Axisymmetric vortex rings which propagate steadily through an unbounded ideal fluid at rest at infinity are considered. The vorticity in the ring is proportional to the distance from the axis of symmetry. Recent theoretical work suggests the existence of a one-parameter family, [npar ]2 ≥ α ≥ 0 (the parameter α is taken as the non-dimensional mean core radius), of these vortex rings extending from Hill's spherical vortex, which has the parameter value α = [npar ]2, to vortex rings of small cross-section, where α → 0. This paper gives a numerical description of vortex rings in this family. As well as the core boundary, propagation velocity and flux, various other properties of the vortex ring are given, including the circulation, fluid impulse and kinetic energy. This numerical description is then compared with asymptotic descriptions which can be found near both ends of the family, that is, when α → [npar ]2 and α → 0.

A higher order theory for two-dimensional turbulent boundary-layer flow of a compressible fluid past a plane wall is formulated, for moderately large values of the Reynolds number, by the method of matched asymptotic expansions. The parameters (γ − 1) M2∞ and the molecular Prandtl number are assumed to be of order unity. The analysis deals with the set of Reynolds equations of mean motion (which are underdetermined without an additional set of closure hypotheses) and assumes that the non-dimensional fluctuations in velocity, temperature and density are of order U*, (friction velocity divided by free-stream velocity a t some designation point), while fluctuations in pressure are of order U2*.The first-order results of the present study lead to asymptotic laws for velocity and temperature distributions which correspond to the law of the wall, logarithmic law and defect law, and also to skin friction and heat-transfer laws. It turns out that the first-order defect law depends upon the gradient of entropy and stagnation enthalpy and the law of the wall is independent of viscous dissipation. The second-order terms of the present work (accounting for mean convection due to turbulent mass flux, viscous dissipation in the inner flow and displacement effects in the outer flow) describe the necessary corrections to first-order terms due to low Reynolds number effects. In the overlap region the second-order results, for the law of the wall and the defect law, show bilogarithmic terms along with logarithmic terms.

Measurements have been made of the mean and fluctuating pressure distributions on long circular cylinders, having smooth and rough surfaces, at Reynolds numbers of 1·11 × 105 and 2·35 × 105 in both uniform and turbulent streams. The presence of free-stream turbulence a t these Reynolds numbers was found to suppress coherent vortex shedding on the smooth cylinder and give rise to a complex pressure field in which the mean pressure distribution was almost independent of Reynolds number over the small range of Reynolds numbers tested. The pressure distributions on the rough cylinder were found to be completely different in uniform and turbulent streams; the presence of turbulence gave rise to an increase in the level of vortex shedding energy, and produced mean pressure distributions similar to those obtained on smooth cylinders at Reynolds numbers of the order of 107.

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

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

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

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

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

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

F    Dynamo theory    Prof. P. H. Roberts

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

There were just over a hundred participants, the great majority of whom were accommodated at Trinity College, Cambridge, for the duration of the meeting.

Experiments have been carried out to examine the spectrum of internal gravity waves excited in a stratified incompressible fluid during stabilization following the buoyant rise of a miscible fluid. The rise time of the buoyant fluid to its stabilization height in the stratified fluid was observed to be about 0.85 of the Brunt-Väisälä period for the stratified fluid. The motion of specific fluid elements in the wave field was observed using neutrally buoyant marker particles, and the particle trajectories were found to be in close accord with theoretical predictions. Observations on the internal waves generated by the forced oscillation of a spherical body suspended in the stratified medium showed the wave pattern to be well behaved and similar to that described by Mowbray & Rarity. However, the gravity wave field generated by the motion of the buoyant fluid was observed to be inhomogeneous and transient in nature. Wave periods from one to four times the Brunt-Väisälä period were clearly observed and at later times it appeared that the motion tended towards vertical oscillations a t the Brunt- Väisälä frequency.

