bluuebunny/crossref_metadata_2025 huggingface dataset:
The stability and transition of a two-dimensional jet
Authors: Sato Hiroshi
Published: March 2006

A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.

The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.

Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.
A shallow-liquid theory in magnetohydrodynamics
Authors: Fraenkel L. E.
Published: March 2006

The non-linear and linear ‘shallow-water’ theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength. These equations have a ‘magneto-hydraulic’ character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical co-ordinate.

Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridional plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating.

The approximate non-linear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the non-linear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance from their ‘undisturbed positions’.
Free convection in the tilted open thermosyphon
Authors: Leslie F. M.
Published: March 2006

Approximate solutions are found for the fluid flow and heat transfer in a heated cylinder, closed at the bottom and opening at the top into a reservoir of cool fluid, which has been tilted at a small angle to the vertical.

One solution is found for large Rayleigh number when the boundary layer does not fill the tube, and another for small Rayleigh number when the boundary layer fills the tube. In both cases tilting causes a small increase in heat transfer which is proportional to the square of (l/a) tan ϕ, where l/a is the length–radius ratio and ϕ the angle of tilt.
The viscous secondary flow ahead of an infinite cylinder in a uniform parallel shear flow
Authors: Toomre Alar
Published: March 2006

A simple method is presented in this paper for calculating

the secondary velocities, and

the lateral displacement of total pressure surfaces (i.e. the ‘displacement effect’)

in the plane of symmetry ahead of an infinitely long cylinder situated normal to a steady, incompressible, slightly viscous shear flow; the cylinder is also perpendicular to the vorticity, which is assumed uniform but small. The method is based on lateral gradients of pressure, these being calculated from the primary flow alone. Profiles of the secondary velocities are obtained at several Reynolds numbers ahead of two specific cylindrical shapes: a circular cylinder, and a flat plate normal to the flow. The displacement effect is derived and, rathe surprisingly, is found to be virtually independent of the Reynolds number.
Interaction of a shock wave with a mixing region
Authors: Riley N.
Published: March 2006

The interaction of a simple wave, in steady supersonic flow, with a two-dimension mixing region is treated by applying Fourier analysis to the linearized equations of motion. From asymptotic forms for the Fourier transforms of physical quantities, for large wave-number, the dominant features of the resulting flow pattern are predicted; in particular it is found that a shock wave, incident on the mixing region, is reflected as a logarithmically infinite ridge of pressure. For two particular Mach-number distributions in the undisturbed flow, numerical solutions are obtained, showing greater detail than the results predicted by the asymptotic approach. A method is given whereby the linear theory may be improved to take into account some non-linear effects; and the reflected wave, for an incident shock wave, is then seen to consist of a shock wave, gradually diminishing in strength, followed by the main expansion wave.
Centrifugal waves
Authors: Phillips O. M.
Published: March 2006

When a hollow circular cylinder with its axis horizontal is partially filled with water and rotated rapidly about its axis, an almost rigid-body motion results with an interior free surface. The emotion is analysed assuming small perturbations to a rigid rotation, and a criterion is found for the stability of the motion. This is confirmed experimentally under varying conditions of water depth and angular velocity of the cylinder. The modes of oscillation (centrifugal waves) of the free surface are examined and a frequency equation deduced. Two particular modes are considered in detail, and satisfactory agreement is found with the frequencies observed.
Measurements of the effect of molecular diffusivity in turbulent diffusion
Authors: Mickelsen William R.
Published: March 2006

Townsend (1954) has shown that turbulent vorticity may rotate and strain a diffusion wake, thereby increasing the contribution of molecular diffusion to the total mean dispersion over short diffusion times. To test whether any such effect occurs at longer diffusion times, the lateral dispersion of both helium and of carbon dioxide in air were measured downstream from a continuous point source in the turbulence produced by a grid in a wind tunnel. The data show that, for long diffusion times, accelerated molecular diffusion is negligible, so that molecular diffusion makes only an independent contribution to the total dispersion.
An empirical torque relation for supercritical flow between rotating cylinders
Authors: Donnelly R. J. , Simon N. J.
Published: March 2006

