ALBERT

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: In studying the stability of the boundary layer with surface mass injection, a generalized version of the Orr–Sommerfeld equation was derived which takes account of the transverse velocity component in the mainflow. The new terms in the generalized Orr–Sommerfeld equation are inversely proportional to the Reynolds number. The resulting eigenvalue problem was solved numerically for a wide range of values of the mass injection intensity. It was found that the critical Reynolds number (based on the distance from the leading edge) decreases with increasing mass injection. The deviations between the critical Reynolds numbers from the generalized and conventional Orr–Sommerfeld equations have a different sign at low injection intensities from that at high injection intensities. © 1971, Cambridge University Press. All rights reserved.

Description: It is demonstrated that the basic stratification in a fluid region subject to thermal forcing may be predicted rather simply for a fairly wide class of boundary conditions. Explicit solutions are derived in certain cases. A useful experimental method for maintaining a stratified system with arbitrarily specified vertical variation of density emerges from the analysis. A preliminary laboratory experiment has demonstrated the efficiency of this method. The restrictions on the validity of the theory involve a limitation on the thermal forcing of the fluid, which may be expressed as an upper limit on the thermal conductance of the boundary of the region. Furthermore, the buoyancy frequency characterizing the solution must be sufficiently large to give rise to a boundary-layer-type flow pattern. © 1971, Cambridge University Press. All rights reserved.

Description: The interactions between electrical tractions at the interface of a liquid jet and instability phenomena are studied with emphasis on effects due to interfacial charge relaxation. Charge relaxation causes the oscillatory growth of a perturbation. When viscous effects are small, small fields tend to decrease the growth rate of the axisymmetric mode, up to a point, and precipitate instability of the non-axisymmetric modes. Still larger field strengths increase the growth rates of asymmetric as well as axisymmetric modes. Instabilities characterized by highfrequency oscillations appear to persist even though the charge relaxation phenomena may be quite rapid. When, on the other hand, viscous effects predominate the only unstable disturbance form is the axisymmetric one, although the manner of growth may be oscillatory. © 1971, Cambridge University Press. All rights reserved.

Description: Centrifugally driven circulations in a rapidly rotating cylinder of fluid heated differentially in the vertical are considered. Boundary-layer solutions obtained previously are extended to include large diameter/height aspect ratios and a centrifugal acceleration of the same magnitude as that of gravity. The ratio of convective to conductive heat transfer is small in the region of parameter space considered. The effect of the circulations on the asymptotic stability of a fluid heat from below and subjected to Coriolis force is then considered. Away from the side wall of the cylinder the basic state circulation increases the critical Rayleigh number at which gravitational instabilities occur; however, a destabilization near the side wall is possible. © 1971, Cambridge University Press. All rights reserved.

Description: A symposium on aerodynamic noise was held at Loughborough University from 14 to 17 September 1970 under the sponsorship of the Royal Aeronautical Society and the British Acoustical Society. The objective of the meeting was to focus attention on unsolved theoretical and experimental problems which will require attention over the next few years. Areas which were covered included jet noise, nonlinear acoustics, rotor noise, and diffraction theory. The symposium was successful in bringing together several new themes in aerodynamic noise research. The most significant of these were the existence of a degree of order in turbulent jet flows, and the dominant effect of inflow conditions on rotor noise radiation. In addition an improved and unified basis for jet noise theory seems to be evolving. © 1971, Cambridge University Press. All rights reserved.

Description: Measurements of turbulence energy diffusion and the spectral distributions of stress components in the core of turbulent pipe flow are presented. The results tend to confirm the proposal of Bradshaw (1967a, b) that an inertial subrange in the spectra can exist at quite modest laboratory Reynolds numbers. They also illuminate the inconsistencies in Laufer's (1954) measurements of dissipation and suggest that the fitting of a ⊟5/3 power law to the spectra may well provide the most accurate method of determining dissipation for Re ≳ 105. © 1971, Cambridge University Press. All rights reserved.

Description: A von Kármán vortex street generated in the usual way was subjected to a deceleration, thereby changing the ratio of longitudinal to lateral spacing between the vortices. Distortion of the individual vortices followed which resulted in annihilation of concentrated vortex regions and creation of a stationary wake flow. This wake flow was itself dynamically unstable and developed into a new vortex street of a different frequency from the initial one. The breakdown of the initial vortex street is qualitatively explained by considering the convection of a concentrated vortex region due to the motion imposed by all the other vortices. © 1971, Cambridge University Press. All rights reserved.

Description: The theory of Hunt & Stewartson (1965) for MHD flow in a rectangular duct with conducting walls parallel and non-conducting walls perpendicular to the magnetic field is applied to the problem of electrically driven MHD flow in a rectangular annulus. It is assumed that the Hartmann number M is sufficiently great for secondary flow effects to be negligible. The experiment described here satisfied the conditions of the theory and thus provides a sensitive experimental check on Hunt & Stewartson's theory. The theory is found to agree with the experiments to within the accuracy of the asymptotic theory. © 1971, Cambridge University Press. All rights reserved.

Description: Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively. © 1971, Cambridge University Press. All rights reserved.

