bluuebunny/crossref_metadata_2025 huggingface dataset:
Line source distributions and slender-body theory
Authors: Moran John P.
Published: March 2006

A systematic procedure is presented for the determination of uniformly valid successive approximations to the axisymmetric incompressible potential flow about elongated bodies of revolution meeting certain shape requirements. The presence of external disturbances moving with respect to the body under study is admitted. The accuracy of the procedure and its extension beyond the scope of the present study—e.g. to problems in plane flow - are discussed.
Two-dimensional reattaching jet flows including the effects of curvature on entrainment
Authors: Sawyer R. A.
Published: March 2006

The analysis given previously for predicting the average pressure and length of the region of recirculating flow enclosed by a low-speed turbulent jet, issuing parallel to a flat plate, has been modified to take into account the different rates of entrainment by the two edges of the curved jet, the initial mixing region and the pressure forces near reattachment. There is improved correlation between theory and experiment. The analysis has been applied to the flow due to a jet emerging at an angle to a flat plate, and gives good prediction of the length and average pressure of the recirculation region for a particular value of an entrainment-ratio parameter.

Curvature has a considerable effect on the rates of entrainment, but a first-order mixing-length theory indicates that this need not necessarily be accompanied by a marked deviation in jet velocity profile from that of a plane jet.
Non-equilibrium flow inside a wavy cylinder
Authors: Ryhming Inge L.
Published: March 2006

The small-perturbation theory for steady, inviscid, non-equilibrium flow is extended to include the transonic speed range. The resulting transonic small-perturbation equation for the velocity potential is solved for flow inside a wavy cylinder. It is shown that this solution gives a representation of transition through sonic speed at the narrowest sections of the cylinder. This extension of the theory is applied to non-equilibrium nozzle flow.
The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers
Authors: Brenner H. , Cox R. G.
Published: March 2006

Assuming that the Stokes flow past an arbitrary particle in a uniform stream is known for any three non-coplanar directions of flow, then the force on the body to O(R), for any direction of flow, is given explicitly in terms of these Stokes velocity fields. The Reynolds number (R) based on the maximum particle dimension is assumed small. For bodies with certain types of symmetry it suffices merely to know the Stokes resistance tensor for the body in order to calculate this force. In this case the resulting formula is identical to that of Brenner (1961) and Chester (1962). However, for bodies devoid of such symmetry, their formula is incomplete—there being an additional force at right angles to the uniform stream which remains invariant under a reversal of the flow at infinity. As this additional force is a lift force, it follows that the Brenner-Chester formula furnishes the correct drag on a body of arbitrary shape; moreover, this drag is always reversed to at least O(R) by a reversal of the uniform flow at infinity.

Exactly analogous formulae are derived using the classical Oseen equations, and it is shown that although this gives both the correct vector force on bodies with the above types of symmetry and the correct drag on bodies of arbitrary shape, it gives in general an incorrect lift component for completely arbitrary particles.

Finally, the singular perturbation result for the force on an arbitrary body is extended to terms of O(R2log R). This higher-order contribution to the force is given explicitly in terms of the Stokes resistance tensor, and has the property of being reversed by a reversal of the flow at infinity, regardless of the geometry of the body.

These results are collected in the Summary at the end of the paper.
Calculation of potential flow with separation in a right-angled elbow with unequal branches
Authors: Lichtarowicz A. , Markland E.
Published: March 2006

A method developed by Roshko (1953a, b) has been applied to the case of flow round a right-angled elbow, in which the fluid separates at the inner corner at a speed which may be higher than the velocity of the separated flow far downstream. Numerical results have been evaluated and are presented for values of the ratio of the channel widths, downstream to upstream of the elbow, equal to 0·1 and 1·0.

The elbow may be regarded as one half of a two-dimensional representation of an axisymmetric plate valve, on which experiments indicate considerable variation in performance, the discharge coefficient being found to depend to a marked extent on the length of the valve land.
The pitching motion of a circular disk
Authors: Kim W. D.
Published: March 2006

The interaction of a pitching circular disk with the motion induced by the disk in the surrounding fluid is investigated in this paper. MacCamy's (1961) method of simplifying the three-dimensional problem of a circular disk to the two-dimensional problem is found to apply in the present analysis. The integral equation is solved numerically to determine the dependence of pressure, added moment of inertia, and damping coefficient on the frequency of the oscillation.
The stability of plane Poiseuille flow of a dusty gas
Authors: Michael D. H.
Published: March 2006

