In a previous paper, Phillips (1960) showed that two or three trains of gravity waves may interact so as to produce a fourth (tertiary) wave whose wave-number is different from any of three primary wave-numbers k1, k2, k3, and whose amplitude grows in time. Such resonant interactions may produce an appreciable modification of the spectrum of ocean waves within a few hours. In this paper, by a slightly different method, the interaction is calculated in detail for the simplest possible case: when two of the three primary wave-numbers are equal (k3 = k1).

It is found that, when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2. Generally, if θ denotes the angle between k1 and k2, the rate of growth of the tertiary wave with time is a maximum when θ [eDot ] 17°; the rate of growth with horizontal distance is a maximum when θ [eDot ] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.

The energy flux in a finite-depth gravity-wave spectrum resulting from weak non-linear couplings between the spectral components is evaluated by means of a perturbation method. The fifth-order analysis yields a fourth-order effect comparable in magnitude to the generating and dissipating processes in wind-generated seas. The energy flux favours equidistribution of energy and vanishes in the limiting case of a white, isotropic spectrum. The influence on the equilibrium structure of fully developed wave spectra and on other phenomena in random seas is discussed briefly.

An experimental investigation is described in which principal emphasis is given to revealing the nature of the motions in the non-linear range of boundary-layer instability and the onset of turbulence. It has as its central purpose the evaluation of existing theoretical considerations and the provision of a sound physical model which can be taken as a basis for a theoretical approach. The experimental method consisted of introducing, in a two-dimensional boundary layer on a flat plate at ‘incompressible’ speeds, three-dimensional disturbances under controlled conditions using the vibrating-ribbon technique, and studying their growth and evolution using hot-wire methods. It has been definitely established that longitudinal vortices are associated with the non-linear three-dimensional wave motions. Sufficient data were obtained for an evaluation of existing theoretical approaches. Those which have been considered are the generation of higher harmonics, the interaction of the mean flow and the Reynold stress, the concave streamline curvature associated with the wave motion, the vortex loop and the non-linear effect of a three-dimensional perturbation. It appears that except for the latter they do not adequately describe the observed phenomena. It is not that they are incorrect or may not play a role in some aspect of the local behaviour, but from the over-all point of view the results suggest that it is the non-linear effect of a three-dimensional perturbation which dominates the behaviour. A principal conclusion to be drawn is that a new perspective, one that takes three-dimensionality into account, is required in connexion with boundary-layer instability. It is demonstrated that the actual breakdown of the wave motion into turbulence is a consequence of a new instability which arises in the aforementioned three-dimensional wave motion. This instability involves the generation of ‘hairpin’ eddies and is remarkably similar in behaviour to ‘inflexional’ instability. It is also shown that the results obtained from the study of controlled disturbances are equally applicable to ‘natural’ transition.

This paper analyses the development, according to the Navier-Stokes equations, of the two-dimensional flow in the neighbourhood of the rear stagnation point on a cylinder which is set in motion impulsively with constant velocity. The local flow is idealized to the extent that the cylindrical boundary is taken to be an infinite plane bounding a semi-infinite domain of fluid. The velocity field is taken to be a linear function of the co-ordinate measured parallel to the boundary, and the initial flow is taken to be the (unique) irrotational form of such a field, namely, inviscid flow away from a stagnation point. Thereafter this irrotational flow is maintained as the outer boundary condition at a large distance from the boundary.

It is suggested that, outside a viscous layer on the boundary, the asymptotic flow for large times is described by a similarity solution of the inviscid form of the governing equation, with a length scale normal to the boundary which increases exponentially with time. This inviscid solution has a steady velocity of slip along the boundary which is equal but opposite to that of the initial flow, so that the flow in the viscous layer ultimately becomes the well-known stagnation flow towards a boundary. The suggestion is supported by a numerical solution of the initial-value problem.

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.

A general theory is put forward for the effect of wall proximity on the Stokes resistance of an arbitrary particle. The theory is developed completely for the case where the motion of the particle is parallel to a principal axis of resistance. In this case, the wall-effect correction can be calculated entirely from a knowledge of the force experienced by the particle in an unbounded fluid, providing (i) that the wall correction is already known for a spherical particle and (ii) that the particle is small in comparison to its distance from the boundary. Experimental data are cited which confirm the theory. The theory is extended to the wall effect on a particle rotating near a boundary.

