bluuebunny/crossref_metadata_2025 huggingface dataset:
On turbulent boundary-layer separation
Authors: Sandborn V. A. , Liu C. Y.
Published: March 2006

An experimental and analytical study of the separation of a turbulent boundary layer is reported. The turbulent boundary-layer separation model proposed by Sandborn & Kline (1961) is demonstrated to predict the experimental results. Two distinct turbulent separation regions, an intermittent and a steady separation, with correspondingly different velocity distributions are confirmed. The true zero wall shear stress turbulent separation point is determined by electronic means. The associated mean velocity profile is shown to belong to the same family of profiles as found for laminar separation. The velocity distribution at the point of reattachment of a turbulent boundary layer behind a step is also shown to belong to the laminar separation family.

Prediction of the location of steady turbulent boundary-layer separation is made using the technique employed by Stratford (1959) for intermittent separation.
The decay of free motion of a floating body: force coefficients at large complex frequencies
Authors: Crapper G. D.
Published: March 2006

The results of Ursell (1964) are confirmed by proving that there are no additional contributions to Ursell's integral from singularities of the integrand at infinity. The method consists of proving that the asymptotic expansion of a force coefficient Δ(N) is uniformly valid in a finite sector of the complex N-plane. This in turn requires that the kernel of an integral equation remains small in this sector.
Long waves in a uniform channel of arbitrary cross-section
Authors: Peregrine D. H.
Published: March 2006

Equations of motion are derived for long gravity waves in a straight uniform channel. The cross-section of the channel may be of any shape provided that it does not have gently sloping banks and it is not very wide compared with its depth. The equations may be reduced to those for two-dimensional motion such as occurs in a rectangular channel. The order of approximation in these equations is sufficient to give the solitary wave as a solution.
The motion of a spherical liquid drop at high Reynolds number
Authors: Harper J. F. , Moore D. W.
Published: March 2006

The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent. We investigate the boundary layers associated with the inviscid first approximation to the flow, which is shown to be Hill's spherical vortex inside, and potential flow outside. The boundary layers are shown to perturb the velocity field only slightly at high Reynolds numbers, and to obey linear equations which are used to find first and second approximations to the drag coefficient and the rate of internal circulation.

Drag coefficients calculated from the theory agree quite well with experimental values for liquids which satisfy the conditions of the theory. There appear to be no experimental results available to test our prediction of the internal circulation.
The Rayleigh—Jeffreys problem with boundary slab of finite conductivity
Authors: Nield D. A.
Published: March 2006

Linear perturbation analysis is applied to the problem of the onset of convection in a horizontal layer of fluid heated uniformly from below, when the fluid is bounded below by a rigid plate of inlinite conductivity and above by a solid layer of finite conductivity and finite thickness. The critical Rayleigh number and wave-number are found for various thickness ratios and thermal conductivity ratios. Both numbers are reduced by the presence of a boundary of finite (rather than infinite) conductivity in qualitative agreement with the observation of Koschmieder (1966).
Classical cellular convection with a spatial heat source
Authors: Watson Pauline M.
Published: March 2006

This paper considers the problem of the stability of an infinite horizontal layer of a viscous fluid which loses heat throughout its volume at a constant rate. The variation of the critical Rayleigh number, Rt, and the cell aspect ratio, a, with the rate of heat loss, is calculated with two sets of boundary conditions corresponding to two free and two rigid boundaries. In both cases we find that, as the rate of heat loss increases, Rt decreases, showing that the layer becomes more unstable, and a increases, showing that the cells become narrower. We also consider the possibility that a double layer of cells is formed for large values of the rate of heat loss, by the stable layer at the top, and find that this does not occur while the temperature of the upper surface of the layer is less than that of the lower.
The oscillations of a fluid droplet immersed in another fluid
Authors: Miller C. A. , Scriven L. E.
Published: March 2006

From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.

When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.
On standing internal gravity waves of finite amplitude
Authors: Thorpe S. A.
Published: March 2006

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.

Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.

Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.
Lee waves in a stratified flow Part 1. Thin barrier
Authors: Miles John W.
Published: March 2006

The lee-wave amplitudes and wave drag for a thin barrier in a two-dimensional stratified flow in which the upstream dynamic pressure and density gradient are constant (Long's model) are determined as functions of barrier height and Froude number for a channel of finite height and for a half-space. Variational approximations to these quantities are obtained and validated by comparison with the earlier results of Drazin & Moore (1967) for the channel and with the results of an exact solution for the half-space, as obtained through separation of variables. An approximate solution of the integral equation for the channel also is obtained through a reduction to a singular integral equation of potential theory. The wave drag tends to increase with decreasing wind speed, but it seems likely that the flow is unstable in the region of high drag. The maximum attainable drag coefficient consistent with stable lee-wave formation appears to be roughly two and almost certainly less than three.
On the existence and structure of inertial boundary currents in a stratified ocean
Authors: Spiegel S. L. , Robinson A. R.
Published: March 2006

An investigation is made of the properties of inertial boundary currents in a stably stratified, inviscid, non-diffusive ocean. The Boussinesq and β-plane approximations are adopted. The formalism developed by Robinson (1965) is used: the equations are transformed so that density replaces the vertical coordinate as an independent variable, and after a suitable non-dimensionalization of variables, the various fields are expanded as power series in the downstream co-ordinate η. The motion is shown to conserve potential vorticity. The equations and boundary conditions are obtained to order η2. Solutions are obtained in the region of formation of the coastal jet (i.e. the case of no mass flux through the plane η = 0) for several cases in which the potential vorticity function depends on stream function and density in a simple way. For these cases, it is found that in a constant depth ocean, a boundary current can exist only if the geostrophic drift at the boundary-layer edge is westward at all depths. This constraint is relaxed if the depth increases rapidly enough in the downstream (northward) direction. For slopes just in excess of the critical value, a deep onshore counter-current is predicted. Solutions of the first-order problem, using realistic values of the various parameters, have been computed and are found to be in qualitative agreement with observed features of the Florida Current.

In an appendix, it is shown that the constraint of westward geostrophic drift at all levels must hold in a flat-bottomed ocean for arbitrary potential vorticity distributions consistent with stable stratification.
Convection in a box: on the dependence of preferred wave-number upon the Rayleigh number at finite amplitude
Authors: Davis Stephen H.
Published: March 2006

Using Stuart's shape assumption and a condition of maximum heat transport it is found that the preferred number of finite roll cells present in Bénard convection in athree-dimensional rectangular boxtends todecreasewith increasing supercritical Rayleigh number in contradiction to the behaviour in an infinite layer but in accordance with experimental observation.

This ‘end effect’ might explain the similar observation of wave-number decrease in the Taylor instability between rotating cylinders.
The distortion of turbulence by irrotational plane strain
Authors: Tucker H. J. , Reynolds A. J.
Published: March 2006

The experiments reported here extend those of Townsend which form the basis of his model of free turbulence. Here straining is carried to a strain ratio of 6:1, while Townsend's straining went only to 4:1. Two kinds of distorting ducts are used to produce the uniform mean strain applied to initially nearly isotropic grid turbulence.

The results differ from Townsend's in that: (i) a considerably higher degree of anisotropy is achieved, Townsend's measure of anisotropy attaining values up to 0·6, rather than the maximum of 0·42 he found; (ii) there is no evidence that an equilibrium structure is attained; and (iii) the strained turbulence rapidly becomes less anisotropic when the straining ceases.

