bluuebunny/crossref_metadata_2025 huggingface dataset:
Linear stability of the dissipative, two-fluid, cylindrical Couette problem. Part 1. The stably-stratified hydrodynamic problem
Authors: Schneyer G. P. , Berger S. A.
Published: March 2006

The stability of a two-fluid vortex is studied as a step towards understanding the separation and containment problems in a gaseous-core nuclear rocket. In particular, the linear hydrodynamic stability of two incompressible, immiscible, viscous fluids occupying separate annular regions of a cylindrical Couette apparatus is considered. Neglecting surface tension and gravity, a conservative assumption, the governing equations for arbitrary jumps in fluid properties are derived and numerical solutions to the resultant eigenvalue problems obtained. Results are presented for the effect on neutral stability of density and viscosity jumps, varying gap widths, and differing fluid-fluid interfacial positions. The solutions are limited, however, to the case of stably stratified fluids and a stationary outer cylinder.

Two separate modes (multiple eigenvalues) have been discovered for all cases in which two fluids, differing in any property, are present. A rationale is presented for this phenomenon as well as for most of the other observed results.

While most results are believed to be manifestations of the Taylor cylindrical Couette instability phenomenon, evidence is presented for the existence of additional ‘hidden’ eigenvalues attributable to the classical Kelvin–Helmholtz and/or the recently reported Yih viscosity-stratification instability phenomena.
Viscosity-induced instability of a one-dimensional lattice of falling spheres
Authors: Crowley Joseph M.
Published: March 2006

When a layer of particles moves through a viscous liquid it experiences forces which tend to disrupt the layer into clusters of particles separated by open channels. A theoretical description of this process is presented and a viscous instability is predicted. The spatial growth of the instability is approximated by eγz, where
\[
\gamma = {\textstyle\frac{3}{2}} a/d^2,
\]
where a is the particle radius and d is the average distance between particles. This result implies that any initial irregularity in a uniform particle distribution will be amplified by viscous forces alone. Significant amplification will occur when the particle has drifted a small multiple of the separation distance, if this separation is not much greater than the particle diameter. Thus, any initially uniform particle layer will form clusters as it drifts through a viscous fluid. The distance in which this clustering occurs will be unaffected by changes in the particle velocity, as long as the Reynolds number remains small. The preferred form of irregularity will consist of small clusters separated by individual particles which trail some distance behind. Experimental verification of these conclusions is presented.
On the energy deficiency in self-preserving convective flows
Authors: Turner J. S.
Published: March 2006

When similarity solutions are used to describe convective plumes or thermals, there is always found to be a discrepancy between the work done by buoyancy forces and the kinetic energy of mean motion. It is the main purpose of this paper to set down the ratio of these quantities for a wide variety of forms of buoyant elements and environmental stabilities. For consistency, the remaining fraction of the energy must appear as turbulent kinetic energy and eventually be dissipated, but these processes are not investigated in detail. The results are shown to have some relevance to the problem of convectively driven mixing across a density interface, where the largest scales of motion are dominant, and to the understanding of the transition zone between two self-preserving states of turbulent convection.
Rotation of a cylinder about an eccentric parallel axis in a viscous fluid
Authors: Wang Chang-Yi
Published: March 2006

A circular cylinder in an infinite fluid rotates rigidly about a fixed axis which is parallel to, but does not coincide with, its geometric axis. It is found that, depending on the relative magnitude of the Reynolds number R and eccentricity ε, the flow may have two, one or no boundary layers. General solutions for R [Lt ] are obtained. It is found that owing to eccentricity there exist both a flow periodic in the circumferential direction and a non-periodic flow which is a function only of the radial distance from the centre of the cylinder. The non-periodic flow is caused by the nonlinear Reynolds stress and contributes to the torque experienced by the cylinder. The high Reynolds number case,
\[
1 \Lt R \Lt\epsilon^{-\frac{3}{2}},
\]
is solved by matched asymptotic expansions. The stream function can be represented by Hankel functions of order 1/3 and a slight decrease in torque is found. In the low Reynolds number case, R [Lt ] 1, the torque is increased owing to eccentricity when R < 0·145 and decreased when R > 0·145. A physical explanation is presented.
Nonlinear motions induced by moving thermal waves
Authors: Young Richard E. , Schubert Gerald , Torrance Kenneth E.
Published: March 2006

The motion induced in a layer of Boussinesq fluid by moving periodic thermal waves is obtained by numerically solving the complete nonlinear two-dimensional momentum and temperature equations. Three sets of boundary conditions are treated: rigid upper and lower boundaries with symmetrical heating; free upper boundary and rigid lower boundary with heating only at the top; free upper and lower boundaries with symmetrical heating. The nonlinear streamline patterns show that, when the velocity fluctuations are larger than the phase speed of the thermal wave and the mean flow, the convection cells have shapes governed by fluctuating nonlinear interactions. Significant mean velocities can be created even without the characteristic tilt in the convection cells expected on the basis of linear theory. Nonlinear interactions can lead to a mean shear even in the absence of motion of the thermal source. When the viscous diffusion time across the fluid layer is less than or of the same order as the period of the thermal wave, the order of magnitude of the induced mean velocity does not exceed that of the phase speed of the wave, even for intense thermal forcing.
Semi-infinite vortex trails, and their relation to oscillating airfoils
Authors: Weihs D.
Published: March 2006

