bluuebunny/crossref_metadata_2025 huggingface dataset:
A non-central feeding hydrostatic thrust bearing
Authors: Chow Chuen-Yen
Published: March 2006

In a conventional hydrostatic thrust bearing, the lubricant is supplied from the centre and flows radially outwards. It has been found that the load capacity of such a bearing decreases with increasing angular speed of the rotor. The bearing fails when a critical rotor speed is reached at which the load capacity becomes zero. In this paper a modified feeding system is suggested in which the lubricant is supplied from a ring-shaped groove situated between the exits at the centre and the edge of the bearing. Analysis shows that, by reversing the flow direction near the centre, the load capacity of a bearing of proper geometry can be made to decrease at a much slower rate than that of a conventional bearing as the rotor speed increases. The non-central feeding system might be considered for use in high-speed thrust bearings.
A novel form of the MHD Rayleigh–Taylor instability
Authors: Robinson I. S.
Published: March 2006

When a mutually perpendicular horizontal magnetic field and electric current are imposed upon an electrolytic liquid in a tank covered by a lighter nonconducting fluid, the interface can become unstable. In the experiment described in this paper the tank floor is stepped so that only part of the interface becomes unstable. The result is a distinctive arch-like configuration of the electrolyte, which finally disintegrates to break the current path, but which will continue to repeat itself.
Initial flow field over an impulsively started circular cylinder
Authors: Bar-Lev M. , Yang H. T.
Published: March 2006

The initial flow field of an incompressible, viscous fluid around a circular cylinder, set impulsively to move normal to its axis, is studied in detail. The nonlinear vorticity equation is solved by the method of matched asymptotic expansions. Analytic solutions for the stream function in terms of exponential and error functions for the inner flow field, and of circular functions for the outer, are obtained to the third order, from which a uniformly valid composite solution is found. Also presented are the vorticity, pressure, separation point and drag. These quantities agree with the numerical computations of Collins & Dennis. Extended solutions developed by Padé approximants indicate that higher than third-order approximations will yield only minor improvements.
The large amplitude motion of a liquid-filled gyroscope and the non-interaction of inertial and Rossby waves
Authors: Scott W. E.
Published: March 2006

An attempt is made to explain theoretically two curious phenomena involving the motion of the liquid in a spinning, gyrating, liquid-filled gyroscope. One of the phenomena is the periodic breakdown of the free-surface wave form of the spinning liquid in the gyroscope when it gyrates at angles larger than about 1°. The other is the resonant amplitude growth rate of the liquid-filled gyroscope at these angles, for then the small angle stability theory of Stewartson (1959) fails to make the correct predictions.

The analysis exploits the experimental fact that the axis of rotation of liquid in the rotor of a spinning gyrating gyroscope does not remain coincident with the axis of rotation of the rotor when the gyroscope gyrates at amplitudes greater than the above-mentioned 1°. It is shown that this lack of coincidence generates Rossby waves and modifies the inertial wave frequencies that would ordinarily occur in a right circular cylinder. There is no nonlinear interaction between these Rossby and inertial waves; hence the free-surface breakdown remains unexplained. However, the modification of the inertial wave frequencies does seem to account for the curious amplitude growth rate.
Observations on turbulent-drag reduction in a dilute suspension of clay in sea-water
Authors: Gust Giselher
Published: March 2006

Hot-wire anemometer measurements have been made in a dilute sea-water/claymineral suspension. For fully developed turbulent flows in an open channel with a smooth mud (from the North Sea) bottom, mean streamwise velocity profiles were measured for Reynolds numbers between 5400 and 27 800 (i.e. non-eroding and eroding flow rates) and compared with Newtonian flows under the same experimental conditions. For the clay-mineral suspensions, measurements of the kinematic viscosityv, Kármán's constantkand the mean streamwise velocity$\overline{u}$of the logarithmic layer seemed to verify a Newtonian flow structure. Although the distributions of concentration showed no substantial increase towards the wall, it was found that beneath this Newtonian core there existed a viscous sublayer whose thickness was enhanced by a factor of 2–5. The friction velocityu*determined by the gradient method in the viscous sublayer was reduced by as much as 40 %. The mean flow structure exhibited an additional new layer in the region 10 <y+< 30.