In this paper some approximate results are presented for the problem of the stability of plane Poiseuille flow of a dusty gas, following a formulation given recently by Saffman (1962). It is assumed that the mass concentration of the dust, f, is small, and results are obtained by making a perturbation of the curve of neutral stability for a clean gas, using the approximate solutions given by Stuart (1954). The perturbation equation is expressed in terms of integrals by introducing the adjoint wave function, the calculation of which is described. The integral coefficients were evaluated by numerical integration using a Mercury computer, and the results are illustrated for f = 0.05 by a set of perturbed neutral stability curves at different values of the time relaxation parameter SR varying from 0 to 500. These results, whilst not of great numerical accuracy, are sufficient to show qualitatively how the curve of neutral stability is modified by the presence of the dust.
The stability of elastico-viscous flow between rotating cylinders. Part 1
Authors: Thomas R. H. , Walters K.
Published: March 2006

Consideration is given to the flow of an idealized elastico-viscous liquid contained in the narrow channel between two concentric cylinders, the motion being due to the relative rotation of the cylinders. It is shown that the presence of elasticity in the liquid lowers the value of the critical Taylor number at which instability occurs. The secondary motion arising at the onset of instability has the usual cellular pattern, the cell length being decreased by the presence of elasticity in the liquid.
Viscous flow through a grating or lattice of cylinders
Authors: Keller Joseph B.
Published: March 2006

Viscous flow perpendicular to a line (or ‘grating’) of evenly spaced identical cylinders is considered in the case when the spacing between the cylinders is much smaller than their cross-sectional dimensions. Lubrication theory is used to find the pressure drop across the grating and hence the force on each cylinder. A square array (or ‘lattice’) of closely packed cylinders is similarly treated.
Analysis of a rotating flow at small Reynolds number
Authors: Bentwich Michael
Published: March 2006

A solution of the Navier-Stokes equations is obtained for the flow resulting from the steady rotation of a semi-infinite right circular (solid) cylinder about its vertical axis. Incompressible viscous fluid is assumed to fill the space outside the cylinder on one side of a horizontal solid plane. In the proposed method of solution the pertinent physical quantities are expressed as series in positive powers of the Reynolds numberRewith space-dependent coefficients. It is shown that the coefficients of (Re)Mcan be obtained by solving linear partial differential equations which depend on the coefficients of (Re)i, wherei<M. A truncated solution, which holds for smallRe, is obtained by solving for the first two coefficients. These results indicate that at the flat end of the cylinder the pressure distribution is nearly constant, yet along the adjacent bounding plane it rises with the radial direction.
The reflexion and diffraction of shock waves
Authors: Pack D. C.
Published: March 2006

When I was first asked to give a general lecture to the 5th British Theoretical Mechanics Colloquium (held at Liverpool University, 2–5 April 1963), I put up the title ‘Shock Waves’, thinking that I might run over a wide field of present-day research, pointing out some of the unanswered problems. In the intervening months, however, I came across the new book by Dr Bradley (1962) of Liverpool University,The Physics and Chemistry of Shock Waves, and as recently as last November there appeared a summarizing article by Pain bt Rogers (1962) of London University inReports on Progress in Physics. The first of these deals in great detail with the modern physical and chemical aspects of the subject—with real gas effects and experimental techniques-and the second summarizes the general classical properties and gives an extended account of recent work, for example, on real gas effects and on magnetohydrodynamics. I also bore in mind that magnetogasdynamics received a majestic treatment at the 4th Colloquium from Dr Shercliff. In the end, then, I decided to confine my remarks to one particular problem, namely the reflexion and diffraction of shock waves, and to concentrate mainly on developments during the past ten years. This paper is the text of the lecture.
A steady-state technique for local heat-transfer measurement and its application to the flat plate
Authors: Ginoux J. J.
Published: March 2006

A steady-state technique of heat-transfer measurement has been developed based on the method of Seban, Emery & Levy (1959) whereby energy is dissipated by the Joule effect in a thin metal sheet on the surface of a model. For the present application, use was made of very thin but mechanically resistant films of metal of very nearly constant thickness, obtained by a simple mirror-silvering technique. The present investigation was prompted by the desire to make very local measurements of heat transfer for application in regions where large variations in convective heat flux and therefore in temperature could be expected.

