbles of arbitrary shape under the action of surface tension, buoyancy and solid surfaces are considered. In the absence of thermal conduction in the bubbles and compressibility of the liquid a conserved ‘ action ‘ is defined for each of the modes of oscillation. The equation governing the decay of the action with time is found by carrying the analysis to second order. The geometrical configuration of the bubbles, in which the oscillations take place, evolves in time under convection by an underlying ‘basic’ flow for which the governing equations are derived. The bubble pulsations influence the development of the basic motion. Later in the work a source of gas bubbles is brought in and its effects on the oscillations discussed. The results of the interaction of pulsating bubbles with the liquid surface are also briefly considered. The determination of the amplitude of oscillations induced by the splitting up of bubbles and by the generation of bubbles from the gas source is described. Finally, several applications of the theory to specific problems are given. © 1981, Cambridge University Press. All rights reserved.

Description: This note has two aspects. The first concerns comparisons between the theoretical predictions of Smith (19796) and the calculations of Fornberg (1980) for flow past a circular cylinder: there is quite good agreement overall, in such quantities as the drag, the front stagnation point pressure, the eddy pressure and the skin friction. The second aspect concerns comments and reservations on the calculated eddy lengths and discrepancies at higher Reynolds numbers. © 1981, Cambridge University Press. All rights reserved.

Description: A differencing scheme known as the ‘Fluid-in-Cell’ method has been used in the numerical simulation of a choked jet of air impinging on a flat plate. Before sonic conditions are applied at the nozzle exit, the field of interest is at rest at the ambient pressure and temperature. The instantaneous application of sonic conditions at the nozzle exit is tantamount to the sudden appearance of a normal shock wave whose strength is determined by the experimental conditions. The results of the simulation describe the decay of the initial shock wave and its reflection at the plate; the formation of a second shock wave and its merging with the reflected shock giving rise to a detached shock wave which oscillates; and the growth and subsequent motion of a toroidal vortex that is generated between this shock wave and the plate. The results show clearly how the flow field which has been observed in physical experiments under stable operating conditions is developed. © 1981, Cambridge University Press. All rights reserved.

Description: Experimental results are presented concerning the spontaneous oscillations observed when a high-viscosity fluid jet fiows vertically against a flat surface. The two jet shapes investigated were the axisymmetric jet and the plane jet. The minimum distance from the jet orifice to the flat surface for which these oscillations are observed, termed the ‘buckling height’, was determined experimentally. The frequency of the subsequent oscillations was also determined. Both were measured as functions of fluid and flow variables. It is found that surface tension effects are the dominant factors influencing the buckling height, while the rate of oscillation is affected by both surface tension effects and by viscous, gravity and inertia effects. The major results are presented in non-dimensional form. Photographs of the buckling phenomenon are provided for representative jet geometries. It is also established experimentally that there is an upper limit to the flow Reynolds number above which buckling does not occur. © 1981, Cambridge University Press. All rights reserved.

Description: Simpson, Chew & Shivaprasad (1981 a, b) describe many experimentally determined features of a separating turbulent boundary layer. For the same flow, experimental results for the transverse velocity component are presented here. A specially designed directionally sensitive laser anemometer was constructed and used to make measurements in the separated region. Cross-wire hot-wire anemometer measurements were obtained upstream of separation and in the outer region of the separated flow and are in good agreement with the laser anemometer results. It was found that u'2 = v'2 in the outer 90 % of the shear layer both upstream and downstream of separation. Features of w'2 profiles in the backflow are related to features of the streamwise velocity component. This behaviour is consistent with the large-scale-structures flow model of a separating boundary layer presented by Simpson et al. (1981, b).Large-scale structures supply the mean streamwise backflow. These large-scale structures also transport the turbulence energy to the backflow from the outer flow by turbulent diffusion since advection and production of turbulence kinetic energy are negligible there compared with the dissipation rate. Because of continuity requirements fluid motions toward the wall must be deflected and contribute to streamwise and transverse motions near the wall. © 1981, Cambridge University Press. All rights reserved.

