The general analysis of Part 1 is applied to the calculation of the second-order viscous and thermal boundary layers for the axisymmetric stagnation point, unsymmetric plane stagnation point, and cusped leading edge at ideal incidence. The second-order effect upon heat transfer is found usually to be of the order of one-third of that for skin friction.

The ship is assumed to be a slender body of revolution with its axis in the mean free surface and making periodic oscillations of small amplitude. The theory presented here is a generalization of the well-known slender-body theory of incompressible aerodynamics in which the fluid is externally unbounded. One version of that theory goes as follows: approximate the body by axial line-distributions of known point singularities (sources and multipoles), whose strength is to be determined; by means of the Fourier convolution theorem express the velocity potentials of these line distributions in terms of the Fourier transforms (in the axial direction) of the point-singularity potentials; expand these Fourier transforms in powers of the radius and retain only the leading terms (it is here that the slender-body assumption is introduced); by means of the Fourier convolution theorem interpret the resulting expressions. By this procedure it is not only shown that near the body the potential is two-dimensional harmonic in every plane normal to the axis; but also the interaction between sections is shown to be involved in the ‘constant’ term and to depend in an explicit manner on the coefficient functions, which can be found without difficulty by applying the prescribed boundary conditions. This foregoing procedure can be justified when the body is slender and sharply pointed.

In the present paper the same procedure is adapted to an oscillating surface ship a t zero speed. The fluid is now bounded by the ship, and also by a horizontal plane (the mean free surface) on which a wave boundary condition must be applied. The point singularities are now wave sources and wave-free potentials, each satisfying the free-surface condition. The Fourier transforms of these singularities are found and are expanded near the axis; the expansions near the axis are the only parts of the argument that present any serious difficulty. When only the leading terms are retained and the results are interpreted by the convolution theorem, explicit two-dimensional potentials are again obtained.

It is assumed that the ratios (ship-radius/ship-length) and (ship-radius/wavelength) are small whereas the ratio (ship-length/wave-length) may have any value. Expressions are given which are valid according as this last ratio is not large or not small. The potential on the body is found, and forces, moments and wave damping are calculated. It is believed that the expansions can be extended with little trouble to certain other ranges of those ratios, to other cross-sections and to the boundary condition for ships moving a t non-zero speeds.

The solitary wave is a well-known example of a flow of permanent type which can be studied by shallow-water theory. A scheme due to K. O. Friedrichs (see Friedrichs & Hyers 1954) can give both an approximation to the exact wave form and a framework for a proof of its existence. It is used here to approximate the solution of a related problem, that of a steady flow near the critical speed over a small obstacle in the bed of a stream. The flow can be considered as a perturbation of a solitary wave since it will approach a solitary wave as the obstacle shrinks in height.

This paper is a contribution to the mathematical analysis of the stability of steady axisymmetric parallel flows of uniform fluid in the absence of rigid boundaries. A jet at sufficiently high Reynolds number for the angle of viscous spreding to be small is the typical example of the primary flows considered, and the theoretical velocity profile far downstream in such a jet is kept in mind continually. It is obvious from experience that such jets are unstable, presumably to infinitesimal disturbances, but there is little observational data about the critical Reynolds number or the mode of disturbance that grows most rapidly at a given Reynolds number.

The typical small disturbance considered is a Fourier component with sinusoidal dependence on both ax and nϕ (x, r, ϕ are cylindrical polar co-ordinates). There is no analogue of Squire's theorem for two-dimensional primary flows, and both α and n are essential parameters of the disturbance. We have concentrated on the stability characteristics in the limit of large Reynolds number, and have aimed in particular at determining the (integral) value of n at which the growth rate is a maximum in these simpler circumstances.

A number of general results for inviscid fluid are established, many of them analogues of corresponding results for two-dimensional primary flow. A necessary condition for the existence of amplified disturbances is that $Q(r)=rU^{\prime}|(n^2+\alpha ^2 r^2)$ should have a numerical maximum a t some point in the fluid; this condition is satisfied for all n in the case of a cylindrical shear layer or ‘top-hat’ jet profile (for which a complete solution of the disturbance equation can be obtained), and for n > 1 in the case of a ‘far-downstream’ jet profile. The wave speed cr of a neutral disturbance is equal to the value of U either a t the point where dQ/dr = 0 or at r = 0. In the latter case the eigen-function (if one exists) is singular at the axis in general; the former case is presumably relevant to the ‘upper branch’ of the curve of neutral stability (for given n). The Reynolds stress due to the disturbance acting across a cylindrical surface is examined. Here, as in some other contexts, it is useful to consider components of velocity parallel and perpendicular to a circular helix on which the phase of the disturbance wave is constant. For a neutral disturbance the component of disturbance velocity parallel to the local wave helix is infinite a t the critical point where U = cr, (corresponding to the known singularity for a three-dimensional disturbance to two-dimensional flow), and there is a peak in the Reynolds stress there.