The measurements indicate that turbulent-drag reduction occurs for the experimental clay-mineral suspension at non-eroding and also at eroding velocities. Agglomeration of suspended clay-mineral particles is suggested as possible explanation of this phenomenon.
An experimental study of the effects of Prandtl number on thermal convection in a rotating, differentially heated cylindrical annulus of fluid
Authors: Fein Jay S. , Pfeffer Richard L.
Published: March 2006

This paper presents the results of experimental studies of the behaviour of different fluids (mercury, Pr = 0·0246; water, Pr = 7·16; and 5 cS silicone oil, Pr = 63) when each is contained in a rotating cylindrical annulus and subjected to an imposed radial temperature difference across the annulus. The results are summarized in the form of two-parameter regime diagrams (thermal Rossby number RoTvs. Taylor number Ta) at different Prandtl numbers Pr. The fluids are in contact above and below with rigid insulating boundaries. The range of thermal Rossby and Taylor numbers surveyed is 10−3 < RoT < 10, 105 < Ta < 109.

The regime diagram for water with a rigid upper lid in contact with the fluid resembles in certain respects the more familiar regime diagram for water with a free upper surface obtained by Fultz (1956) and by Fowlis & Hide (1965). Significant differences do, however, occur and are discussed in a separate paper by Fein (1973). The regime diagram for silicone oil possesses no lower symmetric regime within the range of thermal Rossby and Taylor numbers surveyed; that for mercury possesses no upper symmetric regime within the range surveyed. In mercury, turbulence is observed at high Rossby numbers. An experimental traverse across the regime diagram in which the imposed temperature difference is held constant at ΔT = 5 °C and the rotation rate Ω is changed monotonically reveals a highly conductive temperature structure in mercury and a highly convective temperature structure in both water and silicone oil. The regular wave regime, which appears at all three Prandtl numbers, is found to shift towards higher Taylor and Rossby numbers with decreasing Prandtl number. As a result, a single point in two-dimensionless-parameter space (RoT = 5 × 10−1, Ta = 1 × 106) lies in the upper symmetric regime for 5 cS silicone oil (Pr = 63), in the regular wave regime for water (Pr = 7·16) and in the lower symmetric regime for mercury (Pr = 0·0246).
Numerical simulation of three-dimensional Bénard convection in air
Authors: Lipps Frank B.
Published: March 2006

A numerical model is developed to simulate three-dimensional Bénard convection. This model is used to investigate thermal convection in air for Rayleigh numbers between 4000 and 25000. According to experiments, this range of Rayleigh numbers in air covers three regimes of thermal convection: (i) steady two-dimensional convection, (ii) time-periodic convection and (iii) aperiodic convection. Numerical solutions are obtained for each of these regimes and the results are compared with the available experimental data and theoretical predictions.

At the Rayleigh number Ra = 4000 the present model is able to produce experimentally realistic wavelengths for the two-dimensional convection. The small amplitude wave disturbances at Ra = 6500 have period τ = 0·24. When they become finite amplitude travelling waves, the period is τ = 0·27. These values are in good agreement with theoretical and experimental results. A detailed study of the form of these waves and of their energetics is given in appendix A. As the Rayleigh number is increased to Ra = 9000 and 25 000, the convection manifests progressively more complex spatial and temporal variations.

The vertical heat transport and other mean properties of the convection are calculated for the range of Ra considered and compared with experimental and theoretical data. A detailed comparison is also made between the mean properties of two- and three-dimensional convection at the larger values of Ra. It is found that the heat flux Nu is nearly independent of the dimensionality of the convection.
On the normal modes of parallel flow of inviscid stratified fluid
Authors: Banks W. H. H. , Drazin P. G. , Zaturska M. B.
Published: March 2006

The overall pattern of normal modes of parallel flow of inviscid stratified fluid is examined. For a given flow and wavenumber the modes are divided into five classes, some of which may be empty: (i) a finite class of non-singular unstable modes; (ii) a conjugate finite class of non-singular damped stable modes; (iii) a finite class of singular stable modes, each of these having a branch point and being the limit of unstable modes; (iv) a discrete class of modified internal gravity waves, these being non-singular stable modes (if the density decreases with height everywhere); (v) a continuous class of singular stable modes. The modified internal gravity waves are described asymptotically for large values of the Richardson number. These asymptotic results are related to and extended by numerical calculations for a sinusoidal basic velocity profile and a Bickley jet. The wave speeds for small values of the Richardson number are found to depend only upon the local behaviour of the mean flow near an overall simple maximum or minimum of the velocity profile. Finally some difficulties in the use of the Howard formula for perturbation at a curve of marginal stability are elucidated.
Unsteady flow in a branch
Authors: O'Brien V. , Ehrlich L. W. , Friedman M. H.
Published: March 2006

