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The existence of a steady vortex ring close to Hill's spherical vortex is established, and an approximate description of its boundary is given. The vorticity in the ring is proportional to the distance from the axis of symmetry. The core propagates steadily in an unbounded fluid at rest at infinity. The boundary of the vortex ring is close to an interior stream surface of Hill's vortex.

We consider the motion of the mass of fluid ejected through a sharp-edged orifice by the motion of a piston. The vorticity formed by viscous forces within the separated flow at the sharp edge rolls up to form a concentrated vortex which, after a development period, consists of a core of very fine scale turbulence surrounded by a co-moving bubble of much larger scale turbulence. This bubble entrains outer fluid, mixes with it, and deposits the majority into a wake together with some small fraction of the total vorticity of the ring. Enough fluid is retained to account for the slow growth of the whole fluid mass. A theory which takes account of both the growth process and the loss of vorticity is proposed. By comparison with experimental measurements we have determined that the entrainment coefficient for turbulent vortex rings has a value equal to 0.011 ± 0.001, while their effective drag coefficient is 0.09 ± 0.01. These results seem to be independent of Reynolds number to within experimental accuracy.

The strength (initial circulation) and spacing of vortices in the wake of a circular cylinder have been obtained for conditions under which the body undergoes lateral vibrations. The vibrations of the cylinder were at all times synchronized with those in the wake, thereby suppressing the natural Strouhal frequency in favour of a common synchronized or ‘locked-in’ frequency for the body-wake system. All experiments were performed at a Reynolds number of 144 or 190. An inverse relation between the initial circulation K and the length lF of the vortex formation region was obtained for cylinder oscillations of up to 50% of a diameter, at vibration frequencies both above and below the Strouhal shedding frequency. The initial circulation K of the vortices was increased by as much as 65%, at lF = 1·6 diameters, from the stationary-cylinder value of K corresponding to lF = 3·2d. An increase in the rate A of vorticity generation of 80% from the stationary-cylinder wake value was obtained with the cylinder vibrating at 30% of a diameter and 110% of the Strouhal frequency. Both flow-visualization and hot-wire results show that the lateral spacing of the vortex street decreases as the vibration amplitude of the cylinder is increased, but that the longitudinal vortex spacing is independent of changes in amplitude. The longitudinal spacing, however, varies inversely with the vibration frequency. The street approaches a single line of vortices of alternating sign as the amplitude of vibration approaches values near a full cylinder diameter, and secondary vortex formation at these large amplitudes is associated with the vanishing lateral spacing of the street. Observation of the wake has elucidated the mechanism of vortex formation; the entrainment processes in the formation region have been observed at small intervals over a cycle of the cylinder's motion.

A two-dimensiona l laser velocimeter whose focal volume can be rapidly traversed through a flowfield has been used to overcome the problem introduced by excursions of the central vortex filament within a wind-tunnel test section. The operation of the instrument is reviewed and data are presented which accurately define the trailing vortex from a square-tipped rectangular wing. Measured axial and tangential velocity distributions are given, both with and without a vortex dissipator panel installed. From the experimental data, circulation and vorticity distributions are obtained and the effect of turbulence injection into the vortex structure is discussed. Nomenclature wing span wing chord lift coefficient radial coordinate from vortex center wind-tunnel mainstream velocity axial velocity component tangential velocity component maximum tangential velocity stream wise ordinate, aft from trailing edge normal ordinate, above upper wing surface from trailing edge normal location of vortex center angle of attack, deg slope of velocity distribution in core region vorticity

The condition that the absolute vorticity be conserved is applied to the model of a symmetric vortex embedded in a constant zonal current. It is shown that the radial variation of the rotational component of the wind must then be according to the first-order Bessel function. The speed of propagation of the model vortex is found as a by-product of this result. Finally it is shown that the maximum wind in such a model can be determined from the depth of the corresponding pressure-height field and the size of the vortex.