It is shown from the form of the disturbance equation that there is an upper limit to the value of n (≠ 0) for a neutral (inviscid) disturbance with cr equal to the value of U at the point where Q′ = 0. In the case of a jet with a ‘far-downstream’ profile, only the value n = 1 satisfies this restriction; thus only the sinuous mode n = 1 can yield amplified disturbances in an inviscid fluid. A numerical investigation shows that for this profile the wave-number of the neutral disturbance with n = 1 is α = 1·46.

A jet of dyed water is directed into a large tank of water. As the flow rate is increased several successive modes of breakdown of the dyed column are observed:

1. tiny puffs of dye near the nozzle;

2. axisymmetric condensations well away from the nozzle;

3. sinuous undulations of long wave length far from the nozzle;

4. formation of foot-shaped pockets of dye;

5. confused breakup near the nozzle with rapid spread of dye.

In experiments on the stability of a submerged axisymmetric jet at Reynolds numbers large compared with unity, Reynolds (1962) observed axisymmetric ‘condensations’ which appear to grow spontaneously, whereas Batchelor & Gill (1962) have shown that infinitesimal disturbances of this type do not grow in an inviscid fluid. Here it is shown that axisymmetric disturbances do not grow in a slightly viscous fluid either, and the solution for which the rate of damping is smallest is found. It is suggested that the growth of small but finite disturbances is responsible for the condensations observed. The order of magnitude of the disturbance velocity at which non-linear effects could produce growth of disturbance is found to depend on the wave-number of the disturbance. The smallest velocity which a ‘finite’ disturbance may have is found to be of order $R^{-\frac{2}{3}$, and corresponds to a disturbance whose wave-number is of order $R^{\frac{1}{3}$, R being the Reynolds number based on the local radius and maximum velocity of the jet. On the assumption that some disturbances whose velocity is of this order will grow, deductions are made as to the size, position, wave-number, and point of appearance of condensations. The deductions appear to agree with the experimental results.

Standing surface waves in an inviscid incompressible fluid of finite depth are considered, taking into account the effect of capillary forces. Perturbation solutions for the surface profile, velocity potential, frequency of oscillation, and pressure are found to third order in the amplitude of the waves. A graph is given showing the regions in which the frequency of oscillation increases with amplitude and those in which it decreases with amplitude. These regions are defined as a function of the depth of the fluid and a parameter called the relative capillarity. A graph is also given showing the surface profile of a wave.

In earlier papers Phillips (1960) and Longuet-Higgins (1962) have investigated phase velocity effects and possible resonances associated with the interactions of gravity waves. In this note the problem is discussed from a different viewpoint which demonstrates more clearly the energy-sharing mechanism involved. Equations governing the time dependence of the resonant modes are obtained, rather than the initial growth rate as has been found previously.

This paper describes an experiment in which a cylindrical vortex, formed in a long tube, was used to study the ‘vortex breakdown’ that has been previously reported in investigations of the flow over slender delta wings. By varying the amount of swirl that was imparted to the fluid before it entered the tube, it was found that the breakdown was the intermediate stage between the two basic types of rotating flows, that is, those that do and those that do not exhibit axial velocity reversal. In addition, it was shown that an unusual flow pattern was established after the breakdown and that certain features of this pointed to it being a ‘critical’ phenomenon. The tests were concluded by measuring the swirl angle distribution a short distance ahead of the breakdown and comparing these results with the prediction of Squire's theory (1960).

The phenomenon examined is the abrupt structural change which can occur at some station along the axis of a swirling flow, notably the leading-edge vortex formed above a delta wing at incidence. Contrary to previous attempts at an explanation, the belief demonstrated herein is that vortex breakdown is not a manifestation of instability or of any other effect indicated by study of infinitesimal disturbances alone. It is instead a finite transition between two dynamically conjugate states of axisymmetric flow, analogous to the hydraulic jump in openchannel flow. A set of properties essential to such a transition, corresponding to a set shown to provide a complete explanation for the hydraulic jump, is demonstrated with wide generality for axisymmetric vortex flows; and the interpretation covers both the case of mild transitions, where an undular structure is developed without the need arising for significant energy dissipation, and the case of strong ones where a region of vigorous turbulence is generated. An important part of the theory depends on the calculus of variations; and the comprehensiveness with which certain properties of conjugate flow pairs are demonstrable by this analytical means suggests that present ideas may be useful in various other problems.