Description: The supersonic flight of a slender projectile in a fluid which may undergo internal non-equilibrium transformations is examined by a systematic perturbation scheme. In the frozen limit, the classical results of Whitham and his celebrated ‘rule’ are recovered. Unlike the classical theory, however, the shape of the nose shock can be expressed explicitly in terms of known functions when the relaxation decay length becomes of the same order as a characteristic length scale. The theoretically predicted shock angle, expressed as a function of the radial distance, is found to be in excellent agreement with the experimental measurements of Wegener and Klikoff. © 1971, Cambridge University Press. All rights reserved.

Description: Steady supersonic two-dimensional flows governed by the Navier–Stokes equations are considered. For flows past a thin body, the Oseen theory is shown to fail at large distances. An investigation of the equations bridging the linear and non-linear zones is made. From this, it follows that the resulting equations are a system of Burgers and diffusion equations. The Whitham theory is shown to result under the inviscid limit of our analysis. Various other limits are also obtained. An explicit expression for flows past a thin airfoil is given, and the flow past a double wedge is exhibited in terms of known functions. © 1971, Cambridge University Press. All rights reserved.

Description: By assuming that an uncharged drop situated in a uniform electric field E retains a spheroidal shape while oscillating about its equilibrium configuration, two approximate equations of motion are derived for the deformation ratio γ expressed as the ratio a/b of the major and minor axis of the drop. Solutions of these equations of motion indicate that the stability of a drop of undistorted radius R and surface tension T depends upon E(R/T)½ and the initial displacement of γ from its equilibrium value. The predictions of the two equations are compared to assess the accuracy of the spheroidal assumption as applied to such a dynamical situation. The analysis is used to determine the stability criterion of a drop subject to a step function field. Finally, the limit of validity of the spheroidal assumption is discussed in terms of Rayleigh's criterion for the stability of charged spherical drops. By applying Rayleigh's criterion to the poles of a spheroidal drop, the stage at which the drop departs from spheroidal form to form conical jets was approximately determined. © 1971, Cambridge University Press. All rights reserved.

Description: Previous results concerning the effects of axial velocity on the motion of vortex filaments are reviewed. These results suggest that a slender-body force balance between the Kutta–Joukowski lift on the vortex cross-section and the momentum flux within the curved filament will give some insight into the behaviour of the filament. These simple ideas are exploited for both a single vortex filament and a vortex pair, both containing axial flow. The stability of a straight vortex filament containing an axial flow to long wave sinusoidal displacements of its centre-line is investigated and the stability boundary obtained. The effect of axial flow on the stability of a vortex pair is explored. It is shown that to lowest order (in the ratio of vortex core radius to distance between the vortices) the effect of axial flow is to reduce the self-induced rotation of a single filament and that this effect can be considered as a change in effective core radius. To the next order, travelling waves appear in the instability, the instability mode for the vortex pair becomes non-planar but the amplification rate of the instability is not affected. © 1971, Cambridge University Press. All rights reserved.

Description: Experiments are described in which velocities were measured ahead of a semi-infinite Rankine body moving parallel to a uniform magnetic field in a conducting fluid. The flow disturbance in front of the body is found to increase in length as N½, where N is the interaction parameter. In most of the experiments this parameter was varied from 4 to about 50. Measurements made along the axis of symmetry in the flow show that there is a relatively short region of stagnant fluid directly ahead of the body. The major part of the disturbance is found to consist of a much longer region in which the flow undergoes transition from conditions in the free stream to conditions near the body. Velocity profiles across the flow in this region show that for increased N, at a fixed distance ahead of the body, the velocity defect increases and the disturbance becomes more confined radially. Although the radial gradients in the flow increase with N, they are found to be much smaller than would be expected in a flow containing thin current layers. A physical model of the flow which has currents and pressures consistent with these results is discussed. © 1971, Cambridge University Press. All rights reserved.

Description: Interactions between short gravity waves and larger-scale flows are investigated in the two-scale approximation. The effect of the wave field on the mean flow is described by an interaction stress tensor and a surface mass transfer. The results are applied to Phillips’ and Longuet-Higgins’ model of short waves breaking on the crests of long carrier waves. It is found that the work done on the long waves by the interaction stresses (corresponding to Longuet-Higgins’ ‘maser’ mechanism of wave generation) is almost exactly balanced by the loss of potential energy arising from the mass transfer. The residual energy transfer leads to attenuation of the long waves, independent of their propagation direction relative to the short waves. Damping factors are estimated from the upwind–downwind ratios of radar backscatter cross-sections. It is found that interactions with waves shorter than 35cm yield attenuation rates about an order of magnitude smaller than the observed growth rates due to the wind. © 1971, Cambridge University Press. All rights reserved.

Description: The relaxation of an initially non-uniform gas to equilibrium is studied within the framework of the kinetic theory of gases. The macroscopic gas properties are taken to depend on one spatial dimension as well as the time. The amplitude of the non-uniformity is assumed to be small with a length scale large compared with the mean free path, and the Krook model of the Boltzmann collision integral is employed. By applying multi-time scale perturbation methods to this reduced problem, uniformly valid analytical solutions for the macroscopic velocity, density and temperature are obtained. The macroscopic equations appropriate to each stage of the relaxation process are obtained in a straightforward and unambiguous manner. The distribution function obtained is shown to be a re-expansion of the Chapman–Enskog solution of the Krook equation, with additional terms accounting for the relaxation of the initial conditions to a near equilibrium form. The results indicate that the uniformly valid frst approximation to the macroscopic velocity, density and temperature can be obtained from the Navier–Stokes equations, but that no purely macroscopic set of equations will suffice for the determination of higher approximations. © 1971, Cambridge University Press. All rights reserved.