A series of observations on experimentally produced vortex rings is described. The flow field, ring velocity and growth rate were observed using dye and hydrogen-bubble techniques. It was found that stable rings are formed and grow in such a way that most of their vorticity is distributed throughout a fluid volume which is larger than and moving with the visible dye core.As the vorticity diffuses out of this moving body of fluid into the outer irrotational fluid, it has two effects. It causes some of the fluid, with newly acquired vorticity, to be entrained into the interior of the bubble, while the rest is left behind and accounts for the appearance of ring vorticity in a wake. It was found that the velocity of translationUof these stable rings varies ast−1, at high Reynolds number, wheretis the time measured from the start of the motion at a virtual origin at downstream infinity. A simple theoretical model is presented which explains all of these features of the observed stable flow. Rings of even higher Reynolds number become unstable and shed significantly more vorticity into the wake. Under some circumstances a new more stable vortex emerges from this shedding process and continues with less vorticity than before. Eventually, the ring motion ceases as all of its vorticity is deposited into the wake and is spread by viscous diffusion. Observations of the interaction between two nearly identical rings travelling a common path showed that, contrary to popular belief, rings do not pass back and forth through one another, but that the rearward one becomes entrained into the forward one. Only when the rearward ring has a much higher velocity than its partner can it emerge from the joining process and leave a slower-moving ring behind.

The vortex sponge, devised by John Bernoulli in 1736 in an attempt to explain light in mechanical terms, consists of an ideal fluid interlaced by fine vortex tubes oriented in all directions. When vortex tubes follow the fluid, the medium behaves like an elastic solid because of momentum transfer effects arising from the fine-grained vorticity. This character of the medium is altered by displacements which bend the tubes so that they move laterally. The motion of curved tubes relative to the fluid can result in macroscopic vorticity with accompanying rotation of the bulk medium. The mathematical expressions of these effects have the form of Maxwell's curl equations for free space.

This paper describes the coupled diffusion of heat and vorticity in a high temperature, low speed, gaseous vortex. The coupling is caused by both a radial convection of mass resulting from the unsteady heating of the gas and the temperature dependence of the gas properties. The numerical results were obtained by simultaneously solving the fluid heat conduction, continuity, and Navier-Stokes' equations on a digital computer using a finite difference technique. Solutions are presented for two vortex heating situations : (i) initially hot core with cold, irrotational, outer region and (ii) initially cold core with hot, irrotational, outer region. In order to better understand the coupling processes, two gas models are considered : (i) perfect gas with constant thermodynamic and transport properties and (ii) perfect gas with temperature dependent thermodynamic and transport properties. It was found that compressibility increases the rate of vorticity diffusion in the cold core vortex and heat diffusion in the hot core vortex over that of an incompressible vortex. Temperature dependent gas properties further increase the rates of diffusion of both heat and vorticity only in the cold core vortex.

In a vortex we often observe an approximately axially symmetric core of spiralling fluid in which the vorticity is high. A review is given here of recent work on the structure of such vortex cores. It begins with a discussion of the equations of motion and the appropriate boundary conditions, and continues with a description in general terms of the vigorous but highly responsive character of the flow, of the effects of compressibility and turbulence and of the phenomenon of energy separation. In many vortex cores, including those of trailing vortices far downstream behind a body, and those of leading-edge vortices above slender wings, the variations of velocity and pressure in the axial direction are small compared with those in the radial direction. Such cores, called quasi-cylindrical, are described in some detail. When this simplifying feature is absent, as in some naturally occurring vortices and some vortex-tube flows, only rough calculations can be made and the more rigorous work is confined to similarity solutions and expansions based on these. Finally, a discussion is given of the phenomena of stagnation in the axial flow, spiral instability and finite transition, any of which may destroy the typical core of high vorticity or prevent its maintenance.

It is well known that the classical potential flow of a perfect fluid past a bluff body is not a satisfactory representation of the actual flow, and that, in particular, it fails to give any indication of the resistance or drag which is experienced by the body. If the body is of a long cylindrical shape with axis normal to the axis, and may be conceived as proceeding in two dimensions only. In this case it is found experimentally that thin sheets of vorticity spring from the two sides of the body and enclose a dead-water region behind it, but owing to their essential instability these vortex sheets rapidly disintegrate and give rise to a succession of independent vortices which pass down stream in the form of a double vortex row. The characteristics of such a vortex street in its ultimate form far behind the body have been examined by Karman, who derives an expression for the drag of the body in terms of the strength of the individual vortices and of the breadth of the vortex street. More recently the results of a comprehensive series of experiments on the flow past a flat plate have been published by Fage and Johansen, and these results have been compared with Karman's theory. In view, however, of the fact that the experimental results are of necessity obtained from experiments in a channel of finite breadth, the present investigation has been undertaken to determine the constraint of the channel walls on the characteristics of the vortex street. 2. Karman's Theory of the Vortex Street . Karman considers the behaviour in a perfect fluid of a double row of point vortices in the configuration shown in fig. 1, deducing the relationships which are necessary for the stability of the system, and the corresponding drag force on the body to which the vortex street is due. It appears that the only stable configuration is that in which each vortex of one row is opposite the point mid-way between two successive vortices of the other row.

