bluuebunny/crossref_metadata_2025 huggingface dataset:
Viscous effects on the position of cavitation separation from smooth bodies
Authors: Arakeri Vijay H.
Published: March 2006

Flow visualization by the schlieren technique in the neighbourhood of a fully developed cavity on two axisymmetric headforms has shown the existence of laminar boundary-layer separation upstream of cavitation separation, and the distance between the two separations to be strongly dependent on Reynolds number. Based on present results, a semi-empirical method is developed to predict the position of cavitation separation on a smooth body. The method applies only in the Reynolds-number range when the cavitating body possesses laminar boundary-layer separation under non-cavitating conditions. Calculated positions of cavitation separation on a sphere by the method show good agreement with experimentally observed positions.
The steady streaming induced by a vibrating cylinder
Authors: Riley N.
Published: March 2006

In this paper we apply the techniques of higher-order boundary-layer theory to study the steady streaming induced in the neighbourhood of a cylinder which vibrates harmonically, perpendicular to its generators, in an unbounded fluid. The theoretical predictions are compared with the results of experiments performed at high streaming Reynolds numbers. Improved agreement between theory and experiment is achieved although unresolved discrepancies remain.
Search for the final period of decay of the axisymmetric turbulent wake
Authors: Freymuth Peter
Published: March 2006

Profiles of mean and fluctuating turbulent velocities and temperatures, of the time derivative of velocity and of intermittency have been measured in the wake of an unheated as well as a heated sphere in the low Reynolds number range 600–2500 at distances 80–1800 sphere diameters downstream. The wind-tunnel experiments exhibit a strong dependence on Reynolds number but they do not indicate the attainment of or an approach to the final period of turbulent decay which has been explored theoretically by Phillips and by O'Brien. The lowest turbulence Reynolds number obtained was Rλ, = 1·4.
The buckling and stretching of a viscida
Authors: Buckmaster J. D. , Nachman A. , Ting L.
Published: March 2006

We consider the deformation of a thin thread of viscous liquid (viscida) as its ends are slowly moved together. Equations are deduced which are capable of describing the motion of the thread when the displacement of the axis from a straight line is either on the scale of the thread thickness (problem 1) or on the much larger scale of the thread length (problem 2). In the former case it is shown analytically that an arbitrary initial displacement evolves in such a way that, as the appropriately scaled time τ becomes large, the first mode of the disturbance emerges in a dominant role with an amplitude that is proportional to τ½ and independent of the initial amplitude. This provides the initial condition for problem 2, for which a numerical description is obtained.

In addition, we analyse the situation when the ends of the viscida are slowly pulled apart. In this case the high frequency end of the spectrum dominates as an arbitrary disturbance decays.
Compressibility effects in the hydrodynamic theory of Brownian motion
Authors: Zwanzig Robert , Bixon Mordechai
Published: March 2006

The velocity correlation function of a Brownian particle in a viscous compressible fluid is studied in the limit of very small compressibility. The main effect of compressibility is an initial rapid change of the particle mass from a real inertial mass to a virtual mass.
The ‘sliced-cylinder’ laboratory model of the wind-driven ocean circulation. Part 2. Oscillatory forcing and Rossby wave resonance
Authors: Beardsley Robert C.
Published: March 2006

The response of the ‘sliced-cylinder’ laboratory model for the wind-driven ocean circulation is studied here in part 2 for the case of an oscillatory ‘wind’ stress. The model consists of a rapidly rotating right cylinder with a planar sloping bottom. This basin geometry contains no closed geostrophic contours, so that low frequency topographic Rossby wave modes possessing mean vorticity exist in the sliced-cylinder model by the physical analogy between topographic vortex stretching and the β effect for large-scale planetary flows. The interior flow in the laboratory model is driven by the time-dependent Ekman-layer suction produced by the periodic relative angular velocity of the upper lid. The frequency of the forcing is sufficiently small that the interior motion is quasi-geostrophic with the horizontal velocities being independent of depth. Simple two-dimensional analytic and numerical models are developed and compared very favourably with the laboratory results. The observed horizontal velocity field exhibits both (i) westward intensification and decreased horizontal scale when the forcing frequency is decreased, and (ii) a significant resonant magnification when the forcing frequency is tuned to the natural frequency of one of the lower inviscid topographic Rossby wave modes. The observed westward phase speed of the driven motion is accurately predicted and shows little dependence on the amplitude of the forcing. The instantaneous and mean Lagrangian fluid particle trajectories were measured in the laboratory model. The general derivation by Moore (1970) of the governing equations for the mean Lagrangian motion are extended to incorporate forcing and Ekman-layer dissipation. The results suggest that the mean Lagrangian flow should be significantly reduced near resonant frequencies, since the mean Eulerian motion is partially offset by the Stokes drift associated with the topographic Rossby wave modes. This result is consistent with the small observed amplitude of the mean Lagrangian motion. Also presented are the results for a laboratory experiment conducted using a combined steady and oscillatory ‘wind’ stress.
The ‘Richardson’ criterion for compressible swirling flows
Authors: Lalas Demetrius P.
Published: March 2006

