Wave diffraction due to a step change in bottom topography is considered for the case of two superimposed fluids of different, but constant, densities. The interface lies below the upper surface of the step. Shallow water theory is shown to be applicable only if the ratio of a non-dimensional frequency parameter to the departure of the density ratio from unity is sufficiently small. An approximate solution of the full equations, obtained by a method applied by Miles (1967) to surface wave diffraction, yields results limited only by the condition that the frequency parameter be small.

Time-dependent motion of a rotating stratified fluid is analyzed within the quasigeostrophic approximation. A few examples of mechanically driven flow are analyzed. It is found that the motion is characterized by the ratio B of the stability frequency and the Coriolis parameter. Thus the ratio of the horizontal and vertical characteristic scale is in general O(B). In particular the decay process caused by a horizontal boundary will penetrate a distance B−1L into the fluid, L denoting the horizontal scale of the motion.

Turbulent flow along the axis of a 90° corner is considered. Experimental measurements of velocities, wall shear stresses and $\overline{u^{\prime 2}}$, the turbulent normal stress in the streamwise direction, have been made. The data are considered both from the point of view of momentum integral and similarity analyses. A suitable momentum integral equation is easily derived but difficult to use due to the form of the terms which must be measured. Similarity techniques give a series of correlations which describe the flow over a narrow range of Reynolds number for which data are available. There is experimental evidence for secondary currents in this type of flow but their analysis presents considerable difficulty.

A theoretical and experimental investigation has been made of the initial ionization processes in a strong shock wave in hydrogen. The relaxation length for ionization, which is principally determined by the rate of excitation, was measured and from a comparison with the theory an estimate was obtained for the cross-section for atom-atom excitation collisions.

Detailed theoretical calculations showed that the electron temperature approaches to within 1 % of the atomic temperature in a distance that is small compared with the total relaxation length for ionization. This enabled considerable simplification, for it indicated that a single-temperature model could be used in calculating the theoretical relaxation profile over the experimental range of operating conditions. An electromagnetic shock tube, with a Philippov pinch to create the driver plasma, was employed to produce shock waves of the required velocity. The ionization behind the shock front was studied by means of a double-frequency Mach-Zehnder interferometer, with a ruby laser and a K.D.P. crystal as the light source. A close agreement between the theoretical and experimental electron density profiles, behind the shock front, was obtained for small relaxation lengths, when the cross-section for the atom-atom excitation collisions was assumed to be about 7 × 10−2 times that of the corresponding cross-section for electron-atom excitation collisions.

Rigorous estimates of amplification rates, wave speeds and sufficient conditions for linear stability are derived for the manifold of solutions of the Orr—Sommerfeld problem governing parallel motion in the boundary layer and in round pipes. The estimates for channel flow (part I) are improved and compared with numerical results for the neutral stability of Jeffery—Hamel flow.

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.

The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.

Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

Under the assumption that the horizontal scales of the flow of a stratified fluid are much greater than the vertical scale, it can be shown that the pressure distribution in the fluid is nearly hydrostatic and that the solution for steady flows can be reduced to the solution of a non-linear partial differential equation with only horizontal co-ordinates as the space variables. The theory built on the basic assumption is the shallow-water theory for stratified fluids. Transformations are explicitly given with which a class of solutions for steady three-dimensional flows of a fluid of arbitrary stratification, continuous or discontinuous, issuing from a large reservoir can be found from a corresponding solution for a homogeneous fluid, provided a free surface is present and the shallow-water theory is applicable. A few examples of exact solutions according to the shallow-water theory are given and the parallel flow in a horizontal canal issuing from a large reservoir with the same horizontal bottom, which has some bearing on previous works on stratified flows, is discussed. But it is emphasized that the class is a very special one and that there are other solutions not belonging to this class. The conditions under which a solution belonging to this class is valid are discussed.

Experiments are described in which an oscillating horizontal cylinder was moved through a density stratified fluid at Reynolds numbers of between 1 and 100 based on the diameter and mean velocity of the cylinder. The phase configurations of the internal waves which were generated are shown to compare very well with Lighthill's theory for waves in dispersive systems.

The results of experiments designed to investigate the shock-induced boundary layer on a semi-infinite flat plate are described.

Those for the laminar boundary layer are shown to be in agreement with a theory due to Lam & Crocco (1958) which describes two distinct domains, one near the shock where the flow is quasi-steady in a shock-fixed co-ordinate system and an unsteady region in which the flow characteristics approach the familiar steady state asymptotically. Experimental results are also presented for the non-laminar boundary layer. In particular the transition to turbulence in this unsteady boundary layer is discussed in some detail.

‘Establishment times’ for steady boundary layers are given for both laminar and turbulent flows, and their relevance to the testing times available in shock tubes is discussed. The measured heat transfer rates are compared with existing theories.

A lifting-surface theory is presented for a cascade in subsonic shear flow by applying Fourier integral methods to the expressions of the perturbed flow field. The pressure distribution on the blade surface is determined by means of the socalled singularity method. Some numerical examples are presented and discussed in comparison with the results according to the lifting-line theory.

A significant difference is found in the effect of compressibility between a shear flow and a uniform flow. In shear flows with the maximum Mach number close to one, no such great local lift force is found near the sonic station as would be predicted by the linearized subsonic uniform flow theory. The correlation between the local lift and the local effective angle of attack at high Mach number span-stations shows a great deviation from that according to the uniform flow theory.