Semi-infinite double rows of vortices are used to study the periodic wake of both oscillating and stationary two-dimensional bodies immersed in a uniform incompressible stream. Analytical expressions for the induced Velocities on the body, for trails with constant spacing, which are valid for small values of the oscillation amplitude are presented while, for the general case of vortex shedding, an iterative procedure for the representation of trails of variable spacing is developed and used. Vortex streets due to oscillating bodies are obtained as a function of three non-dimensional parameters: the Strouhal number (initial spacing ratio), a non-dimensional vortex strength and the downstream spacing ratio. Criteria establishing when such trails are expected to widen, become narrow or stay of constant width are presented, as well as expressions for the induced velocities.

The trails and their induced velocities enable the calculation of the vortex strength from measurable quantities. Thus they can serve as a method for estimating the hydrodynamic forces on the airfoil due to large amplitude oscillations, such as those observed in the propulsive movements of fish and cetaceans, as well as the small amplitude oscillations due to hydroelastic interactions.
On trapped oscillations in a slightly viscous rotating fluid
Authors: Stewartson K.
Published: March 2006

Some properties of trapped oscillations in an inviscid rotating fluid were studied by Stewartson (1971, 1972). In this paper the modifications necessary to allow for a small viscosity in the fluid are discussed. It is found that concentrated disturbances of the kind used in ray theory are only possible if the rays are thought of as emanating from two sources outside the fluid and at opposite ends of the ray. Otherwise the ray must have a lateral tail, which vitiates the notion of a ray. The double-source character of rays is preserved at reflexions except when at glancing incidence. In this case a substantial part of the ray strength is lost and may give rise to a lateral tail if uncompensated. The focusing of the rays at the limit cycle is also discussed and shown to lead to a catastropic loss of energy, so that a free oscillation if set up rapidly decays. A simple forced oscillation is considered and shown to lead to weak resonances in the bands of trapped oscillations.
Sedimentation in a dilute dispersion of spheres
Authors: Batchelor G. K.
Published: March 2006

The dispersion considered consists of a large number of identical small rigid spheres with random positions which are falling through Newtonian fluid under gravity. The volume fraction of the spheres (c) is small compared with unity. The dispersion is statistically homogeneous, and the axes of reference are chosen so that the mean volume flux across any stationary surface is zero. The problem is to determine the mean value of the velocity of a sphere (U). In §3 there is described a systematic and rigorous procedure which overcomes the familiar difficulty presented by the occurrence of divergent integrals, essentially by the choice of a quantity V whose mean value can be found exactly and which has the same long-range dependence on the position of a second sphere as U so that the mean of U – V can be expressed in terms of an absolutely convergent integral. The result is that, correct to order c, the mean value of U is U0(1 – 6.55 c), where U0, is the velocity of a single sphere in unbounded fluid. The only assumption made in the calculation is that the centres of spheres in the dispersion take with equal probability all positions such that no two spheres overlap; arguments are given in support of this assumption, which is expected to be valid only when the spheres are identical. Calculations which assume a simple regular arrangement of the spheres or which adopt a cell model of the hydrodynamic interactions give the quite different result that the change in the mean speed of fall is proportional to $c^{\frac{1}{3}}$, for reasons which are made clear.

The general procedure described here is expected to be applicable to other problems concerned with the effect of particle interactions on the average properties of dispersions with small volume fraction of the particles.
Statistical hydromechanics of disperse systems. Part 2. Solution of the kinetic equation for suspended particles
Authors: Buyevich Yu. A.
Published: March 2006

To solve the kinetic equation for particles of a monodisperse two-phase mixture the method of successive approximations is developed; this resembles in its main features the well-known Chapman-Enskog method in the kinetic theory of gases. This method is applicable for a mixture whose state differs slightly from the equilibrium, i.e., when time and space derivatives of the dynamic variables describing the mean flow of both phases of the mixture are sufficiently small. Accordingly, the solution obtained is valid when the time and space scales of the mean flow exceed considerably those for random pseudo-turbulent motion of particles and a fluid. The conservation equations for determination of all the dynamic variables are formulated in approximations which have the same meaning as those of Euler and Navier & Stokes in hydromechanics of one-phase media.
Laminarization of turbulent pipe flow by fluid injection
Authors: Pennell W. T. , Eckert E. R. G. , Sparrow E. M.
Published: March 2006

The effects of fluid injection on the structure of an initially fully developed, low Reynolds number, turbulent pipe flow have been studied by means of a hot-film anemometer. Measurements were made of the axial turbulence intensity field and of the time-mean streamwise velocity distribution, both in the porous-walled pipe and in the solid-walled hydrodynamic development section. Oscilloscope traces showing the timewise pattern of the local velocity fluctuations were also monitored. The Reynolds number of the air flow at the inlet of the porous pipe was varied from 3090 to 6350, and the Reynolds number of the injected air ranged from 60 to 160.