The stability of a compressible non-dissipative swirling flow to adiabatic infinitesimal disturbances of arbitrary orientation is considered. The resulting sufficient condition for stability is the general form of the effective Richardson criterion for swirling flows, first obtained, for axisymmetric modes only, by Howard. In addition, upper bounds to the growth rate of unstable modes are obtained and some extensions of the semicircle theorem to azimuthal disturbances are stated.
Experiments on the instability waves in a supersonic jet and their acoustic radiation
Authors: Mclaughlin Dennis K. , Morrison Gerald L. , Troutt Timothy R.
Published: March 2006

An experimental investigation of the instability and the acoustic radiation of the low Reynolds number axisymmetric supersonic jet has been performed. Hot-wire measurements in the flow field and microphone measurements in the acoustic field were obtained from different size jets at Mach numbers of about 2. The Reynolds number ranged from 8000 to 107000, which contrasts with a Reynolds number of 1·3 × 106for similar jets exhausting into atmospheric pressure.

Hot-wire measurements indicate that the instability process in the perfectly expanded jet consists of numerous discrete frequency modes around a Strouhal number of 0·18. The waves grow almost exponentially and propagate downstream at a supersonic velocity with respect to the surrounding air. Measurements of the wavelength and wave speed of theSt= 0·18 oscillation agree closely with Tam's theoretical predictions.

Microphone measurements have shown that the wavelength, wave orientation and frequency of the acoustic radiation generated by the dominant instability agree with the Mach wave concept. The sound pressure levels measured in the low Reynolds number jet extrapolate to values approaching the noise levels measured by other experimenters in high Reynolds number jets. These measurements provide more evidence that the dominant noise generation mechanism in high Reynolds number jets is the large-scale instability.
On the flow near a weak shock wave downstream of a nozzle throat
Authors: Messiter A. F. , Adamson T. C.
Published: March 2006

In a transonic nozzle flow in which the velocity is slightly supersonic in some neighbourhood of the nozzle throat, a shock wave may be present either very close to the throat or else somewhat further downstream. In the latter case, relatively simple series solutions in general provide an asymptotic description of the fluid motion except very close to the shock wave. These outer solutions are reviewed for symmetric two-dimensional flow, and it is shown that the shock-wave jump conditions are not satisfied. A correction is then derived in the form of an inner solution for a small region immediately behind the shock. The resulting solution exhibits the singularities in the pressure gradient, streamline curvature and shock-wave curvature which are expected to occur at the intersection of a normal shock wave and a curved wall. An extension to axisymmetric flow is also given.
Steady flow in rapidly rotating variable-area rectangular ducts
Authors: Walker John S.
Published: March 2006

Steady liquid flow in a variable-area rectangular duct rotating rapidly about an axis perpendicular to its centre-line is treated. This problem is significant because the idea of driving a liquid through a rotating system has been largely overlooked by rotating-fluid dynamicists and because it closely resembles the flow inside the impellers of centrifugal pumps and hydraulic turbines. For a prototype formed by joining a semi-infinite constant-area duct and a semi-infinite duct with straight diverging walls, the flow in the diverging duct is carried entirely by large, O(E1/3) velocities in a boundary layer of thickness O(E1/3) adjacent to one of the (side) walls parallel to the axis of rotation, where E is the (small) Ekman number. With a vertical axis of rotation this high-velocity boundary layer is adjacent to the side wall on the right when facing in the flow direction. For a diverging or converging duct placed between two semi-infinite constant-area ducts, large, O(E1/3) velocities occur in side-wall boundary layers on both sides of the variable-area duct and on the left and right sides of the upstream and downstream constant-area ducts respectively. The existence of high-velocity side layers in rapidly rotating rectangular ducts should be relatively easy to prove experimentally and actual measurements of their velocity profiles would provide a good test of the present theory.
The starting flow down a step
Authors: Honji H.
Published: March 2006

The incompressible starting flow past a downstream-facing right-angled step was investigated at Reynolds numbers R (based on step height) less than 500 by means of flow-visualization techniques. The distance between the step and the point of reattachment on the downstream wall was found to increase linearly with time at intermediate stages of the flow development. The recirculation region formed behind the step was composed of three vortex domains at earlier stages of its development when R [gap ] 140. It was observed for R [gap ] 200 that once the starting vortices had been shed downstream a steady recirculation region was established. The evolution of filaments of tracer in the flow down the step was found to vary considerably with R. The filaments were observed to develop faster in the flow down the step than in that past a symmetric model without a downstream wall when R [gap ] 140.
On the stability of slowly varying flow: the divergent channel
Authors: Eagles P. M. , Weissman M. A.
Published: March 2006

