ALBERT

Description: This list includes age measurements carried out from December 1968 to October 1969. Archaeologic samples are from Italian and Swat (W Pakistan) territories. All geologic samples come from Italian territory. Chemical techniques remain the same (Bella and Cortesi, 1960; Alessio, Bella, and Cortesi, 1964).

Description: Radiocarbon dates obtained since August 1969 are included in this report. The procedures followed and equipment used have been described previously (Radiocarbon, 1966, v. 8, p. 522-533).

Description: The C14 dates given below are a continuation of the work presented in our previous list (Radiocarbon, 1969, v. 11, p. 451-462), and results obtained mainly during 1969 are described. A new 3.3 L copper counter was put into routine operation besides the 2.7 L stainless steel counter employed heretofore, yielding background counting rates of 8.5 and 5.5 cpm, respectively, when filled with dead CO2 at ca. 1.8 atm.

Description: The following list comprises measurements made since those reported in Radiocarbon, 1969, v. 11, p. 130–136. No changes have been made in measurement technique or in the method of calculating the results described in Radiocarbon, 1965, v. 7, p. 156–161. It was necessary during 1968 to replace all the geiger counters used in the anti-coincidence rings, but the long term stability of background and standard count rates implicit in the use of a 20-week rolling mean has been maintained.

Description: This list contains the results of measurements made during 1967 and 1968. Samples are counted in the form of acetylene gas, as previously, and ages computed on the basis of the Libby half-life, 5568 ± 30 yr. The error listed, always larger than the one-sigma statistical counting error commonly used, takes into account variable laboratory factors, but does not include external (field or atmospheric) variations.

Description: This list includes selected dates of archaeologic samples from DDR, Czechoslovakia, Hungary and the Soviet Union made between 1966 and 1969. As in previous lists the major portion of dates are concerned with Neolithic and Early Bronze age period in Middle and SE Europe.

Description: The C14 measurements reported here were made in this laboratory between October 1968 and October 1969.

Description: The following list presents results obtained during 1968-69 on a series of samples chosen to investigate temporal variations of C14 concentrations in the atmosphere during the past century. Together with data presented previously (Radiocarbon, 1969, v. 11, p. 45-52) they constitute a study of annual variations of C14 activities at N temperate latitudes.

Description: The following list covers samples measured since the last list of atmospheric samples (Radiocarbon, 1967, v. 9, p. 471–476) was written, to determine the increase of the C14/C12 ratio due to explosion of nuclear devices.

Description: The laboratory has been operating more than five years, supported entirely from Venezuelan government funds. Approximately 530 samples have been dated to the end of 1969. Of these, 40% are archaeologic specimens, the majority (141 samples) from Venezuela. Materials from most other countries of Latin America have been processed. The rest of the samples are from research programs of the Radiocarbon Laboratory itself. Thirty per cent of the total are measurements on carbonate species extracted from Venezuelan ground water aquifers.

Description: The present date list covers mainly the datings done from 1964 to 1968. Each sample is measured in one of the two counting units described earlier (Nydal, 1965). The background of the counters has been somewhat reduced during the past few years. Counter 2 with an effective volume of 1.5 L (1.9 L total volume), has a background of 0.9 counts/min, and a recent standard net count of 19.2 counts/min. Counter 3, with an effective volume of 1.1 L (1.3 L total volume), has a background of 2.4 counts/min and a recent standard net count of 14.2 counts/min.

Description: This date list includes those series of samples completed in this laboratory as of November 1969. The b.p. ages are based upon a.d. 1950, and are calculated with a half-life value of 5568 yr. All samples were counted at least twice for periods of not less than 1000 minutes each. Errors quoted are derived from measurement of samples, background, and modern-age calibration, but do not include any half-life error. All samples were pretreated with 3N HCl, and some, where noted, were given additional pretreatment with 2% NaOH for the removal of possible humic contaminants.

Description: The C14 dates given below have been obtained by counting CH4 at 3 atm pressure in a 0.6 L stainless steel counter. Details of procedure are given in the previous lists. Dates are reported in terms of the Libby half-life, 5570 ± 30 years; the errors quoted are based on the standard deviations in counting rate of samples and standards.

Description: The list given below contains the C14 dates obtained for only archaeologic samples, during 1966 and 1967; for the study of the geologic samples dated during the same period, cf. our previous list Gif-III (Radiocarbon, 1969, v. 11, p. 327-344). The same instruments and techniques have been employed.

Description: A third radiocarbon counting system has been established in the Chemistry Department, University of Glasgow, since April, 1968. Operating conditions for the previous systems have remained essentially as described by Baxter et al. (1969).

Description: This list contains mainly radiocarbon dates for Africa. Some results for samples from the ocean floor and from islands in the Atlantic and Indian oceans, as well as a few series of geophysical samples have also been included. The results are grouped into three categories: geologic, archaeologic, and geophysical, and arranged according to the country of provenance, passing roughly from N to S. Descriptions and comments are based on information supplied by the collectors and submitters and on the publications cited. In several cases insufficient information was available, but since it is improbable that it will be forthcoming, the dates have been included.

Description: The following date list includes samples processed by the Illinois State Geological Survey Radiocarbon Dating Laboratory from September 1968 through November 1969. Detailed descriptions of sample methods are published elsewhere (Kim and Ruch, 1969; Kim, Ruch, and Kempton, 1969).

Description: The radiocarbon dates obtained since August, 1968, are reported here. Wood, charcoal, and peat samples are pretreated with dilute NaOH and dilute H3PO4 before conversion to the methane used as counting gas; marls and lake cores are treated with acid only. The reported dates have been calculated using 5568 years as the half-life of C14, 1950 as the reference year. Samples are run at least once in each of two 0.5 liter counters at 3 atm pressure for a minimum total of 15,000 counts. The standard deviation quoted includes only the 1σ of the counting statistics of background, sample, and standard counts.

Description: This list contains a selection of dates from analyses carried out during the past few years. Samples are grouped in geologic-palynologic, and archaeologic sections according to main problem. When influence of human activity on pollen diagrams has been observed it has been explicitly indicated. The descriptions and comments have been written in collaboration with collectors and submitters. For the samples from Switzerland, general reference has also been made to Welten (1958a) and to sections on palynology, geology, and archaeology in work edited by the Schweizerische Gesellschaft für Ur- und Frühgeschichte (1968-1970).

Description: Measurements have continued with both the 1 L and 6 L counters. Results are not corrected for C13 fractionation. Errors quoted refer only to the standard deviation calculated from a statistical analysis of sample and background count rates and the Libby half-life of 5570 ± 30 yr. Pretreatment has been continued as described previously (Shotton, Blundell, and Williams, 1969).

Description: The measurements reported in this list were made in the Louvain C14 Dating Laboratory in 1968 with the 0.6L CH4 proportional counter. The counter generally operates at 3 atm pressure. Samples too poor to provide 2 L methane are measured at 1000 mm Hg pressure. Ages are calculated on the basis of a C14 half life of 5570 yr and are quoted with 1σ counting error. The description of each sample is based on information provided by the submitters.

Description: This date list presents results of samples measured at ISOTOPES during 1968 and 1969 and several measurements made previously for which complete sample data has recently been received.