It is shown that, for vortex trails in a nonviscous fluid, the mean longitudinal spacing a and mean transverse spacing h are invariant. It follows from this and other known facts, that the periodicity of a vortex trail is unstable, not the ratio h/a.In a viscous fluid, a is invariant but h increases. This fact explains various known experimental observations.A rough explanation is given for the order of magnitude of the Strouhal number S, based on the inertia opposed by the wake to transverse oscillations.Finally, using the theory of logarithmic potential, it is shown that, even in a nonviscous fluid, the concentration of vorticity is limited to circles whose diameter is about a/4.

Physical arguments are followed by mathematical developments to show how aerodynamic sound is generated as a result of the movement of vortices, or of vorticity, in an unsteady fluid flow. Changes in circulation or area of a vortex ring give rise to a dipole sound field, the former being illustrated by oscillating flow about a fixed sphere, and the latter by a simple model for the aeolian tone attributable to the stretching of vortex rings. Because in a free flow there can be no change of the total vortex strength (circulation times area), there is no net dipole strength, but each moving element of vorticity still causes local dipolelike flow; each element of moving vorticity acts with some equal and opposite movement elsewhere in the flow so that together they form an oblique quadrupole, although the total effect must be reducible to an assembly of lateral quadrupoles. A cardinal result is that the vorticity in a slightly compressible fluid can be considered to induce the whole flow field, both the hydrodynamic part and the acoustic part. With vorticity taken as the common basis, a slightly compressible flow is compared to the corresponding incompressible one, which may be used in the evaluation of the sound‐radiation formula. The theory is particularly well‐structured to estimate sound from flows described in terms of vorticity: the sound field is determined for two rectilinear vortices spinning about an axis between them, and its basis for similarity methods is demonstrated in application to free shear flow and jet flow.

The motion and decay of circular vortex rings with an inner viscous core is considered by systematic matching of inner and outer asymptotic expansions. The governing Navier-Stokes equations are reduced to a coupled integro-differential system. A method of construction of solutions for the integro-differential system is presented. The initial vorticity distribution may be non-similar. Also presented is a method for introducing a time shift which makes the first term in the series solution for the vorticity to be the “best” approximation. The analysis is then applied to the motion and decay of a pair of coaxial vortex rings.

A mathematical model of unsteady flow around a cylinder gives lift, drag, and velocity fluctuations in reasonable agreement with experimental results. Vorticity is represented by element Rankine vortices, which are superimposed on the potential flow around a cylinder to form the boundary layers and separated shear layers. The solution advances timewise from assumed initial conditions. At each time step, every element vortex is moved through a distance equal to the product of its instantaneous translational velocity and a small time increment, determined after stability tests. Concentrated clouds of vorticity form and are shed from the cylinder, with accompanying fluctuations in lift and drag. The total number of element vortices in the model is minimized by replacing each cloud by a single equivalent vortex. Because feedback from the formation region to the separation points is present in the model, it is potentially well-suited to studies of flow around oscillating cylinders.

The basic mechanisms for air-entraining vortices to form at a water intake are examined and vorticity arising in boundary layers is considered in detail. Theoretical expressions are developed for calculating the maximum strength of air-entraining vortex which may form at a water intake. Experiments confirm that it is possible to calculate the strength of vortex from a knowledge of approach conditions and intake geometry. For the conditions of these tests the efficiency of vortex formation is approximately unity. A kinetic energy evaluation for the approach flow and intake flow sections indicates the organizational nature of the process and gives values for the additional pressure losses at an intake which are attributable to vortex formation.

An analytic solution for the velocity field of a vortex street generated in a viscous fluid is developed. A method is presented for the determination of the true transverse spacing of vortices. Experimental geometry and velocity data, obtained by hot-wire techniques, are presented.The experimental results verified the validity of the analytic solution. The vortices of a real viscous vortex street were found to resemble very closely the exponential solution of the Navier-Stokes equations for an isolated axisymmetric rectilinear vortex. Three basic regions of vortex street behaviour were apparent at each Reynolds number investigated-a ‘formation region’ in which the vortex street is developed and large dissipation of vorticity occurs, a ‘stable region’ in which the vortices display a stable periodic laminar regularity, and an ‘unstable region’ in which the street disappears and turbulence develops. Geometry and velocities were determined.

