In the theory of thermal convective instability between two horizontal planes there are many solutions that are periodic in the horizontal co-ordinates, while in experiment convection is observed to take place in cellular patterns. It is often assumed, or decided after insufficient argument, that the periodic solutions of the mathematical model ‘explain’ or correspond to these patterns, but a completely satisfactory discussion of this correspondence has not been given. Indeed, with certain mathematical solutions ambiguities arise as to what cell centres and cellular boundaries are. A detailed discussion has recently become especially necessary because attempts are being made to predict which particular cellular pattern will occur in given experimental conditions.

In this paper the topic is studied afresh and the question is asked: what features, in the mathematical model, correspond to what an experimentalist observes in cellular convective motion? In answer a definition of a cell is formulated which relates certain surfaces in the flow field of the mathematical model to steady vertical cellular boundaries that are observed in experiment, and which shows where the cell centres lie. In particular the classical hexagonal cellular pattern, of the mathematical model, is shown to be the prototype pattern of what is experimentally observed. On the other hand the square and so called ‘rectangular’ cases of linearized theory are shown not to correspond truly to square and rectangular cells at all. The new formulation is especially relevant to theoretical work on the prediction of cell shape and direction of flow in cells, since precise knowledge of the shapes of the cellular boundaries and locations of cell centres is essential if predictions are to be compared with observation.

Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.

This paper concerns the boundary layer on a semi-infinite flat plate in a uniform stream of conducting fluid, with magnetic field in the stream direction. It is found that when β, the square of the ratio of the Alfvén speed to the undisturbed fluid speed, is slightly less than unity, two solutions of the Greenspan-Carrier equations governing the motion exist for ε(= σμν) less than unity.

The stability of a steady, plane, one-dimensional trans-Alfvénic shock to small transverse disturbances in the velocity and magnetic field is discussed. In the undisturbed flow the magnetic field and velocity are normal to the plane of the shock which is treated as a discontinuity in an inviscid gas of zero thermal conductivity. However, the electrical conductivity of the ambient gas is taken to be finite, i.e. the magnetic diffusivity is very much larger than both the viscous diffusivity and the thermal diffusivity.

A small uniform transverse perturbation of the magnetic field is imposed at zero time. A specific super-Alfvénic normal shock is chosen. Computed values of the magnetic field, in and near the shock, are given for various later times. Two diffusing Alfvén waves of the amplitudes predicted by infinite conductivity theory are shown to propagate away from the shock region, leaving behind the expected steady-state shock profile.

Similar computations are carried out for a specific trans-Alfvénic normal shock. An analytic asymptotic solution, valid for large times, is also obtained. This result agrees with the computations carried out. The shock profile of the transverse quantities is one which grows linearly with time. Outside this magneto-hydrodynamic shock region which surrounds the trans-Alfvénic hydrodynamic shock (discontinuity), steady states are reached in each of which the transverse velocity and transverse magnetic field are uniform. An incident Alfvén wave, consisting of a weak diffusing current sheet, produces these same effects. The resolution of an arbitrary transverse fluctuation, in which both the magnetic field and the velocity have limiting values at large distances from the shock, is discussed. The solution for large times is found.

It is shown that the trans-Alfvénic normal shock and its two limiting cases, the null switch-on and null switch-off shocks, are unstable to general transverse disturbances although there exist particular disturbances of this kind which will not destroy them. An integral condition is obtained which, together with the relevant boundary conditions, determines the profiles of the transverse quantities in the trans-Alfvénic normal shock whenever a steady state is reached. This removes the puzzling arbitrariness of these profiles.

A solution of the Navier-Stokes equations is obtained for the flow resulting from the steady rotation of a semi-infinite right circular (solid) cylinder about its vertical axis. Incompressible viscous fluid is assumed to fill the space outside the cylinder on one side of a horizontal solid plane. In the proposed method of solution the pertinent physical quantities are expressed as series in positive powers of the Reynolds number Re with space-dependent coefficients. It is shown that the coefficients of (Re)M can be obtained by solving linear partial differential equations which depend on the coefficients of (Re)i, where i < M. A truncated solution, which holds for small Re, is obtained by solving for the first two coefficients. These results indicate that at the flat end of the cylinder the pressure distribution is nearly constant, yet along the adjacent bounding plane it rises with the radial direction.

At present, solutions for natural convection problems involving discontinuous changes in wall temperature do not exist. This paper reports results of an extensive experimental investigation of the particular case of a vertical flat plate with a step-change in wall temperature. Temperature profiles and local heat transfer coefficients determined with a Mach-Zehnder Interferometer are presented for a wide range of the pertinent parameters. The flow field associated with this problem was studied by means of a series of Tellurium dye experiments in water. Photographs are presented showing representative flow patterns, and the stability of the flow is considered as a function of the ratio of the excess temperature below a discontinuity to the excess temperature above.