Description: Both the 2-L counter, described in GSC I (Radiocarbon, 1962, v. 4, p. 13–26), and the 5-L counter (GSC IV, Radiocarbon, 1965, v. 7, p. 24–46) were operated routinely during the past year. Approximately half the determinations reported were obtained from each counter. The 2-L counter was operated at 2 atm except for August and September, 1968, when it was operated for the first time at 1 atm. This allowed for the counting of most small samples without the necessity of mixing with dead gas. The 5-L counter was operated at 1 atm, except for the same period when it was operated at 4 atm. Carbon dioxide is used as the counting gas, and both counters are of the proportional type.

Description: The present list covers samples measured between 1963 and July, 1967. Three previous lists have been published in Leningrad I (Sovetskaya arkheologiya, 1961, p. 3); Leningrad II (The Absolute Geochronology of the Quaternary Period, 1963); and Leningrad III (New Methods in Archaeological Investigations, 1963, p. 9–56). The samples measured include charcoal from cultural layers and hearths, wood from barrows [kurgans] and cemeteries, wooden tools from peat deposits, and mounds, as well as peat and animal tissue.

Description: This list reports the first age determinations carried out by the Freiberg Radiocarbon Dating Laboratory. The preparation of samples and radiocarbon dates were done by the first two authors, who constructed the apparatus; sample descriptions and interpretations of dates were made by the third author.

Description: The laboratory has continued to concentrate on soil and water dating, using the benzene method as outlined in Scharpenseel and Pietig (1969a).

Description: The present date list describes the first stage of a co-operative study on the validity of dating secondary soil carbonates in arid and semi-arid environments of Australia. Because of the complex nature of the physical and chemical variables in a soil environment, many additional samples are being dated from stratigraphically controlled sites before final evaluation of carbonate reliability is possible.

Description: This radiocarbon dating laboratory was established to complement the research activities in the field of archaeology and geology in Korea. The benzene liquid scintillation counting method (Noakes, Kim, and Stipp, 1965; Noakes, Kim, and Akers, 1967) is employed in this laboratory because of its compatibility with other dating methods and future possibility in application of this procedure to the biomedical research field. The chemical process for converting carbon from a sample to benzene used by this laboratory is briefly outlined below.

Description: The dating equipment in the Palaeoecology Laboratory has remained essentially as described in Belfast I (this volume). Rewiring of the counter has increased the detection efficiency slightly. Background count corrected to 1606 mb is now 11.0 counts/min and the net count rate for 95% of the NBS oxalic acid standard is 56.0 counts/min. All charcoal samples have been pretreated in accordance with the schedule given in Belfast I.

Description: The dating equipment in the Queen's University Palaeoecology Laboratory was installed to provide data for research projects, initially dealing with the development of agriculture, in the departments of Botany and Archaeology.

Description: The following list shows the age measurements made at Algiers by the Service des Applications Nucléaires using the C14 method. The laboratory was created in 1965 to answer the increasing demand of the archaeologic research in N Africa, and made its first measurements in 1967. The electronic apparatus employed was completed and calibrated in 1968 and the first tests were performed during the same year. Regular dating-test samples started in June, 1968. The electronic circuits are of commercial origin or made in the C14 Laboratory.

Description: Taylor (1953, 1954a) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Fick's law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a2 〉 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Fick's law does not hold. © 1970, Cambridge University Press. All rights reserved.

Description: A Boussinesq fluid is heated from below. The applied temperature gradient is the sum of a steady component and a low-frequency sinusoidal component. An asymptotic solution is obtained which describes the behaviour of infinitesimal disturbances to this configuration. The solution is discussed from the viewpoint of the stability or otherwise of the basic state, and possible stability criteria are analyzed. Some comparison is made with known experimental results. © 1970, Cambridge University Press. All rights reserved.

Description: The acoustic field radiated by a multipole point source positioned near to the surface of a solid sphere is calculated at both low and high frequencies. It is shown that the scattered field at low frequencies is always dipole, but at high frequencies is of the same type as the incident field. The application of the results to the acoustic field radiated by turbulence near a sphere is briefly discussed. © 1970, Cambridge University Press. All rights reserved.

Description: Experiments conducted elsewhere show that a mean fluid motion can be induced in a channel by a travelling thermal wave. An analysis is carried out, linearized under the assumption that the induced motion is slower than the speed of the heat source. The expression for the mean motion is obtained for any Prandtl number and circular frequency of the thermal wave, to complete the results presented by Davey (1967) for low and high frequency ranges. In the problem of the flow between two parallel plates, it is found that with a temperature profile symmetric about the centre of the channel, the induced flow does not exert a net shear force on either plate, while with a non-symmetric one, the plates are subjected to equal and opposite forces. For the problem that the upper surface of the fluid is free and thermally insulated, an approximated result can be deduced from that of the previous problem by a simple transformation. It should agree with the result of Davey, obtained through a more elaborate procedure, except in the low frequency range when the surface deformation becomes important. In agreement with the experiments, our analysis indicates that the induced mean motion is always in a direction opposite to that of the thermal wave, and its magnitude increases rapidly with decreasing Prandtl number. According to the theory, some of the previous experiments were not conducted under the optimum situations, and improved experimental conditions are suggested. © 1970, Cambridge University Press. All rights reserved.

Description: An experimental investigation of a large long air bubble moving into stationary water in a horizontal channel of rectangular cross-section is presented and three well-defined flow régimes for the water discharged beneath the bubble are described. The influence of surface tension on the bubble velocity is explained using the hypothesis that the radius of curvature of the two-phase interface close to the upper wall does not vary greatly with channel depth and is close to the theoretical value for a channel of such depth that the bubble is just motionless. © 1970, Cambridge University Press. All rights reserved.

Description: Energy and linear limits are calculated for the Poiseuille–Couette spiral motion between concentric cylinders which rotate rigidly and rotate and slide relative to one another. The addition of solid rotation can bring the linear limit down to the energy limit with coincidence achieved in the limit of infinitely fast rotation. If the differential rotation is also added, the solid rotation rate need be only finite to achieve near coincidence. Sufficient conditions for non-existence of sub-linear instability are derived. The basic spiral character of the instability is analysed and the results compared with the experiments of Ludwieg (1964). © 1970, Cambridge University Press. All rights reserved.

Description: The exact solution of the Stokes equations for the creeping motion of two spheres of arbitrary size and arbitrarily oriented with respect to a shear field is obtained by use of spherical bipolar co-ordinates. Numerical results are given for two special cases: (1) the free motion of two equal-sized spheres in simple shear flow and (2) the free motion of a sphere near a wall in the rotational shear field between two parallel disks rotating at different rates. The sphere trajectories calculated for the first of these problems are found to agree fairly well with those observed experimentally. © 1970, Cambridge University Press. All rights reserved.

Description: The stability of the laminar flow in a rectangular channel with aspect ratio 1:8 was investigated experimentally, with and without artificial excitation. The critical Reynolds number based on the hydraulic diameter and the average velocity was found to be 2600. Behaviour of damped and growing waves, using artificial excitation, was examined in detail. In particular the progress of growing disturbances was followed. Breaking was found to be the ultimate fate of a growing wave. Spectra of growing and damped waves were also obtained. Measurements were made for wavelengths, wave speeds and amplification or damping rates. The neutral stability boundary in the αr, R plane was determined. In the damped region, comparison of several aspects of the behaviour of the measured disturbances with the plane Poiseuille theory for spatial decay yielded good agreement. Three-dimensionality and non-linear subcritical instability were briefly examined. Neutral subcritical waves at low Reynolds numbers appeared possible when the exciter amplitude was quadrupled. The possible bearings of the present study on the stability of plane Poiseuille flow are suggested. © 1970, Cambridge University Press. All rights reserved.