This paper is concerned with vortex rings, in an unbounded inviscid fluid of uniform density, that move without change of form and with constant velocity when the fluid at infinity is at rest. The work is restricted to rings whose cross-sectional area is small relative to the square of a mean ring radius. An existence theorem is proved for distributions of vorticity in the core that are arbitrary, apart from the condition imposed by the equation of motion and certain smoothness requirements. The method of proof relies on the nearly plane, or two-dimensional, nature of the flow in the neighbourhood of a small cross-section, and leads to approximate but explicit formulae for the propagation speed and shape of the vortex rings in question.

One of the best known results in finite wing theory, which is quoted in aerodynamic text books, states that the distance between two rolled up vortices behind a wing with an elliptic spanwise load distribution is sπ/2, where s is the wing semi-span. Within the framework of linear wing theory this result is correct. But it is of interest to enquire how this result is affected when the standard linear model becomes less valid, for example, at higher wing lift coefficients.In this note, it is assumed that a continuous trailing vortex sheet rolls up into two discrete vortices. These two discrete vortices are assumed to have cores of finite radius; inside the cores the flow is taken to be solid body rotation, while the flow outside the cores is the standard irrotational vortex field. It is assumed that this rolling up process takes place far downstream of the finite wing which generates the trailing vorticity.

Axially symmetric thermal convection in a rotating environment with a friction layer is investigated by numerically integrating an appropriate system of equations. The study examines the influence of the friction layer upon dry adiabatic thermal convection and the resulting vortex formation process by making comparative computer runs, with and without addition of the friction layer, for four values of ambient-vertical vorticity. The analysis shows that the addition of a friction layer has a significant influence on the resulting velocity distributions. Friction-induced radial convergence adds to the central updraft of the rising thermal and increases its rate of rise. This friction layer enhancement increases with larger rotation rates and consequently overcomes the correspondingly increased rotational suppression. The vortex height is observed to be greater for the friction layer runs. Additionally, the vortex is observed to have a smoother but less intense profile of tangential velocity with th...

With the object of gaining a better understanding of the mechanism involved in the formation of vortices in buoyancy-driven systems interacting with ambient vorticity, a mathematical approach based upon the hydrodynamical instability theory is employed. For simplicity, the mathematical model consists of an infinite horizontal line heat source with uniform ambient vorticity supplied. A governing stability equation analogous to the Orr-Sommerfeld equation is derived and solved by a Newton-Raphson iterative technique employing special matching conditions just outside the plume and symmetry conditions enabling analysis in a half plane. The solution predicts a standing-wave-type instability of the class suggestive of incipient vortices. Neutral stability curves are generated and a critical Reynolds number based on plume thickness is obtained for Prandtl numbers of 5/9 and 2. Based on neutrally stable disturbances, a representative perturbed flow streamline pattern is shown. The theory predicts a critical elevation above which unstable modes exist. An optimization analysis was performed in the domain of unstable disturbance modes which suggested an overall selection process. It was found that at one particular elevation in the plume, one particular mode of disturbance grows much more rapidly than all others at any location. This property is then related to the vortex periodicity in the observed phenomenon. An attempt is made to relate the analytical results to relevant field observations of various classes of geophysical vortices.

Four siphons of square section with sides 24 mm (0.95 in.), 38 mm (1.5 in.), 50 mm (1.2 in.) and 62 mm (2.4 in.) were tested in a model stilling pond. The dimensionless critical submergence over the siphon decreased with increase in siphon size and for a given siphon, the improvement in critical submergence varied with the baffle position along the horizontal and vertical axis of the stilling pond. The critical submergence increases in the distance of the baffle from the siphon; results with all siphons suggest that the best location for the vortex suppression is with the baffle situated at one diameter from the siphon. The optimum submersion of the baffle to prevent vortex formation was greatest for the 24 mm siphon and decreased with increase in the siphon size. A theoretical equation suggests the functional relationship between the vorticity responsible for air entrainment and variables.