An empirical relationship for the torque transmitted by fluid friction to an outer cylinder as a function of the angular velocity of the inner cylinder has been obtained by analysis of experimental data published by Wendt, Taylor and Donnelly. With one exception it is found that the torque G has the functional form
$G = a \Omega^{-1}_1 + b\Omega ^{1 \cdot 36}_1,$
where Ω1 is the angular velocity of the inner cylinder and a and b are constants determined from the data. The formula applies to a range of values of Ω1 above the onset of instability extending to about 10 times the critical angular velocity. The experiments also show that the finite amplitude analysis recently advanced by Stuart gives the correct variation of torque over a short range above the critical speed. At speeds well beyond critical it is found that G varies approximately as Ω1.51, and that the variation of torque with gap width can be expressed as a simple power law with exponent about 0.31. In an appendix Dr G. K. Batchelor shows that these latter relations are consistent with the supposition that the flow is steady and consists of inviscid cores surrounded by boundary layers.
A note on fluctuating heat transfer at small Péclet numbers
Authors: Illingworth C. R.
Published: March 2006

The hot-wire anemometer, used for recording speed variations in turbulent flow, involves in its working principle the unsteady heat transfer from a hot fixed surface to a fluctuating air stream moving past the surface. If the wire is maintained at a constant (high) temperature, the rate of loss of heat from the wire changes with the velocity of the incident stream, and the compensating rate of gain of heat, produced by the Joule heating effect of the electric current, changes, correspondingly. The accompanying change of current can be measured, and used to calculate the varying velocity of the air stream. The hot wire may have a diameter as low as 10−4 in. and the Reynolds number of the flow is then of the order of 0.05 for each ft. per sec of velocity. With low velocities, of the order of 10 or 20 ft./sec, the flow past the wire is in the range of small Reynolds number, and the exact equations of flow may be approximated by simpler equations in the manner of Oseen's theory (Lamb 1932). The approximate equations are not easy to solve when the flow is compressible, as it will be in the presence of the large temperature differences imposed by the heat of the wire. If, however, the temperature differences are assumed to be small, the approximate energy equation is no longer linked with the equations of continuity and momentum, and it may be solved without knowledge of the velocity field. The purpose of this note is to give the solution for the temperature field when a warm circular cylinder or a warm sphere is held at rest in a fluctuating stream.
On the motion of thin airfoils in fluids of finite electrical conductivity
Authors: McCune James E.
Published: March 2006

A two-dimensional, small-perturbation theory for the steady motion of thin lifting airfoils in an incompressible conducting fluid, with the uniform applied magnetic field perpendicular to (and in the plane of) the undisturbed, uniform flow field, is described. The conductivity of the fluid is assumed to be such that the magnetic Reynolds number,Rm, of the flow is large but finite. Within this assumption, a theory based on superposition of sinusoidal modes is constructed and applied to some simple thin airfoil problems.

It is shown that with this particular field geometry the Alfvén wave mechanism is important in making possible very deep penetration into the flow field of currents and their associated vorticity. It is also shown that the current penetration for an airfoil is much larger than for a wavy wall of wavelength equal to the airfoil chord.

A value ofRm= 5 is found to be a good approximation to infinity in this study; in fact, use of the present technique for values ofRmof the order of unity is permissible. These results provide an indication of what is meant by ‘large’ magnetic Reynolds number in two-dimensional magneto-aerodynamics.
The instability of uniform viscous flow under rollers and spreaders
Authors: Pearson J. R. A.
Published: March 2006

When a thin film of viscous fluid is produced by passing it through a small gap between a roller or spreader and a flat plate, it often presents a waved, or ribbed, surface. An analysis is given here in terms of lubrication theory to show why in many cases flow leading to a uniform film is unstable. Account is taken of surface tension which proves to be a stabilizing factor. The most unstable values of the wave-number, n (characterizing the disturbance), are calculated as functions of the dimensionless variable T/μU0, and of the geometry of the system; T is the surface tension, μ the viscosity and U0 a representative velocity of the fluid. For the particular case of a spreader in the form of a wide-angled wedge, these predictions are compared with experimental observations. Agreement is obtained for values of T/μU0 between about 10 and 0.1, but for smaller values of T/μU0 it is clear that other considerations, involving only viscous and pressure forces, determine the nature of the secondary flow.
Instability of a viscous liquid of variable density in a vertical Hele-Shaw cell
Authors: Wooding R. A.
Published: March 2006

Approximate equations of motion, continuity and mass transport are given for a viscous liquid of variable density moving very slowly between vertical and impermeable parallel planes. These equations are used to calculate approximate stability criteria when the liquid is at rest under a vertical density gradient. The results are applicable to the problem of the stability of a viscous liquid of variable density to two-dimensional disturbances in a porous medium.