Description: Instability of the alternate-bar type in straight channels has long been identified as the cause of fluvial meandering. The condition of inerodible sidewalls, however, does not allow a meandering channel to develop. Herein a stability analysis of a sinuous channel with erodible banks allows for delineation of a ‘bend' instability that does not occur in straight channels, and differs from the alternate-bar instability. © 1981, Cambridge University Press. All rights reserved.

Description: In this paper we describe a simple way of modelling boundary-layer effects in analytical flow noise studies. We develop an exact analogy between the real flow and one in which there is a step velocity profile. This profile is intended to model a boundary layer in an idealized way and we recognize it in our Green's function for the problem. We insist that the Green's function is bounded, a step that makes it non-causal and similar to those used in recent jet-noise analogies. We derive an expression for the induced pressure which consists of surface and volume terms, just as in Lighthill's theory, but, because both contain elements to be evaluated in future time, we argue that the turbulence must be able to respond to linear surface stimulus and avoid the otherwise inevitable violation of causality. This novel feature distinguishes our analysis from applications of Lighthill's theory to boundary-layer-induced noise. The response of the turbulence may be large when the surface is driven at low boundary-layer Strouhal number. But it is negligible at high Strouhal number, and in that limit the surface terms are found to depend only on the instantaneous boundary geometry and its rate of change. This leads to a simple expression for ‘boundary-layer fluid loading’, in which the finite boundary-layer scale emerges explicitly. To illustrate the physical consequences of this result, we use it to estimate the impedance of a baffled piston vibrating beneath a boundary layer. Potential theory predicts that flow should destabilize the piston motion while experiments usually indicate the reverse. We find that the boundary layer is responsible for the discrepancy and that experimentally observed behaviour is predicted quite reasonably by our model. © 1981, Cambridge University Press. All rights reserved.

Description: The velocity-probability-distribution flatness and skewness factors for u and I are reported for the separating turbulent boundary layer described by Simpson, Chew & Shivaprasad (1981). Downstream of separation the skewness factor for u is negative near the wall, whereas it is positive upstream of separation. The flatness factor for u downstream of separation differs from the upstream behaviour in that it has a local maximum of about 4 at the minimum mean velocity location in the backflow. Both upstream and downstream of separation the skewness factor for I has a profile shape and magnitudes that are approximately the mirror image or negative of the skewness factor for u. The flatness factor for I seems to be affected little by separation. Examination of the momentum and turbulence-energy equations reveals that the effects of normal stresses are important in a separating boundary layer. Negligible turbulence-energy production occurs in the backflow. Turbulence-energy diffusion is increasingly significant as separation is approached and is the mechanism for supplying turbulence energy to the backflow. The backflow appears to be controlled by the large-scale eddies in the outer region flow, which provides the mechanism for turbulence-energy diffusion. The backflow behaviour does not appear to be significantly dependent on the far downstream near-wall conditions when the thickness of the backflow region is small compared with the turbulent shear layer thickness. © 1981, Cambridge University Press. All rights reserved.

Description: This paper is a summary of the Third Beer-Sheva Seminar on magnetohydrodynamic (MHD) flows and turbulence, held in Israel in March 1981 with 67 participants from 9 countries. Reviews and research papers were presented on fundamental MHD and turbulence studies, both theoretical and experimental, including two-phase phenomena, and on applications of MHD to electrical generation (especially in two-phase systems), electromagnetic pumps, flow-couplers and flowmeters, thermonuclear fusion and a range of metallurgical problems, many involving free surfaces. © 1981, Cambridge University Press. All rights reserved.

Description: The results of experiments involving instability, known as fingering, in a circular Hele Shaw cell with inward and outward flow are presented. The width of fingers in this situation is examined, and an approximate equation for the growth of fingers is proposed. The equation rα - cos (n) is shown to fit the shape of long fingers. © 1981, Cambridge University Press. All rights reserved.

