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.

The vorticity transport equation is solved for radial incompressible flow between disks by a finite-difference method with discretization based on the method of Allen & Southwell. The solution permits detailed characterization of the flow for the Reynolds number range investigated, 1 ≤ Re ≤ 300. Above Re = 60 separation is observed with the bubble size increasing rapidly with Re. The streamwise and transverse pressure and velocity gradients are examined to interpret the observed phenomena.

The rheology of particulate dispersions which are strongly influenced by particular types of non-hydrodynamic forces is analysed within the framework of suspension mechanics. Interactions between particles in a homogeneous shear flow without inertia are governed by viscous, electrostatic, London-van der Waals and Brownian forces. The balance among these provides the fluid with a microstructure described quantitatively at dilute concentrations by a pair distribution function and qualitatively by a characteristic interaction length. The bulk rheology follows from the microstructural variables through suitable averaging.

In dilute electrostatically stabilized suspensions of small rigid spheres for which London-van der Waals attractions and hydrodynamic interactions can be ignored, the theory predicts a Newtonian low-shear limit. The analytic expression for the viscosity contains a ϕ2-coefficient which can be quite large and agrees well with experimental data. At higher flow strengths a scale analysis of the pair conservation equation indicates a shear-and strain-thinning rheology, representing a breakdown of the fluid microstructure. Without flow the interaction length attains a maximum determined by the balance between Brownian motion and electrostatic repulsion. A weak shear merely perturbs this balance but generates stresses proportional to the fifth power of the length. With increasing shear rate this length and consequently the shear viscosity are reduced until viscous interactions completely dominate. Asymptotic solutions for an intermediate regime in which Brownian motion and hydrodynamic interactions are both negligible reveal power-law extensional and shear viscosities with non-zero normal stresses.

The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. The primary motivation for this work was to check numerical solution schemes by comparing the numerical results with laboratory measurements obtained with a rotating laser-Doppler velocimeter. The laser-Doppler technique is capable of high accuracy with small space and time resolution, and disturbances of the flow are virtually negligible. A series of measurements was made of the zonal flow over a range of Ekman numbers (1·06 × 10−3 ≤ E ≤ 3·30 × 10−3) and Rossby numbers (0·10 [les ]|ε| [les ] 0·33) at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.

In the absence of molecular diffusion there exists a space-independent transformation which transforms the probability density of dynamically passive scalars undergoing chemical reaction and advection into the probability density of scalar fields undergoing advection alone. In two well-known limits the equation for the probability density of non-reacting scalars is linear and parabolic in physical space. In such cases it is shown that the equation for the probability density of reacting scalars is likewise linear and parabolic in physical space, although hyperbolic in concentration space. The general solution of such an equation is obtained and the particular case of a second-order, decaying, single-species reaction is displayed.

Local velocity measurements performed in a convecting layer of fluid show that the velocity field can be described by a dominant fundamental velocity mode mixed with an increasing proportion of second and third harmonics as ε, the reduced distance to the convective threshold Rc, is increased from 0 to ∼ 10. The spatial and thermal dependences of the amplitudes of these different modes are reported and compared with theoretical predictions.

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.

Experiments on the interaction between solitary shallow-water waves propagating in the same direction have been performed in a rectangular channel. Two methods were devised to compensate for the dissipation of the waves in order to compare results with Hirota's (1971) solution for the collision of solitons described by the Kortewegde Vries equation. Both qualitative and quantitative agreement with theory is obtained using the proposed corrections for wave damping.

The structure of the Stewartson E¼-layer (E being the Ekman number) in a compressible gas contained in a rapidly rotating cylinder is investigated for the case in which the end plates of the cylinder are thermally insulated. It was found by Matsuda & Hashimoto (1976) that the E¼-layer could not have a relevant structure in the ordinary configuration in which the E¼-layer meets the end plates through its Ekman extension of thickness E½. In this paper the E¼ × E¼ square region, in addition to the Ekman extension, is considered. The heat generation due to the radial fluid motions in the Ekman extension causes the temperature fields in the E¼ × E¼ square region through which heat is conducted to the side wall of the cylinder. Numerical calculations were made to obtain the temperature fields, which are shown in several figures.

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.

The axisymmetry of a body which is diffracting water waves may be exploited to give a line integral equation to be solved for the scattered wave field and forces on the body. Each term in a previously established surface integral equation is shown to be expressible as a Fourier series, which is then integrated once analytically. The resulting one-dimensional equation is shown to possess singularities, previously ignored by Black (1975). This equation, with series transformations and subtraction of singularities such that all series are quickly convergent and that it has to be solved only along a curve, reduces computational effort by some three orders of magnitude. Results obtained by this method give good agreement with previous analytical and experimental results, even if a rather coarse numerical approximation is used.