Description: The effect of blowing through a porous rotating disk on the flow induced by this disk is studied. For strong blowing the flow is almost wholly inviscid. First-order viscous effects are encountered only in a thin layer at some distance from the disk. The results of an asymptotic analysis are compared with numerical integrations of the full equations and complete agreement is found. © 1971, Cambridge University Press. All rights reserved.

Description: An experimental investigation was conducted in a circular pipe to examine the influence of a transverse magnetic field on the structure of turbulent shear flow of a conducting fluid (mercury). In the present paper, part 1, mean velocity profiles, turbulence intensity profiles, velocity fluctuation spectra, axial pressure drop profiles, and skin friction data are presented which quantitatively exhibit the Hartmann effect and damping of the velocity fluctuations over a broad range of Reynolds numbers and magnetic fields. The results of heat transfer experiments will be reported by the authors in the following paper, part 2. © 1971, Cambridge University Press. All rights reserved.

Description: Experiments were conducted in an ascending laminar flow through a vertical pipe under combined free and forced convection at constant heat flux through the wall. Mean velocity and temperature profiles were measured with a hot-wire probe. This velocity profile which is deformed by the buoyancy forces, enabled us to compute the reduced acceleration parameter. The profiles obtained showed that the value of the parameter at which reverse transition takes place is approximately the same as that found in isothermal boundary-layer flow. By measuring the autocorrelation function of the velocity after the reverse transition it was shown that the flow in the boundary layer becomes laminar as well as fluctuating and that it oscillates with a predominating period. © 1971, Cambridge University Press. All rights reserved.

Description: The 23rd EUROMECH colloquium on finite-amplitude and diffusive effects in acoustics was held in Rapperswil (Switzerland), 5–7 April 1971. There were 41 participants from 7 countries, and the authors were the chairmen of the meeting organization. References quoted in this report give the titles of the talks and sources for further details of the work described at the meeting; there will be no other publication of the proceedings. The subject matter of this meeting was more strongly restricted than is indicated by its title, inasmuch as papers motivated solely by sonic-boom research problems were not included. Included in particular were problems of acoustic damping by relaxation, dust, moisture, etc.; damping in ducts; effects of turbulence; acoustic streaming; and thermo-acoustic effects. © 1971, Cambridge University Press. All rights reserved.

Description: Waves generated at the stern of a ship must travel through the ship's wake. The effect of the mean flow in the wake refracting the waves is calculated by using a much simplified model. It is found that the waves diverging from the stern of a ship may differ considerably from the bow waves, in qualitative agreement with observation. © 1971, Cambridge University Press. All rights reserved.

Description: The velocity fields in and around a deformed drop suspended in an arbitrary (albeit Stokesian) unbounded flow field are solved. The usefulness of the solution is demonstrated by solving the drag force and lateral migration of a drop suspended in an unbounded Poiseuillian field. It is demonstrated that, due to the deformation of the drop, there exists a radial component of the settling velocity. The direction of the radial migration depends primarily on the product UHR (the Hadamard-Rybczynski terminal settling velocity) by U0 (the maximum Poiseuillian velocity). A positive product results in a lateral migration away from the location of maximum velocity; the converse also holds. © 1971, Cambridge University Press. All rights reserved.

Description: Benjamin (1970) has calculated the upstream influence in open-channel flow, and has argued that a similar effect occurs when a body is moved along the axis of a tube of uniformly rotating fluid. In the present paper Benjamin's work is extended to the case of interfacial waves in a two-fluid system. It is shown that there are certain special flows for which the upstream influence vanishes. © 1971, Cambridge University Press. All rights reserved.

Description: Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods. In this paper, the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored. Accuracy of simulation is tested empirically using a simple scalar-convection test problem and the Taylor–Green vortex-decay problem. It is demonstrated empirically that the Galerkin (Fourier) equations involving Np degrees of freedom, where p is the number of space dimensions, give simulations at least as accurate as finite-difference simulations involving (2N)p degrees of freedom. The theoretical basis for the improved accuracy of the Galerkin (Fourier) method is explained. In particular, the nature of aliasing errors is examined in detail. It is shown that ‘aliasing’ errors need not be errors at all, but that aliasing should be avoided in flow simulations. An eigenvalue analysis of schemes for simulation of passive scalar convection supplies the mathematical basis for the improved accuracy of the Galerkin (Fourier) method. A comparison is made of the computational efficiency of Galerkin and finite-difference simulations, and a survey is given of those problems where Galerkin methods are likely to be applied most usefully. We conclude that numerical simulation of many of the flows of current interest is done most efficiently and accurately using the spectral methods advocated here. © 1971, Cambridge University Press. All rights reserved.

Description: A new theory is developed for the wake far downstream of a cylindrical body of height h, placed with its generators perpendicular to the flow on a surface above which there is a boundary layer of thickness δ. If the streamwise (x) velocity in the wakeis (U + u), then assuming (h/δ) is small enough that the velocity profile in the boundary layer may be regarded as U = αy, and assuming |u| ≪ U, linear differential equations governing u are derived. It is found that a constant along the wake is [formula omited] This result can be used to find an order of magnitude estimate for u, because I is related to the forces on the body producing the wake by the approximate formula [formula omited] where C1 is that component of the couple on the body produced by pressure and viscous stresses in the x direction. For the particular case of a small hump on the boundary of height h and length b, such that h ≪ b, the above relation is shown to be exact. The perturbation velocity in the wake is found to have a similarity solution [formula omited] the physical implications of which are discussed in detail. The relevance of the theory to the problem of transition behind a trip wire is also mentioned. © 1971, Cambridge University Press. All rights reserved.