Description: Based on theoretical analysis and laboratory data, we proposed a unified two-parameter wave spectral model as [Formula omitted] with β and m as functions of the internal parameter, the significant slope η of the wave field which is defined as [Formula omitted] is the mean squared surface elevation, and λ0, n0 are the wavelength and frequency of the waves at the spectral peak. This spectral model is independent of local wind. Because the spectral model depends only on internal parameters, it contains information about fluid-dynamical processes. For example, it maintains a variable bandwidth as a function of the significant slope which measures the nonlinearity of the wave field. And it also contains the exact total energy of the true spectrum. Comparisons of this spectral model with the JONSWAP model and field data show excellent agreements. Thus we established an alternative approach for spectral models. Future research efforts should concentrate on relating the internal parameters to the external environmental variables. © 1981, Cambridge University Press. All rights reserved.

Description: The motion of a very thin vortex filament is investigated using the localized induction equation. A family of vortex filaments which move without change of form are obtained. They are expressed in terms of elliptic integrals of the first, second and third kinds. In general they do not close and have infinite lengths. In some particular cases they take the form of closed coils which wind a doughnut. There exist a family of closed vortex filaments which do not travel in space but only rotate around a fixed axis. Our solutions include various well-known shapes such as the circular vortex ring, the helicoidal filament, the plane sinusoidal filament, Euler's elastica and the solitary-wave-type filament. It is shown that they correspond to the travelling wave solution of a nonlinear Schrodinger equation. © 1981, Cambridge University Press. All rights reserved.

Description: The stability of pipe flow to axisymmetric disturbances is studied by direct numerical simulation of the incompressible Navier-Stokes equations. There is no evidence of finite-amplitude equilibria at any of the wavenumber/Reynolds number combinations investigated, with all perturbations decaying on a time scale much shorter than the diffusive (viscous) time scale. In particular, decay is obtained where amplitude-expansion perturbation techniques predict equilibria, indicating that these methods are not valid away from the neutral curve of linear stability theory. © 1981, Cambridge University Press. All rights reserved.

Description: This paper complements an earlier paper by Bentwich & Miloh in which the matched asymptotic expansion type of solution is presented for an unsteady low-Reynolds-number flow past a sphere when a constant rectilinear velocity is suddenly imparted to the sphere. It is shown that the matching procedure proposed in the earlier paper is incomplete. The present paper represents a complete procedure for successful matching; the drag of the sphere is calculated up to the term of 0(Re2 In Be) using the new procedure. © 1981, Cambridge University Press. All rights reserved.

Description: The head of an intrusive flow advancing along the interface between two fluids is studied experimentally when the two layers are of equal depth and the density of the intrusion is the mean of the two densities. The dependence of the flow on the interface thickness and the depth of the intrusion is determined. When the interface is very thin the flow is similar to the nominally inviscid gravity currents observed by Britter & Simpson (1978). © 1981, Cambridge University Press. All rights reserved.

Description: The thermal response of hot-film anemometers to fluid velocity fluctuations is investigated for low-Prandtl-number fluids. A quartz-coated hot-film probe with a diameter of 0⋅002 inches and an aspect ratio of 20 was oscillated in a horizontal plane while immersed in a steadily rotating tank of mercury. The probe was oscillated sinusoidally from 2 to 1200 Hz with a vibrator. The amplitude of velocity fluctuation was regulated to about 20 % of the mean flow within a Peclet-number range of 0⋅1–1⋅0. The findings concur with the theoretical results obtained numerically and published earlier by Malcolm & Verma (1973) and compare well with some results of other researchers. The results confirm that the sensitivity of the hot-film probes is inhibited at low Peclet numbers, even at quite low frequencies in liquid metals owing to their very low Prandtl numbers. Two important effects are noticed: (a)The amplitude of fluctuation is attenuated and the degree of attenuation depends upon a non-dimentional quantity αf/U2, for the range of Peclet numbers considered, whereL α is the thermal diffusivity, f is the frequency of the fluctuations and is the free-stream velocity, in compatible units. The amplitude is attenuated by 10 % and 90 % at αf/U2 values of 0⋅02 and 4⋅0 respectively. (b)There is a phase lag in the hot-film probe signal with respect to the true velocity of the fluctuation which is somewhat the same as that in potential flow at low frequencies, but is considerably higher than that in potential flow at higher frequencies. The measured lag does not level off asympotically at high frequencies as noted in the numerically obtained results for potential flow. Corrections may be made to unsteady velocity measurements in low Prandtl number fluids to account for the above effects with some confidence depending upon the value of the αf/U2 quantity. The results of the investigation are of more general interest, in that the hot-film sensor can be considered as a model of a long circular cylinder in a flow at low to moderate Peclet numbers. © 1981, Cambridge University Press. All rights reserved.