An exact stability analysis for the liquid between parallel planes is also given, and expansions in powers of the disturbance wave-number are obtained for the critical Rayleigh number at neutral stability. The previous approximate results are found to correspond to the leading terms of the series expansions. For the most unstable type of disturbance, the velocity distribution closely resembles plane Poiseuille flow, which was the form assumed in the approximate equations.

An asymptotic expansion is derived for the critical Rayleigh number at neutral stability in a long vertical channel, or duct, the cross-section of which is a thin rectangle. The typical neutral disturbance possesses a ‘boundary layer’ at each end of the cell cross-section, and this has a small stabilizing effect.

The critical Rayleigh number for a long vertical channel of rectangular cross-section is found experimentally by comparing the density gradient of the liquid in the channel at neutral stability with the corresponding density gradient in a vertical capillary tube. There is better agreement with the exact theory than with the approximate theory, the experimental result being about 4% higher than the value predicted by the ‘exact’ asymptotic expression, and about 10% higher than the value predicted by the simple approximate theory.
The influence of hole dimensions on static pressure measurements
Authors: Shaw R.
Published: March 2006

The pressure measured at a static pressure hole differs slightly from the true static pressure, by an amount which depends on the hole size and shape. The present investigation extends the range of previous work to determine the error in static pressure measurement in incompressible turbulent flow. The observed static pressure was always greater than the true static pressure. The results are presented in dimesionless form as a function of the Reynolds number based on hole diameter and friction velocity.
Remarks on transition in a round tube
Authors: Smith A. M. O.
Published: March 2006

This article has a twofold purpose: (1) to analyse the available theoretical and experimental knowledge concerning flow in the inlet region of a smooth round tube, and (2) to point out that the e9 amplification factor method apparently predicts natural transition correctly over a significant fraction of the entire inlet lenght of the tube. The successful prediction indicates, but does not prove, that flow in a smooth round tube becomes turbulent at higher Reynolds numbers because transition occurs in the inlet lenght—not in the fully developed Poiseuille régime. The close agreement between theory and a test result obtained by Pfenninger indicates that the e9 method is valid for a wide variety of flows having x Reynolds numbers of transition ranging from 570,000 to 40 million. The results are applicable to both plane and axially symmetric flows.
The linearized flow of a dissociating gas
Authors: Clarke J. F.
Published: March 2006

The equations for planar two-dimensional steady flow of an ideal dissociating gas are linearized, assuming small disturbances to a free stream in chemical equilibrium.

As an example of their solution, the flow past a sharp corner in a supersonic stream is evaluated and the variations of flow properties in the relaxation zone are found. Numerical illustrations are provided using an ‘oxygen-like’ ideal gas and comparisons made with a characteristics solution. The flow past a sharp corner can be studied in a conventional shock tube and it may be possible to verify the present theory experimentally. In particular it may prove feasible to use the results to obtain a measure of the reaction rates in the gas mixture.
An experimental study of the forces generated by the collapse of transient cavities in water
Authors: Jones I. R. , Edwards D. H.
Published: March 2006

The paper describes an experimental investigation of the pressures developed at the seat of collapse of cavities in water. Single transient cavities, generated by a spark discharge, are allowed to collapse on the end of a piezoelectric pressure-bar gauge which measures the variation of thrust with time. It is shown that both the peak force and duration of the cavity collapse pulse are functions of the cavity lifetime. From an estimate of the minimum radius attained by the cavity and the peak force, the peak pressure on collapse is found to be at least 10,000 atm. Streak schlieren photographs of the collapse process show that a shock wave is radiated into the water at the moment of collapse and that the cavity rebounds. At the collapse of the rebound cavities the pressures developed are comparable with those developed by the collapse of the initial cavity, and these probably contribute materially to cavitational surface damage.
Curvature of attached shock waves in steady axially symmetric flow
Authors: Bianco E. , Cabannes H. , Kuntzmann J.
Published: March 2006

An electronic computer has been employed to calculate the ratio between the initial radii of curvature of the attached shock wave and the body for an axially symmetrical body in a uniform supersonic stream. The results are obtained with 4 exact digits for more than 200 cases. They extend results obtained previously (Cabannes 1951) by means of numerical integration.
The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk
Authors: Rogers M. H. , Lance G. N.
Published: March 2006

The flow produced by an infinite rotating disk when the fluid at infinity is in a state of solid rotation is investigated numerically. When the fluid at infinity is rotating in the same sense as the disk, physically acceptable solutions exist in all cases. When the fluid at infinity is rotating in the opposite sense to that of the disk, the only physically acceptable solutions appear to be those in which there is a uniform suction present acting through the disk.
Dynamics of a dissociating gas Part 2. Quasi-equilibrium transfer theory
Authors: Lighthill M. J.
Published: March 2006