Description: The flow past a circular cylinder moving close to a free surface at high Froude number is investigated by the method of matched asymptotic expansions. In contrast with the linearized solution in which the dimensionless depth of immersion h = h′g/U′2 is kept constant, in the present analysis h → 0 as Fr → ∞. The inner flow model is that of a non-separated non-linear gravity-free flow past a doublet, while the linear outer solution is that of a singularity a t the free surface. At deep submergence the solution coincides with the linearized solution. At moderate immersion depths the linearized solution is still valid, provided that the depth is replaced by an effective depth, larger than the actual one. For a body close to the free surface the non-linear solution differs significantly from the linearized solution. © 1971, Cambridge University Press. All rights reserved.

Description: The modified Oseen method is extended to provide a description of the boundary layers which accompany certain swirling flows over a rigid boundary in a rotating container. By comparison with known results it is shown that a refined procedure has errors of the order of 1% when the inviscid flow is a rigid body rotation; it is anticipated that, for the more interesting flows, the error is of the order of 30%. © 1971, Cambridge University Press. All rights reserved.

Description: This paper describes some laboratory and numerical experiments made on the longitudinal dispersion in an open channel flow. Particular attention has been paid to the initial stages of the process. Physical arguments suggest that the streamwise dispersion of a line of marked fluid elements across a two-dimensional turbulent shear flow occurs in three distinct stages. These stages are identified by a change in the form of the distribution of marked fluid elements in the flow direction. The skewed distribution of the first stage is readily identified by a constant value (approximately 1-1) for the ratio of the peak velocity (V,) of the distribution to the mean-flow velocity [formula omited] experiments using dyed fluid, made at this stage of the process, have revealed six identifiable features of the suggested distribution. The distributions suggested for the second and the third stage are consistent with the experimental findings of Elder (1959) for the second stage and Taylor (1954) for the third stage. An attempt has been made to simulate the process numerically using a Markovian model. The results of the simulation confirm features suggested by physical arguments and are in agreement with the open channel experiments. The Lagrangian autocorrelation function is found to be related to the Lagrangian velocity-history of marked fluid released from extreme positions on the flow cross-section. The correlation function, as expressed in terms of the velocity-history function provided by the numerical simulation, is [formula omited] where u* is the friction velocity and U(y′) is the temporal mean velocity at a (non-dimensionalized) distance y′ from the flow boundary. In an open channel flow at a Reynolds number (based on friction velocity and channel depth) of 500, the numerical simulation provides the value of b = 0·536. The results of an experiment, in which the three-dimensional motion of small neutrally buoyant spheres was recorded in many small discrete time intervals, corroborate the theoretical suggestions and simulation results. © 1971, Cambridge University Press. All rights reserved.

Description: Some experiments are described in which steady-state shearing flows are developed in stratified brine solutions contained in a cyclically continuous tank of rectangular cross-section. Over the range of overall Richardson numbers studied, the results suggest that whenever turbulent layers are present on either side of a region of fluid with a gravitationally stable density gradient, they cause erosion of this region to occur. The erosion leads to the formation of two homogeneous layers separated by a thin layer of strong density and velocity gradients. The gradient Richardson number, computed by using the velocity and density gradients in this transition layer, tends to have a value of order one. If we define an overall Richardson number Ri* by averaging the velocity and density gradients over the entire depth of fluid in the tank, we find that the non-dimensional buoyancy flux, Q, is functionally related to Ri* by an equation of the form Q = C1(Ri*)⊟1 where C1 is a constant, approximately, and Ri* ranges in value between one and thirty. To check the effect of a large variation of the molecular diffusivity coefficient on flow conditions, we ran a limited number of experiments with thermally stratified fluid. Over a restricted range, 1·0 ≪ Ri* ≪ 5·0, velocity profiles very similar to those measured in the brine-stratified experiments at like values of Ri* were obtained. This suggests that the coefficient of molecular diffusion is not an important parameter in either type of experiment. Other experiments, made in the same apparatus, describe the entrainment by a turbulent, homogeneous layer of an initially quiescent layer of fluid with a linear density gradient. The depth of the turbulent layer, D, increases with time, t, according to the relation. [formula omited] This result is consistent with that found by Kato & Phillips (1969), although the turbulent layer in the present experiment is generated in a different manner. © 1971, Cambridge University Press. All rights reserved.

Description: A thermal boundary layer is established by heating a vertical plate in a dielectric liquid. An alternating voltage is applied between the heated plate and another plate which is not parallel to the heated plate. This voltage produces a non-uni-form electric field which in turn produces electrical forces acting on the gradients in dielectric permittivity which result from the temperature gradients. These electrical forces alter the boundary layer. In this paper approximate equations are developed which allow one to calculate the boundary-layer,thickness, velocity, and Nusselt numbers for the boundary layer in the presence of the non-uniform electric field. Numerical calculations show that the heat-transfer coefficient can be either increased or decreased by the non-uniform field, depending on whether the field is strongest at the top or bottom of the plates and also on the field strength. Experiments were performed which demonstrate the change in heat transfer caused by the non-uniform field. © 1971, Cambridge University Press. All rights reserved.