Description: At low Reynolds numbers, three-dimensional features are frequently observed in the vortices shed behind a basically two-dimensional circular cylinder. This paper deals with the dependence of the configuration of the vortices on various end constructions. The cylinder is towed at a uniform speed in a water tank and simple flow visualization is used. It is found that the three-dimensional structure of the wake depends strongly on the flow configuration at each end of the cylinder. The boundary condition imposed on the nascent vortex lines determines the subsequent behaviour of the shed vortices. Consequently, the vortex street can be rendered more nearly two-dimensional by allowing the vortices to link outside the boundary as they approach that boundary normally. This is the case for the water-air interface when the water surface is clean. In the case of a contaminated water surface or of a solid surface acting as a boundary to the vortex street, the vortices link between themselves underneath the water surface and a strong interaction takes place behind the end of the cylinder. The subsequent effect is a bowing of the vortices towards the end of the cylinder. The free-end effect at the bottom end of the cylinder induces a strong bowing of the vortices towards that end and causes the wake to contract. It follows from the effect of surface contamination that the study of vortex wakes by the spreading of some surface contaminants might not necessarily show the true behaviour of the wake below the surface. It is postulated that slantwise shedding arises from a difference in the two end effects. © 1981, Cambridge University Press. All rights reserved.

Description: The development of a laminar boundary layer on a twisted helical blade is described. An appropriate co-ordinate system is developed in which the boundary-layer equations have a relatively simple form. The choice of blade geometry and the free-stream conditions result in a constant-pressure flow. This permits the flow to be considered the analogue, in a rotating frame, of the zero-pressure-gradient flat-plate boundary layer in a stationary frame. The boundary-layer equations are solved using a double series expansion in powers of distance from the leading edge and the cosine of the blade twist angle. Chordwise and span wise velocity profiles are calculated. The variation in the skin friction coefficients is calculated as a function of position on the blade. © 1981, Cambridge University Press. All rights reserved.

Description: An exact solution to the Navier-Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented. By means of a similarity transformation the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically. For the two-dimensional flow in a channel, a single solution is found to exist when the Reynolds number R is less than 310. When R exceeds 310, two additional solutions appear and form a closed branch connecting two different asymptotic states at infinite R. The large R structure of the solutions consists of an inviscid fluid core plus an 0(R-½) thin boundary layer adjacent to the moving wall. Matched-asymptotic-expansion techniques are used to construct asymptotic series that are consistent with each of the numerical solutions. For the axisymmetric non-swirling flow in a tube, however, the situation is quite different. For R 〈 10.25, two solutions exist which form a closed branch. Beyond 10.25, no similarity solutions exist within the range 10.25 〈 R 〈 147. Once R exceeds 147, multiple solutions reappear, which form two closed branches that connect four different asymptotic states at infinite R. The possibility of an axisymmetric flow with swirl is considered, and two sets of swirling solutions are found to exist for all R 〉 0. These solutions, however, do not evolve from the R = 0 state nor do they bifurcate from the non-swirling solutions at any finite value of R. © 1981, Cambridge University Press. All rights reserved.

