A study of the unique role of impurities in the initial stages of ionization relaxation in shock-heated argon, using a sampling mass spectrometer to determine the ionic products of the reaction, is described. The ions are extracted from the shock tube through a small orifice in the end wall after they have diffused through the dense thermal layer adjacent to the wall from the ionizing gas behind the reflected shock wave. The ion diffusion is analysed in detail to assess the possibility that the sampling process alters the reaction products. It is shown that this is unlikely because the impurities are in dilute concentration and the reaction is studied in its initial stages. This mode of sampling is compared with others.

The experiments were conducted in argon at temperature of 16,600 °K and pressure of 16 mmHg with an estimated impurity level of 300 parts per million. A surprisingly large number of different ions were detected during the initial stages of ionization. O+ and H+ were found in much greater amounts than any of the other products, each being about five times more abundant than A+. The results suggest that H2O is probably quite generally the most important impurity in thermal-ionization experiments, and that ionization ‘incubation’ is due to dissociation of molecular impurities (especially H2O) before ionization commences. Possible explanations of the well-known efficiency of small amounts of impurities in initiating ionization are discussed.

The turbulent flow of a weakly conducting liquid between parallel plates in the presence of a transverse magnetic field is investigated. The form of the mean velocity profile is determined by a series of constraints resulting from the boundary conditions and the Navier–Stokes equations and by the Malkus postulates on the spectrum of the mean vorticity gradient. The width of the transition regions near the walls is derived in terms of the governing dimensionless numbers and this expression is checked, in the asymptotic laminar case, against the well-known Hartmann result. A graphical method, exploiting the relation between the boundary region thickness and the smallest scale of motion defined by the Malkus theory is proposed to determine the scale of the velocity profile, i.e. the flow rate in terms of the pressure gradient and the magnetic field strength.

An experiment has been carried out to verify the existence of the resonant interaction between trains of gravity waves, predicted by Phillips (1960). As suggested by Longuet-Higgins (1962), two trains of waves in mutually perpendicular directions were generated in a rectangular wave tank. The ratio σ1/σ2 of the wave frequencies was varied (1·4 < σ1/σ2 < 2·1). When σ1/σ2 [eDot ] 1·7357 it was expected that a resonant interaction would take place, generating a wave of frequency (2σ1−σ2). The amplitude of the third wave was expected to increase almost linearly in the direction of wave propagation. The shape of the response curve as a function of σ1/σ2 was also predicted.

In the present experiments rather large wave amplitudes had to be used, and the theoretical shape of the response curve was distorted by non-linear detuning. Nevertheless the peak amplitude of the resonant wave was found to increase with distance in very nearly the manner predicted.

These experiments were carried out in 1961 but publication was deferred pending a similar but more accurate investigation by McGoldrick, Phillips, Huang & Hodgson (1966). Much of the theoretical discussion given in the present paper is relevant to their work.

A fluid is contained between rigid horizontal planes which move relative to each other with constant horizontal velocity. A gravitationally unstable temperature difference is maintained at the boundaries, and the heat flux and momentum flux (stress) transmitted by the fluid are measured. The Nusselt number, Nu, and the dimensionless momentum flux, Mo, are obtained for small mean rates of shear. The Rayleigh number, R, and the Prandtl number, σ, are both large in these experiments. The data are consistent with the following relations: \[ Nu \propto Mo \sigma^{\frac{1}{2}} \propto R^{\frac{1}{3}}. \]

Kraichnan's mixing-length theory of turbulent thermal convection is extended to the present situation, and the above experimental dependence of Nu and Mo on R and σ is obtained. The agreement between mixing-length theory and experiment provides strong support for Kraichnan's concise treatment of turbulent convection. The importance of this result is that heat and momentum fluxes may be calculated in this way for a variety of flows in geophysics.

This study of the boundary layer of steady, incompressible, plane, crossed-fields m.h.d. flow at large Reynolds number Re and magnetic Reynolds number Rm begins with a review of Hartmann's case, where a boundary layer occurs whose thickness is proportional to (Re Rm)−½. Following this clue, it is shown that in general the boundary layer is a ‘local Hartmann boundary layer’. Its profiles are always exponential and it is determined completely by local quantities. The skin friction and the total electric current in the layer are proportional to the square root of the magnetic Prandtl number, i.e. to (Rm/Re)½. Thus the exterior-flow problem, the solution of which precedes a boundary-layer solution, generally involves a current sheet at the fluid-solid interface.

This inviscid-flow problem becomes tractable if (Rm/Re)½ is small enough to permit a linearized solution. The flow field about a flat plate at zero incidence is calculated in this approximation. It is pointed out that the thin-cylinder solutions of Sears & Resler (1959), which pertain to Rm/Re = 0, can immediately be extended to small, non-zero values of this parameter by linear combination with this flat-plate solution.