The introduction to Part 1 (Lighthill 1957) proposed a paper in three parts, in the nature of a broad introductory discussion of the dynamics of a pure dissociating gas, which might act as a source of ideas for later, more comprehensive investigations. Part 1 treated ‘equilibrium flow’, Part 2 was to treat transport properties under ‘quasi-equilibrium’ conditions, and Part 3 flows in which large departures from equilibrium occur. When Part 1 was submitted for publication, much of Part 2 had already been written. At this moment, however, the author suddenly became obliged to abandon the project for two years, in order that his contributions to two collective works, Laminar Boundary Layers and Incompressible Aerodynamics, could be produced in time to avoid serious delay in the appearance of the contributions of other authors.
On the tendency towards hexagonal cells in steady convection
Authors: Palm Enok
Published: March 2006

This paper attempts to explain theoretically the observed results that (1) the cells in steady convection approach a hexagonal form, and (2) the occurrence of ascent or descent in the middle of the cell depends on how the kinematical viscosity varies with temperature. The theory is based on non-linear equations and, of course, a variable coefficient of viscosity.

It is found that, due to the variation of viscosity with temperature, the non-linear terms contain a second-order term which is destabilizing. This second-order term regulates the development and leads to a final motion composed of regular hexagons with ascent or descent in the middle of the cell according as the viscosity decreases or increases with temperature.

The influence of a variable viscosity on Rayleigh's result concerning the initiation of convection is obtained as a by-product.
An improved perturbation theory for shock waves propagating through non-uniform regions
Authors: Friedman M. P.
Published: March 2006

This paper considers the problem of the propagation of a shock wave down a non-uniform tube. Linearized solutions to the problem (Chester 1954) do not hold when the velocity behind the shock is near or at the sonic speed. By retaining appropriate non-linear terms of the flow equations, a solution is obtained which holds for all conditions behind the shock, and reduces to the linearized solution for conditions away from sonic.

The behaviour of supersonic or subsonic flow entering regions of expanding or contracting area changes is discussed. It is found that subsidiary shocks may be formed; these can be located and described using the present solution. Explicit solutions are given for the cases of supersonic or subsonic flow entering a region of linearly expanding or contracting area. The point of shock formation as well as the path of the subsidiary shock is obtained for the case in which the area contracts.
Laminar convection in uniformly heated vertical pipes
Authors: Morton B. R.
Published: March 2006

An exact solution is presented in this paper for the problem of laminar convective flow under a pressure gradient along a vertical pipe, the walls of which are heated or cooled uniformly; the solution is based on the assumption that velocity and buoyancy profiles far from the pipe entrance do not change with height, and entry-lengt effects are ignored. Two different types of behaviour are found accordingly as the pressure gradient and buoyancy forces act together or in opposition near the centre of the pipe.

When an upflow is heated (or a downflow cooled) the velocity near the walls is increased relatively and that near the axis decreased until, for sufficiently large Rayleigh numbers, definite velocity and thermal boundary layers are formed.

In the case of cooled upflow (or heated downflow) there is an increase in the velocity across the whole profile for small Rayleigh numbers. As the Rayleigh number is increased the velocity and buoyancy increase, slowly at first and then rapidly, and the solution ‘runs away’ at a Rayleigh number of about 33. For higher Rayleigh numbers, laminar Poiseuille flow of an increasingly complicated profile is theoretically possible, but is unlikely to be found in practise.
On the collapse of an empty cavity in water
Authors: Hunter C.
Published: March 2006

This paper is concerned with the spherically symmetric collapse of an empty cavity in water. The effects of viscosity and surface tension are neglected, but the compressibility of the water is allowed for and a suitable equation of state for the water assumed. The object of this is to clarify the effect of compressibility on the flow by considering it in isolation and thus to describe the formation of a pressure pulse by the collapse and its subsequent propagation.

The exact flow equations are integrated numerically and it is found that very high flow speeds develop in the neighbourhood of the collapse point. The radius of the cavity is found to be proportional to (-t)nfor smallt, wheret= 0 is the instant of collapse, andnis some positive number less than unity.

