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 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.

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.

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.

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.

A generalization of the isotropic theory of Batchelor & Proudman (1954) is developed to estimate the effect of sudden but arbitrary three-dimensional distortion on homogeneous, initially axisymmetric turbulence. The energy changes due to distortion are expressed in terms of the Fourier coefficients of an expansion in zonal harmonics of the two independent scalar functions that describe the axisymmetric spectral tensor. However, for two special but non-trivial forms of this tensor, which represent possibly the simplest kinds of non-isotropic turbulence and specify the angular distribution but not the wavenumber dependence, the energy ratios have been determined in closed form. The deviation of the ratio from its isotropic value is the product of a factor containing R, the initial value of the ratio of the longitudinal to the transverse energy component, and another factor depending only on the geometry of the distortion. It is found that, in axisymmetric and large two-dimensional contractions, the isotropic theory gives nearly the correct longitudinal energy, but (when R > 1) over-estimates the increase in the transverse energy; the product of the two intensities varies little unless the distortion is very large, thus accounting for the stress-freezing observed in rapidly accelerated shear flows.

Comparisons with available experimental data for the spectra and for the energy ratios show reasonable agreement. The different ansatzes predict results in broad qualitative agreement with a simple strategem suggested by Reynolds & Tucker (1975), but the quantitative differences are not always negligible.

Experiments have shown that the two-dimensional flow near a forward stagnation line may be unstable to three-dimensional disturbances. The growing disturbance takes the form of secondary vortices, i.e. vortices more or less parallel to the original streamlines. The instability is usually confined to the boundary layer and the spacing of the secondary vortices is of the order of the boundary-layer thickness. This situation is analysed theoretically for the case of infinitesimal disturbances of the type first studied by Görtler and Hämmerlin. These are disturbances periodic in the direction perpendicular to the plane of the flow, in the limit of infinite Reynolds number. It is shown that the flow is always stable to these disturbances.

The mechanisms involved in the spreading of one liquid on the surface of another are considered for the case of a positive spreading coefficient. Taking into account the effects of gravity, capillary pressure and the spreading coefficient, the general equations determining the spreading are derived. These are then solved for the situation where the spreading liquid is introduced onto the surface of a uniformly flowing substratum at a position upstream of a barrier on the substratum surface. In this manner a steady-state situation is achieved for which the variation of spreading-liquid thickness with position is obtained as a function of the deposited volume of spreading liquid and the velocity of the substratum.

The rate at which a particle in suspension can capture diffusing solute molecules can be increased by stirring the fluid. It is shown that the increase is closely related to the total power expended, per unit volume of fluid, by the stirring device. The increase in diffusion current to a particle is related to the local rate of deformation of the fluid Ω (which, in turn, determines the dissipation) by a function F(Ωa2/D), where a is the particle radius and D the diffusion constant for the molecule in question. The function F has been determined experimentally by investigating the corresponding problem in heat transfer. In a fluid of viscosity η, stirring which is vigorous enough to double the mean diffusion current to a particle must entail a power dissipation, per unit volume, not less than 500ηD2/a4. For particles a few microns in size, or smaller, in water, effective stirring is not feasible. The results can be used also to predict the effect of stirring on the coagulation of similar particles. To double, by stirring, the rate at which particles of radius b form dimers requires a stirring power proportional to b−6, and is not feasible in water if b is less than 10−5 cm.

Structure functions of turbulent temperature and velocity fluctuations are measured both for the atmosphere, in the surface layer over land, and for the laboratory, in the inner region of a thermal boundary layer and on the axis of a heated jet. Even-order temperature structure functions, up to order eight, generally compare favourably with the analysis of Antonia & Van Atta over the inertial subrange. The Reynolds number dependence of these structure functions, as predicted by the analysis, is in qualitative agreement with the measured data. Odd-order temperature structure functions depart significantly from the isotropic value of zero, particularly at large time delays. This departure is reasonably well predicted, over the inertial subrange, by postulating a simple ramp model for the temperature fluctuations. Assumptions involved in this model are directly tested by measurements in the heated jet. The ramp structure does not seriously affect either the even-order temperature structure functions or the mixed velocity-temperature functions, which include even-order moments of the temperature difference.

Turbulent transport of fluctuating turbulent energy, turbulent momentum flux, temperature variance, turbulent heat flux, etc. in the upper part of the atmospheric boundary layer is usually dominated by buoyant transport. This transport is responsible for the erosion of the overlying stably stratified region, resulting in progressive thickening of the mixed layer. It is easy to show that a classical gradient transport model for the transport will not work, because it transports energy in the wrong direction. On the other hand, application of the eddy-damped quasi-Gaussian approximation to the equations for the third moments results in a transport model which predicts realistic inversion rise rates and heat-flux profiles. This is also a gradient transport model, but like molecular transport in solutions, a flux of one quantity depends on gradients of all relevant quantities. Transport coefficients are modified by the heat flux, so that the vertical transport is severely reduced near the inversion base. A simple Lagrangian model of transport of an indelible scalar in a stratified flow indicates that the form of the modified transport coefficients results from a marked anisotropic change in the Lagrangian time scale in stratification.