The linear stability of a slowly varying flow, the flow in a diverging straight-walled channel, is studied using a modification of the ‘WKB’ or ‘ray’ method. It is shown that ‘quasi-parallel’ theory, the usual method for handling such flows, gives the formally correct lowest-order growth rate; however, this growth rate can be substantially in error if its magnitude is comparable to that of the rate of change of the basic state. The method used clearly demonstrates the dependence of the growth rate, wavenumber, neutral curves, etc., on the cross-stream variable and on the flow quantity under consideration. When applied to the divergent channel, the method yields a much wider ‘unstable’ region and a much lower ‘critical’ Reynolds number (depending on the flow quantity used) than those predicted by quasi-parallel theory. The determination of the downstream development of waves of constant frequency shows that waves of all frequencies eventually decay.
On transition in a pipe. Part 2. The equilibrium puff
Authors: Wygnanski I. , Sokolov M. , Friedman D.
Published: March 2006

Conditionally sampled hot-wire measurements were taken in a pipe at low Reynolds numbers (2700 > Re > 2000) corresponding to the onset of turbulence as a result of a large perturbation in the flow. This type of transition gives rise to a turbulent puff which maintains itself indefinitely at around Re = 2200. The structure of puffs was investigated in some detail and was found to be very different from the structure of fully developed turbulent pipe flow. Nevertheless, it is independent of the character of the disturbance which created it. The purpose of the study was to gain some insight into the mechanism of transition in a pipe.
Experimental results for the transpired turbulent boundary layer in an adverse pressure gradient
Authors: Andersen P. S. , Kays W. M. , Moffat R. J.
Published: March 2006

An experimental investigation of the fluid mechanics of the transpired turbulent boundary layer in zero and adverse pressure gradients was carried out on the Stanford Heat and Mass Transfer Apparatus. Profiles of (a) the mean velocity, (b) the intensities of the three components of the turbulent velocity fluctuations and (c) the Reynolds stress were obtained by hot-wire anemometry. The wall shear stress was measured by using an integrated form of the boundary-layer equation to ‘extrapolate’ the measured shear-stress profiles to the wall.

The two experimental adverse pressure gradients corresponded to free-stream velocity distributions of the type u∞ ∞ xm, where m = −0·15 and −0·20, x being the streamwise co-ordinate. Equilibrium boundary layers (i.e. flows with velocity defect profile similarity) were obtained when the transpiration velocity v0 was varied such that the blowing parameter B = pv0u∞/τ0 and the Clauser pressure-gradient parameter $\beta\equiv\delta_1\tau_0^{-1}\,dp/dx $ were held constant. (τ0 is the shear stress at the wall and δ1 is the displacement thickness.)

Tabular and graphical results are presented.
Stokes flow past a particle of arbitrary shape: a numerical method of solution
Authors: Youngren G. K. , Acrivos A.
Published: March 2006

The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear integral equations of the first kind for a distribution of Stokeslets over the particle surface. The unknown density of Stokeslets is identical with the surface-stress force and can be obtained numerically by reducing the integral equations to a system of linear algebraic equations. This appears to be an efficient way of determining solutions for several external flows past a particle, since it requires that the matrix of the algebraic system be inverted only once for a given particle.

The technique was tested successfully against the analytic solutions for spheroids in uniform and simple shear flows, and was then applied to two problems involving the motion of finite circular cylinders: (i) a cylinder translating parallel to its axis, for which the local stress force distribution and the drag were determined; and (ii) the equivalent axis ratio of a freely suspended cylinder, which was calculated by determining the couple on a stationary cylinder placed symmetrically in two different simple shear flows. The numerical results were found to be consistent with the asymptotic analysis of Cox (1970, 1971) and in excellent agreement with the experiments of Anczurowski & Mason (1968), but not with those of Harris & Pittman (1975).
The density of organized vortices in a turbulent mixing layer
Authors: Moore D. W. , Saffman P. G.
Published: March 2006

It is argued on the basis of exact solutions for uniform vortices in straining fields that vortices of finite cross-section in a row will disintegrate if the spacing is too small. The results are applied to the organized vortex structures observed in turbulent mixing layers. An explanation is provided for the disappearance of these structures as they move downstream and it is deduced that the ratio of average spacing to width should be about 3·5, the width being defined by the maximum slope of the mean velocity. It is shown in an appendix that walls have negligible effect.
The rotating rod viscometer
Authors: Beavers G. S. , Joseph D. D.
Published: March 2006

This paper reports the development of practical methods of viscometry to characterize non-Newtonian fluids in slow flow. It is shown that measurements of the free surface near rods rotating in STP and polyacrylamide are accurate, reproducible, and in excellent agreement with a theory of rod climbing. Results are presented that establish the theory and experiment as a viscometer for determining the values of certain (Rivlin-Ericksen) constants that arise in the theory of slow flow. The variation of these constants with temperature in our sample of STP has been explicitly and accurately determined. The experiments in STP show that there is a range of rotational speeds for which STP may be well described by the fluids of grade four. Depth-averaged equations are derived from the equations governing steady axisymmetric flow of any incompressible simple fluid. From the depth-averaged equations, we prove a theorem about the variation of the torque required to turn the rod.
Coherent structures in turbulence
Authors: Davies P. O. A. L. , Yule A. J.
Published: March 2006