Near the tube wall, the initial effect of injection is a significant reduction of the axial turbulence level and an increase in the thickness of the viscous and buffer layers. The degree by which turbulence is reduced in this region is more or less proportional to the ratio of the injection to entrance Reynolds numbers. In the core region of the flow, which is centred about the tube axis, there is also an initial reduction in the magnitude of the axial component of turbulence which is thought to be due to injection-induced acceleration of the flow. There is also an annular region, which separates the wall and core regions, in which the turbulence intensity initially increases. In the downstream portion of the porous tube the entire flow undergoes a re-transition to fully developed turbulence.
Creeping flow around a deforming sphere
Authors: Lin S. P. , Gautesen A. K.
Published: March 2006

The flow of an incompressible viscous fluid past a deforming sphere is studied for small values of the Reynolds number. The deformation is assumed to be radial but is otherwise quite general. The case of S = O(l), where S is the Strouhal number, is investigated in detail. In particular, the drag is obtained up to O(R2 In R), where R is the Reynolds number.
Asymptotic analysis of turbulent channel and boundary-layer flow
Authors: Bush William B. , Fendell Francis E.
Published: March 2006

Asymptotic expansion techniques are used, in the limit of large Reynolds number, to study the structure of fully turbulent shear layers. The relevant Reynolds number characterizes the ratio of the local turbulent stress to the local laminar stress, so that a relatively thick outer defect layer, in which, to lowest order, there is a balance between turbulent stress and convection of momentum, may be distinguished from a relatively thin wall layer, in which, to lowest order, there is a balance between turbulent and laminar stresses. The two cases examined are channel (or pipe) flow and two-dimensional boundary-layer flow with an applied pressure gradient, upstream of any separation. Attention, for these two cases, is confined to the flow of incompressible constant property fluids. Closure is effected through the introduction of an eddy-viscosity model formulated with sufficient generality for most existing models to be special cases. Results are carried to higher orders of approximation to indicate what properties for the friction velocity, integral thicknesses, and velocity profiles, and what conditions for similarity are implied by current eddy-viscosity closures.
The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles
Authors: Hinch E. J. , Leal L. G.
Published: March 2006

The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate γ/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, γ/D [Lt ] 1, has been known for many years. After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r−3) [Lt ] γ/D, and for an intermediate regime relevant for extreme aspect ratios where 1 [Lt ] γ/D [Lt ] (r3 + r−3). The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r ∼ 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior continuum mechanical and experimental results.
An equatorial boundary layer
Authors: Dowden J. M.
Published: March 2006

The singularity of the Ekman layer at the equator of a rotating gravitating sphere makes it difficult to satisfy a prescribed stress boundary condition at the surface of a layer of liquid on the sphere. The equations of motion are investigated for a homogeneous ocean with vertical and lateral eddy viscosities. The horizontal Coriolis terms are not neglected. A linear equation for the boundary layer is obtained and a solution of the equation for the boundary-layer part of the velocity field is found in closed form. This is valid in a parameter range which includes the previous solutions of Stewartson and Gill as limiting cases.
A rotating spherical liquid drop in an electric field
Authors: Sozou C.
Published: March 2006

It is shown that the equilibrium shape of an incompressible dielectric fluid drop rotating with constant angular velocity in the presence of a uniform external electric field of appropriate magnitude along the axis of rotation is spherical. For an inviscid fluid drop, the stability of this spherical configuration to small deformations is investigated by means of Chandrasekhar's virial method. We find that a rotating drop in the presence of an electric field parallel to the axis of rotation is, in some respects, more stable than when either only the electric field or only rotation is present. This is due to the fact that the application of an electric field parallel to the axis of a rotating drop, or of rotation parallel to an electric field in which a drop is immersed, shifts the instability mechanism to another normal mode.
Turbulent convection in a horizontal layer of water
Authors: Chu T. Y. , Goldstein R. J.
Published: March 2006

Overall heat transfer and mean temperature distribution measurements have been made of turbulent thermal convection in horizontal water layers heated from below. The Nusselt number is found to be proportional to Ra0·278 in the range 2·76 × 105 < Ra < 1·05 × 108. Eight discrete heat flux transitions are found in this Rayleigh number range. An interferometric method is used to measure the mean temperature distribution for Rayleigh numbers between 3·11 × 105 and 1·86 × 107. Direct visual and photographic observations of the fluctuating interferogram patterns show that the main heat transfer mechanism is the release of thermals from the boundary layers. For relatively low Rayleigh numbers (up to 5 × 105) many of the thermals reach the opposite surface and coalesce to form large masses of relatively warm fluid near the cold surface and masses of cold fluid near the warm surface, resulting in a temperature-gradient reversal. With increasing Rayleigh numbers, fewer and fewer thermals reach the opposite bounding surface and the thermals show persistent horizontal movements near the bounding surfaces. The central region of the layer becomes an isothermal core. The mean temperature distributions for the high Rayleigh number range are found to follow a Z−2 power law over a considerable range, where Z is the distance from the bounding surface. A very limited agreement with the theoretically predicted Z−1 power law is also found.
Nonlinear resonant wave motion of a radiating gas
Authors: Eninger James E. , Vincenti Walter G.
Published: March 2006

