The effects of convection and refraction dominate the heart-shaped pattern of jet noise. These can be corrected out to yield the small ‘basic directivity’ of the eddy noise generators. The observed quasi-ellipsoidal pattern was predicted by Ribner (1963, 1964) in a variant of the Lighthill theory postulating isotropic turbulence superposed on a mean shear flow. This had the feature of dealing with the joint effects of the quadrupoles without displaying them individually. The present paper reformulates the theory so as to calculate the relative contributions of the different quadrupole self and cross-correlations to the sound emitted in a given direction. Some minor errors are corrected.

Of the thirty-six possible quadrupole correlations only nine yield distinct non-vanishing contributions to the axisymmetric noise pattern of a round jet. The correlations contribute either cos4θ, cos2θ sin2θ or sin4θ directional patterns, where θ is the angle with the jet axis. A separation into parts called ‘self noise’ (from turbulence alone) and ‘shear noise’ (jointly from turbulence and mean flow) may be made.

The nine self-noise patterns combine as $A\; cos^4\theta(1)+A\; cos^2sin^2\theta(\frac{7}{8}+\frac{7}{8}+\frac{1}{8}+\frac{1}{8})+A sin^2 \theta (\frac {12}{32} & + & \frac{12}{32}+\frac{7}{32}+\frac{1}{32})\\ & = & A(cos^2\theta+sin^\theta)^2 = A;$ this is uniform in all directions as it must be, arising from isotropic turbulence. The two non-vanishing shear-noise correlation patterns combine as $B\;cos^4\theta (1) + B\;cos^2\theta sin^2\theta(\frac{1}{2})=B(cos^2\theta+sin^2\theta)^2 = A;$

The overall ‘basic’ pattern (self noise plus shear noise) thus has the form A + B(cos2θ + cos4θ)/2; this is a slight change from the previous result. The dimensional constants A and B are of comparable magnitude; the pattern in any plane through the jet axis thus resembles an ellipse of modest eccentricity.

Frequency spectra are also discussed, following the earlier work. Since the self noise depends quadratically on turbulent velocity components and the shear noise only linearly, there is a relative shift of the self noise to higher frequencies. This in conjunction with refraction figures in the explanation of the deeper pitch of jet noise radiated at small angles to the axis.

Finally, the predictions are shown to be compatible with recent experimental results.

The paper examines the influence of a piston motion on the gaseous motion set up by an explosion. This is a problem of relevance not only to explosions in which the remnants of the original explosive are present, but also by the application of the hypersonic similarity law, to the steady flow at high Mach number past a blunt-nosed object with increasing thickness. In the vicinity of the piston perturbation theory breaks down and matching techniques have to be employed. A solution is given in terms of a small parameter which essentially depends on the ratio of the speed of the piston to that of the shock. Some particular cases are discussed in which the piston motion plays a similar role to the counter pressure of the undisturbed gas.

The linear, normal mode instability of three-dimensional laminar and turbulent shear layers is studied. The flows consist of two streams semi-infinitely extended in the y-direction, flowing obliquely in the x-z plane. It is found that the stability of the flows depends on the main flow velocity components in the direction of the disturbance wave-number vector. Numerical calculations are performed to obtain the neutral stability curves. Under the usual parallel unperturbed flow assumption, the neutral stability curves pass through the origin of the α-R diagram, where α is the wave-number and R is the flow Reynolds number. It is also found that eddy viscosity has a destabilizing effect for small Reynolds numbers but a stabilizing effect at larger Reynolds numbers. Because any linear perturbation trajectory eventually leaves the unstable region of the α-R plane, it is probable that a Lin-Benney or Taylor-Goertler secondary instability ensues. Suitable components of the existing turbulence grow and develop into a large eddy system which causes rapid entrainment, giving rise to a turbulent burst. A first-order non-parallel correction is made to the neutral stability curves. The new curves have both upper and lower branches, and there exist minimum critical Reynolds numbers.

A disk moves edgewise in a vertical plane with a sufficiently small constant velocity in a uniformly stratified inviscid fluid under gravity. The resultant hydrodynamic thrust on the disk is estimated. Configurations of some associated phase surfaces are shown. A transient Green's function is presented for the case when the Boussinesq approximation is not made, and also for an analogous case of inertial waves in a rotating fluid.

