Various theories seeking to relate the velocity statistics of Lagrangian particles to the statistics of the Eulerian flow in which they are embedded are examined. Mean particle drift, mean-square particle velocity and the frequency spectrum of velocity are examined for stationary, homogeneous and joint-normally distributed Eulerian fields. Predictions based on a third-order weak-interaction expansion, the successive approximation procedure of Phythian (1975), the quasi-normal approximation of Saffman (1969), the parametrized model of Saffman (1962), and a new procedure based on a statistical estimator of the kinematic equation are compared with simulations of particle motion in one-dimensional flow. Only the statistical estimator produces both acceptable mean-drift and frequency-spectrum predictions.

Liquid metal interfaces, stressed by a high-frequency, alternating magnetic field are commonly observed to undulate. Even a planar interface stressed from above by a uniform magnetic field takes on an appearance that is very different from what is observed if the same layer is heated from below with about the same thermal input as associated with the eddy currents. This behaviour affects internal mixing and the transport of heat and material from interfaces. In applications where the interface is used to form glass or other materials, the undulations can be disasterous. A goal of this paper is to identify the circumstances under which this motion can be avoided. A theoretical model is developed for fluid motions, coupled to a magnetic flux density (having magnitude B0 and angular frequency ω) through a force density that is time averaged over one period of the alternating field. This theory, which does not include thermal effects, predicts a threshold for onset of instability determined by the ratio of layer thickness to skin depth and by the parameter M = B02/μηω where μ = 4π × 10−7 and and η is the viscosity. The instability has an internal nature in that it is predicted even when the liquid is bounded by rigid insulating materials. Threshold measurements are reported that agree with the predictions over more than an order-of-magnitude variation in frequency, including low frequencies, for which the finite depth of the liquid layer is important. However, observed growth times are far shorter than predicted. It is concluded that the observed motions are in fact thermally driven, but take on an appearance dictated by the hydromagnetics. A previously developed lumped parameter model, which includes thermally driven motion, does predict growth times on the order of those observed. In the lumped parameter model the critical field strength grossly affects the nonlinear saturation velocity. The critical M sets an upper limit on the extent to which a liquid metal can be levitated, depressed or transported magnetically at a given frequency without incurring interfacial undulations and an augmentation of mass and heat transfer.

Analytical and computational techniques are developed to investigate the stability of converging shock Avaves in cylindrical and spherical geometry. The linearized Chester-Chisnell-Whitham (CCW) equations describing the evolution of an arbitrary perturbation about an imploding shock wave in an ideal fluid are solved exactly in the strong-shock limit for a density profile ρ(r) ∼ r−q. All modes are found to be relatively unstable (i.e. the ratio of perturbation amplitude to shock radius diverges as the latter goes to zero), provided that q is not too large. The nonlinear CCW equations are solved numerically for both moderate and strong shocks. The small-amplitude limit agrees with the analytical results, but some forms of perturbation which are stable at small amplitude become unstable in the nonlinear regime. The results are related to the problem of pellet compression in experiments on inertial confinement fusion.

The two- and three-dimensional stability properties of the family of coherent shear-layer vortices discovered by Stuart are investigated. The stability problem is formulated as a non-separable eigenvalue problem in two independent variables, and solved numerically using spectral methods. It is found that there are two main classes of instabilities. The first class is subharmonic, and corresponds to pairing or localized pairing of vortex tubes; the pairing instability is most unstable in the two-dimensional limit, in which the perturbation has no spanwise variations. The second class repeats in the streamwise direction with the same periodicity as the basic flow. This mode is most unstable for spanwise wavelengths approximately 2/3 of the space between vortex centres, and can lead to the generation of streamwise vorticity and coherent ridges of upwelling. Comparison is made between the calculated instabilities and the observed pairing, helical pairing, and streak transitions. The theoretical and experimental results are found to be in reasonable agreement.

The essence of fluid-mechanical mixing of diffusing and reacting fluids can be traced back to kinematics, connectedness of material volumes and transport processes occurring across deforming material surfaces. Descriptions based on kinematics of homoeomorphic deforming material surfaces (tracers) are restricted solely to continuous motions and conveniently analysed by transport equations in Lagrangian frames.