The equations for nonlinear Bénard convection with rotation for a layer of fluid, thickness d, are derived using the Glansdorff & Prigogine (1964) evolutionary criterion as used by Roberts (1966) in his paper on non-rotational Bénard convection. The parameters of the problem in this case are the Rayleigh number R = αgΔθd/vK, the Taylor number T = 4d4Ω23/v2 and the Prandtl number Pr = v/K, where α is the coefficient of volume expansion, g the acceleration due to gravity, Δθ the temperature difference between the horizontal surfaces, v the kinematic viscosity, K the thermal diffusivity and Ω3 the rotation rate about the vertical direction. The asymptotic solution for two-dimensional cells (rolls) is investigated for large Rayleigh numbers and large Taylor numbers. For rolls the convection equations are found t o be independent of the Prandtl number. However, the solutions depend upon the Prandtl number for another reason. The rotational problem differs from the non-rotational one in that the Rayleigh number and the horizontal wavenumber a of the convection are now functions of the Taylor number. These are taken to be R ∼ ρTα′ and α ∼ ATβ, where α′ and β are positive numbers. Thermal layers develop as R becomes large with ρ or T becoming large. The order in which ρ and T are allowed to increase is important since the horizontal wavenumber a also increases with T and the convection equations can be reduced in this case. A liquid of large Prandtl number such as water has v [Gt ] K. Since R ∼ O (1/vK) and T ∼ O(1/v2), ρ will be greater than T for a given (large) Δθ and Ω3. Similarly, for a liquid of small Prandtl number such as mercury v [Lt ] K, and T is greater than ρ for a given Δθ and Ω3. For rigid-rigid horizontal boundaries with ρ large and then T large the ρ thermal layer has the same structure as for the non-rotating problem. As T → ∞ three types of thermal layers are possible: a linear Ekman layer, a nonlinear Ekman layer and a Blasius-type thermal layer. When the horizontal boundaries are both free the ρ thermal layer is again of the same structure as for non-rotating BBnard convection. As T → ∞ a nonlinear Ekman layer and a Blasius-type thermal layer are possible.

When T is large and then ρ made large the differential equations governing the convection are reduced from eighth order to sixth order owing to a becoming large as T → ∞. There are Ekman layers as T → ∞, when the horizontal boundaries are both rigid. The ρ thermal layers now have a different structure from the non-rotating problem for both rigid-rigid and free-free horizontal boundaries. The equation for small amplitude convection near to the marginal case is derived and the solution for free-free horizontal boundaries is obtained.

The transport properties of a diffusive interface with diffusivity ratio $\kappa_S/\kappa_T = {\textstyle\frac{1}{3}}$ have been measured, using salt and sugar as the diffusing components. The flux ratio is constant and equal to (κS/κT)½. The normalized salt flux is related to the density anomaly ratio Rρ = βΔS/αΔT by the power law F*T = 2·59Rρ−12.6 over four decades. Optical measurements show that the vertical gradients of concentration of salt and sugar within the interface are those required if molecular diffusion is to account for the whole flux of each component.

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.

Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.

The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.

A study is made of the natural convective flow which is induced in an infinite expanse of gas by the presence of a vertical hot flat plate from which hot gas of the same chemical type is being blown. The transpiration rate is assumed to be such that a self-similar boundary-layer type of solution is available. It differs from previous analyses in the following respects. Most important, the density is not assumed to be constant a t any stage in the description of the flow field. Also the form of the induced flow in the outer domain is calculated and proves to be substantially independent of the blowing rate in this case; the induced outer flow is found to be of large lateral extent. Finally, the variable-gas-property problem is carried to second order and solutions are obtained by using an ‘exact’ form of Howarth-Dorodnitsyn variable. The opportunity is taken t o make some comments about the comparison between theory and experiment for finite flat plates without transpiration.

An analysis is made of the axially symmetric flow of a rotating inviscid incompressible fluid into a point sink a t sufficiently low values of the Rossby number. Based on the experimental observations, a theoretical flow model involving a surface of velocity discontinuity which separates the central flowing core from the surrounding stagnant region is proposed. A family of solutions is obtained after posing the problem as one involving a free streamline which is the line of velocity discontinuity in the axial plane. It is found that the flow possesses a minimum flow force as well as a minimum energy flux. Corresponding to such a state, a unique intrinsic Rossby number R′ based on the properties of the flowing core with a value of 1/[npar ]8 is determined. A discussion is made of the flow field development induced by a sudden start of a sink discharge. A theoretical model involving a blocking wave propagating upstream is proposed. The speeds of blocking waves are found to be higher than the maximum group velocity of the infinitesimal waves for R > 0.06. On the other hand, for R < 0.03, the waves are linear and dispersive in nature.

A photographic study has been made of Kelvin-Helmholtz waves on thin liquid sheets and it has been found that, contrary to two-dimensional theory, wave growth is critically dependent upon sheet velocity and distance from the origin. This is attributed to boundary-layer separation and to subsequent vortex growth.

An infinitely long circular cylinder is fixed with its generators horizontal so that it is half-immersed, with its axis lying in the free surface of water. A regular train of water waves is incident on the cylinder from an arbitrary horizontal direction, and is partly reflected and partly transmitted under the cylinder. In the present paper we are concerned with the vertical component of the wave acting on the cylinder. It is assumed that the fluid is inviscid, that the fluid motion is irrotational, and that the depth of water is infinite. The equations of motion are linearized, and surface tension is neglected.

