We present here a brief description of a numerical technique suitable for solving axisymmetric (or two-dimensional) free-boundary problems of fluid mechanics. The technique is based on a finite-difference solution of the equations of motion on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The overall solution is achieved via a global iterative process, with the condition of balance between total normal stress and the capillary pressure at the free boundary being used to drive the boundary shape to its ultimate equilibrium position.

In this paper numerical results are presented for the buoyancy-driven rise of a deformable bubble through an unbounded quiescent fluid. Complete solutions, including the bubble shape, are obtained for Reynolds numbers in the range 1 ≤ R ≤ 200 and for Weber numbers up to 20. For Reynolds numbers R ≤ 20 the shape of the bubble changes from nearly spherical to oblate-ellipsoidal to spherical-cap depending on Weber number; at higher Reynolds numbers ‘disk-like’ and ‘saucer-like’ shapes appear at W = O(10). The present results show clearly that flow separation may occur at a smooth free surface at intermediate Reynolds numbers; this fact suggests a qualitative explanation of the often-observed irregular (zigzag or helical) paths of rising bubbles.

We consider the deformation of a bubble in a uniaxial extensional flow for Reynolds numbers in the range 0.1 [les ] R [les ] 100. The computations show that the bubble bursts at a relatively early stage of deformation for R [ges ] O(10), never reaching the highly elongated shapes observed and predicted at lower Reynolds numbers. We also compute the deformation of the bubble under the assumption of potential flow, and conclude that the potential-flow solution provides a good approximation to the real flow in this case for R [ges ] O(100).

Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. We present here a computational procedure, termed WINGA2, for solving the Euler equations for transonic flow around aircraft upon a 0–0 mesh generated by transfinite interpolation. An explicit time-marching finite-volume technique solves the flow equations and features a non-reflecting far-field boundary condition and an internal mechanism for temporal damping together with a model for artificial viscosity. The method's convergence to a steady state is studied, and results computed on the CYBER 205 vector processor are presented. The Euler equation model is found to predict the existence of a tip vortex created by flow separating from the downstream region of the tip of the ONERA M6 wing where the radius of curvature approaches zero.

A theoretical model is developed giving a moderately detailed description of the flow in a turbulent bore, the velocity profiles, the shear stresses, the energy dissipation, etc. An analysis of the flow conditions at the toe of the turbulent front indicates significant differences from the usual description based on the finite-amplitude shallow-water equations, and it is shown that the present model gives a closer description of the actual physical conditions. Finally, numerical results are presented that illustrate how the model works, and test its validity on an example with known properties.

The effect of a shallow isolated topography on a linearly stratified, three-dimensional, initially uniform flow in the x-direction is considered. The Green-function solution for the velocity disturbance due to this topography, which is equivalent to that due to a dipole at the origin, is shown to be without swirl, i.e. the velocity disturbance lies strictly in planes passing through the x-axis. Thus this disturbance can be described in terms of a stream function. The asymptotic forms of the wavelike portion of the stream function and the vertical displacement field are obtained. The latter is in agreement with the limited versions due to Crapper (1959). The Gaussian curvature of the zero-frequency dispersion surface is obtained analytically as a step in the stationary-phase calculation. The model is extended to determine the vertical displacement field for an arbitrary shallow topography far downstream. For topographies that are even functions of x and y it is shown that the details of the topography affect the displacement field only in the vicinity of the x-axis. Elsewhere, the amplitude of the displacement is proportional to the net volume of the topography.

The full elliptic Navier–Stokes equations have been solved for entrance flow into a curved pipe using the artificial compressibility technique developed by Chorin (1967). The problem is formulated for arbitrary values of the curvature ratio and the Dean number. Calculations are carried out for two curvature ratios, a/R = 1/7 and 1/20, and for Dean number ranging from 108.2 to 680.3, in a computational mesh extending from the inlet immediately adjacent to the reservoir to the fully developed downstream region.

Secondary flow separation near the inner wall is observed in the developing region of the curved pipe. The separation and the magnitude of the secondary flow are found to be greatly influenced by the curvature ratio. As observed in the experiments of Agrawal, Talbot & Gong (1978) we find: (i) two-step plateau-like axial-velocity profiles for high Dean number, due to the secondary flow separation, and (ii) doubly peaked axial-velocity profiles along the lines parallel to the plane of symmetry, due to the highly distorted secondary-flow vortex structure.

A geometrical condition sufficient for uniqueness in two-dimensional time-periodic linear water-wave problems is derived, and some examples are given. The technique can be seen as a generalization of work by John (1950), who established uniqueness for surface-piercing bodies that have the property that vertical lines down from the free surface do not intersect the body.

