Supersonic laminar boundary-layer equations near the plane of symmetry of a cone at incidence are treated by the similarity method. Numerical integration of differential equations governing such a flow is performed, taking into consideration the temperature dependence of the Prandtl number Pr and viscosity μ throughout the boundary layer. On the leeward side, a detailed consideration of the solutions shows the existence of two solutions up to a critical incidence beyond which it appears that no solution may be found. Calculations carried out for a set of values of the external flow Mach number show up a significant effect of this parameter on the behaviour of the boundary layer.

A series of observations on experimentally produced vortex rings is described. The flow field, ring velocity and growth rate were observed using dye and hydrogen-bubble techniques. It was found that stable rings are formed and grow in such a way that most of their vorticity is distributed throughout a fluid volume which is larger than and moving with the visible dye core.

As the vorticity diffuses out of this moving body of fluid into the outer irrotational fluid, it has two effects. It causes some of the fluid, with newly acquired vorticity, to be entrained into the interior of the bubble, while the rest is left behind and accounts for the appearance of ring vorticity in a wake. It was found that the velocity of translation U of these stable rings varies as t−1, at high Reynolds number, where t is the time measured from the start of the motion at a virtual origin at downstream infinity. A simple theoretical model is presented which explains all of these features of the observed stable flow. Rings of even higher Reynolds number become unstable and shed significantly more vorticity into the wake. Under some circumstances a new more stable vortex emerges from this shedding process and continues with less vorticity than before. Eventually, the ring motion ceases as all of its vorticity is deposited into the wake and is spread by viscous diffusion. Observations of the interaction between two nearly identical rings travelling a common path showed that, contrary to popular belief, rings do not pass back and forth through one another, but that the rearward one becomes entrained into the forward one. Only when the rearward ring has a much higher velocity than its partner can it emerge from the joining process and leave a slower-moving ring behind.

Some exact solutions of the steady magnetohydrodynamic equations for a perfectly conducting inviscid self-gravitating incompressible fluid are discussed. It is shown that there exist solutions for which the free surface of the liquid is that of a planetary ellipsoid and rotates with constant angular velocity about its axis. The stability of the equilibrium configuration is not investigated.

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters, J a ‘typical’ Richardson number, α the (real) wavenumber and c the complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first finds c given α and J, the second finds α and J when c ≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes, but not always, the stability boundary in the α, J plane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

The mean distribution of concentration of a solute over the cross-section of a tube through which a solvent is flowing is given approximately by the solution of an equation of the heat-conduction type (Taylor 1953, 1954). One solution of this equation is a Gaussian curve but observed distributions are normally not Gaussian. On the basis of an equation derived by Chatwin (1970) it is shown here that the deviation of an observed distribution from the Gaussian curve with the same mean and variance is not determined by the heat-conduction equation, although the practical importance of this remark may be small if the initial distribution of solute is greatly elongated along the tube axis.

A noise generation mechanism for a nearly ideally expanded supersonic jet is proposed. It is suggested that the dominant part of the noise of a supersonic jet is generated at two rather localized regions of the jet. These regions are located at distances quite far downstream of the nozzle exit. Large-scale instabilities of the jet flow are believed to be responsible for transferring the kinetic energy of the jet into noise radiation. An analysis based on a simple mathematical model reveals that two large-scale unstable waves are preferentially amplified in a supersonic jet. The rapid growth of these waves causes the oscillations of the jet to penetrate the mixing layer at two locations and to interact strongly with the ambient fluid there. This gives rise to intense noise radiation. Theoretical results based on the proposed noise generation mechanism are found to compare favourably with experimental measurements. A simple scaling formula is also derived.