We shall find it convenient to use the fact that the required vertical force component can be inferred from the solution of a related problem, which we shall call the generalized heaving problem. In this latter problem a certain normal velocity is prescribed on the cylinder so that water waves which travel obliquely outwards are generated. There are no waves incident from infinity. When the prescribed velocity has the same phase everywhere on the cylinder the waves travel normally outwards, and in this case the generalized heaving problem reduces to the ordinary heaving problem, on which much information is already available. The generalized problem is solved here by a method which is a generalization of the known method (Ursell 1949) for ordinary heaving (when the wave crests are parallel to the cylinder axis). Generalized-added-mass coefficients and generalized-wave-making parameters for generalized heaving are computed for a range of wavenumbers and angles of travel, and are extended to larger wave-numbers by means of asymptotic analysis. Reciprocity relations (the Haskind relations) are then used to obtain the vertical force component in the original transmission problem from the wave-making parameters of the generalized heaving problem.

A consistent slender-body approximation is developed for the flow past a fish- like body with arbitrary combinations of body thickness and low-aspect-ratio fin appendages, but with the fins confined to the plane of symmetry of the body. Attention is focused on the interaction of the fin lifting surfaces with the body thickness, and especially on the dynamics of the vortex sheets shed from the fin trailing edges. This vorticity is convected by the (non-lifting) flow past the stretched-straight body, and departs significantly from the purely longitudinal orientation of conventional lifting-surface theory. Explicit results are given for axisymmetric bodies having fins with abrupt trailing edges, and calculations of the total lift force are presented for bodies with symmetric and asymmetric fin configurations, moving with a constant angle of attack.

If the air in a resonance tube half-filled with water is driven at an acoustic resonant frequency, surface waves of much lower frequency may also be excited. Experiments on this phenomenon are reported and compared with the theory of spatial resonance.

This paper describes an experimental and theoretical study of thermal convection in a sloping porous layer. The saturated layer is bounded by two parallel impermeable planes maintained at different temperatures. Several types of flows were observed: a unicellular movement and a juxtaposition of longitudinal coils or of polyhedral cells.

A theoretical analysis has been made using the standard bases of the linear theory of stability and by taking into account some assumptions suggested by experimental observations. The critical conditions for the transition between unicellular and polycellular flows has been determined. For flow in longitudinal coils or with polyhedral cells the average heat transfer depends mainly on the filtration Rayleigh number and on the slope of the layer.

The experimental study was made in a Rayleigh number range 0–800 and for various slopes (0–90°). For both the transition criterion and the heat transfer, a good fit was observed between the experimental and theoretical results. For maximum slope, i.e. 90°, a correlation which connects the Nusselt number with both the Rayleigh number and the vertical extent of the model is proposed.

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

Strongly nonlinear heat transport across a porous layer is studied using Howard's (1963) variational method. The analysis explores a bifurcation property of Busse's (1969) multi-a solution of this variational problem and complements the 1972 study of Busse & Joseph by further restricting the fields which are allowed to compete for the maximum heat transported a t a given temperature difference. The restriction arises, as in the case of infinite Prandtl number convection studied by Chan (1971), from letting a parameter tend to infinity from the outset; here, however, the parameter which is assumed infinitely large (the Prandtl-Darcy number) is actually seldom smaller than O(107).

The theoretical bounding heat-transport curve is computed numerically. The maximizing Nusselt number (Nu) curve is given a t first by a functional of the single-a solution; then this solution bifurcates and the Nusselt number functional is maximized for an interval of Rayleigh numbers (R) by the two-a solution. The agreement between the numerical analysis and recent experiments is striking. The theoretical heat-transport curve is found to be continuously differentiable but has piecewise discontinuous second derivatives.

The results of an asymptotic (R → ∞) analysis following Chan (1971) are in qualitative agreement with the results of numerical analysis and give the asymptotic law Nu = 0.016R. This law is consistent with the result of the porous version of the well-known dimensional argument leading to the one-third power law for regular convection. The asymptotic results, however, do not appear to be in good quantitabive agreement with the numerical results.

In this paper an exact solution is presented for the problem of unsteady laminar convective flow under a pressure gradient along a vertical pipe. We have obtained the solution of the problem on the basis of the assumption that the velocity and buoyancy profiles far from the pipe entrance do not change with the height, and the entry lengths have been ignored. The wall of the pipe is heated or cooled uniformly. We have discussed both the cases, when buoyancy forces act together with the pressure gradient or in opposite direction.