The flow in the neighbourhood of the collapse point can be described by means of a similarity solution, and the value ofnis determined by regularity properties of the similarity solution (the value ofndepends on the equation of state assumed for the water). The similarity theory, which is valid only at very high pressures and velocities, can be continued beyond the instant of collapse to describe the formation of a shock wave after the collapse is completed, and the initial propagation of this shock.
On the effect of the molecular diffusivity in turbulent diffusion
Authors: Saffman P. G.
Published: March 2006

It is shown that the dispersion of a substance, with molecular diffusivity κ, in a stationary, homogeneous, turbulent velocity field can be formulated in terms of a ‘substance auto-correlation function’, this being a generalization of the well-known Lagrangian correlation between the velocity of a fluid particle at different times. It is found that the interaction between the molecular diffusion and the turbulent motion reduces the dispersion from the value it would have if the processes of molecular and turbulent diffusion were independent and additive. The conflict, between the results obtained in this paper and previous results which implied that the interaction increases the dispersion, is resolved. The ratio, of the contributions to the dispersion from the interaction term and the turbulent diffusion term, is obtained for comparatively large times by the use of intuitive arguments, and is found to be inversely proportional to the Prandtl number and a Reynolds number of the turbulence.
The compressible viscous heat-conducting vortex
Authors: Mack Leslie M.
Published: March 2006

The plane, steady, laminar vortex flow of a viscous, heat-conducting perfect gas is treated. Simple relations are obtained for the flow quantities in the irrotational vortex, for arbitrary Prandtl numbers. When the Prandtl number is ½, the irrotational vortex is also isentropic. When the temperature dependence of the viscosity coefficient is taken into account, the vortex flow is rotational. An exact solution for the rotational vortex is obtained which is suitable for numerical evaluation by successive approximations. Distributions of velocity, temperature, pressure, density, and stagnation temperature through the rotational vortex are given for a typical case.
Mass transport in the boundary layer at a free oscillating surface
Authors: Longuet-Higgins M. S.
Published: March 2006

In a previous paper (1953b) it was shown theoretically that just below the boundary layer at the surface of a free wave the mass-transport gradient should be exactly twice that given by Stokes's irrotational theory. The present paper describes careful experiments which confirm the higher value of the gradient.

The results have an implication for any oscillatory boundary layer at the free surface of a fluid; such a boundary layer must generate a second-order mean vorticity which diffuses inwards into the interior of the fluid.
The slow motion of a magnetized sphere in a conducting medium
Authors: Barthel James R. , Lykoudis Paul S.
Published: March 2006

This paper considers the slow motion of a sphere, permanently and uniformly magnetized in one direction, in a viscous electrically conducting medium. The line of the magnetic poles is assumed to be parallel to the direction of the motion of the sphere. The velocity and pressure fields are calculated by two iterations. The distortion of the magnetic field is also calculated. An expression is obtained for total drag due to the viscous, pressure and magnetic forces.
On the disintegration of water drops in an air stream
Authors: Dodd K. N.
Published: March 2006

A Theory is developed, based on the very limited available experimental evidence, to predict the distortion and disintegration of a water drop when it is exposed to a stream of air with continuously increasing relative velocity. The theory is applied to water drops situated in the path of a solid sphere moving through the air.
Deposition of a viscous fluid on a plane surface
Authors: Taylor G. I.
Published: March 2006

Two mechanisms by which a viscous fluid can be deposited on a plane surface are described. Measurement of the thickness of the deposit are compared with calculated values. It is found that the two agree within rather wide limits of experimental error provided the effect of surface tension can be neglected, and the conditions under which this is legitimate are discussed.
Experiments on the effect of suction on the flow due to a rotating disk
Authors: Gregory N. , Walker W. S.
Published: March 2006

The stability and transition Reynolds numbers for the flow due to a disk rotating in still air were increased by low rates of suction through either a woven wire-cloth or a slitted surface. Observations on the slitted disk at rotational speeds between 550 and 1250 r.p.m. showed that the critical value of the Reynolds number r2ω/ν for instability increased from about 135,000 without suction to nearly 250,000 for a value of the suction parameter a of 0·4. The corresponding values for transition increased from about 275,000 to about 400,000. A given increase in stability Reynolds number required about 75% more suction than that theoretically predicted for uniform distributed suction, a satisfactory result in view of the limitations of the apparatus.