The paper is in two parts. Part 1 provides a review of current knowledge on the topic of uniqueness, and gives a simple form of John's proof. Part 2 describes the recent progress, in which the ideas of John's proof are extended so that uniqueness can be demonstrated for surface-piercing bodies that do not satisfy John's geometrical criterion. In fact the new technique proves uniqueness for a large class of problems involving floating bodies, submerged bodies and multiple-body systems. However, the present work still does not constitute a general proof of uniqueness for all configurations.

Spectral models for turbulent pressure fluctuations are developed by directly Fourier transforming the integral solution to the Poisson equation for a homogeneous constantmean-shear flow. The turbulence-turbulence interaction is seen to possess the well-known k−7/3 inertial subrange and to dominate the high-wavenumber region. The turbulence–mean-shear contribution is seen to be dominant in the energy-containing range and falls off as $k^{-\frac{11}{3}}$ in the inertial subrange. The subrange constants and the mean-square pressure fluctuation are evaluated using a spectral model for the velocity. A spectral analysis of the velocity contamination of a pressure probe is also presented. Results are compared with spectral measurements with a static-pressure probe in the mixing layer of an axisymmetric jet.

A theoretical study is made of the stability of pipe flow with superimposed rigid rotation to finite-amplitude disturbances at high Reynolds number. The non-axisymmetric mode that requires the least amount of rotation for linear instability is considered. An amplitude expansion is developed close to the corresponding neutral stability curve; the appropriate Landau constant is calculated. It is demonstrated that the flow exhibits nonlinear subcritical instability, the nonlinear effects being particularly strong owing to the large magnitude of the Landau constant. These findings support the view that a small amount of extraneous rotation could play a significant role in the transition to turbulence of pipe flow.

High-speed schlieren cinéphotography of the firing of a high-energy plasma-jet igniter reveal turbulent structures similar in appearance to laboratory models of thermals or turbulent puffs. Measurements of the growth rates of these features together with those of their impulse and thermal energy confirm this similarity. A simple model based on the entrainment assumption gives a good description of the motion of the element and also of the decay of the internal temperature.

Turbulent airflow over a Stokes water-wave train of small amplitude is studied numerically based on the two-equation closure model of Saffman & Wilcox (1974) together with appropriate boundary conditions on the wave surface. The model calculates, instead of assuming, the viscous sublayer flow, and it is found that the energy transfer between wind and waves depends significantly on the flow being hydraulically rough, transitional or smooth. Systematic computations have yielded a simple approximate formula for the fractional rate of growth per radian \[ \zeta = \delta_{\rm i}\frac{\rho}{\rho_{\rm w}}\left(\frac{U_{\lambda}}{c}-1 \right)^2, \] with δi = 0.04 for transitional or smooth flow and δi = 0.06 for rough flow, where ρ is density of air, ρw that of water, Uλ wind speed at one wavelength height and c the wave phase velocity. This formula is in good agreement with most existing data from field experiments and from wave-tank experiments. In the case of waves travelling against wind, the corresponding values are δi = −0.024 for transitional and smooth flow, and δi = −0.04 for rough flow.

When immersed in a non-uniform electrolyte solution, a rigid charged sphere migrates toward higher or lower concentration of the electrolyte depending on the relative ionic mobilities and the charge borne by the sphere. This motion has a twofold origin: first, a macroscopic electrolyte gradient produces an electric field which acts on the charged sphere (electrophoresis); secondly, the electrolyte gradient polarizes the cloud of counterions surrounding the charged sphere by making the cloud thinner on the high-concentration side (chemiphoresis). In this paper, we compute the terminal velocity of a non-conductive sphere through a slightly non-uniform solution of a symmetrically charged binary electrolyte. The analysis proceeds through an expansion in the small parameter λ (defined as the ratio of the counterion-cloud thickness to the particle radius). Results to O(λ) are presented. The only property of the sphere's surface that affects the velocity is its zeta potential ζ when the electrolyte gradient vanishes; no information concerning the dependence of ζ upon ionic strength is needed. While the chemiphoretic effect always directs the particle toward higher electrolyte concentration, the electrophoretic contribution can move the particle in either direction depending on the sign of βζ, where β is a normalized difference in mobilities between cation and anion of the elecytrolyte; thus particle movement could be directed toward either higher or lower electrolyte concentration depending on the physical properties of the system. With slight algebraic rearrangement, our results are also applicable to conventional electrophoresis (particle motion in an applied electric field) and show excellent agreement with the numerical calculations of O'Brien & White (1978).