Comparison between theory and experiment has been made in the simple case of a flat plate with constant heat flux for which a rigorous computation could be made based on the theory of Chapman & Rubesin (1949). The model was so conceived that the heat losses were small enough to be neglected. Therefore no corrections, which are often inaccurate, were needed for the experimental results, contrary to what is generally done when using other techniques for heat-transfer measurements. The excellent agreement between theory and experiment gives complete confidence in the method. The theoretical analysis showed that the measurements are simply related to the results that could be obtained in the case of an isothermal surface, because of the constant ratio that exists between the corresponding heat-transfer coefficients.
Mixing-layer flows in a saturated porous medium
Authors: Wooding R. A.
Published: March 2006

The steady two-dimensional seepage flow by gravitational convection, of a fluid of density ρ1surrounded by a fluid of density ρ0(≠ρ1) at rest, leads to a potential problem from which the shape of the interface can be determined. When the two fluids are slightly miscible the interface is replaced by a mixing layer, and it is shown that the first-order Prandtl equations for the flow in the layer possess an exact similarity solution. The profile across the layer is of the same form as the profile of the laminar incompressible half-jet with one fluid at rest, and a formula is obtained for the scale of mixing-layer thickness as a function of distance downstream. Three examples are discussed.

(a) Flow of fluid of density ρ1from a horizontal line source. When (ρ1> ρ0) a stagnation point exists above the source, and the fluid ultimately descends in a vertical column of width proportional to the source strength. At the stagnation point, the mixing-layer thickness is finite and is proportional to the square root of the radius of curvature of the interface. At a sufficiently great distance downstream, the thickness increases as the square root of the distance, as in the straight laminar half-jet. These results have been tested experimentally in a Hele-Shaw cell.

(b) Symmetrical flow of an ascending column of fluid (ρ1< ρ0) about an obstacle in the form of a finite horizontal strip. The column reforms after passing the obstacle, and the mixing-layer thickness returns to the value corresponding to an unobstructed vertical half-jet. The flow has been produced experimentally.

(c) Flow in a lens of fresh water overlying salt water, with inflow due to precipitation, as in a two-dimensional Ghyben-Herzberg lens. Here the potential flow solution is calculated approximately by means of Dupuit-Forchheimer theory. In the steady-state solution the thickness of the mixing layer between fresh and saline water is found to be finite and, as in (a), proportional to the square root of the radius of curvature of the lens.
Computer solutions to the problem of vibrational relaxation in hypersonic nozzle flows
Authors: Stollery J. L. , Park C.
Published: March 2006

This report is an extension of an earlier note in which a simple method of estimating the distribution of vibrational temperature along a hypersonic nozzle was described. Results were presented for hyperbolic, axisymmetric nozzles with reservoir conditions 1000[les ]p0[les ] 4000 p.s.i.a., 1000[les ]T0[les ] 3000°K. The problem was subsequently programmed for the Ferranti Mercury computer at the University of London computing centre, and the results of these computations are given here. The vibrational temperatures are compared with those of the previous simple method. The distributions of pressure and temperature through the nozzle are also given and a simple method of estimating the vibrational temperature is described.
Surface tension and buoyancy effects in cellular convection
Authors: Nield D. A.
Published: March 2006

The cells observed by Bénard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Bénard's experiments are briefly discussed.
Rotational stagnation point flow
Authors: Hayes Wallace D.
Published: March 2006

The constant-density inviscid rotational flow in the neighbourhood of a general stagnation point on a wall is investigated. In all but very special cases, the solution is non-analytic and the vorticity at the wall is infinite; the stagnation streamline is tangent to the wall at the stagnation point; stagnation points of saddlepoint type cannot exist.

The boundary-layer equations corresponding to the inviscid solutions studied are presented.
Rotating and stratified fluid flow
Authors: Trustrum Kathleen
Published: March 2006

For flows in either rotating or stratified fluids, a technique is developed for solving initial-value problems using an Oseen approximation to the non-linear inertial terms in the equations of motion. The resulting equations for either application are similar. The solutions bear a strong qualitative resemblence to observed flows of both kinds, being characterized at small Rossby or Froude numbers by a blocked flow upstream of an obstacle and waves on the downstream side.
Some perturbation solutions in laminar boundary layer theory Part 2. The energy equation
Authors: Fox Herbert , Libby Paul A.
Published: March 2006

Solutions for two types of problems involving the energy equation for flows with velocities described by the Blasius solution are presented. The first type arises in flows with arbitrary initial distributions of stagnation enthalpy and with surfaces downstream of the initial station either with constant wall enthalpy or with zero heat transfer. Exact solutions in these cases are obtained for constant ρμ, and Prandtl number of unity; they are given in terms of complete orthogonal sets of functions which can be used to obtain first- and higher-order corrections for the effects of variable ρμ, non-unity Prandtl number, and deviations of the velocity field from that described by the Blasius solution.