Description: The thermally driven motion of water contained in a rotating annulus of square cross-section and having a free surface is investigated by numerical integration of the three-dimensional non-linear Navier–Stokes equations. The nature of steady wave flow is examined in detail and a comparison made with the corresponding axisymmetric solution in parameter space. The steady wave solution proves to be consistent kinematically, dynamically and energetically with Lorenz's hypothesis that the wave can be attributed to the baroclinic instability mechanism. The deviaboric The deviation from the zonal mean. The wave motion is almost completely independent of the side boundary layers which make little contribution to the characteristics and energetics of the deviakoric flow. These side layers are approximately axisymmetric and appear qualitatively indistinguishable from their counterparts in the axisymmetric solution. However, significant Ekman layer features appear in the deviatoric wave structure. Away from the boundaries the dynamical balance of terms is hydrostatic and quasi-geostrophic with changes of vertical vorticity influenced by stretching and viscous diffusion. Heat conduction is completely unimportant except in the side boundary layers. The angular momentum transport by the deviatoric motion is largest at the free surface and is mainly against the angular momentum gradient. A strong outward deviatoric flux of momentum is found in the Ekman layer. The dissipation of deviatoric kinetic energy occurs in the Ekman layer and jet whilst most of the dissipation of the mean kinetic energy occurs in the boundary layer of the inner wall. The large differences between the axisymmetric and zonal mean states indicate that linear baroclinic instability analysis of the axisymmetric state is not strictly relevant to an understanding of the wave formation. The character of the wave suggests that the mean environment with which the deviatoric wave interacts is the wave-present zonal mean state. Only a non-linear finite amplitude baroclinic instability analysis (as yet undeveloped) could possibly explain the wave formation. © 1971, Cambridge University Press. All rights reserved.

Description: An apparatus that approximates a two-dimensional, infinite train of peristaltic waves yields measurements of mean flow, of mean pressure rise, and of pressure-time pulses at fixed locations. In addition, visual observations of ‘reflux’ and ‘trapping’, using dyed fluid, are shown. The inertia-free range extends up to a Reynolds number of about 1. In this range, the theory of Shapiro, Jaffrin & Weinberg (1969) is confirmed with respect to mean pressure vs. mean flow, pressure vs. time, reflux, and trapping. The controversy regarding the criterion of material reflux is settled in favour of the Lagrangian time-mean velocity rather than the Eulerian time-mean velocity. Experiments at higher Reynolds numbers show that the second-order expansion theory of Jaffrin (1971) is valid up to a Reynolds number of about 10. © 1971, Cambridge University Press. All rights reserved.

Description: Davey, Di Prima & Stuart's (1968) double amplitude expansion for disturbances in flow between concentric cylinders is formulated in matrix notation. The stability of the secondary equilibrium (Taylor-vortex) flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation. Qualitative confirmation is found of instabilities to the Taylor-vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number. © 1971, Cambridge University Press. All rights reserved.

Description: An axisymmetric flow of a rotating fluid into a point sink was studied experimentally. The type of motion is mainly controlled by the value of the Rossby number R, a ratio of inertial and Coriolis forces. Experimental investigation shows that at a sufficiently large value of R the fluid motion resembles potential flow. However, as R falls below a critical number the withdrawal of the fluid starts to be selective. The flow field then divides into two regions; namely, a central flowing core and an almost stagnant region surrounding it. It is observed that at a Rossby number below the critical value the flow field, induced by a sudden start of discharge at the sink, experiences several distinct stages during the course of each run. At the initial moment the flow exhibits a feature of potential flow. During the second stage, it develops into a state of selective withdrawal with an inviscid profile of a flowing core, which is the main interest of the present study. In the third stage, due to the unavoidable influence of the free or solid surface at the upstream, flow undergoes another change. The flowing core becomes a fast-spinning jet, in which the viscous force becomes important. It is also found that once selective withdrawal begins, the angular velocity and flow rate of the flowing core differ substantially from the basic rotation and the actual discharge at the sink. During this second stage of flow development, the flowing core tends to adjust itself such that the intrinsic Rossby number R′(= Wmax/2Ωcδc) based on the properties of the flowing core, virtually remains constant for all values of R below the critical value. This constant value of R′ is found to be about 0·36. The critical value of R which marks the beginning of the selective withdrawal is found to be in the neighbourhood of 0·26. © 1971, Cambridge University Press. All rights reserved.

Description: Complementary variational formulations are developed for the scattering of a gravity wave by a circular dock. These formulations, which are based on assumed distributions of the radial velocity and the potential, respectively, on the projection of the cylindrical boundary, yield lower and upper bounds to an impedance parameter that determines the difference between the scattered wave for the dock and the corresponding wave for a circular cylinder. Numerical results, using trial functions based on the incident wave, are compared with the results implied by a Galerkin solution (Garrett 1971). The maximum errors in the variational approximations to the total scattering cross-section are found to be of the order of 2% for a typical depth/radius ratio, draft/depth ratios of 0, ½ and 1, and all wavelengths. The axisymmetric component of the scattering cross-section is found to be very close to the value for scattering by a circular cylinder (dock extending to bottom). The intensity of the scattered wave on the forward axis for long wavelengths and a certain range of the geometric parameters is significantly less than that for a circular cylinder, and may vanish for critical combinations of these parameters. © 1971, Cambridge University Press. All rights reserved.

