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.

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.

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.

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.

Experiments are performed to measure the upward velocity U of the cavity resulting from the draining of a vertical liquid-filled tube of radius b, which rotates about its axis at various speeds. With no rotation the cavity moves so that the Froude number U/(gb)½ is constant, where g is the gravitational acceleration. For high angular velocities Ω, however, the cavity appears to adopt a constant value of the Rossby number U/Ωb, implying that gravitational forces are dominaked by centrifugal forces. The cavity velocity finally achieved is found to be the same as the maximum group velocity of infinitesimal waves, so that U/Ωb = 0.52. Approximate theoretical models which satisfactorily describe the development of the motion are constructed.

The nonlinear hydromagnetic dynamo problem is investigated for the case of convection in a layer of an electrically conducting fluid heated from below. It is shown that two-dimensional convection rolls in conjunction with a longitudinal mean flow are capable of amplifying a magnetic field in the form of a wave propagating in the longitudinal direction. The action of the Lorentz forces causes a reduction of the amplitude of convection with the consequence that the energy of the magnetic field cannot grow beyond an equilibrium value which is determined as a function of the parameters of the problem. The analysis is based on an expansion in powers of the longitudinal wavenumber β of the magnetic field and applies in the case of large values of the magnetic Prandtl number.

The movement of a heavy spherical particle in vertically oscillating liquid is studied by means of an analog computer. Using the Oseen-Tchen-Houghton force-balance model, it is shown that in the case of asymmetrical oscillations the particle may be made quasi-stationary with a finite amplitude of oscillation. It is suggested that under such conditions a mean movement of the particle in the upward direction is also possible.

A series of hot-wire measurements has been carried out in a plane wake to investigate the structure of the turbulence boundary and the relation of its instantaneous position to the behaviour in the core of the flow. The principal measured quantities are as follows: mean velocity profile; intermittency factor; burst rate; mean of the longitudinal component of velocity conditioned upon various specified interface positions; autocorrelation of the intermittency signal; probability densities a t the half-intermittency point for the time between bursts and the duration of a burst; probability density for the longitudinal velocity component and its time derivative at various points across the wake; probability density a t the half-intermittency point for the same quantities in the turbulent and irrotational zones separately. In addition, the profile of the second moment of the probability density for the time between bursts has been obtained indirectly and part of the theory of Phillips (1955) has been shown to be applicable in the intermittent region.

The present measurements appear to indicate that the turbulence boundary in the wake resembles that in other plane flows more closely than has been supposed hitherto. The theory of normally distributed random noise was found to explain many of the observed statistical properties of the turbulence boundary.

An experimental study of fully developed uniform-density turbulent flow in a circular pipe containing one or two rods located off-centre is described. The friction factor in both cases was found to be approximately 5 % higher than the simple pipe friction factor. The shear stress distribution on the rod surface was determined using calibrated boundary-layer fences. The normalized shear stress distributions were independent of Reynolds number in the range 3·7 × 104 to 2·15 × 105. Mean-velocity measurements were obtained to check the validity of the universal law of the wall near the rod surface. Secondary-flow velocities were measured by a hot-wire anemometer and integrated to yield the secondary-flow stream function. Secondary-flow velocities of the order of 1 % of the mean velocity were observed. In the gap between the two pins, however, the secondary-flow velocities were only ½% of the mean velocity. It is demonstrated that the secondary flow cannot be neglected if a force balance is used to determine the shear stress distribution on the rod surface.

An experimental study of the structure of turbulence in a conical diffuser having a total divergence angle of 8° and an area ratio of 4: 1 with fully developed flow at entry is described. Theresearch has been done for pipe entry Reynolds numbers of 152 000 and 293000 of profiles of the mean pressure, mean velocity, turbulence intensities, correlation coefficients and the one-dimensional energy spectra, but owing to similar behaviour for these two Reynolds numbers, data are presented for a Reynolds number of 293 000.

The results show that the rate of turbulent energy production approximately reaches a maximum value at the edge of the wall layer extending to the point of maximum u1-fluctuation. It is found that, within the layer, $\overline{u^2_1}$ varies linearly with the distance from the wall and the linear range grows with distance in the downstream direction.

The turbulent kinetic energy balance indicates that the magnitude of the energy convective diffusion due to kinetic and pressure effects is comparable with that of the energy production.