In this paper some approximate results are presented for the problem of the stability of plane Poiseuille flow of a dusty gas, following a formulation given recently by Saffman (1962). It is assumed that the mass concentration of the dust, f, is small, and results are obtained by making a perturbation of the curve of neutral stability for a clean gas, using the approximate solutions given by Stuart (1954). The perturbation equation is expressed in terms of integrals by introducing the adjoint wave function, the calculation of which is described. The integral coefficients were evaluated by numerical integration using a Mercury computer, and the results are illustrated for f = 0.05 by a set of perturbed neutral stability curves at different values of the time relaxation parameter SR varying from 0 to 500. These results, whilst not of great numerical accuracy, are sufficient to show qualitatively how the curve of neutral stability is modified by the presence of the dust.

Numerical integration of the boundary-layer equations associated with flow past a semi-infinite flat plate in the presence of an aligned magnetic field has shown that the solutions are not unique if ε < 1 for certain values of β < 1, where ε is an intrinsic property of the fluid and β a property of conditions at infinity. An analytic explanation of this phenomenon is given here. The main properties as β → 1 of the unique solutions when ε > 1 are elucidated. Further, the equations associated with flow past a solid boundary in which the magnetic field is zero are solved numerically. The solutions appear to be unique but, on the other hand, the maximum value β0, of β, for which they can be found, tends to zero with ε.

The classical Kelvin-Helmholtz problem of the instability of the vortex sheet between two uniform streams is extended to allow for non-uniformity in the streams. A small-wavelength approximation shows that the most unstable disturbances have a growth rate proportional to the greatest discontinuity of velocity at the vortex sheet. The solution for all wavelengths is found for two cases when the variation in the stream velocity is small compared with the stream velocity itself. One of these cases indicates that a transverse variation in the stream velocity can increase the instability for long wavelengths, but only to a small extent.

In an earlier paper (Gallagher & Mercer 1962) the numerical results for the first eigenvalue of the problem of plane couette flow were given. The higher eigen-values are now examined and are found to be in agreement with those of Southwell & Chitty (1930), but to disagree with those of Grohne (1954).

An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

Consideration is given to the flow of an idealized elastico-viscous liquid contained in the narrow channel between two concentric cylinders, the motion being due to the relative rotation of the cylinders. It is shown that the presence of elasticity in the liquid lowers the value of the critical Taylor number at which instability occurs. The secondary motion arising at the onset of instability has the usual cellular pattern, the cell length being decreased by the presence of elasticity in the liquid.

Free-surface oscillations of a liquid relative to an equilibrium state of uniform rotation about the vertical of an axisymmetric container are considered for small $\alpha = \omega ^2a|g$, where ω is the angular velocity of rotation and a the cylinder radius. A variational approximation is used to obtain explicit results, with an error of O(α2), for axisymmetric gravity and inertial waves in a flat-bottomed circular cylinder; these results are found to be in agreement with observations reported by Fultz (1962). The first-order (in ω) effects of rotation on asymmetric waves in a circular cylinder also are determined.

The paper is discussion of measurements of the statistical properties of the pressure field at the wall of turbulent attached shear flows. These measurements have been made only in part by the author. A preliminary discussion is given of the important limitations imposed by the imperfect space resolution of contemporary pressure transducers. There follows a discussion of the appropriate scales of the pressure field. It is shown that measurements of the longitudinal cross-spectral densities lead to similarity variables for the space-time covariance of the pressure and for the corresponding spectra. The existence of these similarity variables may be due to the dispersion of the sources of pressure by the mean velocity gradient. Such a mechanism is illustrated by a simple model. Lateral cross-spectral densities also lead approximately to similarity variables.

Computations based directly upon detailed pressure-velocity correlation measurements by Wooldridge & Willmarth reveal that an important part of the pressure at the wall of a boundary layer is contributed by source terms which are quadratic in the turbulent velocity fluctuations; the interaction of the mean strain rate with normal velocity fluctuations, being in effect limited to a region very near the wall, supplies a dominant contribution only at high frequencies and its scales, downstream convective speed and convective memory are markedly smaller than those of the observed wall pressure.

The inner part of the Law of the Wall region (y* [les ] 100) seems to be substantially free of pressure sources and within that region (a) the pressure can be given in terms of its boundary value, and (b) the local velocity field is dependent upon but unbale to affect appreciably the turbulent pressures.

The object of this work was to investigate experimentally the structure of the early shear layer of high-speed jets and its relation to the mechanism of noise generation. Of special interest was the question of the existence of periodic fluctuations in the velocity field. The experimental investigation is divided in three parts.