It is found to be possible to predict the variation of the total turbulence energy using rapid-distortion theory with a correction for decay. However, the individual components cannot be accurately predicted in this way.
The measurement of fluctuating skin friction in air with heated thin-film gauges
Authors: Bellhouse B. J. , Schultz D. L.
Published: March 2006

A thin-film, heated-element probe is calibrated to measure fluctuating skin friction. The probe is mounted flush with the surface of a flat plate which is oscillated in its own plane, in the direction of the air stream. The relationship between fluctuating heat transfer from the film and fluctuating skin friction is discussed theoretically using Lighthill's (1954) technique.
A method of producing a shear flow in a stratified fluid
Authors: Thorpe S. A.
Published: March 2006

When the end of a long closed horizontal tube containing a stratified fluid is raised, a laminar accelerating flow begins. The flow is two-dimensional in the central portion of the tube, and, in this region, it is predictable, at least until the onset of instability. The occurrence of instability, its nature, and the subsequent transition to turbulence, are described qualitatively. The apparatus may be used for the study of a variety of other internal hydraulic phenomena with applications to meteorology and oceanography.
Experiments on the pressure drop created by a sphere settling in a viscous liquid. Part 2. Reynolds numbers from 0·2 to 21,000
Authors: Feldman Guili A. , Brenner Howard
Published: March 2006

The pressure drop ΔP created by the motion of a ‘small’ spherical particle settling along the axis of a large-diameter circular cylinder filled with a quiescent liquid was measured in the particle Reynolds number range (based on diameter) from Re = 0·2 to 21,000. For Re < 125 it was found that ΔPA/D = 2·0 (A = cylinder cross-sectional area; D = particle drag), in agreement with existing theory in the Stokes and Oseen regimes. Beyond Re = 125 a fairly abrupt transition occurs, the ΔPA/D ratio decreasing asymptotically towards 1·0, the limiting value predicted by elementary momentum principles for an ‘unbounded’ fluid, with increasing Re. At Re ≈ 6000 the transition is essentially complete.
Stability of a Hartmann boundary layer under the influence of a parallel magnetic field
Authors: Abas S.
Published: March 2006

Stability to infinitesimal disturbances—when a parallel magnetic field is imposed—is investigated for the flow in the boundary layer set up by two-dimensional motion between parallel planes of a viscous, incompressible, electrically conducting fluid under the influence of a transverse magnetic field. The flow is assumed to take place at low magnetic Reynolds number. The usual asymptotic methods are employed for the solution, but, apart from the Tollmientype power series solution, an exact solution of the inviscid equation is obtained in terms of the hypergeometric function and its analytic continuation. Curves of neutral stability for two-dimensional disturbances are calculated and the results for critical Reynolds number modified to take into account three-dimensional disturbances. The parallel magnetic field is found to have a strong stabilizing influence.
On source-sink flows in a rotating fluid
Authors: Hide R.
Published: March 2006

An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the (x, y)-plane, themrigid side walls and the two rigid end walls being in contact with the fluid. Here (x, y, z) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container atQcm3/sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocityu= −∇ϕ − ∇ ×A, is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects onuof uniform rotation of the whole system with angular velocity Ω about thez-axis when the normal component ofuon the side walls is independent ofz.

In the simplest conceivable case,D≡zu−zlis infinite (butD/Qremains finite). End effects are then negligible anduis everywhere independent ofz.The solenoidal component ofu, − ∇ ×A, corresponds tojgyres, one for each of thejirreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within them-ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇2ϕ = 0, depends onQbut not on Ω,jandv(the coefficient of kinematic viscosity), ∇ ×Adepends on all these quantities but vanishes identically whenjΩ = 0. WhenjΩ ≠ 0 butv→ 0, ∇2A+ 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form.

End effects cannot be ignored whenDis finite. WhenDis independent ofxandyand equal toD0(say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3(v/Ω)½(δ [Lt ]D0), the end effects that then arise are only confined to these boundary layers whenj= 0. Whenj≠ 0 boundary-layer suction influences the flow everywhere; thus ∇2Aand ∇ϕ (but not ∇ ×A) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇2Abeing the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δs(typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δs. WhenDis not everywhere independent ofxandy,non-viscous end effects arise which produce relative vorticity in the main body of the fluid even whenj= 0.

Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.
Wave-induced shear instability in the summer thermocline
Authors: Woods J. D.
Published: March 2006

The fine structure of the summer thermocline in the Mediterranean Sea around Malta, investigated by a new temperature-gradient meter and by photographs of dye tracers, is summarized. The principal internal feature of the thermocline is a series of thin, laminar-flowsheetsof high static stability, separated by weakly turbulentlayersof only moderate density gradient and a few metres thick. A mechanism for generating the patches of turbulence observed on these thermocline sheets is established by comparing dye photographs with a theory by Miles & Howard (1964).
The development of three-dimensional wave-packets in unbounded parallel flows
Authors: Gaster M. , Davey A.
Published: March 2006

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.
An overlooked aspect of the wind-driven oceanic circulation
Authors: Pedlosky Joseph
Published: March 2006

The wind-driven circulation of a simple model of the oceanic circulation (linear and homogeneous) is investigated in detail to delineate the role of the Ekman layer mass flux in driving hitherto overlooked components of the oceanic circulation.

The role of upwelling boundary-layer regions in driving interior geostrophic circulations is discussed in detail. Several interesting circulations hidden in the earlier transport theories are described.
On wakes in stratified fluids
Authors: Janowitz G. S.
Published: March 2006

The general nature of the flow at large distances from a two-dimensional body moving uniformly through an unbounded, linearly stratified, non-diffusive viscous fluid is considered. The governing equations are linearized using the Oseen and Boussinesq approximations, and the boundary conditions at the body are replaced by a linearized momentum-integral equation. The solution of this linear problem shows a system of jets upstream and a pattern of waves downstream of the body. The effects of viscosity on these lee waves are considered in detail.
Finite amplitude convection with changing mean temperature. Part 1. Theory
Authors: Krishnamurti Ruby
Published: March 2006

When a horizontal layer of fluid is heated from below and cooled from above with the mean temperature and physical parameters of the fluid constant, the two-dimensional roll is known to be the stable solution near the critical Rayleigh number. In this study, with the mean temperature changing steadily at a rate η, the Rayleigh number and the velocity and temperature fields governed by the Boussinesq equations are expanded in two parameters: η, and the amplitude ε. Hexagons are shown to be the stable solution near the critical Rayleigh number. The direction of the motion depends upon the sign of η. A finite amplitude instability is possible with an associated hysteresis in the heat flux as the critical Rayleigh number is approached from below or from above.
Finite amplitude convection with changing mean temperature. Part 2. An experimental test of the theory
Authors: Krishnamurti Ruby
Published: March 2006

It has been found in part 1 (Krishnamurti 1968) that when the mean temperature of a fluid layer is changing at a constant rate η, hexagonal flows are stable in a range of Rayleigh numbers near the critical. The direction of flow depends upon the sign of η. The static state is unstable to finite amplitude disturbances at Rayleigh numbers below the critical point predicted by linear theory.

The validity of this theory is tested in an experiment in which the heat flux is measured as a function of η and Rayleigh number. The horizontal plan form is determined from the side by continuously exposing a photographic film moving in a vertical direction as tracers in different regions of the fluid are illuminated. Finite amplitude instability and hexagonal cells are indeed observed.
On the equations of motion for mixtures of liquid and gas bubbles
Authors: Wijngaarden L. Van
Published: March 2006

On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.

It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.
Two-dimensional bubbles in slow viscous flows
Authors: Richardson S.
Published: March 2006

The representation of a biharmonic function in terms of analytic functions is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory. The relevant conditions to be satisfied at a free surface, where there is a given surface tension, are derived.

A method for dealing with the difficulties of such a free surface is demonstrated by obtaining solutions for a two-dimensional, in viscid bubble in (a) a shear flow, and (b) a pure straining motion. In both cases the bubble is found to have an elliptical cross-section.

