ALBERT

Description: The following list covers some old measurements not included in previous lists and most of the samples measured at the Uppsala C14 laboratory since the last list (Radiocarbon, 1967, v. 9, p. 454-470); samples utilized for determining the increase of the C14/C12 ratio clue to explosion of nuclear devices are omitted

Description: We list measurements carried out between June and November 1968. Archaeologic samples are from Italian and Swat (W Pakistan) territories; all geologic samples are from Italian territory.Chemical preparation of samples, measurement technique, and the modern standard are unchanged (Bella and Cortesi, 1960; Alessio, Bella, and Cortesi, 1964; Alessio et al., 1968).For each sample of CO2 the counting rate was corrected according to mass-spectrometrically measured C13/C12 ratio. Isotopic analyses were carried out with a 6 in., 60°-sector, double-collecting mass spectrometer, designed and built by G. Boato at Ist. di Fisica, Univ. of Genoa (Boato et al., 1960) and now in use at Ist. di Geochim., Univ. of Rome. C13/C12 ratio is reported as δ-value, the deviation in parts per mil of the C13/C12 ratio of sample from the PDB standard:

Description: This list is a continuation of Univ. of Pennsylvania Dates VII (Radiocarbon, 1965, v. 7, p. 179-186). It includes results for samples of Sequoia gigantea and for Pinus aristata, most of which were tree-ring dated at the Lab. of Tree-Ring Research, Univ. of Arizona.All sequoia and bristlecone pine samples have been corrected for deviations in C13/C12 ratios. The δC13 values listed represent the deviations (multiplied by 2) of the samples measured from the δC13 value of our 100-yr old standard oak sample which is also the reference value (adjusted for zero age) for the calculation of δC14. In our previous publication (Radiocarbon, 1965, v. 7, p. 179-186), δC13 values were erroneously reported as negative deviations from our oak standard. For the calculation of the Δ's, however, they were used in the correct sense. This mistake has been corrected in this list and one notes that the sequoias and bristlecone pines tend to be enriched slightly in C13 as compared with the oak standard.

Description: The C14 dates given below are a continuation of the work presented in our previous list (Radiocarbon, 1968, v. 10, p. 333-345), and have been obtained by counting CO2 at ca. 2 atm pressure in a 2.7 L stainless steel counter. Results obtained mainly during 1968 are described.Dates have been calculated on the basis of the C14 half-life of 5568 yr and 95% of NBS oxalic acid as modern standard.

Description: The C14 measurements reported here were made in this laboratory between September 1967 and October 1968.

Description: The following list includes a selected number of measurements made during 1967-1968 in the Natural Radiocarbon Laboratory of the Centre de Recherches Radiogéologiques de Nancy. This list is a continuation of Nancy Natural Radiocarbon Measurements I (Radiocarbon, 1968, v. 10, p. 119-123). The dating method, counting technique, and equipment are described in that list. All measurements were made in a proportional counter with a capacity of 1.16 L, normally filled with CO2 under a pressure of 736 mm Hg. Ages are calculated using a C14 half-life of 5568 yrs with 1950 as reference yr. Modern standard used following samples Ny-118 is 95% of NBS oxalic acid activity. The SC14 mentioned later in the date list are calculated according to Broecker and Olson (1959).

Description: The laboratory renewed operations in January, 1968 after a year's inactivity due to the absence of the head, who was on leave at the University of Bonn. Synthesized benzene continues as the dating medium, but various improvements have been made on the chemical method in order to increase capacity. Combustions are no longer carried out for normal materials. Instead, charcoal samples are used directly (after the usual pretreatment) and wood, plants, cloth, etc. are carbonized in a nitrogen atmosphere. Charring is a considerably more rapid procedure than combustion since it eliminates the CO2 collection, carbonate precipitation, and filtration steps. The charcoal then is reacted with molten lithium metal. Also, it was found that carbonates are attacked directly by hot lithium to produce carbides and the preliminary generation of CO2 gas is not indispensable. The carbide, cooled to room temperature, is reacted with old water (IVIC-317, Radiocarbon, 1967, v. 9, p. 240), the acetylene separated from hydrogen in a double liquid nitrogen-cooled trap, and benzene produced with a chromium activated silica-alumina catalyst.

Description: This list continues Gakushuin VI (Radiocarbon, 1967, v. 9, p. 43-62), the same instruments and techniques having been employed.Age calculations are based on the Libby half-life of C14, 5570 ± 30 years, and the modern activity given by 0.95 Aox, i.e., 95% of the activity of NBS oxalic acid standard. The errors quoted are the standard deviation obtained from the number of counts only. When observed activity is less than 2σ above background, infinite date is given with a limit corresponding to the activity of 3σ, and when it is greater than 0.95 Aox −2σ, modern date is given with a limit = 0.95 Aox —3σ. For shell samples, dates are computed without any correction for environmental and biological isotopic fractionation.The description and comments are essentially those of the submitters.

Description: This list reports certain measurements made from 1965 to 1967. These samples are devoted to a special study of organic matter in soils. The work in large part is the subject of a thesis defended in Paris by S. M. Nakhla, 1968.

Description: We continue to give dates based on the half-life of 5568 years according to the decision of the Sixth Pullman Conference (Internatl. Conf., Pullman, 1965). The year 1950 has been used as a reference year for converting the dates to A.D./B.C. scale. A value corresponding to 95% net counting rate of the NBS oxalic acid has been used as the modern reference standard.

Description: The following dates are samples measured since publication of Kiel III (Radiocarbon, v. 10, p. 328–332). Age calculations are based on 95% of the activity of NBS oxalic-acid standard as modern value of A.D. 1950. Results are calculated with Libby half-life and reported in yr before 1950. Error corresponds to 1σ variation of sample net counting rate as well as modern standard and background, but does not include the uncertainty in C14 half-life and in secular C14 variations. Dates are not corrected for isotopic fractionation.

Description: The peculiarities of the geologic structure of the Caucasus, of Georgia in particular, and the existence of numerous rich archaeologic monuments on the territory of the Georgian SSR have made it necessary for the Scientific Laboratory to date both geologic and archaeologic samples.

Description: The Radiocarbon Dating Laboratory of the Geological Survey of Canada routinely operates two proportional counters; one 2 L and one 5 L. The 2 L counter is operated entirely at 2 atm. and the 5 L counter mainly at 1 atm. On occasion the 5 L counter is operated at 4 atm. Detailed descriptions of these two counters have recently been published (Dyck, 1967a). A I L counter has been fabricated and is now undergoing preliminary testing.

Description: This date list is comprised of archaeologic and geophysical samples. The latter are in continuation of our investigations of bomb-produced radiocarbon in atmospheric carbon dioxide reported in Tata V. We continue to count samples in the form of methane; the techniques used have been described elsewhere (Agrawalet al., 1965).

Description: The following list presents dates on a portion of the samples measured at ISOTOPES during 1967 and 1968 and measurements made previously for which either complete sample data has been recently received, or, in some cases, deferred due to the editorial load in preparing the definitive list.

Description: Age calculations in this list include samples dated since January, 1967. Samples are converted to benzene and counted in a liquid scintillation spectrometer. Operations are essentially the same as those described in Radiocarbon, 1966, v. 8, p. 46–53 and Radiocarbon, 1967, v. 9, p. 38–42. Modern reference is 95% the activity of NBS oxalic acid standard, not age-corrected wood as reported in Radiocarbon, 1966, v. 8, p. 46–53.

Description: The usual procedure for the preparation of carbon dioxide from the oxalic-acid standard supplied by the National Bureau of Standards is wet oxidation by means of potassium permanganate in acid solution. The procedure is straightforward, but suffers from a certain difficulty in determining the end point of the reaction.