This account of the Colloquium on Coherent Structures in Turbulence held at Southampton from 26-29 March 1974 presents a brief summary of two invited lectures and 42 formal presentations on turbulent shear-flow structure. A number of shorter contributions and discussions are also outlined. The present position of the study of turbulent shear-flow structure is reviewed and some new experimental techniques are discussed.
Electro-convection in a dielectric liquid layer subjected to unipolar injection
Authors: Lacroix J. C. , Atten P. , Hopfinger E. J.
Published: March 2006

The problem of electric charge convection in a dielectric liquid layer of high ionic purity, when subjected to unipolar injection, is in many ways analogous to that of thermal convection in a horizontal fluid layer heated from below, although no formal analogy can be established. The problem treated is intrinsically more nonlinear than the thermal problem. We consider two asymptotic states of convection: one where the whole motion is dominated by viscosity, and one where inertial effects dominate. In each state, two or three spatial regions are distinguished. From the approximate equations that hold in the different regions, information about the variation of the different quantities with distance from the injector is obtained, and further approximations permit us to establish the dependence of the current density ratioI/I0(called theelectric Nusselt number) on the stability parameterT=M2R= εϕ0/Kρν, and on 1/R= ν/Kϕ0, which is an equivalent Prandtl number (ε is the permittivity, ρ the fluid density,Kthe mobility, ν the kinematic viscosity, and ϕ0the applied voltage). In the viscous state, the analysis givesI/I0∞T½; in the inertial state the lawI/I0∞ (T/R)1/4=M½is obtained. SinceMis independent of the applied voltage, the latter law shows the saturation in the electric Nusselt number observed in earlier experiments. The transition in the states is associated with a transition number (MR)T[gap ] 30, which is an electric Reynolds number, related to an ordinary Reynolds number of about 10.

The experimental results, obtained in liquids of very different viscosities and dielectric constants, verify these theoretical predictions; further, they yield more precise numerical coefficients. As for the transition criteria, the experiments confirm that the viscous and inertial effects are of the same order whenRe[gap ] 10. It was also possible to determine roughly the limits of the viscous and inertial states. The viscous analysis remains valid up to a Reynolds number of about 1; the inertial state can be considered valid down to a Reynolds number of 60. Schlieren observations show that the motion has the structure of very stable hexagonal cells at applied voltages just above the critical voltage, which are transformed into unstable filaments when the voltage is increased further. At even higher voltages, the motion finally breaks down into turbulence. It may be of interest to point out that, whenM< 3, the electric Nusselt number approaches 1, which is equivalent to the situation in thermal convection at low Prandtl numbers.
The free surface on a liquid filling a trench heated from its side
Authors: Joseph Daniel D. , Sturges Leroy
Published: March 2006

In this paper we compute the motion and the shape of the free surface on a liquid in a trench heated from its side. The analysis is based on Joseph's Lagrangian theory of domain perturbations, which is developed in general and through simple examples, chosen so as to make the comparison of the Lagrangian method with Stokes's Eulerian theory very clear. The perturbation problems are resolved analytically by application of biorthogonality conditions to a powerful set of biharmonic eigenfunctions.
Pattern formation in a suspension of swimming microorganisms: equations and stability theory
Authors: Childress S. , Levandowsky M. , Spiegel E. A.
Published: March 2006

A model for collective movement and pattern formation in layered suspensions of negatively geotactic micro-organisms is presented. The motility of the organism is described by an average upward swimming speedUand a diffusivity tensorD. It is shown that the equilibrium suspension is unstable to infinitesimal perturbations when either the layer depth or the mean concentration of the organisms exceeds a critical value. For deep layers the maximum growth rate determines a preferred pattern size explicitly in terms ofUandD. The results are compared with observations of patterns formed by the ciliated protozoanTetrahymena pyriformis.
A viscous internal wave in a stratified fluid whose buoyancy frequency varies with altitude
Authors: Gordon D. , Klement U. R. , Stevenson T. N.
Published: March 2006

A viscous incompressible stably stratified fluid with a buoyancy frequency which varies slowly with altitude is considered. A simple harmonic localized disturbance generates an internal wave in which the energy propagates along curved paths. Small amplitude similarity solutions are obtained for two-dimensional and axisymmetric waves. It is found that under certain conditions the wave amplitude can increase with height. The two-dimensional theory compares quite well with experimental measurements.
Numerical prediction of incompressible separation bubbles
Authors: Briley W. Roger , Mcdonald Henry
Published: March 2006