The nature of a shear flow over a wavy boundary of small amplitude is investigated. It is found that if the viscosity is small, the nature of the flow is highly dependent on the wave amplitude. If the wave amplitude is truly infinitesimal, the flow is described by the Orr-Sommerfeld equation and in the neighbourhood of the critical layer viscous stresses are important even in the limit of vanishing viscosity. However, if the wave is sufficiently large, viscous stresses may be neglected even in the critical layer. An approximate solution of the inviscid equations of motion is obtained to describe the flow over a small but finite wave in the limit of infinite Reynolds number.

The motion generated in an initially quiescent, incompressible, stratified, and/or rotating fluid of infinite extent when a spherical source begins to breathe fluid in and out periodically is considered. The properties of the resulting flow may be understood in terms of the inertial-internal waves which may propagate energy in the fluid. At all points located a finite distance from the source, except those points falling on certain conical surfaces which are tangent to the source and which contain the group velocity vector for waves at the source frequency, the flow is ultimately a steady oscillation at the source frequency. The manner in which the flow depends on source frequency is discussed in detail.

A comprehensive set of exact solutions to the first-order boundary-layer equations has been computed using the finite difference computer programme of Sells, with and without wall cooling. The effects of Prandtl number, wall cooling and Mach number on separation point location were studied. Values of displacement thickness, skin friction coefficient and Stanton number are displayed graphically for the supersonic flow over a circular concave arc, for a subsonic cooled cylinder and for the case of a linearly retarded velocity distribution. The influence of pressure gradient on recovery factor was studied. Velocity and temperature profiles are shown for four cold wall cases. The exact computer results show the errors in many of the more approximate methods available for the case where Ue = U∞(1 - X/L). The importance of second-order effects and the applicability of a first-order solution are discussed briefly.

Measurements of the fluctuating turbulent stresses (uiuj)’ in the mixing layer of a two-dimensional jet are presented. These include the intensity, spatial correlation and the wave-number frequency spectra of the fluctuating stresses.

The acoustic spectrum perpendicular to the axis of a jet can be approximated with the aid of assumptions about jet similarity and a knowledge of the four-dimensional wave-number frequency spectrum at zero wave-number. In the present paper the four-dimensional spectrum was inferred from measured two-dimensional spectra and the self-noise spectrum perpendicular to a round jet estimated. Good agreement was obtained with measured acoustic spectra.

The linearized problem of the instability of a layer of liquid flowing down an inclined plane was formulated by Yih (1954) and was solved by Benjamin (1957). It was found that the instability of such a film flow is initially due to long surface waves of infinitesimally small amplitudes. In the present study, a closed-form expression for the non-linear development of these long surface waves is obtained. It is shown that in the neighbourhood of the neutral curve an exponentially growing infinitesimal disturbance may develop into supercritically stable wave motion of small but finite amplitude if the surface tension of the liquid is sufficiently large. Theoretically obtained amplitudes of such waves are consistent with Kapitza's (1949) observation. The approach used in this analysis is a modification of the method used by Reynolds & Potter (1967), who extended the method of Stuart (1960) and Watson (1960) in their study of the non-linear instability of plane Poiseuille and Poiseuille-Couette flow.

A generalized linear theory is developed for a compressible shear flow by extending the Fourier integral representations for the pressure field disturbed by pressure dipoles, and it is applied to a transonic shear flow past a thin aerofoil.

A method of determining the pressure distribution on the aerofoil surface is shown. Some numerical examples are presented and discussed.

It is found that the three-dimensional effect due to non-uniformity of the Mach number appears in the greatest degree at lower supersonic span-stations. Even within the scope of the linearized theory not such a great lift force arises near the sonic station as would be expected from the linearized uniform flow theory.

The two-dimensional stratified flow over an obstacle placed in a channel of finite height is examined to determine the extent to which Long's model provides an adequate description of real flows. A simple numerical method of solving Long's model for obstacles of arbitrary shape is used to calculate predicted streamline patterns which are compared with experimental observations of the flow over two bluff obstacles. If only a few lee-wave modes are excited there is qualitative agreement between theory and experiment, but, if the flow is subcritical with respect to several lee-wave modes, the effects of turbulence become dominant and the inviscid model is no longer useful. The theory predicts that the drag on an obstacle can increase with decreasing speed owing to the momentum transfer to lee-wave motion. Direct measurement of drag indicates that there are conditions under which the drag does increase with decreasing speed, but under these conditions the wake is dominated by turbulence and no lee waves can be detected.

A computational method is presented for the analysis of two-dimensional flow through a non-uniform gauze. The method, based upon the linearized theory due to Elder (1959), permits solutions for most practical cases to be obtained using relatively simple numerical techniques. Comparison with experimental data shows that the computed solutions are satisfactory provided the restrictions inherent in the linearized theory are observed.

The gain and loss integrals in the Boltzmann equation for a rigid sphere gas are evaluated in closed form for a distribution which can be expressed as a linear combination of Maxwellians. Application to the Mott-Smith bimodal distribution shows that the gain is also bimodal, but the two modes in the gain are less pronounced than in the distribution. Implications of these results for simple collision models in non-equilibrium flow are discussed.

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.