Measurements of mean velocity, turbulent stress and static pressure were made in the mixing region of a jet of air issuing from a slot nozzle into still air. The velocity was low and the two-dimensional flow was effectively incompressible. The results are examined in terms of the unsimplified equations of fluid motion, and comparisons are drawn with the common assumptions and simplifications of free jet theory. Appreciable deviations from isobaric conditions exist and the deviations are closely related to the local turbulent stresses. Negative static pressures were encountered everywhere in the mixing field except in the potential wedge region immediately adjacent to the nozzle. Lateral profiles of mean longitudinal velocity conformed closely to an error curve at all stations further than 7 slot widths from the nozzle mouth. An asymptotic approach to complete self-preservation of the flow was observed.

In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).

The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

The instability of small gas bubbles moving uniformly in various liquids is investigated experimentally and theoretically.

The experiments consist of the measurement of the size and terminal velocity of bubbles at the threshold of instability in various liquids, together with the physical properties of the liquids. The results of the experiments indicate the existence of a universal stability curve. The nature of this curve strongly suggests that there are two separate criteria for predicting the onset of instability, namely, a critical Reynolds number (202) and a critical Weber number (1.26). The former criterion appears to be valid for bubbles moving uniformly in liquids containing impurities and in the somewhat more viscous liquids, whereas the latter criterion is for bubbles moving in pure, relatively inviscid liquids.

The theoretical analysis is directed towards an investigation of the possibility of the interaction of surface tension and hydrodynamic pressure leading to unstable motions of the bubble, i.e. the existence of a critical Weber number. Accordingly, the theoretical model assumes the form of a general perturbation in the shape of a deformable sphere moving with uniform velocity in an inviscid, incompressible fluid medium of infinite extent. The calculations lead to divergent solutions above a certain Weber number, indicating, at least qualitatively, that the interaction of surface tension and hydrodynamic pressure can result in instabilities of the bubble motion.

A subsequent investigation of the time-independent equations, however, shows the presence of large deformations in shape of the bubble prior to the onset of unstable motion, which is not compatible with the approximation of perturbing an essentially spherical bubble. This deformation and its possible effects on the stability calculation are therefore determined by approximate methods. From this it is concluded that the deformation of the bubble serves to introduce quantitative, but not qualitative, changes in the stability calculation.

When plane pressure waves in a duct reach an open end, they establish a complicated three-dimensional wave pattern in the vicinity of the exit which tends to readjust the exit pressure to its steady-flow level. This adjustment process is continually modified by further incident waves, so that the effective instantaneous boundary conditions which determine the reflected wave depend on the flow history. In the analysis of a nonsteady-flow problem by means of a wave diagram, it has been customary to assume that the steady-flow boundary conditions are instantaneously established. While this simplifying assumption appears reasonable, the resulting errors have been undetermined. It is the purpose of the present investigation to obtain improved boundary conditions. The results of a previous study of the reflection of shock waves from an open end have now been extended to other waves of finite amplitude. The reflected waves computed by means of the new procedure are in good agreement with experimental data observed in a shock tube for a variety of flow conditions. The pressure variations in a reflected wave lag behind those derived in the conventional manner by the time in which a sound wave travels about one or two duct diameters. Such lags are small, but may occasionally become significant. As a consequence of the lag, certain discontinuities of the incident wave do not reappear in the reflected wave. This improved understanding of the reflection process has made it possible to clarify some previously unexplained experimental observations.

This paper is concerned with some statistical properties of the displacement of a marked fluid particle released from a given position in a turbulent shear flow, and in particular with the dispersion about the mean position after a long time. It is known that the dispersion takes a simple asymptotic form when the particle velocity is a stationary random function of time, and that analogous results are obtainable when the particle velocity can be transformed to a stationary random function by suitable stretching of the velocity and time scales. The basic hypothesis of the paper is that, in steady free turbulent shear flows which are generated at a point and have a similar structure at different stations downstream, the velocity of a fluid particle exhibits a corresponding Lagrangian similarity and can be so transformed to a stationary random function.

The velocity and time scales characterizing the motion of a fluid particle at time t after release at the origin are determined in terms of the powers with which the Eulerian length and velocity scales of the turbulence vary with distance x from the origin. The time scale has the same dependence on t for all jets, wakes and mixing layers (and also for decaying homogeneous turbulence) possessing the usual kind of Eulerian similarity. The dispersion of a particle in the longitudinal or mean-flow direction (and likewise that in the lateral direction in cases of two-dimensional mean flow) is found to vary with t in such a way as to be proportional to the thickness of the shear layer at the mean position of the particle. The way in which the maximum value of the mean concentration of marked fluid falls off with t (for release of a single particle) or with x (for continuous release) is also found.

This paper presents the application of the methods developed in a previous paper (Part I, Crane & Pack 1957) to the mixing of two parallel streams for both laminar and turbulent flows. The effects of both high velocity and large temperature difference are treated together. The method used consists in developing the stream function in a double series of powers of two parameters, the first being the Mach number and the second depending on the temperature difference of the streams. Analytical expressions are found for the terms up to the second order in the series for the stream function when the streams do not differ too greatly in velocity and temperature. However, when one of the streams is at rest the analytical method is no longer sufficiently accurate, and for this case numerical solutions are given.

For laminar mixing the most important effect is that of ’change of scale’, as was found in Part I for a laminar jet at large distances from the orifice. For turbulent half-jets the effect of ’change of scale’ and the effect of the perturbation terms due to the Mach number of the flows are approximately equal and opposite, leaving the form of the velocity profile sensibly unchanged from that in incompressible flow. This last result is confirmed by comparison with some experiments of Laurence (1955) on a two-dimensional jet at M = 0·7. Lastly, the effect of temperature differences is shown to be relatively unimportant even when these are fairly considerable.

Hodograph methods are applied to determine the flow at high subsonic and sonic velocities past two-dimensional, thin, symmetrical bodies. The boundary value problem for the determination of the stream function Ψ, which in the present theory is a solution of Tricomi's equation, is simplified by the assumption of a free stream breakaway at sonic velocity from the shoulder of the body. A solution is obtained in terms of Bessel functions.

In § 2 and 3 the flow past a wedge of small angle is discussed and expressions are obtained for the pressure on the nose, the drag coefficient and the width of the wake. A comparison with the corresponding results in the case of sonic velocity derived by the more complex analysis of Guderley & Yoshihara (1950) shows that the present simpler theory yields very similar values for the pressure over the nose.

In § 4 the flow at sonic velocity past a profile which is a first-order perturbation upon a wedge profile is analysed on the basis of the same free streamline theory. The flow pattern is obtained past an arbitrarily specified body by an application of the Hankel inversion theorem and an expression is deduced for the drag.