The drag on a sphere moving horizontally through stably stratified salt water is measured in laboratory experiments. The increment ΔCD in drag coefficient due to the stratification is obtained as a function of a stratification parameter κ and, in principle, the usual Reynolds number R. In these experiments, where R ranges from 150 to 5000, ΔCD is insensitive to R. But, as a function of κ, ΔCD has both positive and negative values attributable respectively to lee-wave drag and to suppression of turbulence in the wake. An observed delay in flow separation also apparently results from the lee-wave drag.

The nonlinear electromagnetic stirring induced in a hemispheroidal container by the axisymmetric discharge of an electric current is investigated. The electric current is discharged into the fluid from a circular electrode which is at the centre of the equatorial plane of the container, the remaining part of the equatorial plane being a free surface. The equations of the problem are solved semi-analytically and results are presented for several sets of data. In the case of a point electrode when the current exceeds a critical value we have velocity breakdown. Here it is shown that, as the size of the area through which the current is discharged increases, the intensity of the flow field decreases, and thus for a larger electrode a larger amount of current can be discharged without velocity breakdown. When, however, the current is sufficiently large the solution becomes unstable, and this indicates velocity breakdown. Finally in an Appendix the solution for the case of a point discharge in a semi-infinite fluid is expressed in analytic (series) form.

Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected). Next, by applying the same technique, the momentum-flux tensor of the entire mixture is obtained. An equation expressing the fact that the average force on a massless bubble is zero leads to a third relation. Complemented with mass-conservation equations for liquid and gas, these equations appear to constitute a completely hyperbolic system, unlike the systems with complex characteristics found previously. The characteristic speeds are calculated and shown to be related to the propagation speeds of acoustic waves and concentration waves.

Longitudinal and transverse components of the velocity were simultaneously measured for various vertical locations at 20 off-centreline positions in turbulent spots artificially generated in a zero-pressure-gradient laminar boundary layer. Global ensemble-averaged velocity diagrams and vertical vorticity contours were computed in similarity coordinates ξ = (x − x0)/(U∞(t−t0)) and ζ = z/(U∞(t−t0)) for different heights η = y/δ* above the plate. This global averaging technique inadequately describes the spot, which is not a single large vortex structure. A discriminative averaging technique was developed to construct a ‘statistically most-probable’ spot with sufficient resolution to include some of the largest substructures detected in visual studies. Four eddies were identified in vertical slices of the central region of the spot, while several rows appeared in the plan view. These features of the velocity and vorticity contours of the statistically most-probable spot exhibit similarities with structures observed in flow visualizations.

Turbulent mixing and combustion are investigated in a gaseous shear layer formed between two streams: one containing a low concentration of hydrogen in nitrogen and the other containing a low concentration of fluorine in nitrogen. The resulting temperature field is measured simultaneously at eight points across the width of the layer using fast-response cold-wire thermometry. The results show the presence of large, hot structures separated by tongues of cool fluid that enter the layer from either side. The usual bell-shaped mean-temperature profiles therefore result from a duty cycle whereby a fixed probe sees alternating hot and cool fluid, which results in the local mean. The adiabatic flame temperature is not achieved in the mean, at any location across the layer. For fixed velocities, it is found that, in general, two different mean-temperature profiles result from a given pair of reactant compositions if the sides of the layer on which they are carried are exchanged (‘flipped’). This finding is a direct consequence of the asymmetric entrainment of fluid into the layer. Results are compared with the predictions of Konrad and discussed in the context of the Broadwell–Breidenthal model. By comparison with the liquid result of Breidenthal, the amount of product formed in the layer at high Reynolds number is found to be dependent upon the Schmidt number. Results for a helium–nitrogen layer are discussed briefly.

The longitudinal dispersion due to the shear effect of a current is examined theoretically in the idealized two-dimensional case. This study reveals the process whereby the dispersion reaches a stationary stage after the release of the dispersing substance as an instantaneous line source in steady and in oscillatory currents. In addition, the relation between the stationary dispersion coefficients in steady and oscillatory currents is given analytically. Analysis of the dispersion during the initial stage needs a clear definition of the vertical average of the variance. We can understand the problem of the negative dispersion coefficient, which is obtained by the usual vertical average, through introduction of a new vertical average.

The flow field induced by a jet in incompressible cross-flow is analysed and the results compared with those obtained in a reacting water-jet experiment. It is argued that the axial vortex pair in the flow arises from the jet momentum normal to the free stream, the momentum flux being equivalent to a normal force, i.e. to a lift.