Description: The Radiocarbon Dating Laboratory was founded in 1965 by the Department of Geology, University of Lyon, to study the Late Quaternary geology of the Rhone-Alps Region, and to contribute to hydrogeologic and archaeologic studies. It has been installed in the basement of the Nuclear Physics Institute. Preparation began in 1966 and first dates obtained in June 1967.

Description: The following list of dates contains most of the measurements obtained since our last list. Procedures of measurements and calculation are as previously described in Radiocarbon, 1964, v. 6, p. 194–196; 1966, v. 8, p. 286–291. A new 1.1 L counter, all metal and quartz, built in the laboratory has been in use since 1967.

Description: This list includes most of the dates produced from September 1967 to April 1968. The laboratory continues to use the benzene method of the previous date list (Radiocarbon, 1968, v. 10, p. 8–28); however, the counter and chemical treatment equipment were transferred to a field laboratory in order to avoid any possibility of contamination. Two new benzene synthesis lines of our own construction were added to the commercial unit. Their operation is excellent and over-all costs were nominal.

Description: The following list of dates is compiled from samples prepared since publication of our last date list (Radiocarbon, 7, v. 9, p 316–332) and includes determinations through June, 1968. Equipment and operating procedures are the same as described in OWU-III.

Description: Dates listed below are based on measurements made up to May 1968, and cover a period during which the technique of gas proportional counting using CO2 was gradually replaced by liquid scintillation counting using benzene. The gas counting measurements were carried out by the method and techniques previously described (Barker and Mackey, 1968) the only modifications being the replacement of some old electronic units by more stable solid-state equipment; proportional counting results are indicated in the text by (P) at the end of the relevant sample descriptions. Liquid scintillation counting, which is now the preferred method in this laboratory, is carried out using a Packard Tri-Carb liquid scintillation spectrometer model 3315/AES fitted with selected low-noise quartz-faced photomultipliers. Normally 3 ml of benzene is prepared from each sample. This is dissolved in 12 ml of scintillation grade toluene containing 5 gm/liter of scintillator (PPO) and the solution is measured in a standard low-potassium glass vial at a temperature of 0°C. Photomultiplier E.H.T., amplifier, and channel width settings are optimized for C14, and measurements are carried out at ca. 65% efficiency of detection for C14 to eliminate interference from any tritium which may be present in the benzene. Under these circumstances the background is approx. 8.6 cpm and the modern (95% Aox) is approx. 24.0 cpm. Samples are counted in groups of 3 to 5 together with background and modern reference samples and are measured for at least one week, the instrument being set to cycle at 100 min intervals. In this period, the counts accumulated are such that the background is always measured to a statistical accuracy of better than 1% and most other samples to a higher accuracy than this. Background and modern counts used in the calculation of each result are only those relevant to the period of measurement of that particular sample. Statistical analysis of groups of replicate measurements made under these conditions over a very long period of time has demonstrated the excellent long-term stability of the equipment and indicates that the technique is quite capable of achieving results of very high statistical accuracy when required.

Description: The radiocarbon dating laboratory of the Illinois State Geological Survey has been established to satisfy a growing need for radiocarbon dates for an active Pleistocene research program. Because of the age and type of material dated, the benzene liquid scintillation counting method is employed in this laboratory. The detailed chemical procedure for converting carbon to benzene has been published by Noakes, Kim, and Stipp (1965) and Noakes, Kim, and Akers (1967); however, the procedures for benzene synthesis and sample counting are briefly explained below to clarify this laboratory's procedure.An organic sample, such as peat, organic silt, or wood, is burned and the CO2 evolved is absorbed in NH4OH. SrCl2 solution is added to precipitate the carbonate, and the solution plus precipitate is boiled and cooled before filtration of SrCO3. The SrCO3 is acidified with dilute H3PO4 to liberate CO2 in a closed system, and the CO2 is converted to C2H2, as reported by Barker (1953). In this method, 2.4 gm of dry packed lithium, obtainable from the Lithium Corporation of America, is used for each liter of CO2 that is converted to C2H2. Trimerization of the C2H2 to form C6H6 is accomplished using a vanadium-alumina catalyst.To the C6H6 synthesized from the sample carbon, 2 cc of toluene containing 100 mg Butyl-PBD, 2-(4-tert-Butylphenyl)-5-(4-Biphenylyl)-1,3,4-Oxadiazole, are added, and this mixture is made to a total volume of 10 cc with spectrograde C6H6. A modified Packard Instrument Co. liquid scintillation spectrometer (Model 3375) is used for measurement of C14 activity.Ages are calculated from a C14 half-life of 5568 years, and the standard deviation (1σ) is based only on counting errors; however, if calculated error is less than 200 years, 200 years is chosen as one standard deviation (1σ).

Description: The following list includes selected geologic and pollen dated samples measured since 1965. After moving to new laboratory quarters, we increased shielding to 470 g/cm2 on all sides and 660g/cm2 at the top. Background of our Houtermans-Oeschger-type counters filled with 700 mm Hg of acetylene is now:

Description: Measurements have continued with the 6 L counter. A counter of similar design but with 1 L volume has recently been brought into operation. Despite replacement of the teflon parts, the 1.5 L Oeschger-type counter has not worked consistently and has not been used for dating. Results are not corrected for δ C13. Errors quoted refer only to the standard deviation calculated from a statistical analysis of count rates and the Libby half-life of 5570 ± 30 yr.Alkali pretreatment is used for all samples of charcoal, peat, wood, and plant material provided they are of sufficient quantity. It is now standard practice to boil the sample in 5% HCl solution and filter, both before and after boiling it in 2% NaOH solution. Between each treatment it is washed in boiling distilled water and is finally oven dried at 110°C. Concentrations of acid and alkalis are varied to suit each sample but the sequence of the pretreatment is always the same. In some cases a humate extract is obtained by precipitation with weak acid from the alkali solution filtrate.The collagen fraction is obtained from samples of bone using the method of Krueger (1966) by treatment with dilute acid under reduced pressure. The gas evolved during the treatment is usually discarded but is occasionally retained and dated as the mineral fraction.After mechanical cleaning, samples of shell are dissolved in stages with 6N HCl to divide them into two or three fractions corresponding to the outer, middle, and inner layers of the shell. Normally the outer fraction is discarded when three fractions have been prepared.

Description: The dates detailed below are a selection of C14 dates obtained from February 1966 to December 1967 for geologic samples. The method is essentially the same as described previously in Radiocarbon, 1966, v. 8 p. 74-95. All samples were subjected to pretreatment, differing in individual cases, to remove contamination. On one of our three installations, modern transistorized equipment replaced the original electronics.All dates reported have been calculated on the assumed half-life of 5568 yr for C14, and of 1950 as the reference year.

Description: This date list includes those series of samples completed in this laboratory as of August 1968. 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 min each. Errors quoted are derived from measurement of sample, background, and modern age calibration, but do not include any half-life error. All samples were pretreated with 3N HC1, and some, where noted, were given additional pretreatment with 2% NaOH for the removal of possible humic contaminants.

Description: The radiocarbon dates obtained since December 1967 are included in this report. The procedures followed have been described previously (Radiocarbon, 1966, v. 8, p. 522–533).

Description: The following list comprises measurements made since those reported in NPL V.No changes have been made in measurement technique or in the method of calculating results described in NPL III.

Description: The following list comprises selected measurements made during 1967–68. The method is essentially the same as previously described. A 0.6 L proportional gas-counter at 3 atm CH4pressure is used. Ages are given relative to A.D. 1950 and half-life of 5570 yr has been assumed. The quoted error is the experimental standard deviation and includes the uncertainty on the unknown sample, the modern standard and the background.

