A recapitulation is first given of a recent theory of homogeneous turbulence based on the condition that the Fourier amplitudes of the velocity field be as randomly distributed as the dynamical equations permit. This theory involves the average infinitesimal-impulse-response functions of the Fourier amplitudes and employs a new kind of perturbation method which yields what are belived to be exact expansions of third- and higher-order statistical moments of the Fourier amplitudes in terms of second-order moments and these response functions.

In the present paper the theory is applied in lowest approximation (called the direct-interaction approximation) to stationary isotropic turbulence of very high Reynolds number. The characteristic wave-number $k_0 = \epsilon |v^3_0$ and Reynolds number $R_0 = v_0 k^{-1}_0 |v$ where v0 is the r.m.s. velocity in any given direction, ε is the power dissipated per unit mass, and ν is the kinematic viscosity, are introduced. For $R^{\frac {1}{3}}_0 \gg 1$ it is found that the inertial and dissipation ranges extend over wave-numbers k satisfying k0 [Lt ] k [Lt ] R0k0. The time-correlation and average infinitesimal-impulse-response functions of the Fourier amplitudes in these ranges are evaluated. They are found to be asymptotically identical and given by J1(2v0kτ)/v0kτ, where τ is the time interval.

The energy spectrum in these ranges is determined by a non-linear integral equation, involving the time-correlation and response functions, which is suitable for solution by iteration. The solution is of the form E(k)/v0v = (k/kd) $^{- \frac {3} {2}}$f(k/kd), where E(k) is the three-dimensional spectrum function $k_d = R^{\frac {2}{3}}_0 k_0$ is a wave-number characterizing the dissipation range, and f(k/kd) is a universal function. In the inertial range, E(k) = f(0) (ε V0)½$ k^{- \frac {3} {2}}$, asymptotically. Theparameter f(0) can be obtained by quadratures, without solving the integral equation for E(k). Spectral energy transport throughout the inertial and dissipation ranges is found to proceed by a cascade process essentially local in wave-number space; the direct power delivered by all modes below k to all modes above k′ [Gt ] k is of order ε(k/k′)$ ^{ \frac {3} {2}}$ if k and k′ both lie within the inertial range. The mean-square velocity derivatives of all orders are found to be finite. For $R^{\frac {1}{3}}_0 \gg 1$ the skewness factor of the distribution of the nth-order longitudinal velocity derivative is found to have the asymptotic form $A_n R^{-\frac {1}{6}}_0$ where An is a universal constant.

The theory is compared with experiment and is found to be slightly better supported than the Kolmogorov theory. However, it is stressed that extreme caution must be exercised in interpreting the experimental evidence as support for either theory.

An analysis is given of the relations between the Kolmogorov theory, Heisen-berg's heuristic theory, the analytical theories of Heisenberg and Chandrasekhar, the theories of Proudman & Reid and Tatsumi, and the present theory.

The analysis used by Taylor (1954) and based on the Reynolds analogy has been extended to describe the diffusion of marked fluid in the turbulent flow in an open channel. The coefficient of longitudinal diffusion arising from the combined action of turbulent lateral diffusion and convection by the mean flow is computed to be 5·9uτh, where h is the depth of fluid and uτ the friction velocity. This is in agreement with experiments described herein. The laterla diffusion coefficient is found by experiment to be 0·23uτh, which is three times larger than the value obtained by the assumption of isotropy. The same analysis can be used to describe the longitudinal dispersion of discrete particles, both of zero buoyancy and of finite buoyancy, and comparison is made with observations by Batchelor, Binnie & Phillips (1955) and Binnie & Phillips (1958).

An examination was made of the laminar travelling waves that disturbed the glassy surface in an open water channel, fitted with a trumpet entry and inclined downwards over the range 1 to $2 \frac{3}{4}^{\circ}$. The Reynolds number at their first appearance was observed, and measurements were obtained of their velocities and lengths, the latter being highly irregular. Comparison was made with the theoretical values after corrections had been introduced to allow for the higher velocities of flow near the walls due to the increased depth caused by surface tension. At greater discharges the onset of turbulence occurred in a random manner, leading to the production of intermittent bores moving more slowly than the laminar stream. Transition was found to have usually begun at a Reynolds number R = 2500, R being defined as the discharge per unit width divided by the kinematic viscosity. Except close to the inlet, the channel was continuously occupied by turbulent water when R was in the range 4000 to 4500.