A method is presented for performing detailed computations of thin incompressible separation bubbles on smooth surfaces. The analysis consists of finite-difference solutions to the time-dependent boundary-layer or Navier-Stokes equations for the flow in the immediate vicinity of the bubble. The method employs the McDonald-Fish turbulence model, to predict the development of the time-mean flow field, as influenced by the free-stream turbulence level. It also employs a viscous-inviscid interaction model, which accounts for the elliptic interaction between the shear layer and inviscid free stream. The numerical method is based on an alternating-direction implicit scheme for the vorticity equation. It employs transformations, to allow the free-stream boundary to change in time with the shape of the computed shear layer, and to ensure an adequate resolution of the sublayer region. Numerical solutions are presented for transitional bubbles on an NACA 663-018 airfoil at zero angle of incidence with chordal Reynolds numbers of 2·0 × 106 and 1·7 × 106. These have a qualitative behaviour similar to that observed in numerous experiments; they are also in reasonable quantitative agreement with available experimental data. Little difference is found between steady solutions of the boundary-layer and Navier-Stokes equations for these flow conditions. Numerical studies based on mesh refinement suggest that the well-known singularity at separation, which is present in conventional solutions of the steady boundary-layer equations when the free-stream velocity is specified, is effectively removed when viscous-inviscid interaction is allowed to influence the imposed velocity distribution.
Ergodic properties of inviscid truncated models of two-dimensional incompressible flows
Authors: Basdevant C. , Sadourny R.
Published: March 2006

The equilibrium spectra of two-dimensional numerical model flows are studied from the viewpoint of microcanonical ensemble averages. The method leads to accurate numerical verification of the ergodic, or mixing, hypothesis in the case of systems constrained to a finite number of degrees of freedom.
On trapped waves over a continental shelf
Authors: Huthnance John M.
Published: March 2006

Any straight continental shelf of monotonic depth profile is shown to have as its entire complement of barotropic trapped modes (i) an infinite discrete set of ‘continental-shelf waves’, (ii) a single ‘Kelvin wave’, and (iii) an infinite discrete set of ‘edge waves’. The decomposition of energy density and fluxes into modal constituents is discussed.
On vortex strength and drag in bluff-body wakes
Authors: Griffin Owen M. , Ramberg Steven E.
Published: March 2006

In a recent paper (Griffin & Ramberg 1974) the authors studied the vortex-street wakes behind forced vibrating rigid cylinders. All experimental conditions were within the regime of wake capture or synchronization between the vibration and vortex frequencies. Both mean and fluctuating velocities in the wake together with the length of the vortex formation region were measured as functions of vibration amplitude and frequency at a Reynolds number of 144. The viscous vortex strength, age and spacing at this Reynolds number were then obtained by matching a model for the vortex street with the mean and r.m.s. velocity profiles obtained from hot-wire measurements. These results are employed here to determine the steady drag force on the vibrating cylinder by means of the von Kármán drag formulation. The drag coefficients determined in this way are in agreement with the recently published direct force measurements of Tanida, Okajima & Watanabe (1973) and Griffin, Skop & Koopmann (1973) at Reynolds numbers of 80, 500-900 and 4000. From these results a direct relation is drawn between the increased drag on resonantly vibrating structures and changes in the vortex strength, spacing and formation in their wakes.
Turbulent convection in water over ice
Authors: Adrian R. J.
Published: March 2006

Measurements have been made of the vertical velocity and simultaneous temperature fluctuation at fixed points in turbulent, statistically steady convection in water over an ice surface. Convection was confined to a horizontal layer of constant depth that lay under a stably stratified layer. Measurements of vertical velocities less than 1 mm/s were accomplished using a laser-Doppler velocimeter with a frequency-shifted reference beam. Vertical profiles of all relevant one-point moments, including joint temperature-velocity moments, up to third order were obtained. In the stable layer internal gravity waves were observed, and certain aspects of their behaviour were found to agree with existing theoretical analyses. Typical r.m.s. values of the temperature and velocity fluctuations were nearly equal to the appropriate ‘convection’ scales in the convection layer while in the conduction layer immediately above the ice surface they were more nearly equal to the ‘molecular’ scales. All of the observations were consistent with the existence of buoyant plume-like masses which dominated the structure of the flow. An analysis of the data using a conditional averaging technique to determine the mean properties of these masses showed that below the stable layer they accounted for over 80% of the mean-square fluctuations in temperature and velocity.
Viscous shear flow past small bluff bodies attached to a plane wall
Authors: Kiya Masaru , Arie Mikio
Published: March 2006

Numerical solutions of the Navier-Stokes equations are presented for two-dimensional viscous flow past semicircular and semielliptical projections attached to a plane wall on which a laminar boundary layer has developed. Since the major axis is in the direction normal to the wall and is chosen to be twenty times as long as the minor axis in the present case, the flow around the semielliptical projection will approximately correspond to that around a normal flat plate. It is assumed that the height of each obstacle is so small in comparison with the local boundary-layer thickness that the approaching flow can be approximated by a uniform shear flow. Numerical solutions are obtained for the range 0·1-100 of the Reynolds number, which is defined in terms of the undisturbed approaching velocity at the top of the obstacle and its height. The geometrical shapes of the front and rear standing vortices, the drag coefficients and the pressure and shear-stress distributions are presented as functions of the Reynolds number. The computed results are discussed in connexion with the data already obtained in the other theoretical solutions and an experimental observation.
On double-roll convection in a rotating magnetic system
Authors: Robert P. H. , Stewartson K.
Published: March 2006