Description: This paper presents incompressible laminar boundary-layer results on both the leeside and windside of a prolate spheroid. The results are obtained by an implicit finite difference method of the Crank–Nicolson type. Particular attention has been given to the determination of separation and of embedded streamwise vortices. No restriction on the angle of attack or the thickness ratio is imposed, nor are there invoked any of the common assumptions such as similarity, conical flow and others. The results suggest an embedded vortex region existing between the regular boundary-layer region and the separated region. At higher angle of attack, the vortex region becomes so thick that it itself may be more appropriately called ‘separated’ also. The latter possibility leads to questions of applicability for existing theories on three-dimensional separation. © 1970, Cambridge University Press. All rights reserved.

Description: The effect of a density stratification on the steady, mechanically driven motion of a viscous fluid in a rotating cylinder with axis aligned with the rotation and gravity vectors and with parallel top and bottom surfaces that slope with respect to the plane perpendicular to the rotation vector is studied by a linear theory. Primary attention is given to a study of the alteration of the characteristics of the flow of a homogeneous fluid by the addition of a weak stratification. It is found, for example, that in the range [formula omitted], where E = v/ΩL2 and σS = vαgΔT0/κΩ2L, and with a homogeneous boundary condition on the perturbation temperature, the interior velocity is parallel to the direction perpendicular to the plane determined by the vector normal to the top surface and the rotation vector. The circulation closes in an inviscid, but heat-conducting, boundary layer of thickness E¾(σS)−½ on the side wall. Thus, with stratification, the steady flow in this configuration differs markedly from the corresponding flow in a cylinder where the top and bottom surfaces lie in planes perpendicular to the rotation vector. The difference is caused by the fact that in the container with sloping surfaces the basic stratification interacts with the geostrophic flow whereas, in the other case, the interaction is with the much smaller Ekman layer suction velocities. © 1970, Cambridge University Press. All rights reserved.

Description: The general problem studied is the propagation of an oblique shock wave through a two-dimensional, steady, non-uniform oncoming flow. A higher-order theory is developed to treat the refraction of the incident oblique shock wave by irrotational or rotational disturbances of arbitrary amplitude provided the flow is supersonic behind the shock. A unique feature of the analysis is the formulation of the flow equations on the downstream side of the shock wave. It is shown that the cumulative effect of the downstream wave interactions on the propagation of the shock wave can be accounted for exactly by a single parameter Φ, the local ratio of the pressure gradients along the Mach wave characteristic directions at the rear of the shock front. The general shock refraction problem is then reduced to a single non-linear differential equation for the local shock turning angle θ as a function of upstream conditions and an unknown wave interaction parameter Φ. To lowest order in the expansion variable θΦ, this equation is equivalent to Whitham's (1958) approximate characteristic rule for the propagation of shock waves in non-uniform flow. While some further insight into the accuracy of Whitham's rule does emerge, the theory is not a selfcontained rational approach, since some knowledge of the wave interaction parameter Φ must be assumed. Analytical and numerical solutions to the basic shock refraction relation are presented for a broad range of flows in which the principal interaction occurs with disturbances generated upstream of the shock. These solutions include the passage of a weak oblique shock wave through: a supersonic shear layer, a converging or diverging flow, a pure pressure disturbance, Prandtl–Meyer expansions of the same and opposite family, an isentropic non-simple wave region, and a constant pressure rotational flow. The comparison between analytic and numerical results is very satisfactory. © 1970, Cambridge University Press. All rights reserved.

Description: The problem of a piston moving in a cylinder is studied experimentally using flow visualization techniques. A vortex motion is observed at the piston face and cylinder wall interface as the cylinder wall moves toward the piston. Non-dimensional scaling parameters for the vortex size and stability are determined and semi-empirical theories for the size of the vortex are presented. © 1970, Cambridge University Press. All rights reserved.

Description: A solution satisfying the usual radiation conditions is found to the problem of an internal wave propagating towards a corner. It is found that, far from the corner, and the characteristic emanating from the corner, the solution is asymptotically equivalent to the solution found by plane wave reflexions from an infinite wall. The present solution shows that, by imposing the radiation condition, a singularity predicted by the ray theory along the corner characteristic is absent. A further singularity in the present solution along the same characteristic is shown to be due to an inability of the usual linear internal wave equations to fully describe the motion. The solution is for restricted corner angles. © 1970, Cambridge University Press. All rights reserved.

Description: The Stokes creeping flow, induced by the passage of a uniform current parallel to the axis of a stationary non-conducting ellipsoid of revolution in an incompressible viscous fluid occupying, apart from the ellipsoidal region, the whole space, is investigated. The magnetic field, which is due to the distortion of the uniform current by the ellipsoid, is zero all over the surface of the ellipsoid. The induced flow field is symmetric with respect to the axis, and also with respect to a plane through the centre perpendicular to the axis of the ellipsoid. The case of a non-conducting circular disk, with its plane perpendicular to the direction of the undisturbed current, is deduced from that of a planetary ellipsoid. © 1970, Cambridge University Press. All rights reserved.

Description: The axisymmetric motion of an inviscid, rotating liquid over a prescribed stream surface, say S, is constructed from assumed values of the velocity and azimuthal vorticity on S. The hypothesis of unseparated flow, which implies continuity of the vorticity on S, is shown to imply that: (a) the azimuthal vorticity and azimuthal circulation (relative to the basic flow) must be simply proportional to the perturbation stream function in the exterior of S; (b) the exterior field exhibits a dipole behaviour far upstream of the body, thereby satisfying Long's hypothesis of no upstream disturbance. © 1970, Cambridge University Press. All rights reserved.

Description: This paper describes some experiments in rotating flows in which solitary waves were observed. In one set of experiments the waves were generated on a swirling flow whose circumferential velocity distribution resembled that of the Rankine combined vortex. This flow was established by stirring the liquid in a large cylindrical container, in much the same way as one stirs a cup of tea, and it was often found at the cessation of the stirring that a wave had been generated. This wave propagated along the vortex core and was reflected at the bottom of the container and at the free surface of the liquid and displayed the remarkable permanence characteristic of solitary waves. It appears that, to a first approximation, the speed of the waves may be calculated simply from the depression of the free surface of the liquid at the centre of the vortex. These waves are the rotating-fluid counterpart to the solitary waves in fluids of great depth recently discussed by Benjamin (1967b) and by Davis & Acrivos (1967). In a second set of experiments, solitary waves were generated in a long cylindrical tube and are analogous to the familiar solitary wave of open-channel flows. The theory indicates that these waves are possible in any swirling flow in which the angular velocity is distributed non-uniformly. Thus, a long liquid-filled tube was started rotating about its axis with a uniform angular velocity, and waves were generated before the fluid had reached a state of uniform rotation. Using the known velocity distribution for a tube of infinite length, comparisons have been made between the observed wave forms and the theoretical calculations of Benjamin (1967a). There is good agreement between the observed wave forms and the theoretical predictions. © 1970, Cambridge University Press. All rights reserved.