A detailed treatment of inviscid hypersonic flow past a circular cone is given, for small and moderate yaw angles, within the fremework of shock-layer theory. The basic problem of non-uniform validity associated with the singularity of the entropy field is examined and a valid first-order solution is obtained which provides an explicit description of a thin vortical layer at the inner edge of the shock layer. Analytic formulas for pressure and circumferential velocity are given consistent to the second-order approximation including the non-linear yaw effect.

The study of the entropy field (which is not restricted to the hypersonic case) also provides corrections to previous work on the yawed cone and confirms the validity of the linear yaw effect on pressure field in the Stone theory.

A related investigation of three-dimensional flow fields is presented with special reference to the flow structure near the surface of a pointed, but otherwise arbitrary body. The inviscid streamline pattern on the surface is given by the geodesics originating from the pointed nose as a leading approximation of shock-layer theory. Associated with this streamline pattern is a vortical sublayer which exists generally at small as well as at large angle of attack. At the base of the sublayer, enthalpy and flow speed remain essentially uniform.

It is the purpose of this paper to report an investigation concerning certain factors affecting transition from laminar to turbulent flow in boundary layers. The objectives of the investigation were to clarify further the effects of very small degrees of leading-edge bluntness, and to devise a means for estimating the effect of roughness on boundary-layer transition.

A surface distribution of stress is imposed on an ocean enclosed by two continental boundaries; the resulting transport circulation is studied between two latitudes of zero surface wind-stress curl, within which the curl reaches a single maximum. Under the assumption that turbulent transfer of relative vorticity has a minimum effect on the mean circulation, inviscid flow patterns are deduced in the limit of small transport Rossby number. Inertial currents, or naturally scaled regions of high relative vorticity, occur on both the eastern and the western continental coasts. Limits on the relative transports of the currents are obtained and found to depend on the direction of variation of the wind-stress curl with latitude, relative to that of the Coriolis accelerations. The most striking feature of the inviscid flow is a narrow inertial current the axis of which lies along the latitude of maximum wind-stress curl. All eastward flow occurs in this midlatitude jet.

A feature of the flow which cannot remain essentially free of turbulent processes is the integrated vorticity relationship, since the imposed wind-stress distribution acts as a net source of vorticity for the ocean. Heuristic arguments are used together with this integral constraint to deduce the presence and strength of the turbulent diffusion which must occur in the region of the mid-latitude jet. It is further inferred that the turbulent meanders of the jet must effect a net meridional transport of relative vorticity.

The convective diffusion of matter from a stationary object to a moving fluid stream is distinct from pure heat transfer because of the appearance of a finite interfacial velocity at the solid surface. This velocity is related to the rate of mass transfer by a dimensionless group B in such a way that for −1 < B < 0 the transfer is from the bulk to the surface while for 0 < B < ∞ the transfer is from the surface to the main stream. In this paper, asymptotic solutions to the two-dimensional laminar boundary-layer equations are developed for the case B [Gt ] 1, and for rather general systems. It is shown that in most instances the asymptotic expressions for the rate of mass transfer become accurate when B > 3 and that the transition region between the pure heat-transfer analogy (B ∼ 0) and the B [Gt ] 1 asymptote may be described by a simple graphical interpolation. These results may easily be extended to three-dimensional surfaces of revolution by the usual co-ordinate transformations of boundary-layer theory.

The incompressible, inviscid, parallel flow of a layer of fluid over a slightly denser fluid, in the presence of gravity, is investigated for stability. Two idealized piece-wise-linear steady velocity profiles are examined analytically, and a comparison with related experimental results is made.

The dimensionless parameter U/(gh)½, where U is flow velocity, g the acceleration of gravity, and h the thickness of the upper layer, appears to have a critical value between 0.2 and 0.7, below which stable flows can persist. At the onset of instability disturbances of wavelength about h or 2h are predicted, with more violent disturbances of longer wavelength occurring at higher values of U/(gh)½ For continuous steady velocity profiles this instability phenomenon is found to be relatively insensitive to the ratio of the densities of the two fluids.