The solutions obtained can be shown to be unique only if certain restrictive assumptions are made, and if these are relaxed the same methods may give further solutions. Experiments on three-dimensional inviscid bubbles (Rumscheidt & Mason 1961; Taylor 1934) demonstrate that angular points appear in the bubble surface, and an analysis is presented to show that such a discontinuity in a two-dimensional free surface is necessarily a genuine cusp and the nature of the flow about such a point is examined.
A theory of circulation control by slot-blowing, applied to a circular cylinder
Authors: Dunham J.
Published: March 2006

Lift can be generated on a circular cylinder with its axis normal to an air flow by blowing a sheet of air tangentially round the upper surface from a narrow slot or slots. This lift force may be estimated by matching the external inviscid flow field with separation points calculated by Spalding's unified boundary-layer theory. The theory reproduces experimental results reasonably well, except in certain special conditions fully discussed.
Convection on a non-uniformly heated, rotating plane
Authors: Koschmieder E. L.
Published: March 2006

This investigation is concerned with the convective motions in a shallow layer of silicone oil on a plane, circular copper plate which is heated at the rim and cooled at the centre and at the same time rotated around a vertical axis. The oil is in touch with a glass lid, which is cooled uniformly. With sufficient heating axially symmetric concentric rings develop. The details of the motion can be described as a superposition of a circulation due to the horizontal temperature gradient and a circulation of opposite sense due to the centrifugal force, with the motions due to the vertical instability. There seems to be a conversion into rolls whose axes point radially, if the centrifugal circulation becomes too strong.
A two-dimensional sink in a density-stratified porous medium
Authors: List E. J.
Published: March 2006

A solution is offered for the flow induced by a two-dimensional line sink in a saturated, density-stratified porous medium. It is found that fluid is selectively withdrawn from a thin layer at the elevation of the line sink and not from the entire medium. The velocity distributions predicted by the theory are checked by experiments in a Hele–Shaw cell and good agreement found.
Growth of spectral components in a wind-generated wave train
Authors: Sutherland Alex J.
Published: March 2006

Power spectral density measurements were made of the water surface displacement in a wind-generated wave train in the Stanford wind, water-wave research facility. Growth rates of different components were determined from the measured spectra. The resulting values were compared with those predicted by the viscous Reynolds stress mechanism of energy transfer from wind to wave which was proposed by Miles. Where the growth was exponential the theory could be made to predict growth rates successfully at wave frequencies less than 3·5 c/s. At higher frequencies the theory predicted values an order of magnitude larger than those measured. Limited regions of linear growth were found at the lowest wind speed for components with frequencies less than 2·5 c/s. The scatter in the data did not permit a quantitative comparison with theory to be made for this range.
Role of viscosity stratification in the stability of two-layer flow down an incline
Authors: Kao Timothy W.
Published: March 2006

The stability of the flow of two layers of viscous liquids down an incline is investigated. The problem is governed by two competing long-wavelength modes associated with the surfaces. One mode, calling it the first mode, is faster than the second one. When the two layers have the same coefficient of dynamic viscosity, the second mode was found earlier by the author to be the governing mode for most cases. The situation is greatly altered when viscosity is not the same for the two layers. The second mode is dramatically stabilized for the range of viscosity ratio, m, less than unity, and the first mode is now generally the governing mode in that range. The overall effect is stabilizing compared with m = 1.

A relative stability index is also introduced to compare the result with that of the homogeneous case. It is found that the presence of the upper layer is generally destabilizing compared with that of a homogeneous fluid of the same total depth.
Electrocapillary discharge of a liquid
Authors: Michelson D.
Published: March 2006

The current I flowing in the steady electrocapillary discharge of saturated aqueous and alcohol solutions of sodium chloride was found to be proportional to e−α/V, where V is the applied potential and α is a constant under fixed experimental conditions.
Shear flow instabilities in rotating systems
Authors: Busse F. H.
Published: March 2006

A theoretical description is given for infinitesimal non-axisymmetric disturbances of a shear flow caused by differential rotation. It is assumed that the deviations from the state of rigid rotation are small corresponding to the case of small Rossby number. The shear flow becomes unstable at a finite critical value of the Rossby number, at which the inertial forces overcome the friction in the Ekman boundary layers and the constraint imposed by the variation in depth of the container. The governing equation is closely related to the Rayleigh stability equation in the theory of hydrodynamic stability of plane parallel flow. Experimental observations by R. Hide show reasonable agreement with the theoretical predictions.
Asymptotic solutions for the distribution function in non-equilibrium flows. Part 1. The weak shock
Authors: Narasimha Roddam
Published: March 2006