Description: Experiments are carried out to establish the relationship between the nature of the flow instability and the inclination angle of the plate. The angular dependence of the Rayleigh number characterizing the onset of instability is also determined. An electrochemical flow visualization technique is utilized to expose the patterns of fluid motion. It is found that for inclination angles of less than 14° (relative to the vertical), waves are the mode of instability. On the other hand, for inclination angles in excess of 17°, the instability is characterized by longitudinal vortices. The range between 14° and 17° is a zone of continuous transition, with the two modes of instability co-existing. © 1970, Cambridge University Press. All rights reserved.

Description: The stability of a sand bed in an alluvial channel is investigated by a two-dimensional mathematical model, based on the vorticity transport equation. The model takes account of the internal friction and describes the non-uniform distribution of the suspended sediment. It turns out that the inclusion of the friction and of a definite model of the sediment transport mechanism leads to results rather different from those obtained previously by potential-flow analysis. © 1970, Cambridge University Press. All rights reserved.

Description: The Hunt–Stewartson technique of estimating fluid velocity and magnetic flux profiles in rectangular ducts is generalized for the entire secondary boundary layer. © 1970, Cambridge University Press. All rights reserved.

Description: A linearized analysis of the two-dimensional double vortex sheet model of a jet shows that inviscid jet instabilities occur over a wide range of frequencies at all jet Mach numbers. No particular frequency for maximum growth rate exists unless finite shear layer thickness effects are considered. It is suggested that the model describes the essential characteristics of a real jet disturbed by long wavelength perturbations. The idea is advanced that the jet flow constitutes a broad band amplifier of high gain. Disturbances can grow rapidly to a size when nonlinear effects bring about significant interaction with the mean flow. By seeding the jet with disturbances of a type that are highly amplified it is argued that gross features of the flow may be affected and that the jet may be rendered less noisy at high Mach number. It is argued that some of these ideas are supported by the observation that a supersonic jet diffuses at an unusually rapid rate when subject to the oscillatory condition known as ‘screech’. © 1970, Cambridge University Press. All rights reserved.

Description: An effort is made to understand turbulence in fluid systems like the oceans and atmosphere in which the Richardson number is generally large. Toward this end, a theory is developed for turbulent flow over a flat plate which is moved and cooled in such a way as to produce constant vertical fluxes of momentum and heat. The theory indicates that in a co-ordinate system fixed in the plate the mean velocity increases linearly with height z above a turbulent boundary layer and the mean density decreases as z3, so that the Richardson number is large far from the plate. Near the plate, the results reduce to those of Monin & Obukhov. The curvature of the density profile is essential in the formulation of the theory. When the curvature is negative, a volume of fluid, thoroughly mixed by turbulence, will tend to flatten out at a new level well above the original centre of mass, thereby transporting heat downward. When the curvature is positive a mixed volume of fluid will tend to fall a similar distance, again transporting heat downward. A well-mixed volume of fluid will also tend to rise when the density profile is linear, but this rise is negligible on the basis of the Boussinesq approximation. The interchange of fluid of different, mean horizontal speeds in the formation of the turbulent patch transfers momentum. As the mixing in the patch destroys the mean velocity shear locally, kinetic energy is transferred from mean motion to disturbed motion. The turbulence can arise in spite of the high Richardson number because the precise variations of mean density and mean velocity mentioned above permit wave energy to propagate from the turbulent boundary layer to the whole region above the plate. At the levels of reflexion, where the amplitudes become large, wave-breaking and turbulence will tend to develop. The relationship between the curvature of the density profile and the transfer of heat suggests that the density gradient near the level of a point of inflexion of the density curve (in general cases of stratified, shearing flow) will increase locally as time goes on. There will also be a tendency to increase the shear through the action of local wave stresses. If this results in a progressive reduction in Richardson number, an ultimate outbreak of Kelvin–Helmholtz instability will occur. The resulting sporadic turbulence will transfer heat (and momentum) through the level of the inflexion point. This mechanism for the appearance of regions of low Richardson number is offered as a possible explanation for the formation of the surfaces of strong density and velocity differences observed in the oceans and atmosphere, and for the turbulence that appears on these surfaces. © 1970, Cambridge University Press. All rights reserved.

Description: A simple model is suggested to describe flow in a Hele-Shaw cell when the Hele-Shaw parameter Λ is not necessarily small. The averaged flow is potential with a conservative body force proportional to the local velocity. The elementary ramifications of this are deduced and comparisons made with experiment. In particular no separation is predicted if Λ is less than O(1), in agreement with experiment. Furthermore, the separation cavities occurring for large Λ are completely stagnant. The theory predicts attached viscous shear layers in the wake of a lifting body, reminiscent of certain MHD problems. These tails were not observed experimentally. © 1970, Cambridge University Press. All rights reserved.

Description: A cylindrical tank, full of fluid, is oscillating with frequency ω and rotating with angular velocity Ω about its axis of symmetry. It is assumed that the amplitude of oscillation, δ, is small and the viscosity is low such that boundary layers exist. Analysis shows that the unsteady boundary layer is of thickness [ε/(1 − 2Ω/ω)]½ on the top and bottom plates and of thickness ε½ on the side walls, where ε = v/2ω. The interior unsteady flow shows source-like behaviour at the corners. The steady flow field is caused by the steady component of the non-linear centrifugal forces coupled with an induced steady rotation of the interior. This rotation, of order δ2ω, is prograde when Ω/ω 〈 0·118 and retrograde otherwise. Maximum retrograde rotation occurs at Ω/ω = 0·5. A steady boundary layer of thickness [ε/(1 − 2Ω/ω)]½ exists on the top and bottom plates, and of thicknesses [formula ommitted] on the side walls. Experimental measurements of the interior induced steady rotation compare well with theory. © 1970, Cambridge University Press. All rights reserved.

Description: This and the subsequent part of the Journal of Fluid Mechanics will be devoted to the publication of papers on turbulence. Each of these papers was the subject of an oral report at a Symposium on Turbulence sponsored by the Boeing Company and organized by the Flight Sciences Laboratory of the Boeing Scientific Research Laboratories, a t Seattle, Washington, U.S.A., from 23 to 27 June 1969. The Organizing Committee consisted of Dr A. Goldburg, Prof. L. S. G. Kovasznay (Chairman), andDr Y.-H. Pao (Secretary). At this Symposium invited participants listened both to review lectures and to contributed papers, and tried by discussion to assess the present position in the study of turbulence.

Description: Meecham and his co-workers have developed a theory of turbulence involving a truncated Wiener–Hermite expansion of the velocity field. The randomness is taken up by a white-noise function associated, in the original version of the theory, with the initial state of the flow. The mechanical problem then reduces to a set of coupled integro-differential equations for deterministic kernels. We have solved numerically an analogous set for Burgers's model equation and have computed, for the sake of comparison, actual random solutions of the Burgers equation. We find that the theory based on the first two terms of the Wiener–Hermite expansion predicts an insufficient rate of energy decay for Reynolds numbers larger than two, because the equations for the kernels contain no convolution integrals in wave-number space and therefore permit no cascade of energy. An energy cascade in wave-number space corresponds to a cascade up through successive terms of the Wiener-Hermite expansion. Pictures of the Gaussian and non-Gaussian components of an actual solution of the Burgers equation show directly that only higher-order terms in the Wiener–Hermite expansion are capable of representing shocks, which dissipate the energy. Higher-order terms would be needed even for a nearly Gaussian field of evolving three-dimensional turbulence. ‘Gaussianity’, in the experimentalist's sense, has no bearing on the rate of convergence of a Wiener–Hermite expansion whose white-noise function is associated with the initial state. Such an expansion would converge only if the velocity field and its initial state were joint-normally distributed. The question whether a time-varying white-noise function can speed the convergence is treated in the paper following this one. © 1970, Cambridge University Press. All rights reserved.