Description: The circular hydraulic jump commonly forms on a horizontal plate struck by a vertical jet of liquid. New observations of this phenomenon are described. A previously unreported instability of the jump is examined. This is shown to arise when the local Reynolds number Rj just ahead of the jump exceeds a critical value of 147. Prior to this instability, the flow behind the jump contains a closed eddy, the length of which decreases to zero as R j increases towards its critical value. Physical explanations for this flow structure and instability are proposed. Accurate measurements of liquid depths were made using a light-absorption technique, in which a laser was shone through water containing a strong dye. Liquid depths ahead of and behind the jump were so determined and depth profiles of the jump in the stable regime were obtained. As the outer depth was increased, the jump closed in on the jet and eventually disappeared: this extinction of the jump is also investigated. © 1981, Cambridge University Press. All rights reserved.

Description: An experimental investigation is described of the velocity field in a steady, spillingtype breaker, generated on a steady current by a submerged hydrofoil. Velocities have been measured with a laser-doppler system, and analysed with respect to mean and r.m.s. values as well as Reynolds stresses. The results indicate that the turbulent flow field downstream of the initiation of the separation at the surface resembles that in a self-similar turbulent wake. © 1981, Cambridge University Press. All rights reserved.

Description: A simple derivation of recent shallow-flow equations with bed curvature is presented. These equations are applied to steady flow over a high overflow spillway crest, to obtain the head-discharge relationship and the crest-pressure distribution in good agreement with experiment. It is inferred that the equations are valid for quite large negative curvature. On the other hand, their application to steady flow over a spillway toe indicates their validity for positive curvature is more limited. © 1981, Cambridge University Press. All rights reserved.

Description: The rolling-up of the trailing vortex sheet produced by a wing of finite span is calculated as a series expansion in time. For a vorticity distribution corresponding to a wing with cusped tips, the shape of the sheet is found by summing the series using Padé approxi-mants. The sheet remains analytic for some time but ultimately develops an exponential spiral at the tips. The centroid of vorticity is conserved to high accuracy. © 1981, Cambridge University Press. All rights reserved.

Description: The effect of rotation on the flow around an obstacle in a Hele Shaw cell is considered. For the flow outside the strip-shaped boundary layer near the obstacle, it is shown that the rotating and non-rotating cases are related by a simple transformation. The flow in the boundary layer near the obstacle is computed numerically and its relation to the Stewartson E¼ and [Formula Omitted] layers is briefly discussed. © 1981, Cambridge University Press. All rights reserved.

Description: A nonlinear theory is presented for the resonance of long gravity waves trapped on an uneven bottom when a long packet of short swells is incident. By allowing the trapped wave to be comparable in amplitude to the incident swells, the transient evolution of trapped waves is studied from initial growth through maturity to final decay, for swell packets of finite duration. For a totally submerged ridge, it is found that, while the trapped waves are resonated by second-order periodic modulations of the swell envelope, energy transfer to short swells and nonlinear emission of long waves act as damping mechanisms to limit the amplification. Bottom friction is not qualitatively crucial and affects the results only quantitatively. For a closed beach, breaking of short swells is dealt with empirically but resonant modulation is still the primary factor in exciting surf beats. © 1981, Cambridge University Press. All rights reserved.

Description: Previous work by the present authors on the onset of convection in a layered porous medium heated from below is extended to an investigation of the heat transported by convection at slightly supercritical Rayleigh numbers. The two-dimensional convection patterns and associated values of the critical Rayleigh number, cell width and slope of the Nusselt-number graph are calculated for two- and three-layer configurations over a wide range of layer depth and permeability ratios. The results show that the commonly studied problem of a homogeneous layer bounded above and below by impermeable boundaries is a special case, in that the slope of the Nusselt-number graph at the critical point is nearly independent of cell width. For a homogeneous layer with a permeable upper boundary, and for multi-layered systems, the slope of this graph depends strongly on cell width. © 1981, Cambridge University Press. All rights reserved.