A method due to Chester is applied to the theoretical study of resonant wave motion of a radiatively active gas. The inviscid non-conducting grey gas is confined between two infinite parallel walls, and a one-dimensional wave motion is driven by a sinusoidally varying input of black-body radiation from one of the walls. For sufficiently weak driving radiation, the motion can be described by the general solution of the classical wave equation (with the functional form of the solution still undetermined) plus a particular solution due to the driving radiation. When the driving is done at or near a resonant frequency, however, nonlinearities and perturbations in spontaneous emission from the gas must be taken into account before application of the boundary conditions. Such application then leads to a nonlinear integral equation governing the undetermined function in the general solution of the wave equation. This equation is solved numerically by the method of parametric differentiation.

In a frequency range around resonance and for a sufficiently weak relative level of spontaneous emission, the nonlinearities give rise to shock waves (numberingNat theNth resonant frequency) that are repeatedly reflected at the walls. The perturbations in spontaneous emission give rise to damping, however, and for sufficiently high levels of emission the shock waves disappear. Specific results for various values of optical thickness and various relative levels of spontaneous emission are presented at frequencies in a range around the first resonance.
An experimental and analytical study of instability of asymmetric jetstream-like currents in a rotating fluid
Authors: Dunst Martin
Published: March 2006

Considering the barotropic instability problem of the mean westerly current in the atmosphere we have performed a series of experiments in a rotating vessel (using water and a barotropic model) to study the behaviour of a zonal asymmetric basic current with respect to small perturbations. In the centre of a rotating cylindrical vessel (of large diameter and rotation rate ω) a smaller cylinder was installed, the rotation of which relative to the vessel, at a rate Δω, generates a nearly two-dimensional field of mean relative motion within a sharply defined region. The dominant zonal velocity component $\overline{v}$ shows monotonic radial decrease within this so-called friction zone. Now what happens if the relative rotation of the inner cylinder, the source of momentum, suddenly vanishes, i.e. δΩ = 0? The main result is that the basic zonal current $\overline{v}$, which now has an asymmetric radial profile ($\overline{v} = 0 $ at the inner cylinder and the outer edge of the friction zone), breaks down into vortices, the number of which, the integer wavenumber n, is a function of the parameter ε = δω/ω alone: n = n(ε); increasing ε eff effects a decrease of n. For a theoretical discussion of the experimental results we assume this to be a problem of barotropic instability and base our analytical considerations on the two-dimensional non-divergent vorticity equation, frictional forces being neglected. By applying a perturbation method and prescribing a realistic asymmetric basic current we can derive the relation ν = {[¾π/ln (ε + 1)½]2 + 1}½, which yields the real wavenumber v as a function of the parameter ε = δω/ω. The analytical results are in good agreement with the experiments.
Mean motions and impulse of a guided internal gravity wave packet
Authors: Mcintyre Michael E.
Published: March 2006

Second-order mean fields of motion and density are calculated for the two-dimensional problem of an internal gravity wave packet (the waves are predominantly of a single frequency o and wavenumberk) propagating as a wave-guide mode in an inviscid, diffusionless Boussinesq fluid of constant buoyancy frequencyN, confined between horizontal boundaries. (The same mathematical analysis applies to the formally identical problem for inertia waves in a homogeneous rotating fluid.)

To leading order the mean motions turn out to be zero outside the wave packet, which consequently possesses a well-defined fluid impulse [Iscr ]. This is directed horizontally, and is given in magnitude and sense by\[ {\cal I} = \alpha{\cal M};\quad\alpha = \frac{2c_{\rm g}(c-c_{\rm g})(c+2c_{\rm g})}{c^3-4c^3_{\rm g}}. \]Here [Mscr ] is the so-called ‘wave momentum’, defined as wave energy divided by horizontal phase velocity c ≡ ω/k, andcg=c(N2–ω2)/N2, the group velocity.

If the wave packet is supposed generated by a horizontally towed obstacle, [Mscr ] appears as the total fluid impulse, but of this a portion [Mscr ]-[Iscr ] in general propagates independently away from the wave packet in the form of long waves. When the wave packet itself is totally reflected by a vertical barrier immersed in the fluid, the time-integrated horizontal force on the barrier equals 2 [Iscr ] (and not 2 [Mscr ] as might have been expected from a naive analogy with the radiation pressure of electromagnetic waves.)
Waves generated in rotating fluids by travelling forcing effects
Authors: Rao G. V. Prabhakara
Published: March 2006

The two-dimensional wave pattern produced in a homogeneous rotating fluid by a forcing effect oscillating with a frequency σ′0 and travelling with a uniform speed U along a line inclined to the axis of rotation at an arbitrary angle α is studied following Lighthill's technique. It is shown how the far field changes with α and σ′0.