Description: Derivatives of velocity and temperature in the wind over the ocean were found to be quite variable. Probability distribution functions of squared derivatives were consistent with lognormality predictions by Kolmogoroff, Obukhoff and Yaglom. Kurtosis values for velocity derivatives ranged from 13 to 26 and from 26 to 43 for temperature derivatives. Universal inertial subrange constants were evaluated from dissipation spectra and were found to be 40 to 300% larger than most values reported previously. Evidence for local anisotropy of the temperature field is provided by non-zero values of the measured derivative skewness. © 1970, Cambridge University Press. All rights reserved.

Description: Recent theoretical results for the turbulent flow of polymer solutions in round tubes have been extended to deduce the similarity laws for the boundary-layer flow of drag-reducing polymer solutions. The analysis shows directly how the drag reduction depends on the elastic properties of the fluid and thereby defines the levels of elasticity necessary to achieve significant reductions in drag. Calculations employing available physical property measurements of highly elastic (0·1%) and moderately elastic (0·01%) polymer solutions indicate that, for boundary layers on large objects, drag reduction may not occur at polymer concentrations that are economically attractive. For example, at a Reynolds number of 109 the reduction in drag is predicted to be 60% and 10% for the concentrated and dilute polymer solutions respectively. Some savings in polymer however, may be realized by special injection techniques or fluid systems with specially tailored properties. © 1970, Cambridge University Press. All rights reserved.

Description: An analysis is presented for the interaction of capillary and gravity waves in a liquid layer of finite depth. The method of multiple scales is used to obtain a third-order expansion uniformly valid for all times. Although this expansion is valid for a wide range of wave-numbers, it breaks down at two critical wave-numbers if the liquid depth is larger than √3/kc, kc = (ρg/T)½, where g is the gravitational acceleration, and ρ and T are the liquid density and surface tension, respectively. For a deep liquid, the singularities are at kc/√2 and kc/√3 respectively, as found by Wilton (1915), and Pierson & Fife (1961). A second-order expansion valid for wave-numbers near the first critical value (corresponding to a wavelength of 2·44 cm in deep water) is obtained. This expansion shows that two different wave profiles could exist at or near the first critical wave-number. One of these profiles is gravity-like while the other is capillary-like. © 1970, Cambridge University Press. All rights reserved.

Description: Results for three interrelated problems are obtained by making use of solutions of boundary-value problems obtained in a different context. The first one concerns a thin rigid circular disk rotating in a slow stream of viscous fluid, both when the fluid is conducting and when it is non-conducting. For the case of a conducting fluid formulae are given for both small and large Hartmann numbers. The second problem concerns a disk performing simple harmonic rotary oscillations about its axis of symmetry in a non-conducting viscous fluid which is at rest at infinity. The last problem is that of an arbitrary axisymmetric solid oscillating about its axis of symmetry in a bounded viscous fluid, and the solution is illustrated by the case of an oscillating disk. © 1970, Cambridge University Press. All rights reserved.

Description: A problem with possible physiological applications concerns the escape of a viscous fluid through a small hole in a wall. The solution presented here is for a line source of sinusoidally pulsating strength located at the origin x = y = 0, where the plane y = 0 is a rigid wall and the fluid is at rest at y = + ∞. The linearized Navier—Stokes equations are solved, and results in the form of streamline plots are shown and discussed. © 1970, Cambridge University Press. All rights reserved.

Description: A quantitative shallow water simulation technique for aerodynamic noise study is investigated. The technique is employed for the case of a free jet issuing from a nozzle with variable wall roughness. The experimental results obtained are in remarkably good agreement with the theory. It is concluded that there is great potential for further work in even less well understood areas of aerodynamic noise research. © 1970, Cambridge University Press. All rights reserved.

Description: The spin-up of a homogeneous rotating fluid bounded at the top and/or bottom by a permeable medium has been proposed by Bretherton & Spiegel (1968) as a model for the spin-up in natural flows where turbulent processes transmit the direct effect of the boundaries deeper into the fluid than does the laminar Ekman layer. The theoretical analysis for the spin-up of a laterally unbounded fluid bounded by a permeable medium below is presented here. In addition, an experimental study of the process is presented. Theory and experiment agree reasonably well with a maximum difference of about 8% in the predicted and measured spin-up times. The effects of the side-wall boundary have been studied theoretically by Howard (1969). Experimental observations in the side-wall boundary layer confirm qualitatively the results of Howard's theory. © 1970, Cambridge University Press. All rights reserved.

Description: An expansion scheme is developed to describe the steady axisymmetric flow of a thin tubular liquid film of varying radius; the necessary small parameter is provided by the ratio between the characteristic film thickness and the characteristic tube radius. The co-ordinate system used is an orthogonal one based on the fluid interface and the fluid streamlines. The differential equations that arise thus treat the metric as an unknown set of variables. The method is restricted to situations dominated by viscous forces. Reference is made to numerical solutions that have been obtained in connexion with an industrial polymer-film-blowing process. © 1970, Cambridge University Press. All rights reserved.

Description: In the present paper the stability of plane Couette flow is studied. The stream function and distribution of vorticity and the Reynolds stresses for the linearized solution are computed. The stability effect of the non-linear terms are also discussed and it is found that for small amplitudes, the non-linear terms are destabilizing. A neutral curve determining the necessary amplitude in order to get instability, is found. The convergence of the expansion in the latter case is, however, somewhat uncertain and the result should therefore only be considered as a first, rough approximation. © 1970, Cambridge University Press. All rights reserved.

Description: A solid long slender body is considered placed in a fluid undergoing a given undisturbed flow. Under conditions in which fluid inertia is negligible, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio of the cross-sectional radius to body length. Specific examples are given for the resistance to translation of long slender bodies for cases in which the body centre-line is curved as well as for those for which the centre-line is straight. © 1970, Cambridge University Press. All rights reserved.

Description: Observations are reported on the growth of vortices in the vortex sheets bounding the jet emerging from a sharp-edged two-dimensional slit and from a sharp-edged circular orifice. A regular periodic flow is observed near the orifice for both configurations when the Reynolds number of the jet lies between about 500 and 3000. The two-dimensional jet produces a symmetric pattern of vortex pairs with a Strouhal number of 0·43. Vortex rings are formed in the circular jet with a Strouhal number of 0·63. Computer experiments show that a growing pair of vortices in two parallel vortex sheets produces a symmetric pattern of vortices upstream from the original disturbance. © 1970, Cambridge University Press. All rights reserved.