Electrically and thermally conducting inviscid fluid rotating about a vertical axis is confined between two horizontal plates maintained a t different temperatures, the upper plate being the cooler. The fluid is permeated by a horizontal magnetic field that corotates with the fluid. In an earlier paper (Roberts & Stewartson 1974) the fluid is supposed to be in a state of near-marginal instability to convective overturning and the nonlinear evolution of single rolls is discussed. Inertial terms are neglected. However, if q < 2 and λ < 2/3½, where q and λ may be defined by equation (2.3) below, the principle of the exchange of stabilities holds and there is also a degeneracy in the linear stability problem. There are now two distinct unstable rolls equally possible and their nonlinear interaction leads to a violation of the governing equations. This difficulty has already been noted by Taylor (1963) and it is resolved in this paper by adding a geostrophic motion (the Taylor shear) parallel to the magnetic field and by restoring the inertial terms in the governing equations. We consider particularly instabilities in which one roll predominates and find that, if λ is sufficiently small, each of the rolls that can occur is stable with respect to the other, i.e. an initially weak roll of the other type dies out relative to it. This means that we can expect the fluid motion to consist of single rolls at large times. On the other hand when λ is near 2/3½ both rolls are unstable with respect to the other. The Taylor shear does not then die out and the two rolls become comparable in magnitude and modify each other's structure. At intermediate values of λ one of the rolls is stable in this way and the other unstable.

The study is motivated by a desire to understand better the dynamical means by which a large mass of conducting fluid can create its own magnetism. It is argued that these instabilities suggest the existence of a mechanism of self-adjustment preventing λ from either increasing or decreasing indefinitely and noted that, very roughly, λ is of order unity in the earth's core.
Long-wavelength peristaltic pumping at low Reynolds number
Authors: Manton M. J.
Published: March 2006

An asymptotic expansion is found for the low Reynolds number flow induced in an axisymmetric tube by long peristaltic waves of arbitrary shape. Expressions are determined for the relationship between the mean pressure gradient and the volume flux, for the mean rate of working by the wall of the tube and for the shear stress a t the wall. A necessary and sufficient condition for the occurrence of trapping (that is, regions of separated flow near the axis of the tube in a reference frame moving at the wave speed) is obtained. It is shown that reflux (that is, a mean flux in the negative axial direction in a layer of fluid adjacent to the wall when the net mean flux is positive) occurs whenever there is an adverse mean pressure gradient, independently of the shape of the wave. A n estimate of the amount of reflux is derived.
Kelvin–Helmholtz wave growth on cylindrical sheets
Authors: Crapper G. D. , Dombrowski N. , Pyott G. A. D.
Published: March 2006

A first-order analysis of Kelvin-Helmholtz wave growth on cylindrical sheets is carried out. It is demonstrated that the growth rate of both symmetric and antisymmetric waves increases significantly with reduction of the radius of the core.
The two-dimensional elliptical cap bubble
Authors: Hills J. H.
Published: March 2006

A study of the closed wakes behind large gas bubbles rising between two parallel plates reveals that they are elliptical in shape, rather than circular. Fitting an are of an ellipse, rather than a circle, to the curved upper surface of such a bubble eliminates a discrepancy between theory and experiment noted by Collins (1965a). Collins’ analysis of the effect of finite channel width is extended to the elliptical case, and the stability of the closed wake is examined.
Diffraction of oblique waves by an infinite cylinder
Authors: Bai Kwang June
Published: March 2006

This paper presents a numerical method for solving linearized water-wave problems with oscillatory time dependence. Specifically it considers the diffraction problem for oblique plane waves incident upon an infinitely long fixed cylinder on the free surface. The numerical method is based on a variational principle equivalent to the linearized boundary-value problem. Finite-element techniques are used to represent the velocity potential; and the variational principle is used to determine the unknown coefficients in the solution throughout the fluid domain. To illustrate this method, reflexion and transmission coefficients and the diffraction forces and moment are computed for oblique waves incident upon a vertical flat plate, a horizontal flat plate and rectangular cylinders, where the comparison is made with the existing results by others. Also considered is the associated sinuous forced-motion problem, where comparison is made with the results for a circular cylinder obtained by Bolton & Ursell (1973).
Transition to turbulence in oscillating pipe flow
Authors: Merkli P. , Thomann H.
Published: March 2006

Published results on transition in a Stokes layer indicate a wide range of transition Reynolds numbers. As thermal effects in a resonance tube (Merkli & Thomann 1975) depend on the state of the boundary layer, the transition Reynolds number was determined, and a critical Reynolds numberAc≈ 400 was found. The observations were made with hot wires and with flow visualization by means of smoke, and provide new details on turbulence in a Stokes layer. With this knowledge an explanation of the large discrepancies between some stability theories and the experiments is suggested. The main point is that turbulence occurs in the form of periodic bursts which are followed byrelaminarimtionin the same cycle and donotlead to turbulent flow during the whole cycle.