This paper is the first part of an investigation of the molecular velocity distribution function in non-equilibrium flows. In this part, the general features of the distribution are discussed and illustrated by a detailed study of its asymptotic expansions in different velocity domains for a weak shock, employing a simple relaxation model for the collisions. Using the strength of the shock as a small parameter, the Chapman–Enskog distribution is derived, in a less restrictive way than in previous analyses, as the first two terms in a suitable ‘inner’ asymptotic expansion of the distribution in velocity space, valid only for not very high velocities. It is shown that the consideration of two further limits, called the intermediate and outer, is necessary for a complete description of the distribution in velocity space. The uniformly valid composite expansion demonstrates the slow approach to equilibrium of fast molecules. The outer solution depends on integrals over the flow and is in general ‘global’, in contrast to the inner solution which is essentially local; this introduces certain asymmetries on a fine scale even in a weak shock. It is shown, for example, that fast molecules moving towards the hot side accumulate by collisionless streaming, whereas those moving towards the cold side attenuate like a molecular beam and represent essentially a ‘precursor’ of the hot side. A simple approximation for the distribution in the precursor is derived, and found to contain, in the outer limit, a large perturbation on the local Maxwellian; this results in an approach to equilibrium like $\exp (-|x|^{\frac{2}{3}})$ on the cold side.

A heuristic extension of the argument to the true Boltzmann equation leads to the result that for molecules with an interparticle potential varying as the inverse m-power of the distance, the approach to equilibrium through the precursor is like $\exp (-|x|^l)$, where l = m/(m + 2).
Further experiments on steady separated flows past bluff objects
Authors: Acrivos Andreas , Leal L. G. , Snowden D. D. , Pan F.
Published: March 2006

Detailed experimental results are presented for the steady separated flow past a variety of bluff objects. Included in this work are measurements of: rear stagnation point pressure coefficients; the pressure distribution along the base of a backward-facing step; velocity profiles within the circulating wake-bubble as well as in the outer flow; lengths, maximum widths and vortex centre locations of the corresponding wake-bubbles; and the local shear stress along the surface of a circular cylinder. In every case, the experimental data were found to be consistent with the theoretical model proposed by Acrivos et al. (1965) for the steady separated flow past bluff objects in the limit of large Reynolds numbers R.
Laboratory studies of wind–wave interactions
Authors: Wu Jin
Published: March 2006

The present study consists of wind profile surveys, drift current measurements and water surface observations for a wide range of wind velocities in a wind–wave tank. It is confirmed that the velocity distribution essentially follows the logarithmic law near the water surface and the velocity-defect law toward the outer edge of the boundary layer. The wind stresses and surface roughnesses calculated from these distributions are divided into two groups separated by the occurrence of the wave-breaking phenomenon. For low wind velocities the surface roughness is dictated by ripples, and the wind-stress coefficient varies with U0−½, where U0 is the free-stream wind velocity. The surface roughness is proportional to the average height of the basic gravity wave at higher wind velocities; the stress coefficient is then proportional to U0. In addition, it is found that Charnock's expression (k ∝ u*2/g) holds only at high wind velocities, and that the constant of proportionality determined from the present experiment correlates very well with field observations. A new technique, involving the use of various-sized surface floats to determine the drift current gradient and the surface drift current, has been developed. A good agreement is shown between the gradients obtained from the measured currents and those determined from the wind stresses. Finally, the wind-stress coefficient is shown to be larger than the friction coefficient for turbulent flow along a solid rough surface; the difference is shown to be the wave drag of the wind over the water surface.
On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales
Authors: Tam K. Kuen
Published: March 2006

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.
Laminar natural convection about an isothermally heated sphere at small Grashof number
Authors: Fendell Francis E.
Published: March 2006

The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.
An integral-transform method for shock–shock interaction studies
Authors: Arora N. L.
Published: March 2006

An analytic solution is obtained for the head-on collision of strong shock waves with supersonically moving, axisymmetric, slender bodies. A new method of the solution is developed using integral transforms. Pressure distributions on the surface of a cone illustrate the effects of shock strength and body speed. The results are compared with those of Blankenship (1965), which were obtained by numerical methods.