The second type of problem pertains to flows with power-law descriptions of the wall enthalpy. Again the basic solutions are obtained for Prandtl number of unity and the effect of non-unity Prandtl number is treated as a perturbation.
Directional spectrum of wind-generated ocean waves
Authors: Uberoi Mahinder S.
Published: March 2006

Two sets of published data on an area 2700ft. by 1800ft. of sea surface in the North Atlantic are analysed by an optical computer which gives directly the directional spectrum. The results are compared with (i) those of other investigators obtained laboriously by using a digital computer, (ii) the frequency spectrum, and (iii) an empirical prediction.
The dynamics of spheroidal masses of buoyant fluid
Authors: Turner J. S.
Published: March 2006

It is shown how a simple property of the spherical vortex model can be used to investigate the dynamics of a buoyant, expanding thermal. No details of the vortex motion are required, only the fact that the flow round the buoyant region is potential. The main result is a demonstration that there is a relation between the two constants C and α arising in the dimensional analysis (where in the usual notation w = C (Δr) ½) and r = αz), which have up till now been measured separately and treated as independent. The analysis has been extended to spheroidal thermals by calculating the virtual mass for the appropriate outline, and it has also been generalized to include thermals in which the total buoyancy is increasing with time.

Using these risults, and an earlier experimental verification that the mean angles of spread are nearly the same under various conditions of stability, it is suggested that the whole of the mean behaviour of a thermal can be calculated in two nearly independent steps. first, the density difference or Δ as a function of height may be calculated using purely kinematic equations of conservation. Secondly, the velocity is obtained from the local values of Δ and radius r, using a mean value of C, since this has now also been shown to vary little over a wide range of conditions.
A further note on the bathtub vortex
Authors: Kelly D. L. , Martin B. W. , Taylor E. S.
Published: March 2006

Using identical equipment, an attempt was made to reproduce the unexpected flow reversals obtained by Sibulkin (1962) in his study of the vortex motion associated with draining a liquid from a vessel through an orifice in its bottom. Reversal of the initial direction of rotation was found only for counter-clockwise filling with large initial heads and settling times, unless during settling or draining of the vessel a shock was applied to the system, in which case the tendency to reversal was generally more pronounced with clockwise filling. Re-reversal of the motion was also observed in a number of cases. An alternative explanation is proposed, based on a consideration of the surface waves generated by a shock.
The stability of elastico-viscous flow between rotating cylinders. Part 2
Authors: Thomas R. H. , Walters K.
Published: March 2006

Further consideration is given to the stability of the flow of an idealized elasticoviscous liquid contained in the narrow channel between two rotating coaxial cylinders. The work of Part 1 (Thomas & Walters 1964) is extended to include highly elastic liquids. To facilitate this, use is made of the orthogonal functions used by Reid (1958) in his discussion of the associated Dean-type stability problem. It is shown that the critical Taylor number Tc decreases steadily as the amount of elasticity in the liquid increases, until a transition is reached after which the roots of the determinantal equation which determines the Taylor number T as a function of the wave-number ε become complex. It is concluded that the principle of exchange of stabilities may not hold for highly elastic liquids.
Depression of an infinite liquid surface by an incompressible gas jet
Authors: Olmstead W. E. , Raynor S.
Published: March 2006

The problem of small angle depressions in a liquid surface due to an impinging two-dimensional potential jet is considered. Using conformal mapping methods and finite Hilbert transforms, the problem is formulated as a non-linear singular integral equation. The integral equation is approximated by a set of non-linear algebraic equations which are solved numerically by a method of repeated linear corrections. In addition, an asymptotic solution (for low jet velocity) is derived.

From the numerical solutions of the integral equation, the liquid-surface profiles and the free streamlines of the jet are calculated for four cases. These results verify the appearance of lips on the liquid surface which have been observed experimentally by others.
Convection in a radiating gas
Authors: Gille John , Goody Richard
Published: March 2006

We have made a theoretical and experimental study of the effect of radiative transfer on the onset of convection by means of comparative experiments in dry air and ammonia contained between horizontal aluminium plates maintained at different temperatures.

The first effect of radiative transfer is to distort the linear, diffusive profile in the initial, pre-convective state. A non-grey theory is developed and is shown to predict the non-linear profiles profiles with satisfactory accuracy. The change of radiative flux with pressure of ammonia is also predicted and confirmed. This constitutes the first successful attempt to confirm the methods used in atmospheric studies in controlled experiment.

A non-grey theory of the onset of convection is developed, separating effects associated with the distortion of the static state and the dissipative effect of radiative transfer on temperature fluctuations.

Experiments indicate that the critical Rayleigh number in ammonia can be greatly increaesd over the best value for air (Rac = 1786); the maximum observed being Rac = 4870. Theory and experiment are shown to agree within reasonable limits.