When compact objects or black holes move through a fluid medium, or when turbulent plasma and magnetic fields so conspire, a gas flow is set up which closely resembles the flow of water down a plug-hole (Section 1). A similar hypothesis, but in reverse, was suggested by Jeans in 1928, and would nowadays be referred to as the white hole concept. The dynamics of the flow (Section 2) lead to expressions for the rotational velocity of the fluid far away from (2.1) and near to (2.2) the origin of the vorticity. Rotation curves derived from the model (Section 3) are closely akin to actual galactic rotation curves, but observational data on the latter are not precise enough to permit a delineation to be made between (i) flow around a singularity and (ii) flow around a non-singular sink or source. The other acceptable model, that of (iii) a spreading line vortex, is ruled out by comparison with astrophysical observations (Section 4). The basic analysis for all the models shows that the old problem of the winding-up of spiral arms can be avoided, since the galactic flow system is in a steady state. Section 5 identifies Jeans' speculation as being a hypothesis compatible with singular vortex flow and so with observation, but perhaps not with the usual interpretation of general relativity metrics, even though the requisite dual space does complete the topology in a mathematically satsifying manner. Section 6 concludes that the predictions of the hypothesis of vortex flow agree with the shape, dynamics and structure of galaxies.

We consider the weak (small Rossby numbers) non-linear interaction between a quasi-geostrophic vortex and the Ekman flow which is produced by a uniform wind stress acting at the top of the mixed layer. The differential advection of geostrophic vorticity by the undisturbed Ekman components tends to tilt the axis of the vortex away from the vertical but, because of the strong rotational constraint, the vortex maintians vertical ‘rigidity’ by developing vertical velocities which balance the advective tendency (and frictional rotation). In this approximate vorticity balance for a ‘frictional’ component of the total motion, the vertical integral implies a net vertical velocity (‘Ekman suction’) at the free surface. To maintain the proper boundary condition an equal and opposite velocity must be applied to a second (geostrophic) component of the motion. This coupling condition has been determined as a function of the surface wind stress, the gradient of geostrophic vorticity, and the depth of the homogeneous mixed layer. If the base of the mixed layer were rigid the vortex would move as a unit with the velocity of the mean Ekman drift. The suction velocity formula is then generalized to include the special case ( Charney 1955) of a global wind variation and variable Coriolis parameter ( Sverdrup 1947). The effect of the local wind in supplying kinetic energy to the deeper ocean is also discussed, and suggestions are made for including the baroclinicity of the friction layer into the geostrophic dynamics.

The vortex equations of magneto-hydrodynamics are derived. Three vorticities, viz. a material, an electrical and a magnetic vorticity are introduced. In their integral form, i.e. as equations for the material and the electrical circulation, the equations are applied to a rotating plasma in a thermo nuclear device.

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/r(of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/ris considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.

Planar lifting-surface theory is applied to the problem of the loads induced on an infinite aspect-ratio wing by a straight, infinite, free vortex in a subsonic, compressible, freestream; the vortex lies in a plane parallel to the plane of the wing, at a given distance below it, and at an arbitrary angle with the wing centerline. The solution is obtained in the form of an aerodynamic influence function for an infinite aspect-ratio wing in an oblique, sinusoidal gust. The lifting-surface theory kernel function for this wing is presented. This kernel includes as limits the linear, aerodynamic kernel functions for the problems of steady, three-dimensional flow and unsteady, two-dimensional flow, and for the problems of incompressibl e, two-dimensional flow and transonic, three-dimensional flow. An approximate analytic expression is given for the lift influence function, suitable for practical use in the routine calculation of vortex-induced airloads.

The symmetric vortex separation that is observed on the leeward side of slender bodies of revolution at high angle of attack in the subsonic to moderately supersonic-velocity range is analyzed by means of the “lumped-vorticity” approximation suggested by Edwards and Hill. The equivalent unsteady two-dimensional problem of indicial motion of a cylinder in an incompressible fluid with symmetric vortex wake is also considered. Body vortex position and normal force per unit length are presented for a cylinder and a slender cone at high angle of attack and compared with available experimental information.

The general theory of the bath-tub vortex is reviewed, including both the effects of the secondary flow and the existence of an initial angular velocity. The effect of the secondary flow is demonstrated experimentally by measuring the flow angles at the air intake using a baffle plate set parallel and perpendicular to the Earth axis. The experimental proof was obtained by measuring the fluid vorticity through observation of the behavior of a cross shaped float.