Description: The statistical initial-value problem for a class of weakly coupled waves whose linear dispersion relation is Ω ∞ ± |k| is examined. It is found that in two and higher dimensions a natural asymptotic closure is possible. The redistribution of energy is achieved by means of two mechanisms; the first by a resonance between collinear wave vectors; the second by a local transfer between adjacent rays. The entropy functional is ∫ log n(k) dk and corresponds to particles obeying Bose–Einstein statistics. © 1971, Cambridge University Press. All rights reserved.

Description: Expansions are obtained for the large Prandtl number structure of the laminar natural convection boundary layer, together with its linear stability characteristics, for the case of a uniform-heat-flux semi-infinite vertical plate. The primary source of instability is shown to arise from a temperature-coupling effect associated with the inner heated region of the boundary layer. Based upon an empirical correlation between the results of linear stability theory and experimentally determined regimes of the turbulent-transition process, it is shown that the flow can be expected to become turbulent before the outer vorticity region of the laminar boundary layer is fully established. The results are generalized to the isothermal plate case. © 1971, Cambridge University Press. All rights reserved.

Description: The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 ≤ N ≤ 0·0092 the flow is unstable at all R. For 0·0092 ≪ N ≤ 0·0233 the flow is unstable at 0 ≪ R ≪ Ruc where Ruc decreases as N increases. For 0·0233 ≪ N ≪ 0·0295 the flow is unstable at Rlc ≪ R ≪ Ruc where Rlc increases with N. Lastly for N 〉 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 ≤ M ≤ 0·52, and Rc is given by the corresponding Rlc. For M 〉 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos⊟1(0·52/M) to the direction of the basic flow. © 1971, Cambridge University Press. All rights reserved.

Description: The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented. © 1971, Cambridge University Press. All rights reserved.

Description: The effects of sinusoidal protrusions on steady laminar free convection between vertical walls is investigated in this paper. Numerical results are presented for various values of the size and spacing of the protrusions. In particular optimum values are found which yield maximum wall heat-transfer coefficients. © 1971, Cambridge University Press. All rights reserved.

Description: This paper is intended to evaluate the wall effects in the pure-drag case of plane cavity flow past an arbitrary body held in a closed tunnel, and to establish an accurate correction rule. The three theoretical models in common use, namely, the open-wake, Riabouchinsky and re-entrant-jet models, are employed to provide solutions in the form of some functional equations. From these theoretical solutions several different rules for the correction of wall effects are derived for symmetric wedges. These simple correction rules are found to be accurate, as compared with their corresponding exact numerical solutions, for all wedge angles and for small to moderate ‘tunnel-spacing ratio’ (the ratio of body frontal width to tunnel spacing). According to these correction rules, conversion of a drag coefficient, measured experimentally in a closed tunnel, to the corresponding unbounded flow case requires only the data of the conventional cavitation number and the tunnel-spacing ratio if based on the open-wake model, though using the Riabouchinsky model it requires an additional measurement of the minimum pressure along the tunnel wall. The numerical results for symmetric wedges show that the wall effects in-variably result in a lower drag coefficient than in an unbounded flow at the same cavitation number, and that this percentage drag reduction increases with decreasing wedge angle and/or with decreasing tunnel spacing relative to the body frontal width. This indicates that the wall effects axe generally more significant for thinner bodies in cavity flows, and they become exceedingly small for sufficiently blunt bodies. Physical explanations for these remarkable features of cavity-flow wall effects are sought; they are supported by the present experimental investigation of the pressure distribution on the wetted body surface as the flow parameters are varied. It is also found that the theoretical drag coefficient based on the Riabouchinsky model is smaller than that predicted by the open-wake model, all the flow parameters being equal, except when the flow approaches the choked state (with the cavity becoming infinitely long in a closed tunnel), which is the limiting case common to all theoretical models. This difference between the two flow models becomes especially pronounced for smaller wedge angles, shorter cavities, and with tunnel walls farther apart. In order to gauge the degree of accuracy of these theoretical models in approximating the real flows, and t o ascertain the validity of the correction rules, a series of definitive experiments was carefully designed to complement the theory, and then carried out in a high-speed water tunnel. The measurements on a series of fully cavitating wedges at zero incidence suggest that, of the theoretical models, that due to Riabouchinsky is superior throughout the range tested. The accuracy of the correction rule based on that model has also been firmly established. Although the experimental investigation has been limited to symmetric wedges only, this correction rule (equations (85), (86) of the text) is expected to possess a general validity, at least for symmetric bodies without too large curvatures, since the geometry of the body profile is only implicitly involved in the correction formula. This experimental study is perhaps one of a very few with the particular objective of scrutinizing various theoretical cavity-flow models. © 1971, Cambridge University Press. All rights reserved.