Description: The following results represent measurements carried out since our 3rd date list was prepared. The entire CO2purification technique (Radiocarbon, 1962, v. 4, p. 81–83) has been changed to “wet purification.” No measurements of C12/C13ratios have been made.

Description: This study was started during the summer of 1965 because of the discrepancy observed between dates obtained from wood charcoal and charred corn samples collected from the same archaeologic sites. These results are listed in Table 1.

Description: Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.

Description: The modifying effect of base bleed on the steady separated flow past a two-dimensional bluff body is considered. Detailed experimental results are presented for Reynolds numbers R between 50 and 250 and for bleed coefficients b in the range 0 to 0·15. The streamline pattern near the object is found to be strongly affected by small changes in the rate of bleed, with the recirculating closed wake disappearing altogether for b 〉 0·15. Nevertheless, the qualitative dependence on R of the physical dimensions of the near-wake region and the associated streamwise pressure profile appear to be unaffected by base bleed.

Description: This paper is concerned with the propagation of small amplitude gravity waves over a flow with non-uniform velocity distribution. For such a flow Burns derived a relation for the velocity of propagation in terms of the velocity distribution of the mean flow. This result is derived here in another way and some of its implications are discussed. It is shown that one of these is hardly acceptable physically. Burns's result holds only when a real value of the propagation velocity is assumed; the mentioned difficulties vanish if complex values are allowed for, implying damping or growth of the waves. Viscous effects which are the cause of damping or growth are important in the wall layer near the bottom and also at the critical depth, which is present when the wave speed is between zero and the fluid velocity at the free surface.In § 2 the basic equations for the present problem are given. In § 3 exchange of momentum and energy between wave and primary flow is discussed. This is analogous to what happens at the critical height in a wind flow over wind-driven gravity waves. In § 4 the viscous effects at the bottom are included in the analysis and the complex equation for the propagation velocity is derived. Finally in § 5 illustrations of the theory are given for long waves over running flow and for the flow along a ship advancing in a wavy sea. In these examples a negative curvature of the mean velocity profile is shown to have a stabilizing effect.

Description: As they propagate through a gas, fluctuating pressure signals of moderate amplitude and of ultrasonic frequency are affected by amplitude dispersion, by relaxation damping and, particularly in ‘shock layers’, by diffusive damping. We derive a ‘high frequency’ theory including all these effects, for disturbances of arbitrary wave form excited by a wide variety of boundary conditions. By introducing a phase variable α, and taking account of non-linearity, we show how the signal propagates along the rays of linear acoustics theory, with constantly changing wave profile.Relaxation dampens the signal, as for linear acoustics, and also diminishes amplitude dispersion. A criterion for shock formation is given, and the importance of non-linearity for signal attenuation exhibited. As shocks form, α surfaces coalesce and diffusive mechanisms are accentuated. Whitham's area rule is shown to be relevant for unsteady three-dimensional flows in relaxing gases, and is used to compute the attentuation of an ultrasonic beam. Supersonic relaxing flow over a wavy wall is also analyzed, and focusing effects are discussed.

Description: An electrically conducting fluid is contained above a horizontal plane. A uniform vertical magnetic field is applied externally and the plane is maintained at a uniform temperature except for a point or a line heat source. Density variations are ignored except where they give rise to buoyancy forces.(i) The point heat source. Non-linear effects are small sufficiently far from the source. The resulting buoyancy forces interact with the magnetic forces to maintain a radial inflow towards the heat source. This fluid then escapes vertically as a jet, its structure now depending on the additional influence of viscosity. The perturbations of the temperature distribution and the magnetic field due to the motion are obtained. Finally, the effects of these perturbations back on to the fluid velocity are considered. The most striking features of the perturbations are (a) the action of the jet as a line source of heat for the fluid in the outer regions, (b) the large (compared to other perturbations) eddy in the jet.(ii) The line heat source. The temperature distribution and magnetic field are weakly perturbed only if the thermal and electrical conductivities are sufficiently small. Similar results are obtained, as in (i) above, provided ε (a dimensionless number characterising the strength of thermal convection: see (1.32), (3.24)) is less than ¼. However, even for small ε, the effects of thermal convection cannot be ignored. Hence, superimposed on the jet is an eddy (driven by buoyancy forces) whose flux of fluid increases indefinitely with its height above the plane. When ε 〉 ¼, the results suggest that numerous eddies will be formed.

Description: A thin circular disk translates slowly in its own plane transverse to the axis of rotation of parallel plane boundaries filled with viscous incompressible liquid. It is shown that the indeterminateness of the geostrophic flow is removed by constraints imposed by the dynamics of free shear layers (Stewartson layers), which surround a Taylor column whose boundary is not a stream surface. Fluid particles cross the Taylor column at the expense of deflexion through a finite angle. A comparison is made with the flow past a fat body (Jacobs 1964), where the geostrophie flow is determined without appeal to the dynamics of the shear layers. The problem is also considered for a disk in an unbounded fluid, and it is shown that to leading order there is no disturbance.

Description: Integral constraints are derived for steady recirculating flows of nearly incompressible fluids, arising from the action of a small amount of viscosity and heat conduction. These constraints are then combined with the inviscid nondiffusive incompressible flow equations to show that two-dimensional flows containing closed nested streamlines, or three-dimensional flows with closed nested stream surfaces, are isothermal. In the former case it is shown that the vorticity is constant, and in the latter case there is an analogous result when the flow is axially symmetric and confined to axial planes. For a circular cell free convection problem, the interior temperature and vorticity are determined from the boundary conditions by an approximate integration of the boundary layer equations.

Description: Self-similar flow patterns are studied which arise when a cylindrically symmetric strong shock or detonation wave propagates outwards into a gas at rest in which the ambient density varies as the inverse square of the distance from the axis of symmetry along which flows a line current of either zero or finite constant strength. The electrical conductivity of the gas on either side of the wave is supposed perfect and the discontinuities discussed are either gasdynamic or magnetogas-dynamic in nature. It is shown that self-similar solutions exist for piston driven gasdynamic detonation and shock waves. Whilst no self-similar solutions may occur for magnetogasdynamic detonation waves, it is demonstrated that magnetogasdynamic shock waves do possess such solutions for which detailed flow patterns are presented.

Description: The development of reflected waves are studied when two shocks of unequal strength collide and when a shock collides with a constant temperature wall. Both these problems are examined using the Monte Carlo technique developed by Bird (1967). Some limitations upon this technique are suggested and a modified time advance parameter used.

Description: The Seventh Shock Tube Symposium was held 23–25 June 1969 in Toronto, Canada. Sponsored by the University of Toronto Institute for Aerospace Studies and the United States Air Force Office of Scientific Research, the meeting drew nearly 300 attendees from some 20 countries. Of the 40 invited and contributed papers, several described new shock-tube techniques while the majority presented recent experimental results and related theory in the fields of shock structure, atomic and molecular physics, radiation, plasma flows, shock waves in solids, and boundary layers. This report summarizes the principal advances presented and attempts a projection of future directions in shock-tube research. The full proceedings will be published by the University of Toronto Press; the programme was published in the Bulletin of the American Physical Society, June 1969, p. 754.

Description: An approximate method due originally to Whitham is applied to the study of acoustic waves propagating in a non-grey radiating and absorbing gas, assumed in local molecular equilibrium. The method, which has general applicability in the study of non-equilibrium wave phenomena, replaces the exact governing equation by a set of lower-order equations that can be solved analytically in many cases. The use of the method is demonstrated by reconsidering the onedimensional problems of (i) harmonic waves driven by a harmonic variation in either position or temperature of a planar wall and (ii) the discrete wave produced by the impulsive motion of a constant-temperature wall. The method greatly simplifies the mathematics for these problems, and comparaison of the results with those of earlier investigators shows the approximate method to be accurate. Moreover, the method allows us to obtain a more systematic and complete analytical solution of the second problem than has been obtained by more conventional methods.