Connectedness of material volumes restricts the mixing topology and generates bicontinuous structures characterized by intermaterial-area and striation-thickness distributions. Upper bounds for area generation and material-line elongation are related to mean values of viscous dissipation and govern the average reaction rate in diffusion-controlled reactions. Two concepts are introduced: micromixing, related to local flows, rate of stretching and local viscous dissipation, and macromixing, associated with connectedness of isoconcentration surfaces, vorticity and average viscous dissipation.

Several small-scale flows can be used to typify the interplay between fluid mechanics, mass and energy transport, and chemical reactions: elliptically symmetrical stagnation flows, vortex decay, and swirling flow with uniform stretching. It is proposed that complex fluid motions might be interpreted in terms of integrated behaviour of populations of small-scale flows distributed in space and time to simulate mixing behaviour.

The objective of this work is to present the foundations of a continuum mixing description making reference to earlier approaches to demonstrate computational applicability and practical significance.

An inequality pertaining to the energetics of the boundary layer in turbulent pipe flow and turbulent thermal convection is generalized for the double-diffusive convection problem, where a semi-infinite layer of cold, fresh and light water overlies another hot, salty and dense layer. The smallest possible salt/heat-flux ratio equals the ratio of the square roots of the respective diffusivities. The bound is asymptotically realizable according to a variational principle. A bound on the relative fluxes is predicted when another solute is added (‘multiple diffusion’).

Small-amplitude time-dependent motions of a uniformly rotating, density-stratified, Boussinesq non-dissipative fluid in a rigid container are examined for the case of the rotation axis parallel to gravity. We consider a variety of container shapes, along with arbitrary values for the (constant) Brunt-Väisälä and rotation frequencies. We demonstrate a number of properties of the eigenvalues and eigenfunctions of square-integrable oscillatory motions. Some of these properties hold generally, while others are shown for specific classes of containers (such as with symmetry about the container axis). A full solution is presented for the response of fluid in a cylindrical container to an arbitrary initial disturbance. Features of this solution which are different from the cases of no stratification or no rotation are emphasized. For the situation when Brunt-Väisälä and rotation frequencies are equal, characteristics of the oscillation frequencies and modal structures are found for containers of quite general shape. This situation illustrates, in particular, effects which are possible when rotation and stratification act together and which have been overlooked in previous investigations that assume that the vertical length scale is much smaller than the horizontal scales.

Laboratory experiments in which a stabilizing buoyancy flux is imposed on zero-mean-shear turbulence generated by an oscillating grid are discussed. The buoyancy flux is imposed at the top of a water column either as a constant heat flux or by the continuous addition of fresh water at the top of a salt solution. Two types of experiments were carried out. In the first, the oscillating grid was positioned near the buoyancy input plane at the top of the water column to represent an input of turbulence kinetic energy near the surface. In this case a mixed layer was formed which extended from the surface down to a finite depth, and was bounded below by a stable thermocline. The mixed-layer depth remained constant in time but, contrary to earlier suggestions, was not found to be proportional to the Monin-Obukhov length. Instead the depth of the mixed layer was found to depend on the rate of decay of the turbulent kinetic energy with depth through the mixed layer. A second set of experiments was carried out with the grid positioned well below the surface to represent turbulence produced by bottom stirring. At low values of the buoyancy flux the fluid column remained well mixed, but once the buoyancy flux exceeded a critical value a stable stratification built up near the surface. Under certain conditions, a steady stratification was produced in which the diffusive flux balanced the turbulent flux in the mixed layer below. At larger values of the buoyancy flux, the diffusive flux is not large enough to remove the buoyancy from the surface and a ‘runaway’ stratification develops. The criterion for the formation of surface stratification is also found to depend on the rate of decay of the turbulent kinetic energy with depth, and the implications of this work for the formation of fronts produced by tidal stirring are discussed.