Description: Upon review of past experimental results and theoretical efforts it is apparent that the mechanism by which combustion noise is generated is not well understood. A theory of combustion noise is developed in this paper which follows rigorously from the principles of fluid mechanics. Lighthill's approach, used in his studies of aerodynamic noise, is closely followed in the present work. The sound radiated from open, turbulent flames is found to depend strongly upon the structure of such flames; at present their structure is not well known. However, meaningful bounds and scaling rules for the sound power output and spectral content are derived based upon the present limited knowledge. A framework is developed which explains past experimental work and the origin of combustion noise. © 1971, Cambridge University Press. All rights reserved.

Description: We investigate the behaviour of infinitesimal perturbations introduced into an unstably stratified horizontal Couette flow. We assume that the fluid is Boussinesq and contained in an infinite conducting rectangular channel which is uniformly heated from below. The sidewalls are rigid and the Couette flow is generated by moving them with equal and opposite velocities along the channel. The top and bottom are assumed to be free so that we can separate variables. Without shear, the preferred modes of convection closely resemble transverse ‘finite rolls’ (Davies-Jones 1970). Shear increases the critical wavelength so that the preferred modes become longitudinally elongated cells, or even longitudinal rolls in some cases. The critical Rayleigh number increases quite rapidly at fist with Reynolds number, but at higher Reynolds numbers it levels off to a constant value (which cannot be greater than the shear-independent Rayleigh number at which longitudinal disturbances fist become unstable). We also find that the disturbances are tilted in the same direction as the shear, and that the marginally stable ones transfer kinetic energy from the mean flow to the perturbations. Except at low Reynolds numbers, the long wave perturbations gain more energy through the conversion of mean flow kinetic energy than through the release of potential energy, even though the instability is convective in origin. © 1971, Cambridge University Press. All rights reserved.

Description: The steady axially symmetric incompressible flow past a sphere is investigated for Reynolds numbers, based on the sphere diameter, in the range 0·1 to 40. The formulation is a semi-analytical one whereby the flow variables are expanded as series of Legendre functions, hence reducing the equations of motion to ordinary differential equations. The ordinary differential equations are solved by numerical methods. Only a finite number of these equations can be solved, corresponding to an approximation obtained by truncating the Legendre series at some stage. More terms of the series are required as R increases and the present calculations were terminated at R = 40. The calculated drag coefficient is compared with the results of previous investigations and with experimental data. The Reynolds number at which separation first occurs is estimated as 20·5. © 1971, Cambridge University Press. All rights reserved.

Description: A doubly-infinite sloping flat plate, initially at rest in a slightly diffusive viscous density-stratified fluid, starts to move impulsively with a constant velocity along the line of greatest slope. The resulting flow is found to be an unsteady motion superimposed on a steady diffusion-induced flow, which is present throughout. Laplace transform methods give solutions which are valid either in an essentially non-diffusive outer layer or in a diffusive inner layer. The impulsive start sets up oscillations in the outer layer. These gradually die out, and a steady diffusive flow develops. A glass plate was towed vertically through stratified brine, into which aluminium particles were introduced. The flow velocities deduced from the particle motions confirmed the theoretical predictions. © 1971, Cambridge University Press. All rights reserved.

Description: Recently Cheng & Akiyama (1970) published a numerical analysis of laminar flow in curved channels of square and rectangular section. Experimental results are presented here for flow in curved channels of square section. The channels were toroidal in shape, and the flow was driven electromagnetically. Various ratios of the channel dimension d to the channel radius of curvature, R, were used to investigate the dependence of friction factor, f, on the Dean number K, and the Reynolds number, Re. For 5 × 102 ≪ K ≪ 7 × 104 the formula (fRe) = 1·51 K½ was found to fit all the results, although R/d was varied from 17·5 down to the low value of 1·75. At lower values of K the analysis of Cheng & Akiyama was approximately validated. © 1971, Cambridge University Press. All rights reserved.

Description: The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are εt and ε½(x+a1rt), where t is the time, x the distance in the direction of flow, ε the growth rate of linearized theory and (⊟a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory. © 1971, Cambridge University Press. All rights reserved.

Description: Stability curves are computed for both spatially and temporally growing disturbances in a stratified mixing layer between two uniform streams. The low Froude number limit, in which the effects of buoyancy predominate, and the high Froude number limit, in which the effects of density variation are manifested by the inertial terms of the vorticity equation, are considered as limiting cases. For the buoyant case, although the spatial growth rates can be predicted reasonably well by suitable use of the results for temporal growth, spatially growing disturbances appear to have high group velocities near the lower cutoff wave-number. For the inertial case, it is demonstrated that density variations can be destabilizing. More precisely, when the stream with the higher velocity has the lower density, both the wave-number range of unstable disturbances and the maximum spatial growth rate are increased relative to the case of homogeneous flow. Finally, it is shown how the growth rate of the most unstable wave in the inertial case diminishes as buoyancy becomes important. © 1971, Cambridge University Press. All rights reserved.

Description: The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results. © 1971, Cambridge University Press. All rights reserved.

Description: Using a hot wire in a turbulent boundary layer in air, an experimental study has been made of the frequent periods of activity (to be called ‘bursts’) noticed in a turbulent signal that has been passed through a narrow band-pass filter. Although definitive identification of bursts presents difficulties, it is found that a reasonable characteristic value for the mean interval between such bursts is consistent, at the same Reynolds number, with the mean burst periods measured by Kline et al. (1967), using hydrogen-bubble techniques in water. However, data over the wider Reynolds number range covered here show that, even in the wall or inner layer, the mean burst period scales with outer rather than inner variables; and that the intervals are distributed according to the log normal law. It is suggested that these ‘bursts’ are to be identified with the ‘spottiness’ of Landau & Kolmogorov, and the high-frequency intermittency observed by Batchelor & Townsend. It is also concluded that the dynamics of the energy balance in a turbulent boundary layer can be understood only on the basis of a coupling between the inner and outer layers. © 1971, Cambridge University Press. All rights reserved.