This paper is concerned with the behaviour of buoyant plumes driven by a source of bubbles. It is shown experimentally that, when a bubble plume rises through a stratified environment, fluid can be transported vertically for some distance and then some of this fluid can leave the plume and spread out horizontally at its own density level. A simple plume model which regards the plume as a single entity is discussed in order to make a first assessment of the effects of gas expansion and bubble slip velocity in this stratified case. However, the experiments reveal a more complicated plume structure in which the bubbles remain in the centre part of the plume, and only the outer part of the plume spreads out into the environment at certain levels. On the basis of these observations a double-plume model is proposed which regards the plume as being composed of two parts: an inner circular plume (which contains all the bubbles of gas) and an outer annular plume.

The Stewartson E½- and E¼-layers in a rapidly rotating compressible fluid are considered within the framework of linearized equations and the boundary-layer method. The fluid is contained in a cylinder made of a thermally insulated side wall and conducting top and bottom end plates. The end plates and the side wall rotate with slightly different angular velocities. The case of an incompressible fluid was discussed by Stewartson, who found that the flow is restricted to the side-wall boundary layers. In the case of a compressible fluid, however, the solutions are strongly dependent upon the thermal boundary conditions assumed on the side wall. In particular, if the wall is insulated the fluid in the inner core is dragged along too since it is coupled strongly to the flow in the side-wall Stewartson layers. The critical parameter governing the solutions is found to be $(\gamma -1) PrG_0 E^{-\frac{1}{3}}/4\gamma$, where γ is the ratio of specific heats, Pr the Prandtl number, G0 the square of the rotational Mach number and E the Ekman number.

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 methods described by Bluman & Cole (1974) are used to derive the infinitesimals of the general invariance group of the unsteady, two-dimensional, stream-function equation for the case where the kinematic viscosity v is equal to a constant and the case where v = 0. The infinitesimals in each case involve ten independent parameters, seven of which appear explicitly and three of which are contained implicitly in three arbitrary functions of time. The various finite groups and similarity transformations which may be derived from the infinitesimals are discussed through examples. Two of the arbitrary functions of time are non-trivial and represent invariance of the stream-function equation under a transformation to a co-ordinate system which moves in a non-uniform irrotational fashion. A general similarity form is derived for which the equations dx/dt = u(x, y, t) and dy/dt = v(x, y, t) for the particle paths may be reduced to an autonomous system. This form is general enough to suggest the hypothesis that, under certain restrictions, the entrainment processes of unsteady flows dominated by two-dimensional large-scale motions may be displayed diagrammatically on a phase-plane plot of particle trajectories.

A charged particle suspended in an electrolyte solution attracts ions of opposite charge and repels those of like charge. The surface charge and the resulting distributed charge in the fluid comprise an electrical double layer. When a shear flow deforms the diffuse part of the double layer from equilibrium, stresses are generated which make the effective viscosity of the suspension greater than it would be if there were no charges present. In this paper these stresses are calculated for a dilute dispersion of spheres which have small surface charges and which are surrounded by thin double layers. The viscosity is predicted to be Newtonian in extensional flow but shear-thinning with non-zero normal-stress differences in shear flow. For more complex flows a constitutive equation couples the bulk stress directly to the micro-structural deformation responsible for non-Newtonian effects.

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.

This paper contains a simple and entirely conventional application of lifting-line theory to a close-hauled, high aspect ratio sail, set upright on a flat water surface. In seeking an optimum spanwise loading distribution however, an unconventional criterion is used, which leads to an unexpected result.

By seeking to maximize the forward thrust for a given rolling moment, rather than maximizing the lift-drag ratio, an optimum loading distribution is obtained which departs dramatically from elliptic loading. It offers the prospect of a significant improvement in extreme yacht performance if the associated mechanical and control problems can ever be solved.

A transformation is described which relates the sound generated by low Mach number flow to the flow vorticity. For compact flow fields the apparent sound source is of quadrupole type and linear in the vorticity and therefore also linear in the flow velocity. This scheme is applied to the sound generated by the interaction of two identical thin vortex rings. Then a flow field with a number of compact vortices is discussed. It is found that each vortex can be replaced acoustically by a dipole related to the impulse of the vortex, plus the quadrupole just mentioned plus a spherically symmetric sound source related to the energy of the vortex. An application to low Mach number free-space turbulence shows that the generated sound is related to the vorticity correlation tensor.

This paper describes a detailed study of the structure of turbulence in boundary layers along mildly curved convex and concave surfaces. The surface curvature studied corresponds to δ/Rw = ± 0·01, δ being the boundary-layer thickness and Rw the radius of curvature of the wall, taken as positive for convex and negative for concave curvature. Measurements of turbulent energy balance, autocorrelations, auto- and cross-power spectra, amplitude probability distributions and conditional correlations are reported. It is observed that even mild curvature has very strong effects on the various aspects of the turbulent structure. For example, convex curvature suppresses the diffusion of turbulent energy away from the wall, reduces drastically the integral time scales and shifts the spectral distributions of turbulent energy and Reynolds shear stress towards high wavenumbers. Exactly opposite effects, though generally of a smaller magnitude, are produced by concave wall curvature. It is also found that curvature of either sign affects the v fluctuations more strongly than the u fluctuations and that curvature effects are more significant in the outer region of the boundary layer than in the region close to the wall. The data on the conditional correlations are used to study, in detail, the mechanism of turbulent transport in curved boundary layers.