A pulsating sphere, which performs a sequence of virtually impulsive changes in its radius with time, is completely surrounded by an inviscid, incompressible fluid whose velocity field is generally rotational. This paper indicates how it is possible, by means of Helmholtz's theorem, to relate the corresponding vorticity and velocity fields immediately before and after such expansions or contractions.

The method is then applied to the case of a spherical mass of fluid initially in uniform rotation in which a spherical core undergoes a single sudden expansion, followed after a short interval by an equally rapid contraction back to the original radius. An interesting meridional flow is thereby induced, which tends to decrease the angular velocity of rotation of the fluid near the poles at the outer surface, relative to that of the equatorial fluid. It is perhaps significant that this is in qualitative agreement with the variation of angular velocities observed at the surface of the sun.

The stability of a heavy top, containing a cylindrical cavity partly full of liquid, for small displacements from the sleeping position is studied. It is shown theoretically that instability can occur when any one of the periods of free oscillation of the liquid, which are doubly infinite in number, is sufficiently near to the period of nutation of the empty top. In experiments carried out by Prof. Ward, only the two principal instabilities could be distinguished.

In the laboratory experiments by Fultz & Riehl (1957) and by Hide (1958) on heated rotating liquids contained in the annulus between two cylinders, it has been observed that a strongly marked jet stream appears on the free surface of the liquid under certain conditions of rotating and heating. This jet stream meanders around the annulus in a regular wave-like pattern alternately approaching the outer and inner cylindrical boundaries. The present paper puts forward an analytical theory for this jet, or Rossby regime, in the course of which exact solutions are presented of certain fundamental non-linear partial differential equations. The assumption is made that the flow is geostrophic at the first approximation and that the heat transfer across the stream lines of geostrophic flow (that is, the isobars) is due to molecular conduction. From a calculation of the heat flow it appears that this leads to values of the heat transfer which are too small, so that the ageostrophic terms must be of importance in the actual heat transfer; never-theless, the exact solution obtained here probably reveals the mechanism of the change from one wave pattern to another and certainly provides an explanation for the observed upper limit to the number of waves in a given geometrical configuration, as discussed by Hide. It has been established that the mean zonal flow and mean zonal temperature field are dependent upon the amplitude function of a finite amplitude wave solution. In this exact solution it is found that the amplitude and phase functions of the wave patterns are themselves interdependent and that the shape of the wave depends on the quantity of heat and angular momentum being transferred. It is shown that the wave pattern consisting of an integral number m lobes or petals can exist only in a restricted range of the Rossby number S—this is well-known from the experimental work of Fultz and Hide.

A low speed parallel flow, whose velocity fluctuates sinusoidally in magnitude about a constant mean, has been produced in a boundary layer wind tunnel. Hot-wire measuring techniques have been developed to permit an investigation of the turbulent boundary layer developing on a flat plate with this free stream condition. The response of the layer at a Reynolds number $R_{\delta *} = (\overline {U}_\infty \delta^*)|v = 3\cdot 6 \times 10^3$ was measured for free stream fluctuation amplitudes up to 34% of the mean velocity and frequencies ranging from 0 to 48 cycles/sec. Here $\delta^* = \int ^\infty _0 \left(1 - \frac {\overline{U}} {\overline{U}_\infty} \right)dy$ is the boundary displacement thickness, $\overline {U}(x, y)$ (x, y) the mean velocity in the boundary layer, $\overline {U}_\infty$ the mean velocity in the free stream, and v the kinematic viscosity.

Measurements were made of the mean velocity, the amplitudes of in- and out-of-phase components of the first harmonic of the periodic fluctuations, and the intensity of higher harmonics and turbulence. It was found that non-linear effects, even at the largest fluctuation amplitudes, were so small that they were obscured by experimental errors.

This paper deals with the propagation of a sound pulse into a gas which initially has solid-body rotation and constant temperature, the initial pressure and density increasing outwards like ex, where x is the square of a certain dimensionless radial co-ordinate. The perturbations are due to a source-like disturbance on the axis of symmetry, which begins to act at time t = 0: most attention is paid to source strengths which vary in time like a Dirac pulse or a step function, but the following remarks apply generally.

Immediately behind the wave front the perturbation velocity and temperature decay like e-½x, while the (absolute) perturbation pressure and density grow like e½x (the relative pressure and density increments, which are referred to local conditions in the undisturbed state, then also decay like e-½x). The rotation also introduces oscillations in flows which, with the same disturbance at the origin and no rotation, would vary monotonically with time at a given point.