This paper presents the results of experiments on the resonant interaction of gravity waves. Two mutually-orthogonal primary wave trains are generated in a tank and their interaction products studied at various positions on the surface. Under suitable conditions, the growing resonant third-order interaction product is identified; its amplitude is shown to be a linear function of the interaction distance. The band-width of the response decreases with increasing distance, as is characteristic of the phenomenon of resonance. The ratio of the frequencies of the primary waves at resonance is very close to that predicted theoretically; the growth rate of the third component is close to, though about 20% higher than, the predicted value. Conditions far from resonance are also studied; it is found that the growing tertiary wave is absent in this case.

These results offer the first unambiguous experimental demonstration of resonant wave interactions.

It is found that when the average kinetic energies of normal velocity components in decaying, grid-generated turbulence are equilibrated by a symmetric contraction of the wind tunnel, this equality can persist downstream. A second result is further confirmation of the fact that the best power-law fit to the inverse turbulent energy during the early part of decay is near (x–x1)1·25, for both rod grids and disk grids. The Kolmogorov decay law $\sim (t - t_1)^{\frac{10}{7}}$ is re-derived by a spectral method which is essentially equivalent to the original. Finally, a crude theoretical estimate of component energies in the straight duct after a weak contraction seems to support the experiments.

A general numerical method of characteristics applicable to problems in magneto-fluid dynamics as well as ordinary fluid dynamics is described. The method can be applied to unsteady three-dimensional flows of chemically reacting, non-equilibrium, multi-component media. Dissipative phenomena must be neglected in order to make the governing equations of change hyperbolic, because the method can be applied only to quasi-linear, hyperbolic, partial differential equations. Practical restrictions on computation time usually require unsteady problems to be limited to cases with short transient times although theoretically the method applies to all unsteady flows. In steady flow the local velocity must be greater than the largest local wave speed. The characteristic and compatibility equations are derived for the most general case of magnetofluid dynamics. A new finite-difference network and its corresponding equations are developed similarly. Specialization of the general method to consider simpler problems is outlined. Preliminary numerical results of calculations using the method are presented. The practicality and feasibility of utilizing the general numerical method of characteristics on presently available, electronic digital computers is evaluated in the light of recent experience in calculating multi-dimensional flows with the method.

This paper deals with the growth of small disturbances in a separated laminar boundary layer for high Reynolds numbers as a function of the dimensionless flow parameters. Using a hot-wire technique, the experiments show that spatially growing disturbances are only affected by the Strouhal number. Thus the basic equations of the process become relatively simple. The experiments show good agreement with theoretical results obtained by means of hydrodynamic stability theory for spatially growing disturbances.

In a recent study related to transition in the wake flows behind circular cylinders held transversely to an air stream, Bloor (1964) has reported the observation of velocity ‘spikes’ and attributed these to the close proximity to the hot wire of vortex centres on the opposite side of the von Kármán vortex street. Further observations of spikes are reported here, and the characteristics of their distribution indicate that other explanations of their form must be found. Some idealized flows are considered, and it is concluded that observations of spikiness within the hot-wire output may be accountable in terms of large-scale distributions of vorticity within the flow convected past the wire, the distributions being reasonable representations of a separated flow. The observations also provide some evidence that small vortices of Strouhal frequency exist on the inside of the coherent separated shear layer, and this may assist in the understanding of the feed-back mechanism where by the von Kármán street establishes itself as a self-perpetuating phenomenon.

When a semi-infinite flat plate is immersed parallel to an unbounded, plane, steady, asymmetric, constant shear flow of an incompressible viscous fluid, an interaction occurs between the surface-generated vorticity and the external vorticity. A physical assumption was made in a previous paper (Mark 1962) concerning this problem that the pressure field in a thin layer adjacent to the top-side of the plate may be accurately approximated by the undisturbed constant-pressure field—that which exists before the insertion of the plate into the flow. This means that the vorticity interaction is assumed to have no effect on the undisturbed pressure field. That this is a valid first approximation far downstream along the plate where the interaction is intense is given rigorous support in this paper.

The flow below the plate is examined on a heuristic basis. It is found that there is a strong possibility for the flow to separate from the lower surface near the leading edge of the plate. However, far downstream the flow settles down to a Stokes-type flow near the plate.

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962b) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here.

The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.

The piston problem for a viscous heat-conducting gas is studied under the assumption that the piston Mach number ε is small. The linearized Navier–Stokes equations are found to be valid up to times of the order of ε−2 mean free times after the piston is set in motion, while at large times the solution is governed by Burgers's equation. Boundary conditions for the large-time solution are supplied by the matching principle of the method of inner and outer expansions, which is also used to construct a composite solution valid both for small and for large times.