Description: It is shown that there exist undamped solutions for perturbations of finite amplitude of plane Poiseuille flow, which are periodic in the direction of the axis of the channel. The shift in the ‘neutral curve’ as a function of the amplitude λ* of the disturbance is shown in figure 2. The solution is obtained by a perturbation method in which the eigenfunctions and the eigenvalue c are expanded in power series of the amplitude λ, as shown in (14), (15), (16) and (17). Near the neutral curve for a finite amplitude disturbance, the curvature of the mean flow shows a tendency to become negative (figure 5).

Description: The prototype spin-up problem between infinite flat plates treated by Greenspan & Howard (1963) is extended to include the presence of an imposed axial magnetic field. The fluid is homogeneous, viscous, and electrically conducting. The resulting boundary initial-value problem is solved to first order in Rossby number by Laplace transform techniques. In spite of the linearization the complete hydromagnetic interaction is preserved: currents affect the flow and the flow simultaneously distorts the field. In part 1, we analyze the impulsively started time dependent approach to a final steady Ekman–Hartmann boundary layer on a single insulating flat plate. The transient is found to consist of two diffusively growing boundary layers, inertial oscillations, and a weak Alfvén wave front. In part 2, these one plate results are utilized in discussing spin-up between two infinite flat insulating plates. Two distinct and important hydromagnetic spin-up mechanisms are elucidated. In all cases, the spin-up time is found to be shorter than in the corresponding non-magnetic problem.

Description: A NATO Advanced Study Institute on the topic of transition from laminar flow to turbulence was held at Imperial College, London, from 1 to 6 July 1968. Each morning's session was started with a one-hour general lecture, and was followed by five or six half-hour lectures interspaced with discussion periods. The main lecturers were C. C. Lin (general survey), S. Rosenblat (stability of time-dependent flows), L. S. G. Kovasznay (turbulent, non-turbulent interfaces), L. E. Scriven (free surface effects) and A. A. Townsend (shear turbulence). The idea of the meeting was to bring forth and to discuss current ideas in the subject, both from the point of view of developments out of laminar flow and from that of developments into real turbulence. To this end speakers were chosen to introduce a variety of topics ranging from laminar-flow instabilities (with emphasis on aspects at present imperfectly understood), through non-linear effects to the processes affecting turbulence itself.Many ideas recurred throughout the meeting, both at lectures and in discussion periods. This is true, for example, of several relevant points forcefully made by C. C. Lin. For this reason the present account does not attempt to describe the proceedings of the meeting in chronological order, but rather takes an overall view of the subject matter and points to the areas of agreement and of controversy in relation to various problems.

Description: A non-linear theory of internal gravity waves of finite amplitude is developed in terms of conservation equations averaged with respect to the phase. The theory overcomes the failure of linear ray theory in regions in which waves are trapped and establishes the conditions under which finite amplitude waves may propagate. It gives a geometrical representation of the degeneration of waves into quasi-turbulence and predicts the dependence of the energy density on its parameters.

Description: A complete and detailed study of a radiatively driven plane acoustic wave in a non-grey radiating and absorbing gas is carried out on the assumption of local molecular equilibrium. Specifically, the response of the gas in a semi-infinite space to a step input of radiation from a stationary black wall is investigated. The problem is physically interesting because radiative heat addition is the only driving mechanism, and this mechanism is unique and fundamental to the field of radiative gas dynamics. The solution shows that the heat addition gives rise initially to a compression-expansion wave in the gas, with the wave front controlled by radiation. This wave-front disturbance, though caused initially by the direct effect of radiative transfer, eventually outruns the region of appreciable heating near the wall and becomes a modified-classical disturbance that propagates away from the wall at the isentropic speed of sound. The radiative heat addition continues directly to affect the gas near the wall and in this manner drives the modified-classical wave indirectly by causing the formation of an ‘effective gas piston’. The solution thus exhibits a linearized phenomenology corresponding to that observed in the non-linear leading wave associated with the nuclear fireball.

Description: In this paper time dependent expansions of monatomic gases with spherical symmetry are discussed. For the particular case of Maxwellian molecules closed expressions for the moments up to second order are obtained in regions of the flow where the inviscid solution is no longer valid. These solutions are derived in a general form using the particle path function as a parameter. The structure of the inviscid solution is such that this simplification can be made. The novelty of the present approach is that solutions already derived in previous papers can be obtained from the general solution in various limits; both the results for steady flow and the expansion of a fixed mass of gas can be derived in this manner. Finally, a particular example is constructed in order to illustrate the general theory.

Description: The vibrational relaxation frequency measurements were made on mixtures of carbon dioxide and the light gases using a shock tube and a Mach-Zehnder interferometer. The temperature range covered was 350–1200 °K. It was discovered that in the case of helium the effectiveness of the carbon dioxide-helium collision increases with increasing temperature while in the case of hydrogen and deuterium the collision number displays an anomalous temperature behaviour. At about 1000 °K all the three light gases are almost equally efficient in exciting the vibrational modes of carbon dioxide.

Description: Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

Description: A disturbance of finite amplitude λ, which is periodic in the direction of the axis of the channel, is superimposed on plane Poiseuille flow, and the subsequent development of the disturbance is studied. The disturbance is represented by an expansion in the eigenfunctions of the Orr-Sommerfeld equation with coefficients which are functions of the time, and an accurate numerical solution of the truncated system of non-linear ordinary differential equations for the coefficients is obtained.It is found that even for Reynolds numbers R less than the critical value Rc, the flow breaks down when λ exceeds a critical value λc(R). This is shown in figure 11 for the case when the initial disturbance is represented by the first mode of the Orr-Sommerfeld equation. The development of this type of disturbance is illustrated in figures 1, 3 and 13 and, for the case of a higher-order mode initial disturbance, in figure 14. Near the time of breakdown, the curvature of the modified mean flow changes sign (figure 15), but a disturbance may die down even after a reversal in the sign of the curvature has taken place (see figure 2).The stability of plane Poiseuille flow to disturbances of finite amplitude is affected by the characteristics of the higher-order modes of the Orr-Sommerfeld equation. As shown in figures 4, 10, and 12, and in figures 5, 6, and 7, these modes are either of a ‘boundary type’, characteristic of the region near the wall, or of an ‘interior type’, characteristic of the centre of the channel. The modes in the transition zone, where the two types merge, are easily amplified through mutual constructive interference, even though individually they have high damping coefficients. It is these transition modes which are mainly responsible for the breakdown through finite amplitude effects.

Description: This paper presents the practical and rigorous solution of the potential flow problem associated with the oscillation of a shallow-draft cylinder of infinite length on a free surface. The problem is three-dimensional to the extent that the amplitude of the cylinder oscillation is periodic along its axis as well as with time. The complementary problem associated with the interaction of the fixed cylinder with an incident wave train aligned at some oblique angle with respect to the cylinder axis is also treated. The use of a Green's function reduces the problem to an integral equation which is solved numerically. Numerical results are computed for pressure amplitude distributions, force coefficients, added mass and damping coefficients, transmission and reflexion coefficients and wave height ratios.

Description: The reattachment of a supersonic jet with a turbulent boundary layer abruptly expanding into an axisymmetric parallel diffuser has been experimentally investigated using a surface-flow technique. Measurements were made in the started condition, where the blowing pressure is sufficiently high to establish an oblique shock system in the diffuser. The proposed reattachment criterion correlates present measurements in terms of the diffuser area ratio, and also those of other workers for unconfined flow in terms of the free stream Mach number after separation. As already reported for unconfined flow, it is found that disturbances downstream of reattachment do not affect the upstream region.