Stratford's method of computing the development of a turbulent boundary layer in a strong adverse pressure gradient, based on division of the layer into an inner equilibrium layer and an outer layer of almost constant total-head, is simple and is based on current knowledge of the properties of turbulent flows, but the correspondence with experiment is worse than would be expected. The causes of this are investigated, first by testing the basic assumptions against the measurements of Schubauer & Klebanoff and then by calculating from the theory properties of the layer at intermediate stages of the development towards separation. It is shown that the theoretical condition for zero wall stress (a relation between pressure rise and pressure gradient) is always satisfied twice, first at an intermediate stage of development with wall stress about one-fifth of the initial value and second at the real position of zero stress. This behaviour involves a rapid decrease of pressure gradient in the neighbourhood of separation, caused by rapid thickening of the layer, and it is shown that the pressure distribution near separation depends only on the pressure rise to separation and the characteristics of the initial boundary layer and not on the geometry of the flow. With more measurements of boundary layers separating in strong adverse pressure gradients, these characteristic distributions could be determined as a one-parameter family. Their use for the prediction of position and pressure rise to separation in flows with specified boundaries without prior knowledge of the pressure distribution is discussed.

A one-dimensional steady solution of the equations of motion of a cold plasma in a magnetic field is obtained. The plasma is of semi-infinite extent, bounded by a plane interface which separates it from a vacuum or medium at rest. The particles approach from infinity, are reflected at the front, and return to infinity in the opposite direction. At infinity, the magnetic field is parallel and anti-parallel to the plasma streams, and is inclined at an angle to the normal to the interface. The front is a current sheet across which the lines of force are bent, with the component of the magnetic field in the plane of the front changing direction. The inertia of the electrons is neglected, and the characteristic frequency associated with the front is the ion gyro-frequency.

The shape of a bubble of one liquid inside a denser body of liquid which rotate rigidly together is determined, the effect of gravity being neglected. When the anular velocity of the liquids is zero the bubble assumes a spherical form, and with increasing angular velocity the bubble flattens at the equator and the length increases. It is found that the length of the bubble is asymptotically proportional to the four-thirds power of the angular velocity. If the speed of the rotation is held constant and the volume is increased, then the bubble elongates, the radius approaches a limiting value, and the bubble length increases almost linearly with the volume. This result suggests a method whereby the interfacial surface tension can be measured.

In the second part of the paper the stability of a long bubble subjected to small amplitude axisymmetric disturbances sinusoidal in the axial direction is investigated. The relation between the wave-number and angular velocity for neutral stability is elliptic. When account is taken of the decrease in the radius of the undisturbed bubble with increase in the angular velocity, it is found that the bubble is stable to all wave-lengths provided the radius attains at least 63% of the limiting value. A criterion is then found for the minimum length of the bubble consistent with stability.

The steady incompressible flow of a non-viscous conducting fluid about thin airfoils and slender bodies is studied for the case of a uniform applied magnetic field aligned with the undisturbed fluid stream. Solutions are found, subject to the restriction of small perturbations to the applied field. This condition determines an upper limit on the range of conductivity to which the solution are applicable. Certain results for larger values of conductivity are presented and discussed.

The lift on airfoils is calculated, including the possibility of magnets and externally driven currents inside the airfoil, and a magnetohydrodynamic analogue to the Kutta condition is discussed. Drag formulae are presented for airfoils and slender bodies, and the distribution of internal currents and magnets for zero drag is shown. Optimum drag airfoils and bodies are discussed briefly.

In previous theoretical treatments of the stability of the compressible laminar boundary layer, the effect of the temperature fluctuations on the viscous (rapidly varying) disturbances is accounted for incompletely. A thorough re-examination of this problem shows that temperature fluctuations have a profound influence on both the inviscid (slowly varying) and viscous disturbances above a Mach number of about 2·0. The present analysis includes the effect of temperature fluctuations on the viscosity and thermal conductivity and also introduces the viscous dissipation term that was dropped in the earlier theoretical treatments.