Description: Calculations of the primary electroviscous effect have previously been restricted to spherical particles. Here we examine a suspension of randomly orientated rod-shaped particles of length I. Two competing mechanisms are present in a linear velocity field. The flow relative to each rod is largest at the rod ends. When I is large the consequent large distortion of the charge cloud increases the electroviscous effect. On the other hand, the given total charge on a uniformly charged rod is spread more thinly as I increases, and this tends to reduce the electroviscous effect. The balance of these two mechanisms is examined. © 1981, Cambridge University Press. All rights reserved.

Description: The dispersion relation and component phase speeds of surface gravity wavefields and modulated wavetrains are calculated. A parametric study is performed for a range of nonlinearity and spectral bandwidths. It is found that the amount of departure from linear theory increases with the ratio of nonlinearity to spectral bandwidth. The calculated results are compared quantitatively with laboratory and ocean measurements of wavetrains and wavefields with and without wind. The good agreement between theory and experiment suggests that the nonlinearity-dispersion balance is a likely candidate to account for the observed discrepancy between linear theory and data, as well as for the difference in behaviour between laboratory and oceanic wave measurements. © 1981, Cambridge University Press. All rights reserved.

Description: The paper considers the problem of calculating the statistical characteristics of a passive scalar dispersed by a homogeneous turbulence field. In many turbulent shear flows the time-scale for the evolution of the scalar field is intrinsically related to that of the turbulent velocity field. This is by no means always the case, however, and it is at this more general situation that the present work is aimed. An approximate transport equation for the rate of dissipation of scalar variance is proposed which, it is argued, must contain (at least) two sink terms one of which responds to the time scale of the velocity field while the other reflects that of the scalar field itself. The model has been applied to the limited number of homogeneous scalar flows for which data are available and achieves satisfactory agreement as judged by the evolution of the mean-square scalar variance. © 1981, Cambridge University Press. All rights reserved.

Description: The dispersion of a spot of contaminant in a high-Péclet-number laminar flow is studied by means of the ray method developed by Cohen & Lewis (1967). This method is free from the usual severe restrictions on the time range. Thus, it is possible to investigate strong shear-distortions of the concentration distribution. Also, the effects of rigid boundaries can be allowed for simply by including reflected rays. Three examples are studied in detail: stagnation flow, a point vortex and plane Poiseuille flow. © 1981, Cambridge University Press. All rights reserved.

Description: In this paper, an experimental determination of the three-dimensional velocity distribution due to the physiological pulsatile flow of a Newtonian, incompressible fluid at various locations in a curved tube of circular cross-section is presented. Our results show four interesting features of the pulsatile flow development in the curved tube. First is the presence of a reversed flow along the inner wall of the tube during the diastolic (deceleration) phase of the pulsatile flow cycle. Second is that the flow does not appear to be fully developed in the curved tube through the cross-section whose L/a ratio is equal to 169, the final location at which measurements were made in this study, where L is the axial length and a is the radius of the curved tube. A third feature observed is the vacillation of the peak axial velocity across the horizontal diameter of the tube from the upstream to the downstream region in the curved tube. In the upstream region (L/a = 34), the maximum axial velocity measured occurred nearest to the outer wall. The maximum axial velocity shifted towards the inner wall in the middle of the tube (L/a = 102), while in the downstream region (L/a = 169), the maximum axial velocity measured was again near the outer wall. Finally, trapped vortical motions are observed to occur at the inner wall of the tube in the downstream region. © 1981, Cambridge University Press. All rights reserved.

Description: The displacement height appears in the logarithmic velocity profile for rough-wall boundary layers as a reference height for the vertical co-ordinate. It is shown that this height should be regarded as the level at which the mean drag on the surface appears to act. The equations of motion then show that this also coincides with the average displacement thickness for the shear stress. A simple analytical model, experimental results and dimensional analysis are all used to indicate how the displacement height depends upon the detailed geometry of the roughness elements. © 1981, Cambridge University Press. All rights reserved.