A further, unexpected result of the present investigation is the discovery of vortex patterns superimposed on the normal laminar acoustic motion.
The development of turbulence structure in a uniform shear flow
Authors: Mulhearn P. J. , Luxton R. E.
Published: March 2006

Measurements of the streamwise development of a variety of velocity correlations in a uniform shear flow are given and discussed. It is concluded that, in the region in which the Reynolds stresses are approximately constant, the turbulence structure is practically independent of all the initial conditions except the initial length scale. It appears that the preferential amplification of some eddy structures by the mean shear effectively destroys information about the initial conditions, apart from the initial length scale, after a total strain of about 1.5.
Buoyant convection from horizontal surfaces at const ant pressure
Authors: Traugott S. C.
Published: March 2006

A two-dimensional horizontal flow is discussed, which is induced by other, buoyancy-driven flows elsewhere. It is an adaptation of the incompressible wall jet, which is driven by conditions a t the leading edge and has no streamwise pressure gradient. The relation of this flow to the classical buoyancy-driven boundary layers on inclined and horizontal surfaces is investigated, as well as its possible connexion with a two-dimensional buoyant plume driven by a line source of heat. Composite flows are constructed by patching various such solutions together. The composite flows exhibit$Gr^{\frac{1}{4}}$scaling (Grbeing the Grashof number).
Statistical characteristics of a turbulent jet
Authors: Ribeiro M. M. , Whitelaw J. H.
Published: March 2006

Velocity probability distributions and autocorrelation functions were measured in the self-preserving region of a round free jet at 57 diameters. On-line digital-sampling procedures were used to interpret the signals from a crossed hot-wire probe. Particular attention was paid to the probabilities of the axial and radial velocity components and of the angle between them at radial locations corresponding to the centre-line and the location of maximum shear stress and at an edge location r/x = 0·087.

The results show, for example, that the probability of the axial velocity on the centre-line is slightly non-Gaussian and that, in general, the observed deviations of the probabilities of u depend upon the difference in behaviour of the corresponding distributions for positive and negative ν; outward transport (positive ν) is associated with near-Gaussian u distributions whereas inward transport (negative ν) is associated with skewed u distributions. The probability of the fluctuating vector (u, v) becomes more asymmetric with increasing radius with the dominant direction corresponding to positive $\overline{uv}$. The measured auto-and cross-correlations are shown to be largely independent of radius.
A turbulence velocity scale for curved shear flows
Authors: So Ronald M. C.
Published: March 2006

Assuming the turbulence length scale to be unaffected by streamline curvature, a turbulence velocity scale for curved shear flows is derived from the Reynolds-stress equations. Closure of the equations is obtained by using the scheme of Mellor & Herring (1973), and the Reynolds-stress equations are simplified by invoking the two-dimensional boundary-layer approximations and assuming that production of turbulent energy balances viscous dissipation. The resulting formula for the velocity scale has one free parameter, but this can be determined from data for non-rotating unstratified plane flows. Consequently there is no free constant in the derived expression. A single value of the constant is found to give good agreement between calculated and measured values of the velocity scale for a wide variety of curved shear flows.

The result is also applied to test the validity and extent of the analogy between the effects of buoyancy and streamline curvature. This is done by comparing the present result with that obtained by Mellor (1973). Excellent agreement is obtained for the range −0·21 [les ]Rif[les ] 0·21. Therefore the present result provides direct evidence in support of the use of a Monin–Oboukhov (1954) formula for curved shear flows as proposed by Bradshaw (1969).
An asymptotic solution for slightly buoyant laminar plumes
Authors: Wesseling P.
Published: March 2006

When the buoyancy forces are small compared with the inertia forces, heated plumes in laminar flows which are uniform at upstream infinity approximately satisfy a linearized version of the Boussinesq equations, here called the Oseen–Boussinesq equations. An analytic solution is constructed for arbitrary Prandtl number and arbitrary direction of the unperturbed flow in the case of a plume produced by a point source. The two-dimensional case of the plume from a line source is considered briefly. A Stokes-type paradox occurs: it is found that a line-source solution that vanishes at infinity does not exist.
Thermal convection in gas–droplet mixtures with phase transition
Authors: Kambe T. , Takaki R.
Published: March 2006

Thermal convection in a three-component fluid consisting of an inert carrier gas, a condensable vapour and small liquid droplets dispersed throughout the gaseous components has been investigated both theoretically and experimentally. The theoretical study is concerned with the stability of a horizontal fluid layer subject to gradients of both temperature and droplet density. The stability is characterized by four parameters: two material constants, that is, a modified Prandtl number P and a constant Q proportional to Dm − κ (Dm is the mutual mass diffusivity of the two gaseous constituents, κ the thermometric conductivity of the gas phase), a modified Rayleigh number R and a parameter S defined as the ratio of the droplet density gradient to the gas density gradient. It is shown for positive R that, irrespective of the value of R, the system is stable for S > S∞ (S∞ is a constant dependent on P and Q) and unstable for S < Q (Q is normally less than S∞) and that for the intermediate range Q < S < S∞ a transition from stability to instability occurs via an oscillatory state as R is increased through a critical value depending on S. It is shown that the stability is governed largely by both vapour diffusion through the inert gas and droplet growth or decay due to phase changes.