Some important results of the present study are: (1) the rate of conversion of energy from the mean flow to the disturbance flow through the action of viscosity in the vicinity of the wall increases with Mach number; (2) instead of being nearly constant across the boundary layer, the amplitude of inviscid pressure fluctuations for Mach numbers greater than 3 decreases markedly with distance outward from the plate surface. This behaviour means that the jump in magnitude of the Reynolds stress in the neighbourhood of the critical layer is greatly reduced; (3) at Mach numbers less than about 2, dissipation effects are minor, but they become extremely important at higher Mach number, since for neutral disturbances they must compensate for the generally destabilizing effects of items (1) and (2); (4) the minimum critical Reynolds number for an insulated flat plate boundary layer decreases with increasing Mach number in the range 0 ≤ Me ≤ 3.A full list of symbols is given at the end of this paper.. Since the wave-number varies like 1/M2e when Me [Gt ] 1, the minimum critical Reynolds number is likely to increase sharply at hypersonic speeds.

Numerical examples illustrating the effects of compressibility, including neutral stability characteristics, are obtained and are compared with the experimental results of Laufer & Vrebalovich (1960) at Mach 2·2, and of Demetriades (1960) at Mach 5·8.

The flow past circular cylinders of finite length, supported at one end and lying with their axes perpendicular to a uniform stream, has been investigated in a supersonic stream at Mach number 1.96 and also in a low-speed stream. In both stream it was found that the flow past the cylinders could be divided into three regions: (a) a central region, (b) that near the free end of the cylinder, and (c) that near the supported end. The locations of the second and third regions were found to be almost independent of the cylinder length-to-diameter ratio, provided that this exceeded 4, while the flow within and the extent of the first region were dependent on this ratio. Form-drag coefficients determined in the central region in the supersonic flow were in close agreement with the values determined at the same Mach number by other workers. In the low-speed flow the local form-drag coefficients were dependent on length-to-diameter ratio and were always less than that of an infinite-length cylinder at the same Reynolds number.

The motions of single disks suspended in a liquid undergoing Couette flow have been studied in detail. The angular rotation and the axial spin of the disks were found to be in good agreement with the theory of Jeffery, provided the equivalent axis ratio re was used instead of the measured axis ratio r. It was found that non-interacting disks move in constant orbits without a tendency to drift into orbits corresponding to a minimum dissipation of energy. Discontinuous changes in orbit were observed to occur following two-body collisions.

Thin-airfoil theory is applied to steady, plane potential flow about vented or cavitating hydrofoils of arbitrary profile when there are two free-streamlines detaching from the foil and bounding the single cavity that extends downstream of the trailing edge. Cavity-termination models employed are the closed, the partly closed and the open models for which the thickness of the implied ’wake’ following the cavity ranges from zero to maximum for the open model. The general solution for given wetted-surface profile, cavity length and particular cavity termination is constructed by superposition of the profile's cusp-closure solution (angle of attack α+) plus the particular flat-plate solution to give the desired angle of attack α. Four related integrals involving the wetted-surface contour slope distribution lead to drag, lift, cavity pressure and α+vs cavity length. A comparison of theoretical and experimental lift and drag for a cavitating hydrofoil shows good agreement until the theoretical cavity closure nears the trailing edge.

The two-dimensional steady flow of an incompressible viscous liquid round a circular cylinder is described in terms of matched expansions valid asymptotically at low Reynolds number, the velocity field at large distances being a combination of a uniform simple shear and a uniform stream, relative to axes moving with the centre of the cylinder (but not rotating with it).

An infinite number of terms are computed and summed. There is a transverse force on the cylinder, independent of its rate of rotation to the approximation considered here. At moderate and large distances the balance between convection and diffusion of vorticity is dominated by the shear, being quite different from that in a uniform stream alone.

The rate of longitudinal diffusion of a substance released from an instantaneous line source in a simple shear alone is enhanced. Round a maintained line source for which the local convection velocity is parallel to that of the shear the concentration falls only algebraically with distance in all directions, though it is largest in twin wakes extending directly both upstream and downstream. If there is superposed a lateral convection, however small, at sufficiently large distances the concentration is exponentially small outside a wake centred on half a parabola.

The two-dimensional perturbation velocity round any obstacle held in an unbounded simple shear at any Reynolds number is, at a sufficient distance, an irrotational cross flow decreasing as the inverse two-thirds power, associated with twin shear layers extending upstream and downstream. If there is a uniform lateral motion at large distances this conclusion is completely altered.