This study was motivated by the focal tendency of atherosclerotic plaque to appear near arterial junctions. A two-dimensional bifurcation was selected to provide preliminary information on branch flow dynamics. Superseding previous steady flow estimates in similar branches, digital solutions for two flux waves have been obtained to reveal temporal and spatial distributions of shear and separation in pulsatile flow. Marked differences from steady flow are illustrated as well as variation due to flux-wave shape.
The dispersion relation for a nonlinear random gravity wave field
Authors: Huang Norden E. , Tung Chi-Chao
Published: March 2006

The dispersion relation for a random gravity wave field is derived using the complete system of nonlinear equations. It is found that the generally accepted dispersion relation is only a first-order approximation to the mean value. The correction to this approximation is expressed in terms of the energy spectral function of the wave field. The non-zero mean deviation is proportional to the ratio of the mean Eulerian velocity at the surface and the local phase velocity. In addition to the mean deviation, there is a random scatter. The root-mean-square value of this scatter is proportional to the ratio of the root-mean-square surface velocity and the local phase velocity. As for the phase velocity, the nonzero mean deviation is equal to the mean Eulerian velocity while the root-mean-square scatter is equal to the root-mean-square surface velocity. Special cases are considered and a comparison with experimental data is also discussed.
Diffusion of weak magnetic fields by isotropic turbulence
Authors: Kraichnan Robert H.
Published: March 2006

The diffusion of slowly varying, weak magnetic fields by a statistically isotropic and stationary velocity field in a perfectly conducting fluid is studied by Eulerian analysis. The characteristic wavenumber and variance of the velocity field are k0 and 3v20, thus defining the eddy-circulation time τ0 = 1/v0k0. The velocity field is assumed constant on intervals of duration 2τ1 and statistically independent for distinct intervals. Thus the correlation time is τ1. The α-effect dynamo mechanism in the quasi-linear approximation is corroborated. Both the quasilinear and the direct-interaction approximations give identical diffusion of magnetic and passive scalar fields in reflexionally invariant turbulence. This result is found to be exact for τ1/τ0 → 0 but is demonstrated to be incorrect in general for finite τ1/τ0 because of effects of helicity fluctuations. The nature of the failure of the direct-interaction approximation is exhibited by an exactly soluble model system. Analysis based on a double-averaging device shows that longrange, persistent helicity fluctuations in reflexionally invariant turbulence give an anomalous negative contribution to the magnetic diffusivity which depends on the helicity covariance function. We term this the α2 effect. The magnitude of the effect depends sensitively on the turbulence statistics. If the characteristic scales of the helicity fluctuations are sufficiently larger than τ0 and 1/k0, the magnetic diffusivity is negative, implying unstable growth, while a passive scalar field diffuses normally. On the other hand, a crude estimate suggests that the α2 effect is small in normally distributed turbulence.
Slender-body theory for slow viscous flow
Authors: Keller Joseph B. , Rubinow Sol I.
Published: March 2006

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.
The propagation of dislocations in Rayleigh-Bénard rolls and bimodal flow
Authors: Whitehead J. A.
Published: March 2006

When Rayleigh-Bénard convection is generated under random conditions, the finite amplitude rolls and bimodal flow are observed to possess randomly placed dislocations where the rolls fit together poorly. The dislocations move into the small wavelength convection, and hence provide a size-adjustment mechanism. It is observed that the dimensionless speed of the movement is smaller for larger Prandtl number fluid.
Experiments to investigate transport processes in the near wakes of disks in turbulent air flow
Authors: Humphries W. , Vincent J. H.
Published: March 2006

Experiments reported previously determined the detention time of airborne smoke particles momentarily trapped in the wake bubble behind a flat disk normal to smooth air flow. The dimensionless groupH, the product of the detention timetdand mainstream air velocityUdivided by the disk diameterD, was found to be 7.44 for all combinations ofU, Dand the Reynolds number, a result that was consistent with a suggested physical model for particle transport across the bubble boundary. The work is now extended into the regime of turbulent free-stream flow, whereHis seen to decrease with an increasing level of turbulence while the base pressure coefficient becomes more negative. At the same time, the length of the bubble decreases, as does the bubble shape factor (the ratio of bubble volume to surface area, non-dimensionalized with respect toD). A simple theoretical relationship betweenHand the base pressure coefficient is argued, and is found to be in good agreement with experiment.

An important conclusion from this work is that the free-stream turbulence parameter$\Lambda \equiv l_fk^{\frac{1}{2}}_f/DU $(wherelfandkfare the length scale and the kinetic energy of the free-stream turbulence respectively) controls the properties of the flow about the disk.

