This paper considers the case of a one-dimensional piston moving outwards with a speed proportional to rα and driving a strong shock into a non-uniform ambient gas whose density is initially proportional to r−k, k > 0. This problem is connected with that studied by Grundy & McLaughlin (1977), who effectively discussed the case α = 0. We discover further important uses of the Sedov similarity solutions and find kc, the upper limit to k for the shock path to be asymptotically similar to the piston path.

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.

In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.

This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

An experimental investigation was conducted at selected locations in the wall region of a three-dimensional turbulent boundary layer relaxing in a nominally zero external pressure gradient behind a transverse hump (in the form of a 30° swept, 5 ft chord, wing-type model) faired into the side wall of a low-speed wind tunnel. The boundary layer (approximately 3·5 in. thick near the first survey station, where the length Reynolds number was 5·5 × 106) had a maximum cross-flow velocity ratio of 0·145 and a maximum cross-flow angle of 21·9° close to the wall. The hot-wire data indicated that the apparent dimensionless velocity profiles in the viscous sublayer are universal and that the wall influence on the hot wire is negligible beyond y+ = 5. The existence of wall similarity in the relaxing flow field was confirmed in the form of a log law based on the resultant mean velocity and resultant friction velocity (obtained from the measured skin friction).

The smallest collateral region extended from the point nearest to the wall (y+ ≈ 1) up to y+ = 9·7, corresponding to a resultant mean velocity ratio (local to free-stream) of 0·187. The unusual feature of these profiles was the presence of a narrow region of slightly decreasing cross-flow angle (1° or less) that extended from the point of maximum cross-flow angle down to the outer limit of the collateral region. A sublayer analysis of the flow field using the measured local transverse pressure gradient slightly overestimated the decrease in cross-flow angle. It is concluded that, in the absence of these gradients, the skewing of the flow could have been much more pronounced practically down to the wall (limited only by the resolution of the sensor), implying a near-wall non-collateral flow field consistent with the equations of motion in the neighbourhood of the wall.

The streamwise relaxation of the mean flow field based on the decay of the cross-flow angle was much faster in the inner layer than in the outer layer. Over the stream-wise distance covered, the mean flow in the inner layer and the wall shear-stress vector relaxed to a two-dimensional state in approximately 10 boundary-layer thicknesses whereas the relaxation of the turbulence was slower and was not complete over the same distance.

Fully developed air-flows through an equilateral triangular duct of 12·7 cm sides were investigated over a Reynolds number range of 53 000 to 107 000. Based on equivalent hydraulic diameter, friction factors were found to be about 6% lower than for pipe flow. Mean axial velocity distributions near the wall were describable by the inner law of the wall (when based on local wall shear stress) but the constants differ slightly from those for pipe flow. As expected, the secondary flow pattern was found to consist of six counter-rotating cells bounded by the corner bisectors. Maximum secondary velocities of about 1 ½% of the bulk velocity were observed. The effects of secondary currents were evident in the cross-sectional distributions of mean axial velocity, wall shear stress and Reynolds stresses, and very prominent in the turbulent kinetic energy distribution. For the flow prediction, the vorticity production terms were expressed by modelling the Reynolds stresses in the plane of the cross-section in terms of gradients in the mean axial velocity and a geometrically calculated turbulence length scale. The experimental and predicted characteristics of the flow are shown to be in good agreement.

The present investigation has been performed over a wide range of the dimensionless parameters characterizing the process of propagation of pressure perturbations in a gas-liquid mixture; these are the Reynolds number, and a dispersion parameter responsible for the relation between the values of dispersion and signal intensity. The values of the above parameters were changed mainly by varying the initial perturbation. The results obtained show a complete agreement between the Burgers-Korteweg-de Vries model and the real process of propagation of long-wave perturbations in a liquid with gas bubbles. In addition to signal propagation with the formation of monotonic and oscillatory shock waves, the propagation of signals in the form of solitary waves (solitons) and wave packets was observed experimentally. Data have been obtained on signal damping, energy dissipation and the influence of mixture viscosity on the signal evolution.

A method of multiple-scale expansion is applied to the theory of incompressible isotropic turbulence in order to close the infinite system of cumulant equations. The dynamical equation for the energy spectrum derived from this method is found to give positive-definite solutions at all Reynolds numbers. At large Reynolds numbers the spectrum takes the form of Kolmogorov's $-\frac{5}{3}$ power spectrum in the inertial subrange, whose extent increases indefinitely with Reynolds number. The spectrum in the energy-containing range satisfies an inviscid similarity law, so that the rate of energy decay or of viscous dissipation is also independent of the viscosity. In the higher wavenumber region beyond the inertial subrange the spectrum takes a universal form which is independent of its structure at lower wavenumbers. The universal spectrum is composed of three different subspectra, which are, in order of increasing wavenumber, the $k^{-\frac{5}{3}}$ spectrum, the k−1 spectrum and the exp [−σk1·5] spectrum, σ being a constant. Various statistical quantities such as the energy, the skewness of the velocity derivative, the microscale and the microscale Reynolds number are calculated from the numerical data for the energy spectrum. Theoretical results are discussed in detail in comparison with experimental results.

The unsteady hydrodynamic interaction of two bodies moving in a shallow fluid is examined by applying slender-body theory. The bodies are assumed to be in each other's far field and the free surface is assumed to be rigid. By matched asymptotics, the inner and outer problems are formulated and a pair of coupled integro-differential equations for determining the unknown cross-flows is derived. The degree of coupling is shown to be related to a bottom-clearance parameter. Expressions are given for the unsteady sinkage force, trimming moment, sway force, and yaw moment. Numerical calculations for two weakly coupled cases are presented. One corresponds to the interaction of a stationary body with a passing one, the other to the interaction of two bodies moving in a steady configuration. Theoretical results are compared with existing experimental data.

Split-film electrochemical transducer theory is briefly outlined so as to show its ability to separate statistical properties of each component of the surface velocity gradient in regions as different as a boundary layer, a separated zone and a three-dimensional separation line. A lot of measurements of the flow field near a cylinder which are not yet available elsewhere are carried out everywhere on the cross-section of a yawed cylinder. A direct verification of the independence principle and Wild's thermal analogy in separated zones is executed. The periodic and random parts of each component of the surface velocity gradient are separated. Cross-spectral analysis between any pair of points shows that the periodic surface flow is coherent upon any cross-section and in a large region in the spanwise direction. The stationary wave system is investigated in space and time. The reconstitution, instant by instant, of the averaged integral wall streamlines leads to an understanding of the synchronization of the natural oscillations. The four main stagnation or separation lines quickly move from one extreme position to the other, and at certain times contrary vortices stretching in the direction of the generators can appear. In the rearward stagnation region random small-scale fluctuations are probably turbulent, but in the intermediate region high-level random fluctuations in the spanwise direction are much more coherent and probably contribute to keeping the flow in phase at large distances.

The wake configuration of a sphere has been determined by means of the surface oil-flow method, the smoke method and the tuft-grid method in a wind tunnel at Reynolds numbers ranging from 104 to 106. It was found that the wake performs a progressive wave motion at Reynolds numbers between 104 and 3·8 × 105, and that it forms a pair of stream wise line vortices at Reynolds numbers between 3·8 × 105 and 106.

The phenomenon of steady cellular convection in a rotating fluid in which the stratification is statically unstable is well known. If the temperature field also varies in the horizontal direction a thermal wind is generated which can diminish the amplitude of the cells, or if inertial effects are strong enough, confine them to a narrow vertical band. On the other hand, in an enclosed container resonance can occur which increases the amplitude of the cells beyond the scope of a linear theory.