For all σ′0 < 2Ω, except for σ′0 = 2Ω sin α (Ω being the angular velocity of the fluid), the forcing effect excites two systems of waves. When σ′0 → 2Ω sin α one of these systems spreads out, influencing the upstream side while the other shrinks in the downstream direction. This upstream influence is to the left or to the right of the line of motion of the forcing effect (the forcing line) according as σ′0 − 2Ω sin α[lg ] 0 and increases as σ′0 − 2Ω sin α decreases. For σ′0 > 2Ω there is only a single system propagating downstream. As α varies these systems undergo a kind of rotation retaining the main features. α ≠ 0 or ½π makes the pattern asymmetric about the forcing line while a non-zero σ′0 splits the steady-case identical wave systems into two, which are otherwise coincident.

When σ′0 = 2Ω sin α the forcing effect excites straight unattenuated waves of fixed frequency travelling both ahead and behind in a ‘column’ parallel to the forcing line and enclosing it. Also there are two other systems, which propagate without penetrating into an upstream wedge. It is shown that this ‘column’ is the counterpart of the ‘Taylor column’.
Pipe flow measurements of turbulence and ambiguity using laser-Doppler velocimetry
Authors: Berman N. S. , Dunning J. W.
Published: March 2006

The laser-Doppler ambiguities predicted by George & Lumley have been verified experimentally for turbulent pipe flows. Experiments were performed at Reynolds numbers from 5000 to 15000 at the centre-line and near the wall. Ambiguity levels were measured from power spectral densities of FM demodulated laser signals and compared with calculations based on the theory. The turbulent spectra for these water flows after accounting for the ambiguity were equivalent to hot-film measurements at similar Reynolds numbers. The feasibility of laser-Doppler measurements very close to the wall in shear flows is demonstrated.
Transient internal waves produced by a moving body in a tank of density-stratified fluid
Authors: Graham E. W.
Published: March 2006

The internal waves produced by a moving body are generally longer in the direction of motion than the corresponding surface waves. This difference is accentuated when the density variation is slight and the body velocity is large in which case a very long towing tank may be required for the simulation of a steady-state condition. The following theoretical study of transient waves is intended as a step in relating test conditions and requisite towing-tank sizes.

A source–sink pair travelling for a finite time is used to represent the restricted motion of a body in a tank. The approximate length and volume of the body are fixed, but its precise shape (somewhat irregular and slightly time dependent) is assumed to be of secondary importance and is not calculated here. The density-stratified fluid is assumed to have a constant Brunt–Väisälä frequency.

A solution in the form of a triple sum over the tank eigenfunctions applies quite generally for the internal wave system (neglecting surface waves and the potential-flow-type solution near the body). Examples covering the large-scale structure of the flow field have been solved for two values of an approximate similarity parameter. The value of the similarity parameter indicates how closely steady-state conditions are approached. The first (larger) value chosen produces a well-defined quasi-steady state near the body with transient fluctuations of the order of ± 10%. The second (smaller) value gives a poorly defined quasisteady state with fluctuations of the order of ±50%. More elaborate studies varying the tank length, width and depth could be made by programming the calculations.

The effect of a collapsing wake has not been considered here, but might possibly be treated by similar methods.
Nearly spherical constant-power detonation waves as driven by focused radiation
Authors: George Y. H. , Moore F. K.
Published: March 2006

An analysis is made of the flow within a three-dimensional explosion, or spark, created in a gas absorbing energy from a steady conical beam of radiation with nearly spherical symmetry. The radiation, typically from an array of lasers with a common focus, is assumed to be very intense, and absorbed immediately behind an outwardly advancing strong shock. The resulting self-similar flow has previously been studied for spherical symmetry; somewhat improved calculations for that case are presented here.

Departures of the laser power from spherical uniformity, which would result from practical problems of arrangement, are conveniently represented by an ascending series of Legendre polynomials in the polar angle. For non-uniformities of small amplitude, first-order perturbations of the flow field are analysed in detail. Self-similarity is shown to be retained, for zero counter-pressure and power constant with time.

For the first five harmonics in power distortion, the resulting fourth-order system of equations is solved numerically for profiles of velocity components, density and pressure, and for shock shape. Results are presented graphically. These solutions are singular near the focus, but are nevertheless fully determined. In the limit of large wavenumber, the core of the flow has vanishing tangential velocity and pressure perturbations, and hence the governing equations are only of second order, except presumably in a boundary layer appearing near the shock.

Study of the nonlinear case of large wavenumber along the axis of symmetry shows that the singularity at the focus reflects the existence of a ‘forbidden zone’ whose extent depends on the degree of asymmetry. It is argued that this zone is one within which diffusional processes must dominate.
On the motion of turbulent thermals
Authors: Escudier M. P. , Maxworthy T.
Published: March 2006

An analysis is presented of the motion of a turbulent thermal in an unstratified environment. Although based upon the entrainment hypothesis introduced by G. I. Taylor (see Morton, Taylor & Turner 1956), the analysis differs from previous work in that it is not limited to small density differences between the thermal and its surroundings. Also, the influence of the virtual mass of the unsteadily moving fluid, ignored by previous investigators, is included and shown to be of significance for any density difference.