This work confirms the validity of methods which are of great importance to astrophysics and atmospheric physics.
On the stability of viscous flow between rotating cylinders Part 2. Numerical analysis
Authors: Harris D. L. , Reid W. H.
Published: March 2006

A simple numerical method is presented for solving the eigenvalue problem which governs the stability of Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. When the cylinders rotate in opposite directions, these eigenfunctions exhibit an exponentially damped oscillatory behaviour for sufficiently large values of − μ, where μ = Ω2/Ω1. In terms of the stream function which describes the motion in planes through the axis of the cylinders, this means that weak, viscously driven cells appear in the outer layes of the fluid which, according to Rayleigh's criterion, are dynamically stable. For μ = − 3, for example, four cells are present, the amplitudes of which are in the ratios 1·0:0·0172:0·013:0·00125.
The initial-value problem for long waves of finite amplitude
Authors: Long Robert R.
Published: March 2006

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.
Generation of secondary vorticity in a stratified fluid
Authors: Marris A. W.
Published: March 2006

The theory is presented for the generation of secondary vorticity in a stratified fluid. The analysis is a generalization of original work of Hawthorne (1951). It is shown that for an incompressible, inviscid and non-diffusive flow, a flowwise vorticity component will be generated in a curved stream when a density gradient exists in the direction of the bi-normal to the streamline.
A new technique for the measurement of low fluid velocities
Authors: Gaster M.
Published: March 2006

A new instrument for measuring the velocities of particles suspended in a flowing fluid is described. The instrument is linear and is therefore capable of measuring the mean velocity in a fluctuating stream, even when these fluctuations are greater than this mean value. This particular instrument was developed for free convection work where the velocities to be measured were in the range + k 0.2 in./sec to − 0.2 in./sec, but there seems to be no reason why this range could not be considerably extended.
The slow motion of a sphere in a rotating, viscous fluid
Authors: Childress Stephen
Published: March 2006

The uniform, slow motion of a sphere in a viscous fluid is examined in the case where the undisturbed fluid rotates with constant angular velocity Ω and the axis of rotation is taken to coincide with the line of motion. The various modifications of the classical problem for small Reynolds numbers are discussed. The main analytical result is a correction to Stokes's drag formula, valid for small values of the Reynolds number and Taylor number and tending to the classical Oseen correction as the last parameter tends to zero. The rotation of a free sphere relative to the fluid at infinity is also deduced.
On the inclined non-inertial sinkage of a flat plate
Authors: Moore Desmond F.
Published: March 2006

An approximate analytical solution is derived for the inclined non-inertial sinkage of a flat plate onto a smooth surface, for the case of small angles. The result is a modification to the existing solutions for parallel sinkage, in the form of a polynomial in the dimensionless angle of inclination. Even the smallest angle of inclination of plate to surface produces a large increase in the sinkage rate otherwise obtained. The results are confirmed experimentally.
Secondary flow in an elastico-viscous fluid caused by rotational oscillations of a sphere. Part 1
Authors: Frater K. R.
Published: March 2006

When the angular amplitude of oscillation of a sphere in an infinite mass of elastico-viscous fluid is fairly small, so that velocity and stress components may be expanded as power series in this amplitude, the purely periodic primary motion has associated with it a secondary flow which has a steady component as well as a component of double the primary frequency. An expression for the stream function of the steady secondary flow is obtained for all possible frequencies and the results are illustrated by considering in detail a particular fluid. It is shown that the streamline projections on a plane containing the axis of rotation are strongly dependent on the parameters measuring the elasticity of the fluid and on the frequency. The circulatory secondary flow can be in the opposite sense to that in a Newtonian fluid, in either the whole or part of the elastico-viscous fluid.
Inertial effects on clusters of spheres falling in a viscous fluid
Authors: Bretherton F. P.
Published: March 2006

A compact cluster of 3 to 6 rigid equal spheres is falling under gravity in a viscous liquid. The small effects of intertia on a horizontal regular polygonal configuration are that the polygon expands as it falls and small perturbations from this configuration die out, although when the polygon is large enough it becomes weakly unstable. This is an extension of the analysis of Hocking (1964, which was applied to the experiments of Jayaweera, Mason & Slack (1964).
Note on fluctuating flow near a stagnation point
Authors: Srivastava A. C.
Published: March 2006

The problem of the flow near a stagnation point when the main stream outside the boundary layer fluctuates in magnitude but not in direction about a steady mean is discussed. The velocity distribution is found in the two limiting cases of small and large values of the frequency of the oscillation. The corresponding two approximate solutions give similar results in an overlapping range of frequency.
Resonant interactions between waves. The case of discrete oscillations
Authors: Bretherton F. P.
Published: March 2006

The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.
Heat transfer in a radial liquid jet
Authors: Chaudhury Z. H.
Published: March 2006