In the case when the upflow is heated (or a downflow is cooled) the velocity and thermal boundary layers are formed for sufficiently large Rayleigh numbers. In the second case which has been discussed in detail (when the upflow is cooled or the downflow is heated) we have found the critical value of the Rayleigh number R = R, beyond which the velocity profile and the temperature profile become unsteady and turbulent in all the cases. In the case of the elliptical cylinder R, increases up to 1730 as the ellipticity is increased while in the case of the co-axial pipes this Rayleigh number increases as the gap c between the cylinders is decreased (if c = a/b = 1·2 then R, = 60762, but decreases to 1 when c = 4). Besides this, the time required to reach steady state increases as the Rayleigh number increases in both circular and elliptical pipes; it also increases when the eccentricity is decreased. The cases discussed by Morton (1960 and Dalip Singh (1965 are particular cases of the results derived below.

In this investigation we have dealt with the following ducts: (i) circular tubes, (ii) elliptical tubes and (iii) co-axial tubes. The general solutions for both velocity and temperature fields have been found for the case when the pressure gradient is an arbitrary function of time, with an arbitrary heat source also present. Particular cases when both the parameters are absohte constants have been discussed in detail.

We have made use of finite transforms very frequently; especially for the case of an elliptical tube, a new transform involving Mathieu functions developed by Gupta (1964 has been used. A few new infinite series have been summed with the help of this transform.

Various non-dimensional quantities (for both the cylinders) such as the Nusselt number, volume flux and rate of heat transfer have been found when the pressure gradient and source of heat generation are absolute constants.

Experimental investigations of the heat transfer by combined free and forced convection to water in a horizontal tube with a uniform heat flux have revealed many interesting features of the transition from laminar to turbulent flow. Fluctuations in fluid temperature and overall pressure drop variations facilitated the study of the transition regime.

The first phase of the experiments studied the extreme cases of combined forced- and free-convection transition regimes. For forced convection without heat transfer (isothermal flow) the results compared well with the existing data, predicting a transitional Reynolds number of about 2300. For free convection with no flow, the records of temperature fluctuations during transient heating indicated the transition regime.

In the second phase, extensive measurements were made for the combined forced- and free-convection case. A plot of the Reynolds number against the Rayleigh number Rag enabled the present results (L/D = 150) to be compared with those of Petukhov & Polyakov (for X/D ratios 40 and 100).X = length from the start of flow; L = effective length of heated test section; D = inner diameter of tube. The results show that beyond transition two types of turbulence - hydrodynamic and thermal- can be distinguished depending on whether the fluctuations are pre- dominantly in velocity or in temperature. For large vaIues of Re and Ruq, these two regions merge into turbulent, combined free and forced convection.

In the transition regime two types of flow are encountered: unstable flow characterized by pulses and a stable flow with stable fluctuations. While a curve in the plot of heat input vs. flow velocity demarcates the two regimes, the main features of transition are described by means of an intermittency factor defined as the ratio of thermal fluctuation at any point to a corresponding magnitude of thermal fluctuation on the demarcating curve.

The iso-intermittency lines drawn in the transition envelope assist in describing the structure of transition. The concentration of these lines at the central area of the envelope indicates the region of maximum coupling between thermal and hydrodynamic effects.

A qualitative study of inlet turbulence confirmed the prediction of Mori et al. for air. The results presented here along with those of Mori et al. and Petukhov & Polyakov indicate that thermally induced secondary flows attenuate the fluctuations in low inlet turbulence flows, while they restabilize the flow as the inlet turbulence is increased. Also, the effect of inlet turbulence on transition to an unstable flow regime was studied. In general, an increase in the inlet turbulence brought about an increase in the critical Rayleigh number.

Linearized two-dimensional potential flow theory is applied to an airfoil with an upper surface spoiler. The spoiler wake is modelled as a cavity of empirically given constant pressure, and a sequence of conformal transformations maps the linearized physical plane, with a slit on the real axis representing the airfoil plus cavity, onto the upper half of the plane exterior to the unit circle. The complex acceleration potential is used, and its real part is specified on the real axis, repre- senting the cavity boundary, while its imaginary part is specified on the unit semicircle, representing the wetted surface of the airfoil and spoiler. Solutions are found for both the steady-state lift and the transient lift after spoiler actuation for airfoils of arbitrary camber, thickness and incidence, with and without a simple flap, and with spoilers of arbitrary position, height and angle. The empirical cavity pressure is arbitrary for the steady-state solution, but is assumed to have the free-stream value for the transient solutions. Comparisons are made with the results of wind-tunnel experiments, and, for the steady-state solutions, with predictions of an earlier theory. The agreement of the present theoretical predictions with the experimental results is generally good, and is in most cases somewhat better than that of the earlier theory.