The influence of drag-reducing polymers on the time-averaged velocity gradient $\overline{S_x}$ and on the two components of the fluctuating velocity gradient a t the wall, sx and sz, has been studied for turbulent flow in a 1 in. pipe. The time-averaged velocity gradient is related to the time-averaged wall stress by Newton's law of viscosity. The view is taken that a comparison of the turbulence structure with and without polymers should be made at the same $\overline{S_x} $. Therefore all turbulence measurements have been normalized with wall parameters. On this basis we find that the changes in the intensity of sx, and in the shape of the spectral density functions for sx and sz are not as striking as the decrease in pressure drop caused by the addition of a polymer. The root-mean-square value of sz shows a significant decrease with an increase in drag reduction but the most spectacular change in turbulence structure seems to be associated with the transverse correlation coefficient. We conclude that the increase in drag reduction is accompanied by an increase of the size of longitudinally oriented eddies in the viscous sublayer. One possible explanation for this increase in size is that the influence of the polymer additives is such as to increase the viscous resistance in the transverse direction more than in the direction of mean flow.

Experimental measurements of the advancing head of a gravity current are described. The head, of depth d in mean cross-section and advancing at mean speed U, has an elevated nose whose mean height h satisfies h/d = 0·61 Re-0·23±0·01 through the range 300 < Re < 10000 (Re = Ud/ν). A shifting pattern of lobes and clefts ranges across the head, in which the mean lobe width $\overline{b}$ is found experimentally to satisfy $\overline{b}/d = 7.4\,Re^{-0.39\pm 0.02}$. By moving the floor in the direction of the current, or by introducing a thin dense layer beneath it, the lobes and clefts can be suppressed and a purely two-dimensional flow obtained. It is concluded that the lower boundary plays an essential role in determining the substructure of the head and that this originates in convective instability as the head rides over less dense fluid.

We consider the propagation of a weak nonlinear wave whose energy is concentrated in a narrow band of wavenumbers in a fluid which is both dispersive and dissipative. We use the small amplitude equations of Whitham's theory of slowly varying wave trains, modified slightly to include dissipation, to show that the modulation of the wave may be described by a nonlinear Schrödinger equation. For long waves which are purely dispersive we obtain the Kortewegde Vries equation, and for long waves which are dissipative we obtain Burgers’ equation by suitable transformations of the nonlinear Schrödinger equation. We mention the problem of Stokes waves in deep water and comment briefly upon invariant far-field theory.