Description: This paper attempts to emulate the great study by Goldstein (1929) ‘On the vortex wake of a screw propeller’, by looking for a dynamical theory of how another type of propulsion system has evolved towards ever higher performance. An ‘undulatory’ mode of animal propulsion in water is rather common among invertebrates, and this paper offers a preliminary quantitative analysis of how a series of modifications of that basic undulatory mode, found in the vertebrates (and especially in the fishes), tends to improve speed and hydromechanical efficiency. Posterior lateral compression is the most important of these. It is studied first in ‘pure anguilliform’ (eel-like) motion of fishes whose posterior cross-sections are laterally compressed, although maintaining their depth (while the body tapers) by means of long continuous dorsal and ventral fins all the way to a vertical ‘trailing edge’. Lateral motion of such a cross-section produces a large and immediate exchange of momentum with a considerable ‘virtual mass’ of water near it. In ç 2, ‘elongated-body theory’ (an extended version of inviscid slender-body theory) is developed in detail for pure anguilliform motion and subjected to several careful checks and critical studies. Provided that longitudinal variation of cross-sectional properties is slow on a scale of the cross-sectional depth s (say, if the wavelength of significant harmonic components of that variation exceeds 5s), the basic approach is applicable and lateral water momentum per unit length is closely proportional to the square of the local cross-section depth. The vertical trailing edge can be thought of as acting with a lateral force on the wake through lateral water momentum shed as the fish moves on. The fish's mean rate of working is the mean product of this lateral force with the lateral component of trailing-edge movement, and is enhanced by the virtual-mass effect, which makes for good correlation between lateral movement and local water momentum. The mean rate of shedding of energy of lateral water motions into the vortex wake represents the wasted element in this mean rate of working, and it is from the difference of these two rates that thrust and efficiency can best be calculated. Section 3, still from the standpoint of inviscid theory, studies the effect of any development of discrete dorsal and ventral fins, through calculations on vortex sheets shed by fins. A multiplicity of discrete dorsal (or ventral) fins might be thought to destroy the slow variation of cross-sectional properties on which elongated-body theory depends, but the vortex sheets filling the gaps between them are shown to maintain continuity rather effectively, avoiding thrust reduction and permitting a slight decrease in drag. Further advantage may accrue from a modification of such a system in which (while essentially anguilliform movement is retained) the anterior dorsal and ventral fins become the only prominent ones. Vortex sheets in the gaps between them and the caudal fin may largely be reabsorbed into the caudal-fin boundary layer, without any significant increase in wasted wake energy. The mean rate of working can be improved, however, because the trailing edges of the dorsal and ventral fins do work that is not cancelled at the caudal fin's leading edge, as phase shifts destroy the correlation of that edge's lateral movement with the vortex-sheet momentum reabsorbed there. Tentative improvements to elongated-body theory through taking into account lateral forces of viscous origin are made in ç4. These add to both the momentumandenergyof the water's lateral motions, but mayreduce the efficiencyof anguilliform motion because the extra momentum at the trailing edge, resulting from forces exerted by anterior sections, is badly correlated with that edge's lateral movements. Adoption of the ‘carangiform’ mode, in which the amplitude of the basic undulation grows steeply from almost zero over the first half or even two-thirds of a fish's length to a large value at the caudal fin, avoids this difficulty. Any movement which a fish attempts to make, however, is liable to be accompanied by ‘recoil’, that is, by extra movements of pure translation and rotation required for overall conservation of momentum and angular momentum. These recoil movements, a potentially serious source of thrust and efficiency loss in carangiform motion, are calculated in ç 4, which shows how they are minimized with the right distribution of total inertia (the sum of fish mass and the water's virtual mass). It seems to be no coincidence that carangiform motion goes always with a long anterior region of high depth (possessing a substantial moment of total inertia) and a region of greatly reduced depth just before the caudal fin. The theory suggests (ç5) that reduction of caudal-fin area in relation to depth by development of a caudal fin into a herring-like ‘pair of highly sweptback wings’ should reduce drag without significant loss of thrust. The same effect can be expected (although elongated-body theory ceases to be applicable) from widening of the wing pair (sweepback reduction). That line of development of the carangiform mode in many of the Percomorphi leads towards the lunate tail, a culminating point in the enhancement of speed and propulsive efficiency which has been reached also along some quite different lines of evolution. A beginning in the analysis of its advantages is made here using a ‘twodimensional’ linearized theory. Movements of any horizontal section of caudal fin, with yaw angle fluctuating in phase with its velocity of lateral translation, are studied for different positions of the yawing axis. The wasted energy in the wake has a sharp minimum when that axis is at the ‘three-quarter-chord point’, but rate of working increases somewhat for axis positions distal to that. Something like an optimum regarding efficiency, thrust and the proportion of thrust derived from suction at the section's rounded leading edge is found when the yawing axis is along the trailing edge. This leads on the present over-simplified theory to the suggestion that a hydromechanically advantageous configuration has the leading edge bowed forward but the trailing edge straight. Finally, there is a brief discussion of possible future work, taking three-dimensional and non-linear effects into account, that might throw light on the commonness of a trailing edge that is itself slightly bowed forward among the fastest marine animals. © 1970, Cambridge University Press. All rights reserved.

Description: In a laboratory study, the class of instabilities of the laminar Ekman layer called type II (or class A) are found to have the spectral characteristics of narrow-band noise (Q ∼ 5). The unperturbed laminar profile resembles very closely the ideal Ekman solution. The frequency of the spectral peak varies with the Reynolds number as predicted theoretically by Lilly (1966), but the measured frequencies are only 60% of the predicted value. The critical Reynolds number for this instability is found to be 56·7, in good agreement with Lilly's analysis. The measured boundary layer profile of the magnitude of the spectral peak has the behaviour predicted by Lilly. A sudden onset of turbulence is found at a Reynolds number of 148. © 1970, Cambridge University Press. All rights reserved.

Description: Experiments on fully developed cavity flows were carried out with the prime initial objective of investigating the effects of the addition of small quantities of ‘turbulent drag reducing additive’ upon the cavity-surface boundary-layer instability and transition reported in the previous paper (Brennen 1970). However, in most instances, the additives were found to cause an unforeseen instability in the wetted surface flow around the headform. Upon convection, the resulting disturbances dramatically disfigured the cavity surface, thus negating the original purpose. This new phenomenon warranted investigation and became the principal subject of this paper. © 1970, Cambridge University Press. All rights reserved.

Description: A constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow. This equation is non-linear in the kinematic variables and gives rise to ‘fluid memory’ effects attributable to the droplet surface dynamics. Furthermore, it has the same form as the corresponding expression for a dilute suspension of Hookean elastic spheres (Goddard & Miller 1967), and reduces to a relation previously proposed by Schowalter, Chaffey & Brenner (1968) when time-dependent effects become small. Numerical solutions are also presented for the case of a small bubble in a steady extensional flow for the purpose of estimating the range of validity of the small deformation analysis. It is shown that, unlike the drag of a bubble which, in creeping motion, is known to be relatively insensitive to its exact shape, the macroscopic stress field in an emulsion is not well described by the present analysis unless the shapes of the deformed bubbles agree closely with those given by the first-order theory. Thus, the present rheological equation should prove of value in a qualitative rather than a quantitative sense. © 1970, Cambridge University Press. All rights reserved.

Description: The breakdown of shallow water waves into forms exhibiting several secondary crests is analyzed by numerical computations based on approximate equations accounting for the effects of non-linearity and dispersion. From detailed results of two cases it is shown that when long waves are such that the parameter σ = v*L*2/h*3 is of moderate magnitude, either due to initially steep waves generated at a wave-maker or due to forced amplification by decreasing depth, waves periodic in time do not remain simply periodic in space. Numerical results are compared with experiments for waves propagating past a slope and onto a shelf. © 1970, Cambridge University Press. All rights reserved.

