The steady, spherically symmetric flow of a compressible gas is considered. The gas is both viscous and heat-conducting. In the limit of very high Reynolds number (= α−1, α → 0) and correspondingly low pressure at infinity, the structure of the whole flow field is discussed. The five regions that arise by virtue of the limit α → 0 are briefly considered. Special care is given to the matching across the overlap domains and the first region (close to, but outside, the sonic point) and the fifth (where the pressure adjusts to its ambient value) are carefully examined. It is argued that the application of appropriate matching principles, together with judicious use of numerical solutions, allows an arbitrary pressure and temperature to be assigned to the background gas.

Experiments have been carried out to investigate the effect of rotation of the whole system on decaying turbulence, generally similar to grid turbulence, generated in air in an annular container on a rotating table. Measurements to determine the structure of the turbulence were made during its decay, mean quantities being determined by a mixture of time and ensemble averaging. Quantities measured (as functions of time after the turbulence generation) were turbulence intensities perpendicular to and parallel to the rotation axis, spectra of these two components with respect to a wavenumber perpendicular to the rotation axis, and some correlation coefficients, selected to detect differences in length scales perpendicular and parallel to the rotation axis. The intensity measurements were made for a wide range of rotation rates; the other measurements were made at a single rotation rate (selected to give a Rossby number varying during the decay from about 1 to small values) and, for comparison, at zero rotation. Subsidiary experiments were carried out to measure the spin-up time of the system, and to determine whether the turbulence produced any mean flow relative to the container.

A principal result is that increasing the rotation rate produces faster decay of the turbulence; the nature of the additional energy sink is an important part of the interpretation. Other features of the results are as follows: the measurements with-outrotation can be satisfactorily related to wind-tunnel measurements; even with rotation, the ratio of the intensities in the two directions remains substantially constant; the normalized spectra for the rotating and the non-rotating cases show surprising similarity but do contain slight systematic differences, consistent with the length scales indicated by the correlations; rotation produces a large increase in the length scale parallel to the rotation axis and a smaller increase in that perpendicular to it; the turbulence produces no measurable mean flow.

A model for the interpretation of the results is developed in terms of the action of inertial waves in carrying energy to the boundaries of the enclosure, where it is dissipated in viscous boundary layers. The model provides satisfactory explanations of the overall decay of the turbulence and of the decay of individual spectral components. Transfer of energy between wavenumbers plays a much less significant role in the dynamics of decay than in a non-rotating fluid. The relationship of the model to the interpretation of the length-scale difference in terms of the Taylor-Proudman theorem is discussed.

The model implies that the overall dimensions of the system enter in an important way into the dynamics. This imposes a serious limitation on the application of the results to the geophysical situations at which experiments of this type are aimed.

The paper includes some discussion of the possibility of energy transfer from the turbulence to a mean motion (the ‘vorticity expulsion’ hypothesis). It is possible, on the basis of the observations, to exclude this process as the additional turbulence energy sink. But this does not provide any evidence either for or against the hypothesis in the conditions for which it has been postulated.

This paper describes the measured response of grid turbulence to three limiting types of uniform homogeneous strain: plane straining, axisymmetric elongation and axisymmetric flattening. Straining was achieved by allowing the turbulence to be convected through suitable distorting ducts; the maximum strain ratios were 5·8, 6·0 and 2·3, respectively. An attempt is made, using rapid-distortion theory, to specify an effective strain which accounts for the initial anisotropy of the grid turbulence; in the experiments, this effect was most important for the third species of strain. The maximum effective strain ratios were calculated as 4·05, 7·2 and 2·75, respectively. The rapid-distortion results are able to describe several features of the response of the turbulence with good accuracy: (i) the variation of total turbulence energy through the experimental ducts; (ii) the tendency of one component (that in the direction of the (larger) negative strain) to contain one-half of the turbulence energy after only moderate straining; and (iii) the changes in dimensionless structure parameters composed of ratios of component intensities. The first kind of prediction requires that the concurrent decay be specified in a simple way; (ii) and (iii) require that the initial anisotropy be taken into account. The predictions (iii) are generally less accurate than the others. The degree of success achieved by the rapid-distortion hypo-thesis is rather surprising, since the strain rates in the experiments were not as large as those for which the theory might have been expected to be valid. It is concluded that successful models of turbulence must provide a vorticity amplification essentially like that of rapid-distortion theory. However, the simple distorting flows considered here may not provide severe tests of more refined models, since many features of the response have already been accounted for.

We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possibly useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of ‘modal equations’ is analysed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.

