This paper involves a numerical simulation of the final stages of transition to turbulence in plane channel flow at a Reynolds number of 1500. Three-dimensional incompressible Navier–Stokes equations are numerically integrated to obtain the time evolution of two- and three-dimensional finite-amplitude disturbances. Computations are performed on the CYBER-203 vector processor for a 32 × 51 × 32 grid. Solutions indicate the existence of structures similar to those observed in the laboratory and characteristic of the various stages of transition that lead to final breakdown. In particular, evidence points to the formation of a A-shaped vortex and the subsequent system of horsehoe vortices inclined to the main flow direction as the primary elements of transition. Details of the resulting flow field after breakdown indicate the evolution of streaklike formations found in turbulent flows. Although the flow field does approach a steady state (turbulent channel flow), the introduction of subgrid-scale terms seems necessary to obtain fully developed turbulence statistics.

An incompressible inviscid fluid contained in a channel in a gravitational field admits soliton-like disturbances where the velocity potential depends upon all three coordinates as well as time, yet its centre of mass can be at rest. These solitons were recently discovered by Wu, Keolian & Rudnick. The calculations are carried out with the multiple-scales approach. Consequences of mass conservation and radiation are discussed.

A modulated cross-wave of resonant frequencyω1, carrier frequencyω =ω1 {1 + O(ε)}, slowly varying complex amplitude O(ε½b), longitudinal scale b/ε½ and timescale 1/εω is induced in a long channel of breadth b that contains water of depth d and is subjected to a vertical oscillation of amplitude O(εb) and frequency 2ω, where 0 < ε [Lt ] 1. The complex amplitude satisfies a cubic Schrödinger equation, generalized to incorporate weak damping and the parametric excitation. A solution is obtained that describes the standing solitary wave observed by Wu, Keolian & Rudnick (1984). The results depend on both d/b and l*/b, where l* is the capillary length (l* = 2.7 mm for clean water), and solitary waves are impossible if d/b < 0.325 for l*/b = 0 or if l*/b > 0.045 for d/b [gsim ] 1. The corresponding cnoidal waves (of which the solitary wave is a limiting case) are considered in an appendix.

This paper is a summary of the Fourth Beer-Sheva Seminar on Magnetohydrodynamic (MHD) Flows and Turbulence held in Israel during 27 February–2 March 1984 with 67 participants from 13 countries. Reviews and contributed papers were presented on laminar and turbulent single-phase and two-phase MHD flows, turbulent and two-phase flows without magnetic fields, and applications of MHD in power generation, in nuclear fission and fusion and in metallurgy.

Motions of a single vortex filament in a background flow are studied by numerical simulation of a set of model equations. The model, which in essence is due to Hama, treats the self-interaction of the filament through the so-called ‘localized-induction approximation’ (LIA). Interaction with the prescribed background field is treated by simply advecting the filament appropriately. We are particularly interested in elucidating the evolution of sinuous vortices such as the ‘wiggle’ seen by Breidenthal in the transition to three-dimensionality in the mixing layer. The model studied embodies two of the simplest ingredients that must enter into any dynamical explanation: induction and advection. For finite-amplitude phenomena we make contact with the theory of solitons on strong vortices developed by Betchov and Hasimoto. In a shear, solitons cannot exist, but solitary waves can, and their interactions with the shear are found to be key ingredients for an understanding of the behaviour of the vortex filament. When sheared, a soliton seems to act as a ‘nucleation site’ for the generation of a family of waves. Computed sequences are shown that display a remarkable morphological similarity to flow-visualization studies. The present application of fully nonlinear dynamics to a model presents an attractive alternative to the extrapolations from linearized stability theory applied to the full equations that have so far constituted the theoretical basis for understanding the experimental results.