The heat transfer in a radial liquid jet is investigated. In the region where a similarity solution of the momentum equation is available solutions of the energy equation describing the effects of viscous dissipation, initial heating and wall heating are obtained in closed form. Two examples illustrating the work are discussed. In the second of these an approximate method, based on the heat flux equation, is used to describe the initial development of the thermal boundary layer.
Second-order boundary-layer effects in hypersonic flow past axisymmetric blunt bodies
Authors: Davis R. T. , Flügge-Lotz I.
Published: March 2006

First- and second-order boundary-layer theory are examined in detail for some specific flow cases of practical interest. These cases are for flows over blunt axisymmetric bodies in hypersonic high-altitude (or low density) flow where second-order boundary-layer quantities may become important. These cases consist of flow over a hyperboloid and a paraboloid both with free-stream Mach number infinity and flow over a sphere at free-stream Mach number 10. The method employed in finding the solutions is an implicit finite-difference scheme. It is found to exhibit both stability and accuracy in the examples computed. The method consists of starting near the stagnation-point of a blunt body and marching downstream along the body surface. Several interesting properties of the boundary layer are pointed out, such as the nature of some second-order boundary-layer quantities far downstream in the flow past a sphere and the effect of strong vorticity interaction on the second-order boundary layer in the flow past a hyperboloid. In several of the flow cases, results are compared with other theories and experiments.
The vortical layer on an inclined cone
Authors: Munson Albert G.
Published: March 2006

The problem of flow over a circular cone inclined slightly to a uniform stream is solved using the technique of matched asymptotic expansions. The outer expansion is equivalent to Stone's solution of the problem. The inner expansion, valid in a thin layer near the body, represents Ferri's vortical layer. The solution to first order in angle of attack so obtained is uniformly valid everywhere in the flow field. In the second-order expansion an additional non-uniformity appears near the leeward ray. This defect is removed by inspection. The first-order solution is in agreement with that of Cheng, Woods, Bulakh and Sapunkov. Formulas are given that may be used to render Kopal's numerical result uniformly valid to second order in angle of attack.
Energy transfer between external and internal gravity waves
Authors: Ball F. K.
Published: March 2006

In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail.
On the harmonic oscillations of a rigid body on a free surface
Authors: Kim W. D.
Published: March 2006

The present paper deals with the practical and rigorous solution of the potential problem associated with the harmonic oscillation of a rigid body on a free surface. The body is assumed to have the form of either an elliptical cylinder or an ellipsoid. The use of Green's function reduces the determination of the potential to the solution of an integral equation. The integral equation is solved numerically and the dependency of the hydrodynamic quantities such as added mass, added moment of inertia and damping coefficients of the rigid body on the frequency of the oscillation is established.
A free-streamline solution for stratified flow into a line sink
Authors: Kao Timothy W.
Published: March 2006

An analysis is made of the two-dimensional flow under gravity of an inviscid non-diffusive stratified fluid into a line sink, involving a velocity discontinuity in the flow field. The fluid above the discontinuity is stagnant and hence is not drawn into the sink. At sufficiently low values of the modified Froude number, this is the only physically possible mode of flow, and is the cause of flow separation in many industrial and natural processes. A proper mathematical solution for flows with a stagnant zone has so far been lacking. This paper presents such a solution, after posing the problem as one involving a free-streamline, which is the line of velocity discontinuity. The solution to be given here is obtained by an inverse method. It is also found herein that the modified Froude number has a value of 0·345 for all separated flows of the kind in question.
The Mach wave field radiated by supersonic turbulent shear flows
Authors: Williams J. E. Ffowcs , Maidanik G.
Published: March 2006

Theoretical studies of aerodynamic noise suggest that the sound field of supersonic flows will be dominated by eddy Mach waves. Recent experimental evidence supports this view. In supersonic turbulent boundary layers, and rocket exhaust flows, turbulence occurs in regions of high mean velocity gradient. At low speed, such gradients are known to amplify the sound emitted by turbulence. This paper deals with the corresponding Mach wave problem. The exact equations of sound radiation by turbulence are rearranged in a form where the equivalent sources, derivatives of the turbulence stress tensor, are shown to be dominated by one term. That term is formed from the product of the mean velocity gradient and the rate of change of density. It seems that its resemblance to the dominant source of sound in low speed shear flows is largely fortuitous. In the Mach wave case, the theory is designed to include effects of both temperature gradients and density perturbations, and the approximations of the estimate are of a type that would not be expected to be valid away from the Mach wave condition. The basic theory is used to make an estimate of the sound radiated from supersonic boundary layers, and an approximate equation relating the radiated pressure to the surface pressure is derived. Experimental evidence is then examined to show that the equation is in excellent agreement with observation. The theory is then applied to annular shear flows of the rocket exhaust type. Again an approximate equation relating near and far field pressures is established, and the paper concludes with suggestions for experiments that could check the result.
Theory of heat transfer to a shock-tube end-wall from an ionized monatomic gas
Authors: Fay James A. , Kemp Nelson H.
Published: March 2006

