An analytical solution is obtained for the flow field due to the impinging of a plane shock wave of arbitrary strength by a thin wing moving in the opposite direction. The planform and the thickness distribution of the wing can be arbitrary and the speed of the wing can be either supersonic or subsonic relative to the undisturbed stream ahead of the shock or to that behind the shock. The solution is a generalization of the previous solution of Ting & Ludloff for the diffraction of shock wave by a two-dimensional stationary airfoil to a three-dimensional wing moving with supersonic or subsonic speed relative to the stream ahead of or behind the shock. The solution is employed for the analysis of the changes in aerodynamic forces when an airplane encounters a blast wave or a shock wave of another airplane. It is also used to study the diffraction of a shock wave or an N-wave advancing over flat terrains.

The problem considered is that of absorbing two-dimensional water waves in a channel by means of a moving termination at the end of the channel. The problem is formulated for a semi-infinite channel and solutions are determined according to a linearized theory. The motion of the termination that is needed for absorption is determined in the form of a linear operation on the measured surface elevation at a fixed point in the channel so a self-actuating wave-absorbing system can be devised. A theoretical method of studying the stability of such a system is presented. A system of this type was built and experiments with it are described. Wave absorption is demonstrated both for monochromatic waves and for wave pulses. The absorption of a wave pulse is compared with the absorption of the same pulse by a fixed beach making a ten degree angle with the horizontal direction.

An effort is made to understand turbulence in fluid systems like the oceans and atmosphere in which the Richardson number is generally large. Toward this end, a theory is developed for turbulent flow over a flat plate which is moved and cooled in such a way as to produce constant vertical fluxes of momentum and heat. The theory indicates that in a co-ordinate system fixed in the plate the mean velocity increases linearly with height z above a turbulent boundary layer and the mean density decreases as z3, so that the Richardson number is large far from the plate. Near the plate, the results reduce to those of Monin & Obukhov.

The curvature of the density profile is essential in the formulation of the theory. When the curvature is negative, a volume of fluid, thoroughly mixed by turbulence, will tend to flatten out at a new level well above the original centre of mass, thereby transporting heat downward. When the curvature is positive a mixed volume of fluid will tend to fall a similar distance, again transporting heat downward. A well-mixed volume of fluid will also tend to rise when the density profile is linear, but this rise is negligible on the basis of the Boussinesq approximation. The interchange of fluid of different, mean horizontal speeds in the formation of the turbulent patch transfers momentum. As the mixing in the patch destroys the mean velocity shear locally, kinetic energy is transferred from mean motion to disturbed motion. The turbulence can arise in spite of the high Richardson number because the precise variations of mean density and mean velocity mentioned above permit wave energy to propagate from the turbulent boundary layer to the whole region above the plate. At the levels of reflexion, where the amplitudes become large, wave-breaking and turbulence will tend to develop.

The relationship between the curvature of the density profile and the transfer of heat suggests that the density gradient near the level of a point of inflexion of the density curve (in general cases of stratified, shearing flow) will increase locally as time goes on. There will also be a tendency to increase the shear through the action of local wave stresses. If this results in a progressive reduction in Richardson number, an ultimate outbreak of Kelvin–Helmholtz instability will occur. The resulting sporadic turbulence will transfer heat (and momentum) through the level of the inflexion point. This mechanism for the appearance of regions of low Richardson number is offered as a possible explanation for the formation of the surfaces of strong density and velocity differences observed in the oceans and atmosphere, and for the turbulence that appears on these surfaces.

The boundary layers associated with gravity waves in a fluid with a linear variation of density are discussed in order to examine the steady Eulerian velocities induced by the Reynolds stress. For the case of a standing wave, the induced steady motion is shown to decay in an outer boundary layer which represents a balance between buoyancy and diffusion when the wave slope is sufficiently small but when viscous decay effects are even smaller. When the wave slope is larger, it would appear that two outer regions must be considered. Results for progressive waves are discussed only briefly, as they are qualitatively similar to the surface wave case.

A linearized analysis of the two-dimensional double vortex sheet model of a jet shows that inviscid jet instabilities occur over a wide range of frequencies at all jet Mach numbers. No particular frequency for maximum growth rate exists unless finite shear layer thickness effects are considered. It is suggested that the model describes the essential characteristics of a real jet disturbed by long wavelength perturbations. The idea is advanced that the jet flow constitutes a broad band amplifier of high gain. Disturbances can grow rapidly to a size when nonlinear effects bring about significant interaction with the mean flow. By seeding the jet with disturbances of a type that are highly amplified it is argued that gross features of the flow may be affected and that the jet may be rendered less noisy at high Mach number. It is argued that some of these ideas are supported by the observation that a supersonic jet diffuses at an unusually rapid rate when subject to the oscillatory condition known as ‘screech’.

The equations governing the free-surface flow of a tubular film of liquid are derived from physical arguments, which throw some light on the formal process described in part 1. The solutions of the equations are discussed, in particular with reference to the film-blowing process for the manufacture of thin sheets of thermoplastic material. The qualitative adequacy of a model based on the dominance of viscous forces is demonstrated, and the effect of surface tension, air drag and non-isothermal flow is discussed briefly.

Small perturbations of a parallel shear flow U(z) in an inviscid, incompressible, stably stratified fluid of density ρ(z) are considered, for which the principal measure of stability is the Richardson number, R. For an arbitrary velocity and density profile we discuss the problem of determining whether a curve of neutral stability has adjacent unstable regions in an (α, R) plane, where α is the wave-number of the disturbance. Neutral curves bounding unstable regions are then obtained for a triangular jet flow in conjunction with various density distributions. A comparison is also made between the stability characteristics of jet and shear flows with corresponding density structures.

