Integral-transform theory is used to solve the problem of sinusoidal initial perturbations in a planar stagnation counterflow, with extension to the axisymmetric case. The two incompressible fluids are assumed to be inviscid, so that across the initially plane interface the velocity shear is a velocity slip which grows linearly with the distance from the stagnation point. The solution for the interface displacement shows a competition between an amplification produced by the shear instability and a decay produced by the stretching effect of the accelerating unperturbed flow. At early times the interface perturbation grows exponentially with time, but eventually the stretching process reduces the interface displacement to a magnitude comparable to the initial perturbation.

The origin of meandering and braiding of alluvial rivers is re-analysed in terms of stability theory. The flow is described by a two-dimensional model, and the transportation of sediment is separated into bed-load transport and transport of suspended sediment, by use of the improved knowledge of sediment transport mechanisms achieved in recent years. The paper explains why it is important to distinguish between the sediment transported as bed load and that in suspension.

The analysis is able to predict whether a river remains stable or tends to meander or braid.

The results of the stability analysis are compared with laboratory experiments and data from natural rivers, and the agreement is satisfactory.

This study considers steady-state, finite amplitude thermal convection in a non-Newtonian fluid. Pseudoplastic (power-law) fluids are considered for a power-law exponent n in the range 1 ≤ n ≤ 9. Finite-difference solutions are obtained for two-dimensional periodic convective modes in a horizontally infinite fluid layer heated from below. The results show that the patterns of convective motions differ only slightly from those in a fluid of constant viscosity for n [lsim ] 3 while for n [gsim ] 3 significant differences are observed. An average viscosity is introduced which provides a good correlation of heat transfer across the layer with the Rayleigh number for the complete range of n considered.

Grid turbulence convected by a free stream past a rigid surface moving at the same speed as the free stream is analysed by boundary-layer theory and spectral methods. The turbulence is assumed to be weak, i.e. $u^{\prime}_{\infty}/\overline{u}_{\infty}\ll 1$ and its Reynolds number to be large, i.e. $u^{\prime}_{\infty}/\overline{u}_{\infty}\gg 1$ where u′∞ is the r.m.s. turbulent velocity. Two regions are found to exist. The outer, source region B(s) has a thickness of the order of the integral scale L∞. Here the normal component of turbulence decreases and the lateral and streamwise components are amplified. The inner, viscous region B(v) has thickness $[x\nu/\overline{u}_{\infty}]^{\frac{1}{2}} $, where x, v and $\overline{u}_{\infty} $ are the streamwise co-ordinate, kinematic viscosity and mean velocity respectively. Here the turbulent velocity decays to zero at the surface. Spectra variances and cross-correlations are calculated and found to compare well with measurements of turbulence near moving walls by Uzkan & Reynolds (1967) and Thomas & Hancock (1977).

The results of this theory are shown to have a number of applications including the prediction of turbulence near wind-tunnel walls and near flat plates placed parallel to the flow.

A set of constitutive equations is derived to describe the time-dependent flow of a dilute suspension of identical rigid particles of arbitrary shape which are influenced by Brownian couples. The form of the orientation probability distribution for small departures from isotropy is found. The case of weak flows with strong Brownian effects is studied in detail, and the viscoelastic approximation and second-order-fluid limit of the constitutive equation are derived for a general particle shape. A full numerical solution is given for ellipsoids. The general nearly spherical particle is also considered and constitutive equations for general flow strengths are obtained for this case.

Flow visualization and laser-Doppler anemometry have been used to provide a detailed description of the velocity characteristics of the asymmetric flows which form in symmetric, two-dimensional, plane, sudden-expansion geometries. The flow and geometry boundary conditions which give rise to asymmetric flow are indicated, and the reason for the phenomenon is shown to lie in disturbances generated at the edge of the expansion and amplified in the shear layers. The spectral distributions of the fluctuations in velocity are quantitatively related to the dimensions of the two unequal regions of flow recirculation. It is also shown that the intensity of fluctuating energy in these low Reynolds number flows can be larger than that in corresponding turbulent flows.

The behaviour of the interface between two immiscible fluids of different density and viscosity in a centrifuge is investigated both analytically and experimentally. The study includes the interfacial shape and behaviour during both steady and transient operation of the centrifuge. The interface shape is investigated for rigid-body rotation and during slow steady acceleration or deceleration of the centrifuge. The dependence of the interface shape on time is investigated during a rapid spin-up to a slightly different value of the angular velocity, which then remains constant. Some aspects of interface stability are also reported.

