Measurements of the velocity, pressure, and turbulence behind a series of normal plates in the uniform test section of an air tunnel are described, the oscillation of the wake being prevented in all but one test through use of symmetrically located tail plates. By a combination of experimental and computational techniques, details of the pattern of flow over a wall on a plane boundary in an infinite fluid are closely approximated. A significant difference is indicated between the characteristics of such a flow and those of the flow past an isolated plate with oscillating wake.

This paper contains a theoretical investigation of the displacement effect of a pitot tube in a shear flow. Viscosity is neglected throughout so that the vorticity field alone is considered.

It is first shown that a two-dimensional approach does not produce a large enough displacement effect because it does not include the stretching of vortex tubes that takes place around a three-dimensional pitot tube. Then the three-dimensional problem is considered. A solution is obtained in the plane of symmetry for a sphere in a shear flow. This solution is found by making an assumption about the rate of stetching of vortex tubes perpendicular to the plane of symmetry and then considering the shear flow as a small perturbation of a uniform flow. A solution in the plane of symmetry is sufficient to obtain the displacement effect, which is found to be of the same order as the experimental result obtained by Young & Maas (1936) for a conventional pitot tube. The sphere may be considered to represent a conventional pitot tube (of slightly smaller diameter), so it is concluded that a large part of the displacement effect of a pitot tube may be accounted for without the inclusion of viscosity, i.e. by consideration of the vorticity field alone.

To a first approximation, the vorticity in the plane of symmetry is found to depend only on the distance from the centre of the sphere.

An outline of shear flows past some two-dimensional bodies is given in an appendix. The bodies considered are a circular cylinder and a two-dimensional ‘pitot-tube’ consisting of two parallel semi-infinite plates.

This paper considers the changes that occur in the character of short-crested sea waves when they are refracted by a shallowing depth of water. Besides a change in mean wave-length and direction there is also a change (usually an increase) in the mean length of the crests. If the waves approach obliquely they become skew, that is, the crests become staggered one behind another.

When a short-crested sea is superposed on a long-creasted swell, refraction tends to amplify the longer waves more than the shorter ones. This also produces an increase in the mean length of the crests.

Numerical examples are given.

Frictionless flows with finite voticity are usually made determinate by the imposition of boundary conditions specifying the distribution of vorticity ‘at infinity’. No such boundary conditions are available in the case of flows with closed streamlines, and the velocity distributions in regions where viscous forces are small (the Reynolds number of the flow being assumed large) cannot be made determinate by considerations of the fluid as inviscid. It is shown that if the motion is to be exactly steady there is an integral condition, arising from the existence of viscous forces, which must be satisfied by the vorticity distribution no matter how small the viscosity may be. This condition states that the contribution from viscous forces to the rate of change of circulation round any streamline must be identically zero. (In cases in which the vortex lines are also closed, there is a similar condition concerning the circulation round vortex lines.)

The inviscid flow equations are then combined with this integral condition in cases for which typical streamlines lie entirely in the region of small viscous forces. In two-dimensional closed flows, the vorticity is found to be uniform in a connected region of small viscous forces, with a value which remains to be determined—as is done explicitly in one simple case—by the condition that the viscous boundary layer surrounding this region must also be in steady motion. Analogous results are obtained for rotationally symmetric flows without azimuthal swirl, and for a certain class of flows with swirl having no interior boundary to the streamlines in an axial plane, the latter case requiring use of the fact that the vortex lines are also closed. In all these cases, the results are such that the Bernoulli constant, or ‘total head’, varies linearly with the appropriate stream function, and the effect of viscosity on the rate of change of vorticity at any point vanishes identically.

After an extensive survey of mean-velocity profile measurements in various two-dimensional incompressible turbulent boundary-layer flows, it is proposed to represent the profile by a linear combination of two universal functions. One is the well-known law of the wall. The other, called the law of the wake, is characterized by the profile at a point of separation or reattachment. These functions are considered to be established empirically, by a study of the mean-velocity profile, without reference to any hypothetical mechanism of turbulence. Using the resulting complete analytic representation for the mean-velocity field, the shearing-stress field for several flows is computed from the boundary-layer equations and compared with experimental data.

The development of a turbulent boundary layer is ultimately interpreted in terms of an equivalent wake profile, which supposedly represents the large-eddy structure and is a consequence of the constraint provided by inertia. This equivalent wake profile is modified by the presence of a wall, at which a further constraint is provided by viscosity. The wall constraint, although it penetrates the entire boundary layer, is manifested chiefly in the sublayer flow and in the logarithmic profile near the wall.

Finally, it is suggested that yawed or three-dimensional flows may be usefully represented by the same two universal functions, considered as vector rather than scalar quantities. If the wall component is defined to be in the direction of the surface shearing stress, then the wake component, at least in the few cases studied, is found to be very nearly parallel to the gradient of the pressure.

A previous paper drew attention to the collective importance of three physical quantities Q, R, S associated with ideal fluid flow in a horizontal channel. Invariability of these quantities at different cross-sections of the flow implies respectively conservation of flow rate, energy and momentum; and their values determine a wave-train uniquely. The properties of Q, R, S are recalled in the present paper to account for the various effects of lowering a rigid obstacle into a stream. The conditions giving rise to dissimilar types of flow are examined; in particular, the circumstances causing stationary waves on the downstream side are clearly distinguished from those under which the receding stream assumes a uniform ‘supercritical’ state. A well-known result in the theory of the solitary wave is shown to apply to the receding stream even when the extreme conditions for the wave are exceeded; although it fails to account for a region close to the obstacle where the curvature of the streamlines becomes large. In passing, a feature of the theory is shown to bear on the practical problem of producing a uniform stream. Precise calculations are made for the flow under a vertical sluice-gate and under an inclined plane. To account for the region near the bottom edge of the sluice-gate, a method based on conformal transformation is used whereby an unknown curve in the hodograph plane is approximated by an arc of an ellipse. The accuracy of the results is more than sufficient for practical purposes, and they compare favourably with solutions previously obtained by relaxation methods. A number of experiments with water streams are described.