Calculations of the temporal variations of size, velocity and density are presented in non-dimensional form for thermals with initial density ratios covering the practically attainable range. It is shown a posteriori that losses of momentum and buoyancy to a wake are probably of negligible influence in any real case.
Characteristics of the wake behind a cascade of airfoils
Authors: Raj R. , Lakshminarayana B.
Published: March 2006

An analytical and experimental investigation of the near and far wake characteristics of a cascade of airfoils is reported in this paper. The measurement of mean velocity, turbulence intensity and Reynolds stress across the wake at several distances downstream of the cascade indicates that the wake is asymmetrical and this asymmetry is maintained even up to 3/4 chord length. Experiments carried out at three incidences reveal that the decay of the wake defect is strongly dependent on the downstream variation of the wake edge velocity. For a cascade, the decay rate of the wake defect is found to be slower than that of a flat plate, cylinder or symmetrical airfoil (at zero incidence). The level of turbulence and Reynolds stresses are found to be high and some comments are made regarding self-preservation and structure of the flow. Semi-theoretical expressions are given for the wake profile, and decay of the velocity defect, turbulence intensity and Reynolds stress.
Supercritical flow in a divergent channel
Authors: Eagles P. M.
Published: March 2006

For flow of a viscous fluid in a divergent channel of small angle it is shown that small disturbances to the basic Jeffery-Hamel flow may grow, according to nonlinear theory, to produce a secondary (supercritical) flow, in which the main flow winds from side to side in the channel and vortices form, with the whole pattern moving slowly downstream.
Boundary-layer separation at a free streamline. Part 3. Axisymmetric flow and the flow downstream of separation
Authors: Ackerberg R. C.
Published: March 2006

The results of Ackerberg (1970, 1971a), on the two-dimensional boundary-layer separation at a sharp trailing edge where a free streamline is attached, are extended to the axisymmetric case, with and without swirl. When the flow has swirl, the boundary-layer swirl velocity close to the wall may be opposite to that in the external flow; this may help explain the ‘bathtub vortex paradox’ observed by Sibulkin (1962). The solutions for the detached shear layers downstream of separation for two-dimensional and axisymmetric flows, with and without swirl, have been obtained. Some misprints in parts 1 and 2 are corrected in the appendix.
Report on the NATO Advanced Study Institute on magnetohydrodynamic phenomena in rotating fluids
Authors: Moffatt H. K.
Published: March 2006

An Advanced Study Institute sponsored by NATO on “Magnetohydrodynamic Phenomena in Rotating Fluids” was held in Cambridge, England, from 26 June to 4 July 1972. The aim of the meeting was to provide, through invited lectures, a systematic account of those aspects of the geophysics of the earth's core and of astrophysics in which both rotation effects and magnetic effects play an important part. In addition to the invited lectures, shorter contributions and discussions were strongly encouraged. The main Sessions and the Session Chairmen were as follows:

A    The earth's magnetism and planetary magnetism magnetism    Sir Edward Bullard

B    Convection, differential rotation, and magneto-hydrodynamics in the sun    Dr N. O. Weiss

C    Stellar and interstellar magnetism; pulsars    Prof. E. Spiegel

D    Boundary layers and detached shear layers and spin-up problems    Prof. E. R. Benton

E    Waves and instabilities influenced by Lorentz and/or Coriolis forces    Dr R. Hide

F    Dynamo theory    Prof. P. H. Roberts

These Chairmen, together with Professor P. A. Gilman and Dr H. K. Moffatt, who acted as Director of the Institute, constituted the Organising Committee. The summary of the meeting which follows has been substantially written by Dr Weiss (sessions B and C), Dr Hide (A and E), Professor Benton (D) and Professor Roberts (F), whose willing collaboration is gratefully acknowledged.

There were just over a hundred participants, the great majority of whom were accommodated at Trinity College, Cambridge, for the duration of the meeting.
Nonlinear wave propagation on an arbitrary steady transonic flow
Authors: Prasad Phoolan
Published: March 2006

Here, we have studied the propagation of an arbitrary disturbance bounded in space on an arbitrary two- or three-dimensional transonic flow. First we have presented a general theory valid for an arbitrary system ofnfirst-order quasi- linear partial differential equations and then used the theory for the special case of gasdynamic equations. If a disturbance is created in the neighbourhood of a sonic point, only a part of the disturbance stays in the transonic region and it is bounded by a wave front perpendicular to the streamlines. This part of the disturbance is governed by a very simple partial differential equation and the problem essentially reduces to the discussion of one-dimensional waves. The disturbance decays in the neighbourhood of the points where the flow acceler- ates from a subsonic state to a supersonic state and it attains a steady state where the flow is decelerating.
Nonlinear gravity–capillary surface waves in a slowly varying current
Authors: Holliday Dennis
Published: March 2006

