An analytical expression for the fringe shift in the N-wave is derived from the improved linearized theory for a slender supersonic projectile. From this expression an approximate mapping function is found which gives a simple test for N-wave flow. The validity of the fringe-shift expression is quantitatively confirmed by measuring an interferogram of the flow around a sphere. N-wave flow is shown to exist around a small sphere for r greater than about 70 diameters. Measurements of the shock waves from a sphere and a cone-cylinder show that the shocks assume their asymptotic positions for r > 14 diameters for the sphere, and r > 7 diameters for the cone-cylinder.

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.

The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.

The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

The intensity of turbulent motion in the wake of square mesh grids of bars has been examined experimentally. The measurements show that there are considerable departures from homogeneity over surfaces normal to the mean stream. Large variations of intensity have been observed at a distance of 80 meshlengths from the grid, where they are diminishing slowly, if at all. This inhomogeneity is believed to be the cause of the discrepancies between various sets of measurements of energy decay in the ‘initial period’. There is also a possibility of error in other measurements made in grid turbulence and analysed on the assumption of homogeneity. Observations with a grid of 2 in. mesh show that v2 is significantly less than u2, which is consistent with results from other laboratories.

The partial differential equations which describe steady flow of fluid in saturated homogeneous permeable solid material under non-isothermal conditions are stated. From these are derived the equations for flow of liquid (in particular, water) using suitable approximations and making use of empirical laws when necessary.

It is then postulated that the only ‘ponderomotive’ (i.e. massmoving) forces present are those due to thermal expansion effects. Free convection results. An approximate solution of the equations is attempted for plane flow by means of classical perturbation methods, the temperature and stream-function variables being represented by power series in a convection parameter proportional to the Rayleigh number.

A numberical example of the method, with boundary conditions based on a geothermal area at Wairakei, New Zealand, is given. The results show features which are in fair agreement with temperature measurements made in the area, and it appears that the convection parameter η is of the order of 10.

A first-order relationship between changes in area and shock strength is derived for the case of a shock moving through a small area change in a channel. By integration of this relationship the area of the channel is obtained as a function of the shock strength in closed form. This result is interpreted as giving the average strength of a shock at a given time as it moves along a channel of arbitrary shape.

By suitable choices of the shape of the channel, descriptions of converging cylindrical and spherical shocks are obtained. These descriptions are found to be in close agreement with the similarity solutions valid near the points of collapse of the shocks. The reason for such good agreement is examined.

The results of measurements of the vertical distribution of airborne particles, released usually at a height of 500 ft., and sampled for periods of about 30 minutes at downwind distances of 100, 300 and 500 metres, are presented and discussed.

At all distances the frequency distributions of the particle elevation with respect to the site of release are closely similar in shape and size to the frequency distribution of wind inclination at the site of release. This is interpreted as showing that, despite the uncorrelated effects of small eddies, high correlation was maintained in the motion of a particle, for periods of 1 minute or more, by the dominant action of persistent eddies which contributed heavily to the turbulent energy. In contrast to this result, the wind observations showed that the Eulerian auto-correlation coefficient (Rr) fell to about 0.2 in 10 seconds.

In the author's paper “Drift” (Lighthill 1956) there are mistakes which need correction. Equation (28) should read $\phi \sim -\frac {m}{4\pi U_r}\;\; as \;\; r \rightarrow \infty$ (28) since the definition of the disturbance potential ϕ (p. 35) states that the full velocity potential is U(x + ϕ). Hence by (18) equation (29) for the asymptotic form of the secondary flow due to a source should read $v_x = -\frac {Amy} {4 \pi U (y^2 + z^2)} \left(1 + \frac{\chi}{r}\right),\;\;\;\; v_y = -\frac {Am}{4\pi Ur}, \;\;\;\; v_z=0$ (29) where as well as the U's a 4π was accidentally missing from the version printed and a y misprinted as x.