The rheological behaviour of a dilute suspension of spherical particles of condensed phase dispersed in its own slightly rarefied vapour gas is investigated on the basis of suspension theory (Batchelor 1970) and generalized slip-flow theory for a two-phase system of a gas and its condensed phase derived from the Boltzmann equation. The rarefaction of the gas and the phase-change process at the interfaces of the particles have the effect of reducing the Einstein coefficient of ϕ, volume fraction, in the expression for the effective viscosity in the suspension. In the case in which the pure rarefaction effect alone enters the problem, the coefficient is $\frac{5}{2}(1-2.702\,K)$, where K is the Knudsen number, a rarefaction parameter defined by K = l/L, l and L being respectively the mean free path of gas molecules and the radius of a spherical particle. When both the rarefaction and the phase-change process are taken into account, this becomes $\frac{5}{2} (1-3.533\,K)$. These modifications are not small, even at ordinary pressures, when the size of the particles is of the order of microns.

The fluid mechanics of the closure motion of a short collapsible tube segment, subject to a strong flow deceleration as in one of Henderson & Johnson's (1912) experiments, is investigated experimentally and theoretically. Physical similarity to the closure process of the mitral valve is obtained. In the study particular emphasis is placed upon the evolution of the longitudinal profiles of the collapsible tube during its closure motion. It is found that the flexible tube first closes near its upstream end and that this first phase is followed by a propagation process toward the downstream end. The characteristics of this typical sequence and of the longitudinal shape of the collapsible tube are related to hydrodynamic parameters. The results predicted by the theory agree consistently with those obtained from the experiments.

Submerged laminar jet impingement on a plane is studied using computation. Steady-state Navier-Stokes equations for the axisymmetric case are solved numerically. The extent of the infinite flow is approximated by applying the boundary conditions at a finite but sufficiently large distance. The tube-exit velocity profile is assumed to be either a fully developed parabolic profile or a flat profile. For the former case, two different nozzle heights from the target plane are considered. The presence of a toroid-shaped eddy at low values of Reynolds number, Re, leads to some interesting observations such as the manner in which the wall shear stress depends on Re. An increase in the height of the nozzle exit from the target plane decreases the wall shear stress, more so at lower values of Re. A change from the parabolic exit velocity profile to the flat profile leads to a decrease in wall shear stress due to decreased momentum flux. The study was motivated by experiments designed to measure the yield shear strength of the vascular endothelium wherein a small saline jet was used to erode the tissue by normal impingement.

A near-vertical array of thermistors has been towed through the water beneath several hundred well-defined wind rows formed on the surface of a fresh-water loch in winds of 7–9 m s−1. The loch was stably stratified in the mean, temperatures decreasing with depth, and the air warmer than the water. The temperatures at ten levels in the water and the positions of the wind rows were recorded. These measurements were supplemented by photographs of the wind rows and observations of bubbles, using sonar.

The data has been analysed to produce average temperature sections across the wind rows. Near the surface a core of warm water can be distinguished beneath the wind row in the average section (although not in general in the unaveraged records) and colder water between rows, consistent with there being a Langmuir circulation pattern. At depths greater than about 0·4 times the average distances between rows, it is, however, apparent that the temperature anomalies slope to the left of the wind direction as depth increases.

The anomalies (although generally significantly different from the mean) are much less than the standard deviation of the local fluctuations in loch temperature and also much smaller than the temperature changes at the ‘ramps’ described earlier (Thorpe & Hall 1980), here observed to extend to within 0·4 m of the surface. There are thus two co-existing, but possibly independent, thermal structures which can be recognized in the turbulent mixing layer.

We treat the onset of convective motions for the case in which the base-state density profile is evolving in time. The formulation is in terms of random forcing which we take to be thermodynamic in origin, following our earlier work (see Jhaveri & Homsy 1980). Experimental evidence is reviewed which clearly demonstrates the need for such a stochastic formulation. The randomly forced initial-value problem is solved numerically at high Rayleigh numbers in the mean-field approximation for both a step change and linear temporal increase in surface temperature. The numerical results give both an expected value for the onset time for which convection is measurable and the variance of that expected value. The results are in good agreement with available experiments.