It is also shown that the same analysis can, with certain modifications, be adapted for treating the shock-shock interaction on thin two-dimensional aerofoils of arbitrary shape at small incidence.
The unsteady motion of a small sphere in a viscous liquid
Authors: Ockendon J. R.
Published: March 2006

The drag experienced by a sphere moving with velocity dependent on a single time scale t0 in an unbounded viscous liquid is considered under the assumption that the Reynolds number is small. It is shown that, unless t0 is sufficiently large, an asymptotic expansion in the Reynolds number becomes invalid for large times. Moreover, when the expansion is valid for large times, the drag can differ considerably from that predicted by the unsteady Stokes equations.
Stability of towed, totally submerged flexible cylinders
Authors: Paidoussis M. P.
Published: March 2006

A general theory is presented to account for the small, free, lateral motions of a flexible, slender, cylindrical body with tapered ends, totally submerged in liquid and towed at steady speedU. For particular shapes of the ends and length of tow-rope, it is shown that the body may be subject to oscillatory and non-oscillatory instabilities forU> 0; at smallU, these instabilities correspond to those of a rigid body. At higherU, the system generally regains stability in the above modes, but may be subject to higher-mode, flexural oscillatory instabilities. The critical conditions of stability are calculated extensively and the effect on stability of a number of dimensionless parameters is discussed. It is shown that optimum stability is achieved with a streamlined nose, a blunt tail and a short tow-rope.

Some experiments are described which were designed to test the theory. Rubber cylinders of neutral buoyancy were held in vertical water flow by a nylon ‘tow-rope’. Provided the tail was streamlined and the tow-rope not too short, ‘criss-crossing’, non-flexural oscillations developed at very low flow. Increasing the flow, these oscillations ceased and the cylinder buckled like a column; subsequently higher-mode flexural oscillations developed. However, for a sufficiently blunt tail and short tow-rope, the system was completely stable.

The experimental observations are generally in qualitative agreement with theory. Quantitative comparison of the various instability thresholds and stable zones between experiment and theory, based on estimated values of some of the theoretical dimensionless parameters, is also fairly good.
Finite amplitude effect on the stability of a jet of circular cross-section
Authors: Wang D. P.
Published: March 2006

The effect of finite amplitude on the stable and unstable states of a column of an ideal fluid of circular cross-section under the action of surface tension is studied. The method of solution is a formal extension of the linearized theory; it consists of assuming that the exact solution may be expanded in a power series of a small parameter characterizing the amplitude. The calculation is carried out to the point where the first non-trivial term of the finite amplitude effect is obtained. For the stable states, the result shows that the characteristic wavelength of a disturbance which appears to be stationary with respect to an observer is decreased by the finite amplitude effect. For the unstable states, it reveals that the growth rate depends not only on the wavelength and the magnitude but also on the type of disturbance imposed initially. The last result is a direct consequence of the fact that two independent types of initial disturbance, the disturbance of the velocity field and the disturbance of the free surface, may be imposed simultaneously on the jet.
The behaviour of similar solutions in a compressible boundary layer
Authors: Mcleod J. B. , Serrin J.
Published: March 2006

This paper discusses the mathematical properties of similar solutions of the boundary-layer equations in a compressible model fluid, under assumptions first introduced by Stewartson and by Li & Nagamatsu. Assuming a favourable pressure gradient and that backflow is not present, our results include (among other things) a rigorous proof that velocity overshoot occurs in the boundary layer if the wall is heated, and that this is true whether or not suction, blowing or slipping occurs at the wall; while, conversely, velocity overshoot does not occur when the wall is cooled and the amount of slipping at the wall is suitably restricted.
Some unsteady motions of a diffusion flame sheet
Authors: Clarke John F. , Stegen G. R.
Published: March 2006