The viscous, creeping flow through a cylindrical tube of a liquid, which contains rigid, spherical particles, is investigated analytically. The spheres are located on the axis of the cylinder and are equally spaced. Solutions are derived for particles in motion and fixed, with and without fluid discharge. Numerical results are presented for the drag on each sphere and the mean pressure drop for a wide range of sizes and spacings of the spheres. The study is motivated by possible application to blood flow in capillaries, where red blood cells represent particles of the same order of magnitude as the diameter of the capillary itself. The results may also be of interest in other applications, such as sedimentation and fluidized beds. It is shown that there is little interaction between particles if the spacing is more than one tube diameter, and that the additional pressure drop over that for Poiseuille flow is less than 50% if the sphere diameter is less than 0·8 of the tube diameter.

The effects of viscous dissipation are considered for external natural convection flow over a surface. A class of similar boundary-layer solutions is given and numerical results are presented for a wide range of the dissipation and Prandtl numbers. Several general aspects of similarity conditions for flow over surfaces and in convection plumes are discussed and their special characteristics considered. The general equations including the dissipation effect are given for the non-similar power law surface condition.

Velocity and temperature boundary layers developed on a plane wall by ideal shock-tube flow are considered for weak shock and expansion waves. Analytically, the boundary layer consists of three regions, bounded by (1) expansion-wave head, (2) diaphragm location, (3) contact discontinuity, (4) shock. The flow fields (1, 2) and (3, 4) are, essentially, known. In the interaction region (2, 3), these flow fields merge, the governing equations are ‘singular parabolic’ and admit boundary conditions usually associated with elliptic equations. It is convenient to replace the weak expansion wave in the main flow by a line discontinuity. A consistent linearization scheme can now be devised to obtain the solution in the three regions. In (2, 3), the resulting linear singular parabolic equations for the first-order solutions are solved successfully by an iterative finite difference method, normally applied to elliptic equations.

A study is made of the transient flow which results from the impingement of a plane shock on a blunt body. The analysis is based on Taylor-series expansions in the space and time variables of the flow properties and of the shape of the reflected shock. Coefficients of the series are determined numerically so as to satisfy the exact equations of motion and shock jump conditions. Convergence problems are ameliorated by recasting the series into continued fractions. While the analysis does not treat the transition from regular to Mach reflexion, it remains valid for all time in the subsonic region of the flow if the incident shock is sufficiently strong. In any case, it is accurate enough where it is valid to be useful for evaluation of more conventional numerical methods.

Consideration is given to the effects of gravity which arise when a heated sphere, maintained at a steady uniform temperature, is located in a vertical uniform stream. Restricting analysis to a medium of unit Prandtl number (σ), the method of matched asymptotic expansions is employed in obtaining solutions for the velocity, temperature and pressure fields in the limit: G = o(R2), R ↓ 0 (G and R being, respectively, the Grashof and Reynolds numbers). Based on these results, conjectures are formed about the corresponding pure natural convection problem.

Existing analytical treatments of the hypersonic strong interaction problem adopt the two-region structure of the classical boundary-layer theory; however, the uneven heating and external vorticity created by the highly curved, leading-edge shock wave leads to the question of the uniform validity of the boundary-layer approximation near the boundary-layer edge. Recently Bush (1966), using a non-linear viscosity-temperature law (μ ∞ Tω, ω < 1) instead of the linear one (μ ∞ T) used by previous investigators, demonstrated the need to analyze separately a transitional layer intermediate between the inviscid region and the boundary layer. In this paper, an asymptotic analysis of the Navier-Stokes equations in von Mises's variables, allowing a three-region structure, is carried out for the case of μ ∞ T. Results, with the second-order effects associated with heating and vorticity accounted for, show that a separate analysis of the transitional region is not strictly necessary in this case, and hence the equivalence of the two-region approach is confirmed. On the other hand, it is shown that the second-order boundary-layer correction owing to the heating and vorticity effects, not considered by Bush, is necessary in order to determine a uniformly valid temperature distribution in the physical variables. Numerical results for an insulated and a cold flat plate, considerably different from those of others, are obtained.

Measurements of skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 < Re < 10000. These measurements, and observations of hot-wire signals, indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term ‘fully developed turbulent flow’.

The effect of vertical boundaries on convection in a shallow layer of fluid heated from below is considered. By means of a multiple-scale perturbation analysis, results for horizontally unbounded layers are modified so that they ‘fit in’ to a rectangular region. The critical Rayleigh number and critical wave-number are determined. Motion is predicted to have the form of finite ‘rolls’ whose axes are parallel to the shorter sides of the dish. Aspects of the non-linear development and stability of this motion are studied. The general question of convective pattern selection in a bounded layer is discussed in the light of available theoretical and experimental results.