A second approximation for the velocity of a large gas bubble in an infinite liquid is derived from a linear perturbation of the first approximation. Previously known experimental results are in excellent agreement with the resulting velocity, \[ U = 0.652(g\overline{a})^{\frac{1}{2}}, \] where $\overline{a}$ is the apparent radius of curvature of the front part of the bubble, and g the acceleration due to gravity. The shape of this second approximation is seen to be indistinguishable from spherical over a large region near the front stagnation point.

Because of the relative simplicity of the statistical model for molecular interactions, numerically exact solutions of the Krook kinetic equation can be obtained. Comparison of exact and approximate solutions of the model equation allows the evaluation of approximate procedures for solving the Boltzmann equation. Exact and approximate numerical solutions have been obtained for Couette flow with heat transfer and the structure of a plane shock wave. The present paper summarizes the work of the author in obtaining the numerically exact solutions.

The hypersonic strong-interaction regime for the flow of a viscous, heat-conducting compressible fluid past a flat plate is analysed using the Navier–Stokes equations as a basis. It is assumed that the fluid is a perfect gas having constant specific heats, a constant Prandtl number σ, whose numerical value is of order one, and a viscosity coefficient varying as a power, ω, of the absolute temperature. Limiting forms of solutions are studied as the free-stream Mach number M, the free-stream Reynolds number based on the plate length RL, and the interaction parameter x = {(γM2)2+ω/RL}½, go to infinity.

Through the use of asymptotic expansions and matching, it is shown that, for (1−ω) > 0, three distinct layers for which similarity exists make up the region between the shock wave and the plate. The behaviour of the flow in these three layers is analysed.

The investigation to be described here is a wide-ranging experimental study aimed at determining both the details of the flow field and the pressure drop and friction factor characteristics for turbulent flow in eccentric annular ducts. The experiments were performed utilizing three annular ducts of different diameter ratios; in each case the eccentricity was varied from zero (concentric annulus) to unity (walls in contact). To provide the broadest possible perspective, the measurements of the velocity field are presented in three different ways. First, contour maps showing lines of constant velocity are constructed. From these are deduced circumferential distributions of the local shear stress on the bounding walls. Velocity profiles along lines normal to the walls are represented in terms of both law-of-the-wall variables and defect-law variables. Neither of these representations affords complete agreement with the universal circular-tube distributions. In general, the defect law provides a somewhat closer correlation of the results for the eccentric annulus with those for the tube. The experimental findings do not substantiate a prior analytical model which assumes that the universal law of the wall applies on all lines normal to the bounding walls of the annulus. Friction factors, based on static pressure measurements, are shown to decrease with increasing eccentricity. The measured friction factors are in fair agreement with those of analysis. Hydrodynamic development lengths, deduced from entrance-region pressure data, are found to increase with increasing eccentricity. Circumferential pressure variations also increase with eccentricity.

The paper presents the results of measurements of fluctuating lift and drag on a long square cylinder. The measurements include the correlation of lift along the cylinder and the distribution of fluctuating pressure on a cross-section. The magnitude of the fluctuating lift was found to be considerably greater than that for a circular cross-section and the spanwise correlation much stronger.

It was found that the presence of large-scale turbulence in the stream had a marked influence on both the steady and the fluctuating forces. The most significant changes were at small angles of attack (%alpha; < 10°) and included a reduction in base suction and a decrease in fluctuating lift of about 50%.

An improved version of Corrsin & Kistler's method has been used to measure intermittency in favourable and adverse pressure gradients, and the characteristic parameters of the intermittency have been related to the form parameter H of the mean velocity profiles.

It is found that with adverse pressure gradients the centre of intermittency moves outward from the surface while the width of the intermittent zone decreases. The converse is true of favourable pressure gradients, and it seems likely that at sufficiently low values of H the flow over the full depth of the layer is only intermittently turbulent.

A new method of intermittency measurement is presented which makes use of a photo-electric probe. Smoke is introduced into the boundary layer and illuminated by a narrow beam of parallel light normal to the surface. The photoelectric probe is focused on the illuminated region and a signal is generated when smoke passes through the focal point of the probe lens. Comparison of this signal with the output from a hot-wire at very nearly the same point shows the identity of smoke and turbulence distributions.

This is a continuation of our previous work (1965) on the Sears–Resler–Stewartson controversy, in the context of axially symmetric flow. A new approach is presented using boundary-layer arguments, which remove much of the old complexity.

For resistive bodies of shape R(x) we uncover a similarity with plane airfoils of shape F(x) = R2(x). As before, Stewartson's slug flows develop fore and aft. For bodies of very high conductivity the Sears–Resler (steady-state) solution turns out to be one possibility. It pertains to bodies (of much higher conductivity than the liquid) into which the initial magnetic field has not diffused.