An explanation is proposed for the dependence on Reynolds number and other parameters of the number of waves which appear on vortex rings formed by pushing fluid out of a tube. It is shown that the number of waves can be sensitive to the vorticity distribution in the core of the ring. The process of ring formation is discussed and it is concluded that peaked vorticity distributions, limited by viscosity, will occur. Quantitative estimates of the number of waves are made. Agreement with observation is satisfactory.

A class of nonlinear hydrodynamic problems is studied. Physical problems such as shear flow, flow with a sharp interface separating two fluids of different density and flow in a porous medium all belong to this class. Owing to the density difference across the interface, vorticity is generated along it by the interaction between the gravitational pressure gradient and the density gradient, and the motion consists of essentially two processes: the creation of a vortex sheet and the subsequent mutual induction of different portions of this sheet.

Two numerical methods are investigated. One is based upon the well-known Green's function method, which is a Lagrangian method using the Biot-Savart law, while the other is the vortex-in-cell (VIC) method, which is a Lagrangian-Eulerian method. Both methods treat the interface as sharp and represent it by a distribution of point vortices. The VIC method applies the FFT (fast Fourier transform) to solve the stream-function/vorticity equation on an Eulerian grid, and computational efficiency is further improved by using the reality properties of the physical variables.

Four specific problems are investigated numerically in this paper. They are: the Rayleigh-Taylor instability, the Saffman-Taylor instability, transport of aircraft trailing vortices in a wind shear, and the gravity current. All four problems are solved using the VIC method and the results agree well with results obtained by previous investigators. The first two problems, the Rayleigh-Taylor instability and the Saffman-Taylor instability, are also solved by the Green's function method. Comparisons of results obtained by the two methods show good agreement, but, owing to its computational economy, the VIC method is concluded to be the better method for treating the class of hydrodynamic problems considered here.

Brenner's general results for the force and torque experienced by an arbitrary particle simultaneously translating and rotating in a linear shear flow are applied to the spherical cap. The cap's five fundamental resistance tensors are determined and the corresponding tensors for the sphere and circular disk are recovered as special cases. The problem of a freely moving cap in a linear shear is considered and the resulting translational and rotational motions are analysed. The cap's centre of free rotation is found and trajectories of this point are plotted in a few instances. The cap is also found to possess a ‘point of planar motion’ which always moves in a plane perpendicular to the vorticity vector of the undisturbed shear regardless of the initial orientation of the cap. It is shown that the motion of the cap actually serves as a model for the motions of all ‘oblate’ asymmetric bodies of revolution which are moving freely in a linear shear.

Hot-wire anemometer measurements in a plane jet issuing at a harmonically oscillating angle into a moving air stream have been made to aid the understanding of oscillatory jet flows in general and flow past aerofoils with oscillating jet flaps in particular. The rates of velocity decay and jet spreading are shown to be greater and less, respectively, than those for a steady jet parallel to the air stream. The shapes of instantaneous velocity profiles and limited measurements of turbulence intensity are similar to those for a steady jet in a parallel air stream if a correction is made for the small velocity difference across the curved jet. The motion of the jet centre-line indicates that flow path lines are similar to those for a steady jet flap over a significant range of frequency of jet oscillation. Finally, a quasi-steady jet flap theory is proposed for an analytical description of the major flow features.

This paper describes part of a detailed study of the initial region of three annular jets. The configurations are the basic one, without any bullet in the centre, and those with a conical and an elliptical bullet. From the mean velocity and turbulence intensity measurements the initial region can be divided into the initial merging, the intermediate and the fully merged zones. Within these three zones similarity of both the mean velocity and the turbulence intensity profiles has been found. The similarity curves are compared with those for a single jet.

Artificial acceleration and deceleration of the transition process in a two-dimensional wake were attempted. The wake was produced behind a thin aerofoil placed parallel to uniform flow. The sound from a loudspeaker introduced into the wake acted as an artificial disturbance. Various kinds of sound were tested and the effect on the transition was judged by the energy spectrum. Sinusoidal sound of the frequency of the maximum growth rate in the linear region decelerates the transition, whereas sound of a different frequency accelerates it. Sound of two or four frequencies is more effective in accelerating the transition when the frequencies are properly chosen. White noise from the loudspeaker is not effective, but a two-peak noise specially designed for strong nonlinear interaction is the most effective in accelerating the transition process. These results can be explained by two empirical properties of the nonlinear interaction: the growth suppression induced by a large amplitude fluctuation and the stronger interaction between fluctuations of closer amplitudes.