A theory is developed to describe the evolution of the entrainment interface in turbulent flow, in which the surface is convoluted by the large-scale eddies of the motion and at the same time advances relative to the fluid as a result of the micro-scale entrainment process. A pseudo-Lagrangian description of the process indicates that the interface is characterized by the appearance of ‘billows’ of negative curvature, over which surface area is, on average, being generated, separated by re-entrant wedges (lines of very large positive curvature) where surface area is consumed. An alternative Eulerian description allows calculation of the development of the interfacial configuration when the velocity field is prescribed. Several examples are considered in which the prescribed velocity field in the z direction is of the general form w = Wf(x – Ut), where the maximum value of the function f is unity. These indicate the importance of leading points on the surface which are such that small disturbances in the vicinity will move away from the point in all directions. The necessary and sufficient condition for the existence of one or more leading points on the surface is that U [les ] V, the speed of advance of an element of the surface relative to the fluid element at the same point. The existence of leading points is accompanied by the appearance of line discontinuities in the surface slope re-entrant wedges, In these circumstances, the overall speed of advance of the convoluted surface is found to be W + (V2 – U2)½, where W is the maximum outwards velocity in the region; this result is independent of the distribution f.

When the speed U with which an ‘eddy’ moves relative to the outside fluid is greater than the speed of advance V of an element of the front, the interface develops neither leading points nor discontinuities in slope; the amplitude of the surface convolutions and the overall entrainment speed are both reduced greatly. In a turbulent flow, therefore, the large-scale motions influencing entrainment are primarily those that move slowly relative to the outside fluid (with relative speed less than V). The experimental results of Kovasznay, Kibens & Blackwelder (1970) are reviewed in the light of these conclusions. It appears that in their experiments the entrainment speed V is of the order fifteen times the Kolmogorov velocity, the large constant of proportionality being apparently the result of augmentation by micro-convolutions of the interface associated with small and meso-scale eddies of the turbulence.

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/r (of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/r is considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.

Asymptotic and numerical solutions of the unsteady boundary-layer equations are obtained for a main stream velocity given by equation (1.1). Far downstream the flow develops into a double boundary layer. The inside layer is a Stokes shear-wave motion, which oscillates with zero mean flow, while the outer layer is a modified Blasius motion, which convects the mean flow downstream. The numerical results indicate that most flow quantities approach their asymptotic values far downstream through damped oscillations. This behaviour is attributed to exponentially small oscillatory eigenfunctions, which account for different initial conditions upstream.

Previously reported experiments with a self-propelled body submerged in a fluid with a stable vertical density gradient have demonstrated that the turbulently mixed wake first expands more or less uniformly and then collapses vertically while continuing to expand horizontally (Schooley & Stewart 1963). It was also shown that the vertical collapse of the wake generates internal waves. Essentially two-dimensional experiments have also been used to explore some of the build-up and decay characteristics of vertical wake collapse induced by a sub-merged burst of turbulent mixing (Wu 1969; Schooley 1968). The present paper reports new experimental measurements and a linear theoretical analysis of the internal wave field created in stratified water by a burst of submerged turbulent mixing. The forcing function has been obtained in integral form for an initial-value model of wake collapse in terms of a general Brunt-Väisälä frequency profile, using normal mode theory. Numerical results have been determined for the specialized case of a completely mixed circular wake in a constant Brunt-Väisälä profile. These results are compared to the experimental measurements.

An experimental study of the transition from laminar to turbulent flow in a long 0·248in. I.D. pipe is reported for both water and dilute water solutions of polyethylene oxide which exhibit turbulent flow drag reduction (the Toms phenomenon). The drag-reducing solutions, ranging in effectiveness from near zero to the maximum attainable, are observed to undergo transition in a similar way to the Newtonian solvent in that the solutions exhibit intermittency and the growth rates of the turbulent patches are essentially equal to those of the pure solvent. The growth rate of turbulent patches indicates that drag reduction is associated with the small-scale structure of the turbulence near the pipe wall while patch growth is associated with the larger-scale turbulence in the outer flow. For low-disturbance pipe inlet conditions the strong drag-reducing solutions are observed to undergo transition at lower Reynolds numbers than the pure solvent.

Previous analyses of gas and particle motion around bubbles in fluidized beds have concentrated on idealized isolated bubbles. In this paper three non-idealities are considered using the theoretical models of Davidson and Murray. Gas flow patterns are derived for indented and elongated bubbles and for pairs of interacting bubbles. Cloud boundaries are predicted for these situations and some effects on gas-solid contacting are discussed.