At higher rates of suction (0·4 < a < 1·6), the reduction in secondary flow allowed transverse turbulent contamination to spread inwards from the rim. Consequently the vortices associated with secondary-flow instability were not found, though disturbances of larger wavelength appeared. Intermittent turbulent flow was spread over a much larger region of the disk and no laminar flow could be obtained above a Reynolds number of 400,000, Owing to this feature of the flow, it was not possible to extend laminar flow to values of the unit Reynolds number (ratio of stream velocity to kinematic viscosity) corresponding to flight conditions on a swept-back wing. It is concluded that the rotating disk is not a satisfactory tool for the investigation of the effects of suction on secondary-flow instabilities such as arise in the case of a swept-back wing, or for the testing of suction surfaces.
An experimental investigation of turbulent spots and breakdown to turbulence
Authors: Elder J. W.
Published: March 2006

The theory of hydrodynamic stability and the impact on it of recent work with turbulent spots is discussed. Emmons's (1951) assumptions about the growth and interaction of turbulent spots are found experimentally to be substantially correct. In particular it is shown that the region of turbulent flow on a flat plate is simply the sum of the areas that would be obtained if all spots grew independently.

An investigation of the conditions required for breakdown to turbulence near a wall, that is, to initiate a turbulent spot, suggests that regardless of how disturbances are generated in a laminar boundary layer and independent of both the Reynolds number and the spatial extent of the disturbances, breakdown to turbulence occurs by the initiation of a turbulent spot at all points at which the velocity fluctuation exceeds a critical intensity. Over most of the layer this intensity is about 0·2 times the free-stream velocity. The Reynolds number is important merely in respect of the growth of disturbances prior to breakdown.
On the movement of ships in restricted waterways
Authors: Constantine T.
Published: March 2006

In this paper some simple theoretical considerations concerning the movement of ships in restricted waterways are discussed, and it is shown that for a ship towed from the bank, or by any external force, there are three distinct speed ranges: subcritical, critical and supercritical. In the subcritical and supercritical ranges, Bernoulli's equation and the continuity equation are satisfied everywhere by a state of steady motion relative to the ship, but in the critical range these laws require that a quantity of fluid is piled up continuously ahead of the ship in the form of a bore. Experimental confirmation is given by means of photographs of model tests.
Stability and transition of a supersonic laminar boundary layer on an insulated flat plate
Authors: Laufer John , Vrebalovich Thomas
Published: March 2006

Self-excited oscillations have been discovered experimentally in a supersonic laminar boundary layer along a flat plate. By the use of appropriate measuring techniques, the damping and amplification of the oscillations are studied and the stability limits determined at free-stream Mach numbers 1·6 and 2·2. The wave-like nature of the oscillations is demonstrated and their wave velocities are measured using a specially designed ‘disturbance generator’. It is shown empirically that the stability limits expressed in terms of the boundary-layer-thickness Reynolds number are independent of the Mach number and dependent only on the oscillation frequency. The main effect of compressibility is an increase in wave velocity with Mach number. This has the consequence that the disturbances, although possessing the same dimensionless amplification coefficient as in the incompressible case, have less time (per unit distance) to grow in amplitude. Thus, the adiabatic compressible boundary layer is shown to be more stable than the incompressible one. In general, the experiments confirm the basic assumptions and predictions of the existing stability theory and also suggest the desirability of improvement in the theory in certain phases of the problem. Finally, on the basis of these results a rough estimate of the transition Reynolds number is made in the compressible flow range.
On unsteady laminar boundary layers
Authors: Hassan H. A.
Published: March 2006

A transformation is introduced which, for a class of outer pressure distributions, reduces the unsteady incompressible laminar boundary-layer equations in two dimensions to an equation in which the time does not appear explicitly. A formally exact solution of the resulting equation is then presented in the form of a series and it is shown that the solution can be expressed in terms of universal functions.
Note on the swimming of slender fish
Authors: Lighthill M. J.
Published: March 2006

The paper seeks to determine what transverse oscillatory movements a slender fish can make which will give it a high Froude propulsive efficiency,
$\frac{\hbox{(forward velocity)} \times \hbox{(thrust available to overcome frictional drag)}} {\hbox {(work done to produce both thrust and vortex wake)}}.$
The recommended procedure is for the fish to pass a wave down its body at a speed of around $\frac {5} {4}$ of the desired swimming speed, the amplitude increasing from zero over the front portion to a maximum at the tail, whose span should exceed a certain critical value, and the waveform including both a positive and a negative phase so that angular recoil is minimized. The Appendix gives a review of slender-body theory for deformable bodies.
A note on group velocity
Authors: Whitham G. B.
Published: March 2006

The kinematic approach to group velocity given in Lighthill & Whitham (1955) for one-dimensional waves is extended to cover the general three-dimensional case. The ideas have particular bearing on the theory developed by Ursell (1960) for treating steady wave patterns on non-uniform steady fluid flows.