This work has potential applications in several areas of topical technological interest.
The deformation of a nearly straight thread in a shearing flow with weak Brownian motions
Authors: Hinch E. J.
Published: March 2006

As a model of an almost fully extended macromolecule, a small, flexible, inextensible, nearly straight thread in a shearing flow with weak Brownian motions is considered. The hydrodynamic resistance to motion is included using the slenderbody theory for Stokes flow. The variation of the small transverse displacements along the thread is expressed as a truncated sum of Fourier components, with appropriately chosen modal functions. A diffusion equation is derived in the Fourier space and solved. The expected deformation of the thread is then given for axisymmetric and two-dimensional straining flows. The transverse displacement of the ends and the small shortening of the projected length of the thread are both found to be sensitive to the truncation of the Fourier representation, although it becomes clear on physical grounds that the ratio of the shortening to the typical transverse distortion should increase with the number of degrees of freedom. In simple shear flow the deformation increases as the thread aligns with the flow, until the analysis breaks down when the entire thread is no longer in the extensional quadrants. The influence of the 2:1 ratio of the resistance coefficients from the slender-body theory is found to be a small numerical factor.
Hydraulic jumps in an incompressible stratified fluid
Authors: Su C. H.
Published: March 2006

A multi-layer model is used to describe a ‘two-dimensional’ continuously stratified fluid. We use a momentum theorem in each layer to derive an ordinary differential equation describing the vertical structure behind a jump. This equation is compared with the corresponding equation for continuous flow. As one would expect from the classical one-layer theory, they are identical up to second order for weak disturbances. The energy change across a jump is also derived. By requiring that energy be lost through a jump, we calculate when a weak jump is possible in general. Algorithms for computing jumps of arbitrary strength are given. To ensure that the flow after the jump is stable and also for a numerical reason to be stated in § 8, we require that the Richardson number after the jump be equal to or greater than S¼S. Numerical examples are given to show the range of parameters within which jumps are possible; the velocity profiles related to different kinds of jumps also appear. Since hydraulic jumps in a continuously stratified fluid have not yet been observed in any laboratory, it should be of interest to verify these calculations experimentally.
Fluctuating heat transfer from hot wires in low Reynolds number flow
Authors: Davies Huw G.
Published: March 2006

An estimate is obtained of the heat transfer from a constant-temperature hot wire in a two-dimensional low Reynolds number flow (R [Lt ] 1). The flow has a small sinusoidally fluctuating velocity superimposed on the mean velocity. For the fluctuating components it is shown that at low frequencies the heat transfer is in phase with the velocity, the magnitude of the heat transfer being given by the result for steady heat transfer, but at high frequencies the heat transfer lags behind the velocity, the relative magnitude of the heat transfer decreasing as (frequency)−1.
Computation of the flow between a rotating and a stationary disk
Authors: Roberts S. M. , Shipman J. S.
Published: March 2006

A numerical exploration of multiple-cell solutions for the flow between a rotating and a stationary disk is carried out systematically by the continuation method. The paper confirms and extends the work of Mellor, Chapple & Stokes. The results include the discovery of one-, two-, three- and five-cell solution regions which have not been reported before.
The stability of natural convection in a vertical fluid layer
Authors: Mizushima Jiro , Gotoh Kanefusa
Published: March 2006

The stability of natural convection in fluid between two parallel vertical plates is investigated theoretically. The two plates are maintained at different temperatures and a uniform stable temperature gradient β is present in the vertical direction. The Prandtl number of the fluid is fixed at 7.5. An orthonormalization method is used in numerical integrations of the disturbance equations. It is shown how the critical Grashof number varies with β for both stationary and travelling disturbances. It is found that for β ≤ 7.1 × 10−3 the convection is unstable to stationary disturbances and for β ≥ 7.1 × 10−3 it is unstable to travelling disturbances. The critical Grashof number is given by
\[
G_c = \left\{\begin{array}{@{}l@{\quad}c@{\quad}l@{}}
500 & {\rm for} & \beta < 1.0\times 10^{-3},\\
1.3\times 10^6\beta^3 & {\rm for} &\beta > 4.1\times 10^{-2},
\end{array}\right.
\]
and even for intermediate values of β the variation of Gc is rather simple but not monotonic.
Linearized theory of axisymmetric hypersonic source flow past bodies of revolution
Authors: Yasuhara M. , Mitome H. , Nakao F.
Published: March 2006