A semi-analytical numerical study was performed to simulate the development of a vortex ring in a stratified and/or shearing environment. Practical applications of results of this study can be found in turbulence modelling and in studies of plumes and wakes. The objective is to follow exactly the evolution of a vortex ring so that the three-dimensional vortex-stretching mechanisms due to stratification and the shear effects, respectively, can be understood.

The basic formulation consists of the solution of the vorticity equation in a stratified medium. The approach adopted is unique in that discrete vortex elements are used and arbitrary nonlinear interactions are allowed (therefore three-dimensional effects) among various vorticity generators. One of the two fundamental assumptions in this approach is that the vorticity is allowed to be generated only along the density discontinuity. The second assumption is that, while the vorticity carried by the vortex ring is modelled by vortex elements tangential to the vortex loop (which was a vortex ring initially), the vorticity generated by stratification effects is modelled by long vortex lines parallel to the axis of the vortex ring. This limits the validity of the present calculation to high Froude number flow.

Numerical stability is guaranteed by the finite core radius for each discrete vortex element and uniform spacing between them; the former is determined by consideration of the momentum integral over the vortex-ring plane. The latter is determined by a cubic spline interpolation method which conserves the circulation and centroids of the vorticity. The velocity of each vortex element is determined by the discretized Biot-Savart law, and motion of the vortex loop is calculated by a predicator-corrector time integration method.

Calculations were carried out for both momentum-carrying and momentumless vortex rings. A particular two-dimensional case gives good agreement with Kármán's theory. The evolution of the vortex loop reveals a process in which only the vorticity normal to the stratification is conserved; the remaining vorticity is dissipated through a simulated viscous dissipation. Evolution of a vortex loop on a shear layer reveals a vortex-loop rotation rate equal to the velocity shear, and a twisting motion due to the Magnus force which can lead to the turbulence energy cascade phenomenon. Numerical results demonstrate effects of each individual vorticity source and observed phenomena can be explained.

At shock Mach numbers Ms ∼ 16 in pure argon with initial pressures p0 ∼ 5 torr and final electron number densities ne ∼ 1017 cm−3, the translational shock front in a 10 x 18 cm hypervelocity shock tube develops sinusoidal instabilities which affect the entire shock structure including the ionization relaxation region, the electron-cascade front and the final quasi-equilibrium state. By adding a small amount of hydrogen (∼ 0·5% of the initial pressure), the entire flow is stabilized. However, the relaxation length for ionization is drastically reduced to about one-third of its pure-gas value. Using the familiar two-step collisional model coupled with radiation-energy loss and the appropriate chemical reactions, it was possible to deduce from dual-wavelength interferometric measurements a precise value for the argon-argon collisional excitation cross-section SAr Ar* = 1·0 x 10−19 cm2/eV with or without the presence of a hydrogen impurity. The reason for the success of hydrogen, and not other gases, in bringing about stabilized shock waves is not clear. It was also found that the electron-cascade front approached the translational-shock front near the shock-tube wall. This effect appears to be independent of the wall material and is not affected by the evolution of adsorbed water vapour from the walls or by water vapour added deliberately to the test gas. The sinusoidal instabilities investigated here may offer some important clues to the abatement of instabilities that lead to detonation and explosions.

One of the main problems in welding is to produce consistent weld profiles. Simple heat-flow models of the weldpool, which are currently used to predict the shape of the solid-liquid boundary, do not take account of fluid motion which is observed in practice and the effect of such motion could be significant. Electromagnetic j × B forces due to the welding arc have been proposed as a major cause of the motion and we attempt here to develop existing flow models towards more practical welding situations. We consider the steady-state flow of an incompressible viscous conducting fluid in a hemispherical container due to various axisymmetric representations of the distributed current sources which can arise in arc welding. A solution is found for sufficiently small currents that inertial effects may be ignored and no singularities appear in the velocity field. We discover that varying the current distribution can lead to qualitatively different flow patterns, i.e. poloidal flows in opposite directions and breakup into two distinct counter-rotating loops.

Flows of incompressible, electrically conducting liquids along ducts with electrically insulating or weakly conducting walls situated in a strong magnetic field are analysed. Except over a short length along the duct where the magnetic field strength and/or the duct cross-sectional area vary, the duct is assumed to be straight and the field to be uniform and aligned at right angles to the duct. Magnitudes of the field strength B0 and the mean velocity V are taken to be such that the Hartmann number M [Gt ] 1, the interaction parameter N (= M2/Re) [Gt ] 1 (Re being the Reynolds number of the flow) and the magnetic Reynolds number Rm [Lt ] 1.