Description: The question of whether or not waves exist upstream of an obstacle that moves uniformly through an unbounded, incompressible, inviscid, unseparated, rotating flow is addressed by considering the development of the disturbed flow induced by a weak, moving dipole that is introduced into an axisymmetric, rotating flow that is initially undisturbed. Starting from the linearized equations of motion, it is shown that the flow tends asymptotically to the steady flow determined on the hypothesis of no upstream waves and that the transient at a fixed point is O(1/t). It also is shown that the axial velocity upstream (x 〈 0) of the dipole as x → − ∞ with t fixed is O(|x|−3), as in potential flow, but is O(|x|−1) as t → ∞ with |x| fixed. The results extend directly to closed obstacles of sufficiently small transverse dimensions and suggest the existence of a finite, parametric domain of no upstream waves for smooth, slender obstacles. The axial velocity in front of a small, moving sphere at a given instant in the transient régime is calculated and compared with Pritchard's laboratory measurements. The agreement is within the experimental scatter for Rossby numbers greater than about 0·3 even though the equivalence between sphere and dipole is exact only for infinite Rossby number.

Description: A Symposium on ‘The flow of fluid-solid mixtures’ was held at the University of Cambridge from 24 to 28 March 1969, under the auspices of the International Union of Theoretical and Applied Mechanics. There were 104 participants, representing 19 countries, and attendance was by invitation only. Since there will be no publication of the proceedings in full, the following condensed account of the developments described at the various sessions has been prepared for publication by three of the participants, all of whom were involved in the organization of the Symposium.

Description: Three-dimensional steady flow past a body placed in a uniform stream of viscous, thermally conducting fluid is considered within the framework of the Oseen approximation. Asymptotic forms for the fundamental matrix are obtained for both supersonic and subsonic flow. It is shown how the solution to the flow past a body may be obtained from the fundamental matrix, and that the fundamental matrix itself provides the far field flow.

Description: Experiments have been made to investigate the motion generated by a body moving along the axis of a uniformly rotating fluid.Part of the investigation concerns the motion generated in a cylinder whose radial dimensions are much greater than those of the body. Measurements have been made of the velocities of particles on the axis of rotation both ahead of and behind the body, and the results indicate that there is a significant axial motion generated by the body over a wide range of Rossby numbers. A measurement of the instantaneous velocity profile ahead of the body, determined as a function of the radius, agrees fairly well with a low Rossby number calculation of the flow due to a circular disk (Morgan 1951). In addition, the forward influence of the body has been measured as a function of the Rossby number and from these results it is suggested that the body has a finite influence far upstream at all Rossby numbers and that the blocking phenomenon first reported by Taylor (1922) probably occurs for all values of the Rossby number (UΩa) less than a critical value which is about 0·7.Experiments have also been made in a long cylindrical tube which acts as a wave guide. At large distances from the body the separate effects of the various modes can be observed and hence it is possible to measure the flow corresponding to an individual wave-number: these measurements show that, as a result of the Doppler effect, the motion a large distance ahead of the body is different from that far behind (see Lighthill 1967). Moreover, the experiments indicate that no disturbances propagate ahead of the body when its velocity exceeds the maximum group velocity in the fluid, but that disturbances trail behind the body when its velocity is far in excess of the maximum group velocity. Measurements of the maximum group velocity are in good agreement with the theoretical value.

Description: An electron beam technique was used to measure density profiles of strong shock waves in argon with high accuracy. The experimental results are compared with the results of theoretical models. Of the models that are available in enough detail for comparison, the best agreement with experiment is shown by the direct simulation Monte-Carlo method (Bird 1968), assuming an inverse 12th power molecular interaction force law. It is shown that the density maximum slope thickness is not sufficient for a detailed description of the shock wave structure.

Description: The problem of the reflexion of tides in an enclosed sea such as the North Sea at a point at which it either enters the ocean or its width suddenly increases is considered by investigating the reflexion of a Kelvin wave at the open end of a rotating uniform semi-infinite channel.It is shown that for a given channel, if the wave period is less than a pendulum day, then, according to the linearized theory of long waves in a rotating system, the reflexion coefficient increases with the angular velocity of rotation. It is also shown that there is a resonance effect for certain critical channel widths, namely, those at which extra modes within the channel become possible.

Description: The low Reynolds number flow of a variable property gas past an infinite heated circular cylinder is studied when the temperature difference between the cylinder and the free stream is appreciable. The velocity field (and hence the drag on the cylinder) is calculated by the method of matched asymptotic expansions. It is found that the zero-order velocity field calculated on the Stokes approximation satisfies both the no slip condition at the cylinder and the uniform stream condition at infinity which is in strong contrast with the corresponding velocity field for incompressible slow flow past an unheated cylinder where the uniform stream condition at infinity cannot be satisfied. When the temperature of the cylinder is twice the temperature at infinity it is found that the drag on the cylinder is almost twice the drag on a similar unheated cylinder.

Description: The flow produced by a disk performing small oscillations in a rotating system is considered. Results are obtained for the first-order harmonic velocity and the second-order steady velocity. It is then shown that this mathematical model does not always represent an axially bounded fluid in the limit of infinite separation, and to be general one must allow a steady, azimuthal, perturbation velocity with an arbitrary value to exist at infinity.

Description: We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperatures T (translational) and Ti (internal) in the form e = CvfT + CvfTi, and Ti is governed by a rate equation dTi/dt = (T − Ti)/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x 〉 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form [igg(aufrac{partial}{partial t}+1 igg) igg{frac{partial u}{partial t}+ igg(a_1+frac{gamma + 1}{2}u igg)frac{partial u}{partial x}-{extstylefrac{1}{2}}lambdafrac{partial^2u}{partial x^2} igg}=(a_1-a_0)frac{partial u}{partial x}, ] where a1, a0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − (a20/a21)), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given.When ε is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a0 〈 U 〈 a1. If U 〉 a1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.

Description: In this paper, we study the propagation of a shock wave in water, produced by the expansion of a spherical piston with a finite initial radius. The piston path in the x, t plane is a hyperbola. We have considered the following two cases: (i) the piston accelerates from a zero initial velocity and attains a finite velocity asymptotically as t tends to infinity, and (ii) the piston decelerates, starting from a finite initial velocity. Since an analytic approach to this problem is extremely difficult, we have employed the artificial viscosity method of von Neumann & Richtmyer after examining its applicability in water. For the accelerating piston case, we have studied the effect of different initial radii of the piston, different initial curvatures of the piston path in the x, t plane and the different asymptotic speeds of the piston. The decelerating case exhibits the interesting phenomenon of the formation of a cavity in water when the deceleration of the piston is sufficiently high. We have also studied the motion of the cavity boundary up to 550 cycles.

Description: The rotating system of an infinite disk beneath an unbounded fluid can exhibit resonance if the disk performs torsional oscillations at a certain frequency. This effect is examined in detail, and the solution is shown to depend crucially upon the existence of a small, steady departure from the basic rotational state in the far field.

Description: A study is made of the wind-driven circulation of a two-layer ocean within a square basin, with a view to describing the observed separation of western boundary currents. The lower layer is allowed to surface and the line along which the upper-layer depth vanishes is interpreted as the region of the surfacing thermocline. For a representative wind stress the theory predicts the gross features of the Gulf Stream flow, the region adjacent to the surfacing line containing the separated boundary current. By assuming that the effects of friction and inertia are confined to regions of a boundary-layer character, the position of a separated current is shown to depend only on the degree of stratification and certain integral properties of the applied wind stress.

