The evolution of a model of the flow in a layer of fluid suddenly heated from below at a Rayleigh number sufficient for a laboratory flow to ultimately become turbulent is investigated with numerical experiments. The numerical study simulates the flow by means of the mean field equations along the lines of Herring's (1963, 1964) pioneering study but uses a different finite difference technique and concentrates attention on the flow development rather than on the final statistically steady state. These solutions are compared with previous and some new simulations in two dimensions. The solutions confirm Herring's work and in addition show that the mean field equations, and in particular the weak-coupling approximations, describe the gross features of the model sufficiently well for the mean field equations to be used with reasonable confidence in evolutionary studies.

The boundary-layer flow over a semi-infinite vertical flat plate, heated to a constant temperature in a uniform free stream, is discussed in the two cases when the buoyancy forces aid and oppose the development of the boundary layer. In the former case, two series solutions are obtained, one of which is valid near the leading edge and the other is valid asymptotically. An accurate numerical method is used to describe the flow in the region where the series are not valid. In the latter case, a series, valid near the leading edge is obtained and it is extended by a numerical method to the point where the boundary layer is shown to separate.

Expressions are derived for a three-dimensional polar in the pressure ratio, deflexion angle space to represent conditions downstream of a detonation wave with given upstream conditions. An analysis of reflected waves is undertaken and the representation of three wave confluences in a three-dimensional hodograph space studied. These techniques are applied to experimental results available for spin detonation in gaseous mixtures of acetylene-oxygen-argon and carbon monoxide-oxygen. Assumptions made in earlier papers concerning local extinction of combustion at the confluence point are not necessary in this picture.

The stratified shear flow over a two-dimensional obstacle of semi-elliptical crosssection is considered. The shear flow is assumed to be inviscid with constant upstream values of the density gradient and dynamic pressure (Long's model). Two complete sets of lee-wave functions, each of which satisfies the condition of no upstream reflexion, are determined in elliptic co-ordinates for ε ≤ 1 and ε ≥ 1, where ε is the ratio of height to half-width of the obstacle. These functions are used to determine the lee-wave field produced by, and the consequent drag on, a semi-elliptical obstacle as functions of ε and the reduced frequency (reciprocal Froude number) within the range of stable flow. The reduced frequency at which static instability first occurs is calculated as a function of ε.

A two-dimensional stratified flow over an obstacle in a half space is considered on the assumptions that the upstream dynamic pressure and density gradient are constant (Long's model). A general solution of the resulting boundary-value problem is established in terms of an assumed distribution of dipole sources. Asymptotic solutions for prescribed bodies are established for limiting values of the slenderness ratio ε (height/breadth) of the obstacle and the reduced frequency k (inverse Froude number based on the obstacle breadth) as follows: (i) ε → 0 with k fixed; (ii) k → 0 with ε fixed; (iii) k → ∞ with kε fixed. The approximation (i) is deveoped to both first (linearized theory) and second order in ε in terms of Fourier integrals. The approximation (ii), which constitutes a modification of Rayleigh-scattering theory, is obtained by the method of matched asymptotic expansions and depends essentially on the dipole form (which is proportional to the sum of the displaced and virtual masses) of the obstacle with respect to a uniform flow. A simple approximation to this dipole form is proposed and validated by a series of examples in an appendix. The approximation (iii) is obtained through the reduction of the original integral equation to a singular integral equation of Hilbert's type that is solved by the techniques of function theory. A composite approximation to the lee-wave field that is valid in each of the limits (i)-(iii) also is obtained. The approximation (iii) yields an estimate of the maximum value of kε for which completely stable lee-wave formation behind a slender obstacle is possible. The differential and total scattering cross-sections and the wave drag on the obstacle are related to the power spectrum of the dipole density. It is shown that the drag is invariant under a reversal of the flow in the limits (i) and (ii), but only for a symmetric obstacle in the limit (iii). The results are applied to a semi-ellipse, an asymmetric generalization thereof, the Witch of Agnesi (Queney's mountain), and a rectangle. The approximate results for the semi-ellipse are compared with the more accurate results obtain by Huppert & Miles (1969). It appears from this comparison that the approximate solutions should be adequate for any slender obstacle within the range of kε for which completely stable lee-wave formation is possible. The extension to obstacles in a channel of finite height is considered in an appendix.

It has been found experimentally that under a lid of excellent heat conductivity the wavelength of convective motions on a uniformly heated plane increases with increasing supercritical Rayleigh numbers. Under an insulating lid, the wavelength at the critical Rayleigh number is larger than under a well conducting lid. When heating from below is applied rapidly the wavelength is less than in the corresponding stationary case.

An experiment has been conducted modelling a two-dimensional mixed region collapsing in a continuously density-stratified medium. The process of collapse can be divided into three stages. Empirical formulae have been derived to describe the process of the first two stages, during which gravitational effects determine the modelling criterion. The collapse in the final stage is complicated by viscous effects; a ratio between the Froude number and the Reynolds number seems to provide a probable modification of the time scale.

The pattern of internal waves generated by the initial impulsive collapse of this mixed region can be represented by moving rays connecting either wave crests or troughs. These rays move away from the collapse centre and at the same time decrease their slopes from the horizontal. A simpler steady-state wave pattern generated by an oscillating plunger has also been studied. Taken together, these experimental results are interpreted to show that the energy density of the collapse is skewed toward high frequencies and is peaked at 8/10 of the Brunt-Väisälä frequency of the medium.

Among the parameters which determine the erosion damage sustained by the walls of a nozzle in which a mixture of gas and particles is flowing, is the angle between the direction of the particle flow and the wall surface at the moment of impact. In this work an approximate analytical solution is made for a number of gas particle flows to determine broadly the features on which particle trajectory depends and some experimental results are given which confirm the theoretical computations. It is shown that the divergent region of a conical nozzle is unlikely to suffer a severe particle attack but that for parallel flow convergent-divergent nozzles the convex region near the exit may be affected. The choke, on the other hand, is most susceptible to particle attack even by fairly small particles. It may be said, in general, that any particle which enters the choke section with a velocity which, in the absence of effects from the gas would allow the particle to strike the choke wall, will in fact hit the wall at some point along the length of the choke.

An exact solution is obtained for the two-dimensional flow of an elastico-viscous (Walters fluid B’) incompressible fluid past an infinite porous wall under the following conditions: (i) the suction velocity normal to the plate oscillates in magnitude but not in direction about a non-zero mean; (ii) the free-stream velocity oscillates in time about a constant mean.

The response of the skin-friction to the fluctuating stream and suction velocity is studied for variations in the suction parameter A, the elasticity parameter k and the frequency parameter μ. It is found that the back-flow at the wall is enhanced by k. For the same value of A, the amplitude of the skin-friction decreases with increasing k. Also an increase in k and μ leads to a decrease in the phase of the skin-friction. For moderately large A and k, the phase of the skin-friction may be completely negative.

The structure of shock-waves in gas mixtures is studied. The separation of component velocities and temperatures is described. Velocity overshoot is never found to exist. Other effects, namely, temperature overshoot and undershoot and velocity undershoot are shown to exist in a manner which is self-consistent with the derivation of the governing equations.

The local and average mass flow rates for nearly free molecular flow through a two-dimensional slit are determined for several tank pressure ratios. The equilibrium gas in the two tanks and the container walls are assumed to be at the same temperature and the Willis iterative method with the Bhatnager-Gross-Krook model is used for the analysis. The results for an infinite pressure ratio are also presented in order to illustrate the effects of a finite pressure ratio.

The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.