This paper deals with the calculation of the convective heat transfer rate to the end-wall of a shock tube from a monatomic gas heated by a reflected shock. We consider a range of shock strengths for which the equilibrium thermodynamic state is one of appreciable ionization. The resulting boundary-layer problem involves the thermal conductivity and ambipolar diffusion coefficient for a partially ionized monatomic gas. The formulation here is restricted to the case of a catalytic wall and equal temperatures for all species. We ignore the effect of the plasma sheath at the wall. Consideration is given to three limiting cases for which similarity-type solutions of the boundary-layer equations may be found: (1) complete thermodynamic equilibrium behind the reflected shock and within the boundary layer; (2) equilibrium behind the reflected shock, but no gas-phase recombination in the boundary layer; (3) no ionization in either region. Numerical calculations are carried out for argon using estimated values of thermal conductivity and ambipolar diffusion, and compared with shock-tube experiments of Camac & Feinberg (1965). For no ionization, calculations were made with thermal conductivity varying as the ¾ power of the temperature, which fits the estimates of Amdur & Mason (1958) up to 15,000°K. Excellent agreement with experiment is obtained confirming an extrapolation of this power law up to 75,000°K. For ionized cases, based on estimates of Fay (1964), the theory predicts heating rates 20–40% lower than measured values. Some possible reasons for this discrepancy are discussed.
Electrokinetic flow round an elliptic cylinder in a finite fluid
Authors: Cade R. , Rowe Althea
Published: March 2006

The flow round an elliptic cylinder immersed in a finite volume of fluid and influenced by an electric field is determined, under the boundary conditions arising from electrosmosis at the surface of the cylinder. This is a problem in slow viscous flow with difficult boundary conditions, and is solved by numerical analysis, using the relaxation method. After the flow pattern has been determined, the mechanical action on the cylinder is calculated. An experimental application based on the theoretical results is suggested as a method for the determination of electrokinetic potentials.
On long waves on a rotating earth
Authors: Chambers Ll. G.
Published: March 2006

General solutions are obtained for the long gravity wave equation from solutions of the wave equation in three dimensions. The method is applied to the theory of Tsunamis. It is also suggested that these waves are the cause of the shelf oscillations observed at Guadelupe.
The buoyant motion within a hot gas plume in a horizontal wind
Authors: Csanady G. T.
Published: March 2006

A smoke plume that has become nearly horizontal at some distance from the source behaves much like a ‘line-thermal’, for which, using a perturbation method, a solution in laminar flow may be obtained, on the supposition that excess temperatures are small and buoyant movements slow, i.e. that the Rayleigh number of the problem is suitably low. In analogy with some other problems in turbulent flow and turbulent diffusion, the laminar solution is then assumed to approximate what is observed in the turbulent case, provided that the rate of growth of the diffusing cloud is assessed realistically.

The so-calculated pattern of streamlines in a cross-section of the plume agrees qualitatively with the observed behaviour of hot plumes and puffs, consisting of two vortex-like structures of opposite sense of rotation, lying on either side of the plume centre. The bodily upward movement of the plume is found to depend critically on the rate of growth of the plume. Thus when the plume diameter grows faster than linearly with distance (such behaviour characterizes the ‘quasi-asymptotic’ stage of relative diffusion predicted by Batchelor, 1952, for which Richardson's (4/3)-power law of eddy diffusivity holds) the plume tends to reach an asymptotic height. A crude theoretical estimate of the asymptotic height attained shows fair agreement with observations reported elsewhere. Although the plume nearly reaches this asymptotic height in the quasi-asymptotic phase, it retains a small gradient in the final phase which may be of importance at large distances from the source. The small-Rayleigh-number criterion restricts the validity of the solution to ‘weakly buoyant’ plumes.
Diffraction of surface waves on an incompressible fluid
Authors: Williams W. E.
Published: March 2006

The problem considered is that of the diffraction of the field of a point source in a fluid of infinite depth by an infinite vertical cylinder. It is shown that the surface wave component of the velocity potential may be expressed in terms of the solution to a classical electromagnetic (or acoustic) diffraction problem.
A general approach to linear and non-linear dispersive waves using a Lagrangian
Authors: Whitham G. B.
Published: March 2006

