The stability of individual inviscid barotropic planetary waves and zonal flow on a sphere to small disturbances is examined by means of numerical solution of the algebraic eigenvalue problem arising from the spectral form of the governing equations. It is shown that waves with total wavenumber n (the lower index of the Legendre function Pmn which describes the waves’ meridional structure) less than 3 are stable for all amplitudes, whereas those with n ≥ 3 are unstable if their amplitudes are sufficiently large. For travelling waves (m ≠ 0) with n = 3 and 4 and with disturbances comprised of 30 modes, the amplitudes required for instability are approximated by those obtained from triad interactions, and are smaller than those given by Hoskins (1973). For the zonal-flow modes (m = 0) the critical amplitudes are smaller than those predicted by triad interactions, and are close to those obtained from Rayleigh's classical criterion.

A number of initial- and boundary-value problems for the Boussinesq equations are solved by a finite-difference technique, in an attempt to see how a stably-stratified horizontal shear layer rolls up into horizontally periodic billows of concentrated vorticity, such as are frequently observed in the atmosphere and oceans. This paper describes the methods, results and accuracy of the numerical simulations. The results are further analysed and approximately reproduced by a simple semi-analytic model in Corcos & Sherman (1976).

The detailed dynamics of an unstable free shear layer are examined for a gravitationally stable or neutral fluid. This first article focuses on the part of the evolution that precedes the first subharmonic interaction. This consists of the transformation of selectively amplified sinusoidal waves into periodically spaced regions of vorticity concentration (the cores) joined by thin layers (the braids), in which vorticity is also concentrated. The thin layers are the channels along which vorticity is advected into the cores, and the cores provide the strain which creates the braids. For moderately long waves an analysis is given of the braid structure as a function of time. For gravitationally stable shear layers at high Reynolds numbers, the local vorticity reaches such large values as to cause secondary shear instability on a small (length) and short (time) scale. A physical account of the primary instability and its self-limiting mechanism is used as a basis for a computation, which yields growth rates and maximum amplitude as a function of initial layer parameters. The computation supplies the wavelength of waves that grow to achieve the largest (absolute) amplitude. Finally, the model makes it clear that, in the absence of secondary instability, this initial phase of the nonlinear development of the layer contributes only a modicum of additional mixing, especially at high Reynolds numbers.

As part of a general investigation of complex turbulent flows, extensive one-point measurements have been made of the turbulence structure of the mixing layer bounding a normally impinging plane jet with an irrotational core. The ratio of shear-layer thickness to streamline radius of curvature reaches a maximum of about 0.2, the sense of the curvature being stabilizing. Downstream of the impingement region the shear layer returns asymptotically to being a classical plane mixing layer. The most striking feature of the results is that the return is not monotonic: after decreasing in the region of stabilizing curvature, the Reynolds stresses, triple products, energy dissipation rate and other turbulence quantities overshoot the plane-layer values before finally decreasing. Some conclusions are drawn about the nature of the turbulent transport of Reynolds stress, and about the representation of this and other processes in calculation methods for complex turbulent flows. An incidental result of the work is a comprehensive set of measurements in a plane mixing layer.

This paper describes part of a detailed study of the initial region of coaxial jets of three different mean-velocity ratios. Similarity of the mean-velocity and turbulent-intensity profiles within the two mixing regions inside the initial merging zone, and within the mixing region inside the fully-merged zone, has been observed. Similarity with single-jet results has been obtained. Overall pressure measurements and hot-wire and microphone spectra inside coaxial jets yield two pronounced peaks, suggesting the existence of two types of vortices at different frequencies. The higher-frequency or primary vortices are found to be generated in the primary or inner mixing region, and the lower-frequency or secondary ones in the secondary or outer mixing region. It seems that the former are generated further upstream than the latter. Both types of vortices are found in the initial merging zone. However, their growth or decay and their dominance depend on the mean-velocity ratio. Low mean-velocity ratios indicate the dominance of the high-frequency vortices. At high mean-velocity ratios, the low-frequency vortices are dominant. The Strouhal number of the vortices plays an important part in their growth and decay. The complicated flow structure of coaxial jets can be very simply related to, and described by, the much simpler structure of single jets.

The interface between two miscible fluids which have identical mechanical properties but disparate electrical conductivities and are stressed by an equilibrium tangential electric field is studied experimentally and theoretically. A bulk-coupled electrohydrodynamic instability associated with the diffusive distribution of fluid conductivity at the interface is experimentally observed.

The configuration is modelled using a layer of exponentially varying conductivity spliced on each surface to a constant-conductivity fluid half-space. Over-stable (propagating) modes are discovered and characterized in terms of the complex growth rate and fastest growing wavenumber, with the conductivity ratio and an inertia-viscosity time-constant ratio as parameters. In the low inertia limit, growth rates are governed by the electric-viscous time τ = η/εE2. Instability is found also with the layer of varying conductivity bounded by rigid equipotential walls. A physical mechanism leading to theoretically determined fluid streamlines in the form of propagating cells is described.

At relatively high electric fields, large-scale mixing of the fluid components is observed. Photocell measurements of distributions of average fluid properties demonstrate evolution in time on a scale determined by τ.

In three-dimensional Bénard convection regions of rising and sinking fluid are dissimilar. This geometrical effect is studied for axisymmetric convection in a Boussinesq fluid contained in a cylindrical cell with free boundaries. Near the critical Rayleigh number Rc the solution is obtained from a perturbation expansion, valid only if both the Reynolds number and the Péclet number are small. For values of the Nusselt number N ≤ 2 accurate solutions are provided by an expansion in a finite number of vertical modes. For Prandtl numbers p < 1 the form of the solution changes at large Reynolds number and becomes independent of p; in the limit p → 0 there is an effective critical Rayleigh number R* = 1.32Rc, which can also be derived by a perturbation procedure, and the Nusselt number is a function of the Rayleigh number only. Numerical experiments yield solutions for Rayleigh numbers R ≤ 100Rc and p ≥ 0.01. The results are similar to those for two-dimensional rolls and for R ≥ 5Rc the Nusselt number shows only a weak dependence on p. For p > 1 there is a viscous regime with N ≈ 2(R/Rc)1/3; when R/Rc [gsim ] p3/2, N increases more rapidly, approximately as R0.4. At high Rayleigh numbers a large isothermal region develops, in which the ratio of vorticity to distance from the axis is nearly constant.

A thermally driven steady axisymmetric flow of gas of small diffusivity in a vertical circular cylinder rotating rapidly about its axis of symmetry is studied. The side wall is a thermal insulator and the horizontal end plates are perfect conductors. The temperature of the top end plate is kept slightly higher than that of the bottom one.

The boundary-layer method is applied to solve the linearized basic equations and the following results are obtained.

1. The axial velocity in the inner core is fully controlled by the Ekman suction on the horizontal plates and is the same as that in the case of a perfectly conducting side wall.

2. The closed circulation in the side-wall Stewartson E½ layer is strongly suppressed compared with the case of a perfectly conducting side wall.

This situation is reflected in the inner temperature field, which deviates from that in the case of a perfectly conducting side wall. The critical parameter governing the solution is found to be (γ − 1) PrG0E−1/3/4γ, where Pr is the Prandtl number, γ the ratio of specific heats, E the Ekman number and G0 the square of the Mach number based on the peripheral speed of the cylinder.