Although this note was written in ignorance of the fact, all the main ideas presented here are implicit in §§66 and 67 of the book by Landau & Lifshitz (1959). However, these ideas do not seem to be well known to fluid dynamicists, and it was suggested to the author by the editor that a useful purpose would still be served by publishing this note as an expository article amplifying the paragraphs in Landau & Lifshitz. It also serves the original purpose of providing a supplement to Ursell's paper.
On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow
Authors: Stuart J. T.
Published: March 2006

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.

It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.
On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow
Authors: Watson J.
Published: March 2006

In Part 1 by Stuart (1960), a study was made of the growth of an unstable infinitesimal disturbance, or the decay of a finite disturbance through a stable infinitesimal disturbance to zero, in plane Poiseuille flow, and that paper gave the most important terms in a solution of the equations of motion. The greater part of the present paper is concerned with a re-formulation of this problem which readily yields the complete solution. By the same method a solution for Couette flow is obtained. This solution is only a formal one for the present because the conditions imposed in deriving the solution may not be valid for Couette flow; this flow is believed to be stable to infinitesimal disturbances of the type considered.
An experimental study of the explosion generated by a pressurized sphere
Authors: Boyer D. W.
Published: March 2006

An experimental investigation of the explosions of 2 in. diameter glass spheres under high internal pressure has been made. The spheres were initially filled with air or helium at 400 and 326 p.s.i., respectively, and were exploded in air at atmospheric pressure. Experiments on the simulation of high-altitude explosions are also described. Schlieren and spark shadowgraph records of explosion phenomena, and pressure records of the reflexion of the spherical shock wave at various radii, are presented.

An account of some initial experiments on the implosion of 5 in. diameter glass spheres is given. The results were not very satisfactory because of the failure of the spheres to shatter in a desirable manner while under an external pressure of 65 p.s.i.

Numerical solutions to the air and helium sphere explosions are described and the experimental wave phenomena are shown to be in good quantitative agreement with the theoretical predictions, in that they exhibit all the main features that were predicted and are modified only by the physical limitations of the glass diaphragm. A formation process is associated with the spherical shock waves in practice, resulting in initial shock velocities which are lower than the theoretical values.
The influence of radiative transfer on the propagation of a temperature wave in a stratified diffusing atmosphere
Authors: Goody R. M.
Published: March 2006

An approximate solution is presented to the problem of the propagation of a temperature wave through a stratified medium which both diffuses and radiates heat. The solution is a combination of two waves whose relative amplitudes vary with the distance from the lower boundary. Apparent diffusivities computed from phase lag and attenuation coefficients can differ greatly from each other and can vary with height even if the actual diffusivity does not.

An example, using parameters simulating the earth's atmosphere, suggests that analysis of the propagation of the diurnal wave upward from the earth's surface is likely to be inadequate if radiative effects are not considered.
On longitudinal motion in a magnetic field
Authors: Greenspan H. P.
Published: March 2006

An exact solution is presented of the equations and boundary conditions governing the steady longitudinal motion of a semi-infinite non-conducting plate in an oblique magnetic field. The discussion covers the distortion of the boundary layer, the structure of the induced electric and magnetic fields, the current-density distribution, and the behaviour of the fields near the edge of the plate.
Stokes flow of a conducting fluid past an axially symmetric body in the presence of a uniform magnetic field
Authors: Chang I-Dee
Published: March 2006

Low Reynolds number flow of an incompressible fluid past an axially symmetric body in the presence of a uniform magnetic field is studied using a perturbation method. It is found that for small Hartmann number M an approximate drag formula is given by
$ D^ \prime = D^\prime_0 \left(1 + \frac {D^\prime_0} {16\pi \rho vaU}M\right) + O(M^2),$
where D′0 is the Stokes drag for flow with no magnetic effect.
The flow due to a two-dimensional jet issuing parallel to a flat plate
Authors: Sawyer R. A.
Published: March 2006

An experiment has been performed to determine the cavity pressure and cavity length for the flow under a curved two-dimensional turbulent jet discharging parallel to a flat plate at some distance from the surface. The measurements agree quite closely with the predictions of a simple analysis of the flow when a value is assigned to the spread parameter for the curved jet which is somewhat different from that for the plane jet.