The basic property of equations describing dispersive waves is the existence of solutions representing uniform wave trains. In this paper a general theory is given for non-uniform wave trains whose amplitude, wave-number, etc., vary slowly in space and time, the length and time scales of the variation in amplitude, wave-number, etc., being large compared to the wavelength and period. Dispersive equations may be derived from a variational principle with appropriate Lagrangian, and the whole theory is developed in terms of the Lagrangian. Boussinesq's equations for long water waves are used as a typical example in presenting the theory.
Inviscid fluid flow in an accelerating cylindrical container
Authors: Moore R. E. , Perko L. M.
Published: March 2006

The problem of the axially symmetric irrotational flow of an inviscid incompressible fluid with a free surface in a circular cylindrical container accelerating parallel to its axis is considered. A numerical procedure is developed and some interesting motions exhibiting the development of breakers, splashing and sustained oscillations of the surface are obtained by machine computation.
Sound radiation from turbulent boundary layers formed on compliant surfaces
Authors: Williams John E. Ffowcs
Published: March 2006

The paper considers the effect of turbulence-induced surface response on the sound radiated by a turbulent boundary layer. The analysis is confined to an infinite plane homogeneous surface and the conclusions may not be a good indication of the behaviour of more realistic structures. The main result of the analysis is that no fundamentally more efficient source of sound is introduced by the surface motion. The radiation remains quadrupole in character. The surface merely accounts for a reflexion of the turbulence-generated sound, with the reflexion coefficient being identical to that of plane acoustic waves. Dissipation in the surface reduces the magnitude of the image system. A brief discussion of the effect on the particular quadrupoles to be found in a turbulent boundary layer concludes the paper. There it is argued that the radiation will probably be increased by surface motion, but not by an order of magnitude.
Vortex formation in a free boundary layer according to stability theory
Authors: Michalke A.
Published: March 2006

An attempt is made to explain the formation of vortices in free boundary layers by means of stability theory using a hyperbolic-tangent velocity profile. The vorticity distribution of the disturbed flow, as obtained by the inviscid linearized stability theory, is discussed. The path lines of particles which are initially placed along straight lines parallel to thex-axis are calculated. Lines connecting the positions of these particles give an impression of the instant shape of the disturbed flow. With increasing time the boundary layer becomes thinner in certain regions and thicker in others. A special line—originally positioned at the critical layer—shows in the thicker region a tendency to roll up. Also extrema of the vorticity are located there. Finally, these results are compared with those which can be expected from the non-linear Helmholtz equation. Disagreement is found in the neighbourhood of the critical layer. Using the non-linear stability theory of Stuart up to the third-order terms, the vorticity distribution shows the tendency expected from the non-linear equation.
Note on instability of compressible jets and wakes to long-wave disturbances
Authors: Gill A. E. , Drazin P. G.
Published: March 2006

A two-dimensional jet or wake is observed in a frame of reference moving with the fluid at infinity, so that the velocity w(y) in the x-direction tends to zero as y → ± ∞. The fluid is assumed to be an inviscid perfect gas, to undergo-adiabatic changes, and the local speed of sound to be a function of y such that a(y)→ a∞, as y → ±∞. If the disturbance pressure has the form
Wall layers with non-uniform shear stress
Authors: Reynolds A. J.
Published: March 2006

The empirical description of turbulent wall layers across which the shear stress varies is considered. The description given by Townsend for zero-stress layers is found to be inapplicable to uniform pressure flows in pipes and two-dimensional channels, and to a boundary layer developing in the absence of a pressure gradient.
A study of the turbulent separated-flow region occurring at a compression corner in supersonic flow
Authors: McDonald H.
Published: March 2006

The turbulent separated-flow region occurring at a compression corner under certain circumstances at supersonic speed has been examined in the light of recent improvements to base pressure theory (McDonald 1964). This base pressure theory is further extended from what could be termed a single-layer model of the re-attaching boundary layer to a two-layer model, thus enabling the inviscid shock configuration which occurs at the corner to be determined. Application of this analysis to some experimental results indicates a substantial measure of agreement.

While this analysis has been framed for estimating the scale of the corner interaction, the extension can of course be applied to increase the range of initial boundary-layer thicknesses to which McDonald's analysis is applicable. An example of such an application is shown to be in good agreement with experiment.
Surface-pressure fluctuations induced by boundary-layer flow at finite Mach number
Authors: Williams John E. Ffowcs
Published: March 2006

A theory describing boundary-layer surface-pressure fluctuations on a rigid surface is presented in a form that illustrates the main effect of compressibility. The most significant effect is that the correlation area is proportional to the square of mean-flow Mach number so it does not vanish in flow of finite compressibility. Modifications of the wave-number and frequency spectra by this effect are described, and the results applied to the computation of large plate response. That computation incorporates the effect of fluid loading, which enters the response equations as a dissipative term for components at supersonic phase velocity but merely as an added loading for subsonic components.