Description: Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation. © 1971, Cambridge University Press. All rights reserved.

Description: The development of optical methods for the quantitative study of the fluctuating properties of turbulent flows can provide a supplement to conventional hot-body anemometry techniques. In particular, the study of high-speed flows by hot-wire or hot-film anemometry is often difficult owing to the presence of temperature and velocity fluctuations in the flow, thereby complicating the correct interpretation of measured signals. In addition, restrictions are placed on the application of such anemometers by their physical strength, frequency-response characteristics and the introduction of disturbances by the measuring probes into the flow. The operation of an optical detection system depends primarily on the mechanism by which the detected radiation intensity is modulated by the flow. Methods which have been used successfully include scattering or absorption of incident light by tracer constituents or particles, the absorption or emission of infra-red radiation by the flow and quantitative adaptations of the schlieren and interferometer systems which are sensitive to the flow density structure. All these systems detect a summation of signals from different parts of the flow and in consequence it is necessary to consider in detail the relation of the integrated signal to the local properties of the flow. This paper deals in particular with the application of the schlieren principle to an axisymmetric turbulent jet. © 1971, Cambridge University Press. All rights reserved.

Description: The gravitational instability of a horizontal fluid layer with a univariant phase transition is considered. It is found that the layer can be unstable even when the less dense phase lies above the dense phase and can be stable in the opposite case. Applications of the theory to convection with phase transitions in astrophysical and geophysical problems are briefly discussed. © 1971, Cambridge University Press. All rights reserved.

Description: According to the Lighthill acoustic analogy, the sound induced by a region of turbulence is the same as that due to an equivalent distribution of quadrupole sources within the fluid. It is known that the presence of scattering bodies situated near such multipoles can convert some of their intense near field energy into the form of sound waves whose amplitude is far greater than that of the incident field. Calculations are here presented to determine the extent of this conversion, for hard and soft bodies of various shapes, making use of the reciprocal theorem to recast the problem into one of finding the field, near the obstacle, induced by an incident plane wave. If the obstacle is small compared with a wavelength, then its presence is equivalent to an additional dipole (or source) whose greater efficiency as a sound radiator implies that the familiar intensity law I ∝ U8, for far field intensity I against typical turbulence velocity U for an unbounded flow, is replaced by I ∝ U6 (or I ∝ U4) for a hard (or soft) body. For the situation where the scatterer is large compared with wavelength, the prototype problem of a wedge of exterior angle (p/q)π is shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces. This result is shown to hold for the more general ‘wedge-like’ surfaces, whose dimensions are large scale and whose edges may be smoothed out on a small scale, compared with wavelength. The method used involves the matching of an incompressible flow, on the fine scales typical of the edge geometry, to an outer flow determined by the large scale features of the surface. Favourable comparisons are made with previous results pertaining to the two-dimensional semi-infinite duct and to the half-plate of finite thickness. © 1971, Cambridge University Press. All rights reserved.

Description: The behaviour of a strong shock wave, which is initiated by a point explosion and driven continuously outward by an inner contact surface (or a piston), is studied as a problem of multiple time scales for an infinite shock strength,[formula omited], and a high shock-compression ratio, ρs/ρ∞ ∼ 2γ/(γ ⊟ 1) ≡ ε⊟1 〉 1. The asymptotic analyses are carried out for cases with planar and cylindrical symmetry in which the piston velocity is a step function of time. The solution shows that the transition from an explosion-controlled régime to that of a reattached shock layer is characterized by an oscillation with slowly-varying frequency and amplitude. In the interval of a scaled time 1 ≪ t ≪ ε⊟2/3(1+ν), the oscillation frequency is shown to be (1 + ν) (2π)⊟1t⊟½(1⊟ν) and the amplitude varies as t⊟¼(3+ν) matching the earlier results of Cheng et al. (1961). The approach to the large-time limit, ε1/(1+ν)t → ∞ is found to involve an oscillation with a much reduced frequency, ¼π(1+ν)ε⊟½t⊟1, and with an amplitude decaying more rapidly like ε⊟⅘t⊟½(4+3ν); this terminal behaviour agrees with the fundamental mode of a shock/acoustic-wave interaction. © 1971, Cambridge University Press. All rights reserved.

Description: The suggestion made by the authors in a previous paper (Hurle & Jakeman 1969) that the Soret effect could give rise to overstable solutions of the thermosolutal Rayleigh–Jeffreys problem is investigated theoretically and experimentally. Oscillatory instability is shown to occur in initially homogeneous layers of water-methanol mixtures when they are heated from below. This instability triggers a finite-amplitude steady mode. The magnitude and sign of the Soret coefficient was changed by varying the composition of the mixture; as predicted, overstable modes were observed when the sign of the coefficient was such as to produce a stabilizing contribution to the density gradient. The observed critical Rayleigh numbers and temporal frequencies are consistent with theory. © 1971, Cambridge University Press. All rights reserved.