The propagation of nonlinear gravity–capillary surface waves in a deep slowly varying current is investigated using the conservation equations in the eikonal approximation. Graphical comparisons are made between solutions of the wave- slope and wavenumber equations for infinitesimal waves and finite amplitude waves. Finite amplitude effects are shown to be weaker for small amplitude capillary waves than for gravity waves. The ‘wave barrier’ noted by Gargett & Hughes (1972) for infinitesimal gravity waves on a slowly varying current is seen to be removed by finite amplitude effects.
Wind-induced growth of mechanically generated water waves[dagger]
Authors: Wilson W. Stanley , Banner Michael L. , Flower Ronald J. , Michael Jeffrey A. , Wilson Donald G.
Published: March 2006

An experimental study was conducted to measure the growth rates of mechanically generated surface water waves when subjected to a fully developed turbulent channel airflow. The study was designed to test the accuracy of the growth rates predicted by Miles's (1962b) theory. For a series of wave frequencies (from 2·04 to 6·04 Hz at 0·50 Hz increments) and centre-line wind velocities (0·20, 1·12 and 1·84 m/s) wave amplitudes were measured at three stations (2–21, 3–43 and 4·65 m) downwind from a wave generator. In addition, for centre-line velocities of 1–12 and 1·84 m/s, U* (the velocity at the outer edge of the viscous sublayer) and U1, (the shear velocity) were obtained from measured mean velocity and Reynolds stress profiles. The wave amplitude measurements at the wind velocity of 0·20 m/s provided attenuation rate estimates which agreed reasonably well with theoretical attenuation rates based on viscous effects both on the walls and in the bulk of the water. The amplitude measurements at the wind velocities of 1·12 and 1·84m/s provided growth rate estimates which were compared with theoretical growth rates (computed using the wave frequency, U1 and U* predicted by Miles's (1962b) theory. At 1·12m/s Miles's growth rateswere two to five times larger than those measured; at 1·84 m/s Miles's growth rates were about two times larger.
Steady supercritical Taylor vortex flow
Authors: Burkhalter J. E. , Koschmieder E. L.
Published: March 2006

Experiments studying steady supercritical Taylor vortex flow have been made using pairs of long cylinders with two different radius ratios, three fluids of different viscosities and three different end boundaries for the fluid column. The emphasis in these experiments is on the determination of the wavelength of the Taylor vortices and the size of the end rings. The wavelength which one measures in a finite cylinder differs from the wavelengths found theoretically for infinitely long cylinders. Provided that the end effects were properly taken into account, the wavelength of singly periodic Taylor vortices in aninfinitely long cylinder was found to remain constant between T/Tc = 1 and T/Tc, ≈ 80 in experiments with radius ratios η = 0·505 and η = 0·727. Further studies of Taylor vortex flow at very high Taylor numbers, where the vortices are either doubly periodic or truly turbulent, showed that the wavelength increases under these conditions. However, the observed wavelengths were no longer unique but distributed statistically around a wavelength larger than the critical wavelength.
Hypersonic nozzle flow of air with high initial dissociation levels
Authors: Stalker R. J. , McIntosh M. K.
Published: March 2006

An experimental investigation was undertaken of the flow produced by the inviscid expansion of air through a hypersonic nozzle. Stagnation enthalpy levels up to 4 × 107J/kg were used, with initial dissociation levels approaching 90%. By measuring the flow velocity and the frozen dissociation fractions of oxygen and nitrogen, it was found that existing theoretical models served adequately to define the nozzle flow, at least for the purpose of conducting experiments involving reacting inviscid hypersonic flows about blunt bodies.
The near wall jet of a normally impinging, uniform, axisymmetric, supersonic jet
Authors: Carling J. C. , Hunt B. L.
Published: March 2006

The near wall jet produced by directing a uniform axisymmetric jet of air normally onto a large flat plate has been investigated experimentally and theoretically for four jets in the Mach number range 1·64–2·77. Detailed measurements of the surface pressure and shadowgraph and surface flow pictures are presented. The results show that the mechanism which mainly determines the supersonic near wall jet is the jet-edge expansion and its reflexions from the sonic line and the wall-jet boundaries. The near wall jet is found to consist of an alternating series of expansion and recompression regions whose strengths depend on the jet Mach number and decay with distance. At Mach numbers of 2·4 and above, shock waves are observed in the first recompression region and at a Mach number of 2·77 the boundary layer separates locally. Further out, viscous effects become increasingly important and a constant-pressure shear flow is established at a distance which increases with jet Mach number. The application of the method of characteristics in an approximate manner reproduces a number of the features of the near wall jet which are observed experimentally.