Detailed pressure and velocity measurements show that the rate of growth of the curved jet is in fact closely similar to that of the plane jet. This suggests that the adjustment of the spread parameter should be regarded simply as a convenient device for taking into account the quantitative errors of the assumptions made in the analysis.
The method of images in shear flow
Authors: Glauert M. B.
Published: March 2006

The method of images is used to determine the flow perturbation caused by a small point source of fluid in a stream containing a vortex sheet, a number of vortex sheets, or a continous two-dimensional distribution of vorticity. Calculations are carried out for a variety of such shear flows, to illustrate the range of applicability of the method.

The image source distribution provides an alternative approach to that given by Lighthill for calculating small disturbances to a parallel shear flow, and suggests a simple physical picture by which to interpret the complicated effects associated with the distortion and stretching of vortex lines.
Third-order blast wave theory and its application to hypersonic flow past blunt-nosed cylinders
Authors: Swigart R. J.
Published: March 2006

The inviscid flow behind a cylindrical blast wave and its analogy with hypersonic flow past blunt-nosed cylinders is considered. Sakurai (1953, 1954) obtained a solution for the flow field behind a propagating blast wave by expanding the flow variables in power series of 1/M2, where M is the blast wave Mach number, and determining the coefficients of the first two terms in the series. Here the work is extended to include third-order terms. Third-order theory is shown to improve the prediction of shock wave shapes and surface pressure distribution on hemisphere-cylinder configurations at M∞ = 7·7 and 17·18.
The stability of a viscous fluid between rotating cylinders with an axial flow
Authors: Diprima R. C.
Published: March 2006

The stability of a viscous fluid between two concentric rotating cylinders with an axial flow is investigated. It is assumed that the cylinders are rotating in the same direction and that the spacing between the cylinders is small. The critical Taylor number is computed for small Reynolds number associated with the axial flow. It is found that the critical Taylor number increases with increasing Reynolds number.
Behaviour of small regions of different gases carried in accelerated gas flows
Authors: Rudinger George , Somers Lowell M.
Published: March 2006

Small regions in a flow where the density is different from that of the surrounding gas do not exactly follow accelerated motions of the latter, but move faster or slower depending on whether their density is smaller or larger than that of the main flow. This behaviour cannot be quantitatively explained by treating a gas ‘bubble’ as a hypothetical solid particle of the same density, because a gas bubble cannot move relative to the surrounding gas without being transformed into a vortex which absorbs part of the energy of the relative motion.

To illustrate the acceleration effect, the flow velocity behind known pressure waves in a shock tube is compared with the observed velocity of a bubble produced by a spark discharge. The displacement of such a bubble by a wave exceeds that of a flow element by more than 20%, but the bubble density is not known. If the spark discharge is replaced by a small jet of another gas, a pressure wave cuts off a section of this jet which then represents a bubble of known density.

A theory is developed which permits computing the response of such bubbles to accelerations. The ratio of the bubble velocity to the velocity of the surrounding gas depends on the density ratio for the two gases and on the shape of the bubble, but not on the acceleration. Experimental results with H2, He, and SF6bubbles in air, accelerated by shock waves of various strength, are presented and agree well with the theoretical predictions. The results apply regardless of whether accelerations are produced by pressure waves in a non-steady flow or by curvature of streamlines in a steady flow. Various aspects of the experimental observations are discussed.
Taylor instability of finite surface waves
Authors: Emmons H. W. , Chang C. T. , Watson B. C.
Published: March 2006

The instability of the accelerated interface between a liquid (methanol or carbon tetrachloride) and air has been investigated experimentally for approximate sinusoidal disturbances of wave-number range from well below to well above the cut-off. The growth rates are measured and compared with theoretical results. A third-order theory shows the phenomena of overstability which is found in the experimental results. Some measurements of later stages of growth agree moderately well with the available theory and disclose some additional phenomena of bubble competition, Helmholtz instability with transition to turbulence, and jet instability with production of drops.
Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries
Authors: Saffman P. G.
Published: March 2006

This paper is concerned with the dispersion of a material quantity in the steady flow of a viscous fluid through a random network of capillaries (which is a useful model of a porous medium), for the case in which molecular diffusion and macroscopic mixing, due to the randomness of the streamlines, are both important. A Lagrangian correlation function is introduced and the longitudinal and lateral effective diffusivities are thereby calculated for all values ofUl/κ less than some large value. Here,ldenotes the length of a capillary,Uthe mean velocity of the fluid, and κ the molecular diffusivity of the material quantity. The theory is compared with experimental observations of dispersion in flow through granular beds.