1. Optical observation of the jet flow by means of the shadowgraph technique.

2. Measurement of mean Mach number and temperature profiles.

3. Survey by means of hot-wire of the component of the fluctuating velocity field in the mean flow direction.

The shadowgraphs show very interesting features of the breakdown process of free shear layers, but fail to show any propagation of strong acoustic disturbances in the near pressure-field.

Mean profile measurements show that the flow field in the range 1 [les ] x/d [les ] 4 develops in a conical fashion, i.e. the mean profiles in Mach number and temperature can be expressed in terms of a single conical variable η = (2r–d)/2x. The fluctuating velocity field is described in terms of the intensity of turbulence, its spectral distribution, and two-point space-time correlation functions. Similarity laws are given for the power spectra and the space-time correlation functions. On the cylinder r = ½d, the convection speed of the turbulent field is different for the different eddy sizes, varying from the local mean speed for small eddies to ½Uexit for the large eddies. Measurements of the angular correlation function are reported which show no correlation of the fluctuations across the jet diameter.

An investigation is made of the disturbances produced in a closed, gas-filled tube by the oscillations of a piston at one end, when the piston oscillates at near resonant frequencies. Within a well-defined frequency band around each resonant frequency, shock waves appear in the solution; outside this interval the oscillations are continuous, but not purely sinusoidal.

The solution includes the effects of compressive viscosity, and of shear viscosity in the boundary layer at the walls of the tube. For typical laboratory conditions the effect of compressive viscosity is found to be quite small (giving a shock thickness of the order of 10−4in.). The boundary layer effect can be more significant, though the most important modification required of the usual acoustic theory is found to arise from the non-linear terms.

A kinematic model of the flow around and into a buoyant ‘thermal’ is discussed in some detail, and compared with existing laboratory observations. The basic assumption is that the flow is instantaneously the same as it is for Hill's spherical vortex of fixed size moving through a frictionless fluid. The equations describing the motion of a particle have been modified to allow for an expansion in radius proportional to the distance travelled, and a numerical integration of these time-dependent equations has been performed in order to find particle trajectories, with respect to both the instantaneous boundary of the spherical vortex and axes at rest. It is shown that for the expanding vortex there is no quantity corresponding to the ‘drift distance’ or the total forward displacement of particles in the flow round a sphere of constant size; particles in the wake of an expanding vortex have a finite forward velocity at large times.

The model gives very close agreement with the observed behaviour of particles entering a turbulent buoyant thermal, except that in the laboratory measurements the region containing the turbulent fluid resembles an oblate spheroid rather than a sphere. The thin mixing layer over the front and the addition of fluid over a broad region at the rear of a thermal, as well as the detailed particle trajectories, can all be explained as consequences of this mean velocity distribution.

Hypersonic flow past axisymmetric power-law bodies $(y \propto x^m; m = 0, { \frac {1}{10}},{ \frac {1}{4}},{ \frac {1}{3}},{ \frac {1}{2}},{ \frac {3}{4}}\; and \; 1)$ has been studied experimentally and a detailed analysis of the dependence of shock-wave shape on body shape made.

Results indicate a continuous relationship between the exponent of the shock-wave shape and that of body shape over the range considered.

When I was first asked to give a general lecture to the 5th British Theoretical Mechanics Colloquium (held at Liverpool University, 2–5 April 1963), I put up the title ‘Shock Waves’, thinking that I might run over a wide field of present-day research, pointing out some of the unanswered problems. In the intervening months, however, I came across the new book by Dr Bradley (1962) of Liverpool University, The Physics and Chemistry of Shock Waves, and as recently as last November there appeared a summarizing article by Pain bt Rogers (1962) of London University in Reports on Progress in Physics. The first of these deals in great detail with the modern physical and chemical aspects of the subject—with real gas effects and experimental techniques-and the second summarizes the general classical properties and gives an extended account of recent work, for example, on real gas effects and on magnetohydrodynamics. I also bore in mind that magnetogasdynamics received a majestic treatment at the 4th Colloquium from Dr Shercliff. In the end, then, I decided to confine my remarks to one particular problem, namely the reflexion and diffraction of shock waves, and to concentrate mainly on developments during the past ten years. This paper is the text of the lecture.

In Part 1 of this paper a free-streamline wake model was introduced to treat the fully and partially developed wake flow or cavity flow past an oblique flat plate. This theory is generalized here to investigate the cavity flow past an obstacle of arbitrary profile at an arbitrary cavitation number. Consideration is first given to the cavity flow past a polygonal obstacle whose wetted sides may be concave towards the flow and may also possess some gentle convex corners. The general case of curved walls is then obtained by a limiting process. The analysis in this general case leads to a set of two functional equations for which several methods of solution are developed and discussed.

As a few typical examples the analysis is carried out in detail for the specific cases of wedges, two-step wedges, flapped hydrofoils, and inclined circular are plates. For these cases the present theory is found to be in good agreement with the experimental results available.