Consideration is given to the influence of edge effects on rheological measurements. To facilitate this, we discuss theoretically the flow generated by the slow steady rotation of a solid of revolution in an elastico-viscous liquid confined by convenient bath surfaces. The relevant linear partial differential equations are solved numerically. The simple problem of a rotating sphere and a concentric spherical container, for which an exact analytic solution is available, is first discussed to justify the method employed, and to indicate the necessary conditions to obtain a given accuracy. The numerical method is then applied to the case of a cone of finite dimensions rotating in a bath of elastico-viscous liquid. The predicted flow is shown to be in good agreement with experimental observations.

Application of the theory to rheogoniometric situations indicates that edge effects are not likely to be as significant as has been conjectured in the past.

Observations of temperature drop as a function of heat flow in Rayleigh–Bénard convection with curved density profiles show: (1) reversal of slope in the heating curve, (2) oscillations with time, (3) history dependence, and (4) an increase in critical Rayleigh number as the curvature of the density profile is increased. Some of the results are quite similar to the predictions of Busse.

A stream with constant velocity U is impulsively started at time t = 0 past the trailing edge of a semi-infinite flat plate. According to boundary-layer theory, it is found that the flow at a distance x downstream from the trailing edge is unaware of the presence of the plate when x > Ut; at time t = x/U there is then a discontinuity in the velocity normal to the plate. It is the neglect of diffusion parallel to the axis of the plate that introduces the discontinuity, and when the complete Navier–Stokes equations are considered for t ≃ x/U, a solution is found that can be matched with that gained from boundary-layer arguments.

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

A new method is presented for solution of the Boltzmann equation governing the dynamic behaviour of gases. The essence of the method is idealization of the problem domain into subdomains called finite elements. Then, the Galerkin assumed-mode technique is employed as the basis for discretization of the individual finite elements and also for the assembly of the resulting algebraic models for these finite elements to form an algebraic model for the complete problem. The procedure is cast in a systematic matrix notation that makes evident the broad application potential of the analysis method. An illustrative application is presented for the problem of one-dimensional, linearized Couette flow. Numerical predictions of macroscopic flow velocity and viscous shear stress based upon the subject finite element method are compared with alternative analytical and numerical results. Special attributes of the finite element method are discussed in the context of this example problem. Applications to practical problems governed by generalized forms of the Boltzmann equation are projected on the basis of concepts established herein.

An equation for waves on the surface of a flow with shear is deduced and shown to reduce by suitable scaling to the classical equation of Korteweg & de Vries, which describes such motions on a stationary flow. For steady flows the corresponding theory of cnoidal waves is obtained and the results of Benjamin (1962) for a solitary wave recovered.

A theory is proposed in this paper to describe the behaviour of a class of turbulent shear flows as the Reynolds number approaches infinity. A detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers. The theory considers an underdetermined system of equations and depends critically on the idea that these flows consist of two rather different types of regions. The method of matched asymptotic expansions is employed together with asymptotic hypotheses describing the order of various terms in the equations of mean motion and turbulent kinetic energy. As these hypotheses are not closure hypotheses, they do not impose any functional relationship between quantities determined by the mean velocity field and those determined by the Reynolds stress field. The theory leads to asymptotic laws corresponding to the law of the wall, the logarithmic law, the velocity defect law, and the law of the wake.

The phenomenon of second harmonic resonance for capillary-gravity waves is reconsidered here by the asymptotic method of multiple time and space scales. The periodic finite amplitude waves of permanent form found by Wilton in 1915 which correspond to this configuration are shown to be no more than a special case of the more general resonant interaction theory, and owe their existence to a critical choice of initial conditions. It is further suggested that the influence of viscous dissipation will render this solution virtually undetectable in a real liquid.

The axisymmetric motion of an inviscid, rotating liquid over a prescribed stream surface, say S, is constructed from assumed values of the velocity and azimuthal vorticity on S. The hypothesis of unseparated flow, which implies continuity of the vorticity on S, is shown to imply that: (a) the azimuthal vorticity and azimuthal circulation (relative to the basic flow) must be simply proportional to the perturbation stream function in the exterior of S; (b) the exterior field exhibits a dipole behaviour far upstream of the body, thereby satisfying Long's hypothesis of no upstream disturbance.

The disturbance induced in the uniform flow of a viscous, rotating liquid by an axial point force −D is studied under the restrictions that the Ekman number, E = 2Ων/U2, be small and that D = O(1/logE) as E → 0. The method of matched asymptotic expansions is invoked to obtain inner and outer (with reference to the dimensionless axial co-ordinate x refered to the length U/(2Ω)) approximations to the solution of the Oseen equations as E → 0. The outer approximation, E → 0 with Ex fixed, is also an outer approximation to the solution of the Navier–Stokes equations. The mass flow across any transverse plane, which is equal to D/U for an oseenlet in a non-rotating flow, vanishes in this approximation. The corresponding inner limit yields a non-uniform, cylindrical flow far upstream of the force in the inviscid limit, E → 0, if and only if D ∝ 1/(log E + const.). This cylindrical flow is a one-term, inner approximation to the solution of the Navier–Stokes equations and suffices to show that separation implies the failure of Long's hypothesis of no upstream influence for inviscid, rotating flow past a finite body. A two-term inner representation of the solution is related to Stewartson's solution of the Oseen equations for a moving source in an inviscid, rotating fluid.