In the experiments, thermal convection in a three-component fluid consisting of air, water vapour and water droplets was investigated. The cloud of droplets was mainly formed by injecting cigarette smoke into a horizontal layer of air saturated with water vapour. After the injection several phases of motion were observed successively. Among them there were travelling waves and steady cellular convection. Measurements were made of the critical Rayleigh numbers for the onset of the phases, the scale of the steady convection cells and the speed of the travelling waves. It is found that all the qualitative features of the experiment are explained by the theory.
A three-dimensional law of the wall for turbulent shear flows
Authors: Van Den Berg B.
Published: March 2006

An extended law of the wall is derived for three-dimensional flows. It describes the variation of the magnitude and direction of velocity close to the wall. The effects of both the pressure gradient and the inertial forces have been taken into account. The derived wall law is valid only when the deviations from the simple law of the wall are not large. The most important feature of a three-dimensional wall law is the prediction of the rotation of the velocity vector near the wall. Comparison of the flow angle variations predicted by the present wall law with the few available experimental data shows good agreement.
On the wave structure of the wall region of a turbulent boundary layer
Authors: Bark Fritz H.
Published: March 2006

Following the ideas suggested by Landahl (1967, 1975), some model calculations of the fluctuating velocity field in the wall region of a turbulent boundary layer have been carried out. It was assumed that the turbulent stresses are generated intermittently on small scales in time and space owing to bursting-type motions. The Reynolds-stress distribution during bursting periods and the mean velocity profile were assumed to be known, and the linear large-scale response to a random system of bursts was computed using an idealized model for the joint probability distribution in time and space of the occurrence of bursts. Computed energy spectra of the streamwise velocity fluctuations display scales in the spanwise and streamwise directions and time which are in good agreement with measurements by Morrison, Bullock & Kronauer (1971). However, the wavenumber band-widths of the computed spectra are narrower than those of the measured ones. This discrepancy is probably due to the crudeness of the model employed for the Reynolds stress during bursting.
Statistical properties of the kinematics and dynamics of a random gravity-wave field
Authors: Tung C. C.
Published: March 2006

The probability density function and the first three statistical moments of the velocity, acceleration and dynamic pressure are obtained for a Gaussian, stationary, homogeneous, random gravity-wave field in deep water, using infinitesimal wave solutions. It is shown that the velocity, acceleration and pressure are non-Gaussian. While the horizontal accelerations and vertical velocity component are of zero mean and unskewed, the dynamic pressure, vertical acceleration and horizontal velocity components are skewed and have non-zero mean.
The influence of shear on mixing across density interfaces
Authors: Long Robert R.
Published: March 2006

This paper discusses critically observations of mixing processes across density interfaces in laboratory experiments and inferences that have been made and can be made from these observations. Fluxes of heat or salt and entrainment velocities have been found to depend on negative powers of an overall Richardson number Ri* based on the buoyancy jump across the interface, the depth of the homogeneous layer and the intensity of the turbulence near the source. When the Reynolds and Péclet numbers are large, the fluxes or entrainment velocities appear to be proportional to the minus one and minus three-halves powers of Ri* for flows with and without mean shear respectively, and this difference has caused speculation about the accuracy of the experimental data and about the reasons for the two laws if the difference is real. In the present discussion, we accept the accuracy of the two laws and attribute the higher entrainment rate for shear flows to the decrease of r.m.s. velocities near the interface with increasing Ri* in the case of zero shear. A plausible argument yields the unifying result that the entrainment rates in both cases are proportional to Ri−1, where Ri is a Richardson number based on the buoyancy jump and velocities and lengths characteristic of the turbulence near the interface. It is suggested that the $Ri^{-\frac{3}{2}}$ behaviour inferred by Turner is based on an erroneous interpretation of experimental data.

In the course of the discussion, it is shown that the drag coefficient in flow of a stratified fluid over a rough surface is independent of the Richardson number (or density jump across the interface or inversion) and depends only on the ratio of the roughness length to the depth of the homogeneous layer. This has obvious implications for problems of parameterizing the momentum flux near the ground in the atmosphere.
Withdrawal of a stratified fluid from a vertical two-dimensional duct
Authors: Imberger Jorg , Fandry Chris
Published: March 2006

On the basis of a small Froude number theory it is shown that previous horizontalduct models of the withdrawal of fluid from a stratified tank with a free surface and a fixed bottom are inappropriate at large times and that the flow should be modelled by that from an infinite vertical duct. The falling horizontal free surface in the tank is replaced by a vertically moving column of stratified fluid and the tank bottom is modelled by a stagnant pool of fluid below. This model is analysed and a solution uniformly valid in time is presented. The properties of the solution are then compared with existing theories and experiments.