Pressure distributions obtained in the shock layer show that a stagnation bubble can occur and that its occurrence depends on factors such as the flow upstream of the nozzle. The wall-jet region is found to be largely independent of whether or not a bubble occurs in the shock layer.
Electrohydrostatic instability in electrically stressed dielectric fluids. Part 1
Authors: Michael D. H. , Norbury J. , O'Neill M. E.
Published: March 2006

A theoretical investigation is presented of the electrohydrostatic stability of a given volume of incompressible dielectric fluid when stressed by the application of a potential difference between bounding conducting fluids. It is assumed that the dielectric fluid is located in a channel of breadth 2 a and height 2h, with h/a [Lt ] 1, whose walls are semi-infinite solid dielectric sheets of thickness 2h. The dielectric fluid may have a volume which differs from that of the channel, so that the presence of menisci at the interfaces between conducting and non-conducting fluids is taken into account. By a suitable method for approximating the electric stress at the interfaces, the electrostatic potential difference across the dielectric is determined as a function of the pressure difference across the interfaces for prescribed values of the discrepancy of the volume of the dielectric from the volume of the channel per unit length, and criteria are obtained for determining the critical electric field which precipitates the instability of the system. The variation of the critical electric field with the dimensionless volume excess 2δ is also found and it is shown that, for δ < −0·5, instability is associated with a symmetric mode of disturbance in which the critical field occurs at the maximum in a plot of potential difference vs. pressure difference. For δ > −0·5, instability arises from an asymmetric disturbance with the critical field occurring at a bifurcation point in the potential difference/pressure difference plane. Bifurcations are shown to occur only when the equilibrium profiles of the interfaces have extrema at the edges of the channel.
Steady simple shear flow past a circular cylinder at moderate Reynolds numbers: a numerical solution
Authors: Kossack Charles A. , Acrivos Andreas
Published: March 2006

The two-dimensional steady flow of an incompressible viscous fluid past a circular cylinder, placed symmetrically in a simple shear field, has been studied for both the stationary and the freely rotating case by solving numerically the Navier-Stokes equations for values of the Reynolds numberRin the range\[ 0.047 \leqslant R \leqslant 70. \]AtR= 0·047, the present results are in substantial agreement with the analytic small-Rperturbation solution given by Robertson & Acrivos (1970). Inertia effects were found, however, to play a significant role even atR= 1, and hence the calculated flow pattern forR≥ 1 differs significantly from that of the creeping-flow solution. Specifically, for the freely rotating case, the region of closed streamlines decreases rapidly in extent with increasingR, two symmetrically placed wakes are formed on either side of the cylinder and the dimensionless rotational speed of the freely suspended cylinder decreases asR-½. In fact, for a value ofRas low as 70, many of the gross features of the flow are surprisingly similar to those described by an inviscid solution except for the difference in the two sets of pressure profiles and the fact that the numerical results do not as yet exhibit the expected flow separation on the surface of the cylinder.
Optimal swimming of flagellated micro-organisms
Authors: Pironneau O. , Katz D. F.
Published: March 2006

This paper studies flagellar undulations that propel a micro-organism at a given speed while minimizing its expenditure of hydrodynamical energy. The study is in two basic parts. The first part is a qualitative inquiry into the general nature of undulations that are hydrodynamically optimal in the instantaneous sense. The results indicate that an apparent sliding of the entire flagellum along its instantaneous axis is fundamental to such motions, although an additional deformation is necessary to compensate for the presence of the organism's head. Periodic or semi-periodic undulations are clearly suggested, and must consist of travelling waves propagated in the direction opposite to propulsion.

The second part of the paper is a quantitative inquiry as to the values of parameters that optimize given periodic wave shapes in the time-average sense. The trade-off between wave amplitude and the number of wavelengths is of particular interest. Results are obtained for small amplitude sinusoidal waves and finite amplitude sawtooth waves. For the latter, a single wavelength with amplitude roughly one-sixth of the wavelength is optimal. The significance of the twitching movements of the head is investigated. The results are consistent with the qualitative study and emphasize the need to inhibit such motions. The implications of the dependence of resistive-force coefficients upon wave shape are considered, and the physical significance of rotational pitching motions is assessed.
On the Poincaré problem for a compressible medium
Authors: Gans Roger F.
Published: March 2006

By the ‘Poincaré problem’ is meant the determination of the free oscillations of a contained rotating fluid, its velocity being linearized around a state of solid rotation. Compressibility requires one to introduce a basic thermodynamic profile as well as a basic velocity distribution. Here the temperature gradient has been supposed proportional to the adiabatic gradient, by introduction of a proportionality constant α (α = 0 in the isothermal case; α = 1 in the adiabatic case). In this formulation the system is reducible to a single second-order ordinary differential equation and its boundary condition.

It is proved that if α = 1 the oscillation frequencies in the rotating system cannot equal plus or minus twice the rotation frequency. The negative case is pathological in the sense that there are solutions arbitrarily near the forbidden solution, and a solution curve of frequency as a function of rotation rate crosses the forbidden frequency.

The basic system is expanded in terms of a power series in γ − 1, where γ is the ratio of specific heats. The zeroth-order set of equations is solved in terms of confluent hypergeometric functions, and a solvability condition on the first-order set gives frequency shifts as functions of α. Several zeroth-order frequencies have been calculated, together with four first-order frequency shifts.