Remarkable differences in dispersion of a tracer material injected into turbulent pipe flows of water and water containing as little as 2·5 parts per million by weight of a soluble high-molecular-weight drag-reducing polyoxyethylene additive have been measured. Analysis of the tracer response curves in terms of a simple one-parameter model shows that the observed results are compatible with a drag-reduction mechanism based on thickening of the viscous sublayer adjoining the wall. Other experiments, reported briefly, suggest that polymer adsorption on to the wall is responsible for this thickening.

The thermal boundary layer near a heated vertical plate in a poorly conducting liquid is subject to a horizontal d.c. electric field. If the electric field is strong enough, the boundary layer becomes unstable. In this paper a theory is developed to predict the onset of this instability. Experiments measuring the threshold voltage for instability are compared with the theoretical predictions. Other experiments are reported which determine the effect of this instability on the heat transferred from the heated plate.

This paper examines a region of rapidly varied flow in a two-layered density stratified system with one layer flowing, and the other stationary. The analogous phenomenon in open channel hydraulics is the hydraulic jump. In density stratified flows the phenomenon is referred to as a density jump because it is generally accompanied by a change in density of the flowing layer.

It is shown there is a fundamental difference between the hydraulic jump and the density jump in that flow conditions on either side of a density jump are not uniquely related. A density jump with given flow conditions upstream has a range of possible states which may be attained downstream. The rate of entrainment of ambient fluid into a density jump, and the conditions downstream of the jump, are determined by the downstream control and the upsteam conditions. The particular case of a density jump controlled by a broad crested weir downstream is examined in detail.

When a container of fluid of arbitrary shape is heated from below and the temperature gradient exceeds a critical value ([Rscr ]c2) the conduction solution with no motion becomes unstable and is replaced by convection. The convection may have two forms: one with ‘upflow’ at the centre of the container and one with ‘downflow’ there. Here we study the stability of the two forms of convection. Both forms are here shown to be stable to infinitesimal disturbances. When the viscosity varies with the temperature or the conduction profile is not linear, etc., the steady convection can be driven with finite amplitudes |ε| at subcritical values of the temperature contrast ([Rscr ]2 < [Rscr ]c2). This subcritical convection is stable when the convection is strong (|ε| > |ε*| > 0) but is unstable when the convection is feeble (|ε| > |ε*|). Hence, when |ε| > |ε*| and [Rscr ]2 < [Rscr ]c2 either ‘upflow’ or ‘downflow’, but not both, is stable. When [Rscr ]2 > [Rscr ]c2, however, both the ‘upflow’ and the ‘downflow’ can be stable. This contrasts with the corresponding situation which is known to hold when the container is an unbounded layer. In the layer there is only one stable form of convection. The difference between the bounded domain with two forms of convection and the layer with just one stable form is traced to the mathematical property of simplicity of [Rscr ]c2 when viewed as an eigenvalue of the linear stability problem for the conduction solution. It is argued that [Rscr ]c2 is a simple eigenvalue in most domains, but in the layer [Rscr ]c2 can have infinite multiplicity. The explanation of the transition from the bounded domain to the unbounded layer is sought (1) in the chaotic conditions which frequently prevail at the edges of a ‘bounded’ layer and (2) in the fact that in the layer of large horizontal extent, the higher eigenvalues crowd [Rscr ]c2. In the course of the explanation, a new exact solution of the linear Bénard problem in a cylinder with a rigid side wall and a stress-free top and bottom is derived.

Vapour bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)½ where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For \[ \Delta p/\rho = 10^6 {\rm cm}^2/\sec^2 \approx 1\, \hbox{atm/density of water} \] the jet had a speed of about 130m/sec in the first case and 170m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapour are not important.

The motion induced in a layer of Boussinesq fluid by a moving thermal wave is studied in the case where the mean velocity could exceed the wave speed. A nonlinear boundary-layer theory shows that strong streaming is possible for small viscosity. Velocity fluctuations are limited in magnitude by their short time scale, while viscosity alone, assumed to be relatively weak, limits the mean flow.

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

The truncated Burgers models have a unique equilibrium state which is defined continuously for all the Reynolds numbers and attainable from a realizable class of initial disturbances. Hence, they represent a sequence of convergent approximations to the original (untruncated) Burgers problem. We have pointed out that consideration of certain degenerate equilibrium states can lead to the successive turbulence-turbulence transitions and finite-jump transitions that were suggested by Case & Chiu. As a prototype of the Navier–Stokes equations, Burgers model can simulate the initial-value type of numerical integration of the Fourier amplitude equations for a turbulent channel flow. Thus, the Burgers model dynamics display certain idiosyncrasies of the actual channel flow problem described by a truncated set of Fourier amplitude equations, which includes only a modest number of modes due to the limited capability of the computer at hand.

Based on a set of modulation equations derived in a previous paper, the nonlinear evolution of wave envelope in deep water is studied numerically. It is found that the wave envelope tends to disintegrate to multiple groups of waves each of which approaches a stable permanent envelope representing dynamical equilibrium between the amplitude dispersion and the frequency dispersion. Qualitative agreement with the experimental measurements of Feir (1967) is also observed.

A direct numerical integration of the time-dependent, incompressible Navier-Stokes equations is used to treat the nonlinear evolution of perturbed, linearly unstable, nearly parallel shear flow profiles in two dimensions. Calculations have been made for infinite (inviscid) and finite Reynolds numbers. The latter results are compared with laboratory measurements of Sato & Kuriki for wakes behind thin flat plates, and many of the detailed features are in excellent agreement, including mean flow profiles with ‘overshoot’ development, first harmonic energy profiles with off-axis nulls, and first harmonic phase profiles. The linear instability saturates by forming a vortex street consisting of elliptical vortex pairs. The solutions are followed for times up to eleven linear exponentiation times of the unstable disturbance. A new low-frequency non-linear oscillation is found, which explains the features of the above experiment, including the nearly periodic phase inversions in the first harmonic component of the longitudinal velocity. It results from a nutation of the elliptical vortices with respect to the mean flow direction. Inertial range spectral energy properties are also examined. Inviscid solutions have large wave-number spectral energies obeying the approximate power law, Ek ∼ k−μ, where μ lies between 3 and 4.

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R < RE, the buoyancy boundary layer is the unique steady solution of the Boussinesq equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler–Lagrange equations. Analytic lower bounds to RE are obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE < R < RL, in which subcritical instabilities are allowable.

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R < RE, the Ekman layer is the unique steady solution of the Navier-Stokes equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler-Lagrange equations. An analytic lower bound to RE is obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE < R < RL, in which subcritical instabilities are allowable.