Description: An approximate integral-equation approach is used to model the growth and collapse of a vapour cavity in close proximity to an initially plane free surface. By comparison with experiment, it is shown to predict all the salient features of the bubble and free-surface interaction, provided that the complete nonlinear Bernoulli pressure condition is applied on both surfaces. Features observed and predicted include the formation of an accelerating liquid jet in the bubble and a pronounced spike in the free surface during the collapse phase of the bubble's life. If the bubble is initially sufficiently close to the free surface, it will become ‘entrained' in the raised free surface with a veneer of liquid separating the two free surfaces. © 1981, Cambridge University Press. All rights reserved.

Description: The ‘preferred mode’ of an incompressible axisymmetric free jet has been organized through controlled perturbation, and spatial distributions of time-average as well as phase-average flow properties in the near field are documented. The excitation produces noticeable changes in the time-average measures of the jet, although these changes are less dramatic than those for the excitation producing stable vortex pairing. For different stages in the evolution of the preferred-mode coherent structure, the phase-average vorticity, coherent Reynolds stress, and incoherent turbulence intensities and Reynolds stress have been educed through phase-locked hot-wire measurements, over the spatial extent of the structure and without invoking the Taylor hypothesis. For a particular stage of the evolution (i.e. when the structure is centred at x/D ≃ 3) the distributions of these quantities have been compared for both initially laminar and fully turbulent exit boundary layers, and for four jet Reynolds numbers. The relative merits of the coherent structure streamline and pseudo-stream-function patterns, as compared with phase-average velocity contours, for structure boundary identification have been discussed. The structure shape and size agree closely with those inferred from the average streamline pattern of the natural structure educed by Yule (1978).These data as well as τ-spectra show that even excitation at the preferred mode cannot sustain the initially organized large-scale coherent structure beyond eight diameters from the jet exit. The background turbulence is organized by the coherent motions in such a way that the maximum rate of decrease of the coherent vorticity occurs at the structure centres which are the saddle points of the background-turbulence Reynolds-stress distributions. The structure centres are also the locations of peak phase-average turbulence intensities. The evolving shape of the structure as it travels downstream helps explain the transverse variations of the wavelength and convection velocity across the mixing layer. The coherent structure characteristics are found to be independent of whether the initial boundary layer is laminar or turbulent, but depend somewhat on the jet Reynolds number. With increasing Reynolds number, the structure decreases in the streamwise length and increases in the radial width and becomes relatively more energetic, and more efficient in the production of coherent Reynolds stress.

Description: The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k-3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t 〈 tc) and the similarity period (t 〉 tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero. © 1981, Cambridge University Press. All rights reserved.

Description: An analysis is presented for the propagation of water waves past a rectangular submarine trench. Two-dimensional, linearized potential flow is assumed. The fluid domain is divided into two regions along the mouth of the trench. Solutions in each region are expressed in terms of the unknown normal derivative of the potential function along this common boundary with the final solution obtained by matching. Reflection and transmission coefficients are found for various submarine geometries. The result shows that, for a particular flow configuration, there exists an infinite number of discrete wave frequencies at which waves are completely transmitted. The validity of the solution in the infinite constant-water-depth region is shown by comparing with the results using the boundary integral method for given velocity distributions along the mouth of the trench. The accuracy of the matching procedure is also demonstrated through the results of the boundary integral technique. In addition, laboratory experiments were performed and are compared with the theory for two of the cases considered. © 1981, Cambridge University Press. All rights reserved.

Description: High-resolution measurements have been performed of the convective heat current as a function of time when a Rayleigh- Bénard cell is swept through its threshold with a specified time-dependent heat input. The results are interpreted in terms of the amplitude equation which exactly describes the slow variations in space and time of hydrodynamic quantities near the threshold. A phenomenological forcing field is added to this equation, and its form and magnitude are fitted to the onset time of the convective heat current. A deterministic model in which the field is an adjustable constant yields a good fit to the data for both a step and a linear ramp in the heat input. An alternative stochastic model, in which the field is a Gaussian variable with zero mean and a white-noise spectrum, is adequate for the ramp experiments, but cannot fit the step data for any value of the mean-square field. The systematics of the field and onset time versus ramp rate are studied in both the deterministic and stochastic models, and attempts are made to interpret the field in terms of physical mechanisms. When the data for long times are analysed in terms of the amplitude equation, it is found that the state first excited at onset is not the roll pattern which is stable in steady state. Instead, the system goes first to an intermediate state, which we tentatively identify as a hexagonal configuration. The decay of this state is governed by a further adjustable field in the amplitude equation. © 1981, Cambridge University Press. All rights reserved.