Description: Similarity rules are constructed for the load distributions induced on a thin two-dimensional wing at subsonic speeds by sinusoidal gusts whose wave fronts are at an angle to the leading edge of the wing. It is shown that these rules divide into two groups according to the value of a parameter dependent on the Mach number and the angle between the gust front and the wing. The similarity rules for each group relate all the members of the group to a simpler problem whose solution can be found by existing methods. The similarity between the two groups is also discussed in terms of the two methods of solution available and it is shown that each method of solution is applicable in all cases. © 1970, Cambridge University Press. All rights reserved.

Description: The additional pressure drop arising from the presence of a neutrally-buoyant, eccentrically positioned, spherical particle in a Poiseuille flow is calculated to O(a/Ro)5 (a = sphere radius; Ro = tube radius). Similar calculations (of a lesser order of accuracy) are given for non-circular conduits and for ellipsoidal particles. Due to changes in particle orientation resulting from rotation, the instantaneous pressure drop for an ellipsoid of revolution varies periodically with time. This pressure diminution is averaged over one period to obtain the time-average pressure drop. © 1970, Cambridge University Press. All rights reserved.

Description: The acoustic scattering properties of a semi-infinite compliant plate immersed in turbulent flow are considered in the context of Lighthill's theory of aerodynamic noise. The turbulent eddies are replaced by a volume distribution of quadrupoles, and the reciprocal theorem used to transform the quadrupole scattering problem into one of the diffraction of a plane acoustic wave. This problem is solved by the Wiener–Hopf technique for the case when elastic forces in the plate are negligible, so that a local impedance condition relates the plate velocity to the pressure difference across the plate. Strong scattering of the near-field into propagating sound occurs when certain types of quadrupole lie sufficiently close to the plate edge, and we derive explicit expressions for the scattered fields in various cases. When fluid loading effects are small, and the plate relatively rigid, the results of Ffowcs Williams & Hall (1970) are recovered, in particular the U5 law for radiated intensity. A quite different behaviour is found in the case of high fluid loading, when the plate appears to be relatively limp. The radiated intensity then increases with flow velocity U according to a U6 law. In aeronautical situations, surface compliance is negligible in its effect on the scattering process, and the U5 law must then apply provided the surface is sufficiently large. On the other hand, the effect of appreciable surface compliance is to greatly inhibit the near-field scattering from the surface edge. This weaker scattering is likely to be observed in underwater applications, where fluid loading effects are generally so high as to render unattainable the condition for a plate to be effectively rigid. © 1970, Cambridge University Press. All rights reserved.

Description: Numerical analysis has been used to find the neutral stability curve for the flat plate boundary layer in zero pressure gradient when the main terms representing the growth of boundary-layer thickness are either included or excluded. The boundary layer is found to be slightly less stable when the extra terms are included. The calculations give a critical Reynolds number of 500. © 1970, Cambridge University Press. All rights reserved.

Description: A derivation is given of the Eulerian equations of motion directly from the Lagrangian formulation of Hamilton's principle. The circulation round a circuit of material particles of uniform entropy appears as a constant of the motion associated with the indistinguishability of fluid elements with equal density, entropy and velocity. A discussion is given of the Lin constraint, and it is pointed out that, for a barotropic fluid, the variational principle recently suggested by Seliger & Whitham does not permit velocity fields in which the vortex lines are knotted. © 1970, Cambridge University Press. All rights reserved.

Description: The linear spin-up of a homogeneous electrically conducting fluid confined between infinite flat insulating plates is analyzed for the case in which a uniform magnetic field is applied normal to the boundaries. As in part 1 (Benton & Loper 1969), complete hydromagnetic interaction is embraced even within linearized equations. Approximate inversion of the exact Laplace transform solution reveals the presence of several flow structures: two thin Ekman–Hartmann boundary layers (one on each plate), which are quasi-steady on the time scale of spin-up, two thicker continuously growing magnetic diffusion regions, and an essentially inviscid, current-free core, which may or may not be present on the spin-up time scale, depending upon the growth rate of the magnetic diffusion regions. When a current-free core exists, it is found to spin-up at the same rate as the fluid within magnetic diffusion regions, although different physical mechanisms are at play. As a result, a single hydromagnetic spin-up time is derived, independently of the thickness of magnetic diffusion regions; this time is shorter than in the non-magnetic problem. © 1970, Cambridge University Press. All rights reserved.

Description: Experimental studies of drag reduction and polymer degradation in turbulent pipe flow with dilute water solutions of unfractionated polyethylene oxide are described. Drag reduction results indicate that the magnitude of the reduction cannot be correlated on the basis of weight average molecular weight, rather the phenomenon depends strongly on the concentration of the highest molecular weight species present in the molecular weight distribution. Polymer degradation in turbulent flow is found to be severe for high molecular weight polymers causing appreciable changes in drag reduction and molecular weight with the duration of flow. Data indicates that drag reduction exists in the limit of infinite dilution suggesting that the phenomenon is due to the interaction of individual polymer molecules with the surrounding solvent and that the extent of reduction is relatively independent of pipe diameter when a comparison is carried out at equal solvent wall shear stresses. Consideration of the high viscosity obtained with solutions in an irrotational laminar flow field suggests this is due to polymer molecule deformation and that this phenomenon is central to the mechanism of turbulent flow drag reduction. © 1970, Cambridge University Press. All rights reserved.

Description: The paper examines the scattering properties of a cavitated line vortex when excited by line and point sources of sound. It is found that the vortex resonances discovered by Kelvin for incompressible flow are essentially unmodified by fluid compressibility, and that many of the resonant modes radiate energy to infinity. Those resonant modes dominate the vortex response, their amplitude growing algebraically with time in a driven instability of the model flow. The off resonance response is dependent on the value of a normalized frequency parameter (ω/Ω)2|ln ka|. Ω denotes the angular velocity of the steady vortex flow, k the acoustic wave-number, and a the cavity radius. Even off resonance the cavity is an extremely efficient wave scatterer, the scattering efficiency increasing with source order. For example, the scattered energy of a point quadrupole is shown to exceed that of the direct field by a factor of 108 for the typical underwater flow Mach number of 10−2. © 1970, Cambridge University Press. All rights reserved.

Description: Measurements are given of the total pressure and yaw relative to the pipe axis of flow in various pipe bends attached to the end of a long straight pipe, for a pipe Reynolds number of 2·36 × 105. The behaviour of the induced secondary flows is presented in detail. In a 180° bend it is found that, from the start of the bend, the secondary flows increase to a maximum and then decrease to a steady value. This effect is explained by relating the local total pressure gradient to the production of streamwise vorticity. The same flow mechanism is used to predict qualitatively the flow pattern in an S-bend in which it is found that the secondary flows cause a complete interchange of the slow moving wall fluid and the faster central core. Results of a computer program for calculating the secondary flow from a knowledge of the upstream velocity profile are included. Although real fluid effects, other than those causing the upstream velocity profile, are omitted from the computations, the measured flow pattern is reasonably well predicted in a long bend up to a bend angle of about 75°. © 1970, Cambridge University Press. All rights reserved.