An original method is presented which allows to study the vortex dynamics in one particular point of a superconducting sample. The vortex motion is detected by means of a point contact Josephson junction placed in the point under consideration, the critical d.c. Josephson current of the junction being modulated by the passage of vortices. The results of two types of experiments are in reasonably good agreement with theoretical predictions.

A scanning Doppler radar has capability for measurement of the tangential shear of radial velocity or gradient of the radial component of velocity in a direction normal to the radial vector. Tangential shear of sufficient magnitude in a horizontal direction reveals one term of the expression for vorticity in a horizontal plane. A vortex is reasonably well identified by Doppler radar if the shear is sufficiently steady during the time required for half a revolution of a vortex, and is further confirmed by extension of the shear pattern throughout a depth greater than its diameter, and by invariance of the character of the shear with respect to a change in viewing angle. A vortex in solid rotation has a characteristically uniform signature on a Doppler velocity display which may, however, be smoothed down according to the resolution limitations of the antenna beam. Sharp velocity peaks are rounded down to roughly half the magnitude of the velocity interval encompassed within the half-power antenna ...

We consider a new vortex approximation for solving the initial-value problem for the Euler equations in two dimensions. We assume there exists a smooth solution to these equations and that the vorticity has compact support. Then we show that our approximation to the velocity field converges uniformly in space and time for a short time interval.

Preliminarily it was pointed out, for the case of free convection phenomena, that the transition point from laminar flow to vortex streets flow would not be given by the critical Reynolds number. Then we divided the laminar boundary layer by an imaginary wave length along the flow direction, made the ratio of the momentum energy of the imaginarily divided part in the case that the part would move en bloc independently to the increase of the momentum energy of the same part in the case that the part would move still one more wave length, maintaining the laminar state continuously, and supposed that the transition would happen when the ratio reaches a critical number. The critical Grashoff number was given from that ratio, and also the wave length and the frequency of the vortex streets were calculated with the same ratio, under the assumption of the continuity of the vorticity flow before and behind the transition. Nearly good coincidence was observed between the theoretical relations and the experimental data measured by the author with water and ethylene-glycol and Eckert-Soehngen with air.

The interaction of a vortex with a stationary surface was studied both theoretically and experimentally. The flow field examined was that produced by radially inward flow through a pair of concentric rotating porous cylinders that were perpendicular to, and in contact with, a stationary flat plane. The complete Navier-Stokes equations were solved over a range of tangential Reynolds numbers from 0–300 and a range of radial Reynolds numbers from 0 to −13, the minus sign indicating radially inward flow. In order to facilitate the solution, the original equations were recast in terms of a dimensionless stream function, vorticity, and third variable related to the tangential velocity. The general validity of the numerical technique was demonstrated by the agreement between the theoretical and experimental results. Examination of the numerical results over a wide range of parameters showed that the entire flow field is very sensitive to the amount of radial flow, especially at the transition from zero radial flow to some finite value.

As a contribution to the breakdown phenomenon of vortices in a two-dimensional free boundary layer, this paper deals with the question whether a single cylindrical (i.e. two-dimensional) vortex can become unstable. For this reason a single vortex, as it occurs in a free boundary layer, is approximated by an axisymmetrical vortex model. The inviscid stability theory of rotating fluids is then applied to this vortex model. By general stability criteria it was found that a vortex consisting of vorticity of one sign only is stable according to the Rayleigh criterion, but, if the vorticity has an extremum value outside the axis, may become unstable with respect to cylindrical disturbances. Furthermore, stability calculations for three special types of vortex were performed. It was found that they were more unstable with respect to cylindrical disturbances than to three-dimensional ones.

The linear stability of laminar natural convection flow adjacent to a heated, inclined, upwardfacing plate is investigated for disturbances having the form of longitudinal vortices. The stability problem is formulated with account being taken of the fact that the basic flow and temperature fields depend on the streamwise coordinate. One of the demonstrated consequences of retaining the transverse velocity of the basic flow is the so-called bottling effect, wherein the disturbance vorticity and temperature are contained within the respective boundary layers of the basic flow. The calculated neutral stability curves exhibit an altogether different character depending upon whether the streamwise dependence of the basic flow and temperature fields is taken into account or suppressed; the magnitude of the critical Grashof numbers from the two models differs by several orders of magnitude. The results also show that the greater the inclination of the plate from the vertical, the more susceptible is the flow to the vortex-type instability. The relationship of the analytical results to available experimental information is discussed.