The linearized theory of axisymmetric hypersonic source flow past a slender pointed body is treated by using linearized potential theory, assuming that the hypersonic parameter is less than unity. The governing equation in the transformed co-ordinates with the ratio of specific heats γ = 1.5 shows a modified form relative to the equation for parallel flow, whereas that with γ = 2 shows essentially the same form. Some numerical calculations for a cone and a cone-cylinder are presented, and the surface pressure distributions obtained for both γ = 1.5 and 2 show that there is a considerable pressure-distribution difference between source flow and parallel flow. The pressure on a cone in hypersonic source flow approaches zero downstream, while, if the cone has a cylindrical afterbody, the pressure recovers to the free-stream value quickly. This suggests that the static pressure probe for supersonic parallel flow is useful also in source flow.
A rational model for Langmuir circulations
Authors: Craik A. D. D. , Leibovich S.
Published: March 2006

A realistic theoretical model of steady Langmuir circulations is constructed. Vorticity in the wind direction is generated by the Stokes drift of the gravity-wave field acting upon spanwise vorticity deriving from the wind-driven current. We believe that the steady Langmuir circulations represent a balance between this generating mechanism and turbulent dissipation.

Nonlinear equations governing the motion are derived under fairly general conditions. Analytical and numerical solutions are sought for the case of a directional wave spectrum consisting of a single pair of gravity waves propagating at equal and opposite angles to the wind direction. Also, a statistical analysis, based on linearized equations, is developed for more general directional wave spectra. This yields an estimate of the average spacing of windrows associated with Langmuir circulations. The latter analysis is applied to a particular example with simple properties, and produces an expected windrow spacing of rather more than twice the length of the dominant gravity waves.

The relevance of our model is assessed with reference to known observational features, and the evidence supporting its applicability is promising.
The planar turbulent jet
Authors: Gutmark E. , Wygnanski I.
Published: March 2006

Results of hot-wire measurements in a plane incompressible jet are reported. The flow was found to be self-preserving beyond x/d > 40 and measurements were made up to x/d = 120. The quantities measured include mean velocities, turbulence intensities and third- and fourth-order terms, as well as two-point correlations and the intermittency factor. Conditional sampling techniques were used to obtain exclusively data within the turbulent zone of the jet. The results are compared with previous investigations.

This is the third paper in a sequence providing data on turbulent free shear flows.
The focusing of weak shock waves
Authors: Sturtevant B. , Kulkarny V. A.
Published: March 2006

This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, arêtes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.
Velocity measurements close to a rough plate oscillating in its own plane
Authors: Keiller D. C. , Sleath J. F. A.
Published: March 2006

Measurements are described of the fluid velocities close to rough beds oscillating in their own plane. The roughness with which most of the results were obtained consisted of smooth spheres closely packed in hexagonal formation. Some results are also given for beds of gravel. The beds were oscillated with simple harmonic motion in still air and the measurements were made with a hot-wire anemometer.

The measurements very close to the beds of smooth spheres show two maxima in the velocity profile during each half-cycle. One maximum corresponds to a component of velocity which varies nearly sinusoidally with time. The second forms quite a sharp peak and occurs close to ωt= 90°, 270°, where ω is the angular frequency of oscillation andtis time measured from the instant of maximum velocity of the plate. The phase at which this peak occurs shows little variation with distance from the bed. For values of βD> 3·0, where β = (ω/2ν)½, ν is the kinematic viscosity andDis the sphere diameter, the maximum velocity during each half-cycle is found at this peak over at least a certain range of distances from the bed. The variation with height of the nearly sinusoidal component of velocity is quite close to that given by Stokes’ (1851) solution for a flat plate. The peak at ωt= 90°, 270°, however, rises from zero at the bed to a maximum at a distance of about one-eighth of a sphere diameter above the crests and then falls off again.

The measurements with beds of gravel show a variation in velocity similar to that observed by Kalkanis (1957, 1964) and Sleath (1970). Because of the irregularity of the surface it is difficult to draw definite conclusions about the flow in the immediate vicinity of the bed.

A number of tests were carried out, with the beds of spheres, using a wire slanted at 45° to the bed in order to determine the velocity productuw.
An experimental study of turbulent convection in air
Authors: Fitzjarrald Daniel E.
Published: March 2006

An experiment was performed in a 3·5 by 3·5 m variable-height, closed convection box, with conditions ranging from a Rayleigh number of 4 × 104 up to 7 × 109, using air as the working fluid. Heat-flux measurements made at Rayleigh numbers up to 7 × 109 yielded a Nusselt number Nu = 0·13Ra0·30. Velocities and temperatures were measured up to Ra = 1·7 × 107, and Fourier spectra calculated to find the predominant horizontal scales of the motion midway between the boundaries. The predominant scale at Ra ∼ 105 was approximately four times the distance between plates, changing to six as Ra increased to 106. With side walls introduced so that the transverse aspect ratio was equal to five, Fourier spectra indicated considerable smaller scale motions, approximately equal to the layer depth. These motions decreased in size as Ra was increased. The results are discussed in relation to previous experimental and theoretical work.
The influence of jet flow on jet noise. Part 1. The noise of unheated jets
Authors: Mani R.
Published: March 2006