Description: The structure of a strong blast wave under the influence of an expanding inner contact surface is studied asymptotically in the Newtonian limit: $epsilon equiv (gamma - 1)/2gamma ll 1, epsilon dot{y}^2_s gg a^2_{infty}$. The theory treats the interaction of a shock layer and an inner flow region (the entropy wake) and reduces the problem to an ordinary differential equation for the shock radius. The pressure–volume relation of Cheng et al. (1961) is recovered and extended to a higher order of ε.It is shown that, depending on the rate of growth of the contact surface, the shock layer may ‘reattach’ to the surface at large time. In a number of cases, the reattachment is approached in an oscillatory manner which leads to a period of non-uniformity. The associated problem of multiple time scales (treated in sequels to this paper) is identified.

Description: The dispersion relationship for plane hydromagnetic waves in a stratified rotating fluid (α) indicates that the well-known analogy between rotating fluids and stratified fluids in regard to their hydrodynamic behaviour does not extend to magnetohydrodynamic behaviour, and (b) lends credence to a certain conjecture made in a previous paper, namely that effects due to density stratification can be neglected when considering the dispersion relationship for free hydromagnetic oscillations of the Earth's core if the Brunt—Väisälä frequency is much less than twice the angular speed of the Earth's rotation.

Description: Certain aspects of the transport of solid particles by a turbulent airstream are discussed, namely: the conveyance of particles in a horizontal pipe, including those carrying an appreciable electrostatic charge; the mechanism of deposition onto a solid wall; and the behaviour of fine particles in a shear flow, such as that in a round jet.Rough estimates of the effect of the particles on the gaseous turbulence are made, and a primitive physical explanation is offered of the observed velocitylag and pressure drop associated with the transport of particles in a horizontal pipe, under conditions where the influence of the particles’ weight is significant.Attention is drawn to the difficult problem of dynamically scaling a two-phase flow, and to the different types of interaction between the phases which can occur in a pipe according to its size, the gas velocity through it, and the physical characteristics of the particles.The paper is an annotated version of a survey presented to the I.U.T.A.M. Symposium on ‘Flow of fluid-solid mixtures’ held in Cambridge during March 1969.

Description: When a long rectangular tube containing two immiscible fluids is slightly tilted away from the horizontal, a uniformly accelerating flow is produced with shear at the interface. The presence of shear leads to instability, which is characterized by the spontaneous and rapid growth of almost stationary waves if the fluid depths are equal and the density difference small. The conditions for the onset of Kelvin-Helmholtz instability, taking account of the accelerating flow and the presence of a velocity transition region at the interface, are investigated theoretically and comparison made with observations. The time at which instability occurs is quite well predicted by this theory, but the wavelength of the unstable waves is rather greater than predicted in the accelerating flow. The difference between the predictions and observations may be the result of finite amplitude effects or of the development of Tollmien-Schlichting instability before Kelvin-Helmholtz.

Description: The non-linear magnetization characteristics of recently developed ferrofluids complicate studies of wave dynamics and stability. A general formulation of the incompressible ferrohydrodynamics of a ferrofluid with non-linear magnetization characteristics is presented, which distinguishes clearly between effects of inhomogeneities in the fluid properties and saturation effects from non-uniform fields. The formulation makes it clear that, with uniform and non-uniform fields, the magnetic coupling with homogeneous fluids is confined to interfaces; hence, it is a convenient representation for surface interactions.Detailed attention is given to waves and instabilities on a planar interface between ferrofluids stressed by an arbitrarily directed magnetic field. The close connexion with related work in electrohydrodynamics is cited, and the effect of the non-linear magnetization characteristics on oscillation frequencies and conditions for instability is emphasized. The effects of non-uniform fields are investigated using quasi-one-dimensional models for the imposed fields in which either a perpendicular or a tangential imposed field varies in a direction perpendicular to the interface. Three experiments are reported which support the theoretical models and emphasize the interfacial dynamics as well as the stabilizing effects of a tangential magnetic field. The resonance frequencies of ferrohydrodynamic surface waves are measured as a function of magnetization, with fields imposed first perpendicular, and second tangential, to the unperturbed interface. In a third experiment the second configuration is augmented by a gradient in the imposed magnetic field to demonstrate the stabilization of a ferrofluid surface supported against gravity over air; the ferromagnetic stabilization of a Rayleigh-Taylor instability.

Description: The flow fields in two-dimensional, isoenergetic, viscous free mixing with constant β and with initial velocity profiles deviating slightly from those given by wakelike solutions of the Falkner-Skan equation for that β are considered. The similar solutions of the Falkner-Skan equation are investigated in more detail than in the past, e.g. we show that as β → −1 the flows approach the pure jet with the surrounding fluid at rest, and that there are new branch solutions for β 〈 −1. We have investigated the spatial stability of these flows; it is found that for β 〉 − 0·5 the only spatially stable solutions are the trivial ones f′(η) ≡ 1, but for −1 〈 β 〈 − 0·5 there are non-trivial, jet-like solutions which are spatially stable. As to the new branch solutions for β 〈 − 1, all are spatially unstable.

Description: Fluid material line growth in turbulent flow has been measured by tagging lines with small hydrogen bubbles in the nearly isotropic turbulence behind a regular grid in a water tunnel. The average three-dimensional line lengths were inferred by an intersection-counting method carried out on one-plane photographs. The measurements cover ‘small’ time intervals only.

Description: The energy theory, giving a sufficient condition for stability, is developed for the motions in a horizontal, heated layer subject to buoyancy and surface tension effects. The free surface is assumed to be non-deformable (Pearson's 1958 model).It is shown that the equations governing the energy theory are the symmetric part of the time-independent linear theory problem, and that the surface tension terms behave like a bounded perturbation to the Bénard problem. The qualitative behaviour of the optimal stability boundary as a function of its parameters is given. The optimal stability boundary is computed, and compared with previous linear and non-linear stability theories in terms of allowable subcritical instabilities.

Description: This study concerns the hypersonic flow over blunt bodies in two specific cases. The first is the case when the Mach number is infinite and the ratio of the specific heats approaches one. This is sometimes referred to as the ‘Newtonian limit’. The second is the case of infinite Mach number and very large streamwise distance from the blunt nose with a strong shock wave, or the ‘blast wave limit’. In both cases attention is restricted to power law bodies. Experiments are described of such flows at M∞ = 7·55 in air.The Newtonian flow over bodies of the shape y ∞ xm at zero incidence is shown to be divisible into three regions: the attached layer at small x, the free layer and the blast wave region. As m increases from zero, the free-layer region reduces in extent until it disappears at m = 1/(2+j) (j = 1 and 0 for axisymmetric and plane flow respectively). A difficulty arises in a transition solution of the type given by Freeman (1962b) connecting the free layer with the blast wave result. At m 〉 2/(3+j) the attached layer merges smoothly into the Lees-Kubota solution which replaces the blast-wave result in this range.In the blast wave limit, solutions were obtained for flow over axisymmetric power law shapes in the range ½γ 〈 m 〈 ½. Second-order results taking account of the body shape are given. These solutions are compared with experimental results obtained in air at a free stream Mach number of 7·55 and stagnation temperature of 630 °K, as well as with numerical solutions at Mach number of 100. The numerical method is tested by comparing solutions corresponding to the experimental conditions with experiment.

Description: Oscillatory convective instability is shown to occur in a rotating fluid layer when convection is caused by surface-tension gradients at a free surface. The asymptotic equations, valid when the Taylor number approaches infinity, are solved analytically, and the critical Marangoni number is evaluated numerically. Fluids with Prandtl numbers above 0·201 will exhibit only stationary instability. Fluids with smaller Prandtl numbers will exhibit oscillatory instability with the critical Marangoni number varying as M0T½ where M0 depends on the Prandtl number and T is the Taylor number.