For an O(1) change in the product VB0 along the duct across the non-uniform region, it is shown that:

(i) In the non-uniform region the streamlines and current flow lines follow surfaces containing the field lines satisfying $\int B^{-1}ds = {\rm constant}$, the integration being carried out along the field line within the duct; these surfaces are equipotentials and isobarics. This leads to

(ii) a tube of stagnant, but not current-free fluid at the centre of the duct parallel to the field lines around which the flow divides to bypass it. To accommodate this flow,

(iii) the usual uniform field/straight duct flow is disturbed over very large distances upstream and downstream of this region, the maximum length O(duct radius × M½) occurring in a non-conducting duct;

(iv) a large pressure drop is introduced into the pressure distribution regardless of the direction of the flow, the effect being most severe in a non-conducting duct, where the drop is O(duct radius × (uniform field/straight duct pressure gradient) × M½);

(v) in the part of the duct with the lower value of VB0 a region of reverse flow occurs near the centre of the duct and the stagnant fluid.

This paper describes how the collocation technique previously developed by the authors for treating both unbounded (Gluckman, Pfeffer & Weinbaum 1971; Leichtberg, Weinbaum, Pfeffer & Gluckman 1976) and bounded (Leichtberg, Pfeffer & Weinbaum 1976) multiparticle axisymmetric Stokes flows can be extended to handle a wide variety of non-axisymmetric creeping-motion problems with planar symmetry where the boundaries conform to more than a single orthogonal co-ordinate system. The present paper examines in detail the strong hydrodynamic interaction between two or more closely spaced identical spheres in a plane. The various two-sphere configurations provide a convenient means of carefully testing the accuracy and convergence of the numerical solution technique for three dimensional flow with known exact spherical bipolar solutions.

The important difficulty encountered in applying the collocation technique to multi-particle non-axisymmetric flows is that the selection of boundary points is rather sensitive to the flow orientation. Despite this shortcoming one is able to obtain solutions for the quasi-steady particle velocities and drag for as many as 15 spheres in less than 30 s on an IBM 370/168 computer. The method not only gives accurate global results, but is able to predict the local fluid velocity and to resolve fine features of the flow such as the presence of separated regions of closed streamlines. Time-dependent numerical solutions are also presented for various three-sphere assemblages falling in a vertical plane. These solutions, in which the motion of each sphere is traced for several hundred diameters, are found to be in very good agreement with experimental measurements. The concluding section of the paper describes how the present collocation procedure can be extended to a number of important unsolved three-dimensional problems in Stokes flow with planar symmetry such as the arbitrary off-axis motion of a sphere in a circular cylinder or between parallel walls, or the motion of a neutrally buoyant particle at the entrance to a slit or pore.

In this paper we assess the importance as a noise source of the well-ordered large-scale structure of a jet. We propose two simple models of the structure: the first emphasizes those features in common with waves that initially grow on an unstable shear layer but eventually saturate and decay, while the second regards the abrupt pairing of eddies as the most significant event in the jet's development. Our models demonstrate the possibility that forcing at one frequency could increase the broad-band noise of a jet, though, for jets with supersonic eddy convection velocities, the sound propagating in the direction of the Mach angle retains the spectrum of the excitation field. These features are consistent with the available experimental data, and strongly support the view that the large-scale structure of jet turbulence provides the dominant contribution to jet noise.

Bulk-coupled instability at the interface between miscible fluids which have identical mechanical properties but disparate electrical conductivities and are stressed by an equilibrium normal electric field is studied experimentally and theoretically. Observations of fluid motions permit measurement of an instability wavenumber and growth rate.

A model with a layer of diffusive conductivity distribution coupled to uniform bounding half-spaces is developed. Numerical integration of linearized electromechanical equations leads to a set of growing eigenfrequencies with corresponding eigenmodes, pure real for small wavenumber and complex (propagating) for large wavenumber.

The fastest growing wavenumber and corresponding growth rate are characterized as a function of the conductivity ratio and a time-constant ratio, which reflects the importance of inertial and viscous effects. In the viscous-dominated limit, the description agrees with a corresponding surface-coupled theory. The model is evaluated for parameters corresponding to experimental observations.