The present paper and part 2 (adjacent) study the sound field produced by a convected point quadrupole embedded in and moving along the axis of a round plug-flow jet. Only subsonic eddy convection velocities are considered. We examine cold jets here and hot jets in part 2. A principal feature of the study is extensive comparison with jet-noise data. It appears that this simple model problem succeeds in explaining all the major interesting features of jet-noise data, on both hot and cold jets, for jet exit velocities in the low supersonic range. Particular success is achieved in explaining aspects of the data not explainable by the Lighthill acoustic-analogy approach. The picture of jet-noise generation that emerges (at least for jet velocities in the low supersonic regime) is in many respects a striking reaffirmation of the Lighthill point of view. It appears that there is an intrinsic or universal distribution of compact quadrupoles, whose strength and frequency distribution scale with the jet velocity and nozzle diameter as would be expected from simple dimensional reasoning, responsible for jet-noise generation. These quadrupoles are of course convected by the mean flow and satisfactory agreement with the data is obtained by assuming that they are devoid of any intrinsic directionality. There appears to be no significant jet Mach number (compressibility) or jet temperature effect on the scaling of this intrinsic distribution. The essential improvement over the Lighthill analysis is the incorporation of mean-flow shrouding effects on the radiation of the convected quadrupoles. It is perhaps no exaggeration to claim that, with the incorporation of such a shrouding effect, the problem of scaling jet noise with regard to the jet velocity, jet temperature, jet size and the angle from the jet axis appears to be completely resolved. (The ‘scaling’ principle cannot of course be very simply expressed and in fact needs calculations of the sort contained in the present paper to implement it.)
Brownian diffusion of particles with hydrodynamic interaction
Authors: Batchelor G. K.
Published: March 2006

The classical theory of Brownian motion applies to suspensions which are so dilute that each particle is effectively alone in infinite fluid. We consider here the modifications to the theory that are needed when rigid spherical particles are close enough to interact hydrodynamically. It is first shown that Brownian motion is a diffusion process of the conventional kind provided that the particle configuration does not change significantly during a viscous relaxation time. The original argument due to Einstein, which invokes an equilibrium situation, is generalized to show that the particle flux in probability space due to Brownian motion is the same as that which would be produced by the application of a certain ‘thermodynamic’ force to each particle. We then use this prescription to deduce the Brownian diffusivities in two different types of situation. The first concerns a dilute homogeneous suspension which is being deformed, and the relative translational diffusivity of two rigid spherical particles with a given separation is calculated from the properties of the low-Reynolds-number flow due to two spheres moving under equal and opposite forces. The second concerns a suspension in which there is a gradient of concentration of particles. The thermodynamic force on each particle in this case is shown to be equal to the gradient of the chemical potential of the particles, which brings considerations of the multi-particle excluded volume into the problem. Determination of the particle flux due to the action of this force is equivalent to determination of the sedimentation velocity of particles falling through fluid under gravity, for which a theoretical result correct to the first order in volume fraction of the particles is available. The diffusivity of the particles is found to increase slowly as the concentration rises from zero. These results are generalized to the case of a (dilute) inhomogeneous suspension of several different species of spherical particle, and expressions are obtained for the diagonal and off-diagonal elements of the diffusivity matrix. Numerical values of all the relevant hydrodynamic functions are given for the case of spheres of uniform size.
Viscous boundary layers in reversing flow
Authors: Pedley T. J.
Published: March 2006

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.
The dispersion of a buoyant solute in laminar flow in a straight horizontal pipe. Part 2. The approach to the asymptotic state
Authors: Barton N. G.
Published: March 2006

Chatwin (1970) has described the approach to normality of a cloud of solute which is injected into a pipe containing solvent in steady laminar flow. This paper is concerned with the modification of Chatwin's theory when there is a small density difference between the solvent and the dissolved solute. Asymptotic series are derived for the induced density currents and for the distribution of solute in the case when the molecular diffusivity is constant throughout the pipe's cross-section. These series lead to the asymptotic forms of the moments of the distribution, thereby describing some additional deviations from normality caused by buoyancy effects at large times. The theory predicts that the additional dispersion due to buoyancy effects is proportional to the square of the Rayleigh number and depends on the Péclet number of the flow. There is excellent agreement between the results and those previously obtained by the author (1976) from Erdogan & Chatwin's (1967) model of dispersing buoyant solutes. The results confirm that Erdogan & Chatwin's intuitive theory correctly models the significant features of the situation for large Schmidt numbers.
Non-equilibrium ideal-gas dissociation after a curved shock wave
Authors: Hornung H. G.
Published: March 2006