Description: The structure of fully developed turbulence in smooth circular tubes has been studied in detail in the Reynolds number range between 10,700 and 96,500 (R based on centre velocity and radius). The data was taken as longitudinal and transverse correlations of the longitudinal component of turbulence in narrow frequency bands. By taking Fourier transforms of the correlations, crosspower spectral densities are formed with frequency, ω, and longitudinal or transverse wave-number, kx or kz, as the independent variables. In this form the data shows the distribution of turbulence intensity among waves of different size and inclination, and permits an estimate of the phase velocity of the individual waves.Data taken at radii where the mean velocity profile is logarithmic show that the waves of smaller size (higher (k2x + k2z)½) decrease in intensity more rapidly with distance from the wall than the larger waves, and also possess lower phase velocity. This suggests that the waves might constitute a geometrically similar family such that the variation of intensity with wall distance is a unique function with a scale established by (k2x + k2z)−½). The hypothesis fits the data very well for waves of small inclination, α = tan−1(kx/kz), and permits a collapse of the intensity data at the several radii into a single ‘wave-strength’ distribution. The function of intensity with wall distance which effects this collapse has a peak at a wall distance roughly equal to 0·6(k2x + k2z)−½). For waves whose inclination is not small, it would not be expected that the intensity data could collapse in this way since the measured longitudinal component of turbulence represents a combination of two turbulence components when resolved in the wave co-ordinate system.Although the similarity hypothesis is strictly true only for data taken where the mean velocity profile is logarithmic, a simple correction procedure has been discovered which permits the extension of the similarity concept to the sublayer region as well. This procedure requires only that the observed total turbulence intensity at any station in the sublayer be reduced by a factor which depends solely on the y+ distance from the wall (i.e. on the distance from the wall, scaled by the viscid parameters of the sublayer). The correction factor is independent of Reynolds number and applies equally to waves of all sizes. In this way, all of the turbulence waves down to the very smallest of any significance, are found to satisfy slightly modified similarity conditions.From the data taken a t Reynolds numbers between 96,500 and 46,000 wave ‘strength’ is seen to be distributed more or less uniformly over a range bounded at one extreme by the largest waves which the tube can contain (k2x + k2z ≅ (2/a)2, where a is the tube radius) and at the other extreme by the smallest waves which can be sustained against the dissipative action of viscosity (k2x + k2z ≅ (0·04v/Uτ)2, where Uτ is the shear velocity). As the Reynolds number of the flow is lowered, the spread between the bounds becomes smaller. If the data is projected to a Reynolds number of order lo3 the bounds coalesce and turbulence should no longer be sustainable.

Description: The method of matched asymptotic expansions is used to determine the lateral flow of an ideal fluid past a slender body, when the flow is constrained by a pair of closely spaced walls parallel to the long axis of the body. In the absence of walls, the flow field would be nearly two-dimensional in the cross-flow plane normal to the body axis, but the walls introduce an effective blockage in the cross-flow plane, which causes the flow field to become three-dimensional. Part of the flow is diverted around the body ends, and part flows past the body in the inner cross-flow plane with a reduced ‘inner stream velocity’. An integro-differential equation of identical form to Prandtl's lifting-line equation is derived for the determination of this unknown inner stream velocity in the cross-flow plane. Approximate solutions are applied to determine the added mass and moment of inertia for accelerated body motions and the lift force and moment acting on a wing of low aspect ratio. It is found that the walls generally increase these forces and moments, but that the effect is significant only when the clearance between the body and the walls is very small.

Description: Adiabatic disturbances propagating as transverse waves in an inviscid fluid rotating as a Rankine vortex about the axis of its cylindrical container are considered. The propagation of the first mode of the first two harmonic waves has been investigated. Relative to a fixed co-ordinate system, for each harmonic, there are three waves. Two waves are rotating in the same direction as the fluid, one faster and the other slower than the core of the fluid, and one wave rotates in the opposite direction. The latter is stable and relative to the core of the rotating fluid it is the fastest wave. Relative to the container, the other two waves are speeded up by rotation. However, relative to the rotating core, the angular velocity of the fast wave decreases when the fluid is speeded up, and when it is zero the wave breaks down. As the region of potential flow decreases the angular velocity of the slow wave increases and its amplitude decreases, and in the limit of vanishing potential flow, the wave rotates as fast as the fluid and its amplitude tends to zero.

Description: It does not seem to be possible to prove analytically that an incompressible, inviscid free shear layer is less unstable with respect to spatially growing three-dimensional disturbances than to two-dimensional ones. For this reason a numerical calculation for the special case of the hyperbolic tangent velocity profile was performed. It was found that even for spatially growing disturbances the amplification of three-dimensional disturbances is smaller than for two-dimensional ones.

Description: Direct measurements of the energy transfer spectrum in locally isotropic grid turbulence have been used to determine the extent of validity for grid turbulence of the dynamical equation for the three-dimensional energy spectrum in isotropic turbulence. The extent of applicability of the isotropic energy balance is consistent with the usual local isotropy criterion based on energy spectra alone.The present results are in general agreement with some previous measurements by Uberoi, who determined the transfer spectrum assuming the strict validity of the isotropic dynamical equation. The measured energy transfer spectra are quantitatively similar to those calculated by Kraichnan using the direct-interaction approximation.

Description: The problem considered is that of two-dimensional viscous flow in a straight channel. The steady Navier-Stokes equations are linearized on the assumption of small disturbance from the fully developed flow, leading to an eigenvalue equation resembling the Orr-Sommerfeld equation. This is solved in the limiting cases of small and large Reynolds numberR, and an approximate method is proposed for moderateR. The main results are (i) the dominant mode of the disturbance velocity (i.e. that which persists longest) is antisymmetrical; (ii) for largeRthere are two sequences of eigenvalues. Both sequences are asymptotically real asR→ ∞. The members of the first sequence areO(1) asR→ ∞ and are complex for all finiteR. The members of the second sequence areO(R−1) and the imaginary part isO(R−N) for allN. It is the eigenvalues of the second sequence which will dominate the flow at largeR.

Description: In the design of dielectrophoretic liquid orientation and expulsion systems for zero-gravity environments, maximum electromechanical effect of an imposed electric field is obtained by concentrating the field gradients in the neighbourhood of liquid interfaces. In typical configurations, the electric field gradient plays the role of an electromechanical wall, with a stiffness and inertia represented dynamically by electrohydrodynamic surface waves. As an orientation system rotates, the liquid motions are characterized by these waves as they couple to inertial bulk oscillations and centrifugal surface waves resulting from the rotation. A study is made of configurations typified by an equilibrium in which a circular cylindrical column of inviscid liquid undergoes rigid body rotation. The equilibrium is made possible, even though the cylindrical interface is bounded from outside only by its vapour, because the interface is stressed by an essentially tangential axial electric field intensity, with a strong gradient in the radial direction. Dispersion equations are developed for the electrohydrodynamic centrifugal waves of small amplitude. Conditions for incipience of instability and the frequencies of normal modes of oscillation are given. Experimental observations, which demonstrate the destabilizing influence of the rotation and the effect of rotation and electric field intensity on the normal mode frequencies, are in satisfactory agreement with the theory.

Description: The stability of non-Newtonian jets was investigated. A linearized stability analysis shows that a liquid column of a viscoelastic fluid exhibits more rapid growth of axisymmetric wave disturbances than a Newtonian fluid of the same zero shear viscosity. This result is independent of the form of constitutive equation chosen. Experiments in weakly elastic fluids confirm this expectation, whereas data on fluids with more pronounced elastic properties indicate that non-linear phenomena are dominating. The disturbances appear as a series of droplets connected by random lengths of threads, which thin with distance and eventually lead to jet breakup. Even in dilute viscoelastic solutions, jet breakup does not occur by the growth of clearly defined waves.