Dissipative heating is produced by irreversible processes, such as viscous or ohmic heating, in a convecting fluid; its importance depends on the ratio d/HT of the depth of the convecting region to the temperature scale height. Integrating the entropy equation for steady flow yields an upper bound to the total rate of dissipative heating in a convecting layer. For liquids there is a regime in which the ratio of dissipative heating to the convected heat flux is approximately equal to c(d/HT), where the constant c is independent of the Rayleigh number. This result is confirmed by numerical experiments using the Boussinesq approximation, which is valid only if d/HT is small. For deep layers the dissipative heating rate may be much greater than the convected heat flux. If the earth's magnetic field is maintained by a convectively driven dynamo, ohmic losses are limited to 5% of the convected flux emerging from the core. In the earth's mantle viscous heating may be important locally beneath ridges and behind island arcs.

The slow motion of a body in a viscous shearing field is examined. Variational principles are used to derive inequalities which approximate the elements of the shearing matrix M of a body of arbitrary shape, where M is the matrix relating the force, torque and stresslet exerted by the body on the fluid to the relative translational and rotational velocities of the body and the rate of deformation of the undisturbed linear field. An upper bound for the elements of M is obtained by showing that the quadratic form of M increases monotonically with B, the region occupied by the body, while a lower bound for this form is given in terms of the electrostatic properties of a conductor and a dielectric of the same shape as B. Particular attention is paid to bodies of revolution, for which certain more definitive results are obtained: for example, their resistance to a rotation with axial symmetry is always less than twice their resistance to a rotation perpendicular to their axis.

The stability of a baroclinic zonal current to symmetric perturbations on a meridionally unbounded f-plane is considered. The lower boundary is at rest but the upper one moves with a constant velocity in keeping with the velocity of the zonal current. Following Stone (1966) a horizontal length scale O(Ro) is taken, where Ro is the Rossby number, with the Richardson number Ri = O(1). Instability sets in when the wavelength is O(E1/3), where E is the Ekman number based on the distance between the rigid horizontal boundaries, which corresponds to Stone's inviscid value zero, and to McIntyre's (1970) value infinity on a length scale O(E½).

A nonlinear analysis about the point of onset of instability yields the result that for the monotonic mode zonal momentum is convected polewards. The possible implications of this result for the dynamics of Jupiter's atmosphere are discussed.

Some of the consequences of the conservation of magnetic helicity $\int \rm{a.b}\it{d}^{\rm{3}}\rm{r\qquad (a\; =\; vector\; potential\; of\; magnetic\; field\; b)}$ for incompressible three-dimensional turbulent MHD flows are investigated. Absolute equilibrium spectra for inviscid infinitely conducting flows truncated at lower and upper wavenumbers kmin and kmax are obtained. When the total magnetic helicity approaches an upper limit given by the total energy (kinetic plus magnetic) divided by kmin, the spectra of magnetic energy and helicity are strongly peaked near kmin; in addition, when the cross-correlations between the velocity and magnetic fields are small, the magnetic energy density near kmin greatly exceeds the kinetic energy density. Several arguments are presented in favour of the existence of inverse cascades of magnetic helicity towards small wavenumbers leading to the generation of large-scale magnetic energy.

Flow visualization by the schlieren technique in the neighbourhood of a fully developed cavity on two axisymmetric headforms has shown the existence of laminar boundary-layer separation upstream of cavitation separation, and the distance between the two separations to be strongly dependent on Reynolds number. Based on present results, a semi-empirical method is developed to predict the position of cavitation separation on a smooth body. The method applies only in the Reynolds-number range when the cavitating body possesses laminar boundary-layer separation under non-cavitating conditions. Calculated positions of cavitation separation on a sphere by the method show good agreement with experimentally observed positions.

In this paper we apply the techniques of higher-order boundary-layer theory to study the steady streaming induced in the neighbourhood of a cylinder which vibrates harmonically, perpendicular to its generators, in an unbounded fluid. The theoretical predictions are compared with the results of experiments performed at high streaming Reynolds numbers. Improved agreement between theory and experiment is achieved although unresolved discrepancies remain.

Profiles of mean and fluctuating turbulent velocities and temperatures, of the time derivative of velocity and of intermittency have been measured in the wake of an unheated as well as a heated sphere in the low Reynolds number range 600–2500 at distances 80–1800 sphere diameters downstream. The wind-tunnel experiments exhibit a strong dependence on Reynolds number but they do not indicate the attainment of or an approach to the final period of turbulent decay which has been explored theoretically by Phillips and by O'Brien. The lowest turbulence Reynolds number obtained was Rλ, = 1·4.