A study is made of the response of a diffusion flame sheet to perturbations of reactant concentration which are introduced, either as changes in the free streams, or as appropriate initial distributions throughout the field. Species’ and energy conservation requirements are approximated by linearized boundary-layer equations; general solutions are derived for species, enthalpy and temperature distributions, as well as for the flame sheet shape, and a number of specific problems are solved.
The Cauchy–Poisson problem for a viscous liquid
Authors: Miles John W.
Published: March 2006

The axisymmetric, free-surface response of a semi-infinite viscous liquid to either a point impulse or an initial displacement of zero net volume is calculated. The asymptotic disturbance is resolved into three components: (i) a damped gravity wave, which represents a primary balance between gravitational and inertial forces with secondary, but cumulative, modification by viscous forces; (ii) a diffusive motion, which represents a balance between viscous and inertial forces; (iii) a creep wave, which represents a balance between gravitational and viscous forces. Van Dorn has suggested that the results may be relevant to the concentric circular ridges that surround the crater Orientale on the Moon.
Reversion of turbulent to laminar flow
Authors: Patel V. C. , Head M. R.
Published: March 2006

It has been shown experimentally that quite large departures occur from the universal inner-law velocity distribution in the presence of severe favourable pressure gradients in turbulent boundary layers and that these departures are associated with the tendency for the turbulent boundary layer to revert to a laminar state. From the measurements a criterion for the onset of reverse transition has been deduced in terms of the mean shear-stress gradient in the wall region of the flow. Experiments in fully developed pipe and channel flows suggest that the proposed criterion may be quite generally applicable to all fully turbulent shear flows.
The stability of free-surface flows with viscosity stratification
Authors: Craik Alex D. D. , Smith F. I. P.
Published: March 2006

The hydrodynamic stability is examined of liquids in which the viscosity varies with distance below a free surface. It is assumed that the viscosity becomes indefinitely large with distance from the surface, and that there is an ‘effective depth’ within which most of the motion occurs; but, otherwise, the viscosity distribution is arbitrary. The particular cases of wave-generation by a concurrent air flow at a horizontal liquid surface, and the stability of inclined flow under gravity are treated. It is shown that instabilities may occur which are similar to those known for thin uniform liquid films.
The spectrum of temperature fluctuations in turbulent flow
Authors: Grant H. L. , Hughes B. A. , Vogel W. M. , Moilliet A.
Published: March 2006

Temperature and velocity fluctuations have been recorded in the open sea and in a tidal channel, and power spectra have been determined from the records. The one-dimensional spectra of temperature fluctuations are found to have an inertial subrange. At larger wave-numbers the data can be fitted by Batchelor's spectrum function for the viscous-convective range. The spectra are inconsistent with the form proposed by Pao for the viscous-convective range.

Estimates are given for the constants in Batchelor's spectrum function, but these depend upon knowledge of the rate of dissipation of kinetic energy, which is determined from the velocity spectra. There is doubt about the validity of some of the velocity spectra, and in other cases there is reason to suspect that the turbulence is not locally isotropic.
Some observations of the occurrence of turbulence in and above the thermocline
Authors: Grant H. L. , Moilliet A. , Vogel W. M.
Published: March 2006

A hot film flowmeter, a thin film resistance thermometer and several thermistors were mounted on a submarine and used to study turbulence and temperature microstructure in the open sea. The surface layer is found to be continuously turbulent and patches of turbulence have been found in and below the main ocean thermocline.
Turbulent front structure of an axisymmetric compressible wake
Authors: Demetriades Anthony
Published: March 2006

Intermittency measurements have been performed within the first 100 virtual diameters of an axisymmetric compressible wake. The primary measurements, performed with the hot-wire anemometer, concerned the intermittency factor and the frequency of ‘zero crossings’. The intermittent flow covered most of the wake profile but for a narrow region about the axis, and was found to be normally distributed around the average front position. The latter, along with the extent of the front standard deviation, was found to agree numerically with expectations based on low speed wakes and to grow as the 1/3 power of axial distance. A weak periodicity of the front was detected at a wavelength about nine times greater than the longitudinal scale of turbulent velocity fluctuations. This periodicity affects the turbulent spectra.