The thermal cellular convection in rotating rectangular boxes has been investigated both theoretically and experimentally. In the theoretical analysis, a linear stability theory is used to calculate the stability behaviour and the configuration of the three-dimensional convection flow. The numerical results show that the rolls change their orientation for a Taylor number greater than a critical value. In the experimental investigation, the flow patterns were visualized by a special differential interferometer. The experimental results are presented in stability diagrams and interferogram series which demonstrate the influence of rotation as well as initial and boundary conditions on the convective flow. We found that the effects of the Coriolis force and those of centrifugal forces could be studied separately by the choice of different test fluids, e.g. nitrogen is good for the Coriolis-force effect while silicone oil is good for the centrifugal-force effect.

When compared with experimental results, our theoretical model is shown to be good for fluids of small Prandtl number such as nitrogen gas. We also compare our results with the well-known asymptotic behaviour of the critical Rayleigh number and wavenumber.

The instability of an initially flat vortex sheet to a sinusoidal perturbation of the vorticity is studied by means of high-order Taylor series in time t. All finite-amplitude corrections are retained at each order in t. Our analysis indicates that the sheet develops a curvature singularity at t = tc < ∞. The variation of tc with the amplitude a of the perturbation vorticity is in good agreement with the asymptotic results of Moore. When a is O(1), the Fourier coefficient of order n decays slightly faster than predicted by Moore. Extensions of the present prototype of Kelvin-Helmholtz instability to other layered flows, such as Rayleigh-Taylor instability, are indicated.

The two-dimensional steady flow of a fluid over a semicircular obstacle on the bottom of a stream is discussed. A linearized theory is presented, along with a numerical method for the solution of the fully nonlinear problem. The nonlinear free-surface profile is obtained after solution of an integrodifferential equation coupled with the dynamic free-surface condition. The wave resistance of the semicircle is calculated from knowledge of the solution at the free surface.

A numerical investigation of normal-mode perturbations of a finite-amplitude Stokes wave has revealed regions of instability lying near resonance curves given by the linear-dispersion relation. It is found that, for small amplitude, the dominant instability is two-dimensional (of Benjamin-Fier type) but, for larger amplitudes, the dominant instability becomes a three-dimensional perturbation. Results are compared with recent experimental observations of steep wave trains.

A study of the instabilities of finite-amplitude periodic gravity waves has extended recent deep-water results to finite depth. Results are presented for three depths, one greater and two less than the depth for stabilization of two-dimensional long-wave perturbations. The results include those instabilities found by perturbation methods as well as some new types of instability.

The influence of an external flow velocity on the instability of a circular jet has been investigated by means of the inviscid linearized stability theory. The instability properties of spatially growing axisymmetric and first-order azimuthal disturbances show that the external flow inhibits the instability of the circular jet, but increases the unstable frequency range. Similarity considerations lead to the result that, in a first approximation, the disturbed flow field is independent of the external flow velocity, if the axial co-ordinate is contracted by a suitably chosen stretching factor and if the disturbance frequency is reduced by the same factor. It is concluded that the large-scale structure of jet turbulence is modified in the same manner by the external flow.

Results are presented of a series of experiments which were carried out with great care to produce accurate, comparable results in which the relatively small changes in the flow past a circular cylinder associated with varying blockage ratios can be clearly distinguished. The experiments cover a range of blockage from 2 to 16% and of aspect ratios from 4 to 10. End plates were fitted to the cylinders in all experiments.

For blockage ratios less than 6%, it is shown that the effects of blockage on pressure distribution and the drag coefficient are small and that the Strouhal number is unaffected by blockage. For blockage ratios in the range 6–16%, there is considerable distortion of the flow due to blockage and the effects are complex. The pressure distribution is of a different form and the Strouhal number changes. Hoevever, conflicting influences result in a blocked drag coefficient which is not very different from that at no blockage. Reduction in aspect ratio has effects on drag coefficient and on base pressure coefficient which are similar to those associated with increase in blockage ratio.

It is concluded that blockage correction procedures based on the momentum method and on the image method are unsatisfactory in their prediction of the unblocked drag coefficient but the momentum method predicts the unblocked base pressure coefficient quite well.