Description: An analysis is presented of the flow field near a neutrally-buoyant rigid spherical particle immersed in an in compressible Newtonian fluid which, at large distances from the particle, is undergoing simple shear flow. Subject to conditions of continuity of stress at the particle surface and to conditions of zero net torque and zero net force on the sphere, the effect of fluid inertia on the velocity and pressure fields in the vicinity of the particle has been computed to [formula omitted], where R = a2G/v is a shear Reynolds number, a being the sphere radius, G the velocity gradient in the free stream (taken to be a positive number), and v the kinematic viscosity. Some streamlines have been computed and plotted. These illustrate how the fore–aft symmetry of the creeping-motion solution is destroyed when one includes inertial effects. Knowledge of the velocity and pressure fields enables one to compute the effect of inertial forces in suspension rheology. The results include a correction to the Einstein viscosity law to [formula omitted] for a dilute (non-interacting) suspension of spheres. In addition it is found that inertial effects give rise to a non-isotropic normal stress. © 1970, Cambridge University Press. All rights reserved.

Description: The function of a healthy ureter is analyzed in terms of a fluid-mechanical model. To the extent that the Reynolds number is of the order of one, the fundamental equations are shown to reduce to those of the theory of lubrication. It is found that from the point of view of the pressure variation with time (the urometrogram) the important part of the peristaltic wave is the constricting part. For this reason this part of the wave is represented with an algebraic expression of the form h ∼ xn making it possible to find closed form solutions. Using Fourier analysis in defining the complete wave shape of the ureter it was also possible to obtain numerical solutions. For both cases it is shown that there is good agreement between the theoretical and experimental pressure distributions, this not being the case for sinusoidal wave shapes. An approximate equation for the flux is developed and a universal relation is presented connecting the maximum pressure, flux and kinematic behaviour of the ureter. © 1970, Cambridge University Press. All rights reserved.

Description: Numerical space-amplified solutions of the Orr-Sommerfeld equation for the case of a boundary layer on a flat plate have been calculated for a wide range of values of frequency and Reynolds number. The mean flow is assumed to be parallel and given by the appropriate component of the Blasius solution. The results are presented in a form suitable for comparison with experiment and are also compared with calculations of earlier authors. © 1970, Cambridge University Press. All rights reserved.

Description: The time-mean characteristics of the laminar boundary layer near a two-dimensional stagnation point, when the velocity of the oncoming flow relative to the body oscillates are investigated analytically. First, when the amplitude of the oscillating velocity is small compared with the oncoming flow velocity, a series expansion is made and the obtained equations are solved numerically. The equations are also solved approximately in the extreme cases when the frequency is low and high. The obtained approximate solutions are compared with the numerical solutions in terms of skin friction. Next, when the frequency is high, the finite-velocity-amplitude case is treated. Time-mean velocity profiles and skin friction are obtained and compared with the small-amplitude case. © 1970, Cambridge University Press. All rights reserved.

Description: The equations of motion and energy governing a three-dimensional fluctuating flow of an incompressible fluid in the vicinity of a stagnation point on a regular surface have been integrated analytically. The velocity of the oncoming flow relative to the body oscillates in magnitude but not in direction. It has also been shown that the analysis of Lighthill for the two-dimensional fluctuating flow may be extended to the three-dimensional flow (both chordwise and spanwise), namely for each point on the body there is a critical frequency ω0 such that for frequencies ω 〉 ω0 the oscillations are to a close approximation ordinary ‘shear waves’, unaffected by the mean flow; the phase advance in the skin friction is then 45°. For frequencies ω 〈 ω0 the oscillations may be closely approximated by the sum of two parts: one quasi-steady part and the other proportional to the acceleration of the oncoming stream. The phase advance in the skin friction is then tan−1 (ω/ω0). © 1970, Cambridge University Press. All rights reserved.

Description: The non-linear stability of hydromagnetic flows is investigated by applying energy methods. A universal stability estimate, namely a stability limit for motions subject to arbitrary non-linear disturbances, is obtained for bounded or periodic domains. Our analysis is restricted to fluids possessing constant density and electrical conductivity and we do not take into account temperature or Hall effects. This result establishes the existence of an open region of certain stability near the origin of the [formula omitted] Cartesian plane for every fixed Pm (where [formula omitted] and Pm are the Reynolds number, magnetic Reynolds number and magnetic Prandtl number, respectively). The universal stability limit can then be improved by suitably defining a maximum problem using variational techniques, and obtaining the relevant Euler–Lagrange equations. The tentative solution to this problem gives a stability limit which enlarges the universal stability region. Our results are then compared with linear and experimental ones, with special emphasis given to the role played by the magnetic field. © 1970, Cambridge University Press. All rights reserved.

Description: This paper describes theoretical and experimental work carried out at the Cavendish Laboratory of the University of Cambridge. The main object of the work was to develop a new technique for measuring the structure of fluid turbulence. A parallel beam of light is passed through the turbulent region, containing refractive index fluctuations, and analyzed on exit by gratings of periodic transmissivity. Two forms of analysis yield (a) the spatial power spectrum of the refractive index fluctuations in the turbulence, and (b) the velocity distribution within the beam aperture. The method does not disturb the fluid physically, does not depend on the existence of a mean flow velocity, and works well in liquids. One of the limitations of this single-beam method is that it produces information averaged along the path length of the beam in the turbulence, and to overcome this a cross-beam technique, using two beams intersecting at right-angles, has been developed in theory. This method gives the spatial power spectrum of the refractive index fluctuations, as does the single beam method, but the results are characteristic only of the volume of intersection of the beams. The paper first discusses the theory of the single-beam and crossed-beam techniques, and then experimental results obtained with the single-beam method. The turbulent region investigated was a rectangular tank of water, heated from below and cooled from above, producing convective turbulence of high Rayleigh number (4·1 × 108), a system difficult to analyze by conventional methods of measurement, such as the hot-wire anemometer. Spectral density functions (power spectra) of refractive index, and hence in this case temperature fluctuations, have been measured, as have velocity distributions. Statistical analysis of the results also gives useful information about the Eulerian time scale of the turbulent field. © 1970, Cambridge University Press. All rights reserved.

Description: A statically stable, gravitationally stratified compressible fluid containing a parallel shear flow is examined for stability against infinitesimal adiabatic perturbations. It is found that the Miles–Howard theorem of incompressible fluids may be generalized to this system, so that n2 ≥ ¼U′2 throughout the flow is a sufficient condition for stability. Here n2 is the Brunt–Väissälä frequency and U’ is the vertical gradient of the flow speed. Howard's upper bound on the growth rate of an unstable mode also generalizes to this compressible system. © 1970, Cambridge University Press. All rights reserved.

Description: A study of Tollmien–Schlichting waves in the Blasius boundary layer has been carried out under good flow conditions. The maximum r.m.s. amplitude of u, the downstream component of perturbation velocity, was limited to about 0·06% of U0, the free-stream velocity. Measurements of the wave-number and of the distribution of u/U0 normal to the plate agree closely with the theoretical results obtained in parts 1 and 2 of this paper. The experimental critical Reynolds number, Rc, is 400; the theoretical Rc derived from the imaginary part of the eigenvalue is 500 (part 2), but additional amplification carried by the eigenvector removes most of this discrepancy. © 1970, Cambridge University Press. All rights reserved.