Analytic solutions are obtained for non-equilibrium dissociating flow of an inviscid Lighthill-Freeman gas after a curved shock, by dividing the flow into a thin reacting layer near the shock and a frozen region further downstream. The method of matched asymptotic expansions is used, with the product of shock curvature and reaction length as the small parameter. In particular, the solution gives expressions for the reacting-layer thickness, the frozen dissociation level, effective shock values of the frozen flow and the maximum density on a stream-line as functions of free-stream, gas and shock parameters. Numerical examples are presented and the results are compared with experiments.
The maintenance of turbulent shear stress in a mixing layer
Authors: Jones Ian S. F.
Published: March 2006

Wavenumber frequency spectra have been measured in a two-dimensional incompressible mixing layer, using linearized hot-wire anemometers. Spectra of two dimensions (frequency and wavenumber) have been measured for lateral and longitudinal turbulent velocities, and used to construct three-dimensional spectra. The validity of the separation assumption used to construct these spectra was tested. Spectra of the velocity product responsible for the mean shear stress and the lateral gradient of this spectrum have been determined, as has a structure constant for the lateral velocity fluctuations that Phillips (1967) suggested is relevant to the maintenance of the shear stress gradient. Phillips’ (1967) stress model fails the tests proposed in this study.
The distortion of a flexible inextensible thread in a shearing flow
Authors: Hinch E. J.
Published: March 2006

Using the slender-body theory for Stokes flow, the equations of motion are developed for a small flexible inextensible thread. The nearly straight thread is examined analytically, and is shown to straighten rapidly. In a simple shear flow the distortions decay less rapidly, but rapidly enough not to rotate the thread through the plane of flow. Numerical studies of simple shear with more substantially distorted threads show the same qualitative behaviour. Additionally some differences are revealed in the nonlinear regime between the buckling and stretching processes which occur in the compressive and tensile quadrants of the flow.
Nonlinear effects in edge waves
Authors: Whitham G. B.
Published: March 2006

Nonlinear corrections to Stokes's linear edge-wave solution are obtained by means of perturbation expansions in the amplitude. The shallow-water formulation is considered first, but even for small beach angles β the behaviour in the deep water offshore becomes important and this formulation is limited. In the full formulation, amplitude dependence is required in the dispersion relation and in the exponents for the exponential decay away from the shore. There is a non-uniformity in the results as β → ½π, which is corrected by a special perturbation expansion.
Nonlinear edge waves and shallow-water theory
Authors: Minzoni A. A.
Published: March 2006

Nonlinear effects are considered for shallow-water edge waves on beaches with a general depth distribution. The case of uniform depth away from the shoreline is considered in detail. It is shown that the results obtained are in qualitative agreement with those obtained by Whitham (1976) using the full nonlinear theory for a beach of constant slope.
Nonlinear energy transfer and the energy balance of the internal wave field in the deep ocean
Authors: Olbers Dirk J.
Published: March 2006

The source function describing the energy transfer between the components of the internal wave spectrum due to nonlinear interactions is derived from the Lagrangian of the fluid motion and evaluated numerically for the spectral models of Garrett & Munk (1972a, 1975). The characteristic time scales of the transfer are found to be typically of the order of some days, so that nonlinear interactions will play an important role in the energy balance of the wave field. Thus implications of the nonlinear transfer within the spectrum for generation and dissipation processes are considered.
Convective amplification of real simple sources
Authors: Dowling Ann
Published: March 2006

This paper examines the problem of a pulsating compact body moving with constant velocity. This problem is usually incorrectly treated as a convected monopole. The analysis here shows that the motion of such a ‘real’ source introduces additional coupled multipoles, whose combined effects generate previously unexpected convective features. The amplification obtained is not the monopole convective amplification (1 —Mr)−2. It is found to depend on the virtual mass tensor of the body, the minimum effect being (1 —Mr)−3. There is also amplification in the direction perpendicular to the flight path (unless the motion is parallel to a principal axis of the virtual mass tensor).

The field produced by oscillation of a convected compact body of constant geometry is also investigated. Again, this problem is often misrepresented as a moving dipole. Here it is shown that the effect of convection on such a real source is surprising and complicated. It cannot be described completely by Doppler factors, and there is amplification in the direction perpendicular to the source motion.