Description: Disintegration processes occurring with charged liquid jets of distilled water have been examined. High-speed photographic techniques were used to determine the effect of charging on disintegration of the jet, the size distribution of the drops formed, and the velocities of the drops. At relatively small currents, the jet remains undisplaced while the drops formed acquire components of velocity perpendicular to the jet axis. At sufficiently large currents, the jet develops kink or longitudinal instabilities which can cause the jet to be appreciably displaced, and the drops may have relatively large components of velocity perpendicular to the jet axis. The size distributions of the drops formed are significantly different from those resulting in the absence of electrification. Mean drop size decreases with increased charging in all cases. Drop speeds increase with increased charging as a result of both increased electrical repulsion and reduction in size.

Description: The study of sink flow turbulent boundary layers is of particular relevance to the problem of laminarization. The reason lies in the fact that the acceleration parameter which principally determines when a turbulent boundary layer will begin to revert towards laminar is, in these flows, constant from station to station. The paper presents theoretical solutions to this class of boundary layer by making use of the Prandtl mixing-length formula to relate the turbulent shear stress to the mean velocity gradient. Near the wall the Van Driest recommendation for mixing length is adopted and the Van Driest function, A+, is chosen such that the skin friction coefficient does not exceed a certain maximum value.The predicted solutions, which are in good agreement with available experimental data, display a plausible shift from the turbulent towards the laminar solution as the acceleration parameter is increased.

Description: The flow is examined in the neighbourhood of the trailing edge of slender aerodynamic shapes which terminate in either a cusp or a wedge. The manner in which the boundary layer reacts to the rapidly varying pressure field in such regions is analyzed using the method of matched asymptotic expansions. The case of a wedge is examined in greater detail and a criterion for separation to occur is established.

Description: An analysis of the full, compressible, non-adiabatic boundary-layer equations is presented to describe the so-called ‘throat’ formed when a two-dimensional viscous layer, interacting with a supersonic inviscid outer stream, is accelerated or decelerated through sonic velocity defined in some mean sense. The basic analysis differs from previous momentum integral theories in that the dynamics of the viscous layer is described by the exact local expressions for the streamwise gradients of the flow variables that obtain from the boundary-layer conservation equations, rather than on streamwise derivatives of integral properties of these equations. The theory is then used to develop an extensive analogy with the classical analysis of the throat in the inviscid quasi one-dimensional streamtube. The theory shows that a single integral constraint exists at the throat, which relates the velocity and temperature profiles in the viscous layer to the motion of the inviscid outer flow. One consequence of this constraint is that, for a one-parameter family of profile shapes, the solution can be started at the throat station by specifying only a single variable, the free stream Reynolds number based on the physical thickness of the viscous layer at the throat station. For the hypersonic near wake, this simplification permits one to obtain an approximate solution for the downstream flow without first solving the detailed motion in the base recirculation region. The paper ends with a discussion of the numerical results for the Stewartson family of wake-like profiles.

Description: A fluid swirling through an axisymmetrically deformed tube is considered, ignoring viscosity and compressibility. For a tube of radius R, having a longitudinal wall deformation of wave number k, the flow near the wall is blocked, if the Rossby number assumes one of the critical values (λ2n + k2R2)−½, where n is any positive integer, and λn is the nth zero of the Bessel function J1(λ). Rossby number is defined as W/2Rω, in which W and ω are the uniform axial and angular velocities in an undeformed tube. For a convergent-divergent nozzle, the critical Rossby numbers have the same form, with kR = 0. The flow exhibits radically different patterns when each critical Rossby number is crossed.

Description: Collapsing cylindrical metallic shells have been used to compress magnetic fluxes and generate megagauss magnetic fields. Such shells experience large, rapidly growing accelerations and their symmetry can be completely destroyed by Rayleigh-Taylor instabilities. This paper presents a theoretical study of the Rayleigh-Taylor instability for radially accelerated incompressible cylindrical shells submitted to the pressure of much lighter media. Low-amplitude flute perturbations are considered and Fourier-analyzed in the azimuthal angle. A fourth-order linear differential system with time-dependent coefficients is derived, which determines the two interface-displacements. Stability criteria are discussed. When the perturbation wavelength is much greater or much smaller than the shell thickness, the differential system splits into two independent differential equations and results are greatly simplified; analytical solutions are available for some cases.The case of axial field compression (A.F.C.) is discussed as an application. Numerical solutions give the time behaviour of all possible initially given disturbances. The initial perturbations, which are able to reach the axis during their development and which are consequently dangerous in magnetic field compression experiments, have been calculated. Results are consistent with the few experimental data available. They show that the degree of symmetry of cylindrical devices has to be extremely good in order to get successful compressions.Finally, non-linear and compressibility effects have been taken into account for some A.F.C. cases, solving the full non-linear fluid equations numerically.

Description: We consider here the flow due to the oscillation of a slender oblate spheroid in a non-homogeneous, rotating fluid confined between two parallel planes which are perpendicular to the (vertical) axis of rotation. The direction of oscillation of the spheroid is perpendicular to the axis of rotation. By solving a set of dual integrals the steady-state solution is obtained in the two cases when the plates are at an infinite distance from the body and when they are at a large but finite distance. The singular or discontinuous surfaces observed in the case of homogeneous fluid are absent here. Also, the steady-state velocity is no longer independent of the distance along the axis of rotation. The velocity has now a vertical gradient, an important feature in the case of stratified fluid. It is also found that the presence of the plane boundaries increases the force on the body.

Description: The stability and time-dependent deflexions of a thin flexible cylinder with zero bending rigidity set in a viscous stream are examined. The cylinder is fixed at one end and free at the other. Modal shapes are found based on solutions to linearized equations resulting from small deflexion assumptions; the dependence of cylinder motion upon the aerodynamic, elastic and physical size characteristics of the cylinder is exhibited. It is found that the cylinder motion is always unstable and that cylinder amplitude increases without bound as time is increased.

Description: An analysis of a hydrostatic thrust bearing with electrically conducting compressible lubricant under an axial-current-induced pinch is presented. It is shown that the load capacity of the bearing can be increased by the pinch effect and the magnitude of the pinch effect depends on the mass flow rate. It is also shown that a load proportional to the square of the axial current can be sustained even when there is no flow or external pressurization.

Description: The paper discusses free convective flows above a horizontal plate, both theoretically and on the basis of experiments which yield quantitative data. The theory is applicable to the semi-infinite plate and is extended to cover the complete range of Prandtl number values including Pr → 0 and Pr → ∞. Experiments were carried out to demonstrate the existence of a laminar boundary layer above a horizontal plate at intermediate Grashof (respectively Rayleigh) numbers, and its extent along the plate. This layer breaks down into large-eddy instability some distance from the leading edge. The value of the critical Rayleigh number for this to occur, obtained experimentally using semi-focusing colour-Schlieren photography is in reasonable qualitative agreement with previously known data (Tritton 1963a,b).

Description: This note presents a discussion of the roles of axial momentum flux, flow force, angular momentum flux and circulation in determining the strength and hence characterizing the structure of such narrow rotating axisymmetric core flows as swirling jets, vortex jets, sink vortices and vortex wakes. The salient (though sometimes neglected) features of these core flows are that perturbation pressure plays an essential role both in the coupling of axial and azimuthal velocity fields and in the transmission of force along the core, and that flux of angular momentum is invariant only along cores with zero gross circulation. A number of existing solutions are brought into relationship by the discussion, including Long's similarity solution for draining vortices and Reynolds’ dimensional treatment of swirling wakes.