Description: Closed-form expressions for the turbulent mean reaction rate and its covariance with the temperature are derived for premixed and non-premixed combustion. The limit of large activation energies is exploited for a chemical reaction rate that, by virtue of coupling functions, depends on the mixture fraction and a non-equilibrium progress variable only. The probability density function (p.d.f.) formulation with an assumed shape of the p.d.f. is used; a beta-function distribution is assumed for the progress variable. The mean reaction rate is expressed in terms of the mean and the variance of the temperature and, for non-premixed combustion, of the mixture fraction. The reaction kinetics are represented by the non-dimensional activation energy and the laminar flame velocity. For non-premixed systems the possibility of local extinction by flame stretch is considered. © 1981, Cambridge University Press. All rights reserved.

Description: Experimental results are presented for steady, supercritical flow of a liquid in a thin-walled compliant tube which is in a state of partial collapse due to a negative transmural pressure. Particular attention is paid to the effects of longitudinal tension. With a constant external pressure, friction acts to increase the area in the downstream direction. With the tube tilted downward, friction may be so balanced by gravity forces as to result in an asymptotic approach to an equilibrium situation in which the area and velocity remain constant. When the downstream pressure is increased sufficiently, shock-like transitions to subcritical inflated states, with positive transmural pressure, occur. The longitudinal length scale of the shock is on the order of one to several tube diameters. The pressure rise across the shock lies between that for loss-free pressure recovery and that given by the Borda-Carnot sudden-expansion theory. The presence of longitudinal tension causes a train of standing waves of area to appear upstream of a local area disturbance, such as a shock-like transition. The standing waves are superimposed upon the more gradual area changes associated with friction and gravity. The wave amplitude grows in the downstream direction. Theoretical interpretations of the observations are presented in the companion paper (part 2). © 1981, Cambridge University Press. All rights reserved.

Description: In a previous paper (Grimshaw 1979) the resonant over-reflection of internal gravity waves from a vortex sheet was considered in the weakly nonlinear regime. It was shown there that the time evolution of the amplitude of the vortex sheet displacement was balanced by a cubic nonlinearity. For one vortex sheet mode, symmetrical with Tespect to the interface, it was shown that a steady finite-amplitude wave was possible. For the other, asymmetric modes, a singularity develops in a finite time. In the present paper, that analysis is extended by replacing the vortex sheet with a thin shear layer of thickness α2, where α is the amplitude of the shear layer displacement. The effect of this extension is to introduce a linear growth rate term in the amplitude equation, which is otherwise unaltered. The linear growth rate can be computed from a formula due to Drazin & Howard (1966, p. 67). The effect on the modes is that the symmetric mode is linearly damped and requires sustained forcing to be observed, while the asymmetric modes are slightly destabilized by the linear term and, as in the vortex-sheet model, develop a singularity in finite time. © 1981, Cambridge University Press. All rights reserved.

Description: We report theoretical growth rates for the Rayleigh-Béiard instability when the fluid layer is contained by non-slip walls in a cylindrical geometry with diameter D and height L. Our results are for the growth rates of the first two axisymmetric modes as functions of the Prandtl number P and the aspect ratio γ ≡D/2L. We have considered the two extreme cases of ideally insulating and ideally conducting side walls, and found that the growth rate is relatively insensitive to the choice of the thermal boundary conditions on the side walls. Our results are useful in understanding recent experimental measurements of the convective time-scale. © 1981, Cambridge University Press. All rights reserved.