These two model problems serve as a warning that the effect of flight on real sources cannot be anticipated until such real sources are correctly modelled, and that also the influence of source motion is likely to be much greater than has so far been anticipated.
Intrusion into a stratified fluid
Authors: Manins P. C.
Published: March 2006

Preliminary measurements have been made of the debouching of homogeneous fluid from a broad source at its equilibrium depth into a linearly stratified tank of salt water. With c the velocity of the nose of the intrusion, h its half-thickness near the source, N the environmental buoyancy frequency and v the kinematic viscosity of the fluid, it is shown for 100 [lsim ] Re ≡ 2ch/ν [lsim ] 500 that the intrusion becomes practically steady under an inertia-buoyancy balance. The internal Froude number Fr = c/Nh is shown to be of order unity. Forward-propagating disturbances and the ends of the tank are inferred to play an important part in the flow.
An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders
Authors: Kuehn T. H. , Goldstein R. J.
Published: March 2006

An experimental and theoretical-numerical investigation has been carried out to extend existing knowledge of velocity and temperature distributions and local heat-transfer coefficients for naturel convection within a horizontal annulus. A Mach—Zehnder interferometer was used to determine temperature distributions and local heat-transfer coefficients experimentally. Results were obtained using water and air at atmospheric pressure with a ratio of gap width to inner-cylinder diameter of 0·8. The Rayleigh number based on the gap width varied from 2·11 × 104to 9·76 × 105. A finite-difference method was used to solve the governing constant-property equations numerically. The Rayleigh number was changed from 102 to 105 with the influence of Prandtl number and diameter ratio obtained near a Rayleigh number of 104. Comparisons between the present experimental and numerical results under similar conditions show good agreement.
The effects of trailing vorticity on the flow through highly loaded cascades
Authors: Mccune James E. , Hawthorne William R.
Published: March 2006

This paper presents a procedure whereby three-dimensional inviscid flow through a highly loaded turbomachinery cascade of lifting lines can be treated by methods corresponding to classical aerodynamic theory. In contrast to earlier linearized (thin airfoil) three-dimensional theory, the present study allows analysis of the flow corresponding to the large turning and/or large pressure ratios induced by practical rotors or stators. For the sake of simplicity, the present paper is limited to incompressible flow through a highly loaded rectilinear cascade and to the design problem, i.e. given blade loading. Formulae are derived for both the mean and the three-dimensional components of the flow; in particular, the velocities at the blades induced by the trailing vorticity associated with nonuniform blade circulation are determined.
A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust
Authors: Goldstein M. E. , Atassi H.
Published: March 2006

In this paper we develop a uniformly valid, second-order theory for calculating the unsteady incompressible flow that occurs when an airfoil is subjected to a convected sinusoidal gust. Explicit formulae for the airfoil response functions (i.e. fluctuating lift) are given. The theory accounts for the effect of the distortion of the gust by the steady-state potential flow around the airfoil, and this effect is found to have an important influence on the response functions. A number of results relevant to the general theory of the scattering of vorticity waves by solid objects are also presented.
Intermittency of coherent structures in the core region of fully developed turbulent pipe flow
Authors: Sabot Jean , Comte-Bellot Geneviève
Published: March 2006

The present investigation is oriented towards a better understanding of the turbulent structure in the core region of fully developed and completely wall-bounded flows. In view of the already existing results concerning the bursting process in boundary layers (which are semi-bounded flows), an amplitude analysis of the Reynolds shear stress fluctuation u1u2, sorted into four quadrants of the u1, u2 plane, was carried out in a turbulent pipe flow. For the wall side of the core region, in which the correlation coefficient u1u2/u’1u’2 does not change appreciably with the distance from the wall, the structure of the Reynolds stress is found to be similar to that obtained in boundary layers: bursts, i.e. ejections of low speed fluid, make the dominant contribution to the Reynolds stress; the regions of violent Reynolds stress are small fractions of the overall flow; and the mean time interval between bursts is found to be almost constant across the flow. For the core region, the large cross-stream evolution of the correlation coefficient u1u2/u’1u’2 is associated with a new structure of the Reynolds stress induced by the completely wall-bounded nature of the flow. Very large amplitudes of u1u2 are still observed, but two distinct burst-like patterns are now identified and related to ejections originating from the two opposite halves of the flow. In addition to this interaction, a focusing effect caused by the circular section of the pipe is observed. As a result of these two effects, the mean time interval between the bursts decreases significantly in the core region and reaches a minimum on the pipe axis. Investigation of specific space-time velocity correlations reveals the possible existence of rotating structures similar to those observed at the outer edge of turbulent boundary layers. These coherent motions are found to have a scale noticeably larger than that of the bursts.