It has been shown experimentally (Lee et al. 1982) that water drops with injected air bubbles inside them may be forced dynamically to assume the spherosymmetric shape. Linear analysis is unable to predict a centring mechanism, but provides two distinct modes of oscillation. Weakly nonlinear theory (Tsamopoulos & Brown 1987) indicates that centring of the bubble inside the drop occurs when the two interfaces move out of phase. A hybrid boundary element-finite element schemes is used here to study the complete effect of nonlinearity on the dynamics of the motion. The gas inside the liquid shell may be considered either incompressible or compressible by using a polytropic relation. In both cases, the present calculations show that besides the fast oscillation of the shell due to an initial disturbance, a slow oscillatory motion of the centres of the bubble and the drop is induced around the concentric configuration. This occurs in both modes of oscillation and is a direct result of Bernoulli's law. Furthermore, when this slow oscillation is damped by viscous forces, it is anticipated that it will lead to a spherosymmetric shape.

The interaction of two identical circular viscous vortex rings starting in a side-by-side configuration is investigated by solving the Navier–Stokes equation using a spectral method with 643 grid points. This study covers initial Reynolds numbers (ratio of circulation to viscosity) up to 1153. The vortices undergo two successive reconnections, fusion and fission, as has been visualized experimentally, but the simulation shows topological details not observed in experiments. The shapes of the evolving vortex rings are different for different initial conditions, but the mechanism of the reconnection is explained by bridging (Melander & Hussain 1988) except that the bridges are created on the front of the dipole close to the position of the maximum strain rate. Spatial structures of various field quantities are compared. It is found that domains of high energy dissipation and high enstrophy production overlap, and that they are highly localized in space compared with the regions of concentrated vorticity. The kinetic energy decays according to the same power laws as found in fully developed turbulence, consistent with concentrated regions of energy dissipation. The main vortex cores survive for a relatively long time. On the other hand, the helicity density which is higher in roots of bridges and threads (or legs) changes rapidly in time. The high-helicity-density and high-energy-dissipation regions overlap significantly although their peaks do not always do so. Thus a long-lived structure may carry high-vorticity rather than necessarily high-helicity density. It is shown that the time evolution of concentration of a passive scalar is quite different from that of the vorticity field, confirming our longstanding warning against relying too heavily on flow visualization in laboratory experiments for studying vortex dynamics and coherent structures.

The quantitative effects of uniform strain and background rotation on the stability of a strip of constant vorticity (a simple shear layer) are examined. The thickness of the strip decreases in time under the strain, so it is necessary to formulate the linear stability analysis for a time-dependent basic flow. The results show that even a strain rate γ (scaled with the vorticity of the strip) as small as 0.25 suppresses the conventional Rayleigh shear instability mechanism, in the sense that the r.m.s. wave steepness cannot amplify by more than a certain factor, and must eventually decay. For γ < 0.25 the amplification factor increases as γ decreases; however, it is only 3 when γ ė 0.065. Numerical simulations confirm the predictions of linear theory at small steepness and predict a threshold value necessary for the formation of coherent vortices. The results help to explain the impression from numerous simulations of two-dimensional turbulence reported in the literature that filaments of vorticity infrequently roll up into vortices. The stabilization effect may be expected to extend to two- and three-dimensional quasi-geostrophic flows.

Two-layer hydraulics is developed for problems in which the moving layers can have stagnant layers above and below, the two internal wave modes can have comparable speeds and the total depth of the moving layers may vary. The general development allows both Boussinesq and non-Boussinesq problems to be studied. Solutions are presented in the Froude-number plane and the effect of different layer densities on the form of the solution space is shown. The theory is applied to two-layer plunging flows and a variety of controlled solutions are found. Solutions for the 2½-layer theory and the plunging flow theory are demonstrated experimentally. Shear instability is often observed in the divergent section of the channel.